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1. FUNDAMENTAL PRINCIPLES AND METHODS OF PARTICLE DETECTION 1 .l. Interaction of Radiation with Matter* 7 1.1 .l. Introduction In this chapter, we shall discuss the various processes which take place when charged particles and y radiation pass through matter. For any type of charged particle (proton, meson, electron, etc.), there will be a loss of energy as the particle traverses the material, due to the excitation and ionization of the atoms of the medium close to the path of the particle. The loss of energy per cm of path, dE/dx, is generally referred to as the ionization loss. In Section 1.1.2, we give a simplified derivation of the theoretical expression for dE/dx, the well-known Bethe-Bloch formula, including a discussion of the density effect which becomes important at high energies. The ionization loss of a fast charged particle is frequently used as a means of identifying the particle, by observing its track in a cloud chamber, bubbIe chamber, or in photographic emulsion. The ionization loss dE/dx is a function onIy of the veIocity v of the particle (for a given charge), so that a simultaneous measurement of dE/dx and of the momentum p enables one to determine the mass m of the particle. The ionization loss can also be used to determine approximately the energy of the particle, if its identity has been established by other methods. A further important property of the ionization process is that the energy w required to form an ion pair in a gas is approximately independent of the energy and the charge of the incident particle, so that when a particle is stopped in a gas, a measurement of the total number of ion pairs enables one to obtain the energy of the incident particle, provided that the value of w for the stopping gas is known. This property has been widely used in the operation of ionization chambers. 1 In Section 1.1.2, expressions for dE/dx are given for various cases, together with a discussion of the fluctuations of the ionization loss (Landau effect). The recent experiments on the ionization loss of relativistic charged particles will be discussed in some detail. For particles heavier than electrons (e.g., protons, K, T, or p mesons), the ionization loss dE/dx is the most important mechanism of energy loss. As a result, a particle with a given incident kinetic energy T will have a quite well-defined range R, which depends on T, on the mass m and on the t See also, Vol. 4, B, Parts 6, 7, and 8. 1 See also this volume, Chapter 1.2. * Chapter 1.1 is by R. M. Sternheimer. 1
Transcript
Page 1: n

1. FUNDAMENTAL PRINCIPLES AND METHODS OF PARTICLE DETECTION

1 .l. Interaction of Radiation with Matter* 7 1.1 .l. Introduction

In this chapter, we shall discuss the various processes which take place when charged particles and y radiation pass through matter. For any type of charged particle (proton, meson, electron, etc.), there will be a loss of energy as the particle traverses the material, due to the excitation and ionization of the atoms of the medium close to the path of the particle. The loss of energy per cm of path, dE/dx, is generally referred to as the ionization loss. In Section 1.1.2, we give a simplified derivation of the theoretical expression for dE/dx, the well-known Bethe-Bloch formula, including a discussion of the density effect which becomes important at high energies. The ionization loss of a fast charged particle is frequently used as a means of identifying the particle, by observing its track in a cloud chamber, bubbIe chamber, or in photographic emulsion. The ionization loss dE/dx is a function onIy of the veIocity v of the particle (for a given charge), so that a simultaneous measurement of dE/dx and of the momentum p enables one to determine the mass m of the particle. The ionization loss can also be used to determine approximately the energy of the particle, if its identity has been established by other methods. A further important property of the ionization process is that the energy w required to form an ion pair in a gas is approximately independent of the energy and the charge of the incident particle, so that when a particle is stopped in a gas, a measurement of the total number of ion pairs enables one to obtain the energy of the incident particle, provided that the value of w for the stopping gas is known. This property has been widely used in the operation of ionization chambers. 1 In Section 1.1.2, expressions for dE/dx are given for various cases, together with a discussion of the fluctuations of the ionization loss (Landau effect). The recent experiments on the ionization loss of relativistic charged particles will be discussed in some detail.

For particles heavier than electrons (e.g., protons, K , T , or p mesons), the ionization loss dE/dx is the most important mechanism of energy loss. As a result, a particle with a given incident kinetic energy T will have a quite well-defined range R, which depends on T, on the mass m and on the

t See also, Vol. 4, B, Parts 6, 7, and 8. 1 See also this volume, Chapter 1.2.

* Chapter 1.1 is by R. M. Sternheimer. 1

Page 2: n

2 1. PARTICLE DETECTION

charge z of the particle, as well as on the stopping substance. The relation between R and T is known as the range-energy relation. Tables of the range-energy relation for protons of energies T , = 2 Mev to 100 Bev have been recently calculated by Sternheimer for the following materials : Be, C, All Cu, Pb, and air. These range-energy relations differ from the results of Aron et a1.2 in two respects: (1) the density effect correction is included a t the higher energies ( T , 2 2 Bev); (2) recent values of the mean excitation potential I (which enters into the Bethe-Bloch formula) have been used, which are somewhat higher than the value I = 11.52 ev employed by Aron et al. The tables of the range-energy reIations are given in Section 1.1.3, together with a table of the values of dE/dx which were used in the calculation of R ( T ) . Section 1.1.3 also includes a brief discussion of the range straggling.

Section 1.1.4 gives various formulas pertaining to the scattering of heavy particles (heavier than electrons) by atoms.

When electrons pass through matter, they lose energy by ionization in the same manner as any charged particle (see Section 1.1.2). However, in addition, a high-energy electron will produce electromagnetic radiation (bremsstrahlung) in the field of the atomic nuclei.* For electrons above the critical energy E , (e.g., 47 Mev for All 6.9 Mev for Pb), the energy loss due to radiation exceeds the ionization loss, and constitutes the pre- dominant mechanism for the slowing down process. The y quanta from the bremsstrahlung can create electron-positron pairs, which in turn can produce additional y rays. The resulting electromagnetic cascade is called a shower and has been widely observed in cloud-chamber pictures both with incident electrons and y rays. The theoretical expressions for the bremsstrahlung and a discussion of shower production are presented in Section 1.1.5.

The multiple scattering of charged particles is considered briefly in Section 1.1.6.

The penetration of y rays through matter is characterized by an absorp- tion coefficient r which determines the exponential attenuation of the y ray beam. The processes which contribute to r are the photoelectric effect, the Compton scattering, and the pair production. A summary of the theoretical expressions for these three processes is given in Sec- tion 1.1.7.

The discussion of Sections 1.1.4-1.1.7 follows closely the review article

* See also Vol. 4 , A, Section 1.5.2. 1 R. M. Sternheimer, Phys. Rev. 116, 137 (1959). 2 W. A. Aron, B. G. Hoffman, and F. C. Williams, University of California Radiation

Laboratory Report UCRL-121 (1 951); Atomic Energy Commission Report AECU-663 (1951).

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1.1. INTERACTION OF RADIATION W I T H MATTER 3

by Bethe and Ashkin3 on the “Passage of Radiations through Matter.” In 1956, in order to solve certain difficulties connected with the decay

of the strange particles (particularly the K meson), Lee and Yang4 dis- cussed the consequences of a possible nonconservation of parity in the weak interactions (beta decay, strange particle decay, a- and p-meson decay). They suggested a number of experiments to test this hypothesis. These experimentss7 were performed soon after the publication of their paper, and have shown very clearly that parity is not conserved in the weak (decay) interactions, in contrast to the strong interactions which conserve parity to a high accuracy. An important consequence of parity nonconservation is that the electrons (or positrons) from the beta decay of unpolarized nuclei should be strongly longitudinally polarized, i.e., the electron spin should be aligned predominantly antiparallel to the electron direction of motion, while for positron decays, the positron spin should be aligned predominantly parallel to the positron direction of motion. The magnitude of the polarization P is predicted to be v/c in each case, where v is the velocity of the particle (electron or positron). Thus for relativistic electrons or positrons, P should be essentially 100%. It should be noted that the prediction that P = v / c follows only from a particularly simple theory of parity nonconservation, namely the two-component theory of the neutrino. In a separate article,* we have given a discussion of the proposals of Lee and Yang4 concerning parity nonconservation in weak interactions. This article also contains a description of the crucial experiments of Wu et d 6 on the beta decay of oriented nuclei (Co60), and of Garwin and co-workersO on the polarization of the p+ from a+ decay, which together with the work of Friedman and Telegdi,7 were the first experiments that demonstrated the violation of parity conservation in weak interactions. We have also summarized* the two-component theory of the neutrino, which was proposed independently by Lee and Yang,g Landau,lo and Salam.”

A large number of experiments have been performed to establish the longitudinal polarization of the electrons and positrons from beta decay.

3 H. A. Bethe and J. Ashkin, Passage of radiations through matter. In “Experi-

4 T. D. Lee and C . N. Yang, Phys. Rev. 104, 254 (1956). 6 C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys.

6 R. L. Garwin, L. M. Lederman, and M. Weinrich, P h p . Rev. 106, 1415 (1957). 7 J. I. Friedman and V. L. Telegdi, Phys. Rev. 106, 1681 (1957). 8 R. M. Sternheimer, Advances in Electronics and Electron Phys. 11, 31 (1959). 9 T. D. Lee and C . N. Yang, Phys. Rev. 106, 1671 (1957). 10 L. D. Landau, Nuclear Phys. 8, 127 (1957). 11 A. Salam, Nuovo n’mento 1101 6, 299 (1957).

mental Nuclear Physics” (E. SegrB, ed.), Vol. 1, p. 166. Wiley, New York, 1953.

Rev. 106, 1413 (1957).

Page 4: n

4 1. PARTICLE DETECTION

These investigations involve a variety of methods to determine the longi- tudinal polarization : scattering of the polarized electrons on nuclei (Mott scattering) ; scattering on polarized electrons (ferromagnetic 3d electrons of iron in a magnetic field), which is often referred to as Mdler scattering; circular polarization of the bremsstrahlung emitted by the polarized elec- trons; and annihilation of the polarized positrons in various materials. The experiments have in turn led to important developments of the theories presented in Sections 1.1.4 and 1.1.5 on the scattering and inter- action of electrons in matter. These new theoretical results, as well as a review of the experiments on the longitudinal polarization, are presented in the latter part of the article on parity nonconservation."

1.1.2. The Ionization Loss dE/dx of Charged Particles

1.1.2.1. The Bethe-Bloch Formula. The theoretical expression for dE/dx is based on the Bethe-Bloch formula, which has been derived from the work of Bohr,12 Bethe,13 Bloch,I4 and others. The Bethe-Bloch formula for particles heavier than electrons is given by

- 2 p - 6 - u ] (1.1.1) dx

where n = number of electrons per cma in the stopping substance, m = electron mass, p = v/c, where v = velocity of the particle, z = charge of the particle, I = mean excitation potential of the atoms of the sub- stance, Wmsx = maximum energy transfer from the incident particle to the atomic electrons, 6 is the correction for the density effect, which is due to the polarization of the medium, as will be discussed below, and U is a term due to the nonparticipation of the inner shells ( K , L, . . .) for very low velocities of the incident particle. This term is generally called the shell correction term, and will be discussed below [see Eq. (1.1.34)l. The maximum energy transfer W,, is given by

( I . 1.2) Wmsx = 2mv2/(1 - p 2 )

for energies E << (mi2/2m)c2, where mi is the mass of the incident particle. Throughout this chapter, m (without subscript) denotes the mass of the electron.

The Bethe-Bloch formula (1.1.1) is obtained in the following man- ner. The eIectromagnetic field of the passing particle will excite the

* Other aspects of electron polarization are discussed in Vol. IV, A, Chapter 3 5; this volume, Chapter 2.5.

l2 N. Bohr, Phil. Mag. [6] 26, 10 (1913); [6] SO, 581 (1915). l a H. A. Bethe, Ann. Physik [7] 6, 325 (1930). l4 F. Bloch, 2. Physik 81, 363 (1933).

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1.1. INTERACTION OF RADIATION WITH MATTER 5

atomic electrons from their initial ground state to higher excited states, either discrete states or states in the continuum corresponding to ioniza- tion. For close collisions, with small values of the impact parameter b ( b - 10-8 - cm), tho at,omic electrons can he considered as free, i.e., the atomic binding forces are negligible compared to the field of the passing particle during the collision. Small values of b are associated with large energy transfers W . (W = l / b 2 . )

The following approximate derivation of Eq. (1.1.1) follows closely that given by Fermi.16 The atomic electron is assumed to be essentially a t rest before the collision, i.e., its velocity veI is assumed to be small com- pared to the velocity v of the incident particle. The impulse I* given to the electron in the direction perpendicular to the path of the passing particle is given by

(1.1.3)

where F , is the perpendicular component of the force acting on the atomic electron. The factor ( ze2 /b2 ) in Eq. (1.1.3) gives the order of magnitude of FI, while the factor ( b / v ) is of the order of the collision time. An exact calculation shows that II is actuaIly equal to 2ze2/bv. The longitudinal part of the impulse, Ill, is zero.

I , is equal to the momentum pel acquired by the electron. The corre- sponding energy acquired (which is lost by the passing partide) is given by

(1.1.4)

The number of electrons per cm of path in the range of impact parameters from b t o b + db is 2 m b db . Hence the energy loss per cm to electrons in the range (b , b + db) is

( 1.1.5) 4nnz2e4 db

mu2 b - d E ( b ) = (:$) -~ (27rnb db) = ~ -.

The complete rate of energy loss d E / d x is obtained by integrating Ey. (1.1.5) between the limits bmin and b,,,, which are the minimum and maximum values of b for which the above treatment of the energy transfer to the atomic electrons is valid. Approximate values of bmin and bmsx will be obtained below. One thus finds

d E 4?mz2e4 bmsx In -* d x mv2 bmin

- - = - (1.1.6)

l6 E. Fermi, “Nuclear Physics,’’ p. 27. Univ. of Chicago Press, Chicago, 1950.

Page 6: n

6 1. PARTICLE DETECTION

The upper limit b,, arises from the fact that if the time of collision T is large compared to the period of revolution of the atomic electrons in their respective Bohr orbits, the passing particle does not lose any appreciable energy to the electrons, since the perturbation of the electron motion is then essentially an adiabatic process. In the nonrelativistic region, the time of collision r is -b/v. At relativistic energies, the region of space in which the perpendicular component El of the electric field (and hence the impulse I J is large, is contracted by a factor (1 - p2)-1'2 where p = v/c. Hence the time T during which a particular electron experiences the field El is also decreased by the factor (1 - p2)-lI2, so that we have

7 N (b/v)(l - p 2 ) l i Z . (1.1.7)

denotes the mean frequency of excitation of the electrons, b,,, is If determined by

whence 1/Y g (brn"JV)(1 - p 2 ) l i z

b,,, g ( v / F ) ( l - p - 1 ' 2 .

(1.1.8)

(1.1 .Y)

The lower limit bmin arises from the limitation of the classical treatment presented above. Thus if the de Broglie wavelength X is larger than the impact parameter b, the above classical considerations will lose validity. I n this connection, X must be taken as the wavelength of the atomic elec- tron in the center-of-mass system, which approximately coincides with the rest system of the incident particle (for particles heavier than elec- trons). Since we assume that vel <<v, the velocity of the electron in the center-of-mass system is close to v, and the resultant momentum is mv(1 - p2)-1/2. One thus obtains

bmi, E X = h(l - P2)1/2/(mv) ( 1.1.10)

where h is Planck's constant h divided by 2 ~ . Upon inserting Eqs. (1.1.9) and (1.1.10) into Eq. (1.1.6), one finds

(1.1.11)

In the logarithm of Eq. ( l . l . l l ) , f i Y is 1 / ( 2 ~ ) times the mean excitation potential of the atom (denoted previously by I ) . Thus the argument of the logarithm is 27rmv2/[1(1 - p 2 ) ] . Actually, an accurate treatment of the limits b,,, and bmi, gives

( 1.1.12) 2mv2 In --.. dE - 4?mx2e4 dx mu2 1(1 - p2)

Equation (1.1.12) is ident,ical with the Bethe-Bloch formula ( l . l . l ) ,

- _ _ _ _ _

Page 7: n

1.1. INTERACTION OF RADIATION WITH MATTER 7

except for the relativistic term -2p2 and the terms due to the density effect and the shell corrections (- 6 - U ) in the square bracket of (1.1.1). The equivalence of the logarithms of Eqs. (1.1.1) and (1.1.12) can be easily verified by substituting Eq. (1.1.2) for W,,, in Eq. (1.1.1).

1.1.2.2. Dependence of the Ionization Loss on the Particle Velocity. The Density Effect. We will now discuss the genera1 behavior of Eq. (1.1.1) for dE/dx as a function of the particle velocity u. At low energies, - dE/dx decreases rapidly with increasing v. This decrease is essentially due to the fact that with increasing u , the collision times 7 decreases, resulting in a decrease of the excitation of the atomic electrons. The momentum transfer pel is proportional to 7 or to l / v . The resulting energy transfer equals p$/2m, and is therefore proportional to 1 /u2 , as is seen from Eq. (1.1.1). In the relativistic region (v = c), the factor 1/v2 becomes nearly constant, and the velocity dependence is mainly determined by the behavior of the square bracket of Eq. (1.1.1). In view of Eq. (1,1.2), the logarithm of (1.1.1) can be written

2mv2 1(1 - p")' L = 2 111 (1.1.13)

The denominator (1 - p2) results in a logarithmic increase of dE/dx with increasing energy, which is called the relativistic rise. This increase takes place after dE/dx reaches a minimum a t u = 0.96~ (see Figs. 1 and 2). As is seen from Eq. (1.1.1)) a part of this rise is due to the distant col- lisions [the factor (1 - p2)-' in the logarithm of (l)], and a part is due to the close collisions; this part results from the increase of Wm, with increasing energy. As is shown by Eq. (1.1,9), the increase due to the distant collisions arises directly from the relativistic increase of b,,,. It is in this connection that the density effect (represented by the term 6) becomes important. Thus with increasing energy, the cylindrical region around the path of the particle where the atoms are excited will increase in radius (= Lax). However, the atoms close to the path of the particle will produce a polarization which reduces the electric field acting on the electrons at larger distances, thus diminishing the energy loss to these electrons. The existence of this effect was first suggested by Swann16 in 1938. It has been calIed density effect, since it depends strongly on the density of the medium, in as much as the polarization is proportional to n, the number of electrons per cm3. The density effect correction was first evaluated by Fermi" under simplified assumptions. Further calculations

16 W. F. G. S w a m , J . Franklin Znst. 226, 598 (1938). ' 7 E. Fermi, Phys. Rev. 66, 1242 (1939); 67, 485 (1940).

Page 8: n

a 1. PARTICLE DETECTION

of this effect have been carried out by Halpern and Ha11,18 Wick,lg Stern- heimer,20s21 BudinilZ2 and others. I n the extreme relativistic region, the density effect correction 6 is given by

6 = - ln(1 - p2) - ln(12/h2vp2) - 1 (1.1.14)

where v p is the plasma frequency of the electrons and is given by

yP = (ne2/i..m)1/2.

The term - ln(1 - 02) in 6 cancels the term due to (1 - p2) in the logarithm of Eq. (1.1.1). Thus the part of the relativistic rise due to the broadening of the field region is completely canceled by the density effect in the limit of very large energies. There remains the part of the rise due to the term In W,.,, i.e., due to the energy transfer in close collisions. W,,, increases as (1 - p2)-’ for not too high energies [Eq. (1.1.2)]. At very high energies [E > (mi2/2m)c2], W,,, increases approximately as (1 - p2)-1/2. Indeed, Vmax approaches the value E - (mi2/2m)c2, as is shown by the following general formula due to Bhabha.23

where E is the total energy of the particle (including the rest mass). Thus in the extreme relativistic region, the logarithm of (1.1.1) has a term - ln(1 - p2)3 /2 , whereas 6 has a term - ln(1 - p2) , so that the relativis- tic rise is only one-third as large as i t would be without the density effect.

1.1.2.3. Energy Loss due to eerenkov Radiation. It should be noted that the relativistic rise includes the energy loss due to Cerenkov radia- tion. The Cerenkov loss is given by the formula of Frank and T t ~ m m . ~ ~

(1.1.16)

where the integral extends over the frequencies v for which pn > 1, and where n(v) is the index of refraction of the medium. The Cerenkov loss is

l8 0. Halpern and H. Hall, Phys. Rev. 67, 459 (1940); 73, 477 (1948). l9 G. C . Wick, Ricerca sci. 11, 273 (1940); 12, 858 (1941); Nuovo cimento [9] 1,

20R. M. Sternheimer, Phys. Rev. 88, 851 (1952); 91, 256 (1953); 93, 351, 1434

21 R. M. Sternheimer, Phys. Reu. 103, 511 (1956). 22 P. Budini, Nuovo cimento [9] 10, 236 (1953). 23 H. J. Bhabha, Proc. Roy. SOC. A164, 257 (1937). Z4 I. Frank and I. Tamm, Compt. rend. acad. sci. U.R.S.S. 14, 109 (1937).

302 (1943).

(1954).

Page 9: n

1.1. INTERACTION OF RADIATION WITH MATTER 9

zero at low energies, and increases to a small saturation value in the region of the relativistic r i ~ e . ~ " ~ ' The magnitude of (1.1.16) is in all cases small compared to the magnitude of the relativistic rise. This result arises from the fact that there is a large number of absorption lines and con- t i n ~ a , ~ ~ - ~ ~ corresponding to excitation of the electrons in the K , L, M . . . shells, except for the very light atoms (H, He) where, however, there is still a wide spectrum of absorption frequencies corresponding to the continuum above the discrete spectrum for excitation from the 1s shell. As a result, the expression for the atomic polarizability C Y ( V ) con- tains a large number of terms, one term for each absorption frequency (discrete line spectrum or excitation to the continuum). For such a be- havior of C Y ( V ) , the index of refraction %(.) is less than 1 over a considerable region of V . As a result the region of integration of (1.1.16) is considerably restricted, and actually the only region which makes an important con- tribution to (dE/dx)c is the region of v below the first absorption limit, i.e., effectively the optical and near-ultraviolet region, as has been shown by Sterr~heimer.~~ It should also be noted that the width of the spectral lines gives rise to a strong absorption of the cerenkov radiation for values of v close to the frequencies of the atomic transitions, thus result- ing in a further reduction of the Cerenkov energy loss. For condensed materials, - (l/p)(dE/dx)c is of the orderz0 of lop3 Mev/g cm-2 and hence completely negligible compared to the total ionization loss (> 1 Mev/g cm-2).31 For gases, the cerenkov loss is somewhat more impor- tant,29of the order of 0.1 Mev/g for Hz and He, and 0.01 Mev/g cm-2 for gases with medium and large Z . Even for H,, the cerenkov loss accounts for only -15% of the relativistic rise. Comprehensive treat- ments of the stopping power problems in dense materials, including the Cerenkov radiation, have been recently given by fan^,^^ Budini and Taff Bra, 3 5 and Tidman. 3 4

The Bethe-Bloch formula, Eq. ( l . l . l ) , includes the cerenkov loss. l7 Thus, in order to obtain the energy -(dE/dx)d deposited close to the path of the particle, it is necessary in principle to subtract - (dE/dx)c.

26 A. Bohr, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 24, No. 19 (1948). 24 H. Messel and D. M. Ritson, Phil. Mag. [7] 41, 1129 (1950). 17 M. Schoenberg, Nuovo cimento [9] 8,159 (1951); 9,210,372 (1952); M. Huybrechts

28 P. Rudini, Phys. Rev. 89, 1147 (1953). 99 R. M. Sternheimer, Phys. Rev. 89, 1148 (1953); 91, 256 (1953); 93, 1434 (1954). 80 G. N. Fowler and G. M. D. B. Jones, Proc. Phys. SOC. (London) A66, 597 (1953). 81 See also J. R. Allen, Phys. Reu. 93, 353 (1954). 84 U. Fano, Phys. Reu. 103, 1202 (1956). 88 P. Budini and L. Taffara, Nuouo cimenlo [lo] 4, 23 (1956). *4 D. A. Tidman, Nuclear Phys. 2, 289 (1956); 4, 257 (1957).

and M. Schoenberg, ibid. 9, 764 (1952).

Page 10: n

10 1. PARTICLE DETECTION

We have

(1.1.17)

However, as pointed out above, (dE/dx) c is generally negligible compared to the relativistic rise of dE/dx, except for gases of low 2 (H2, He).

While the energy loss due to Cerenkov radiation is very small compared to the total ionization loss, the Cerenkov effecta6 has received an important application in the design of Cerenkov counters," which are based on the property that the Cerenkov radiation is absent unless the velocity of the particle exceeds a critical value vc determined by

v,no/c = 1 (1.1.18)

where no is the index of refraction of the radiating substance (usually a liquid) in the optical region. The Cerenkov counter is used as a velocity selector, and together with a momentum measurement, it enables one to identify charged particles by placing an upper or lower limit on the mass.

1.1.2.4. Evaluation of dE/dx. The Mean Excitation Potential 1. Equa- tion (1.1.1) can be written in the following form:

P B + 0.69 + 2 In - + In Wmax,Mev - 2p2 - 6 - U Pdx: P 2 mic (1.1.19)

where p is the density of the medium in g/cm3, so that -(l/p)(dE/dz) gives the energy loss in Mev/g cm-2; A and 3 are defined by:

A = 21rnx2e4/(mc2p) (1.1.19a) 3 = 1n[mc2(1OE ev)/P]. (1.1.19b)

In Eq. (1.1.19), Wmax,Mev is the value of Wmax [Eqs. (1.1.2), (1.1.15)] in MeV. In order to obtain - (l/p)(dE/dz) in Mev/g cm-2, one must take A as: 0.1536(Z/A0)x~, where 2 = atomic number, A0 = atomic weight of stopping substance. The following expression for 6 has been obtained by Sternheimer :20,21

6 = 4.60GX + C + U ( X I - X)' (Xo < X < X i ) (1.1.20) 6 = 4.606X + C ( X > Xl) (1.1.20a)

where X = loglo(p/m,c), X O and X I are particular values of X which de- pend on the substance. X O is the value of X below which 6 is zero; XI is the value of X above which the high-energy expression, Eq. (1.1.20a), applies.

* See also this volume, Chapter 1.5. a s A review of the applications of the eerenkov effect has been given by J. V. Jelley,

Progr. in Nuclear Phys. S, 84 (1953).

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1.1. INTERACTION O F RADIATION WITH MATTER 11

In the region X > XI, the ionization loss becomes independent of the excitation potential I , as has been first shown by Fermi.I7 In Eqs. (1.1.20) and (1.1.20a), a, s, wd C are constants which depend on the substance and on the value chosen for I . C is defined as

C = -2 ln(I/hv,) - 1. (1.1.21)

The mean excitation potential I has been the subject of numerous investigations. In 1933, Bloch14 showed that, on the basis of the Thomas- Fermi model, I should be proportional t o the atomic number: I = kZ, but he could not determine theoretically the value of the proportionality constant lc, which therefore has to be obtained from experiment. An early determination by Bethe36 of I for air from the range-energy relation of (Y

particles gave I,, = 80.5 ev. In 1940, Wilson3? obtained a value for aluminuni,IA1 = 150 ev. Both of these results giveI/Z 11.5 ev. In 1951, Bakker and Segr&38 measured dE/dx for 340-Mev protons in a number of materials, and obtained values of I of the order of -92 - 102 ev for heavy elements. Measurements of the ranges of 340-Mev protons by Mather and Segr&39 led to similar values of I . For Al, Mather and Segr83g found IA1 = 148 i- 3 ev, in good agreement with Wilson’s37 earlier result. On the other hand, Sachs and Richardson40 from a determination of the absolute stopping power for 18-Mev protons in A1 obtained a substjailtially higher value for IAl, namely I , = 168 k 3 ev. For heavy elements Sachs and Richardson obtained I ,- 142 - 182 ev, which is also considerably higher than the results of Bakker and Segr&38 From measurements of the range in A1 of protons of various energies from 35 to 120 MeV, Bloember- gen and van Heerden4l deduced a value I A 1 = 162 & 5 ev. A similar measurement for 18-Mev protons in A1 by Hubbard and M a ~ K e n z i e ~ ~ gave I A I = 170 ev.43

In 1955, C a l d ~ e 1 1 ~ ~ recalculated the values of I from the data of Sachs and Richardson40 with the inclusion of the low-energy shell corrections [ C , and CL, see Eq. (1.1.34)]. The resulting values of I are somewhat smaller than those originally obtained by Sachs and Richardson, but are still considerably above the Bakker-Segr& values. Thus for Al, C a l d ~ e 1 1 ~ ~ found = 163 ev, and for the heavy elements, I , - 132 - 142 ev.

M. S. Livingston and H. A. Bethe, Revs. Modern Phys. 9, 261 (1937). R. R. Wilson, Phys. Rev. 60, 749 (1941).

R. L. Mather and E. SegrB, Phys. Rev. 84, 191 (1951). ** C . J. Bakker and E. SegrB, Phys. Rev. 81, 489 (1951).

40 D. C. Sachs and R. J . Richardson, Phys. Rev. 83, 834 (1951); 89, 1163 (1953) 41 N. Bloembergen and P. J. van Heerden, Phys. Rev. 83, 561 (1951). 4* E. L. Hubbard and K. R. MacKenzie, Phys. Rev. 86, 107 (1952). 43 See also D. H. Simmons, Proc. Phys. Sac. (London) A66, 454 (1952). 44 D. 0. Caldwell, Phys. Rev. 100, 291 (1955).

Page 12: n

12 1. PARTICLE DETECTION

Caldwell also showed that the various experiments are generally con- sistent with I values which are independent of the velocity of the incident particle. 46 This result was important, since it had bee%previously believed that I might be velocity-dependent, in order to reconcile data from different experiments. 46

Recently, two accurate determinations of I have been made from measurements of the range and stopping power of low-energy protons ( s 2 0 Mev). From measurements of the range of protons of various energies from 6-18 MeV, Bichsel et aL4’ have obtained the following I values for Be, Al, Cu, Ag, and Au: I B e = 63.4 f 0.5 ev, I A I = 166.5 k 1 ev, Icu = 375.6 k 20 ev, I A g = 585 k 40 ev, and I A u = 1037 k 100 ev. The result for Be confirms an earlier determination by Madsen and Venkate~warlu,~~ who obtained IBe = 64 f 5 ev. The large value of I / Z for Be (1/Z iX 16) had been previously predicted by A. B ~ h r ~ ~ (who ob- tained I = 60 ev) on the basis of polarization effects caused by the pres- ence of the two conduction electrons per atom. For Al, Cu, Ag, and Au, the I values of Bichsel et al. give I / Z - 13 ev, which is somewhat smaller than Caldwell’s results,44 but is considerably higher than the values of I / Z obtained by Bakker and Segr&.@

The other recent determination of I values has been made by Burkig and M a ~ K e n z i e . ~ ~ These authors measured the stopping powers relative to aIuminum of a number of metals of 19.8-Mev protons. The resulting values of I are based on the value I A I = 166.5 ev of Bichsel et The values of I of Burkig and MacKenzie for Be, Cu, Ag, Au, and Pb are: IBe = 64 ev, Icu = 366 ev, I,, = 587 ev, IAu = 997 ev, and I,, = 1070 ev. These values of I are in good agreement with the results of Bichsel et aL4’

In 1959, Zrelov and stole to^^^" measured the range R in copper of 660-Mev protons from the Dubna synchrocyclotron. These authors have obtained a value R = 257.6 f 1.2 gm/cm2, which leads to a calculated mean excitation potential ICu = 305 f 10 ev ( I / Z = 10.5 k 0.3 ev). This value of Icu is appreciably smaller than the values obtained by Bichsel et al.47 and by Burkig and M a c K e n ~ i e ~ ~ at lower energies (6-20 Mev). Zrelov and stole to^^^^ have also determined the stopping power relative to copper for H, Be, C, Fe, Cd, and W for 635-Mev protons. For H, Be, and

45 See also W. Brandt, Phys. Rev. 104,691 (1956); 111,1042 (1958); 112,1624 (1958). teJ. Lindhard and M. Scharff, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd.

47 H. Bichsel, R. F. Mozley, and W. A. Aron, Phys. Rev. 106, 1788 (1957). 48 C. B. Madsen and P. Venkateswarlu, Phys. Rev. 74, 648 (1948). 49 V. C. Burkig and K. R. MacKenzie, Phys. Rev. 106, 848 (1957). 4OeV. P. Zrelov and G. D. Stoletov, Zhur. Eksptl. i Teoret. Fiz. 86, 658 (1959):

27, No. 15 (1953).

[translation: Soviet Phys. JETP 9, 461 (1959)l.

Page 13: n

1.1. INTERACTION OF RADIATION WITH MATTER 13

C, the resulting values of I / Z are -14-15 ev (IH = 15, I B e = 61 f 6, I0 = 85 f 8 ev). The value of I B e is in good agreement with the results of earlier experiment^.^'-^^ On the other hand, for Cd and W, the values of I / Z are 9.8 and 9.2 ev, respectively, indicating that for heavy elements, the value of I /Z may be appreciably lower than the results (-13 ev) of references 47 and 49.

The values of the constants a, s, and C for the density effect correction 6 which are given in reference 20 were based on the Bakker-Segr838 values of I , whereas the results for 6 of reference 21 were obtained by means of the higher I values of C a l d ~ e 1 1 . ~ ~ We denote these values of I by I1 and Iz, respectively. The values of 6 for any intermediate 1 value, I = I O (e.g., that of Bichsel et aZ.47), can be obtained by logarithmic interpolation, as follows. Let 61 and 8 2 denote the values of 6 pertaining to I I and 12, respectively.20.21 Then 60 appropriate to I = I0 is given by

60 = 761 + (1 - 7162 where q is defined by

(1.1.22)

(1.1.23)

Of course, Eqs. (1.1.22) and (1.1.23) apply also, within reasonable limits, if I0 is outside the range (IJZ), i.e., for I0 < I1 or I 0 > Iz.

When the stopping material is a compound containing several atomic species, the stopping powers of the individual elements are approximately additive (Bragg’s rule). Thus Eq. (1.1.1) still holds for the compound, provided that the mean excitation potential I in this equation is replaced by the following average potential 1:

with

(1.1.24)

(1.1.25)

Here n, is the number of atoms of the ith type in the compound, with atomic number Z, and excitation potential Ii; fi of Eq. (1.1.25) is the oscillator strength of the atomic electrons belonging to the i th species.

The density effect correction 6 of Eq. (1.1.1) must be replaced by 8 defined bv

(1.1.26) i

where 6i is the value of 6 for the i th constituent.

Page 14: n

14 1. PARTICLE DETECTION

In order to obtain - (l/p)(dE/dz) in Mev/g cm-2, the constant A of Eq. (1.1.19a) must be taken as follows:

(1.1.27)

where Ai is the atomic weight of the i th element. Extensive experiments have been carried out by Thompsons0 t o verify

the additivity of the stopping powers for a number of organic compounds (containing C, H, N, 0, and Cl). I n these experiments, the 340-Mev protons from the Berkeley cyclotron were slowed down to 200 MeV. It was found that the stopping powers of the constituent elements in the compound are additive to within 1%. Small deviations (less than 1%) from additivity were observed; these were attributed to the influence of the chemical binding. Thompson also obtained values of I for the elements C, H, N, and 0, using liquid targets for H, N, and 0.

1.1.2.5. The Restricted Ionization Loss, - (l/p)(dE/dz)~,. Equation (1.1.1) or (1.1.19) gives the average energy loss of the charged particle. These expressions include the possibility of large energy transfers, up to the maximum value W,,, [Eqs. (1.1.2), (1.1.15)]. I n certain applications, one is, however, interested in the restricted energy loss with energy transfers less than a certain fixed value WO. This is true in particular for the grain count in nuclear emulsion* and for the drop count of tracks in cloud chambers. t For the grain count in emulsion, the relevant quantity is the ionization loss with energy transfers less than W O - 5 kev,26 be- cause larger energy transfers generally result in the development of neighboring grains not directly in line with the track, so that the grain count along the track is no longer proportional to the complete ionization loss. A similar phenomenon takes place for cloud chamber tracks. For energy transfers which are larger than -1 kev, a cluster of drops is formed, so that the drop count along the track is not proportional to the complete ionization loss. The restricted energy loss, with maximum energy transfer Wo, is given by

* See also this volume, Chapter 1.8. t See also this volume, Chapters 1.6 and 1.7. 60 T. Thompson, University of California Radiation Laboratory Report UCRL-1910

(1952). See also T. Westermark, Phys. Rev. 93, 835 (1954).

Page 15: n

1.1. INTERACTION OF RADIATION WITH MATTER 15

or in terms of the constants A and B

1 - p) = $ [ B + 0.69 + 2 In - P + In wO,,, - /Y - 6 - u P dx wo mic

(1.1.29)

where W0,Mev is the value of W O in MeV. Whereas the average ionization loss - ( l /p ) (dE/dz) continues to rise

indefinitely with increasing energy (due to the increase of W,,,), the restricted energy loss - ( l /p) (dE/dz) W , levels off to a constant value at high energies, which is generally referred to as the Fermi plateau. This

I I 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 I I l l 1 72

f i MESON MOMENTUM (IN Mev/c)

FIG. 1. The ionization loss of p mesons in 02 as a function of the p-meson momentum. The solid curve represents the restricted energy loss, - ( l / p ) ( d E / d x ) w , with W o = 1 kev, as obtained from Eq. (1.1.28). The crosses represent the experimental data of Ghosh et aL61 The theoretical curve and the experimental drop count (number of ion pairs per cm) have been normalized a t the minimum of ionization.

result arises from the fact that the logarithmic term in 6 exactly cancels the effect of the (1 - ,@) denominator in Eq. (1.1.28), as has been dis- cussed above.

An example of the relativistic increase of - (l/p) (dE/dz) w,, and the Fermi plateau at very high energies is presented in Fig. 1, which shows the restricted energy loss of p mesons in oxygen (at normal pressure) for an assumed value Wo = 1 kev. A measure of the relativistic increase is given by the ratio R = Jplat/Jmin, where JP1,, and Jmi, are the values of - ( l /p)(dE/dz)~, in the plateau region and at the minimum of ionization

61 S. K. Ghosh, G. M. D. B. Jones, and J. G. Wilson, Proc. Phys. SOC. (London) A66, 68 (1952); A67, 331 (1954).

Page 16: n

16 1. PARTICLE DETECTION

(v = 0.96c), respectively. As is seen from Fig. 1, for 0 2 , with W O = 1 kev, we have Jplat = 1.63, Jmin = 1.08 Mev/g cm-2, so that R = 1.51.

Figure 1 also show the data of Ghosh et aLbl obtained from the drop count in an expansion cloud chamber filled with oxygen a t normal pressure (see Section 1.1.2.9). The theoretical curve and the experimental data have been normalized at the minimum of ionization (Jmi. = 44 ion pairs/cm). The equivalent number of ion pairs is indicated on the right-hand scale.

- 1.2 '*4 " z 2 1.11 ' " t l l l l l " 1 1 1 1 1 1 ' ' 1 1 1 1 1 1 1 ' ' I J I I I ' ' " ' 1 ' 1 '

10' lo3 lo" 10' lo6 10' p MESON MOMENTUM (IN Mev/c)

FIG. 2. The ionization loss of p mesons in He as a function of the p-meson momentum. The solid curve represents the restricted energy loss, -(l/p)(dE/dx) wO1 with W O = 1 kev. The dashed curve shows the energy deposited along the track, -(l/p)(dE/dx)d, after subtraction of the estimated energy escape due to Cerenkov radiation, - (I/p) (dE/ds )c [see Eq. (1.1.17)]. The flat part of the curves a t very high momenta ( >lo5 Mev/c) is often referred to as the Fermi plateau.

The theoretical curve is in good agreement with the data, except a t the highest momenta of the experiment ( p , 2 10 Bev/c), where the data give an indication that the relativistic rise may be somewhat smaller than predicted by the theory. However, the uncertainties of the measurements make it impossible to decide at present whether there is a real discrepancy. The Cerenkov loss (dE/dz)c may also be partly responsible for the ap- parent disagreement, since it reduces the energy deposited along the track (dE/dz)d. However, - (l/p)(dE/dx)c is expected to be quite small for oxygen (-0.02 Mev/g ern+ in the region of the Fermi plateau29).

Figure 2 shows the relativistic rise in helium at normal pressure. In this case, Jmin = 1.22, Jplst = 1.79 Mev/g so that R = 1.47. We have made an estimate of the cerenkov energy ~ o s s , ~ ~ and the dashed curve of

Page 17: n

1.1. INTERACTION OF RADIATION WITH MATTER 17

Fig. 2 shows the resultant energydepositedalong the track - (l/p) (dE/dz)d . The difference between the solid and the dashed curves represents the cerenkov loss, - (l/p)(dE/dz)~ [see Eq. (1.1.17)].

The relativistic rise of - (l/p) (dE/dz) w o in gases has been observed in a large number of experiments. A summary of these experimental investiga- tions is given in Section 1.1.2.9.

1.1.2.6. The Most Probable Ionization Loss Eprob. Fluctuations of the Energy Loss. As has been shown by Williams,62 Landau,63 and others, the ionization loss in a thin absorber is subject to appreciable fluctuations, be- cause of the statistical nature of the ionization process. The energy loss in a thin absorber has a considerable spread about the most probable value Eprob. This spread is often referred to as the Landau effect, since Landau was the first to calculate the expected distribution of the energy losses. Further contributions to this problem have been made by S ~ r n o n , ~ ~ and by Blunck and Le i~egang .~~ The distribution is asymmetric, with a long tail on the side of high-energy losses which is due to the infrequent colli- sions with very large energy transfers which result in a relatively large angle scattering of the incident particle. The full width of the Landau distribution at half-maximum is of the order of 20% of Eprob for typical cases.

From Landau’s theory,63 one obtains the following expression for eproh

for a thin absorber (of thickness t g/cm2) :

Eprob = ~

- P2 + 0.37 - 6 - U mv2p - (1.1.30)

Equation (1.1.30) can be written as follows:

P At B + 1.06 + 2 In - + In mic

As an example, a thickness of 6.97 g/cm2 of Be gives a most probable loss eprob = 10 Mev for 3.0-Bev protons. The Landau distribution for this case is shown in Fig. 3. This figure shows the rapid rise of the probability P ( E ) on the side of low-energy losses, and the long tail on the side of large energy losses. The maximum energy transfer of a 3-Bev proton to a single electron, Wmx = 17 MeV, is indicated on the abscissa for comparison with

61 E. J. Williams, Proc. Roy. SOC. A126, 420 (1929). sSL. D. Landau, J. Phys. U.S.S.R. 8, 201 (1944). 6 4 K. R. Symon, quoted by B. Rossi, in “High-Energy Particles,” p. 32. Prentice-

6 s 0. Rlunck and S. Leisegang, Z. Physin‘k 128, 500 (1950). Hall, New York, 1952.

Page 18: n

18 1. PARTICLE DETECTION

the values of Epr& and e~~ (average energy loss). can, of course, be ob- tained by integrating over the distribution, i.e., = lo" eP(e) de. The values of the loss E for which the distribution has half its maximum value a 1 - 8 ~ ~ 0 ~ ~ € 1 = 9.13 Mev and EZ = 11.20 MeV. The fractional spread (€2 - t l ) /~proh = 0.21 is a measure of the width of the distribution. The ratio ( € 2 - eprob)/(Eprob - el) is 1.38, which is a measure of the skewness of the distribution. The average loss in the same thickness of Be is: eAv = 6.97 X 1.593 = 11.10 MeV, where 1.593 Mev/g cm-2 is - ( l /p ) (dE /dx) (see Table 111). The difference between and Eprob is also

0.35 t - 7 0.30 -

gr W - =0.25 - z - - u0.20 - a -

0.15 - i -

m

- I

- 0.10 -

0 -

0.05 - -

0 - 7

n

L

ENERGY LOSS 6 (IN M e V )

FIG. 3. The Landau distribution of energy losses E for 3-Bev protons traversing a thickness 6.97 gm/cm2 of Be, for which €pr& = 10 Mev, EA" = 11.10 Mev, and W,, = 17 Mev.

an indirect measure of the importance of the infrequent large energy transfers.

In similarity to the restricted energy loss, the most probable loss eprob

also levels off to a constant value (Fermi plateau) at very high energies. This result is, of course, due to the fact that the close collisions (which would result in an unlimited increase of dE/dx) do not contribute to eprob,

but only to the tail of the Landau distribution. A summary of the experi- mental determination of Eprob in thin absorbers will be given in Section 1.1.2.9.

1.1.2.7. Ionization Loss of Electrons. Equations (1.1.2.8) and (1.1.2.9) for (dE/dz)w, and Eqs. (1.1.30) and (1.1.31) for eprab are valid for any type of charged particle: electron, meson, etc. These expressions do not iiivolve the close collisions which differentiate slightly between

Page 19: n

1.1. INTERACTION O F RADIATION WITH MATTER 19

electrons and particles heavier than electrons. On the other hand, Eqs. (1.1.1) and (1.1.19) for the average energy loss are applicable only for particles heavier than electrons. These expressions include a term due to close collisions. For electrons, this term is somewhat different, and the average ionization loss is given by:56

where T, is the kinetic energy of the electron. The factor $T, in the logarithm represents the effective maximum energy transfer W,,,. The reason for this result is that the maximum possible energy transfer from the incident electron to the atomic electron is T,. However, since the two electrons are indistinguishable, one can call the incident electron after the collision that which has the highest energy. Since this energy is 2$Te, the effective maximum energy transfer is $T,. Aside from the replacement of W,, by +Te, the square bracket of Eq. (1.1.32) for electrons differs from that for heavy particles [Eq. (1.1.1)] by an amount:

A = (17/8) - In 2 = 1.43 (1.1.33)

a t very high energies (p = 1). In (1.1.33) the term In 2 is due to the fact that the reduced mass of the system (indident particle + atomic electron) is +m for an incident electron, as compared to =m for a heavy par t i~ le .~’ The term 17/8 is due t o the difference between the cross sections for close collisions of electrons as compared to heavy particles. The effect of A on dE/dx is relatively small ( 5 10 %) since the value of the square bracket of (1.1.32) is generally -20.

1.1.2.8. The Shell Correction Term U. We shall nbw discuss the correc- tion U for the nonparticipation of the K , L, . . . electrons a t low energies of the incident particle. This correction has been introduced by Bethe.36 U is given by

(1.1.34)

where CK and CL are the K and L shell corrections, respectively. CK and C L are negligible at high energies, and become appreciable only when the velocity v of the particlgis decreased to a value of the order of the velocity

66 H. A. Bethe, in “Handbuch der Physik” (H. Geiger and K. Scheel, eds.), Vol. 24,

67 H. A. Bethe and J. Ashkin, in “Experimental Nuclear Physics” (E. SegrB, ed.), p. 273. Springer, Berlin, 1933; C. Mgller, Ann. Physik [5] 14, 531 (1932).

Vol. 1, p. 253. Wiley, New York, 1953.

Page 20: n

20 1. PARTICLE DETECTION

of the atomic electrons in the K and L shells, respectively. Thus the shell corrections will enter at a somewhat higher energy for heavy elements than for light elements. As an example, for Cu, 2 C ~ / 2 is less than 0.05 for proton energies above T , = 65 MeV, corresponding to v, = 0.35~. CL be- comes appreciable only at still lower energies. For Cu, ~ C L / Z < 0.05 for T, > 11 Mev (v , > 0.15~). Detailed calculations of CK and CL have been carried out by Walske.68 The corrections for the M , N , and higher shells of heavy atoms are generally negligible, except at very low energies (T, 5 1 MeV). where the present theory becomes unreliable for other reasons Section

,,

(capture and loss of electrons by the incident particle, 1.1.2.10).

I I I I I I I I

see

I

% 1.8 s z 1.6 Q t 1.4 E

1.2

I02 lo3 lo4 lo5 lo6 lo7 1.0’ I ’ 1 1 1 1 1 1 1 ’ ‘ ‘ * l t 1 l ’ ’ ’ ’ 1 1 1 1 1 1 1 1

p MESON KINETIC ENERGY (IN MeV)

FIG. 4. The average ionization loss of p mesons in Be, Al, Cu, Ag, and Au, as a function of the p-meson kinetic energy [Eq. (1.1.19)].

1.1.2.9. Example: - (l/p)(dE/dx) for /I Mesons in Various Materials. Experimental Verification of the Bethe-Bloch Formula at Relativistic Energies. In summary, Eqs. (1.1.1), (1.1.19) and (1.128)-(1.1.32) givethe expressions for the ionization loss for 4 different possibilities: (1) average energy loss of particles heavier than eIectrons; (2) average energy loss of electrons; (3) restricted energy loss (energy transfers less than a fixed value WO) ; (4) most probable loss in a thin absorber.

As an example of the behavior of dE/dx as a function of energy, Figs. 4 and 5 show curves of -(l/p)(dE/dx) versus kinetic energy T, for /I mesons in various solids and gases. In calculating the curves of Fig. 4 for the solids, we used for the excitation potential I of each substance the

68 M. C. Walske, Phys. Rev. 88, 1283 (1952); 101, 940 (1956).

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1.1. INTERACTfON OF RADfATfON WfTH MATTER 21

average of the I values determined by Bichsel et aL4' and by Burkig and M a c K e n ~ i e . ~ ~ The resulting I values are: IBe = 64 ev, Id = 166 ev, Icu = 371 ev, IA= = 586 ev, and I , = 1,017 ev. For Fig. 5, we used the I values given in reference 21 for Hz, He, and air: I H 2 = 19 ev, I H e = 44 ev16Q and Isir = 94 ev (the last corresponding to I = 132 ev). For Ar and Xe, the values of I / Z were obtained by interpolation from the above I values for Cu, Ag, and Au. This gives: IAr = 230 ev, and ZXe = 684 ev.

Figures 4 and 5 show the ionization minimum (for T, - 200-300 MeV) and the relativistic rise at higher energies. The value of - ( l /p) (dE/dz) a t the minimum decreases with increasing 2, on account of the increase of I

p MESON KINETIC ENERGY (IN MeV)

FIG. 5. The average ionization loss of p mesons in Hz, He, air, Ar, and Xe, as a func- tion of the p-meson kinetic energy [Eq. (1.1.19)]. The curves for He, air, Ar, and Xe pertain to normal pressure.

in the denominator of the logarithm of Eq. (1.1.1). For H,, three curves are presented corresponding to different pressures. On account of the density effect, the ionization loss decreases with increasing pressure a t very high energies ( T , 2 10 Bev). It may be noted that, in contrast to Figs. 1 and 2 for - ( l / p ) (dE/dz) w0, which show a plateau at high energies, the curves of Figs. 4 and 5 have an unlimited logarithmic increase, which is, of course, due to the fact that they represent the average energy loss, including all possible energy transfers up to Wmax [cf. discussion following Eq. (1.1.29)].

Figures 4 and 5 also apply for protons, provided that the numbers on the abscissa are multiplied by the factor mp/m,, where mp and m,, are the

69 E. J WilIiams, Proc. Cambridge Phit. Soc. 33, 179 (1937).

Page 22: n

22 1. PARTICLE DETECTION

proton and the p-meson mass, respectively. [ d E / d x for protons of energy T, is equal to dE/dx for p mesons of energy T,(m,/m,).]

The Bethe-Bloch formula has been verified in numerous experimental investigations. A summary of the low-energy work on the ionization loss d E / d x and on the range-energy relation is to be found in the articles of Bethe and A ~ h k i n , ~ Allison and Warshaw,60 and Taylor.61 The experi- mental studies at relativistic energies are discussed in the review articles of Price62 and of Uehling.'j3 Here we shall present a summary of some of the experiments performed at relativistic energies to verify the existence of the relativistic rise and of the Fermi plateau (due to the density effect). An outline of the main features of some of the experimental investigations on the relativistic rise and the density effect is given in Table I.

The most extensive experiments on the energy loss in gases a t relativistic energies have been made by means of expansion cloud chambers,* by obtaining the drop count along the tracks of the particles. The momentum is determined by measuring the curvature in the magnetic field of the cloud chamber. One of the earliest experiments of this type is that of Ghosh and co-workerssl who measured the restricted energy loss ( W O = 1 kev) of 1.1 mesons in 0 2 a t normal pressure. They observed a rise from 44 drops per cm to 60 drops per cm in going from the minimum ionization at p , = 0.4 Bev/c to the Fermi plateau starting at p, = 20 Bev/c. From their determinations of the drop count at p , = 7, 15, and 30 Bev/c, it is clear that the energy loss - (dE/dx)w- , does not increase indefinitely with increasing p,, but instead levels off to a saturation value. This result provides a direct confirmation of the existence of the density effect in gases a t high energies (see Fig. 1).

Aside from the work of Ghosh et u Z . , ~ ~ there have been several other cloud-chamber determinations of the relativistic rise, although these have generally somewhat poorer momentum determinations. Carter and W h i t t e m ~ r e ~ ~ performed measurements in a helium-filled chamber, both for p mesons and electrons, and obtained evidence that high-energy electrons ionize more heavily than minimum, which would confirm the relativistic rise. These authors also obtained direct evidence for the relativistic rise by comparing the ionization due to p mesons with momenta between 70 and 250 Mev/c (14.7 f 0.35 droplets/mm on the photo- graphic film) with the ionization for a group with p , > 1500 Mev/c

* See also in this volume, Chapter 1.6. 60 S. K. Allison and S. D. Warshaw, Revs. Modern Phys. 26, 779 (1953).

A. E. Taylor, Repfs. Progr. in Phys. 16, 49 (1952). 6a B. T. Price, Repis. Progr. in Phys. 18, 52 (1955). 83 E. A. Uehling, Ann. Rev. Nuclear Sci. 4, 315 (1954). G4 R. S. Carter and W. L. Whittemore, Phys. Rev. 87, 494 (1952).

Page 23: n

1.1. INTERACTION OF RADIATION WITH MATTER 23

TABLE I. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss

For each particular method of determination, the experiments are listed in chrono- logical order. For additional details and a more complete list of references,

see text (Section 1.1.2.9).

Type of particle, Method of energy range, and

Author determination material traversed Results

Corson and Cloud chamber Electrons (0.3-60 MeV) Brodeae (1938) (drop count) in cloud chamber

filled with Nz at 1.5 atmos pressure.

Sen Gupta67 Cloud chamber Electrons (2-500 MeV) (1 940)

Hazen66 (1945) Cloud chamber Electrons in air. Two energy groups: 1.4- 2.1 Mev and 30-240 MeV.

Haywardas (1947) Cloud chamber Electrons in He.

Carter and Whittemore'4 (1952)

Ghosh et aL61 (1954)

Kepler et aLa9 (1958)

Cloud chamber p mesons in He (at pressure P = 98 cm Hg). Two momen- tum groups: p = 70- 250 Mev/c, and p > 1500 Mev/c.

p mesons in 0, (at nor- mal pressure); p = 0.3-30 Bev/c.

Cloud chamber

Cloud chamber p mesons with p / p c = 3-80, and electrons with p/pc = 50- 2000, in following

Observation of mini- mum of ionization (at T , - 2 MeV) and relativistic rise for electrons.

Observation of mini- mum of ionization and relativistic rise for electrons.

Observation of relativ- istic rise (of -40% between two energy groups) in agreement with Bethe-Bloch formula.

High-energy electrons (T , > 100 MeV) have 1.4 times mini- mum ionization, in agreement with theory.

Increase of ionization between the two momentum groups in good agreement with theory.

Observation of relativ- istic increase of ion- ization and levelling off to Fermi plateau above 6 Bev/c in reasonable agree- ment with calcula- tions including the density effect.

Observation of relativ- istic rise and Fermi plateau: in good agreement with

Page 24: n

24 1. PARTICLE DETECTION

TABLE I. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued)

~~ ~

Type of particle, Method of energy range, and

Author determination material traversed Results

Parry et al.1° (1953)

Eyeions et al.7'

(1955)

Palmatier, et aZ.I2 (1955)

Lanou and Kray- bi11728 (1959)

Barbersl (1 955)

Proportional counter

Proportional counter

Proportional counter

Proportional counter

Ionization chamber

gases: He (1.3 at- mos); Ar (0.2 at- mos) ; Ar-He mix- ture (each a t 0.2 at- mos); Xe-He mix- ture (each at 0.1 at- mos) . Bev/c) in mixture of argon (P = 774 mm Hg) and ethylene (P = 46 mm Hg).

p mesons (0.3-70

p mesons in Ne.

p mesons (0.2-15 Bev/c) in argon pressures from 2 40 atmos.

at to

p mesons (3.3-140 Bev/c) in a mixture of 95% He and 5% CO2 a t a total pres- sure of 2.7 atmos.

Electrons (1-35 MeV) in Hi, He, and N2 (normal pressure).

theory for He; for other cases, calcu- lated rise is some- what larger than ex- perimental values.

Observation of relativ- istic rise and Fermi plateau in good agreement with theory.

Observation of relativ- istic rise and Fermi plateau in good agreement with theory.

Decrease of the relativ- istic rise with in- creasing pressure, in good agreement with calculations of Stern- heimer20 on the den- sity effect.

Observation of relativ- istic rise in He and saturation of the most probable ion- ization loss a t p/m,c 2 200 (Fermi pla- teau). In this region, the ionization loss is 1.28 & 0.04 times minimum.

Observations in good agreement with cal- culated relativistic rise for Nz, but ex- perimental increase of ionization some- what smaller than calculated increase of IdE/drl for Hz and

Page 25: n

1.1. INTERACTION O F RADIATION WITH MATTER 25

TABLE I. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued)

Type of particle, Method of energy range, and

Author determination material traversed Results

Barbers2 (1956) Ionization chamber

Herefords7 Low-pressure (1 948) counter

Shamos and Low-pressure Hudes88 counter (1951)

McClureS6 Low-pressure (1953) counter

Pickup and Nuclear Vo.yvodics9 emu 1 s i o n

Electrons (1-35 MeV) in H2 and He a t 1 and 10 atmos pres- sure.

Electrons (0.2-9.0 MeV) in H 2 ( P = 7 cm Hg).

Cosmic-ray p mesons at sea level (average momentum = 3.5 Bev/e) and under 140 feet of rock (av- erage momentum = 48 Bev/c); primary specific ionization in H2 filled counter ( P = 2.0 cm Hg).

MeV) in He, He, Ne, and Ar.

Electrons (0.2-1.6

p-decay electrons and relativistic fi mesons

(1950) (grain count) in plate exposed to

He, possibly clue to production of Ceren- kov radiation which does not contribute to ionization.20

Observations in good agreement with cal- culated relativistic rise. At 10 atmos, re- duction of ionization due to density effect is observed for T, 2 10 Mev in HZ and for T. >, 18 Mev in He. The reductions at 35 Mev are in rea- sonable agreement with calculations of Sternheimer.20

rise between 1.7 Mev (minimum) and 9.0 Mev in reasonable agreement with cal- culations.

Relativistic rise of 1.17 k 0.03 between sea

level spectrum and underground spec- trum, in good agree- ment with calcula- tions using density effect correction.

Observed relativistic

Observations in good agreement with Bethe’s theory of primary specific ionization.

First observation of -10% relativistic rise of grain count in

Page 26: n

26 1. PARTICLE DETECTION

TABLE I. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued)

~

Type of particle, Method of energy range, and

Author determination material traversed Results -

Morrishgo (1952)

Nuclear emulsion (blob count)

Daniel et aLS4 Nuclear (1952) emu 1 s i o n

Stiller and Shapiros2 (1 953)

Fleming and Lord93 (1953)

Alexander and JohnstongTa (1957)

(grain count)

Nuclear emulsion (blob count)

Nuclear emulsion (blob count)

Nuclear emulsion (blob count)

sea-level cosmic-ray spectrum; and high- energy electrons, n

mesons, and protons in plate exposed to cosmic rays at high altitudes.

Bev. Electrons (5 Mev-5

n mesons (200 Mev/c-3 Bev/c).

Electrons (y > lo), T mesons (y < loo), and protons (y < 10).

Electrons from p decay (average energy = 34 MeV) and n- mesons (31-230 Mevl.

T mesons (109.1 Mev) from Krz decay, and p mesons (152.7

emulsion between T/m& N 3 and T/mor2 - 20, in rea- sonable agreement with theoretical pre- dictions.

Blob count increases -5% between 5 Mev and 15 MeV, then re- mains constant to 25 Bev; relativistic rise is smaller than value predicted by theory (14%).

Grain count increases by -8% between 500 and 1500 Mev/c, in reasonable agree- ment with calcula- tions of Budini.z*

Ratio R of plateau to minimum blob count, R = 1.14 k 0.03, in good agree- ment with theoreti- cal value 1.14; slow rise of grain count until saturation is reached for > 100, in good agreement with calculations of Sternheimer.20

Relativistic increase of 14% (between mini- mum and plateau ionization), and slow rate of rise, in good agreement with theory.

Ratio R of plateau to minimum grain count, R = 1.133.

Page 27: n

1.1. INTERACTION OF RADIATION WITH MATTER 27

TABLE I. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued)

Type of particle, Method of energy range, and

Author determination material traversed Results

JongejansSTb (1960)

Whittemore and Street104 (1949)

Bowen and Roser1°6 (1952)

Hudson and Hofstadterllo (1952)

Baskin and Winckler O 6

(1953)

Nuclear emuIsion (blob count)

Crystal counter (pulse-height distribution)

Scin tillator (pulse height distribution)

Scintillator

Scintillator

MeV) from K,? de- cay.

Beam pions (5.2-5.7 Bcv/c) from Berkeley Bevatron; secondary pions pro- duced by beam pions

electron pairs (65 < y < 1100).

(1.8 < y 5 3.5);

Cosmic-ray p mesons in AgCl crystal. Two energy groups: T, = 0.3 Bev (minimum ionization), and T, > 1.6 Bev.

Cosmic-ray p mesons (30 Mev-3 Bev) in anthracene crystal.

Cosmic-ray p mesons in NaI (Tl) crystal ( p r > 225 Mev/c).

Cosmic-ray p mesons (80-2200 MeV) in xylene solution (with terphenyl).

The authors have obtained an accurate calibration curve for grain count versus ppc for the region 0.5 < p < 0.95.

Ratio R of plateau to minimum grain count, R = 1.129 f 0.010. The relativ- istic rise of the grain count is slow, with an appreciable in- crease (-4%) tak- ing place between y = 40 and y - 1000 (plateau).

Observed relativistic increase between the two energy groups is in agreement with the prediction of the Bethe-Bloch for- mula, including den- sity effect correction.

No detectable relativ- istic rise of most probable energy loss eprob, in good agree- ment with theory in- cluding the density effect.

Observed pulse-height distribution in good agreement with cal- culations including the density effect.

No relativistic rise is observed, in agree- ment with calcula- t i o n 9 including the density effect.

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28 1. PARTICLE DETECTION

TABLE I. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued)

~~~ ~~

Type of particle, Method of energy range, and

Author determination material traversed Results

Bowen111 (1954)

Millar et uZ.109

(1958)

Paul and Reich"4 (1950)

Goldwasser, MiIIs, and Hanson 119

(1952)

Goldwasser et diZ1 (1955)

Scintillator Accelerator-produced T- mesons (60-220 Mev), p- mesons (245 Mev), and cos- mic-ray p mesons (4 groups with average energies: 0.37, 0.76, 1.47, and 5.23 Bev); NaI(T1) crystal.

Scintillator Cosmic-ray p mesons (two energies: T p = 0.30 Bev and 2.2 Bev); liquid scintil- lation counter filled with triethylben- zene (plus ter- phenyl).

Energy loss in Electrons (2.8 and 4.7

(-0.3 gm/cma) of Be, C, H20, Fe, and Pb.

thin sample Mev) in samples

Energy loss Electrons (9.6 and 15.7 MeV) in thin sam- ples (-1 gm/cm*) of Be, polystyrene, Al, Cu, and Au.

Energy loss Electrons (15.7 MeV) in thin samples of Teflon and Kel-F, and in the corre- sponding gases (same chemical composi- tion) : perfluoro- cyclobutane and chlorotrifluoro- ethylene.

Relativistic increase in reasonable agree- ment with calcula- tions; rise to plateau value may be some- what faster than pre- dicted by theory [ob- served rise: (10.9 i 1.0)% at T, = 50m,c*].

Observed value of €i,rob (0.30 Bev)/e,,,b (2.2 Bev) = 1.016 2c 0.005, in good agree- ment with Bethe- Bloch formula, in- cluding density ef- fect correction.

Observed mean energy loss is in better agree- ment with calculated value if density ef- fect correction is in- cluded.

Observed most proba- ble energy loss (-1 Mev) is in good agreement with Landau formula, in- cluding correction for density effect.

Direct observation of the density effect by comparing energy loss in solid and gase- ous samples of the same substance. The reduction of the ion- ization loss in the solid samples is in good agreement with calculations of the density effect.

Page 29: n

1.1. INTERACTION O F RADIATION W I T H MATTER 29

TABLE I. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued)

Type of particle, Method of energy range, and

Author determination material traversed Results

Hudson121 Energy loss Electrons (150 MeV) in Observed energy loss (1957) thin targets (-2.5 eprob in good agree-

gm/cma) of Li, Be, ment with Landau C, and Al. formuIa including

the density effect correction.

(17.9 f 0.25 droplets/mm). The theoretically predicted value for this high-energy group of p mesons is 18.5 droplets/mm, in good agreement with the experimental result. Hazene6 verified the relativistic rise for electrons in air. Even earlier experiments by Corson and Brodese and by Sen GuptaG7 gave convincing evidence for the relativistic rise by using cosmic-ray electrons. More recently, Hayward68 showed that high-energy electrons have an ionization of 1.4Jni, in helium, where Jmin is the mini- mum ionization.

In a recent experiment, Kepler et ~ 1 . 6 9 have measured the relativistic rise of the ionization loss of p mesons and electrons in He, Ar, and Xe, by obtaining the drop count in an expansion cloud chamber. Measurements were made for He a t -1.3 atmos, for Ar a t -0.2 atmos, for an Ar-He mixture, each a t -0.2 atmos, and for a Xe-He mixture, each a t -0.1 atmos. I n each case, the cloud chamber contained alcohol and water vapor a t a partial pressure of -5 cm Hg. The p-meson momenta extend from minimum ionization (p/m,,c = 3) to p/m,c = 80. The electron momenta extend from p/mc = 50 to z2000. Thus the entire region of the rela- tivistic rise is covered in these measurements, including the Fermi plateau, which starts a t p/mc E! 1000. For the helium experiment, the theoryzkz2 is in very good agreement with the data, if Williams’ valuesg of the excitation potential I for He is used (IH, = 44 ev;69 I for gas mixture = 49.4 ev69). For the argon and the argon-plus-helium experi- ments, the calculated rise is larger by a t least one standard deviation than

66 W. E. Haaen, Phys. Rev. 67, 269 (1945). 68 D. R. Corson and R. B. Brode, Phys. Rev. 63, 773 (1938). 67 R. L. Sen Gupta, Nature 146,65 (1940); Proc. NatE. Znst. SCi. fndiu 9,295 (1943). 88 E. Hayward, Phys. Rev. 72, 937 (1947). 60 R. G. Kepler, C. A. d’Andlau, W. B. Fretter, and L. F. Hansen, Nuovo cimento [lo]

7, 71 (1958). See also A. Rousset, A. Lagarrigue, P. Musset, P. Rancon, and X. Sauteron, Nuovo cimento [lo] 14, 365 (1959).

Page 30: n

30 1. PARTICLE DETECTION

the experimental values. Thus for the Ar-He mixture, a t p/mc % 1700, the observed value of J/Jmin is 1.59 f 0.04, whereas the calculated resultz0 is 1.64. Here J is the observed ionization loss, - (l/p)(dE/dx)w,, and Jmin is the value of J at the minimum of ionization. In this experiment, the effective maximum energy transfer W O (determined by the size of a blob = 40 drops) was 700-1000 ev. For the Xe-He mixture, the dis- crepancy is appreciably larger. For p/mc E 1000, the experimental J/Jmin = 1.58 -t 0.05, whereas the values calculated from the theories of BudiniZ2 and SternheimerZ0 are 1.78 and 1.75, respectively. On the whole, it appears that the experimental points for 4, AF-He, and Xe-He lie on a curve which increases less rapidly with increasing momentum than the theoretical curve obtained from the expression for - (l/p) (dE/dz)w, [Eq. (1.1.28)]. Kepler et ~ 1 . ~ ~ have given various possible reasons for this discrepancy for the heavy gases, in particular: (1) a variation of the maximum energy transfer W O with atomic number 2, due to the 2 dependence of the binding energy of the struck electron; (2) the ratio of the energy loss to excitation and to ionization may depend on the velocity v and on 2, in such a manner as to decrease the slope of the curve of - ( l /p ) (dE /dz )w, versus p / p c at high momenta beyond the ionization minimum; (3) shielding effects of the inner electron shells in the heavy ele- ments are important at very high energies. However, these shielding effects are taken into account, a t least in first approximation, by the density effect term 6 in Eq. (1.1.28). As a check on the Xe-He experiment, Kepler et al.69 have determined the number of drops per cm at Jmin as 28.9 k 0.6 under well-controlled conditions. This experimental value can be compared with the theoretical predictions: 31.8 + 0.4 drops/cm for the Bakker-SegrP excitation potentialszo I , and 29.6 ? 0.4 drops/cm for the higher C a l d ~ e 1 1 ~ ~ values ofz1 I . It is seen that the experimental value is in good agreement with Caldwell's results, which are also favored by several other ionization loss experiments (see Section 1.1.2.4).

Important experiments on the ionization loss a t relativistic energies have been carried out with proportional counters. I n these experiments, the Landau distribution is measured, from which, of course, one obtains the most probable loss Ep& The most accurate determinations are those of Parry et aL7O on the ionization loss eprob of p mesons in argon, and those of Eyeions et d." who obtained Eprob for p mesons in a neon-filled counter. In both cases, a relativistic rise of -50% was found, and the leveling off to

70 J. K. Parry, H. D. Rathgeber, and J. L. Rouse, PTOC. Phys. Soc. (London) A66,

7l D. A. Eyeions, B. G. Owen, B. T. Price, and J. G. Wilson, PTOC. Phys. Soc. 541 (1953).

(London) A68, 793 (1955).

Page 31: n

1.1. INTERACTION OF RADIATION WITH MATTER 31

the Fermi plateau was clearly observed. These two experiments used a cosmic-ray magnetic spectrometer, in which the high-energy particles are passed through a strong magnetic field and the resulting deflection is measured in a hodoscope array of Geiger counters placed below the proportional counter.

Palmatier and co-worker~~~ have investigated the relativistic rise and the Fermi plateau of Eprob for p mesons in a counter filled with argon a t various pressures up to 40 atmospheres. These authors have directly verified the dependence of Eprob on the pressure, i.e., the increase of the density effect with increasing pressure,20 and the resulting decrease of cprob,plat/Eproh,min, the ratio of the plateau to the minimum value of eprob. The calculated values of €,,rob are in reasonable agreement with these experi- mental results.

Lanou and Kraybil1728 have recently carried out a-similar investigation using p mesons of momenta 3.3-140 Bev/c in a proportional counter filled with a mixture of 95% He and 5% COZ at a total pressure of 2.7 atmo- spheres. These authors have observed the relativistic rise of cprob in helium, and have found that the rise saturates at momenta p/m,c 2 200 due to the density effect. In the region of the Fermi plateau, the most probable ionization loss is 1.28 k 0.04 times the value a t the minimum, in agree- ment with the calculations of Sternheimer.20,21

Several other experiments with proportional counters demonstrate the relativistic rise, but were not accurate enough to establish the existence of the Fermi plateau. Among these studies, we may mention the experiments of Kupperian and Palmatier,73 Becker et aZ.,74 Price et aZ.,76 and Eliseiev et aZ.76

Several experimenters have investigated the width A of the Landau distribution as a function of the value of At/ (p21) , which enters as a parameter in Landau’s theory. West77 found that, although the percentage width 100A/Eprob decreases with increasing At/(P21), as required by Landau’s theory, the value of A/cprob is larger than Landau’s result by a factor of -2. This discrepancy for the width of theidistribution can be re-

7 1 E. D. Palmatier, J. T. Meers, and C. M. Askey, Phys. Rev. 97, 486 (1955). 788 R. E. Lanou and H. L. Kraybill, Phys. Rev. 113, 657 (1959). 73 J. E. Kupperian and E. D. Palmatier, Phys. Rev. 91, 1186 (1953). T4 J. Becker, P. Chanson, E. Nageotte, P. Treille, B. T. Price, and P. Rothwell,

Proc. Phys. SOC. (London) A66, 437 (1952). 75 R. T. Price, D. West, J. Becker, P Chanson, E. Nageotte, and P. Treille, Proc.

Phys. soc. (London) A66, 167 (1953). 7 6 G. P. Eliseiev, V. K. Kosmachevsky, and V. A. Lubimov, Doklady Akad. Nauk.

S.S.S.R. 90, 995 (1953); (English translation: NSF-tr-163, Dept. of Commerce, Washington, D.C.)

77 D. West, Proc. Phys. SOC. (London) A66, 306 (1953).

Page 32: n

32 1. PARTICLE DETECTION

moved by improvements in the Landau theory which have been discussed by fan^^^ and H i n e ~ . ~ ~

Igo and co-workerssO have measured the distribution of energy losses of 31.5-Mev protons in a $inch proportional counter filled with an Ar-C02 mixture (96 % Ar, 4 % C02). The pulse-height distribution was in reasonable agreement with the Landau distribution, although slightly wider in the region of the tail for large energy losses.

Barbers1.s2 has measured the specific ionization of electrons in Hz, He, and Nz in an ionization chamber. The electrons were obtained from the Stanford linear accelerator and had energies ranging from 1 to 35 MeV. A collimated beam of electrons was sent through an ionization chamber into a Faraday cup, so that the ratio of the collected ionic charge to the charge collected in the Faraday cup is proportional to the specific ionization. In Barber's first experiment,S' H2, He, and Nz at atmospheric pressure were used. Under these conditions, one does not expect any density effect cor- rection, since the density effect sets in above 35 Mev for gases a t normal pressure.20 At minimum ionization, the number of ion pairs per cm (probable specific ionization) was 7.56 f 0.09,6.15 k 0.08, and 53.2 f 0.7 in H2, He, and N2, respectively (at normal temperature and pressure). These results were compared with the theoretical expression for the restricted energy loss, - ( l / p ) ( d E / d ~ ) ~ , [Eq. (l.l.28)] with W o = 17.4 kev for Hz, 16.4 kev for He, and 70 kev for Nz, as determined from the size of the ionization chamber and the experimental conditions. Barbers1 thus obtained the following values for w, the average energy required to produce an ion pair: 37.8 k 0.7, 44.5 k 0.9, and 34.82",; ev for Hz, He, and N'L, respectively. These results are in reasonable agreement with the values of w obtained by Jesse and S a d a u s k i ~ , ~ ~ Bortner and H ~ r s t , ~ ~ and Bakker and S e g r P (see Section 1.1.2.12). The total number of ion pairs per cm a t the ionization minimum as obtained from the average energy loss, - (l/p)(dE/ds) [Eq. (1.1.1)] without any limitation on the maximum energy transfer, was found to be: 9.19 f 0.18,7.55 k 0.16, and 61.62::; for the three gases. The relativistic increase of the ionization from minimum (at -1.7 MeV) to 35 Mev is 1.17 for H2, 1.20 for He, and 1.24 for Nz. For N P , the calculated increase of the ionization loss agrees within 1% with the observed rise, but for Hz and He, the predicted increase is

U. Fano, Phys. Rev. 92, 328 (1953).

G. J. Igo, D. D. Clark, and R. M. Eisberg, Phys. Rev. 89, 879 (1953). 79 K. C. Hines, Phys. Rev. 97, 1725 (1955).

*1 W. C. Barber, Phys. Rev. 97, 1071 (1955). 82 W. C. Barber, Phys. Rev. 103, 1281 (1956).

W. P. Jesse and J. Sadauskis, Phy8. Rev. 90, 1120 (1953). R 4 T. E. Bortner and G. S. Hurst, Phys. Rev. 90, 160 (1953).

Page 33: n

1.1. INTERACTION OF RADIATION WITH MATTER 33

somewhat higher than the observed value, assuming that the energy loss per ion pair w is independent of the electron energy. In particular, for Hz, the deviation between the calculated and the observed values would correspond to an increase of (3.3 i- 0.7)% in w as the electron energy is increased to 35 MeV. It is possible that the lowering of the rate of rise of the ionizatJion is due to the production of ce rpkov radiation29 which is not reabsorbed t'o form ions in the gas.

In Barber's second experiment,82 the specific ionization of electrons in H2 and He was measured at 1 and 10 atmospheres pressure. At 10 atom- spheres, a sizable density effect is expected for both gases a t 35 MeV, whereas at 1 atmosphere the density effect correction is negligible. The experimental setup was essentially the same as in the first experiments1 (ionization chamber; Faraday cup). The theory is in good agreement with the experimental results which show that at 10 atmospheres, the specific ionization J [Eq. (1.1.28)] a t first increases above the minimum, from -2 to 10-15 MeV, but levels off above 10 Mev for H2 and 18 Mev for He. For Hz at 10 atmospheres, the value of J(35 Mev)/Jmi, is 1.12 upon inclusion of 6 (density effect), as compared to the experimental value: J / J m i , = 1.11. (The calculated value of Jmin is 3.39 Mev/g cm-2.) With- out the density effect, the calculated J / J m i , would be 1.20. Thus the data provide a good confirmation of the existence of the density effect for gases at high pressure. Upon taking into account the experimental uncertainties, BarberB2 finds that the ionization loss is decreased by (8 f 1 ) % by the density effect, as compared to the theoretical reductionz0 of 6.5 %. Similar agreement is obtained for the measurements in He at 10 atmospheres, where the observed decrease of the ionization J at 35 Mev is (3.5 f 1.3) %, as compared to the calculated valuez0 of 3%.

Low-pressure Geiger counters have also been used to determine the relativistic rise of the ionization loss in gases. In a low-pressure counter, one attempts to measure the total number of ionizing collisions:

N = /owm"xP(W) dW

where P(W) dW is the probability of a collision with energy transfer be- tween Wand W + dW. By contrast, a proportional counter measurement gives the total energy deposited: & = J~"""P(W)W dW. The fluctua- tions in & are largely due to the presence of the large energy transfers W of the order of W,.,, which are weighted by the factor W in the integrand for &. For N , on the other hand, there is no factor Win the integrand, so that the large energy transfers are weighted much less heavily than for &, and the fluctuations in N are correspondingly reduced. The condition under which the Geiger counter measures N rather than & is that the incident

Page 34: n

34 1. PARTICLE DETECTION

particles make on the average less than one ionizing collision in traversing the counter. N is proportional to the primary specific ionization J,, of the incident p a r t i ~ l e . ~ ~

Several experiments have been performed using low-pressure Geiger counters. The most extensive recent measurements for electrons in the 0.2-1.6 Mev energy rangepre those of McC1ureS6 who obtained the pri- mary specific ionization J,, of electrons in this energy range for Hz, He, Ne, and Ar. For Hz, the results could be fitted to the theoretical curve of J,, versus p/mc obtained by Bethe.86 Somewhat earlier, Heref~rd,~’ using counter measurements, obtained evidence for the relativistic rise of the ionization loss of electrons by measuring J,, for electrons in hydrogen in the range from 0.2 to 9.0 MeV. Evidence for the relativistic rise for p

mesons in hydrogen has been obtained by Shamos and Hudes.88 The relativistic rise of the ionization loss in photographic emulsion has

been the subject of numerous investigations. The first definite evidence for a -10% rise in the grain count in emulsion was obtained by Pickup and VoyvodicS9 in 1950. The minimum value of the grain count G is ob- tained for a ratio T/moc2 - 3 of the kinetic energy T to the rest energy mgc2 of the particle. The rise starts at T/moc2 N 3 and continues until the plateau value G,,,, is reached for T/moc2 - 10-100. The precise value of T/moc2 at which the plateau is reached has been the subject of some controversy, with some experiments favoring a rapid rate of rise of G to GPIst a t T / m d - 10, while others give evidence of a more gradual rise for which the plateau is reached only for T/moc2 - 50-100. In the experi- ments, following a suggestion of Morrish,So one obtains generally the blob count rather than the grain count. Here a blob is defined as either a single grain or a group of overlapping grains which cannot be resolved. It was found that the blob count is considerably more independent of the observer than the grain count.91 In the region between the minimum and the plateau, the blob count is proportional to the grain count.92 For com- parison of the theory with the grain or blob count observations, one must calculate the restricted energy loss [Eq. (1.1.2S)l with maximum energy transfer W O - 5 kev. The reason is that for energy transfers W >, 5 kev, the delta-ray will have a large enough range to traverse one or more addi-

86 G. W. McClure, Phys. Rev. 90, 796 (1953). 88 H. A. Bethe, in “Handbuch der Physik” (H. Geiger and K. Scheel, eds.), Vol. 24,

*’ F. L. Hereford, Phys. Rev. 73, 982 (1947); 74, 574 (1948). M. H. Shamos and I. Hudes, Phys. Rev. 84, 1056 (1951).

88 E. Pickup and L. Voyvodic, Phys. Rev. 80, 89 (1950). A. H. Morrish, Phil. Mug. [71 43, 533 (1952).

91 See also L. Jauneau and F. Hug-Bousser, J . phys. radium 13, 465 (1952). 92 B. Stiller and M. M. Shapiro, Phye. Rev. 92, 735 (1953).

p. 515. Springer, Berlin, 1933.

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1.1. INTERACTION OF RADIATION WITH MATTER 35

tional grains not directly in line with the path of the particle. If only the grains along the track are counted, or if a blob count is made, in which all of the overlapping grains due to an energetic 6 ray are counted as a single unit, then the observed count will be essentially unaffected by the pres- ence of the high-energy 6 rays, so that a cutoff at WO - 5 kev is indicated, as was first pointed out by Messel and Ritson.26 The resulting values of - (l/p)(dE/dx)wo as obtained by Sternheimer20n21 give a relativistic rise of 14% which saturates slowly and does not level off until T/moc2 - 100. The magnitude of the increase and the gradual character of the relativistic rise are in good agreement with the observations of Stiller and Shapirog2 using cosmic rays, and those of Fleming and Lordg3 using cosmic-ray electrons and accelerator-produced 7- mesons. On the other hand, Budini’s calculations22*2s give a more rapid rate of rise, with GpIst being reached for T/moc2 between 10 and 40, depending on the specific assumptions made about the widths of the spectroscopic lines of the Ag and Br atoms of the emulsion. Budini’s calculations are in reasonable agreement with the results of Daniel et al. 94 which indicate a more rapid rate of rise than those of references 92 and 93. Data on the ionization loss in emulsion have also been obtained by McDiarmidlg6 Michaelis and Violet,g6 Morrishlg7 and others.

Alexander and Johnstong7a have obtained a very accurate calibration curve for the grain density as a function of ppc for ?r and p mesons. In this work, the authors used T and p mesons of constant and precisely known energy from the K,2 and Kp2 decays of K partides at rest. The calibration curve extends from p = 0.5 to0.95, corresponding to 1 < g* < 3, where g* is the grain density normalized to the minimum of ionization:

g* = (dE/dX)w,/[(dE/dX) Wolrnin.

In the range 1 < g* < 1.6, the accuracy of the calibration curve for g* is estimated to be better than 1%. For the ratio of plateau to minimum grain count, the authors have obtained g,*l,, = 1.133.

Recently, J ~ n g e j a n s ~ ’ ~ has measured the relativistic rise of the grain density in Ilford G5 emulsion, using pion tracks of energy -5.4 Bev from the Berkeley Bevatron, and secondaries produced by the pions in the emulsion [pions stopping in the emulsion, with y between 1.8 and 2.5;

98 J. R. Fleming and J. J. Lord, Phys. Rev. 92, 511 (1953). 94 R. R. Daniel, J. H. Davies, J. H. Mulvey, and D. H. Perkins, Phil. Mag. [7] 43,

9 5 I. B. McDiarmid, Phys. Rev. 84, 851 (1951). 06 R. P. Michaelis and C. E. Violet, Phys. Rev. 90, 723 (1953). $7 A. H. Morrish, Phys. Rev. 91, 423 (1953). 978 G. Alexander and R. W. H. Johnston, Nuovo cimenfo [lo] 6, 363 (1957). wb B. Jongejans, Nuovo cimenlo [lo] 16, 625 (1960).

753 (1952).

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36 1. PARTICLE DETECTION

other secondary pions; electron pairs with y between 65 and 1100, where y = (T/moc2) + I]. The momentum of the tracks was determined from multiple scattering measurements. For the ratio GPlat/Gmin, a value 1.129 k 0.010 was obtained, in good agreement with the results of Stiller and Shapirog2 (1.14 2 0.03), Fleming and Lordg3 (1.14 f O.Ol), and Alexander and Johnston97a (1.133 f 0.008). The rise of the grain density G to the plateau value was found to be slow, with an appreciable increase taking place between y = 40 and y - 1000. This result is in good agree- ment with the calculations of Sternheimer,20*21 and with the experiments of references 92 and 93. A calibration curve for g” versus (y - 1) is given in the paper of J0ngejans.97~ This curve was calculated using a value of the excitation potential I = 501 ev for AgBr.

In comparing the theory with these data for emulsion, it must be borne in mind that one would expect large fluctuations of the ionization loss in each grain, since the most probable energy loss Eprcb for a minimum ionizing particle in a 0.2 p AgBr grain is only -50 ev. On the other hand, the threshold value 7 for the energy deposit below which the grain is not ex- posed is of the order of several hundreds of volts. Thus the effect of Landau-type fluctuations on the grain count is expected to be quite important, as was first pointed out by BarkasI9* and subsequently by Brown.99 In view of these difficulties and limitations of the theory, the detailed quantitative agreement which has been obtained for emulsion is surprisingly good.

In connection with the emulsion measurements on electron-positron pairs produced by y rays of very high energy (E , 2 10 Bev), Perkins’** observed that there is a reduction of the ionization loss J below the value of twice minimum ionization (2Jmi,) due to the interference between the electromagnetic fields of the positron and electron. With increasing dis- tance x from the origin of the pair, the ionization J varies form 0 to 2Jni,, as a result of the increase of the distance d between the positron and elec- tron. Several authorslo’ have treated theoretically the problem of the ionization loss of an electron-positron pair, along the lines of Fermi’s cal~ulation’~ of the ionization loss of a single particle. The theory gives the dependence of the ionization J on the distance d. The asymptotic value 2Jmi. is attained when d becomes large compared to c/(2rvP)

W. H. Barkas, in “Colloque Bur la sensibilite des cristaur et des emulsions photographiques,” Paris, September, 1951; see also W. H. Barkas and D. M. Young, University of California, Radiation Laboratory Report URCL-2579, revised (1954).

99 L. M. Brown, Phys. Rev. 90, 95 (1953). loo D. H. Perkins, Phil. Mag. [7] 46, 1146 (1955). 1olA. E. Cudakov, Izvest. Akad. Nauk S.S.S.R. 19, 650 (1955); I. Mito and H.

Ezawa, P r o p . Theoret. Phys. (Kyoto) 18, 437 (1957); G. Yekutieli, Nuovo cimento [lo] 6, 1381 (1957).

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1.1. INTERACTION OF RADIATION WITH MATTER 37

(= 0.51 X lodE cm for emulsion). Upon using the theoretical dependence of J on d, the measurements of J ( x ) enable one to obtain the value of the opening angle of the pair 0 (= d/x), from which in turn one can estimate the energy E , of the parent y ray.lo2 The results obtained by this methodLo3 are in reasonable agreement with the conventional determination of the y-ray energy from the subsequent development of the shower of electrons and y rays (see Section 1.1.5.2).

Besides the experiments on photographic emulsion, crystal counters and scintillators* have also been used to observe the relativistic increase of the ionization loss of p mesons in condensed materials. The first of these experiments was carried out by Whittemore and Streeti0* in 1949, using a silver chloride crystal. These authors compared the ionization pulses produced by p mesons of range > 112 cm of Pb (T, > 1.6 Bev) with those produced by minimum ionization p mesons (T, = 0.3 Bev) , and found a definite relativistic increase. The results were in agreement with the predictions of the Bethe-Bloch formula including the density effect correction.

Experiments with p mesons passing through an anthracene scintillator have been performed by Bowen and Roser,lo6 who obtained no detectable relativistic increase of the ionization loss Eprob above the minimum value. This result is in agreement with the theoretical predictions, since the density effect sets in at a relatively low energy (near the ionization mini- mum) for low atomic number, and is large enough to prevent any rise of €pro,, from occurring. Similar results were obtained by Baskin and W i n ~ k l e r , ~ ~ ~ ~ ~ ~ ~ using a liquid scintillator of low Z (xylene). In these experiments, it is assumed that the light output of the scintillator is proportional to the energy deposited by the incident particle ( p meson). This assumption has been verified by Chou,lo8 who showed that the re- sponse of most scintillators is nearly linear up to 3Jmin-4Jmi,.

Millar, et al. lo9 have exposed a large-area liquid scintillation counter to cosmic-ray p mesons. The counter was filled with triethyl-benzene (plus terphenyl). The most probable loss Eprob and the Landau distribution were obtained both for T, = 0.30 Bev and T , = 2.2 Bev. The value of fprob

* See also in this volume, Chapter 1.4. 102 A. Borsellino, Phys. Rev. 89, 1023 (1953). 108 W. Wolter and M. Miesowice, Nuovo cimento [lo] 4, 648 (1956). lo4 W. L. Whitternore and J. C. Street, Phys. Rev. 76, 1786 (1949). 106T. Bowen and F. X. Roser, Phys. Rev. 86, 992 (1952). 108 R. Baskin and J. R. Winckler, Phys. Rev. 92, 464 (1953). 107 See also A. G. Meshkovskii and V. A. Shebanov, Doklady Akad. Nauk S.S.S.R.

108 C. N. Chou, Phys. Rev. 87, 903 (1952). 109 C. H. Millar, E. P. Hincks, and G. C. Hanna, Can. J . Phys. 36,54 (1958).

83, 233 (1952).

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38 1. PARTICLE DETECTION

at 0.30 Bev is higher by (1.6 & 0.5) % than the value a t 2.2 Bev, in good agreement with Eq. (1.1.30) including the density effect correction 6. The prediction of the Landau theory for the width of the pulse-height distribu- tion (18 % at half-maximum) is in reasonable agreement with the observed width (20.5% at half-maximum in the central area of the counter) when the width due to the counter resolution function (8%) is taken into account.

Hudson and Hofstadterl'O have exposed a thallium-activated sodium iodide crystal [NaI(TI)] t o the cosmic-ray p-meson spectrum and have found that the resulting observed pulse-height distribution is in much better agreement with the theoretical distribution obtained upon inclusion of the density effect correction 6 (as calculated from the paper of Halpern and Hall's) than with a theoretical distribution obtained by setting 6 = 0. In each case, the theoretical curve was obtained by folding the Landau straggling distribution6a with the cosmic-ray p-meson spectrum.

In a later investigation, Bowen'" used a NaI(T1) crystal to measure the energy loss of T- and p mesons of selected energies or energy groups. The T- mesons were produced by the Chicago 450-Mev cyclotron and had well-defined energies extending from 61 to 222 MeV. In addition, 245-Mev p- mesons arising from the decay of 227-Mev T- mesons were used. Moreover, four energy groups of the cosmic-ray p-meson spectrum were studied. These groups were separated in energy by using various thicknesses of iron absorber. The average energies of the p mesons in the four groups were: T, = 368,755,1470, and 5230 MeV. The energies of the T- from the cyclotron were: 61, 85, 118, 163, and 222 MeV. At each of these energies, the most probable loss Eprob was obtained from the observed pulse distribution. Bowen thus obtained values of epr& as a function of T/moc2. The theoretical prediction20321 for 6prab versus Tlmoc2 is in reason- able agreement with these data. Thus from the calculations one obtains an 8.2% increase (relative to minimum ionization) at T,, = 50mpc2, and an asymptotic value of the rise (at very high energies) of 11.4%. The experimental value is 10.9 5 1.0% at T, = 50m,c2. This result may indi- cate that eprob rises to the plateau value somewhat more rapidly than pre- dicted by the theory. It may be noted that the reason why there is a relativistic rise of Eprob for NaI but none for anthracene or xylene is that with increasing 2, the density effect correction 6 sets in a t higher energies, thereby leading to a relativistic rise before the energy loss eprob saturates due to the onset of 6.

The density effect has been extensively studied by observing the energy

11oA. Hudson and R. Hofstadter, Phys. Rev. 88, 589 (1952). l11 T. Bowen, Phys. Rev. 96, 754 (1954).

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1.1. INTERACTION OF RADIATION WITH MATTER 39

loss of electrons in passing through thin foils. The straggling of the energy loss in thin foils was already clearly demonstrated in 1928 by the work of White and Millington,"2 as well as that of Madgwick.'13 More recently, Paul and Reich114 measured the energy loss of 2.8-Mev and 4.7-Mev electrons in foils of Be, C, Fe, and Pb. Chen and Warshaw116 showed that eprob for electrons with energies T , < 2 Mev is correctly given by Landau's theory.63 However, from their data they were unable to discriminate be- tween the Landau distributionKa of energy losses, and the (somewhat wider) distribution of Blunck and Leisegang.66 On the other hand, in the experiments of Kalil and Birkhoff,'ls an accurate comparison could be made with the Blunck-Leisegang distribution, and it was found that while for the heavy elements (e.g., Pb) this distribution is in essential agreement with the observations, for the light elements (e.g., Be) the predicted width of the distribution at half-maximum is too small by a factor of -1.8. However, the discrepancy for light elements was not observed in a more recent experiment by Hungerford and Birkhoff."' Kageyama et ~ 1 . ~ ~ ~ also obtained good agreement with the Blunck- Leisegang distribution for foils of Al, Cu, In, and Pb.

Goldwasser, Mills, and Hansonllg have measured the energy loss of 15.7-Mev electrons in passing through thin samples of Be, polystyrene, Al, Cu, and Au. With the exception of Au, they found that good agree- ment for eproo could be obtained by using the asymptotic value of the density effect correction [Eq. (1.1.2Oa)l. For the case of Au, the expression for 6 for intermediate energies, Eq. (1.1.20), must be used, as was pointed out by Warner and Rohrlich.120 The energy loss distributions of Gold- wasser et aZ.l19 are in essential agreement with those predicted from the Landau theory.6a Goldwasser, Mills, and RobillardlZ1 have obtained a direct demonstration of the density effect, by measuring the energy loss of 15.7-Mev electrons in (solid) Teflon and Kel-F, and then in the gases corresponding to Teflon and Kel-F (i.e., gases having the same chemical composition). It was found that the difference between the values of Cprob

112 P. White, and G. Millington, Proc. Roy. SOC. A120, 701 (1928). 11* E. Madgwick, Proc. Cambridge Phil. SOC. 23, 970 (1927). 114 W. Paul and H. Reich, 2. Physik 127, 429 (1950). 116 J. J. L. Chen and S. D. Warshaw, Phys. Rev. 84, 355 (1951). 116 F. Kalil and R. D. Birkhoff, Phys. Rev. 91, 505 (1953). 117E. T. Hungerford and R. D. Birkhoff, Phys. Rev. 96, 6 (1954). 118 S. Kageyama, K. Nishimura, and Y. Onai, J . Phys. Sor. Japan 8, 682 (1953);

Kageyama, S., and Nishimura, K., J . Phys. SOC. Japan 7 , 292 (1952). 119 E. L. Goldwasser, F. E. Mills, and A. 0. Hanson, Phys. Rev. 88, 1137 (1952). 120 C. Warner and F. Rohrlich, Phys. Rev. 93, 406 (1954). lZ1 E. L. Goldwasser, F. E. Mills, and T. R. Robillard, Phys. Rev. 98, 1763 (1955);

see also A. M. Hudson, Phys. Rev. 106, 1 (1957).

Page 40: n

40 1. PARTICLE DETECTION

in the solid and the gaseous phases is given by the predicted density effect correction 6.

The ionization loss is rapidly becoming an important tool in bubble chamber investigations. The bubble count (number of bubbles per cm of path) is a function only of the velocity of the particle and the temperature of the liquid. Glaser et ~ 1 . ~ ~ ~ were the first to make .a systematic study of the bubble count as a function of the velocity of the particle, by using secondary protons and T+ mesons of momenta between 0.53 and 1.60 Bev/c from the Brookhaven Cosmotron. They found that the bubble density b is approximately proportional to 1/P2. This indicates that the bubble formation is proportional to the number of slow 6 rays (secondary elect,rons). The number of 6 rays per gm/cmZ is given by

where El' is the lower limit and Ez' is the upper limit of the energies of the 6 rays considered in ns; El' and E l are in electron volts. In similarity to the grain count in emulsion or the drop count for cloud-chamber tracks, E 2 is taken as the energy of a S ray that has a long enough range to extend to a visible distance from the track of the incident (primary) particle. One thus obtains E2' = 50 kev. El' is taken as -3 times the mean excitation potential I of the atoms of the liquid, so that the 6 rays with energy E1' can be treated as free electrons during the collision. Thus El' is a t most a few kev, and therefore 126 is not very sensitive to the precise value of E i , since 1/Ez/ << l/El'. From their experimental data, Glaser et have obtained the following empirical expression for the bubble count b :

b = (A /p2) + B ( T ) bubbles/cm (1.1.36)

where A = 9.2 5 0.2 bubbles/cm for protons in propane, and B(T) is a function only of the temperature T. For propane, B C 8 at 56"C, =11 at 57"C, and B rises rapidly above 57°C to a value of =38 at 59.5"C. A is constant between 55°C and 59.5"C1 but decreases rapidly below 55OC. (At 50°C no tracks are visible.) By using fast comparison tracks of known velocity, the velocity can be determined to an accuracy of 5% for proton tracks 10 cm long.

In a more recent paper, Willis et ~ 1 . ' ~ ~ have suggested that instead of counting the number of bubbles along the track, the distribution of

122 D. A. Glaser, D. C. Rahm, and C. Dodd, Phys. Rev. 102, 1653 (1956); see also G. A. Askarian, Zhur. Eksptl. i Teorel. Fiz. 30, 610 (1956); [translation: Soviet Phys. JETP 3, 4 (1056)l.

123 W. J. Willis, E. C. Fowler, and D. C. Rahm, Phys. Rev. 108, 1046 (1957).

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1.1. INTERACTION OF RADIATION WITH MATTER 41

lengths of the spaces between bubbles should be obtained. This method was found to be more accurate than the direct bubble count, because i t avoids errors in the bubble counting method which are due to limited optical resolution of the images of neighboring bubbles and possible bubble coalescence. This procedure for bubble tracks is somewhat analo- gous to the blob count for nuclear emulsion tracks, since an aggregate of neighboring or overlapping bubbles (“blobs”) is effectively treated as a single unit. The bubble count obtained from the distribution of spacings can be fitted by the formula: b = C ( T ) / p 2 , where C ( T ) is a function only of the temperature. Willis et al. point out that a similar dependence to the 1/p2 dependence of n6 is obtained by considering the restricted energy loss with maximum energy transfer WO = 70 kev, above which the 6 rays are assumed to form separate tracks not in line with the track of the incident particle. Equation (1.1.28) for - (l/p) (dE /dx)w, gives an ap- proximate dependence l/p1,8a (in the velocity range of interest) which is experimentally indistinguishable from l/p2.

Blinov et in an experiment similar to that of Willis et ~1.~123 also obtain a l /p2 dependence for the bubble density as a function of velocity.

1.1.2.10. Capture and Loss of Electrons at Very Low Energies. For very low energies, the charged particle may capture an electron. Subse- quently the electron may be lost again. This process is very complicated, and a complete theoretical treatment does not exist a t present. For a review of the literature on this problem, the reader is referred to the review articles of Allison and Warshaw,60 and Bethe and Ashkin.3 A thorough discussion of the problems involved has been given by Bohr.126

From the Thomas-Fermi model, BohrlZ6 has obtained the following expression for the capture cross section uc:

U C - ~TUH~Z~’~(VO/V)~ (V >> V o ) (1.1.37)

where v o = e2/h is the velocity of an electron in the first Bohr orbit of hydrogen (of radius aH), and v is the velocity of the incident particle. Equation (1.1.37) holds for heavy eIements, which have several atomic electrons with velocities vel larger than v . For light elements, where this condition is not fulfilled, Brinkman and K r a m e r P have derived the following formula for uc:

uc = (2’~,/5)a,2zz(v,/v)’2 (1.1.38)

which is expected to hold for ( v / v o ) 2 10.

31, 762 (1956); [translation: Soviet Phys. JETP 4 , 661 (1957)l. 124 G. A. Blinov, Iu. S. Krestnikov, and M. F. Lomanov, Zhur. Eksptl. i Teoret. Fiz.

126 N. Bohr, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 18, No. 8 (1948). 126 H. C. Brinkman and H. A. Kramers, Koninkl. Ned. Akad. Wetenschap., Proc. 33,

973 (1930).

Page 42: n

42 1. PARTICLE DETECTION

The theory of the loss of electrons by a charged particle (ion) is less complicated. The cross section for loss al is of the order of r u H z S 10-16 cm2 for v - V O , i.e., for protons of -25 kev. uz falls off rather slowly with in- creasing v . The capture and loss cross sections of protons are equal (a, = ul ) for air at -20 kev and for hydrogen a t -50 kev. Above this energy, we have ae < a.

For intermediate 2 values, BohrlZ6 has obtained the following estimate of a1:

a1 - 7raH222 /3 (421) . (1.1.39) Thus for protons, q/aC varies as Z 1 / 8 T 6 / 2 for medium 2. The ratio is nearly independent of 2 and increases rapidly with energy. In agreement with this result, i t is found experimentally that at predominates rapidly over ac

as the energy is increased above the value (-25 kev) for which ac = a ~ .

The processes of capture and loss of electrons are very important for the energy loss and range of fission fragment^.^

1.1.2.11. The Stopping Power at Very Low Energies. For very low energies, where the velocity of the particle is less than the velocity of the atomic electrons (T , 5 25 kev for protons), Fermi and Teller127 have obtained the following expression for the energy loss:

(1.1.40)

where Ry is the Rydberg unit, and urn is the maximum velocity of the electrons of the substance if the latter are regarded as constituting a degenerate Fermi gas. Thus urn = (3~~/8?r)'/~(h/m), where TL is the number of electrons per ~ m . ~ Equation (1.1.40) shows that the energy loss in- creases with increasing v in this region, in contrast to the decrease with increasing v at higher energies ( = l / v 2 ) .

Experimentally, good evidence has been obtained for the increase of the stopping power with increasing velocity a t low energies. Warshawl28 has made careful measurements of dE/dx for protons in Be, Al, Cu, Ag, and Au in the energy range from 50 to 400 kev. For all cases, he obtained a maximum of dE/dx in the neighborhood of T, = 100 kev. In the region below the maximum, an extrapolation of Warshaw's results (from -50 to -25 kev) could be well fitted by the Fermi-Teller formula. The maxi- mum of the ionization loss dE/dx , to be denoted by J,, has the value 640 Mev/gmcm-2 for Be, where it occurs a t TP., = 75 kev. For Al, J, = 440 Mev/gm cm+ and T,,, = 72 kev. For Cu, Ag, and Au, J , = 230, 140, and 100 Mev/gm cm-2, respectively. The corresponding values of T,,, are 140, 160, and 160 kev, respectively.

127 E. Fermi and E. Teller, Phys. Rev. 73, 399 (1947). l a* S. D. Warshaw, Phys. Rev. 76, 1759 (1949).

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1.1. INTERACTION OF RADIATION WITH MATTER 43

For somewhat higher energies (T, - 400 kev), above T,,,, BohrlZ6 has given an approximate theory based on the Thomas-Fermi model of the atom, and has obtained the following expression for the energy loss:

dE - 1 6 ~ n Z l ' ~ h e ~ dx mu

- - - (1.1.41)

In this region, -dE/dx goes as l / u , instead of the l / u 2 dependence which prevails at somewhat higher energies. Warshaw lZ8 has also obtained rea- sonable agreement of Eq. (1.1.41) with stopping power data for Cu, Ag, and Au in the range from T, = 350 to 550 kev. For a more detailed discussion of the stopping power measurements at low energies ( T , 2 2 MeV) the reader is referred to the review article of Allison and Warshaw.60

1.1.2.12. The Energy w Required to Produce an Ion Pair in a Gas. When a heavy charged particle passes through a medium, it excites and ionizes the atoms of the material. The ion pairs which are formed by direct action of the particle in the immediate vicinity of its path are called the primary ions. The most energetic of these primary ions, called delta rays, may travel a considerable distance before being themselves stopped by the medium. In the slowing down process, the delta rays produce addi- tional ions called secondary ions. The sum of the primary and secondary ionization constitutes the total ionization produced by the passing charged particle.

It has been found experimentally that the energy w required to produce an ion pair is approximately independent of the energy and charge of the incident particle. Moreover, w does not vary appreciably for different gases, all values being of the order of 25-35 ev/ion pair. Typical values of w, obtained by Jesse and S a d a ~ k i s ' ~ ~ are as follows: 36.3 ev for H2, 42.3 ev for He, 35.0 ev for N2, 26.4 ev for Ar, 34.0 ev for air, and 32.9 ev for COz. Fano130 has proposed a theory which explains both the fact that w is independent of the energy and charge of the incident particle, and also the smallness of the variation of w with the atomic number 2.

The constancy of w as a function of E has been widely used in ionization chambers for the determination of the energy of particles. Thus if a particle is stopped in the gas of an ionization chamber, its initial energy is proportional to the total number of ions produced, which can be elec- tronically measured by means of a linear amplifier. It is necessary to know the value of w for the gas in the ionization chamber; w can be deter- mined by measuring the number of ions produced by a particle whose

lZ9 W. P. Jesse and J. Sadaukis, Phys. Rev. 107, 766 (1957); see also T. E. Bortner and G. S. Hurst, Phys. Rev. 95, 1236 (1954); R. H. Frost and C. E. Nielsen, Phys. Rev. 91, 864 (1953).

u0 U. Fano, Phys. Rev. 70, 44 (1946); 72, 26 (1947).

Page 44: n

44 1. PARTICLE DETECTION

energy is known by other means (e.g., natural a particle or particle originating from an exothermic nuclear reaction).

1.1.3. Range-Energy Relations

The mean range R of a particle of kinetic energy TI is given by

(1.1.42)

where -(l/p)(dE/dx) is the average energy loss as obtained from Eq. (1.1.1) or (1.1.19). In this section, we shall restrict ourselves to particles heavier than electrons, since the range of high-energy electrons is deter- mined by the bremsstrahlung and shower production rather than the ionization loss, as will be discussed in Section 1.1.5 [see Eqs. (1.1.81) and (1.1.82)].

1.1.3.1. Summary of Range-Energy Relations. Range energy relations have been obtained by several authors. In 1937, Livingston and Bethe36 published range-energy relations for protons, deuterons, and a particles in air. In obtaining these results, various experimental data on the ranges of natural a particles were used. The ranges of a particles and protons are related by

R,(T,) = 1.0072 R,(3.971 T,) - 0.20 cm (1.1.43)

where R,(T,) is the proton range for an energy T p ( E *T,), and R,(T,) is the a-particle range for an energy T,; 1.0072 = za2(mp/ma); 3.971 =

(m,/m,) [Eq. (1.1.44)], and the constant term131 -0.20 cm is due to the capture and loss of electrons at low energies which has a somewhat differ- ent effect on a particles and protons. The proton range-energy relation of Livingston and Bethe extends up to 15 MeV. In 1947, Smith1S2 obtained range-energy relations for protons up to 10 Bev, both for air and alumi- num. For air, Smith used the same value of I as Livingston and Bethe:36 Iair = 80.5 ev; for Al, Wilson’s value3’ la l = 150 ev was used. Somewhat later, Aron el aL2 calculated proton range-energy relations for a number of metals and gases, up to 10 Bev, using a value of I = 11.52 ev, which was essentially derived from Wilson’s result for Al. The calculations of Aron et al. as well as those of Smith neglect the density effect, which becomes important for proton energies T, above -2 Bev. The tables of Aron et al. have been extended by Rich and made^.'^^ A summary of the range measurements at various energies has been given by Bethe and Ashkh3

l 3 1 P. M. S. Blackett and L. Lees, Pmc. Roy. SOC. A134, 658 (1932). 182 J. H. Smith, Phys. Rev. 71, 32 (1947). la3 M. Rich and R. Madey, University of California Radiation Laboratory Report

UCRL-2301 (1954).

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1.1. INTERACTION OF RADIATION WITH MATTER 45

There have been several determinations of the range-energy relation for nuclear emulsion. In 1953, V i g n e r ~ n ' ~ ~ obtained a range-energy rela- tion, based on older data, particularly those of R ~ t b l a t , ' ~ ~ and Cuer and Jung. l3-5 Vigneron's results were later extended by Barkas and Young.137 Calculations of the range-energy relation for high energies, including the density effect correction in dE/dx, have been carried out by Baroni et ~ 1 . l ~ ~ Friedlander, Keefe, and M e n ~ n , ' ~ ~ " have made a comparison of the ranges in emulsion and in aluminum for protons of energies 87, 118, and 146 MeV. Recently, Barkas and his c o - w o r k e r ~ ~ ~ ~ have made very extensive measurements of the ranges in Ilford G5 emulsion, taking into account the effect of the water content of the emulsion on the range- energy relation. The water content determines the density of the emulsion. Barkas140 has calculated a new and very accurate range-energy relation for Ilford G5 emulsion for a "standard density" of 3.815 gm/cm3, and has given the correction which must be applied to ranges measured under nonstandard conditions to obtain the corresponding ranges for the stand- ard density (and hence the energy of the particle). In obtaining the values of dE/dx used in calculating the range [Eq. (1.1.42)], Barkas has included both the shell correction U at low energies and the density effect correc- tion 6 a t high energies. The mean excitation potential I was used as an adjustable parameter, to be determined so as to give the best fit of R ( T ) t o the available range measurements. I n this manner, a value I = 331 f 6 ev was obtained, which gives an average I / Z = 12.1 & 0.2 ev for the elements of emulsion (excluding the hydrogen). This value of I/2 is in good agreement with the recent results of Bichsel el aL4' and of Burkig and M a c K e n ~ i e . ~ ~

1.1.3.2. Calculations of the Range-Energy Relations of Protons for 6 Substances. As mentioned above, the range-energy tables of Aron et aL2 do not take into account the density effect correction 6. Moreover, these tables were calculated for an excitation potential I = 11.52 ev, which is somewhat lower than the most recent value, I - 12.5-132 ev, as obtained

l a 4 L. Vigneron, J. phys. radium 14, 145 (1953) la6 J. Rotblat, Nature 167, 550 (1951). 136 P. Cuer and J. J. Jung, Sci. et ind. phof. 22, 401 (1951). 137 W. H. Barkas and D. M. Young, University of California Radiation Laboratory

188 G. Baroni, C. Castagnoli, G. Cortini, C. Franzinetti, and A. Manfredini, Report

la*s M. W. Friedlander, D. Keefe, and M. G. K. Menon, Nuovo cimenfo [lo] 6,

Report UCRL-2579, revised (1954).

BS-9, Istituto di Fisica dell'Universit8, Rome, 1954.

461 (1957). 189 W. H. Barkas, P. H. Barrett, P. Cuer, H. H. Heckman, F. M. Smith, and H. K.

Ticho, Phys. Rev. 102, 583 (1956); Nuovo cimento [lo] 8, 185 (1958). $40 W. H. Barkas, Nuovo cimento [lo] 8, 201 (1958).

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46 1. PARTICLE DETECTION

from references 47 and 49. Sternheimer' has carried out calculations to determine new range-energy relations for some of the commonly used materials, using the higher I values and including the density effect correction. Range-energy relations have been obtained for 6 substances : Be, C, Al, Cu, Pb, and air, for proton energies T , from 2 Mev to 100 Bev. The reason for choosing 100 Bev as the upper limit of the tables is that this enables one to obtain pmeson ranges up to T, - 10 Bev, upon applying a small correction for the fact that the maximum energy transfer W,,, becomes slightly dependent on the mass ratio m,/m a t the highest energies considered. This correction will be given below. The ranges obtained in this work' are -1 to -9% higher for T , = 10 Bev than t,hose of Aron et aL2 and of Smith.132 The largest differences occur for Be (9.2%) and C (6.4%).

Details of the calculations of the ranges are given in reference 1. The values of - ( l /p ) (dE/dz) were calculated from Eq. (1.1.1). The mean excitation potentials I were obtained from the work of C a l d ~ e 1 1 , ~ ~ Bichsel et U Z . , ~ ~ and Burkig and M a ~ K e n z i e . ~ ~ The following I values were used: IBe = 64 ev, Ic = 78 ev, Iair = 94 ev, IAl = 166 ev, Icu = 371 ev, and Ipb = 1070 ev. The density effect correction 6 was evaluated from the calculations of Sternheimer.20*21 The K and L shell corrections CK and CL a t low energies were obtained from Walske's Table I1

TABLE 11. Values of the Constants Used to Obtain the Ionization Loss' Z is the mean excitation potential. A and B are the constants appearing in Eq. (1.1.19).

C, a, s, X O , and XI enter into the expression for the density effect correction 6 [Eqs. (1.1.20) and (1.1.2Oa)l.

Material Z (ev) A (s) B -C a S xo XI

Be 64 0.0681 18.64 2.83 0.413 2.82 -0.10 2 C 78 0.0768 18.25 3.18 0.509 2.67 -0.05 2 A1 166 0.0740 16.73 4.25 0.110 3.34 0.05 3 Cu 371 0.0701 15.13 4.71 0.118 3.38 0.20 3 Pb 1070 0.0608 13.01 6.73 0.0542 3.52 0.40 4 Air 94 0.0768 17.89 10.70 0.126 3.72 1.87 4

gives the values of the constants which were used in the calculation of the ionization loss [Eq. (1.1.19)] and the density effect correction 6 [Eqs. (1.1.20) and (1.1.2Oa)l.

Table I11 gives the values of - (l/p)(dE/dx) for protons. The resulting range-energy relations are presented in Table IV.

Recently Sternheimer140a has derived an expression for the range- energy relation R(T,) for protons as a function of the mean excitation

140s R. M. Sternheimer, Phys. Rev. 118. 1045 (1960).

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1.1. INTERACTION OF RADIATION W I T H MATTER 47

TABLE 111. Values of the Ionization Loss -(l/p)(dE/dz) (in Mev/gm crn-l) for Protons in Be, C, Al, Cu, Pb, and Air1

A1 c u Pb Air

2 3 4 5 6 7 8 9 10 12 14 16 18 20 22.5 25 27.5 30 35 40 45 50 55 60 65 70 75 80 90 100 110 120 130 140 150 160 1 80 200 225 250 275 300 325 350 375

131.9 97.45 78.06 65.59 56.69 50.15 45.03 40.99 37.63 32.44 28.62 25.65 23.30 21.38 19.41 17.80 16.47 15.34 13.53 12.15 11.05 10.15 9.412 8.788 8.254 7.791 7.385 7.026 6.424 5.933 5.527 5.187 4.896 4.644 4.424 4.232 3.908 3.647 3.384 3.173 3.000 2.853 2.730 2.625 2.534

140.6 104.4 83.97 70.74 61.29 54.28 48.81 44.47 40.87 35.29 31.17 27.96 25.42 23 34 21.21 19.46 18.01 16.79 14.82 13.32 12.12 11.14 10.33 9.645 9.062 8.556 8.112 7.719 7.061 6.526 6.079 5.706 5.388 5.112 4.872 4.659 4.304 4.016 3.728 3.497 3.307 3.148 3.013 2.896 2.797

110.8 83.16 67.44 57.19 49.84 44.38 40.09 36.67 33.80 29.35 26.04 23.45 21.39 19.70 17.95 16.52 15.32 14.31 12.67 11.41 10.41 9.584 8.902 8.325 7.831 7.402 7.026 6.693 6.132 5.674 5.292 4.973 4.700 4.464 4.258 4.077 3.768 3.522 3.272 3.072 2.908 2.771 2.655 2.555 2.469

78.93 61.83 51.27 44.08 38.73 34.71 31.50 28.94 26.77 23.38 20.83 18.82 17.22 15.91 14.54 13.42 12.48 11.68 10.38 9.383 8.584 7.925 7.378 6.914 6.514 6.167 5.861 5.590 5.133 4.760 4.449 4.187 3.961 3.767 3.594 3.445 3.192 2.989 2.783 2.616 2.480 2.366 2.268 2.185 2.112

41.14 34.62 29.85 26.36 23.65 21.54 19.81 18.40 17.18 15.23 13.73 12.52 11.54 10.73 9.874 9.163 8.564 8.050 7.203 6.548 6.020 5.581 5.213 4,900 4.629 4.391 4.181 3.996 3.682 3.424 3.209 3.027 2.870 2.734 2.616 2.511 2.333 2.189 2.042 1.924 1.828 1.747 1.678 1.619 1.568

134.0 99.86 80.53 68.00 58.99 52.32 47.11 42.96 39.51 34.15 30.20 27.10 24.66 22.66 20.61 18.93 17.53 16.35 14.44 12.98 11.82 10.87 10.09 9.420 8.852 8.360 7.928 7.546 6.904 6.382 5.950 5.587 5.276 5.007 4.773 4.567 4.221 3.942 3.660 3.434 3.248 3.093 3.961 2.848 2.751

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48 1. PARTICLE DETECTION

TABLE 111. Values of the Ionization Loss - (l/p) ( d E / d x ) (in Mev/gm crn-)) for Protons in Be, C, Al, Cu, Pb, and Air1 (Continued)

400 450 500 550 600 700 800 900 1000 1250 1500 1750 2000 2250 2500 2750 3000 3500 4000 4500 5000 6000 7000 8000 9000 10,000 12,500 15,000 17,500 20,000 22,500 25,000 27,500 30,000 40,000 50,000 60,000 70,000 80,000 90,000 100,000

2.453 2.321 2.215 2.129 2.059 1.950 1.871 1.812 1.767 1.692 1.649 1.623 1.608 1.599 1.595 1.593 1.593 1.597 1.604 1.612 1.621 1.638 1.655 1.670 1.685 1.699 1.728 1.753 1.774 1.792 1.808 1.822 1.835 1.847 1.886 1.915 1.939 1.959 1.976 1,991 2.005

2.709 2.563 2.448 2.355 2.278 2.159 2.074 2.009 1.960 1.879 1.833 1.806 1.791 1.782 1.778 1.777 1.778 1.784 1.793 1.802 1.813 1.834 1.854 1.873 1.890 1.905 1.939 1.968 1.993 2.014 2.033 2.050 2.065 2.077 2.122 2.156 2.183 2.206 2.225 2.242 2.257

2.392 2.268 2.169 2.090 2.022 1.921 1 3 4 9 1.795 1.754 1.687 1.649 1.629 1.618 1.613 1.611 1.613 1.615 1.624 1.635 1.647 1.659 1.682 1.704 1.724 1.743 1.759 1.796 1.827 1.853 1.876 1.895 1.913 1.929 1.944 1.991 2.027 2.056 2.080 2.100 2.118 2.134

2.049 1.945 1.863 1.795 1.741 1.658 1.598 1.555 1.522 1.471 1.443 1.429 1.422 1.420 1.422 1.425 1.429 1.440 1.452 1.465 1.478 1.502 1.524 1.544 1.562 1.579 1.615 1.645 1.671 1.693 1.712 1.729 1.745 1.759 1.804 1.839 1.866 1.890 1.909 1.926 1.941

1.523 1.448 1.390 1.343 1.305 1.246 1.205 1.175 1.153 1.120 1.104 1.099 1.099 1.102 1.108 1.114 1.121 1.135 1.150 1.164 1.178 1.204 1.227 1.248 1.267 1.284 1.321 1.351 1.377 1.399 1.418 1.436 1.451 1.465 1.511 1.546 1.574 1.597 1.616 1.633 1.648

2.666 2.524 2.413 2.323 2.249 2.136 2.055 1.995 1.950 1.877 1.838 1.819 1.809 1.806 1.808 1.812 1.818 1.834 1.851 1.870 1.889 1.924 1.958 1.989 2.017 2.044 2.102 2.151 2.194 2.232 2.265 2.296 2.323 2.348 2.433 2.499 2.552 2.597 2.631 2.661 2.687

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1.1. INTERACTION O F RADIATION WITH MATTER 49

TULE IV. Range-Energy Relations for Protons in Be, C, Al, Cu, Pb, and Air1 The range R is given in grn cm-z.

TP (MeV) Be C A1 c u Pb A i r

2 3 4 5 6 7 8 9 10 12 14 16 18 20 22.5 25 27.5 30 35 40 45 50 55 60 65 70 75 80 90 100 110 120 130 140 150 160 180 200 225 250 275 300 325 350 375

0.0091 0.0180 0.0296 0.0436 0.0601 0.0789 0.0999 0.1232 0.1487 0.2061 0.2719 0.3459 0.4278 0.5175 0.6404 0.7750 0.9212 1.079 1.426 1.817 2.249 2.722 3.234 3.784 4.371 4.995 5.655 6.349 7.840 9.461 11.21 13.08 15.06 17.16 19.37 21.68 26.61 31.91 39.03 46.67 54.78 63.33 72.30 81.64 91.34

0.0084 0.0168 0.0275 0,0406 0.0558 0.0732 0.0926 0.1141 0.1376 0.1904 0.2508 0.3187 0.3937 0.4759 0.5884 0.7116 0.8452 0.9891 1.307 1.663 2.057 2.488 2.954 3.456 3.991 4.559 5.160 5.792 7.148 8.623 10.21 11.91 13.72 15.62 17.63 19.73 24.20 29.02 35.49 42.42 49.77 57.53 65.65 74.12 82.91

0.0115 0.0221 0.0355 0.0517 0.0704 0.0917 0.1155 0.1416 0.1700 0.2337 0.3062 0.3872 0.4766 0.5742 0.7073 0.8526 1.010 1.179 1.551 1.967 2.427 2.928 3.469 4.051 4.670 5.327 6.021 6.750 8.313 10.01 11.84 13.79 15.86 18.04 20.34 22.74 27.85 33.34 40.72 48.61 56.98 65.79 75.02 84.62 94.58

0,0190 0.0335 0.0513 0.0724 0.0967 0.1240 0.1542 0.1874 0.2234 0.3035 0.3943 0.4954 0.6066 0.7276 0.8922 1.071 1.265 1.472 1.927 2.434 2.992 3.599 4.253 4.954 5.699 6.488 7.321 8.195 10.06 12.09 14.27 16.58 19.04 21.63 24.35 27.19 33.23 39.71 48.39 57.66 67.49 77.82 88.61 99.85 111.5

0.0410 0.0676 0.0988 0.1345 0.1746 0.2190 0.2674 0.3198 0.3761 0.5000 0.6385 0.7912 0.9576 1.138 1.381 1.644 1.926 2.229 2,885 3.614 4.411 5.275 6.202 7.192 8.243 9.352 10.52 11.74 14.35 17.17 20.19 23.40 26.80 30.37 34.11 38.02 46.29 55.14 66.98 79.61 92.95 107.0 121.6 136.7 152.4

0.0087 0.0175 0.0287 0.0423 0.0581 0.0761 0.0963 0.1185 0.1428 0.1974 0.2598 0.3299 0.4073 0.4920 0.6078 0.7346 0.8720 1.020 1.346 1.712 2.116 2.557 3.035 3.549 4.097 4.678 5.293 5.940 7.327 8.835 10.46 12.20 14.04 15.99 18.03 20.17 24.73 29.64 36.23 43.29 50.79 58.68 66.95 75.56 84.50

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50 1. PARTICLE DETECTION

TABLE IV. Range-Energy Relations for Protons in Be, C, Al, Cu, Pb, and Air1 (Continued)

400 450 500 550 600 700 800 900 1000 1250 1500 1750 2000 2250 2500 2750 3000 3500 4000 4500 5000 6000 7000 8000 9000 10,000 12,500 15,000 17,500 20,000 22,500 25 , 000 27 , 500 30,000 40 , 000 50,000 60,000 70,000 80,000 90,000 100,000

101.4 122.3 144.4 167.5 191.3 241.3 293.7 348.1 404.0 548.9 698.8 851.7 1007 1163 1319 1476 1633 1946 2259 2570 2879 3493 4100 4702 5298 5889 7347 8784 10,202 11,604 12 , 993 14,370 15 , 737 17,095 22,450 27,711 32 , 899 38,030 43,112 48,152 53 , 158

91.99 111.0 131.0 151.8 173.4 218.6 265.9 314.9 365.3 495.8 630.7 768.2 907.3 1047 1188 1328 1469 1750 2029 2308 2584 3133 3675 4212 4743 5270 6570 7850 9112 10,359 11,595 12,820 14,036 15,243 20,003 24,677 29,286 33,843 38,356 42 , 833 47,278

104.9 126.4 148.9 172.4 196.7 247.6 300.7 355.6 412.0 557.7 707.7 860.4 1014 1169 1324 1479 1634 1943 2250 2555 2857 3456 4046 4629 5206 5777 71 83 8563 9922 11,262 12 , 588 13,901 15 , 202 16 , 494 21,574 26,550 31 , 448 36,284 41,067 45,807 50,509

123.5 148.6 174.9 202.2 230.5 289.5 350.9 414.4 479.4 646.8 818.7 992.9 1168 1344 1520 1696 1871 2220 2566 2908 3248 3919 4580 5232 5876 6512 8077 9610 11,117 12,604 14,072 15,525 16,964 18,391 24,002 29,491

40,214

50,692 55,863

34 , 888

45,477

Pb Air

168.6 202.3 237.6 274.2 312.0 390.5 472.2 556.3 642.2 862.7 1088 1315 1543 1770 1996 222 1 2445 2888 3326 3758 4185 5024 5847 6655 7450 8234 10,153 12,023 13,856 15,657 17,432 19,184 20,915 22,629 29,344 35,883 42,290 48,596 54,820 60 , 975 67,070

93.73 113.0 133.3 154.4 176.3 222.0 269.8 319.2 370.0 500.9 635.7 772.5 910.3 1049 1187 1325 1463 1737 2008 2277 2543 3067 3583 4089 4589 5081 6287 7462 8612 9742 10,853 11,950 13,032 14,102 18,282 22 , 336 26,295 30,177 34,002 37,781 41,519

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1.1. INTERACTION O F RADIATION W I T H MATTER 51

potential I . The expression for R(T,) is obtained by an interpolation of the previously calculated range-energy relations’ for Be, Al, Cu, and P b (see Table IV). The result for R(T,) is accurate to within 1% for values of I between 60 and 1100 ev. The expression for R(T,) can be used to calculate the range-energy relation for any substance, provided an appropriate value of I is assumed.

1.1.3.3. Range-Energy Relations for Particles other than Protons. The Correction Factor Fi. The range R for any other particle i (heavier than an electron) with energy Ti can be obtained from the proton ranges of Table IV by means of the relation

R;(Ti) = - - R, 3 Ti F i zi2 (tl) (mi ) ( 1.1.44)

where z, is the charge of the particle, mi is its mass, mp = proton mass, and R,[(mp/m,) Ti] is the proton range for the appropriate energy (m,/m,) Ti. In Eq. (1.1.44), the factor Fi corrects for the slight dependence of the maximum energy transfer W,,, on mi a t very high energies. Thus Wmax for 1.1, T , and K mesons is slightly smaller than for protons with the same value of yi = Ei/mic2, where Ei is the total energy (including rest mass) of the particle. Hence - (l/p) (dE/dz) is decreased and the range Ri is slightly increased for mesons (Fi > 1). From Eqs. (1.1.1) and (1.1.15), one finds that the change of - (l/p)(dE/dz) is given by

Values of F, for p mesons are given in Table V. These values were obtained’ by numerical integration of Eq. (1.1.42) with - (l/p) (dE/dz) calculated from the appropriate W,,, for p mesons.

Table V shows that the correction for 1.1 mesons is very small ( F , - 1 5 0.01), and that the values of F , are practically independent of 2, being nearly the same for Be and Pb. For 7r and K mesons, the corrections F , and F K are not tabulated, since one will not generally be interested in the ranges of these particles for y i 2 5 , in view of the large probability that they will interact before coming to the end of the range. Actually for a given yi, the corrections are even smaller than for 1.1 mesons. Thus for Pb, F , = 1.0095 for Y~ = 100, and F K = 1.0017 for Y K = 100.

It should be noted that a t very high energies [E >> (mi2/m)c2], spin- dependent effects on the energy loss in close collisions will be present,141 which are not included in the Bethe-Bloch formula.

1.1.3.4. Range Straggling. When a beam of particles loses energy by 141 See, for example, B. Rossi, “ High-Energy Particles,” p. 14. Prentice-Hall, New

York, 1952.

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52 1. PARTICLE DETECTION

TABLE V. Values of the Factor F, Which Enters into the Expression for the p-Meson Range R,, at Very High Energies [Eq. (1.1.44)11

F,, is given for Be and Pb, as a function of 7, = E,/m,,ea.

4 6 8

10 15 20 25 30 40 50 60 70 80 90

100

1.0010 1.0014 1.0017 1,0020 1.0027 1.0034 1,0041 1.0047 1.0058 1.0068 1.0079 1.0089 1.0098 1.0107 1.0115

1.0013 1.0017 1.0021 1.0025 1.0032 1.0039 1.0047 1.0054 1.0066 1 -0077 1.0088 1.0098 1.0107 1.0116 1.0125

ionization, all of the particles do not come to the end of their range and stop after traversing the same thickness of material. Instead there is a distribution of the ranges due to the statistical nature of the ionization loss process. This distribution is a Gaussian. The probability p(R) dR of a particle of well-defined initial energy To having a range between R and R 4- dR is given by

p(R) dR = __ 1 exp [ - ( R a ~ 0 ) 2 ] dR (1.1.46) cYT1'2

where CY' E 2(R - RQ);, = 2Jp(R)(R - Ro)'dR. (1.1.47)

In Eqs. (1.1.46) and (1.1.47), Ro is the mean range142 obtained above [Eq. (1.1.42)] by integration over the average energy loss - (l /p)(dE/dx). An approximate equation for ( R - Ro);, has been obtained by Bohr.12 For sufficiently large initial velocities v of the particle (2mv2 2 I K , where IK is the ionization potential of the K shell), Bohr's formula gives

( R - Ro);, = 4re4z2NZ loTo ( d E / d ~ ) - ~ dT (1.1.48)

where TO is the initial energy of the particle, N = number of stopping atoms per ema, and Z = atomic number of stopping material.

In practice, it is difficult to obtain directly the value of ( R - RO);, from the observed distribution of ranges. Instead one obtains the number-

142 See also H. W. Lewis, Phys. Rev. 86, 20 (1952); U. Fano, ibid. 92, 328 (1953).

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1.1. INTERACTION OF RADIATION W I T H MATTER 53

FIG. 6. Schematic number-distance curve, showing RQ, Rextr and S (cf. Bethe and Ashkin, reference 3, p. 246, Fig. 15b).

distance curve by plotting the number of particles which survive as a function of the thickness traversed. An example of such a curve is shown in Fig. 6. From the Gaussian of Eq. (1.1.46), one finds that the curve of N has half its maximum value, N = +No, at R = Ro, the mean range of the particles which are assumed to be initially homogeneous in energy. More- over, a t R = Ro, the theoretical curve of N versus R has its maximum slope, - (1/ad2). By drawing a tangent to the N versus R curve a t its steepest point ( R = Ro) and obtaining the intersection of the tangent with the R axis (see Fig. 6), one finds the extrapolated range, Re,,,, which is given by

( 1.1.49)

The difference Rextr - RO is defined3 as the straggling parameter S. Thus

S2 = h a 2 = +(R - Ro);, (1.1.50)

and the distribution function p(R) dR [see Eq. (1.1.46)] can be written as f o l l o ~ s : ~

Re,,= = RO + TT 1 112 a.

[ - (&) ( R - R d 2 ] dR. p(R) dR = - exp 1 2 s (1 J.51)

The value of ( R - Ro);, obtained from the measured S by means of Eq. (1.1.50) can be compared with the theoretical expression, Eq. (1.1.48). Good agreement has been obtained in several experiments.

As an example, the calculated percentage straggling of protons in 100S/Ro decreases slowly from 2.29 for RO = 5 cm, to 2.13 a t

149 H. A. Bethe and J. Ashkin, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 1, p. 244. Wiley, New York, 1953.

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54 1. PARTICLE DETECTION

10 cm, 1.78 at 100 cm, 1.57 a t lo3 cm, 1.31 a t lo4 cm, and 0.97 a t lo5 em. Millburn and S ~ h e c t e r ' ~ ~ have found experimentally that SIRo varies very slowly with 2. Thus the value of SIR0 relative to copper is 0.90 for Be, 0.95 for Al, 1.02 for Ag, and 1.06 for Pb.

The range straggling in emulsion has been thoroughly investigated by Barkas et U Z . , ' ~ ~ who used protons and T+, T-, and p+ mesons, These authors have found that the Bohr formula [Eq. (1.1.48)] gives reasonable agreement with their data. They have also discussed various additional straggling effects which are present in range measurements with nuclear emulsion.

Values of the range straggling for six substances (Be, C, Al, Cu, Pb, and air) have been recently calculated by Sternheimer.'45a In these calcula- tions, the following expression for ( R - Ro)i, was used:

Equation (1.1.48a) differs from the Bohr formula [Eq. (1.1.48)] by the in- clusion of the following factors in the integrand: (1) the factor (1 - &P2) / (1 - pz), which is a relativistic correction that was first derived by Lind- hard and S ~ h a r f f ; ~ ~ (2) the factor [l + (2m/mi)y]-' which is derived in reference 145a, and which becomes important only for very high energies (y >, mi/2m); (3) the correction factor K which takes into account the effects of binding on the atomic electrons a t low velocities of the incident particle [v 5 (IK/2m)1'2]. The correction K is similar to the binding effect corrections C K and CL which enter into the Bethe-Bloch formula [Eq. (1.1.34)]. K becomes 1 for sufficiently high energies ( T , 2 100 Mev for Al; 400 Mev for Pb). The expression for K has been obtained by Bethe.3v36

The percentage range straggling t = 100[(R - R O ) ~ " ~ / R ~ decreases with increasing energy until a minimum is reached for T,/mLc2 - 2.5, which is in the same region as the minimum of the ionization loss d E / d x . Beyond the minimum, e increases with energy, as a result of the effect of the factor (1 - p2)-' in the integrand of (1.1.48a). We note that c as defined above is related to S by: t = 100(2/~)"~S/Ro. As an example, for p mesons in Cu, e, decreases from 3.94 a t T , = 10 Mev to a minimum of 2.69 a t 280 MeV, and then increases to 3.07 at T, = 1 Bev, 4.07 at 3 Bev, and 5.74 a t 10 Bev. It is that B is almost independent of 2, showing only a small increase in going from Be to Pb (at a constant

Report UCRL-2234, revised (1953). 144 G. P. Millburn and L. Schecter, University of California Radiation Laboratory

lPs W. H. Barkas, F. M. Smith, and W. Birnbaum, Phys. Reu. 98, 605 (1955). 145B R. M. Sternheimer, Phye. Rev. 117, 485 (1960).

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1.1. INTERACTION O F RADIATION W I T H MATTER 55

energy T). As an example, for 300-Mev protons, Q,(Z)/~,(CU) is 0.885 for Be, 0.895 for C, 0.941 for Al, and 1.113 for Pb.

1.1.4. Scattering of Heavy Particles by Atoms

scattering of heavy particles by nuclei of charge Z e is given by For nonrelativistic velocities, the Rutherford formula for the elastic

2?re422Z2 sin e de 16T2 s in4(p) d@Ll(B.) = (1.1.52)

where d% is the differential cross section, e is the scattering angle, ze is the charge of the particle, and T is its nonrelativistic kinetic energy. Equation (1.1.52) can also be written

(1.1.53)

where TYev is T in units MeV.

the M ~ t t l ~ ~ formula gives for the cross section: For the scattering of identical particles of spin + (protons, electrons),

2T24e4 cos f? sin e de 1 1 -~

a%(@) = T2 [sin4 e + X e L

- 1 cos tz In tan2 e)]. (1.1.54) sin2 e cos2 e The last term in the square bracket (involving ti) arises from the quantum- mechanical exchange phenomena which are a consequence of the identity of the incident particle and the scatterer.

In the inelastic collisions of heavy charged particles with atoms, i t is of interest to obtain the energy distribution of the secondary electrons (6 rays). The angle of ejection $ in the laboratory system is related as follows to the energy W of the secondary electron:

W = 2 mu2 cos2 $ (1.1.55)

where m = electron mass, v = velocity of incident heavy particle. The maximum value of $ is 90’ in which case W = 0. The cross section for ejection of a 6 ray with energy between W and W + dW is

(1.1.56)

148 N. F. Mott, Proc. Roy. Soc. Al26, 259 (1930); see also N. F. Mott and H. S. W . Massey, “The Theory of Atomic Collisions,” 2nd ed. Oxford Univ. Press, London and New York, 1949.

Page 56: n

56 1. PARTICLE DETECTION

The cross section for finding a 6 ray between $ and $ + d$ is

(1.1.57)

For relativistic energies, BhabhaZ3 has shown that the collision cross section for a particle with spin 0 is given by

( 1.1.58)

where W,,, is the maximum possible energy transfer to the atomic elec- trons [Eq. (1.1.15)]. Of course, for p+ 0, Eq. (1.1.58) reduces to the nonrelativistic expression (1.1.56). It may also be remarked that for energy transfers W << W-,, Eq. (1.1.58) gives the same result for d@ as Eq. (1.1.56). In this important case, there are no relativistic corrections to the Rutherford formula.

1.1.5. Passage of Electrons through Matter

Electrons passing through matter lose energy by ionization and excita- tion of the atomic electrons of the medium, in the same manner as heavy particles. The expression for the ionization loss of electrons has been dis- cussed in Section 1.1.2.7 [Eq. (1.1.32)]. However, in addition, for high- energy electrons, there is the possibility of radiation in the field of the nucleus (bremmsstrahlung). Above a certain energy, called the critical energy E,, the radiation loss predominates over the ionization loss, whereas for E < E,, the ionization loss is the dominant mechanism of energy loss.

1.1.5.1. Radiation by Electrons (Brernsstrahlung). The energy distribu- tion and total cross section for the bremsstrahlung depend strongly on whether the field of the nucleus is effectively screened by the atomic electrons. Thus if the effective impact parameter b is small compared to the atomic radius a - U,Z-'/~ (aa = Bohr radius), there is essentially no screening, whereas for b >> a, the screening is virtually complete. The relevant parameter for the screening is E, defined by

(1.1.59)

where Eo is the initial total energy of the electron, E is its final energy, and hv (= Eo - E ) is the energy of the radiated quantum. For 5 >> 1, there is no screening, and the energy distribution of the radiation, as obtained

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1.1. INTERACTION OF RADIATION WITH MATTER 57

by Bethe and Heitler,14' is given by

@(Eo,v) d v = 4Z2r02 ~ d v - [ 1 + (:)' - - -- "1 [In ("") - i] (1.1.60) 137 v 3 Eo mc2hv

where T O = e2/mc2 is the classical electron radius. For 5 = 0, we have complete screening, and @(EO,v) is given by

Equations (1.1.60) and (1.1.61) give only the probability of radiation in the field of the nucleus. There is also some radiation in the field of the atomic electrons. This contribution *el is of the order of 1/Z of the nuclear contribution. For the case of complete screening, Wheeler and Lamb148 have derived the following expression for the energy distribution :

4zr02 d v ([ 1 + (:J2 - _ - - 3 Eo

@,~(Eo,zJ) d v = __ - 137 Y

(1.1.62)

The bremsstrahlung in the field of the atomic electrons, in the limit of no screening, has been investigated by Borsellin0,'~9 Votruba,160 Rohrlich, 161

Nemirovsky,162 and Watson.lK3 Figure 7 shows the energy distribution of the photons from the brems-

strahlung as a function of the photon energy hv divided by the kinetic energy To of the incident electron. The curves pertain to various values of To/mc2 and were taken from the work of Bethe and Heitler.I4' These curves pertain to lead and include the effect of screening. It may be noted that the function plotted is the frequency distribution of the energy radiated hv@ [in units mc25, see Eq. (1.1.66)]. This distribution approaches a constant value as v + 0, in contrast to the probability distribution

147 H. A. Bethe and W. Heitler, Proc. Roy. SOC. A146, 83 (1934); for a discussion of the screening, see also W. Heitler, "The Quantum Theory of Radiation," 2nd ed., p. 168. Oxford Univ. Press, London and New York, 1944.

14* J. A. Wheeler and W. E. Lamb, Phys. Rev. 66, 858 (1939). 149 A. Borsellino, Nuovo cimento [9] 4, 112 (1947); Rev. univ. nuc. Tucumdn Ser. A .

160 V. Votruba, Phys. Rev. 73, 1468 (1948). l61F. Rohrlich, Phys. Rev. 96, 657 (1954); see also J. M. Jauch and F. Rohrlich,

I' The Theory of Photons and Electrons," p. 249. Addison-Wesley, Cambridge, Massachusetts, 1955; F. Rohrlich and J. Joseph, Phys. Rev. 100, 1241 (1955); J. Joseph and F. Rohrlich, Revs. Modern Phys. 30, 354 (1958).

Mat. y $8. ledrica A6, 7 (1947).

162 P. Nemirovsky, J. Phys. U.S.S.R. 11, 94 (1947). 163 K. M. Watson, Phys. Rev. 72, 1060 (1947).

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58 1. PARTICLE DETECTION

itself, which behaves as 1 / v for small frequencies v. Figure 7 shows that for small To, the energy distribution hv4, decreases uniformly as hv is in- creased from 0 to TO. With increasing TO, hv9 decreases less rapidly near Y = 0, with the main decrease occurring close to the maximum value hv = TO. For any finite TO, 9 becomes 0 a t hv = TO. The value of hv4, at v = 0 is independent of To (hva 20mc2h).

h v / T o

FIG. 7. Energy distribution of bremsstrahlung emitted by a fast electron. These curves were taken from the work of H. A. Bethe and W. Heitler [Proc. Roy. SOC. A146, 83 (1934)l; see also Bethe and Ashkin, reference 3, p. 270, Fig. 19. The curves pertain to lead, and include the effect of screening.

The energy loss of the electrons by radiation is given by

- f$)rad = N 1” hv4,(EoJv) d v (1.1.63)

where vo = Eo/h, and N = number of atoms per cm3. Equations (1.1.60) and (1.1.61) show that (P(E0,v) is roughly proportional t o 1 / v for small frequencies v, so that the integrand hvQ(Eo,v) does not have any strong dependence on v, and actually becomes almost independent of Eo and v/To for sufficiently high energies ( T O 2 100mc2) (see Fig. 7). As a result, - (dEo/dz),,d is approximately proportional to the primary energy Eo. Thus we write.

- rs) = NEoQ,,, rad

( 1.1.64)

where 9 . a is the integral of (1.1.63) divided by Eo.

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1.1. INTERACTION OF RADIATION WITH MATTER 59

For mc2 << Eo << 137rn~~Z-"~, screening can be neglected, and one finds

(1.1.65)

where I is defined by

- 1)r02 = Z(Z + 1)5.79 X lo-@ cm2. (1.1.66) @ = 137

For the case of complete screening (Eo >> 137rn~~Z-' /~) , one obtains:

where = I ' [ 4 ln(183Z-1/3) + $1 ( 1.1.67)

I' E Z(Z + {)5.79 X 10-z8 cm2 (1.1.68)

and where { is the correction for the contribution of the atomic elec- t r o n ~ ; ' ~ ~ { is of the order of 1.2-1.4.

The distance Xo over which the electron has its energy decreased by a factor e is called the radiation length. Thus XO is defined by

( 1.1.69) l /Xo = 4NI' ln(183Z-1'3).

For large energies, we have [from Eq. (1.1.67)]:

(1.1.70)

where b = 1/[18 1n(183Z-1/3)] is very small compared to unity (b = 0.012 for air, 0.015 for Pb). Table VI gives values of the radiation length XO for various materials.

TABLE VI. Values of .the Critical Energy E, and Radiation Length XO for Various Substances

This table is taken from Bethe and Ashkin, reference 3, p. 266.

Substance E, (Mev) XO (gm/cm2) ~~ ~

Hydrogen 340 58 Helium 220 85 Carbon 103 42.5 Nitrogen 87 38 Oxygen 77 34.2 Aluminum 47 23 .9 argon 34 .5 19 .4 Iron 24 13 .8 Copper 21.5 12 .8 Lead 6 . 9 5 . 8 Air 83 36.5 Water 93 3 5 . 9

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60 1. PARTICLE DETECTION

Bethe and Heitler14’ have given the following approximate formula for the critical energy:

E, S 1600mc2/Z (1.1.71)

from which they have obtained the following expression for the ratio of the radiation loss to the collision loss:

(1.1.72)

It should be noted that Eqs. (1.1.71) and (1.1.72) are very approximate, as can be seen by comparing the values of E, calculated from Eq. (1.1.71) with the actual values given in Table VI.

With increasing energy of the electron, the radiation becomes increas- ingly peaked forward. Aside from a factor which depends slowly on EO and hv, the angular distribution of the radiation du/dQ is determined by:154

(dEo/dz)r,fi - - EoZ (dEo/dz),,11 - 1 6 0 0 ~ ~ ~ ’ ’

(1.1.73)

where 0 is the angle of emission of the radiation and B is a constant. Thus the average angle of emission is given by

<@> mc2/Eo (1.1.74)

which becomes very small with increasing Eo. Recently tfberal1155 has investigated the bremsstrahlung produced by

fast electrons in single crystals. He has shown that interference phe- nomena are expected to occur which can enhance the radiation and markedly change the y-ray spectrum. A similar effect for pair production (see Section 1.1.7.3) is also discussed by Uberall. The crystal effect is small at low energies, and sets in for q noh/a, where q is the momentum transfer to the target atoms, a is the lattice constant, and no is of the order of 2 or 3. This condition corresponds to an electron energy To - 200 Mev for bremsstrahlung, and y energy hv - 1 Bev for pair production. The interference effect is confined to angles 00 of order 0 0 5 (137Z-’I3) X (mca/Eo) between the primary beam and the line of atoms participating in the interference.

In a second paper, tfberal1165 has discussed the polarization of brems- strahlung emitted from a monocrystalline target. The polarization P is defined as: P = (aL - q)/(al + all), where ul and ~ 1 1 are the cross sec- tionsjor producing radiation polarized perpendicular and parallel, respec-

lS4 A. Sommerfeld, “Atombau und Spektrallinien,” Vol. 2, p. 551. Vieweg, Braun- schweig, Germany, 1939.

lKK H. Vberall, Phys. Rev. 103, 1055 (1956); 107, 223 (1957).

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1.1. INTERACTION O F RADIATION WITH MATTER 61

tively, to the production plane (formed by the incoming electron momen- tum PO and the emitted y-ray momentum k). He has shown that, in typical cases, P is increased by a factor of -1.5 above the value obtained when an amorphous target is used. Moreover, there is a net polarization with respect to the plane formed by the incident direction (PO) and the crystal axis. Thus for a Cu crystal, at T = O", with an incident electron energy EO = 600 MeV, for 0 0 = 20 X rad, the polarization PO of the entire bremsstrahlung cone is E0.15 for z 3 kv/Eo from 0 to 0.2. Between 2 = 0.2 and 0.5, PO decreases slowly to 0. Here eo is the angle between the primary direction and the crystal axis. For O o = 5 X 10-8 rad, Po is 0.31 a t x = 0, and decreases rapidly with increasing x, becoming negative at z = 0.19, with minimum value PO = -0.11 at x = 0.33. Thus by using an appropriate angle 00, i t may be possible to obtain partially polarized y radiation of sufficient intensity to perform high-energy polarization experiments.

1 . I .5.2. Shower Production. As the eIectron proceeds through the material, it will create a shower, which is produced as follows. The elec- tron loses energy by bremsstrahlung, producing a high-energy y ray. The y ray in turn can produce an electron and positron by pair production (see Section 1.1.7.3 below). The pair in turn can radiate energy by brems- strahlung, thereby producing photons, which can then create more pairs. In this way a cascade of photons and high-energy e+ and e- is produced, which is called a shower. The number of e+ and e- present increases at first with increasing thickness t , then attains a maximum a t a certain thickness t,, and decreases for larger t. In this connection, as was men- tioned above, it is convenient to define a critical energy E, by the condi- tion that for EO = E,, the radiation loss, Eq. (1.1.701, is equal to the energy loss by ionization. Values of E , for various materials are given in Table VI.

Figure 8 shows the expected number of electrons n as a function of the thickness t in radiation lengths Xo. In the figure, loglo n is plotted against t for 4 different values of the total energy E O of the primary electron, which is given in units of the critical energy E,. These curves were taken from the work of Rossi and Greisen.lS8 Figure 8 shows that for a shower withiiincident energy EO = 100E,, n increases from n = 1 at t = 0 to a maximum n, G 10 a t t = t, 4, and thereafter decreases to n = 1 at t g 12, and is negligible for t >, 12. The thickness t, at which the maxi- mum is reached, and the value n, at the maximum both increase with increasing energy EO of the primary electron. Thus for EO = 104E,, we have n, 1000, t, E 9. In this case, n becomes negligible only for t 2 30. Many authors have treated analytically the complicated mathe-

166 B. Rossi and K. Greisen, Revs. Modern Phys. 13, 240 (1941).

Page 62: n

62 1. PARTICLE DETECTION

matical problems involved in shower production. A review of these calcu- lations is given in the book by Ro~si . '~ '

Wilson168 has recently treated the problem of the shower development by a Monte Carlo method, in which a large number of electrons are fol- lowed through the material, with a statistical (probability) determination of the bremsstrahlung and pair production processes in each particular ( ( case history." Neglecting scattering, by means of an approximate

5

4

3

C

2 2 s

I

0

- I 0

t

FIG. 8. The number n of electrons in a shower as a function of the thickness traversed t in radiation lengths. These curves were taken from the work of B. Rossi and K. Greisen [Revs. Modern Phys. 13, 240 (1941)l.

theoretical model of shower production, Wilson finds for the mean range r (in units Xo) of an electron of initial energy Eo

(1.1.75) r = ln 2 In (- Eo + 1). E, In 2

The distribution of ranges around r is approximately Gaussian, and the root mean square straggling s (in units X,) is given by

(1.1.76)

Wilson has shown from his Monte Carlo calculations that for an in- cident electron, one is more likely to find 1, 3, 5 . . . than 2, 4, 6, . . . electrons and positrons a t a given thickness in the shower, since electrons

lS7 B. Rossi, " High-Energy Particles," Chapter 5. Prentice-Hall, New York, 1952. R. R. Wilson, Phys. Rev. 84, 100 (1951); 86, 261 (1952).

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1.1. INTERACTION O F RADIATION W I T H MATTER G3

and positrons are formed in pairs. On the other hand, if the shower is initiated by a y ray, one is more likely to find 2 ,4 ,6 . . , than 1 , 3 , 5 . . . electrons and positrons a t any thickness in the shower, Wilson’s shower curves as obtained by the Monte Carlo method are more spread out than those of the general (analytic) shower theory, i.e., the shower penetrates to a greater thickness t than according to the general t,heory. One of the reasons for this difference is that Wilson’s calculations take into account the fact that the low-energy y rays have a relatively long mean free path (see Section 1.1.7.4).

1.1.5.3. Production of Secondary Electrons by Electrons. Scattering of Electrons by Electrons and Nuclei. The cross section for ejection of sec- ondary electrons (6 rays) by an electron passing through matter is given by146

d@e(T.W) . , I re4 1 1 - 1 cos [ ln (‘+)I} = - dW ( wi + (T - W)’ W ( T - W ) T

(1.1.77)

where T is the kinetic energy of the primary electron, and W is the energy transfer, i.e., the kinetic energy of the secondary electron. In Eq. (1.1.77), the second and third (cosine) terms in the curly bracket are exchange terms. For small W , these terms become negligible, and the resulting cross section [re4/(TWZ)] dW is the same as that for primary heavy particles (e.g., protons) [see Eq. (1.1.56)].

For relativistic energies of the incident electron, the electron-electron scattering cross section has been obtained by Mdler,L69 and is given by

1 (T - W ) z + + (T + mcz)2

(1.1.78)

For T < mc2, Eq. (1.1.78) reduces to the nonrelativistic Mott formula, Eq. (1.1.77), in which the cosine factor in the last term is ~ 1 , unless W7 is very close to either 0 or T.

In his calculations,169 MGller summed over both directions of polariza- tion of the two electrons, so that Eq. (1.1.78) represents the average cross section for unpolarized incident electrons. However, recently in connec- tion with the experiments on parity noncon~ervat ion~-~ in beta decay, i t has become of interest to evaluate the cross sections for polarized elec- trons scattered by polarized electrons, in particular for the case that the two spin directions are parallel or antiparallel to each other, and along

159 C. Mdler, Ann. Physik [5] 14, 531 (1932).

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64 1. PARTICLE DETECTION

the relative direction of motion of the electrons. This problem arises because it has been shown from the two-component theory of the neu- trino,*ll that the electrons and positrons from beta decay are expected to be longitudinally polarized10.16't-162 (i.e., with average spin direction along the direction of motion). The value of the polarization P is pre- dicted to be: P = T v/c, where v = velocity of p particle, and the minus sign applies to electrons, while the plus sign applies to positrons. Thus high-energy electrons from p decay (v - c) are almost 100% polarized, with the spin pointing opposite to the direction of motion. A method of determining the longitudinal polarization consists in scattering the &decay electrons on the electrons in a ferromagnetic sample of material. As is well known, if a strong magnetic field is applied to an iron sample, the 3d electrons of iron will be polarized in the direction of the applied field.* Thus if a magnetic field is applied along a direction parallel or antiparallel to the direction of the incident p electrons, one can obtain the polarization P of the incident beam from a comparison of the c o u n h g rates of scattered electrons at a particular angle, for the two field direc- tions. This result arises from the fact that the cross sections for parallel spin directions ( b p and for antiparallel spin directions & are appreciably different from each other, for all values of the incident energy, provided that the energy transfer W is sufficiently large (WIT 2 0.2, where T is the kinetic energy of the incident electron).

The first calculation of the spin-dependent cross sections r # J p and (ba was carried out by Bincer,163 and we shall here briefly summarize his results. For the differential cross sections in the center-of-mass system of the two electrons (to be abbreviated as c.m. system), one obtains

[2 C O S ~ t7 + p ( 3 C O S ~ G + C O S ~ t7) + B 4 ( 1 + C O S ~ s)] eddn 2~284 sin4 t7

d(bp = - -

(1.1.79)

[I + C O S ~ G + B2(2 + 3 C O S ~ t7 - C O S ~ 8) e4dQ 2 ~ ~ 8 4 sin4 t7

dr#Ja = - -

+ 84(5 - 4 C O S ~ G + C O S ~ t7)] (1.1.80)

where 8 is the c.m. scattering angle, B is the c.m. velocity of either electron (in units of c), is the total c.m. energy of either electron, and d o is the element of solid angle in the c.m. system. We have S2 = (y - l) /(y + l),

* See also Vol. 4, A, Chapter 3.5. l60 J. D. Jackson, S. B. Treiman, and H. W. Wyld, Phys. Rev. 106, 517 (1957). Iel L. Wolfenstein and L. A. Page, BUZZ. Am. Phys. Sac. [Z] 2, 190 (1957). 162 R. B. Curtis and R. R. Lewis, Phys. Rev. 107, 543 (1957).

A. M. Bincer, Phys. Rev. 107, 1434 (1957); see also G. W. Ford and C. J. Mullin, Phys. Rev. 108, 477 (1957).

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1.1. INTERACTION O F RADIATION WITH MATTER 65

where y = E/mc2, with E = total laboratory energy of incident electron. Upon defining x = cos' 8, Eqs. (1.1.79) and (1.1.80) can be rewritten as follows :

d + P

(1.1.81) @a

We note that cos e = 1 - 2w

( 1.1.82)

(1.1.83)

where w is the fractional energy transfer, w = W/T. Thus after inte- grating over the azimuthal angle, one obtains

J dn = 27r sin ede = -2nd(cos #) = 47r dw. (1.1.84)

In view of Eq. (1.1.83), we have: x = (1 - 2 ~ ) ~ . Upon substituting these results in Eqs. (1.1.81) and (1.1.82), one finds for the average differ- ential cross section per unit w (for unpolariaed electrons) :

x [ y y i - 2~ + 3 ~ 2 - 2w3 + w4) - (27 - i ) ( w - 2w3 + W4)l. (1.1.85)

Finally, upon using the relations: T = (y - 1)mc2, and W = wT, one can easilyshow that Eq. (1.1.85) is equivalent to Eq. (1.1.78) ford@(T,W).

From Eqs. (1.1.81) and (1,1.82), one obtains

( 1.1.86)

Figure 9 shows the curves of as a function of w, for y = 1,3, and co , as obtained by B i n ~ e r . ' ~ ~ It is seen that decreases rapidly with in- creasing w, independently of y. The minimum value is attained for w = 0.5 (0 = go"), and is given by

&, - ~ ' ( 1 + 62 + x2) - 2 y ( l - Z) + 1 - 5'

& 87' - 2y(4 - 52 + 2') + 4 - 62 + 22" - -

(Y - 1)' = 4(2y2 - 2y + 1)

(1.1.87)

which becomes 0 for y 4 1 (nonrelativistic energies) and + for y 4 co .

Page 66: n

66 1. PARTICLE DETECTION

The dependence of the electron-electron scattering cross section on the relative directions of polarization has been used in a few experiments t o determine the longitudinal polarization of electrons from B decay. 164,18i, The arrangement of the experiment of Frauenfelder et ~ 1 . l ~ ~ is shown

I .3

.8

.6

0, +o

- .4

.2

0 0 .I .2 .3 A .5

FIG. 9. The ratio &/&, for electron-electron scattering, as a function of the relative kinetic energy t,ransfer w. This figure is taken from the work of A. M. Bincer [Phys. Rev. 107, 1434 (1957), Fig. 11, and is reprinted with the permission of the author and the Editor of the Physical Review.

schematically in Fig. 10. The scattered electrons are recorded in coin- cidence by the counters 61 and Cz. The counting rates are compared for opposite directions of the magnetizing current around the Deltamax scattering foil. The scattering angle 0 is usually so chosen that i t corre-

164 H. Frauenfelder, A. 0. Hanson, N. Levine, A. Rossi, and G. De Pasquali, Phys. Rev. 107, 643 (1957).

lo5 N. Benczer-Koller, A. Schwarzschild, J. B. Vise, and C. S. Wu, Phys. Rev. 109, 85 (1958).

W

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1.1. INTERACTION OF RADIATION W I T H MATTER 67

sponds approximately to the maximum relative energy transfer w = 0.5, for which has its smallest value, as discussed above. For w = 0.5, the two final electrons have equal energies in the laboratory system and are emitted symmetrically with respect to the incident direction a t an angle 8, given by

sin2 8, = 2/(7 + 3). (1.1.88)

A description of the experiments on the M3ller scattering of P-decay electrons, as well as a more complete discussion of the theory, can be found in reference 8.

COLLIMATOR

& SOURCE

FIG. 10. Schematic view of the experimental arrangement of Frauenfelder et a1.ln4 used to demonstrate the longitudinal polarization of electrons from the f l decay of P32 and Pr144, by means of the Mflller (electron-electron) scattering.

For the scattering of relativistic electrons by nuclei, McKinley and FeshbachlGB have obtained the following expression :

I ZsP 137 + - s in(p) [ l - sin(+0)] . (1.139)

This expression applies provided that 2/137 is not too large, Le., not for the heaviest nuclei. For P + 1, Eq. (1.1.89) may be rewritten as follows:167

] (2s sin 0 dB) Ze2 cos2(&8) a2 sin(;O)[l - sin(@)]

d @ = ( 2E -) sin4((B8) - { I + - - 137 cos2 (Be) (1.1.90)

where E is the total laboratory energy of the electron. We note that both in (1.1.89) and (1.1.90), 0 is the angle of scattering in the center-of-mass system.

lB8 W. A. McKinley and H. Feshbach, Phys. Rev. 74, 1759 (1948). R. Hofstadter, Revs. Modern Phys. 28, 214 (1956).

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68 1. PARTICLE DETECTION

In connection with the longitudinal polarization of the electrons from beta decay,loJao-lez the Mott scatteringlea of the electrons from a heavy nucleus has also been used to detect the po1arization.t In this type of experiment, the longitudinal polarization of the electrons is first trans- formed into a transverse polarization, for instance, by deflecting the electrons through -90" by means of an electrostatic field. A system of crossed electric and magnetic fields may also be used, t o take advantage of the fact that the focusing condition for the electrons can then be made identical with the condition for turning the spin through 90". After the particles have thus acquired a substantial amount of transverse polariza- tion (spin d perpendicular to momentum p), they are scattered through an angle 0 in the plane perpendicular to the (d,p) plane, which we assume to be horizontal, and the up-down asymmetry of the scattered intensity is observed. That is, the intensity of the electrons scattered through an angle 8 in the upward direction is different from the intensity of the electrons scattered through the same angle 8 in the downward direction. As was first shown by Mottles in 1929, the asymmetry in the scattering of transversely polarized electrons is largest for heavy elements and for large scattering angles (6 - 90"-150").

For a beam with transverse polarization P, the ratio R of the scattered intensities in both azimuthal directions perpendicular to the (d,p) plane (i.e., upward and downward in the example discussed above) is given by

(1.1.91)

where s(0) is a function, first calculated by Mott,le8 which depends on the atomic number of the scatterer, the incident electron energy, and the angle of scattering 8.

The most complete recent calculation of S(0) has been carried out by Sherman,169 who has tabulated s(e) at intervals of 15" for various values of the electron velocity p, for three elements: mercury (2 = 80) ; cadmium (2 = 48); and aluminum (2 = 13). s(0) is given by

[F(B)G*(B) + F*(e)G(0)] (1.1.92) 2PE(1 - p*)1'2

sin e(da/dQ) s(e) =

where 5 = 2/(137p), X is the de Broglie wavelength, P(0) and G(0) are the

t See also Vol. 4, A, Section 3.5.1. 168 N. F. Mott, PTOC. Roy. Soc. Al24,425 (1929); A136,429 (1932); see also the dis-

cussion in N. F. Mott and H. S. W. Massey, "The Theory of Atomic Collisions," 2nd ed., pp. 74-85. Oxford Univ. Press, London and New York, 1949.

lEg N. Sherman, Phys. Rev. 103, 1601 (1956).

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1.1. INTERACTION O F RADIATION WITH MATTER 69

regular and irregular solutions, respectively, of the Schroedinger equation for the electron in the field of the nucleus. These functions are in general complex, and the asterisk denotes the complex conjugate. In Eq. (1.1.92) du/dQ denotes the differential cross section for an unpolarized beam of electrons, which is given by

du/dQ = X2[t2(1 - pz) ]PIz csc2(+0) + ]GI2 sec2(+0)]. (1.1.93)

In addition to the asymmetry function i3(0), Sherman*eB has also tabu- lated the values of the real and imaginary parts of F and G, as well as the

FIQ. 11. The asymmetry factor -S(O) for mercury (2 = 80) as a function of the scattering angle e. The values of S(O) were obtained from the results of Sherman.leg The curves for electron velocities j3 = 0.2, 0.4, 0.6, 0.8, and 0.9 correspond to electron energies T, = 10.5, 46.6, 128, 341, and 661 kev, respectively.

differential cross section du/dQ. As mentioned above, Ii3(0)l is largest for heavy elements and large values of 0. Figure 11 shows the curves of s(0) versus 0 for 2 = 80 and for various velocities p. x(e) is zero for 0 = 0" and 180" for all energies, and a t B = 1 for all angles 0. Several experiments have been carried out at 0 = 90" using electrons with p - 0.6 (T, - 130 kev), for which lS(90")1 has its maximum value. It may be noted that for p = 0.6, IS(0)l increases from 0.271 a t 90" to 0.424 a t 120" and 0.418 at 135'. Nevertheless, it has been found desirable to work a t -90" because of the rapid decrease of the cross section du/dQ with increasing angle.

Page 70: n

70 1. PARTICLE DETECTION

FIQ. 12. The ratio 9 = (du/dn)/(duz/dn) for mercury (2 = 80) as a function of the scattering angle 0. The solid curves of 7 were obtained from the results of Sherrnan.lBg The dashed curve of 9 for fi = 0.6 was calculated from the formula of McKinley and FeshbachlG6 [Eq. (1.1.89)].

Figure 12 shows the values of the ratio q for mercury, as obtained by Sherman, 169 where q is defined by

(1.1.94)

where duR/dQ is the Rutherford cross section, which is given by the factor outside the curly bracket of Eq. (1.1.89) :

(1.1.95)

The dashed curve in Fig. 12 shows the values of q predicted by the formula of McKinley and FeshbachIG6 for = 0.6, i.e., the curly bracket of Eq. (1.1.89). It is seen that the actual values of q differ appreciably from the McKinley-Feshbach result, as would be expected in view of the large 2 value (2/137 = 0.58).

Among the earlier determinations of S(O), we may mention the calcula- tions of Mott, 1 6 * Bartlett and Watson,17o Bartlett and Welton,'" and Mohr and T a ~ s i e . " ~

170 J. H. Bartlett and R. E. Watson, Proc. Am. Acad. Ads Sci. 74, 53 (1940).

l72 C. B. 0. Mohr and L. J. Tassie, Proc. Phys. Soc. (London) A67, 711 (1954); J. H. Bartlett and T. A. Welton, Phys. Rev. 69, 281 (1941).

C. B. 0. Mohr, Proc. Roy. Soc. A182, 189 (1943).

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1.1. INTERACTION OF RADIATION WITH MATTER 71

It should be pointed out that the function S(0) was originally intro- duced by Mott"j8 in connection with the double scattering of an initially unpolarized beam of electrons. In this case, the polarization P after a single scattering through an angle O1 is given by S(B1), and the direction of the spin d after the scattering is perpendicular to the plane of the scattering. After a second scattering, the relative intensity of the beam as a function of the angle cp between the first and second planes of scattering is given by

I(el,ez,cp) = 1 + s (e1)s (e2)co~ (o (1.1.96)

where 0 2 is the angle of the second scattering. Among the more recent double scattering experiments which have attempted to verify the theoretical values of S(0), we may cite the works of Shull et u Z . , ' ~ ~ Ryu et u Z . , ' ~ ~ and Louisell et A review of these investigations has been given by Tolhoek. 176

The Mott scattering has been used in several e x p e r i m e n t ~ ' ~ ~ - ~ 8 ~ on the longitudinal polarization of p-decay electrons, and has shown that the polarization agrees with the predicted value, P = v/c, within the experi- mental errors.8 The same conclusion was obtained from the experiments using the MIdller scattering.'64s165

1.1 -5.4. Range of p Rays in Matter. In some cases, a crude value of the energy of a beam of homogeneous p rays is obtained by measuring the so- called practical range R, in some material, such as aluminum. The practical range is obtained by extrapolating the straight-line (maximum slope) part of the graph of transmission versus thickness traversed, and taking into account the background (cf. Fig. 6). For p rays in aluminum, Katz and P e n f ~ l d l ~ ~ have given the following expressions for R, as a func-

l73 C. G. Shull, C. T. Chase, and F. E. Myers, Phys. Re*. 63, 29 (1943). 17* N. Ryu, K. Hashimoto, and I. Nonaka, J . Phys. Soc. Japan 8, 575 (1953). l76 W. H. Louisell, R. W. Pidd, and H. R. Crane, Phys. Rev. 94, 7 (1954). 178 H. A. Tolhoek, Revs. Modern Phys. 28, 277 (1956). 177 H. Frauenfelder, R. Bobone, E. von Goeler, N. Levine, H. R. Lewis, R. N. Pea-

cock, A. Rossi, and G. De Pasquali, Pkys. Rev. 106, 386 (1957). 178 H. De Waard and 0. J. Poppema, Physica 23, 597 (1957). 179 P. E. Cavanagh, J. F. Turner, C. F. Coleman, G. A. Gard, and B. W. Ridley,

l80 A. I. Alikhanov, G. P. Eliseiev, V. A. Lubimov, and B. V. Ershler, Nuclear Phys.

181 A. de-Shalit, S. Kuperman, H. J. Lipkin, and T. Rothem, Phys. Rev. 107, 1459

182 H. J. Lipkin, S. Kuperman, T. Rothem, and A. de-Shalit, Phys. Rev. 109, 223

183 L. Katz and A. S. Penfold, Revs. Modern Phys. 24, 28 (1952).

Phil. Mag. [8] 2, 1105 (1957).

6, 588 (1958).

(1957).

(1958).

Page 72: n

72 1. PARTICLE DETECTION

tion of the incident electron energy TO (in MeV) :

R, = 412Ton mg/cm2 n = 1.265 - 0.0954 In TO

for 0.01 d To =< 2.5 MeV, and

R, = 530To - 106 mg/cm2

for 2.5 5 TO 6 20 MeV.

(1.1.97)

(1.1.98)

FIG. 13. The practical range RP of low-energy electrons in aluminum as a function of the electron kinetic energy To. This curve was calculated from the range-energy relation given by L. Katz and A. S. Penfold [Revs. Modern Phys. 24, 28 (1952)]. See also Eqs. (1.1.97) and (1.1.98) of text.

Figure 13 shows a plot of Eqs. (1.1.97) and (1.1.98). This curve is in good agreement with the experimental data on the maximum range of electrons from natural beta emitters. Among the earlier works on the range of low-energy /3 rays, we may mention those of Marshall and Ward, lS4 FeatherJ1S5 Flammersfeld, l86 Bleuler and Zunti, l*7 Glendenin, and Hereford and Swann.lE9

lS4 J. Marshall and A. G. Ward, Cun. J. Research A16, 39 (1937). lS6 N. Feather, Proc. Cambridge Phil. SOC. 34, 599 (1938). I** A . Flammersfeld, Nuturwissenschaften 33, 280 (1946). lg7 E. Bleuler and W. Ziinti, Helv. Phys. Actu 19, 137, 375 (1946); 20, 195 (1947). la* L. E. Glendenin, Nucleonics 2, 12 (1948). lag F. L. Hereford and C. P. Swann, Phys. Rev. 78, 727 (1950).

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1.1. INTERACTION OF RADIATION WITH MATTER 73

1.1.6. Multiple Scattering of Charged Particles

When a charged particle penetrates a thick absorber, i t undergoes a large number of small-angle Coulomb scatterings. This process, which is called multiple scattering, was first treated quantitatively by Williams. l90 In addition, the particle may undergo a small number of relatively large- angle scatterings, for which the probability can be directly obtained from the Rutherford scattering formula. We shall here be concerned with the multiple scattering only.

The resultant distribution of the space angle 6 between the incoming and outgoing directions of the particle is given by

(1.1.99)

where < e2 > is the mean square value of 8 and is given by1g1

<e2> = 2OI2 In (2) - = el2 In [ 4 ~ Z ~ / % ~ i V t ($)2]- (1.1.100)

Here Omin is the minimum angle of scattering in a single encounter, Omin Ei X/a, where X is the de Broglie wavelength of the particle and a is the radius of the atom, a - a& 1‘3. In Eq. ( l . l . l O O ) , t is the thickness of material traversed, z is the charge of the particle, m is the electron mass (regardless of the type of scattered particle: proton, meson, etc.), N is the number of atoms of absorber per cm3, and el is that angle for which there is, on the average, only one collision with 8 > 01 throughout the absorber. el2 is given by

Equation (1.1.100) can be written as follows:

e 1 2 = 4 T ~ ~ ( ~ + 1)9e4t/(p42. (1.1.101)

0*1572(2 4- 1)z2t ln[1.13 X 104Z4/3z2A-1tp-2] (1.1.102) A (PO)

< e 2 > =

where pv is in MeV, t is in gm cm-2, and A is the atomic weight in grams- The expression preceding the logarithm in (1.1.102) is el2.

Rossi and GreisenlS8 have given a somewhat different formula for < e2> which has been frequently used in experimental applications. This expression is given by

(1.1.103)

1** E. J. Williams, Proc. Roy. SOC. A169, 531 (1939); Phys. Rev. 68, 292 (1940). 191 H. A. Bethe and J. Ashkin, in “Experimental Nuclear Physics” (E. SegrB, ed.),

Vol. I, p. 285. Wiley, New York, 1953.

Page 74: n

74 1. PARTICLE DETECTION

where Xo is the radiation length in the material [cf. Eq. (l.l.G9) and Table VI], and E, is a constant energy given by

E, = ( 4 ~ X 137)1%c2 = 21.2 MeV. ( 1.1.104)

As was shown by Bethe and Ashkin,lgl Eq. (1.1.103) applies only for relatively large thicknesses t > to, where t o is given by

t o = 6.7(137/2)2A113 gm cm-2. (1.1.105)

For Pb, to = 110 gm cm-2, while for C, to = 8000 gm cm-2. For small thicknesses, Eq. (1.1.103) overestimates the mean square multiple scattering angle. Thus for 3-Bev protons and samples of thickness t = 10 gm cm-2 of C, Cu, and Pb, < 0 2 > u2 = 0.123", 0.248', 0.388' from Eq. (1.1.102) for C, Cu, and Pb, respectively, whereas the corresponding values from Eq. (1.1.103) are: <82>1'2 = 0.159", 0.289", and 0.429', respectively. The factor F by which Eq. (1.1.103) differs from (1.1.102) is: F = 1.29, 1.17, and 1.11 for C, Cu, and Pb, respectively (for t = 10 gm cm-2).

It is often useful to consider the projected angles 8, and 01/, i.e., the projections of the angle 8 on the zy plane perpendicular to the direction of motion of the particle. The distribution of the 8, values is a Gaussian:

(1.1.106)

where < eZ2 > is the mean square value of 8, and is given by <e,2> = ;<e2> (1.1.107)

with < 8 2 > given by Eq. (1.1.102). Thus the denominator of the exponent is the same (= <e2>) for both P(8) and P,(Q [Eqs. (1.1.99) and (1.1.106)]. Of course, the distribution P1/(02/) do, for the projected angle 8, has the same form as P,(8,) do,.

The distribution of the lateral displacement r of the particles has been determined by Ferrni,Ig2 and is given by

(1.1.108)

where t' = t/X, is the thickness traversed in units of radiation lengths X O . More elaborate theories of the multiple scattering have been developed

by Goudsinit and S a ~ n d e r s o n , ~ ~ ~ Snyder and Scott,Ig6 192 E. Fermi, quoted by B. Rossi and K. Greisen, Revs. Modern Phys. 13,265 (1941). 193 S. Goudsmit and J. L. Saunderson, Phys. Rev. 67, 24 (1940). ls4 G. MoliBre, 2. Naturforsch. Sa, 78 (1948). 195 W. Paul and H. Steinwedel, in "Beta- and Gamma-Ray Spectroscopy" (K.

196 H. S. Snyder and W. T. Scott, Phys. Rev. 76, 220 (1949). Siegbahn, ed.), p. 1. Interscience, New York, 1955.

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1.1. INTERACTION OF RADIATION WITH MATTER 75

Lewis,lg7 and Bethe. lg8 These theories treat more accurately the transition from the small-angle region of multiple scattering to the large-angle region where single scattering predominates. This transition region is sometimes called the region of plural scattering.

In a recent investigation, Nigam and c o - w ~ r k e r s ' ~ ~ have made a critical study of the MoliBre theorylS4 of multiple scattering, and have obtained a consistent treatment of the scattering of a charged particle by the field of an atom, up to the second Born approximation. I n this work, Nigam et al. have used the expression of Dalitz2"" for the scattering cross section of a relativistic particle of spin + in a screened atomic field, for which the potential is: V = -(Ze2/r)exp(--r), where K is an arbitrary constant, and r is the distance from the nucleus. It was found that the deviation of the complete expression for the "screening angle " 8, from the value given by the first Born approximation is considerably smaller than was obtained by MoliBre. Moreover, the expression for the distribution function P(e) contains additional terms of order xZ/137, which were not obtained by Moliitre.

Nigam et al. have carried out calculations of the distribution function P(0) for the case of 15.6-Mev electrons scattered by Au and Be, in order t o compare their theory with the experimental results of Hanson el aLZo1 The theoretical results are in good agreement with the data. The two cases considered correspond to electrons of average energy 15.7 Mev scattered by a gold foil of thickness t = 37.2 mg/cm2, and 15.2-Mev electrons scattered by a beryllium sample of thickness t = 491.3 mg/cm2. The experimental distributionsof Hanson etal. havea llewidth O,(exp) = 3.78" for Au and 4.33" for Be. Here 0, denotes the angle (measured from the direction of the incident beam) a t which P(0) has fallen off t o l/e of its value a t e = 0". e, is thus given by < 02> l i 2 for a Gaussian distribution [Eq. (1.1.100)]. It may be noted, however, that the actual multiple scattering d i s t r i b ~ t i o n ~ ~ ~ - ' ~ ~ deviates somewhat from a Gaussian a t all angles. In particular, at large angles (0 >, 28,), the actual P(0) lies above the Gaussian of Eq. (l.l.lOO), and slowly approaches the single-scattering cross section (which decreases only as m e - * ) . For comparison with the values of e,(exp), the theory of Nigam et a l l g 9 gives 0, = 3.80" for Au, and 4.35" for Be, in very good agreement with the data. On the other hand, MoliBre's theorylg4 gives e, = 3.83" for Au, and 4.56" for Be. The result

H. W. Lewis, Phys. Rev. '78, 526 (1950). 1**H. A. Bethe, Phys. Rev. 89, 1256 (1953). 199 B. P. Nigam, M. K. Sundaresan, and T. Y. Wu, Phys. Rev. 116, 491 (1959). zoo R. H. Dalitz, Proc. Roy. Soe. A206, 509 (1951). 201 A. 0. Hanson, L. H. Lanzl, E. M. Lyman, and M. B. Scott, Phys. Rev. 84, 634

(1951).

Page 76: n

76 1. PARTICLE DETECTION

for Be is thus too large by 5%. It may be noted that from the simple expression of Bethe and Ashkin given above [Eq. (1.1.102)], one obtains 0, = 3.94" for Au and 4.33" for Be. The value for Au is too large by 476, while the result for Be agrees exactly with B,(exp). On the other hand, the formula of Rossi and Greisen [Eq. (1.1.103)] would give the values 5.82" for Au and 6.56" for Be, which are both considerably larger than O,(exp).

The multiple scattering has been frequently used for a crude measure- ment of pv for charged particles in nuclear emulsion.202 If the track of the particle is subdivided into sections (cells) of length t , the average angle between successive sections is given by

(1 .l. 109)

where ,6 = v/c , t is measured in microns, pv is in MeV, < O> A~ is in degrees, and K(t,,6) is a slowly varying function of t and p. The theoretical value194*196 of K is between 23 and 24, which is in satisfactory agreement with the experimental results both of the Bristol groupzo3 and of C o r ~ o n , * ~ ~ namely K = 25.1 f 0.6 for P = 1 and t = 100.

1.1.7. Penetration of Gamma Rays

such that the intensity I ( z ) after traversing a thickness 2 is given by For y rays passing through matter, there is an exponential attenuation

I ( z ) = I(O)exp( - Nuz) = I(O)exp( -m) (1.1.110)

where I(0) is the incident intensity (at 5 = 0), N is the number of atoms of absorber per cm3, u is the total cross section for absorption or scattering of the y rays, and r = Nu(cm-') is the absorption coefficient of the radiation.

There are three processes which contribute to 6: (1) the photoelectric effect, which consists of the ionization of atomic electrons by the incident photon. (2) The Compton scattering of the photons by the atomic electrons. In this process, the atomic electrons can generally be considered as free, and the energy transfer to the electron is a function of the scatter- ing angle O of the ? ray and its initial energy hvo. The energy transfer is determined in a straightforward manner from conservation of momentum and energy. (3) The production of an electron-positron pair in the field of a nucleus.

Zo2 P. H. Fowler, Phi.?. Mag. [7] 41, 169 (1950). 2osK. Gottstein, M. G. K. Menon, J. H. Mulvey, C. O'Ceallaigh, and 0. Rochat,

go4 D. R. Corson, Phys. Rev. 80, 303 (1950); 84, 605 (1951). Phil. Mag. [7] 42, 708 (1951).

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1.1. INTERACTION OF RADIATION WITH MATTER 77

The photoelectric effect predominates at low y energieszo6 (hv < 0.05 Mev for All hv < 0.5 Mev for Pb). The Compton effect gives the main contribution at intermediate energies (0.05 < hv < 16 Mev for Al; 0.5 < hv < 4.8 Mev for Pb). The pair production predominates a t high energies (hv > 16 Mev for Al; hv > 4.8 Mev for Pb). We shall now con- sider separately each of these three processes.

For a general review of the subject of the interaction of y rays with matter, the reader is referred to the review articles of Bethe and Ashkinj3 Davisson and Evans,206 and D a v i s ~ o n . ~ ~ ’

1.1.7.1. Photoelectric Effect. For energies far above the K absorption edge and in the nonrelativistic range (hv << mc2) the cross section for the photoelectric effect from the K shell is given byzo8

@,,hot.K =

where 40 is the Thomson scattering cross section:

40 = sT (x)’ = 6.651 X cm2. 3 mc2

(1.1.11 1)

( 1.1.1 12)

In obtaining Eq. (1.1.111), the Born approximation was used, the wave function of the outgoing electron being taken as a plane wave. This procedure is not justified when the kinetic energy of the ejected electron is of the same order or less than the binding energy of the K electrons.2n9 Thus for relatively small y energies hv [i.e., for hv/(Z2Ry) - 1 - 10, where Ry = Rydberg unit], aPhat,~ is considerably smaller than would be given by Eq. (1.1.111). The ratio of the actual @ ) p h o t , ~ to the value of the expression (1.1.111) is 0.12 for hv = ZZRy, 0.43 for hv = 10Z2Ry, and 0.90 for hv = 1000Z2Ry.

For very high y energies, hv >> md, relativistic effects become impor- tant. This problem has been treated by SauterlZ1O Hulme,211 and others. The following formula obtained by Hall is valid in the limit hv >> mc2, and includes the effect of the Coulomb field of the nucleus on

206 See, for example, W. Heitler, “The Quantum Theory of Radiation,” 2nd ed.,

206 C. M. Davisson and R. D. Evans, Revs. Modern Phys. 24, 79 (1952). 207 C. M. Davisson, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.),

208 H. Hall, Revs. Modern Phys. 8, 358 (1936). a** M. Stobbe, Ann. Physik [5] 7, 661 (1930). Z1O F. Sauter, Ann. Physik [5] 9, 217 (1931); 11, 454 (1931). 211 H. R. Hulme, Proc. Roy. Soc. A133, 381 (1931). P1z H. Hall, Phys. Rev. 46, 620 (1934); 84, 167 (1951).

p. 216, Fig. 21. Oxford Univ. Press, London and New York, 1944.

p. 24. Interscience, New York, 1955.

Page 78: n

78 1. PARTICLE DETECTION

the outgoing electron, which becomes appreciable for heavy elements :

3 Z6da mc2 2 1374 hv @phot.K = - - - exp[ -TCY + 2 d ( 1 - In CY)] ( 1.1.113)

where CY = 2/137. Equation (1.1.113) shows that aphot,R decreases quite slowly with increasing v (only as v-l) in the relativistic region, as com- pared to the k 7 1 2 decrease at nonrelativistic y energies [Eq. (1.1.111)].

Aside from the approximate calculations mentioned above, which are based in part on the Born approximation, and on the use of plane-wave or nonrelativistic wave functions, Hulme et ~ 1 . ~ ~ 3 have carried out exact calculations for the photoelectric effect from the K shell, using the appro- priate Dirac wave functions in the field of the nucleus. The calculations of Hulme et al. were carried out for two y-ray energies, hv = 0.354 and 1.13 MeV, and for three values of 2 : 26, 50, and 84. These results have been extensively used to check the validity of various approximation formulas and to obtain smooth curves of versus hv in the intermediate energy region (hv - 1 Mev).

In obtaining the photoelectric absorption coefficient, one must include the contribution of the absorption from the L, M , . . . shells. Latyshev214 has made direct measurements of the photoelectrons ejected from the K and L shells of Pb and Ta, for the ThC’ y rays (hv = 2.62 Mev).

Detailed calculations of the total photoelectric cross section @,,,,, have been carried out by White1215 who used the results of StobbelZo9 Sauterj210

and Hulme et aL213 According to White,21b the ratio [ of the total photoelectric cross section @phot to the K shell contribution aphot.K is -1.15 for heavy elements. White has obtained E for various values of 2, for two y-ray energies: (1) a t the K absorption edge; (2) for mc2/hv = 1.5 (i.e., hv = 0.340 Mev). At the K edge, [ = 1.02 for 2 = 6, 1.11 for 2 = 29, 1.14 for 2 = 50, and 1.167 for 2 = 92. For hv = 0.340 MeV, [ = 1.01 for 2 = 6, 1.07 for 2 = 29, 1.10 for 2 = 50, and 1.138 for 2 = 92. Figure 14 shows the plot of loglo(@ph,t/&,) versus hv for C, All Cu, Sn, and Pb, as obtained from the results of White. For C and All and for the heavier elements a t low photon energies, aphot decreases approxi- mately as Y - ” ~ , as expected from Eq. (1.1.111). On the other hand, for Pb at high energies, between 5 and 50 MeV, aphot is proportional to v-1 [see Eq. (1.1.113)]. White’s calculations include the effect of the L and M

Z 1 3 H. R. Hulme, J. McDougall, R. A. Buckingham, and R. H. Fowler, Proe. Roy.

214 G. D. Latyshev, Revs. Modern Phys. 19, 132 (1947). 216 G. R. White, Natl. Bur. Standards Rept. 1003 (1952); see also Appendix I

by C. M. Davisson, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 857. Interscience, New York, 1955.

Sac. A149, 131 (1935).

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1.1. INTERACTION O F RADIATION WITH MATTER 79

shells, which becomes the sole contribution to the photoelectric effect a t low energies below the K absorption edge. Thus for Sn, the break in the curve of @)phot at 29.25 kev is due to the K edge [@,hot = 1.05 X cm2 on low-frequency side of K edge (L and M shells only); = 8.58 x

cm2 on high-frequency side ( K + L + M shells contribute)]. Similarly, for Pb, the break a t 88.2 kev is due to the K edge, while the discontinuity in the region of 15 kev is due to the L absorption edges (LI, LII, ,5111). Below 13.07 kev (LIrl edge), only the M , N , and 0 shells contribute to the photoelectric effect.

PHOTON ENERGY h v ( IN MeV)

FIG. 14. The cross section *,,hot for photoelectric absorption for C, Al, Cu, Sn, and Pb, as a function of the photon energy hv. These curves were obtained from the results of White.216

The theoretical results presented above are in fairly good agreement with various experiments on the photoelectric e f f e ~ t . ~ ~ ~ s ~ ~ ~ , ~ ~ ~

1.1.7.2. Compton Scattering. The Compton scattering consists of the scattering of y-rays by atomic electrons which can be considered as free (no atomic binding forces) for sufficiently high y-ray energies. From the laws of conservation of energy and momentum, one finds that the fre- quency v of the scattered quantum is given by

(1.1.114) yo

1 + (hvo/mc2) (1 - cos e) v =

where vo is the frequency of the incident quantum, and 0 is the angle of H. A. Bethe and J. Ashkin, in “Experimental Nuclear Physics” (E. Segr6, ed.),

Vol. 1, p. 304. Wiley, New York, 1953.

Page 80: n

80 1 . PARTICLE DETECTION

FIG. 15. Schematic diagram of Compton effect, showing notation used in the text: hvo and hv are the energies of the incident and the scattered quanta, respectively; T is the kinetic energy of the recoil electron.

scattering of the y ray. The notation used is shown in Fig. 15. The kinetic energy T of the electron is given by

T = h(vo - V ) = 2 m ~ ~ ( h v o ) ~ COS' cp

(hvo + mc2)2 - (hvo)2 cos2 cp ( 1.1.1 15)

where (a is the angle between the directions of the outgoing electron and the incident y. The angles cp of the electron and 6 of the y ray are related as follows:

tan cp = Y O :';c:s 0 = (mcT$hv) cot(&@. (1.1.116)

The energy of the scattered photon decreases with increasing 6. The mini- mum value, attained for 6 = 180", is given by

(1.1.117)

The maximum possible angle of the recoil electron is cp = go", in which case the energy of the electron is T = 0, while the scattered y ray con- tinues with its initial energy (hv = hvo) in the forward direction.

The differential cross section for Compton scattering was first obtained by Klein and NishinaZ17 in 1929. The Klein-Nishina formula gives

) (1.1.118)

817 0. Klein and Y . Nishina, 2. Physik 62, 853 (1929).

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1.1. INTERACTION OF RADIATION W I T H MATTER 81

where ro = e2/mc2, k = hv, ko = hvo, and d@c is the cross section for scattering of the y ray through an angle e into the solid angle dQ. Upon substituting Eq. (1.1.114) for k = hv, one obtains

(1.1.119)

where y 3 ko/mc2. Equation (1.1.119) gives the cross section as a function of the angle 8. For small values of y, the distribution follows the 1 + cos2 0 law characteristic of classical electromagnetic theory. As y increases, the distribution becomes increasingly peaked forward, as is generally the case for any high-energy process.

The differential cross section as a function of the energy k is given by

(1.1.120) 1 ?rro2mc2dk [ (;J2 2(y + 1) + (1 + 2r)k I k~

r2k 1 + - -

Y 2 y2ko d@c = k k o

with k

1 + 27 - ko 5 - 5 1 1 (1.1.121)

Detailed calculations of various quantities and spectra pertaining to the Compton scattering have been carried out by Figure 16 shows the spectrum of the scattered quanta [Eq. (1.1.120)] for incident y energies hvo = 0.5, 1, 2, and 3 MeV. The minimum value hv,i. [Eqs. (1.1.117), (1.1.121)] increases slowly with increasing hvo. Thus hvmin = 0.169 Mev for hvo = 0.5 MeV, and hvmi, = 0.236 Mev for hvo = 3 MeV. (The asymptotic value in the limit v o 4 co is mc2/2 = 0.255 Mev.) It is seen from Fig. 16 that, as hv is increased above hvmin, d @ ~ / d ( h v ) first de- creases to a minimum value, and then increases uniformly up to hv = hvo. The minimum of the cross section becomes increasingly more shallow as the primary energy hvo is increased.

The total Compton scattering cross section @C is given by

-

(1.1.122)

where 40 is the Thomson cross section [Eq. (1.1.112)]. For small y (y << l),

zl$ See also R. Latter and H. Kahn, “Gamma-Ray Absorption Coefficients.” Published by The Rand Corporation, Santa Monica, California, 1949; G. Allen, Nat l . Advisory Comm. Aeronaut. Tech. Notes 2026 (1950).

A. T. Nelms, Nail. Bur. Standards Circ. 642 (1953).

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82 1. PARTICLE DETECTION

Eq. (1.1.122) gives 9 c Ei 4 0 0 - 2Y)

whereas for very large y (y >> l), one obtains

(1.1.122a)

(1.1.122b)

Equations (1.1.122) show that the Compton cross section decreases uniformly with increasing energy of the y quantum. Figure 17 shows a plot of %/40 versus h v ~ , which was taken from Bethe and Ashkin, refer- ence 3, p. 322. For comparison, we have also shown the photoelectric

PHOTON ENERGY hv (IN MeV)

FIG. 16. Spectrum of Compton scattered quanta, as obtained from Eq. (1.1.120), for incident photon energies hvo = 0.5, 1, 2, and 3 MeV.

cross section divided by 2, in the same units 40 [i.e., @phot/z+O]. The reason for dividing aphat by 2 is that aphot pertains to the photoelectric effect for the entire atom, so that @.phot/Z represents the photoeffect per atomic electron and is therefore the quantity to be compared with 9 c (Compton scattering per electron). We note that the energy hvo E h ; ~ for which +phot/Z = 9~ increases with increasing 2. Thus hih = 0.02, 0.05, 0.13, and 0.53 Mev for C, All Cu, and Pb, respectively. As hvo is increased above h h , the photoelectric effect rapidly becomes unimportant com- pared to the Compton scattering as a source of y-ray attenuation.

It should be noted that the expression for @c [Eq. (1.1.122)] no longer applies for very low photon energies, where the binding of the atomic electrons must be taken into account. I n this case, the incoherent (Comp- ton) scattering will be reduced, both because of the binding of the atomic

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1.1. INTERACTION O F RADIATION WITH MATTER 83

electrons, and because of the effect of the exclusion principle in preventing transitions to occupied atomic levels. On the other hand, there will also be a substantial amount of coherent scattering from the atom as a whole, so that the total scattering cross section will generally be larger than the value Z@c which would be calculated from Eq. (1.1.122). For a discussion of these effects, the reader is referred to the review article of Davisson.*07

The Klein-Nishina formula has been tested in various experiments, and has been shown to be in good agreement with the experimental dat~.206,207.216

FIG. 17. The Compton total cross section and the photoelectric cross section per electron, iP,hot/Z, for C, Al, Cu, and Pb, as functions of the incident photon energy hvo. The curves of GphOt/Z were obtained from the results of White.216

1.1.7.3. P a i r Production. The theory of the pair production by y rays is closely related to the theory of the bremsstrahlung by a high-energy electron. The general formula obtained by Bethe and Heitler147 using the Born approximation is very complicated and will not be given here. However, the formula simplifies considerably if the energies of both posi- tron and electron are not too high so that screening can be neglected, i.e., if

( 1. I . 123)

where E+ and E- are the total energies of the positron and electron, respectively, and k = hv is the energy of the incident photon. If in addi- tion to Eq. (1.1.123) , all energies involved are large compared to mc2, the

Page 84: n

84 1. PARTICLE DETECTION

energy distribution of the positrons (or electrons) is given by

Q(E+) dE+ = 45 dE+

where 5 is defined by

ka kmc2 ( I. 1.124)

5 (Z2/137)ra2 (1.1.124a)

with TO = e2/mc2 (classical radius of the electron = 2.82 X cm). As is also true for the general Bethe-Heitler formula, Eq. (1.1.124) gives a symmetric energy distribution for the positron and electron. Actually for small velocities v+ and v- of the pair, and for large 2, the Coulomb effect (which is neglected in the Born approximation) becomes important and results in a somewhat asymmetric distribution favoring higher energy positrons.

For very large energies E+, E-, the screening is complete (4 = 0 ) , and Q(E+) is given by

Q(E+) dE+ = ~- [( + + Em2 + +E+E-)ln(183Z-1’s) - +E+E-]. 45dE+ k3

(1.1.125)

Figure 18 shows the energy distribution of the pair particles (positrons or electrons) as obtained from the calculations of Bethe and Ashkin (reference 3, p. 328). For hv up to 10mc2, the curves do not include screen- ing and are valid for all elements; for higher photon energies] the calcula- tions of Bethe and Ashkin were done for Pb and include the effect of screening. It is seen that for small values of hv, the energy distributions are generally quite flat between the minimum and maximum values, T+,mi, = 0 and T+,,,, = k - 2mc2, where T+ is the kinetic energy of the positron. For k/mc2 5 30, the distribution has a broad maximum at T+ = +T+,mx = +lc - mc2. For larger k/mc2, Q(E+) has a broad minimum at +T+,m.x and two subsidiary maxima on each side of the minimum, which implies that for large k/mc2, either the positron or the electron tends to carry off most of the energy of the y ray. For any finite k/mc2, the distribution is zero a t the two ends, T+,mi, = 0 and T+,msx = k - 2mc2. Figure 18 shows that the distributions are symmetrical with respect to T+ = $T++,., = +lc - me2. This is a consequence of the use of the Born approximation for Q(E+), which gives identical spectra for the positron and electron.

In analogy with the bremsstrahlung in the field of the atomic electrons] which has been discussed above [Eqs. (1.1.62), (1.1.68)], there is also the possibility of pair production in the field of the atomic electrons. Wheeler

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1.1. INTERACTION O F RADIATION WITH MATTER 85

and Lamb14* have shown that, for complete screening, the electronic pair production is a fraction [/Z of the production in the field of the nucleus, where [ (-1.2-1.4) is the same quantity that appears in the formula for the bremsstrahlung [Eq. (l.l.68)l. Thus, for the case of complete screening, the total @(E+) dE+ (including the electronic con- tribution) is obtained by replacing Z2 by Z(Z + [) in the definition of b [Eq. (1.1.124a)I. For low energies, where screening can be neglected (hv 5 20 Mev), the pair production in the field of the atomic electrons has been calculated by Borsei l in~, '~~ Votruba, 15* Rohrlich,'j' Nemirov- sky,L62 and Watson.153

10

a

NE N

I b L: I

- 4 '? w"

a 2

-

'0 0.1 0.2 0.3 0.4 0.5 0.6 07 Q0 0.9 I (E; m c 2 ) / ~ v - 2 m c 2 )

FIG. 18. Energy distribution of the positron (or electron) in an electron pair as a function of the positron kinetic energy for various energies hv of the incident y ray. These curves were taken from the calculations of Bethe and Ashkin (reference 3, p. 328, Fig. 38). For hv 5 IOrnc~, the curves do not include screening and are valid for all elements. For higher photon energies, the curves pertain to Pb and include the effect of screening.

The angular distribution of the pair electrons becomes increasingly forward with increasing energy of the primary quantum, in analogy with the bremsstrahlung distribution. In particular, for high y energies k, the average angle between the incident y ray and the direction of motion of the electron (or positron) is given by

< 9 > Emmc2/E (1.1.126)

where E is the energy of the electron (or positron). Equation (1.1.126) is completely analogous to Eq. (1.1.74) for the bremsstrahlung. The pair production in monocrystalline targets has been discussed by ubera11.166

Page 86: n

86 1. PARTICLE DETECTION

The total cross section for pair production aPaiF can be obtained analyti- cally for two limiting cases:

(1) For me2 << hv << 1 3 7 r n ~ ~ Z - ' / ~ (no screening), Eq. (1.1.124) is valid, and one finds

(1.1.127)

(2) For h v >> 1 3 7 m ~ ~ Z - " ~ (complete screening), integration of Eq. (1.1.125) gives

(1.1.128) = $[? ln(183Z-1'3) - A]. For intermediate values of hv, the total cross section @pair must be ob- tained by numerical integration. Figure 19 shows the resulting curves of

@'pair v

@

14

12

10

8

6

4

2

n "I 2 5 10 20 100 200 500 2000 l0,OOO

hv/mc2

FIG. 19. The total cross section for pair production Qlpair as a function of the 7-ray energy, for air and Pb, and for the hypothetical case of no screening. These curves were taken from the calculations of Bethe and Ashkin (reference 3, p. 338, Fig. 41).

for the (hypothetical) case of no screening, and for air and Pb (including screening), as obtained by Bethe and Ashkin (reference 3, p. 338). It is seen that for large energies (hv/mc2 2 50), the values of

for air and Pb fall below the curve for no screening, and slowly approach the asymptotic value [Eq. (1.1.128)] which is 14.1 for air and 11.6 for Pb.

As mentioned above, the Bethe-Heitler theory based on the Born approximation cannot be expected to give accurate results for high Z and low energies of the positron or electron, since the wave functions for these particles will then be appreciably distorted by the Coulomb field

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1.1. INTERACTION OF RADIATION WITH MATTER 87

of the nucleus. Jaeger and Hulme220 have calculated the pair production cross sections at two photon energies (hv = 3mc2 and 5.2mc2), using the exact Dirac wave functions for the pair particles. At hv = 3mc2, they obtained results for 2 = 50, 65, and 82; at hv = 5.2mc2, the calculations were carried out for 2 = 82 only. For the worst case, 2 = 82, hv = 3mc2, the Born approximation cross section is too low by a factor of 2 (0.34 X

cm2 from the exact calculation). For the other 2 values, and hv = 3mc2, the Bethe-Heitler result is also too small, but the deviation decreases rapidly with decreasing 2 or in- creasing hv. Thus for hv = 5.2mc2, 2 = 82, the error is only 16%.

Recent experimental results a t high energies (2 10 Mev)221-227 have shown that the measured pair production cross sections are appreciably lower than the Bethe-Heitler calculated values, and that the deviation is proportional to Z2. Thus Lawson,222 from his measurements at 88 MeV, concluded that the ratio of the experimental to the theoretical cross sec- tion can be approximately represented by

~ ~ ~ ~ / a ~ ~ ~ ~ = 1 - 1.5 x 1 0 - 5 2 2 . (1.1.129)

In view of these results, Bethe et al.2zs have carried out an accurate cal- culation of the pair production, in which the Born approximation was not used. They have found that the correction to the Born approximation result of Bethe and Heitler147 is a reduction of the cross section propor- tional to Z2, as is indicated by the experimental data. Upon applying this correction, Bethe et a1.228 have obtained excellent agreement with the measurements of LawsonZP2 at 88 MeV, and those of DeWire et aLZz4 a t 280 MeV. It may be noted that the correction to the Born approximation is a reduction of the cross section a t high energies, as compared to an increase of the cross section at low energies.220 The correction goes through zero at hv - 6 Mev.226n228

1.1.7.4. Total Absorption Cross Section for y Rays. The total cross section u for the removal of a y-ray photon from the incident beam is given by

u z= a p h o t + 2a.c + @'pair. ( 1.1.130) Z z O J . C. Jaeger and H. R. Hulme, Proc. Roy. SOC. A163, 443 (1936); J. C. Jaeger,

2 2 1 C. D. Adams, Phys. Rev. 74, 1707 (1948). a22 J. L. Lawson, Phys. Rev. 76, 433 (1949). 223 R. L. Walker, Phys. Rev. 76, 527 (1949). 224 J. W. DeWire, A. Ashkin, and L. A. Beach, Phys. Rev. 83, 505 (1951). 2zaC. R. Emigh, Phys. Rev. 86, 1028 (1952). z*BE. S. Rosenblum, E. F. Shrader, and R. M. Warner, Phys. Rev. 88, 612 (1952). 227 A. I. Berman, Phys. Rev. 90, 210 (1953). 228 H. A. Bethe and L. C. Maximon, Phys. Rev. 93, 768 (1954); H. Davies, H. A.

Bethe, and L. C. Maximon, ibid. 93, 788 (1954).

cm2, as compared to 0.67 X

Nature 137, 781 (1936).

Page 88: n

88 1. PARTICLE DETECTION

The complete absorption coefficient c equals Nu. Figure 20 shows the mass absorption coefficient T/P (in units cmZ/gm) for Al, Cu, and Pb. For Pb, we have presented the separate contributions to r / p due to the photo- electric effect, Compton effect, and pair production (dashed curves). The values of r / p were obtained from the tables of White.216 It is seen from Fig. 20 that, as a function of frequency, T has a minimum, which occurs at hv ? 20 Mev for Al, .=8 Mev for Cu, and ~ 3 . 4 Mev for Pb. For Pb, the minimum lies in the region where the Compton effect is pre- dominant. For lower v, the photoelectric effect predominates, and the rapid decrease of apbot with increasing v is responsible for the decrease of

ale c

$' 0.16 E g 0.14 0

- $ 0.12

0.10 2

0.08 ' 0.06 t 0.04 a 0

I-

a 2 O.O2

0 0.1 0.2 0.5 I 2 5 10 20 50 100

PHOTON ENERGY h Y (IN MeV)

FIG. 20. The mass absorption coefficient T/P for Al, Cu, and Pb, as a function of the photon energy hv. For Pb, the separate contributions of the photoelectric effect, Compton effect, and pair production are shown by the dashed curve8. The curves of r / p shown in this figure were obtained from the tables of White.216

the total absorption coefficient c. At frequencies somewhat above the position of the minimum, the pair production becomes the main effect, and is responsible for the rapid increase of with increasing v, until a t very high energies (hv - 5 Bev), T approaches a constant value as a result of the saturation of due to screening. The minimum of r implies that y rays of energies of the order of 5-20 Mev have a relatively long mean free path in matter. The values of ~ / p at the minimum for Al, Cu, and Pb, are as follows, according to the data of White: 0.0217 cm2/gm for Al; 0.0306 cm2/gm for Cu, and 0.041 cm2/gm for Pb. The corresponding values of the maximum mean free path are: Amsx = 46.1 gm/cm2 for A1 (at hv = 20 Mev); A,, = 32.7 gm/cm2 for Cu (at hv = 8 Mev); X,,, = 24.4 gm/cm2 for P b (at hv = 3.4 Mev).

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1.2. IONIZATION CHAMBERS 89

C ~ l g a t e ~ ~ ~ has carried out measurements of the total y-ray absorption cross section a for y energies of 0.411, 0.664, 1.33, 2.62 Mev using radio- active sources, and 4.47, 6.13, 17.6 Mev using y rays from nuclear reac- tions. The measurements were made for a variety of absorbers (poly- ethylene, C, Al, Cu, Sn, Pt, Pb, Bi, and U). In general, the theory is in good agreement with these data, when account is taken of a small correc- tion due to Rayleigh in addition to the three principal effects: the photoelectric effect, Compton scattering, and pair production in the field of the nucleus and the atomic electrons.

The present theory is in reasonable agreement with measurements of the absorption coefficient at various energies up to 280 M ~ v . ~ ~ ~ , ~ ~ ~ , ~ ~ ~

1.2. Ionization Chambers*

1.2.1. General considerations’

An ionization chamber is a device which measures the amount of ionization created by charged particles passing through a gas. The basic processes of ionization of gases by charged particles have been discussed in Chapter 1.1.7 If an electric field be maintained in the gas by a pair of electrodes, the positive and negative ions will drift apart, inducing charges on the electrodes which can be detected as a voltage pulse. Or if a steady flux of particles enter the chamber one can measure the average current caused by the ionization. The latter application will be specifically con- sidered in Section 1.2.7; the other sections of this chapter, however, will be principally devoted to the ionization chamber as a detector of single particles and therefore as a pulse instrument.

1.2.1.1. Essentials of a Pulse Ionization Chamber. Figure 1 shows schematically the essential parts of this very simple device. One of the electrodes, the “collector” (a misnomer, as we shall see), is designed to have a low capacity both to the other electrode and to ground, so that a very small charge will still give a measurable potential change. The small amount of charge is characteristic: if the particle loses 1 Mev in collisions

**o S. A. Colgate, Phys. Rev. 87, 592 (1952). *an W. Franz, Z. Physik 98, 314 (1935); P. Debye, Physik. Z. 31, 419 (1930). t See also Vol. 2, Chapter 4.1 and Vol. 4, B, Chapter 7.5 and Section 9.2.3. 1 General references for Sections 1.2 and 1.3 are: D. H. Wilkinson, “Ionization

Chambers and Counters.” Cambridge Univ. Press, London and New York, 1950; B. B. Rossi and H. Staub, “Ionization Chambers and Counters.” McGraw-Hill, New York, 1949; S. C. Curran and J. D. Craggs, “Counting Tubes.” Academic Press, New York, 1949; see also Vol. 4, A, Section 2.1.5.

* Chapter 1.2 is by Robert W. Williams. -

Page 90: n

90 1. PARTICLE DETECTION

with the gas, the number N of electrons released will be about 30,000, or a charge of 5 X

The passage of a particle creates the ionization, for all practical pur- poses, instantaneously. The positive ions then drift toward the negative electrode with a velocity of the order of (1 cm/sec) X [(760 mm Hg)/p X [ E / ( l volt/cm)] or in a typical case 0.001 cm/psec. The electrons, if they do not suffer attachment and thereby become heavy negative ions, will drift toward the positive electrode with velocities, under comparable con- ditions, of 1-5 cm/psec.

For definiteness assume, as is usually the case, that the collector is the anode. The collector potential is lowered both by the motion of electrons toward it and by the motion of positive ions away from it. It is instructive to calculate explicitly the potential change in a highly idealized case; for

coulomb.

'\

\

I I

=k - FIG. 1. The essentials of a pulse ionization chamber (schematic). The dotted line

illustrates the path of an ionizing particle whose passage leaves pairs of ions in the gas of the chamber.

example, insulated long cylindrical electrodes with cylindrical sheets of positive and negative charge, Q+ = Q- = Q formed a t radius r1 (Fig. 2). Let the initial potential difference between inner and outer electrodes be Vo(b) - VO(U), where V,(r) is the potential at any point in the chamber before the charge sheets have been moved apart, and assume that the negative sheet of charge collapses uniformly toward the central electrode and the positive one expands toward the outer electrode. When the two charge distributions are at r- and r+ respectively we find from elementary calculation that the potential has changed by an amount which is inde- pendent of the magnitude of the initial potential, and which can be written

(1.2.1)

This result, that the potential change of the electrode is proportional t o the fraction of the total potential drop through which the charge has

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1.2. IONIZATION CHAMBERS 91

moved, is not restricted to this special geometry; i t will be discussed in more detail in Section 1.2.2. We use it here to note that the total voltage pulse, if all the ions are ultimately collected, is Q/C as expected, and that the part of this pulse corresponding to electron motion occurs orders of magnitude more rapidly than the part due to ion motion. It is therefore important to know under what circumstances the electrons will remain free as they drift through the gas, and what mechanisms may prevent complete collection of all ions.

FIG. 2. Idealized cylindrical ionization chamber.

1.2.1.2. Behavior of Ions and Electrons in Gases. The positive ions which are formed upon passage of a charged particle through the gas remain nearly in thermal equilibrium with the gas. The presence of the electric field causes them to drift toward the cathode, but the increase in kinetic energy is very small and they rapidly reach a terminal mean velocity for which the energy gained from the electric field is dissipated in molecular collisions. This “drift velocity’’ would be expected to depend on the ratio of field strength to mean free path, and indeed i t is observed experimentally to be, in the ionization-chamber range, a linear function of E / p :

760E P,

w = Ko- (1.2.2)

with E in volts per centimeter and p in mm Hg, K O ranges from about 8 cm/sec for Ne, t o 2.5 cm/sec for Ar, t o 1 cm/sec for Xe.2 Negative ions

Press, London and New York, 1938. 2 A. M. Tyndall, “The Mobility of Positive Ions in Gases.” Cambridge Univ.

Page 92: n

92 1. PARTICLE DETECTION

(not electrons) have about the same drift velocities. A simple theory of drift velocity8 yields the expression

e XE m u P

w = _ _ -

where is the mean free path and u the ion’s speed, assumed the same for all ions.

Since practical limitations (Section 1.2.3) usually restrict the average value of E/p to not more than -1 volt/cm/mm Hg, the minimum time required for ions to cross even a small ionization chamber, say 1 cm, will be -0.2 msec, and in most cases it will be considerably longer. In nearly all cases where ionization chambers are used to count individual particles only the fast portion of the pulse, due to the motion of the electrons, is utilized.

The electrons which are released in the initial ionization prove to remain free, in most gases, until they impinge on an electrode (or other surface). The exceptions are electronegative gases which have an appreciable probability to form negative ions by electron attachment. For 0 2 , the most dangerous offender, the probability of attachment, per collision, is 10V to Water vapor has a similarly large attachment probability, and Clz, NH,, N20, HzS, SO?, NO, and HC1 are all known to be bad.

In a nonattaching gas-Ar, N2, CH4 are among the commonly used ones-the electrons continue to drift toward the anode, but acquire from the field a kinetic energy many times their thermal energy, because the mechanisms of energy transfer to the gas are relatively inefficient. The ratio of mean kinetic energy to thermal energy (#KT), denoted by the “agitation energy” 7, is typically of the order of 100 in a noble gas, where elastic collision and atomic excitation are the only available energy- transfer mechanisms; it is down by an order of magnitude in diatomic gases, and is not much greater than 1 in polyatomic gases. Table I gives some values5 for the drift velocity and agitation energy of electrons in various gases.

Argon is a particularly important gas; it is convenient and is widely used. Its first excitation level is very high, 11.5 volts, and in consequence the electron agitation energy is large. A small amount of polyatomic im- purity gas lowers the agitation energy greatly; Rossi and Staub’ find

8 13. B. Rossi and H. Staub, “Ionization Chambers and Counters,” p. 6. McGraw- Hill, New York, 1949.

D. H. Wilkinson, “Ionization Chambers and Counters,” p. 41. Cambridge Univ. Press, London and New York, 1950.

6Based on R. H. Healey and J. W. Reed, “The Behavior of Slow Electrons in Gases,” Amalgamated Wireless, Ltd., Sydney, Australia, 1941; and tables in Wilkin- eon,‘ and Rossi and Staub,s

Page 93: n

1.2. IONIZATION CHAMBERS 93

about a factor of ten for 10% COZ at E / p = 1. This has the effect of increasing the drift velocity, for two reasons: the electron speed is lowered [see Eq. (1.2.2)], and because of the Ramsauer resonance effect on the cross section of noble gas atoms for electrons the mean free path proves to increase, in the region of interest (which is from -10 ev to -1 ev). Table I includes the drift velocity in 5% and 10% CO,; at E / p = 1 the increase over pure Ar is a factor of ten; there is a comparable increase when the mixture is compared to pure C02, because the mean free path of -1 ev electrons in argon is so great.

TABLE I. Drift Velocity w and Ratio of Agitation Energy to Thermal Energy 9 for Electrons in Various Gases a t Room Temperature; Principally from

Healey and Reed5 Values are approximate and in noble gases are strongly impurity sensitive.

E / p = 0.2 v/cm/mm Hg E / p = 1 v/cm/mm Hg --- w (cm/piiec) tl w (cm/Mec) 9

He Ne Ar HI

N2 coo 0.95 Ar

0.09 Ar 0 . 1 coz

0.05 COz

0 . 5 11 0 . 5 62 0.3 120 0 . 4 2 .7 0 . 4 6 . 5 0 . 1 1 . 5

3 . 3

0 .9

0 . 9 53 1 .5 216 0 .5 285 1 .o 9 .3 0 . 8 21.5 0.55 1 . 5

4 .3

5 .3

0 R. H. Healey and J. W. Reed, “The Behavior of Slow Electrons in Gases.” Amalgamated Wireless, Ltd., Sydney, Australia, 1941.

The greater drift velocity of Ar-C02 mixtures is often of great practical value in obtaining a faster pulse, and these mixtures are widely used. Unpurified tank argon alone will give rise to an electron drift velocity considerably greater than that of Table I, and is satisfactory for many applications where E / p is large (-1) and where accurate pulse height is not essential. In high-pressure cylindrical-geometry chambers, where E / p at the outer electrode is usually quite low, the Ar-C02 mixture may show some attachmentJ6 and pure Ar will give better performance.

There are two additional complications in the motions of ions or elec-

6 Under ordinary conditions COO has negligible electron attachment. Experience with cosmic-ray ionization chambers [H. S. Bridge, W. E. Hazen, B. B. Rossi, and R. W. Williams, Phys. Rev. 74, 1083 (1948)J indicates that a t very low E / p values (-0.01), “pure” Ar has distinctly less attachment than Ar-COZ mixtures.

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94 1. PARTICLE DETECTION

trons in gases which can lead to reduction in pulse height; we now con- sider the first of these, diffusion. The center of gravity of a group of ions which is liberated at a point in an electric field will have a displacement proportional to the time, while the mean distance of the ions from the center of gravity will increase as the square root of the time. The impor- tance of diffusion can be measured by the ratio of the latter distance to the former one. This ratio can be calculated from kinetic theory; the mean free path cancels out, and one has, for a gas at room temperature.’

- diffusion distance = o.18 d+

drift distance (1.2.3)

where q is the agitation energy and V the voltage difference between the point of release of the ions and the final position of the center of gravity. For heavy ions 7 = 1 and the effects of diffusion will generally be small. For electrons 7 may be -100 in noble gases, and diffusion may be impor- tant; electrons may diffuse back to the cathode, or out beyond the bound- aries of the apparatus. The “ C 0 2 effect ” may be utilized to reduce r] and therefore decrease the electron diffusion.

The second effect is recombination, the neutralization of positive and negative ions before they are collected. This is a complex subject, con- sidered in detail by Wilkin~on,~ and we shall only summarize the principal results. The recombination coefficient a is defined by writing the rate of disappearance of ions, when n+ positive ions and n- negative ions per cubic centimeter are present, as an+n-,8 with a depending on agitation energy, and, of course, on the nature of the negative ions; a is cm3/sec for heavy negative ions, and

In air chambers or other chambers where attachment is more or less complete, recombination can be a serious cause of pulse loss, particularly in those current chambers which contain a large density of ionization (Section 1.2.7). Free-electron chambers are better both because of the smaller value of 01 and the shorter time which the electrons spend in the gas. The over-all improvement factor is -lo7; therefore free-electron chambers do not suffer from recombination effects under any circum- stances ordinarily realized. Ionization chambers used as monitors of the direct beams of pulsed machines (synchrotrons, etc.) present special problems. They will be considered in Section 2.7.1.

cm3/sec for electrons.

7 D. H. Wilkinson, “Ionization Chambers and Counters,” p. 37. Cambridge Univ. Press, London and New York, 1950.

* This assumes that an ion, when created, does not recombine preferentially with the other member of the pair. The assumption is surely correct if the negative ions are electrons, bu t it might be expected to break down in very high-pressure air.

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1.2. IONIZATION CHAMBERS 95

1.2.2. Pulse Formation

Quantitative use of the ionization chamber as a particle detector requires consideration of shape and magnitude of the voltage pulse caused by release of ionization. We at first calculate the potential change of the collecting electrode due to the motion of a single charge, assuming that the electrode is essentially insulated (i.e., referring to Fig. 1, that the time constant RC of the grid resistor and the capacity of the collector is large compared to the collection time of an ion). The result which was obtained for a special case in Section 1.2.1 can be obtained for arbitrary two-elec- trode geometry under the assumption (always true in practice, even in a Geiger counter) that the charge released in the gas is small compared to the charges residing on the electrodes which give rise to the initial poten- tial difference Vb - Va.

Consider an electron of charge -e at point r l in the gas. The potential at rl will consist of two parts: the potential Vo(rl) due to the charges on the electrodes, which is, by our assumption, essentially independent of any charges which may be present in the gas, and V8(rl), the potential due to other ionization which may be present in the gas. Then the electro- static energy of the electron consists of two independent parts, the space- charge energy and the energy in the field of the electrodes, and we may consider the latter ~eparately.~ The energy of the system of electrodes, plus the electron, is

+BqiVi = i[-eV,(r,) -k &ova0 - QoVao].

where V , and v b are the electrode potentials, and the subscript 0 refers to initial values. As the electron drifts from rl to r2 the work done on it by the field must be extracted from this system; Q O and VbO must remain fixed, but Val the collector potential, can change by some amount AV. Thus we have

-e[Vo(rz) - Vo(r1)l = +[-eVo(rd + eV&d + QO AVI

(1.2.4)

which is the desired result.

9 The space charge has no direct effect on the size of the pulse due to one electron but it may affect the velocity of the electron and therefore the pulse shape, and second- ary effects of the electron. In pulse ionization chambers these questions are unim- portant because the space charge is so small, but they are crucial in Geiger counter operation (Section 1.3.2).

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96 1. PARTICLE DETECTION

Two examples will serve to illustrate the behavior of pulse ionization chambers utilizing electron collection (the pulse shape of “slow” cham- bers is not usually interesting).

First the parallel-plate chamber of separation d, ion-pair formed at X;O from the positive electrode, at t = 0. AV will have a rapid contribution from the motion of the electron which has drift velocity w-, and a much slower contribution from the positive ion (drift velocity w+).

- e w-t + wft A V = - ( C ) until w-t = x o

- e w+t = (7 + 2) until w+t = d - xo

- e C - -- finally.

Figure 3 illustrates this pulse using drift velocities characteristic of argon. To take advantage of the fast electron-collection pulse one must elimi-

nate the effects of the slow positive ions by incorporating a low-frequency rejection network in the collector circuit of the chamber or, more com- monly, in the amplifier-e.g., a short time-constant T in a resistance- capacitance coupling stage such that RC = r << t+, the collection time of the ions. The rapidly-rising (electron-collection) part of the pulse will not be much affected so long as r >> t- where t- is the collection time of the electrons, but the positive-ion pulse will be reduced by roughly r/t+. For a given form of input pulse the detailed shape of the pulse after passing through the low-frequency rejection network can be obtained by standard transient-response analysis-Wilkinsonl gives several examples. The dotted line in Fig. 3 illustrates the effect of a time-constant r = 5t- (10% pulse-height loss). An approximation which is sufficient for some purposes is to assume that the voltage rise due to the electrons is undis- torted, but that the voltage then returns to zero with the time constant r, and with no positive ion contribution.

A network which gives a more nearly square-topped pulse is illustrated in Fig. 4; it consists of a shorted delay line in series with a resistance equal to its characteristic impedance. It is less convenient than the RC, and causes a 50% amplitude loss, so that i t is usually used only when there is some reason to require a good pulse shape. The pulse-shaping action of this network is illustrated in Fig. 5.

The second example of the voltage pulse in an electron-collection cham- ber is that of cylindrical geometry: a small central electrode, of radius a, is surrounded by a concentric cylinder of radius b. This is a simple, low- capacity arrangement, and its chief virtue is that the electron-collection

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1.2. IONIZATION CHAMBERS

- C

3Q.C d C

nv t

97

,

\

I \ '\ . '. -

0-j D.L.

FIQ. 4. Shorted-delay-line pulse-shaping network. Rk is equal to the characteristic impedance of the delay-line.

t - AV

FIQ. 5. Effect of delay-line pulse shaping on a typical pulse from a cylindrical ioniza- tion chamber; 270 is the round-trip time of the line, and must be greater than t-, the electron collection time, if the pulse is to have a flat top.

FIQ. 3. Idealized voltage pulse from a parallel-plate ionization chamber. An ion pair is released at distance Xa from the negative electrode. The time scale would be approximately right if, for example, d were 2 cm, V were 1000 volts, and P were 1 atmos. The dashed line indicates the effect of a five-microsecond time constant.

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98 1. PARTICLE DETECTION

pulse is reasonably independent of the point at which the electron is released, since most of the potential drop occurs near the central wire, Figure 6 shows the electron pulse in a chamber with b/a = 120, calculated for a single electron [curves (a) and (a’)]; for uniform ionization in the chamber [curves (b) and (b‘)]; and for uniform ionization along a straight line passing through the axis of the chamber [curve (c)]. The pulse shape,

FIG. 6. Electron pulse in a cylindrical ionization chamber: 1- is the drift time from outer to inner electrode; curves (a) and (a’) are for a single electron, (b) and (b’) for a uniform distribution, and (c) for a linear distribution.

AV(t)/AVfina~, is calculated from Eq. (1.2.1), with elapsed time related to drift velocity by

t = J:& and with two assumptions for w: constant (solid curves) and proportional to (E)”q(dotted curves). Inspection of curve (a) shows that a considerable portion of the chamber volume gives rise to pulses of nearly maximum height. It is easy to show that uniform ionization gives a pulse which is a fraction f of the total-charge-collection pulse,

f = b2 / (b2 - u2) - 1/[2 ln(b/u)] This is 0.90 for b/a = 120.

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1.2. IONIZATION CHAMBERS 99

The parallel-plate chamber can be modified so that it gives an electron pulse independent of the point of production,1° by adding a third elec- trode. A grid at a fixed potential, near the collecting electrode, will shield the collecting electrode from the rest of the chamber (Fig. 7) so that only that portion of the electron’s travel which takes place between the grid and the collector will cause a pulse to be induced.

The electrostatic situation near the edge of an electrode will in general be complicated-either the field will be quite distorted and therefore the effective volume, and expected pulse shape, somewhat uncertain, or the field shape can be maintained by “guard

Grid,

Useful Volume

-

electrodes”-extra electrodes,

Collector

h FIG. 7. Schematic diagram of a gridded chamber. Electrons originating in the

shaded area will give pulses of nearly uniform height as they pass between the grid and the collector.

held at the average potential of the collecting electrode, which maintain the symmetry of the field beyond the edge of the collecting electrode. The pulse induced by an electron near the edge of the collector is more complicated in this case; detailed calculations in simple cases are given by Rossi and S t a ~ b . ~

In many applications of ionization chambers one is interested in average current or in total amount of charge collected (Section 1.2.7). For these chambers (as well as for “slow” pulse chambers which respond to the motion of ions) the details of the eIectron pulse are unimportant. The guard electrode helps to define accurately the volume of gas from which ionization is collected. The role of the guard electrode in preventing leakage current is discussed in Section 1.2.7.

10 For the detailed theory of this device, Bee D. H. Witkinson, “Ionization Chambers and Counters,” p. 74. Cambridge Univ. Press, London and New York, 1950.

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100 1. PARTICLE DETECTION

1.2.3. Quantitative Operation and Some Practical Considerations

1.2.3.1. Attachment, Diffusion, Recombination. Those basic phenom- ena of the passage of electricity through gases which affect the operation of ionization chambers are discussed in Section 1.2.1. Their importance may range from very little in the case of large E / p and nonabsolute pulse- height requirements (e.g., a low-pressure parallel-plate chamber used as a counter) to considerable in the opposite extremes (e.g., a high-pressure cylindrical chamber used for proton recoil pulse-height spectrum work). The maximum value of E / p which can be used is determined by the condition that there should be no gas multiplication (Section 1.3.1) even in the region where E / p is largest. For cylindrical or spherical chambers this means that E / p will necessarily be low in the region near the outer electrode; and in general the difficulties mentioned are associated with low E / p . Even with parallel-plate chambers the inconvenience of working with high voltages will often put a practical limit on the available E / p .

Electron attachment can be eliminated by sufficiently rigorous exclusion of electronegative gases, of which O2 and H20 are the most frequent offenders. Noble gases can be purified very effectively by circulation over hot calcium c h i p ~ . ~ J l Purified argon in a clean metal-and-glass chamber with soldered seals has been found to remain free from attachment for years, in an application in which the minimum E / p was about 0.01 v/cm/mm Hg. However, a chamber containing volatile material (e.g., rubber gaskets) must be purified frequently if quantitative performance under low E / p conditions is to be maintained.

The Ar-C02 mixture previously referred to can also be purified with hot ~alciurn,~ and is free of attachment under most conditions. However, there are indications12 that at very low E / p this mixture, unlike pure argon, shows noticeable attachment.

Gases which cannot be purified by such drastic methods, e.g., BF3 (which in pure form does not have serious attachment), must be prepared with great care. Graves and Fromanla describe a suitable technique for preparing BF3 for ionization chambers.

Recombination can be shown to be negligible14 under nearly any cir- cumstances in chambers in which no attachment takes place. This subject will therefore be treated in Section 1.2.7, in connection with current chambers.

11 L. Colli and U. Facchini, Rev. Sci. Instr. 23, 39 (1952). la H. S. Bridge, W. E. Hazen, B. B. Rossi, and R. W. Williams, Phys. Rev. 74,1083

l3 A. C. Graves and D. K. Froman, “Miscellaneous Physical and Chemical Tech-

l4 D. H. Wilkinson, “Ionization Chambers and Counters,” Chapter 111. Cambridge

(1948).

niques of the Los Alamos Project,” p. 154. McGraw-Hill, New York, 1952.

Univ. Press, London and New York, 1950.

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1.2. IONIZATION CHAMBERS 101

Diffusion effects are important only for electrons under conditions of large agitation energy q [Eq. (1.2.3)]. A typical situation in which diffusion may be important occurs when ionization is released adjacent to the wall of the chamber, as when a noncollimated alpha-particle source is incor- porated in the negative electrode as a calibrating standard. It can be assumed that electrons which diffuse back to the negative electrode are lost, and with the help of Eq. (1.2.3) an estimate of the pulse loss from this effect can be made. Addition of a polyatomic gas, with consequent

10 r

0 0.5 1.0 15 2 .o 2.5

E ( e v )

FIG. 8. Attachment probability h upon collision of an electron with an oxygen molecule, as a function of electron energy E. From Wilkinson,6 by permission.

reduction of 7, is of course desirable for applications where diffusion must be minimized. An example of a complete diffusion calculation is worked out by Rossi and Staub.16

1.2.3.2. Checks for Quantitative Operation. Spurious Effects. The most commonly used test for proper ionization-chamber operation is the examination of the pulse height, from some reproducible source, as a function of collecting voltage. If a chamber exhibits a good “plateau”- region in which pulse height is independent of voltage-it is usually considered free from the defects we have outlined. However, examination of the dependence of electron attachment coefficient on electron energy, Fig. 8, shows that in oxygen, at least, there is a region in which increasing electron energy causes an increase in attachment coefficient, which might

l6 B. B. Rossi and H. Staub, “Ionization Chambers and Counters,” p. 27. McGraw- Hill, New York, 1949.

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102 1. PARTICLE DETECTION

compensate the decreased number of collisions which electrons suffer at higher energies. Also, the drift velocity (in impure argon, for example) l6 may not increase with E. The existence of a plateau is therefore not a sufficient indication, in a pulse chamber, that all electrons are being collected.

A source of alpha particles of known energy can be used to release a known amount of ionization in the chamber (Section 1.2.4), say Qo. The pulse height due to the electron motion should then be

Vo = (fQo/C,)G where f is the (average) fraction of total voltage drop through which the electrons move, C, the collector capacity, and G the amplifier gain. A quantitative check system based on this principle is outlined by Bridge and associates.6

A more elaborate method of checking, which does not depend on know- ing the capacity of the chamber, has been used by Hazen and collabora- tors.I6 They provide a polonium source which remains at the potential of the negative electrode but can be moved toward the collecting electrode, reducing the degree of attachment by reducing the path length through which the electrons move. Constant pulse height as the source is moved in is a reliable check in this case.

A still more elaborate method, using a pulsed X-ray source and measur- ing the fraction of total current carried by electrons, is described by Rossi and Staub.”

In the design and construction of pulse ionization chambers a reasonable care must be taken to avoid spurious pulses from high-voltage leakage or breakdown or electrical pickup. Any good insulators can be used (in con- trast to the current chambers described in Section 1.2.7, which require very high quality insulators) ; in particular, glass-Kovar seals are very useful. In most applications it is possible to provide a grounded con- ductor (guard electrode) which separates the high-voltage insulator from the collector insulator, thereby greatly reducing the dificulties caused by leakage across the high-voltage insulator. It is sometimes convenient to have the collector at high voltage, and to connect it to the amplifier through a coupling condenser. The condenser must then be selected very carefully. Ceramic condensers seem to be the most satisfactory.

The signal obtained from a pulse chamber is often of the order of a millivolt or less. Obviously the collector and the amplifier input must be completely shielded. Ordinarily a double shield (i.e., a grounded case

l8 See F. E. Driggers, Phys. Rev. 87, 1080 (1952). l7 B. B. Rossi and H. Staub, “Ionization Chambers and Counters,” p. 58. McGraw-

Hill, New York, 1949.

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1.2. IONIZATION CHAMBERS 103

outside the high-voltage electrode) is necessary; occasionally one must take special precautions-such as connecting all grounds together only a t the first tube of the amplifier-to prevent electrical pickup. The use of a short time-constant in the amplifier eliminates most ac and micro- phonic difficulties, although in low-level work it is often necessary t o operate the preamplifier filaments on dc.

1.2.4. Amount of Ionization liberated

The basic process by which a fast charged particle loses energy in a gas have been discussed in Chapter 1.1,* where it is pointed out that the total energy loss of a particle of charge Z and velocity @ is, within certain limitations, just proportional to Z 2 times a function of @. Experimentally i t is found that a given energy loss gives rise to a number of ion pairs that depends on the gas, but is approximately (to at worst 10%) independent of the nature and speed of the particle. One can understand in a qualita- tive way why this should be so: the primary energy-loss event results either in excitation of the gas molecule, or in ionization. In the latter case the electron may be ejected with considerable energy, but if so it will itself undergo further excitation or ionization collisions, 80 that the energy ulti- mately either goes into ionization or into excitation (whence i t is dissi- pated in collisions or escapes as radiation.I8 The partition of energy between ionization and excitation depends mainly on the behavior of rather slow electrons even though the primary particle may be of very high energy. The energy loss corresponding to the formation of one ion pair W proves to be a few times the ionization potential.

The constancy of W means that the energy of a particle which stops in an ionization chamber can be measured by measuring the quantity of ionization released, or more generally the energy lost in the chamber by particles passing through is directly proportional to the ionization. This is an important and much-used property of the ionization chamber, and for quantitative work i t is clearly necessary to have accurate empirical data on W . The work of Jesse and S a d a u k i ~ ' ~ - ~ ' has provided a large amount of information on W ; i t extends previous work, and where a cross-check

* See also Vol. 4, A, Parts 1 and 4. 16 Xenon and, to a lesser extent, krypton and argon give off a considerable fraction

of this energy as "scintillation" light in the visible and ultraviolet region. See R. A. Nobles, Rev. Sci. Znstr. 27, 280 (1956); C. Figgler and C. M. Huddleston, Phys. Rev. 96, 600 (1954); A. Sayres and C. S. Wu, Rev. Sci. Instr. 28, 758 (1957). Such noble gas scintillations are discussed in Chapter 1.4.

18 W. P. Jesse and J. Sadaukis, Phvs. Rev. 97, 1668 (1955); 100, 1755 (1955); W. P. Jesse, H. Forstat, and J. Sadaukis, ibid. 77, 782 (1950).

20 W. P. Jesse and J. Sadaukis, Phys. Rev. 102, 389 (1956). 21 W. P. Jesse and J. Sadaukis, Phys. Rev. 107, 766 (1957).

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104 1. PARTICLE DETECTION

is possible it agrees with other contemporary data; the relative accuracy between different gases is a few tenths of a per cent, and the absolute accuracy about 1 %. Their principal results are summarized in Table 11,

TABLE 11. Average Energy, in ev, to Make an Ion Pair W for Beta Particles, Po210 Alpha Particles, and Low-Energy Alpha Particles, in Pure Gases*

Mean W,g W , for PoZlO W , for alpha Gas for beta particles alpha particles particles of 1.2 Mev

He Ne Ar Kr Xe Hz Air Nz 0 2

COZ CaHi CzHe CHI CzHz

42.3 36.6

(26.4) 24.2 22.0 36.3 34.0 35.0 30.9 32.9 26.2 24.8 27.3 25.9

42.7 36.8 26.4 24.1 21.9 36.3 35.5 36.6 32.5 34.5 28 .0 26.6 29.2 27.5

42.4 37.4

(26.4) 24.1

37.1 38.1

36.3 29.8 28.5 31 .O 29.0

W. P. Jesse and J. Sadaukis, Phys. Rev. 102, 389 (1956); 107, 766 (1957).

where values of W are listed as the energy loss in electron volts corre- sponding to one ion pair. The table is actually based on the assumption that in argon W is a true constant for different particles; this assumption is strongly supported in reference 21. The constancy of W in argon for alpha particles of varying velocities has been checkedlS for a range of 1 to 9 MeV.

There is evidence from a study of Po210 recoil nuclei20 that these ex- tremely slow and heavy particles have a W , in argon, about four times that of alpha particles. However, fission fragments already exhibit “nor- mal” behaviorzz-they have a W of 36 ev in air, and presumably would show the standard W in argon.

Bakker and SegrtP found W for 340-Mev protons to be 35.3 for Hz and 33.6 Nz, 3% lower than the W,g values of Table 11. BarberZ4 studied the specific ionization of high-energy electrons (1 to 34 Mev). By assuming the validity of the theoretical energy-loss expression he found, for elec-

22 D. H. Wilkinson, “Ionization Chambers and Counters,” p. 21. Cambridge Univ. Press, London and New York, 1950. [See D. West, Can. J. Research A26, 115 (1948); N. D. Lassen, Phys. Rev. 70, 577 (1946).]

** C. J. Bakker and E. Segr6, Phys. Rev. 81, 489 (1951). 24 W. C. Barber, Phys. Rev. 97, 1071 (1955).

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1.2. IONIZATION CHAMBERS 105

trons of -2 MeV, W values of 37.8 0.7 in Hz, 44.5 t- 0.9 in He, and 34.8 5- 0.9 in Nz. At higher energies the Nz value remains constant but the Hz value increased by about 3% at 34 MeV, presumably because Cerenkov radiation begins to carry off some of the energy.

The general conclusion to be drawn is that in pure noble gases, and in hydrogen, W is independent of the energy and nature of the particle, to about 1%, over a wide range. In air and other complex gases one can expect a lessened ionization efficiency at high specific ionizations, i.e., an increase in W which can be of the order of 10%.

However, it has been found’g that in the noble gases of very high ionization potential, notably helium, minute admixtures of impurity reduce the value of W very decidedly: for example, 100 parts per million of Ar in He has a 25% effect. Presumably this is caused by ionizing colli- sions of metastable He atoms with argon atoms (the first excited state of He is higher than the ionization potential of Ar). Argon should be free of this difficulty, since it has a much lower ionization potential.

Statistical fluctuations in the actual number of N of ions formed in a chamber around the average number of fl = AE/W (where energy AE is lost in the chamber) present two quite different problems. If the particle loses all its energy, fluctuations in the rate of energy loss will be correlated, since the total energy is fixed, and the standard deviation in N will be less than 4%; it has been calculated to be about two-thirds of this.25 But if the energy lost in the chamber, AE, is only a small fraction of E , the fluctuations in AE will depend on the number of primary collisions N , and on the energy given to the delta ray in each collision; the standard deviation will be considerably greater than dz. The problem is more characteristic of proportional counters than of ionization chambers, since the former must be used when AE is small; discussion is therefore post- poned to Section 1.3.1.

1.2.5. Noise: Practical Limit of Energy Loss Measurable

The smallest charge which can be detected as a pulse in an ionization chamber is limited by the intrinsic noise of the first stage of the amplifier. Amplifier noise as i t affects the sensitivity of ionization chambers is dis- cussed by ElmoreZ* and Gille~pie.~’ * Depending on conditions (fast or slow rise, large or small chamber capacity), different sources of noise may become the predominant source of noise charge. The optimum signal-to-

* See also Vol. 2, Chapter 12.5. 26 U. Fano, Phys. Rev. 72, 26 (1947). 26 W. C. Elmore, Nucleonics 2, (3), 16 (1948). 27 A. B. Gillespie, “Signal, Noise, and Resolution in Nuclear Counter Amplifiers.”

Pergamon, New York, 1953.

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106 1. PARTICLE DETECTION

noise situation proves to be one in which rise time is comparable with pulse duration. Elmore shows that in a typical short-pulse-duration chamber, if C,,, is the chamber capacity in micromicrofarads, R, the “equivalent shot noise resistance” in ohms, and T the pulse duration in microseconds, the most probable noise charge in electron charges is

Q/e - CPpf dR./7 for the best signal-to-noise ratio. The equivalent resistance R, is defined so that 4kTR, = 2eI,/gm2 where k is the Boltzmann constant, and I , and gm are the plate current and transconductance of the first tube of the amplifier. A typical value for R, is 10000, for 7 , 10 psec, and for C , 30 ppf, so that Q / e - 300 ion pairs equivalent noise. This would correspond to about 9000 ev of energy loss in the chamber. Of course fluctuations of three or four times the most probable noise occur frequently, and the minimum charge which can be detected reliably corresponds to about 5 times this, or an energy loss of 50,000 ev. This is somewhat better than is usually achieved in practice, although Wilkinson28 cites some experience indicating that it may be attainable.

Improvement by further lengthening T is not very effective, even if speed of response can be sacrificed, since grid resistor noise, independent of T , becomes important. For T not restricted, Elmore finds for the opti- mum case the most probable noise is

Q / e = 735[C,,t(R,/R,)11’2

where R, is the effective grid resistance (the grid resistor in parallel with the equivalent noise resistance of grid current). For a high gm tube such as the 6AK5, R, is limited by grid current, so one is limited to a threshold sensitivity which proves to be two or three times better than that of the short-pulse limit. The pulse duration and rise time, for a typical case, might be 50 Msec.

The selection of amplifier rise time and clipping time depends not only, or even principally, on the noise problem, but on the particular application in hand-the speed of counting, necessity for accurate timing, shape of ionization chamber pulses, quantitative preservation of pulse height, etc. If accurate reproduction of pulse shape is not important, signal-to-noise and pulse-height reproduction can both be improved by using a rise time and clipping time which are equal, and somewhat longer than the rise time of the slowest chamber pulse. For a detailed discussion of several specific cases, see Wilkinson,’ Chapter 4.

x8 D. H. Willcinson, “Ionization Chambers and Counters,” p. 142. Cambridge Univ. Press, London and New York, 1950.

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1.2. IONIZATION CHAMBERS 107

1.2.6. Some Types of Pulse ionization Chambers

1.2.6.1. Alpha-Particle Chambers. The typical energy release of a radioactive or induced alpha emission is several MeV, so i t is clear from the foregoing that the ionization chamber is very well suited to the detec- tion of these particles, or measurement of their energy. Typical ranges of these alpha particles in air at one atmosphere are a few centimeters, so that it is easy to contain the entire path of the particle in the chamber. Absolute counting of alpha particles is usually accomplished by placing a thin deposit containing the alpha-active material on the negative elec- trode of the chamber. Ideally this constitutes a ‘ I 27r” counter (assuming the electrode is plane), so that the fractionf of all disintegrations counted is +. However, the finite thickness t of the deposit will introduce a correc- tion; the fraction which escapes can easily be shown to be $(l - t /2R), where R is the range of the particle in the material; since there will be some energy Emin which is the least energy an escaping alpha can have and still be counted, because a nonlinear discriminator of some sort must be set to reject the unwanted small pulses due to noise, etc., the range is reduced by R(Emin), the range of a particle of energy Emin, and the fraction becomes

f = + { 1 - t /2[R - R(EmiJ]}.

A second correction is required to take into account the backscattering of alpha particles which start into the material in a direction away from the gas, but are deflected into the counting volume by multiple scattering. Rossi and S t a ~ b ~ ~ give numerical results of a calculation of this effect for various materials and energies. Typical values would be, for an alpha particle of 3.68 cm range, an increase in f of 8% for gold or 2% for aluminum.

Measurement of alpha-particle energy by means of ionization chambers is considered in Section 2.2.1.2.

1.2.6.2. Proton Recoil Detectors. Neutrons can be studied by observing the ionization of recoil protons from n - p collisions in the counter gas, if it is hydrogenous, or in a hydrogenous foil or lining of the chamber. A proton recoil at an angle 0 with respect to the direction of a neutron of energy EO will have an energy of EO cos2 e (nonrelativistic). I n the energy range from a few hundred kilovolts to a few Mev the recoils can be stopped in the gas (a 5-Mev proton has a range of 34 cm in air at 1 atmosphere) and the recoil chamber can be used as a measure of the energy and abso- lute flux of neutrons; this application is discussed in Section 2.2.2.1.

As a method for counting neutrons on a relative basis the recoil ioniza- 29 B. B. Rossi and H. Staub, ‘‘Ionization Chambers and Counters,” p. 127 McGraw-

Hill, New York, 1949.

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108 1. PARTICLE DETECTION

tion chamber is usually less satisfactory than the proportional counter (Section 1.3.1) because the continuous energy distribution of the protons leads to many undetectable recoils.

A large number of specific designs of proton recoil detectors are dis- cussed in references 3 and 4. More recent work is discussed by Johnson and Trai130 and by Berenson and Sh~rman.3~

1.2.6.3. Boron Trifluoride Chambers and Fission Chambers. Neutron: of all energies can be detected by the energy released in an ionization chamber by a nuclear disintegration; it is sometimes convenient to detect high-energy protons this way also. For neutrons the most widely used reaction is Bl0(n,a) Li7, which releases 2.34 Mev for thermal neutrons, and of course more for fast neutrons. The cross section for this reaction is very large, so that even though natural boron contains only 19% B'O it leads to relatively high efficiency counters.

Among the gaseous compounds of boron BFt is the most stable and satisfactory. It is reasonably free from electron attachment when pure, but the commercial gas often contains impurities which are difficult to remove and which lead to attachment. A method of preparing the pure gas is described by Graves and Froman. l3 Boron enriched in Bl0 is availa- ble from suppliers of stable isotopes.

B'O has a cross section for slow neutrons which varies as l / v up to energies in the kilovolt range. Thus it lends itself to absolute measure- ments of neutron density n (rather than flux nv) in the low-energy range, since the disintegration rate is proportional to nvu, and therefore, to n, since u - l / v .

An ionization chamber filled with BF3 provides a very stable method for a relative measurement of neutron flux, since it can be arranged so that the majority of the pulses are the same height (by using cylindrical geometry or a gridded chamber) and the pulse height is essentially inde- pendent of the applied voltage. However, the pulses are small and for most applications a proportional counter (Section 1.3.1) is more convenient.

For measuring an integrated flux or time-average flux a current chamber (Section 1.2.7) filled with BF3 is very satisfactory.

Another useful reaction for neutron measurement is the fission of heavy nuclei. Fission releases nearly 200 MeV; the fragments have a range of about 2 cm in air, and ionize most heavily at the start of their range. Very thin deposits of fissionable material are necessary to obtain the full energy of the fragment, but the energy release is so large that for counting this is not very critical.

UZ3& has a large slow-neutron fission cross section, and competes in 30 C . H. Johnson and C. C. Trail, Rev. Sci. fnstr. 27, 468 (1956). 31 R. E. Berenson and M. B. Shurman, Rev. Sci. Instr. 20, 1 (1958).

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1.2. IONIZATION CHAMBERS 109

efficiency with boron. U233 and Pu239 are also slow-neutron fissionable, but have higher alpha activities. However, none of these substances is gener- ally available.

At higher neutron energies several heavy elements show convenient fission “thresholds”; some of these are listed in Table 111. Above bismuth

TABLE 111. Approximate Thresholds of Various Heavy Substances for Fission by Neutronsn

Nucleus Threshold

NpZ3’ 0.4 Mev Pa232 0.5 Mev U 2 3 8 1.1 Mev Th234 1.3 Mev Bi2o9 60 Mev

*From B. T. Feld, in. “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 2, p. 347. Wiley, New York, 1953.

all of these elements are alpha-active. Rapid electron collection and a short resolving time are necessary to prevent “pile up”-imitation of a large fission pulse by two or more alpha pulses.

1.2.7. Current Ionization Chambers and Integrating Chambers*

Historically the earliest use of ionization chambers was not as single- particle detectors but as meters for the average rate of ionization occurring in a gas, and such current-or charge-meters still have very wide utility; for example, in detecting and measuring radioactivity, in radiological health measurements, in cosmic-ray intensity studies, and in beam- monitoring at particle accelerators. In general the problems of current chambers are quite different from those of pulse chambers and they will not be discussed here in The current to be measured is usually very small and the use of good insulators is essential. Polystyrene, amber, quartz, and Teflon are satisfactory. Guard electrodes to prevent a direct leakage path between high-voltage electrode and collector (Section 1.2.3.) are essential in this application. Care should be taken that ionization cannot collect on an insulating surface.

The second principal problem of current chambers is recombination. Since the recombination rate of ions is proportional to the square of the ionization density, recombination affects the linearity as well as the

* See also Vol. 4, A, Section 2.1.5. 32 For more information on current chambers see D. H. Wilkinson, “Ionization

Chambers and Counters,” Chapter 5. Cambridge Univ. Press, London and New York, 1950.

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110 1. PARTICLE DETECTION

absolute accuracy of ionization chambers. In many applications repro- ducibility and accuracy are the essential qualities demanded of the current chamber. A pure, nonattaching gas is essential unless the E / p values are everywhere high. A small amount of attachment, which may be quite acceptable in a pulse chamber, will often lead to recombination and there- fore to nonquantitative, nonlinear effects in a current chamber. Recom- bination is particularly serious in chambers used with pulsed accelerators, since the ion density is high during the pulse even though the average ion current may be low.

A well-designed ionization chamber can have a time-constant of weeks, and its accuracy is usually limited by the precision of measurement of current or amount of charge.

The transient response of a current chamber is of course quite different depending on whether or not attachment is taking place. In a free-electron chamber the part of the current carried by electrons will have a response in the microsecond region. Rossi and S t a ~ b ~ ~ describe a chamber which was used to detect changes in gamma-ray flux occurring in less than 1 psec.

1.3. Gas-Filled Counters*

1.3.1. Gas Multiplication; Proportional Counterst

In the discussion of the behavior of electrons in gases (Section 1.2.1) it has been assumed that the electrons released by the initial ionizing par- ticles do not create further ionization after they have been slowed down to the mean agitation energy. They continually gain energy from the electric field, at, an average rate weE, where w is the drift velocity, but i t has been assumed that they lost this energy in elastic collisions or excita- tion collisions with gas molecules. If the field strength is sufficiently great, however, some electrons will acquire an agitation energy greater than the ionization‘potential of the gas molecules, and new ionization will be formed. If-the average rate of ionizing collisions is a per centimeter, the average number of electrons after I centimeters, per original electron, would be n = eaz. This effect is called gas mu2tipZication.f By its use the total amount of ionization from a given initial act can be increased, thereby increasing the available signal. It is, of course, an undesirable

33 B. B. Rossi and H. Staub, “Ionization Chambers and Counters,” p. 106. McGraw- Hill, New York, 1949.

t See also Vol. 4, A, Section 2.1.4. $ See also Vol. 2, Chapter 4.1.

* Chapter 1.3 is by Robert W. Williams.

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1.3. GAS-FILLED COUNTERS 111

effect in an ionization chamber, where its presence destroys the quantita- tive relationship between the energy lost in the chamber and the amount of ionization released.

A counter which is designed to use gas multiplication as an amplifying device is called a proportional counter. It will normally have a very small- radius positive electrode, so that the region of high field and therefore of gas multiplication will be confined to a small volume near the electrode. The most frequently used geometry is cylindrical, with concentric elec- trodes of radii a and b, and with a << b. We will discuss only this case; others which are found useful, such as a small loop of wire inside a sphere, or a grid of parallel wires inside a disk, are fairly obvious extensions. Since the region of multiplication is confined to a small volume near the central (positive) electrode, nearly all the electrons released by the particle to be detected will traverse the entire region of multiplication and therefore will have an average multiplication, which we designate by M , and which is independent of the part of the chamber in which the ionization was released. The signal will therefore be proportional to the initial ionization, provided that the space-charge near the wire remains small enough that the field distribution is substantially unaltered (otherwise the multiplica- tion probability will be reduced). The space-charge limitation means that the largest usable M , in a given counter, will depend on the amount of ionization released by the particle traversing the counter, and will be larger for a small initial ionization.

An actual calculation of gas multiplication is not very useful, since i t depends on cross sections which are not well known, and may be impurity- sensitive. The most practical scheme for designing counters is to extra- polate from empirical data with the help of scaling laws.

Assuming that all multiplication occurs near the central electrode, and that photoelectric processes can be ignored, one can show’ that the multiplication M , a t voltage V O and pressure P, will have the functional form

= f (&,Pa). (1.3.1)

Multiplication depends sensitively on the nature of the gas, and it is clear (Section 1.2.1) that pure simple gases, especially noble gases, will have a relatively low threshold for multiplication, because energy-loss mechanisms are inefficient. However, there are two disadvantages to pure noble gases which often makes it advisable to go to a mixture or a complex gas: the multiplication is a very sensitive function of voltage; and the

1 B. B. Rossi and H. Staub, “Ionization Chambers and Counters,” pp. 78-86. McGrsw-Hill, New York, 1949. Reprinted by permission of the U.S. Atomic Energy Commission.

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112 1. PARTICLE DETECTION

energetic photons from excited atoms may eject photoelectrons from the cathode (or from other molecules) which will start new electron “ava- lanches.” One thus approaches the “breakdown” threshold, which is reached when the expected number of new avalanches per avalanche is

1000

too

M

10

200 600 1000 1400 1800 2200 2600 3000

FIG. 1. Gas multiplication M versus voltage, for a counter with wire diameter 2a = 0.010 in., cylinder diameter 2b = 0.87 in. (A) Tank hydrogen, 99.97% pure, with pressures of 10 (A,) and 55 (Ae) cm Hg. (B) Methane, 85% pure, with pressures of 10 (B1) and 40 (BZ) cm Hg. (C) Tank argon, 99.6% pure, with pressures of 10 (Ct) and 40 (C,) cm Hg. (D) Mixture of 90% hydrogen and 10% methane, with pressures of 10 (D,) and 40 (Dz) cm Hg. (From Rossi and Staub,‘ by permission.)

greater than one. Breakdown counters (Geiger counters, etc.) are con- sidered in Section 1.3.2.

Data from the Los Alamos studies on proportional counters’ are repro- duced in Figs. 1 to 6, where experimentally observed values of multiplica-

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1.3. GAS-FILLED COUNTERS 113

tion M are plotted against voltage. These curves together with Eq. (1.3.1) will facilitate the design of proportional counters of various shapes.

The time response of a proportional counter can be understood by reference to the ionization-chamber equation, Eq. (1.2.4) and its accom- panying discussion. The mean free path of electrons in gases at one atmos-

FIQ. 2. Tank argon, 99.6% pure. Wire diameter 0.001 in.; cylinder diameter 1.56 in. Gas multiplication M versus voltage for a pressure of 6.8 atmos. (From Rossi and Staub,' by permission.)

phere is of the order of cm, and the gas multiplication becomes impor- tant only in the last few mean free paths, so that the extra electrons of the avalanche are liberated very close to the central wire, usually within con- siderably less than one radius. The fraction of potential through which they fall is

where 6 is the distance from the ionizing collisions to the wire. The motion of the electrons toward the wire therefore causes only a small fraction of the total pulse. The bulk of the pulse begins to rise very steeply as the

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114 1. PARTICLE DETECTION

VOLTS

FIG. 3. Spectroscopic nitrogen. Wire diameter 2a = 0.001 in.; cylinder diameter 1.56 in. Gas multiplication M versus voltage for pressures from 0.79 to 4.25 atmos. (From Rossi and Staub,' by permission.)

VOLTS

FIG. 4. Boron trifluoride. Gas multiplication M versus voltage. (A) Wire diameter 2a = 0.010 in.; cylinder diameter 2a = 1.50 in.; pressure p = 10 cm Hg. (3) Wire diameter 2a = 0.001 in.; cylinder diameter, 1.56 in.; pressure p = 80.4 cm Hg. (From Rossi and Staub,' by permission.)

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1.3. GAS-FILLED COUNTERS 115

positive ions which are formed near the wire start to move out of the very strong field there. Figure 7 illustrates the pulse schematically: the period of delay, which may be from 0.1 psec a t low pressure to 1 psec or more for a high-pressure counter, the steep rise as multiplication frees the posi- tive ions, the gradual lessening of ion drift velocities as they move out- ward. A sharply differentiated proportional counter signal will give a very

1 0 0 0 - , I [ I I , I I I 4 - - - - - - - - - - -

- - - - - -

300 600 900 1200 IS00 VOLTS

FIG. 5. Mixture of 98% argon and 2% Con. Wire diameter 2a = 0.010 in.; cylinder diameter 2b = 0.87 in. Gas multiplication M versus voltage for pressures of 10 and 40 cm Hg. (From Rossi and Staub,’ by permission.)

short pulse (perhaps 0.2 psec) but in coincidence work the effect of the delay may be more important than pulse length.

The usefulness of the proportional counter as a quantitative instrument depends on the observed behavior of the multiplication-its linearity, constancy throughout the chamber, fluctuations, and reliability. Accord- ing to the Los Alamos studies’ the “proportional” action for different parts of the counter is excellent provided there is negligible attachment. E / p is usually quite high in a proportional counter, so this difficulty is only serious in a gas like BF3, which often has impurities showing some attachment. However, even BF, counters may be satisfactory from this

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116 1. PARTICLE DETECTION

‘Oo0 ! E 3

1700 2100 2500 2900 3300 3700 4100 4500 VOLTS

FIQ. 6. Mixture of 90% argon and 10% COi. Wire diameter 2a = 0.005 in.; cylinder diameter 2b = 1.56 in. Gas multiplication M versus voltage for pressures of 1.13, 2.15, and 3.5 atmos. (From Rossi and Staub,’ by permission.)

t

FIG. 7. Schematic representation of a proportional-counter pulse. In practice the slow portion of the pulse would usually be eliminated by an RC or other low-frequency rejec,tion network.

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1.3. GAS-FILLED COUNTERS 117

point of view, when only neutron detection and not the energy of the disintegration is of interest.

The linearity of response is limited by space-charge considerations and therefore by total ionization. For a proton or alpha-particle a gas multi- plication of 100 is usually safe, while for a beta ray the multiplication can be made correspondingly higher. Thus Curran et aL2 have used pro- portional counters with multiplication as high as lo4 to measure very weak beta rays down to a few hundred electron volts, and even to detect single photoelectrons from ultraviolet light.

The statistics of the multiplication process have been worked out by Snyder3 for a simplified model. The result is, for large gas multiplication, that the standard deviation is larger by a factor of d2 than that expected from the statistics of the initial ionization process if the latter is Poisson. That is, if N is the average initial ionization caused by the primary parti- cle, the standard deviation in the relative output, AV/V, would be ex- pected to be d T N for an ionization chamber, but is

AV/V = d 2 / N (1.3.2)

for a proportional counter. Experimentally2 the fluctuations seem to be a little bit smaller than this.

Such fluctuations are, therefore, sufficiently small that the propor- tional counter can be used as a reliable instrument for obtaining informa- tion on individual events.

The fluctuations in initial ionization have here been assumed to be Poisson, that is, equal to dw. However, they can be much larger (be- cause of knock-on processes) for a fast particle which does not lose, on the average, a large amount of energy in the counter. Such “Landau” fluc- tuations are thoroughly treated by Rossi4*

The general reliability and stability of proportional counters which con- tain a t least some polyatomic gas is very good. Obviously the high-voltage supply must be well stabilized and the gas composition must remain constant; under these conditions they are very satisfactory. The factor of 100 or so in pulse height, over the ionization chamber, brings the signal from a heavily ionizing particle up to a level that is very convenient electronically, and brings the signal from a relativistic particle into the detectable range. Many of the functions of proportional counters can be performed more efficiently or a t higher resolution in time with scintillation counters (Section 1.4.1.9) ; however, proportional counters require less

*Also refer to Section 1.1.2.6 of this volume. 1s. C. Curran, J. Angus, and A. L. Cockroft, Phil. Mag. [7] 40, 36, 53, 929 (1949). SH. 9. Snyder, Phys. Rev. 72, 181(A) (1947).

B. B. Rossi, “High-Energy Particles,” p. 29. Prentice-Hall, New York, 1952.

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118 1. PARTICLE DETECTION

total energy loss, and are more nearly linear when used with heavily ionizing particles.

Proportional counters are available from some commercial radioactiv- ity-equipment suppliers, with BF, and other fillings, in various cylindrical sizes and as thin-window counters.

1.3.2. Geiger Counters and Other Breakdown Counters

In the preceding section we considered the possibility that a sufficiently large electron-initiated avalanche in a counter with a high-field region might produce, in addition to the n electrons of the avalanche, a photo- electron at the cathode or in the low-field region of the counter. If the probability of doing this is y per avalanche electron, the chance of starting a second avalanche is yn, and the expectation value of the total number of electrons produced from one original electron by the series of avalanches is

M = n + yn2 + y2n3 + . . . which sums to M = n/(l - yn) if y n is less than 1; but which diverges if yn is greater than 1. The latter is the phenomenon of breakdown, and leads to a counter which responds to each electron introduced into it with a signal which is limited only by some other mechanism.

The Geiger counter is a breakdown counter in which the discharge is eventually stopped because the positive-ion space charge that accumulates around the anode modifies the field to the point where yn < 1. As the electrons are collected, leaving the slowly moving positive ions in place, the potential of the anode is nearly unchanged (see preceding section), the charge on the anode is reduced, and one has instead a positive space- charge “sheath” around the anode. The field is thus smoothed out, becoming weaker near the wire, and gas multiplication is reduced. A simple-gas counter rekindles itself , however, and will exhibit a relaxation phenomenon indefinitely unless electrically “quenched.” The source of the new discharge, which takes place after the positive-ion ‘‘ sheath” of space charge has moved away from the anode and the field is partially restored, has been shown to be the ejection of electrons from the cathode by the positive ions as they collide with it. It is easy to make a counter which does not have this defect (and therefore does not require quench- ing), and nearly all counters now used are made this way. One adds to the filling gas, normally argon, a polyatomic gas or vapor such as alcohol, ethylene, or petroleum ether. The ionization potential of the polyatomic molecule is less than that of argon, and the Ar+ ions transfer their charge to the complex molecules. If the molecular gas represents about 10% of the total pressure, the probability of charge transfer is essentially unity.

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1.3. GAS-FILLED COUNTERS 119

The energy available in neutralizing the complex ion at the cathode is now dissipated in molecular dissociation, with the result that the probabil- ity of releasing an electron from the cathode proves to be negligible. Of course this means that the complex gas is consumed as the counter operates, and the counter has a finite life.

The counter with a complex vapor added is called self-quenching, and we restrict ourselves to this type in the general discussion that follows. A very common type of counter is cylindrical, about 1 in. in diameter, with a 0.001-in. diameter central wire and a filling of argon (9 cm Hg) and alcohol (1 cm Hg). Such counters typically have an operating region of about 1000-1200 volts over which the size of all pulses, for a given voltage, is the same independent of initial ionization, and over which the counting rate rises by 3 to 5% (because of increase of effective counting volume and increase of spurious pulses, presumably) ; this is the so-called plateau.

In the self-quenching counter the photoelectrons which maintain the discharge are released from the vapor molecules and not from the cathode -the vapor is fairly opaque to the ultraviolet photons in question, and the photon mean free path is small compared t o the counter radius. The discharge therefore spreads down the central wire with a speed charac- teristic of the mean free path of a photon divided by the time necessary for an avalanche to develop. This is fairly slow, about 10 cm per micro- second. As with the proportional counter, the pulse is mainly due to positive-ion motion, but the important part of that motion takes place more quickly than the discharge-spreading time, so that the observed pulse rise is more or less linear for the first microsecond or two (depending, obviously, on the length of the counter and on whether the discharge starts in the middle or at one end). A 12-in. counter with a low-capacity load, operated 50 volts above threshold will give the order of 5 volts in a microsecond or two; the rate of rise falls off quickly from then on. The counter does not recover sufficiently to give another full-sized pulse until 100-150 psec have elapsed.

Since about lo9 ions are released per count, all resulting in decomposi- tion of the vapor molecules, the counter’s characteristics change with use, the threshold rises and eventually it becomes unusable, either because of too much tar or not enough alcohol; the useful life usually is between lo8 and lo9 counts. Halogen-filled counters do not suffer from this defect, but they exhibit electron capture, with the obvious concomitant disad- vantage of unpredictable time delays. Electronically quenched simple-gas counters are of course a possibility; but if one is going to that much trouble it may be better to use high-multiplication proportional counters.

The principal advantages of the Geiger counter are the simplicity of its associated equipment (because of its large pulse), and its high effi-

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120 1. PARTICLE DETECTION

ciency. The efficiency for a minimum-ionizing charged particle passing through the gas, averaged over the area, is more than 99%; ordinarily the inactive volume represented by the finite thickness of the walls is a more serious effect. The efficiency of a brass-walled Geiger counter for gamma rays is roughly, in per cent, equal to the energy of the gamma ray in MeV, in the vicinity of 1 MeV.

Wilkinson6 gives a large amount of information on Geiger counters, and discusses some examples of nonstandard counters (odd shapes, etc.).

Although the Geiger counter is the only breakdown counter presently in wide use, it should be mentioned that uniform-field breakdown counters have been successfully operated.6 A novel breakdown counter of quite different sort is described by Conversi and Gozzini.7 They have found that glass tubes filled with neon, when placed in a sufficiently strong electric field, will break down and emit a flash of light when ionization is released in them. Applications appear to be restricted to tracing the paths of high-energy particles.

1.4. Scintillation Counters and Luminescent Chambers*

1.4.1. Scintillation Counters

1.4.1 .l. Infroduction. The field of scintillation counting has developed mainly in the past twelve years, and has for some eight years been one of the principal techniques for particle detection. By direct or secondary processes it is now possible to detect every known type of elementary particle by this means. The instrumentation has developed from a 1 cmS crystal mounted on a 931-A photomultiplier, to the use of large tanks,? crystals or plastics as indicated in Fig. 1, coupled with photo- multipliers with cathodes 2, 5, or 16 in. in some cases.$

Scintillators exist in several forms: crystals, liquids, plastic solids, and gases. In all cases, the phenomenon depends on the fact that the suitable ‘ ‘ f l~o r s ’~ give off pulses of light when a charged particle passes through

D. H. Wilkinson, “Ionization Chambers and Counters,” Chapter 7. Cambridge

J. W. Keuffel, Rev. Sci. Znstr. 20, 202 (1949). M. Conversi and A. Gozzini, Nuovo cimento [lo] 2, 189 (1955).

Univ. Press, London and New York, 1950.

t Refer to Section 1.4.1.10 of this volume. 3 See also Vol. 2, Section 11.1.3.

* All of Chapter 1.4 is by George T. Reynolds, except Section 1.4.1.10 __

which is by F. Reines.

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1.4. SCINTILLATION COUNTERS 121

them.* This light is directed to a photomultiplier cathode where it ejects electrons that are then accelerated and multiplied in the dynode structure of the tube. The charge finally collected a t the anode is recorded by suitable circuits. Among the materials found suitable in crystal form are inorganic and organic substances (see Table I). The liquids and plas- tics are, so far, organic. There are several important distinctions between the characteristics of organic and inorganic scintillators, including life-

FIG. 1. Two examples of large tank scintillators. Tanks of the general design fea- tures have been made up to four feet in dimension.

times, linearity of energy response, temperature effects, etc., but the basic difference appears to be due to the fact that the process by means of which light is emitted from an inorganic crystal is due primarily to the crystal structure, whereas organic substances exhibit luminescence by virtue of molecular properties.

1.4.1.2. Scintillation Process in Organic and Inorganic Substances. A discussion of the scintillation process in organic substances has been given by Brooks.’ Briefly, the phenomena involved are as follows, with reference to Fig. 2. A charged particle passing through a scintillator

* See also Vol. 6, B, Part 11. IF. D. Brooks, in “Progress in Nuclear Physics” (0. R. Frisch, ed.), Vol. 5,

pp. 252-313. Pergamon, New York, 1956.

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122 1. PAltTICLE DETECTION

TOLE I. Scintillator Characteristics

Inorganic crystal scintillators Relative Decay time kmSx Density mp

Scintillator pulse height (sea) (A) (@;m/cc) (OC)

Sodium iodide 2 .1 0.25 X lo-' 4100 3.7 651 (thallium activated) Cesium iodide 0.6 1.5 X 10-6 4500 4.5 62 1 (thallium activated) Cesium iodide -5.0 c 4500 4.5 62 1 (cooled to 77°K) Cadmium tungstate 0.2 6 X 5300 7.8 1325 Lithium iodide 0.7 2 X 10-6 4400 4.1 446 (europium activated)

Organic crystal scintillators Relative Decay time Amax Density mp

Scintillator pulse height (mrsec) (A) (gm/cc) ("C) a//3

Anthracene 1.00 32 4400 1.25 217 0 .1 p,p'-Quaterphenyl 0.85 7 4200 318 trans-Stilbene 0.60 6.4 4100 1.16 124 Diphenylacetylene 0.45 5.4 3900 1.18 62.5 Terphenyl 0.40 5.0 4000 1.23 213

Liquid scintillators Relative

Secondary pulse Decay Primary solute solute height to time in Am,, Density

Solvent (gm/l) (gm/l) anthracene mpsec (A) (gm/cc) a/@

Toluene Diphenyl- anthracene (10)

BBO (1) POPOP (1.5) PRD (8) a-NPO (3) DEAMC (1.2) PPO (4) P-TP (9)

POPOP (0. I) a-NPO (0. I) DPH (0.1)

PBD (10) BPOP (0.2) p-Xylene PBD (10)

PCH P-TP (3) DPH (0.01) P-TP (5)

PPO (12)

0.85 0.70 0.67 0.63 0.63 0.59 0.54 0.48 0.66 0.59 0.51 0.77 0.68 0.50 0.48 0.33

0.87 0.09

4300 <2.8 3700

4200

- < 3 . 0 3800 -2.2 3550

- <3.0

8.0

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1.4. SCINTILLATION COUNTERS 123

TABLE I. Scintillator Characteristics (Continued) ~ ~~

Plastic sczntillators

Secondary pulse Decay Relative

Primary solute solute height to time in Density Solvent (gm/l) (gm/l) anthracene mpsec (A) (gm/cc) CY/B

Polystyrene TP (36) 0 .28 5 3 . 0 3550 0 . 9 0.10 TPB (16) 0 .38 4 . 6 4800 0 .08

PVT TP (30) 0 .33 4.0 CY-NPO (1) 0.44 POPOP (1) 0.47 BBO (1) 0.50

PBD (30) 0.45 (20) BBO (1) 0.52 (20) DPS 0.54

BBO (10) 0.44

DPS (1) 0.52 5 3 . 0 -3800 0.10

Explanations: TP: p-terphenyl; TBP: tetraphenylbutadiene; PPO: 2,5 diphenyl- oxazole; NPO: 2-( l-naphtyl)-5-phenyloxazole; PBD : 2 phenyl 5-(4-biphenylyl)- 1,3,4 oxadiazole; POPOP: 1,4, bis 2-(5-phenyloxa~olyl)-benzene; BBO: 2,5,-di-(4- biphenyly1)-oxazole; DPS: diphenylstilbene.

B e! 03

W

I (d) (el

i (C)

i (b)

I (0)

FIG. 2. Energy levels in an organic molecule and luminescence processes. (a) Ex- citation. (b) Internal degradation. (c) and (d) Internal conversion. (e) Fluorescence.

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124 1. PARTICLE DETECTION

looses energy ionizing, exciting, * and perhaps dissociating the molecules in a column near its trajectory, to an extent depending on the rate of energy loss, and therefore the velocity, of the particle. In the case of excitation, the processes are those of section (a) of the figure, where the electronic levels of the molecule are indicated. This excitation can be transferred to other molecules or dissipated in nonradiative processes, such as transitions among vibrational levels (heat). The present under- standing of the process is that transitions from higher excited levels to the la eve1 occur rapidly (10-l2 second) so that the situation of section (c) of the figure is that from which the radiative transitions of section (e) or the nonradiative (quenching) transitions of section (d) occur. Since transitions to 10 levels occur quickly, many aspects of the process are independent of the excitation mechanism, and much has been learned from the study of photoluminescence. However, although the molecular quantum efficiencies of some successful scintillators are as high as 90%, the energy conversion factor or scintillation efficiency in charged particle detection is the order of 4% for anthracene to 10% or somewhat more for certain inorganic crystals.

A great deal has been learned about the luminescence process in organic scintillators by the study of liquid and solid solutions, and these results have aided in the description of the process in crystals. The mechanism by means of which the energy is transferred from the molecules which are initially affected to those which finally radiate is not comp.etely understood, but it appears that the main process is one in which collisions are responsible for the transfer, either by exciton transfer or by a dipole resonance interaction such as that suggested by Forster2 until the energy is quenched or radiated. It appears that even the molecules of crystals are not in permanent strong attraction, and that here also collisions play an essential role. The detected spectrum in a practical scintillator depends on the emission and absorption spectra of the molecule, as well as the size of the sample. The effect of environment is indicated by the case of anthracene, where the absorption spectrum shifts to longer wavelengths by approximately 250 A as the material is changed from vapor to dilute solution to solid. The closer collisions possible in the liquid state result in increased self-quenching, reducing the quantum efficiency and the lifetime.

The role of radiative transfer has been indicated as minor, being most significant in plastic scintillators where it accounts for, a t most, 20 % of the energy transfer.a

* Refer to Chapter 1.1 of this volume. 2 T. Forster, Ann. Phusik [6] 2, 55 (1948). 8 R. K. Swank and W. L. Buck, Phys. Rev. 91,927 (1953).

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1.4. SCINTILLATION COUNTERS 125

The nature of the phenomenon in inorganic crystals is discussed by Swank,4 Birks, and Curran. * The successful inorganic scintillators from ionic crystals, and in particular, the alkali halides, notably NaI with thallium impurity, are of interest. In such crystals, charged particles may raise electrons into the conduction bands or into excited levels. The electron and the hole it leaves in the filled band move rapidly throughout the crystal (as an “exciton”) until captured by an imperfection, giving up energy in the form of vibrational transfer, or until captured by an impurity. A suitable impurity will become excited and then radiate as a scintillator. The alkali halides with thallium added as the impurity have high transparency to their own radiation. The best known are listed in Table I. The thallium impurity concentration does not seem to be critical between approximately 0.1 and 0.2 %. Other inorganic scintillators, including the alkaline earth tungstates are also listed in the table. Most of these crystals are available commercially.

1.4.1.3. Scintillator Characteristics. 1.4.1.3.1. GENERAL PROPERTIES OF SCINTILLATORS. The most commonly used inorganic crystal for high- energy particle detection is NaI(T1). The organic crystal usually taken as standard is anthracene, although it is more difficult to make this reproducible than is stilbene. Anthracene has been taken as about one- half as sensitive as NaI(T1) on the basis of pulse height) but the work of Sangster’ indicates that it may be relatively better than this. The com- parison of various scintillators in the literature often leads to incon- sistencies because of the fact that the spectral output of the scintillator and that of the photomultiplier are not exactly matched. Reflector and container characteristics can markedly influence comparative results. Removal of oxygen from the organic liquid scintillators by bubbling argon or nitrogen, has been found to make a significant increase (approxi- mately 25%) in pulse height in certain cases.

A useful quantity in the description of fluorescence phenomena is the quantum efficiency

number of quanta emitted = number of quanta absorbed‘

The “practical” q of a scintillation counter may be less than the molecu- lar q due to reabsorption. Since reabsorption depends on crystal size, it is a practical consideration.

* Reference is made again to Vol. 6, B, Part 11.

6 J. B. Birks, “Scintillation Counters.” McGraw-Hill, New York, 1953. 8 S. C. Curran, “Luminescence and the Scintillation Counter.” Academic Press,

R. K. Swank, Ann. Rev. Nuclear Sn’. 4, 111-140 (1954).

New York, 1953. R. C. Sangster, J . Chem. Phys. 24, 670 (1956).

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126 1, PARTICLE DETECTION

The scintillation efficiency is defined as energy emitted per fluorescence

energy lost by charged particle per fluorescence'

The role of reabsorption can be seen as follows, with reference to Fig. 3. Let E be the energy lost in the scintillator by a charged particle, w the

Absorption Emission Spectrum Spectrum

~~

FIG. 3. Schematic representation of the absorption and emission spectra of a scintillator.

energy necessary to produce a fluorescence excitation, and qo the molecu- lar quantum efficiency. The number of photons in a scintillation is given by

(1.4.1)

If the ratio of area b to a in Fig. 4 is denoted by K, then of the N photons emitted, KN escape, (1 - K ) N are reabsorbed, and qO(1 - K)N are reemitted. Thus finally N o = NK[1 - (1 - K)qo]-' escape, so that

E N = - qo.

w

(1.4.2)

is the practical q. The effect of reabsorption is also to lengthen the lifetime.

A typical result for anthracene is q o = 0.94 and q = 0.80. Since direct measurements on anthracene indicate that the energy lost by a charged particle in producing one observable photon is about 65 electron volts (corresponding to q) this result indicates that the number of electron

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1.4. SCINTILLATION COUNTERS 127

volts required to release L photon in the initial encounter is about GO elec- tron volts. The scintillation efficiency is determined as follows.

For a photon of wavelength 4400 A, hv = 2.8 volts. Therefore the efficiency is (2.3/60) = 4 or 5%.

Measurements on stilbene, terphenyl, diphenylacetylene, and plastic scintillat ons indicate a performance about 0.5 to 0.6 of anthracene and

c 0

E - dx

MeV/mg /cm2 of Anthrocene

FIG. 4. Energy response of an anthracene scintillator.

so about 100 to 120 volts per photon. The best solutions require about 80 volts per photon. Thus, in an application where a minimum ionizing particle passes through a 1 cm thick plastic scintillator, and the light collection efficiency is lo%, the number of photoelectrons from the cathode of a photomultiplier of cathode efficiency 10% is determined to be

-- lo' X 0.10 X 0.10 165. 120

Calculations similar to this for particular arrangements afford the means for estimating pulse heights and resolutions that can be achieved, accord- ing to principles discussed by Morton.8

8 G. A. Morton, RCA Revs. 10, 525 (1949).

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128 1. PARTICLE DETECTION

1.4.1.3.2. ENERGY RESPONSE. The energy response of. organic and inorganic scintillators differs markedly in the degree of linearity. By energy response is meant the light output as a function of energy loss of the particle. The early work of Jentschke and his groupg has been gen- erally confirmed. Although not strictly proportional to energy loss, the light output from inorganic crystals [NaI(Tl) is the standard example] is very nearly proportional to the energy loss down to -1 Mev for protons and -15 Mev for a’s. For p’s, protons, and a’s of the same energy loss a t these low energies the response of sodium iodide is roughly 1 : 1 : 0 .6 Recent work in this area is discussed by Brolley and Ribe.’”

On the other hand, the organic scintillators are not nearly as propor- tional. In fact, an early description of this lack of proportionality was given in the so called a l p ratio-the ratio of pulse height (i.e., light out- put) per unit energy loss for 5 Mev a’s and relativistic p’s. For organic liquids, plastics, and crystals this ratio lies between 0.08 and 0.10. This lack of uniformity has generally been attributed to “damage” suffered by the molecule when large energy transfers, due to high rates of energy loss by the particle, are involved, It can be shown’’ that without specify- ing any details as to the nature of the damage, the rate of luminescence can be given by

dL - A(dE/dx) dx 1 + B(dE/dk) - _ - (1.4.3)

where dE/dx is the energy loss, and A and B are constants of the material related to energy transfer and molecular “damage.” Brooks’ has reviewed more refined attempts to describe this nonuniformity, including those based on bimolecular processes. Results of comparing data with theory are shown in Fig. 4 where curve (a) is based on Eq. (1.4.3) and curve (b) based on a consideration of the role of bimolecular processes. At high values of d E / d x surface effects may be important.12 Because of the way that the measurements are made these might have been the cause of the apparent flattening of the curve in the region of the a points. With this in mind, curve (b) might be favored. There are still several factors that have not been investigated, including the effect of the extent of the ionized column cross section, and dependence on charge multiplicity. For example, curve (b) is consistent with results of fission products com- pared to a response. Further, the detailed nature of the distribution of

C. J. Taylor, W. K. Jentschke, M. E. Remley, F. S. Eby, and P. G. Kruger, Phys Rev. 84, 1034 (1951); also F. S. Eby and W. K. Jentschke, ibid. 96, 911 (1954).

lo J. E. Brolley, Jr. and F. L. Ribe, Phys. Rev. 98, 1112 (1955). l 1 J. B. Birks, Proc. Phys. SOC. (London) A64, 874 (1951). 12 E. Mateosian and L. C. L. Yuan, Phys. Rev. 90, 868 (1953).

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1.4. SCINTILLATION COUNTERS 129

the energy transfer (6 rays, etc.) might result in more “damage” locally caused by a’s than protons and therefore effects of a different nature.

1.4.1.3.3. LIFETIMES. The lifetimes of inorganic scintillators are gen- erally longer than those of the organic scintillators. Table I gives some of the lifetimes for the common scintillators. In the case of the organic substances, the study of lifetimes adds much to the understanding of the processes involved. For example, an impurity that tends to quench will reduce the lifetime; if there is significant optical reabsorption the life- time will be increased; WrightI3 has shown that surface effects can also change the lifetime. Generally, an exponential decay time is indicated by experiment.

Since most decay times are measured with photomultipliers followed by suitable circuitry, i t is important to keep the usual response time consideration in mind. It can be shown4 that the pulse-height maximum from such a system is given by

(1.4.4)

where N is the total number of excitations; q is the quantum efficiency of the cathode; G is the multiplication factor of the photomultiplier; g is the fraction of photons collected; and y = (RC/r ) , the circuit constant over the decay time of the fluorescence. Thus, if y is large, the pulse height does not depend on r ; otherwise there is a dependence. This has served as a basis for finding long components after initial short compo- nents in organic and NaI crystals.*

Also, since most organic scintillators have lifetimes of the order of 10-9 to lo-* second, the spread in transit times for the photomultiplier is an important limitation, sometimes amounting to as much as 2 x second. Recently developed photomultipliers have succeeded in reducing this time spread to the millimicrosecond region. I4

There are three general methods for measuring lifetimes. 1. In the notation above, the current from the photomultiplier is given

by 1 = NgqGef(t) where f ( t ) is the fraction decaying a t a given time, i.e.,

Z = Z o f ( t ) = (ZO/T)e-?

Thus, the voltage a t the input is

11 (1.4.5) ‘v = V,,’(e-’/RC - e - t / 7 ) = V,,’e- t /r[et /RC+t/T -

- Vo’e-t/r MT+RC)I /RCT - 11 1 - -

* On pulse-height determination, see also Vol. 2, Section 9.6.1. 1 3 G. T. Wright, Proc. Phys. Soc. (London) A68, 241 (1955). 1 4 Proc. 6th Scintillation Counter Symposium, IRE Trans. on Nuclear Sci. NS-6,

No. 3 (1958); E. H. Thorndike and W. J. Shlaer, Rev. Sci. Znstr. 30, 838 (1959).

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130 1. PARTICLE DETECTION

so that for RC small,

The procedure is to connect the photomultiplier directly to the plates of a scope. If RC is small, the anode voltage is proportional to the anode current so that a measurable exponential is obtained. This method has been used by Post who pulsed the photomultiplier high voltage, and by Swank, using pulsed X-rays.

2. Leibson has determined lifetimes using a pulsed X-ray source, modulated by diffraction from an oscillating quartz crystal (frequency u). Then the light output is

sin(ut - e) q J 0

41 + 0272 F ( t ) = (1.4.7)

where JO is the initial intensity and tan e = wr. Thus r can be determined from intensity decrease or phase.

3. A further method utilizes a shorted delay line and diode rectifier giving a waveform of the sort:

Thus, changing the delay line changes the pulse recorded by a following slow amplifier, from which the lifetimes can be deduced.

Work on lifetimes originally reported by Wright have been continued a t Harwell, relating to the difference in decay times of certain organic materials depending on whether the excitation was by P’s or a’s. The basic difference seems to lie in the effect of the specific ionization. Work by Owen14 indicates that although all the scintillators examined have nearly

TABLE 11. Decay-Time Dependence on Excitation

Phosphor

Fastest component showing proton/electron difference

Fast comp. decay time Decay time Intensity ratios in mpsec in mpsec P l e

Stilbene Anthracene Quaterphenyl Toluene solution Borate-xylene solution

6 . 2 370 1.8 33 370 2 .1

4.5 350 2 .1 1 2 . 8 200 1.8

4 .2 200 1.9

a Ratio of amplitude of these components under proton and electron excitation for pulses of equal peak height. This component is estimated to contain approximately 10% of the total energy for electron excitation.

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1.4. SCINTILLATION COUNTERS 131

the same long time decay component, the relative amount of light output found in the long component depends on the nature of the excitation. This is summarized in Table 11, for results using a PoBe source (4.5-Mev y’s and 12-Mev neutrons, which give approximately equal pulse heights), From these results it appears possible to distinguish between excitations due to a’s, neutrons or protons on the one hand, and y’s or 6’s on the other, by comparing the ratios of peak heights to total charge in the photomultiplier output pulses.

34uo

p- Terphenyl

p-p‘ Quaterphenyl

4600 5 )O

Wavelength - i; FIG. 5(a). Emission spectra of scintillator solutions in toluene (from Brooks).

1.4.1.3.4. SPECIAL DISTRIBUTION OF SCINTILLATION. In considering the spectral distribution of the scintillator output one must be concerned with how well the emission is matched to the photomultiplier used. Some typical scintillator emission curves are indicated in Figs. 5(a) and (b).

A. Crystals. The details of the emission spectra of crystals depends on their thickness, as mentioned in earlier discussions of reabsorption. The peak responses recorded for organic and inorganic scintillators are given in Table I in the column labeled Am*=.

B. Liquids. Here the wavelengths are the order of 200 or 300 8 shorter than crystals (thus note the evidence for interaction effects in the solid state) but considerable success has been achieved by use of wavelength shifters (for example diphenylthexatriene and aNPO) as indicated in

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132 1. PARTICLE DETECTION

Table I. Note that DPH added to terphenyl gives A,, at 4500 (10 mg/ liter) and aNPO added to terphenyl gives A,, a t 4100 (20 mg/liter).

C. Plastics. As Table I shows, terphenyl in polystyrene gives A,, a t about 3500 b and the addition of TPB as a shifter changes this to about 4500 b. TPB in Polystyrene or polyvinyltoluene has Amax a t about 4500 A.

Wovelength

FIG. 5(b). Emission spectra of solid solutions in polystyrene (from Brooks).

These spectra determine the nature of the cathode response required of the photomultiplier. Commercial photomultiplier responses vary con- siderably, but fairly typical responses are given in Fig. 6.* The spectra also determine the nature of the reflecting surface required, as discussed in the next paragraphs.

1.4.1.4. Liquid Scintillators. Five gm/liter of terphenylI6 in toluene gives a response of the order of 0.45 that of anthracene. Development of other solvents and solutes to which suitable wavelength shifters have been added have improved this response up to nearly 0.8. A good review of the liquid scintillation counter field will be found in the report of the Liquid Scintillation Counter Symposium held a t Northwestern Univer- sity in August, 1957, to be published by Pergamon Press.

*See also Vol. 2, Section 11.1.3.2. l6 G. T. Reynolds, F. B. Harrison, and G. Salvini, Phys. Rev. 78, 488 (1950).

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1.4. SCINTILLATION COUNTERS 133

The scintillation process in liquids is more complex than in crystals, but correspondingly more can be learned by varying the parameters. Using the concepts discussed earlier concerning the nature of the scintilla- tion process in organic crystals we can describe the process in a liquid solution as follows. The solvent initially receives most of the energy given up by the charged particle to be detected. In the course of collisions,

1111111111111 3 4 5 6 7 8 u 3 4 5 6 7 8 L 3 4 5 6 7 8

L

in 3 4 5 6 7 8 .- 5: Wave’ength angstroms x

s 4 s9 SII Multi-alkali

FIG. 6. Spectral response of various photomultiplier cathodes.

some of this energy is transferred to the solute, or fluor of the solution. Evidently the characteristics of the successful solvent must include ability to keep from quenching the energy for a time long enough for a transfer to the solute. The energy levels of the solute and solvent must be related so that this transfer can occur. Finally, the solute must have a high probability for radiating the energy as light a t a wavelength suitable for detection by standard photomultipliers. The processes involved are indicated in Table 111, in which c is the concentration of solute. In these

TABLE 111. Energy Transfer Processes

Process Description Reaction

probability

S + hvi+ S* s*+ s + h Y *

s* + s+ s + s or S*+ S

s* + s+ s + s* S* + F - + F* + S F* + F + h ~ a F * + S + F + S

F* + F + F + F S* + F - , S + F

or F * + F

Excitation of solvent Fluorescence of solvent (small)

Self-quenching of solvent Energy transfer (solvent) Energy transfer to solute Fluorescence of solute

kl

kz

Internal quenching of solute kb Self-quenching of solute C k 6

Quenching of solvent by solute (small) C k 7

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134 1. PARTICLE DETECTION

terms the quantum efficiency for the solution is of the form

(1.4.8)

where

ka k4 + kti

m=-

and the quantum efficiency for the solvent is neglected. In these equa- tions, the k’s are reaction probabilities and have the following significance: k l , fluorescence of the solvent; ckg, energy tranfer to the solute; h, fluorescence of the solute; k6, internal quenching of the solute; cks, self-quenching of the solute. This shows that the response as a function of concentration rises to a maximum and then falls off. The form of the equation given above can be compared with those given by Furst and Kallmann :18

(1.4.9)

Figure 7 shows some results of studies of relative pulse heights as a function of concentration. The maximum response is seen to be very broad. Some “typical” results indicate that the ratio of transfer to quenching in solvent is about 0.75; the ratio of self-quenching of the solute to emission plus internal quenching of solute is about 0.02; and that internal quenching of solute is small compared to emission. Certain results are known concerning the effects of “secondary solutes” and “ secondary solvents.”

a. The secondary solvent is an intermediate in the transfer process. It may be useful in the “loaded” scintillators (discussed later). The secondary solvent must be soluble and also have good energy transfer to and from the solute and solvent. Napthalene appears to be best (300 gm/liter), but not when used with terphenyl.

b. Secondary solutes are wavelength shif ters and will be discussed later. POPOP, diphenylthexatriene, etc., used in concentrations of about 1 gm/liter are in this category. Possibly nonradiative transfers are involved.

The energy response of liquids is very similar to that of crystals. The a/@ ratio generally decreases as the concentration decreases.

l8 M. Furst and H. Kallmann, Phya. Rev. 86, 816 (1952).

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1.4. SCINTILLATION COUNTERS 135

1.4.1.5. Plastic Scintillators. Plastic scintillators (i.e., solid solutions) were developed in 1950 and 1951 by Schorr and Tornsy, and Koski. Much fine recent work has been done by Swank. Common plastics have been terphenyl in polystyrene and terphenyl in polyvinyl-toluene, with wavelength shifters similar to those used in liquids. Tetraphenyl buta- diene is also used in place of terphenyl and most recent combinations

0.8

0.6

0.4 B- Excitation RCA-6342 MgO Reflector

0.2

0 4 8 12 16 20 24

Grams of fluor per liter of solution

FIG. 7. Light output as a function of concentration in a typical solvent. (All solutions argon-saturated.)

are shown in Table I. The concentration of solute in plastic scintillators is about 5 to 10 times that in liquids. The energy dependence in plastics is very similar to that in liquids. The concentration effects in plastics are similar to those in liquids, but the work of Swank and Buck3 is par- ticularly useful here in that it shows that radiative transfer is relatively more important in plastics, due to the better scintillation properties of the solvent (polystyrene as compared to toluene) and the strong absorp- tion of the solute in this region. Even so, the relat.ve importance of radiative transfer is only about 20% a t most.

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136 1. PARTICLE DETECTION

Various methods of polymerizing the solute are used, including:

1. mixing solute in molten polystyrene solvent; 2. polymerizing a monomer solution by means of a catalyst (benzoyl

3. polymerizing the monomer without the catalyst. peroxide -55 %) ;

Method No. 2 probably gives best results, but No. 3 is used most for ease of handling. A high-temperature (146°C) process results in a shorter time, lower average molecular weights and better response by -lo%, compared to lower temperatures (126°C). A good plastic appears to be : 20 grams of 2 phenyl 5-(4-biphenylyl)-1,3,4 oxadiazole (PBD) in 1000 grams of polyvinyltoluene (PVT) with 1 gram of p , p’ diphenyl- stilbene (DPS) as a wavelength shifter. This gives a pulse-height response equal to 0.54 that of anthracene. Certain commercial plastics are avail- able, some apparently exceeding this performance. The effect of con- centration on performance of plastics is similar to those found in the case of liquids.

1.4.1.6. light Pipes and Reflectors for Scintillation Counters. A crys- tal scintillation counter is normally placed in a metal container. In the case of the hygroscopic alkali halides this container must also seal the crystal from moisture and, of course, in all cases the box must be light- tight. Efforts to determine whether specular or diffuse reflecting surfaces provide the best light collection have not shown any striking differences but it is generally believed that diffuse reflecting surfaces are best. In the case of crystal and plastic scintillators this diffuse surface can be easily applied by “smoking” on MgOz or painting with certain com- mercial preparations such as Tygon. In the case of liquids the problem is more difficult if it is desired to have the liquid in direct contact with the reflector. In this case special procedures to apply titanium dioxide in lacquer to an etched container face could be followed.

Containers for liquid scintillators can be made out of aluminum with thin transparent windows of alite. If the sealing is with Neoprene gaskets it is necessary to boil the Neoprene in toluene for several days to remove impurities. Using this procedure, counters have been constructed that were usable over many months. If an all transparent container is required, it can be fabricated out of Lucite, but then phenylcyclohexane must be used as the solvent of the solution. Acetic acid has proved to be a suitable cement in this case.

When the setup is such that the photomultiplier cannot be put in direct contact with the scintillator, Lucite light pipes have been used. Care must be taken in the selection of the cement used to attach the

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1.4. SCINTILLATION COUNTERS 137

light pipe to the scintillator face or window in order to avoid crazing. A commercial product designated as R 313 has proved useful here.

If the photomultiplier is immersed directly in the liquid i t is necessary to run the cathode near ground potential to avoid deterioration of the cath- ode. If a light pipe link is used, optical coupling with the photomulti- plier face is generally made by the use of some high viscosity silicone.

1.4.1.7. Loaded Scintillators. The usefulness of scintillators can be enhanced for some applications by the process of "loading" the scintilla- tor with selected heavy elements. Neutron detection experiments have suggested loading with boron, cadmium, etc. Some success has been achieved by using :

1. triethylbenzene + methyl borate [BIO (np) reaction]; 2. triethylbenzene + cadmium octoate

or toluene + cadmium propionate 3. toluene + samarium or gadolinium propionate [Sm149 ( n , ~ ) ] .

Generally however, quenching sets in before much of the desired sub- stance is in solution, so that concentrations have until recently been restricted to approximately one percent. Recently studies by Kallmann" and Swank have shown that intermediate solvents such a s naphthalene and biphenyl can be used to extend the amounts and types of compounds that can be successfully dissolved without quenching. Recent work by Hyman16 has resulted in plastic scintillators containing up to 5% by weight of lead, with a response that is about 50% that of an unloaded plastic scintillator.

1.4.1.8. Noble Element Scintillators. In the past several years efforts to prepare gaseous scintillation counters have resulted in certain successes in the use of the noble gases. These counters have provided properties of interest in speed, large light output, linearity, simplicity and flexibility in Z and density. Work has been done by Northrup and noble^,^^-^* Eggler and Huddle~ton , '~ and Sayres and Wu120 among others. Early results were difficult to correlate until the importance of eliminating impurities was fully realized. Other factors that must be taken into account are the size of the container and t,he effect of wavelength shifters.

Consistent and successful gas scintillation counters have been reported by Sayres and Wu20 constructed along the lines shown in Fig. 8. The gases used were helium, argon, krypton, and xenon. Since the gas is

} [Cd113 (n,r>l;

17 H. Kallmann, I R E Trans. on Nuclear Sci. NS-3, No. 4 (1956). 18 J. A. Northrup and R. Nobles, IRE Trans. on Nuclear Sci. NS-3, No. 4 (1956). 19 C. Eggler and C. M. Huddleston, IRE Trans. on Nuclear Sci. NS-3, No. 4 (1956). 10 A. Sayres and C. S. Wu, Rev. Sci. Znstr. 28, 758 (1957); W. R. Bennett, Jr. and

C. S. Wu, Bull. Am. Phys. SOC. [2] 2, No. 1 (1957).

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138 1. PARTICLE DETECTION

susceptible to impurities it is necessary to take special precautions in construction. Teflon is used as gasket material; metal components and gaskets are carefully baked out. In addition a gas circulating pump and calcium purifier were also incorporated. With proper precautions and initial purification Sayres and Wu found no measurable deterioration in performance in their counters over a period of five days.

Most of the light coming from the noble element scintillators lies in the ultraviolet so that wavelength shifters are necessary. Some of the

,Q. phenyl,

as ut

u234 I' alpha source

1 0 I in. u

Scale

m

FIG. 8. A typical gas scintillator. (From Sayres and WulZo Rev. Sci. Instr. 28,759 (1957), Fig. 3.)

early work was confused by the quenching effects of certain wavelength shifters. The most successful method for wavelength shifting has been to deposit thin layers of quaterphenyl or diphenylstilbene (30 to 50 pgm/ emz) on the walls of the container and adjoining photomultiplier face. Difficulties in the interpretation of the role of nitrogen as a wavelength shifter have been removed by the work of Bennett and WuZo which indicates that the observed spectra, rise times, etc., when nitrogen is present are explained on the basis of collision phenomena rather than photoexcitation.

Using a chamber of the sort shown in Fig. 8, Sayres and Wu have made systematic observations using helium, argon, krypton, and xenon as the scintillators count Po210 a particles. In each case the pressure was adjusted to be as high or higher than that needed to stop the a particles in the chamber volume. The results of these tests show that pulse heights reach

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1.4. SCINTILLATION COUNTERS 139

a maximum when the particle is stopped in the chamber, and that the resolution also improves with increasing pressure. Resolutions (in per cent peak width) of 5% to 10% were obtained. The relative performances as well as the effect of wavelength shifting is shown in Table IV.

TABLE IV. Performance of Noble Element Scintillators

Glass phototube Quartz phototube (6292) (K 1306)

Without With Without With

Gas (optimum pressure) quater- quater- quater- quater- phenyl phenyl phenyl phenyl

Xenon (6 psi) 6 105 88 145 83 Krypton (8 psi) <3 50

Argon (10 psi) < 3 15 11 28 Helium (45 psi) 9 38 14 40

-

[ Noise/background] 3 3 3 4

The linearity of these scintillators has been demonstrated for charged particles having high rates of energy loss, including fission fragments. Nobles2‘ using xenon gas and protons, deuterons and a particles from 2-5 Mev found that pulse height versus energy yielded a straight line, but with an intercept at 0.5 MeV. Other work indicates linear response among similar particles, but some “inverse saturation” effects. It is possible that some of these discrepancies are the result of spurious rather than inherent effects.

There are conflicting reports in the literature concerning the lifetimes of the scintillation processes in the noble elements, but it seems clear that these counters are “fast.” In gases there appears to be an inverse pressure dependence’* so that the lifetimes are approximately 75 mpsec at 25-cm pressure, 30 mpsec a t 50-cm pressure, and 14 mpsec a t 100-cm pressure.

The light output relative to NaI(T1) is also the subject of conflicting reports. The work of Northrup14 indicates that pulse heights from argon, krypton, and xenon are only a little less than those from NaI(T1). This is consistent with the work of Sayres and Wu, where comparisons showed gas pulse heights about one-half those of CsI.

Several special applications of gas counters, or counters containing gas mixtures have been suggested and tried. Sayres and Wu constructed a slow neutron detector by placing a metallic boron film (20 pgm/cm2, 97% B’O) in a xenon gas counter. The resulting spectrum of pulse heights when slow neutrons irradiated the counter showed two peaks in the ratio

21 R. A. Nobles, Rev. Sci. Instr. 27, 280 (1956).

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140 1. PARTICLE DETECTION

7:4 as expected for the Li and a particles from the reaction Bl0 (%,a) Li7. In a similar way, a fission counter was constructed by placing a thin U2a5 disk in a xenon chamber and irradiating with slow neutrons. Figure 9 shows the fission spectrum obtained.

1500 -

N

1000 -

500 -

-a's

61.4 Mev

FIG. 9. The fission spectrum of U*36 in 29 psi of xenon, using the K-1306 phototube and quaterphenyl wavelength shifter.

Some very interesting results have been obtained for binary mixtures by Northrup et aZ.,14 particularly with regard to the possibility of neutron detectors utilizing He3. Sayres and Wu have also investigated a mixture of 90% He and 10% xenon and found that, a t proper pressures, the response is as good as that of a pure xenon counter. This counter also serves to show the good discrimination properties possible for ionizing particles and y rays. Results of Po2l0 a and radium y tests are given in Table V.

Scintillations have also been detected from the liquid and solid states of some of the noble elements. The work of NorthrupI4 shows that liquid and solid xenon, krypton, and argon give pulses the order of one-half or better than those obtained from NaI(T1). The techniques are very similar to those involved in the work on gas scintillators. Liquids and solids appear to have scintillation decay times under 10 mpsec.'*

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I .4. SCINTILLATION COUNTERS 141

The great interest in He4 and He3 in neutron scattering and detectioii has resulted in an investigation of liquid helium as a scintillator by Thorndike and Shlaer14 and Wu and her collaborators.22 Using a! particles from P3, Wu et a1.22 found that, under optimum conditions, relating

TABLE V. Results of Poz1" a and Radium y Tests

Scintillator Pulse height Peak effect of a source of y source -

CsI (1 mm thick) 160 75 Anthracene (3 mm thick) 22 77 He + 10% Xe (60 psi total) I05 5 16292 tube with quaterphenyl j

to the light collection aiid range of particles the signal is well above the background, and energy resolution is very satisfact,ory (-20%). In the system used by these workers, the pulse-height ratio between liquid helium aiid CsI (at room t,emperature) is roughly 1 :2. This same work indicated that He vapor also scintillates satisfactorily.

In conclusion, it is possible to say that techniques have beeii developed that allow the practical use of the noble elements as scintillators in the gaseous arid liquid states; and that certain of t,hem are also good scintilla- tors as solids.

1.4.1.9. General Applications of Scintillation Counters. Certain special applications of scintillation counter techniques are discussed elsewhere in this book. It is appropriate a t this point to describe some general applications of the technique.

For some experiments scintillation counters have B marked advantage over the visual techiiiques of cloud chamber, diffusion chamber, bubble chamber, and emulsion, due to t.heir fast, t,iming aspect. This advantage might be exploited in a lifet,ime det,ermination, such as the T+ lifetime measurements of Jakobsen et al.?' or t,he Kf meson lifetime determina- tion of Motley and I ? i t ~ h . ~ * In l i fe the experiments scintillators are arranged to give a fast t,imiiig pulse when the parent particle enters and does not leave the absorber, and another pulse when the decay product leaves the absorber. With current techniques and organic scintillators, timing resolutions of several milliniicroseconds are possible. Similar techniques, involving large area liquid scintillators have been used t80 measure capture h i e s of p - mesniis ill the t,ens of millimicrosecond

22 C. S. \Vu , private coiiiinunication. 23 M. J. Jskohsen, A. G. Schula, and J. Steinberger, P h p . Rev. 81, 894 (1951). 2' H.. Motley a n d V. Fitch, Ph?p. Rev. 106, 265 (1952.

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142 1. PARTICLE DETECTION

range.25 The fast timing and good geometry aspects of scintillation counters have also been used in time of flight applications such as the antiproton work at Berkeley.28 Here it was possible to distinguish between R mesons and antiprotons of the same momentum (1.19 Bevlc) by their time of flight over a 40-ft base line. The times were 40 and 51 mpsec respectively.

Another area where the fast time resolution of scintillators is an advantage is in certain scattering experiments such as those of Yuan and Lindenbaum2’ and Cronin et aL28 In these applications the sacrifice in precise spatial resolution is offset by the advantage of being able to record many events in a single pulse of the beam of the high-energy particle accelerator.

From the above remarks, it is obvious that a very great advantage would result if the tracks of particles could be simultaneously viewed, as in chambers and emulsions, and timed, as in counters. This combina- tion of properties is discussed in Section 1.4.2.

1.4.1 .lo. large Scintillation Counters.* 1.4.1.10.1. GENERAL CHARAC- TERISTICS. Large volume liquid2Y scintillation detectors, arbitrarily defined as >+ cubic meter, are useful in the efficient detection of neutrons and gamma rays as well as charged particles. In the case of neutral radiations this characteristic stems from the detector size which is by definition

2K J. W. Keuffel, F. B. Harrison, T. N. K. Godfrey, and Geo. T. Reynolds, Phys. Rev. 87, 942 (1952).

28 0. Chamberlain, E. Segrh, C. Wiegand, and T. Ypsilantis, Phys. Rev. 100, 947 (1955).

27L. C. L. Yuan and S. J. Lindenbaum, Proc. 4th Conf. on High Energy Nuclear

2 * J. W. Cronin, R. Cool, and A. Abashian, Phys. Rev. 107, 1121 (1957). 29 Large volume detectors using plastic scintillators have been employed by the

Los Alamos Bomb Test Division in the detection of gamma rays. A total absorption spectrometer using a large single crystal of NaI, 9a in. diameter by 9 in. high is described by R. C. Davis, P. R. Bell, G. C. Kelly, and N. H. Lazar, in the Proc. 5th Scintillation Counter Symposium, February, 1956, Washington, D.C., published in IRE Trans. on Nuclear Sci. NS-8, 82 (1956).

We restrict our discussion to liquid detectors because of the relative ease vis-a-vis plastic with which it is possible to incorporate other substances such as neutron absorbers into liquids. Single crystals are limited in size although i t may be that a transparent slurry of smaller crystals in a bath of optically matching liquid could be devised to overcome this limitation.

Physics, Rochester, New York, pp. 98-100 (1954).

- * Section 1.4.1.10 is by F. Reines.

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1.4. SCINTILLATION COUNTERS 143

large or a t least comparable with the absorption mean free path for these radiations in the scintillator. Charged particles can in some cases also be studied more effectively by such detectors because of their more com- plete absorption of the particle energy and decay particles, if any. In addition to particle containment, large detectors have been constructed which have fair energy and time resolution, and neutron capturing ele- ments such as boron and cadmium have been incorporated into the scintillator to provide a distiiictive signal on neutron capture. A limited amount of spatial resolution can be achieved by the use of several tanks or of light shields dividing a common liquid container. The characteristics resulting from large size have been found useful in a variety of problems requiring very high (> 75%) neutron detection e f f i~ i ency~~ such as the measurement of neutron multiplicity from the fission process,31 in the measurement of large-volume low-intensity gamma sources such as the human body,S2 and in free neutrino studies which required a large hydro- geneous target and good positron and neutron detection efficiencies.*

Apart from size, which introduces special problems of construction and transparency of the liquid to scintillation light, large volume scintillation detectors are in principle much the same as the more conventional small detectors. A particle enters the liquid and causes the emission of scintilla- tion photons which traverse the liquid to its boundary and either strike a photomultiplier face or container wall and are absorbed or are reflected back into the liquid. The scintillation light,, diffusing in this manner, is eventually absorbed either by the liquid, the walls, or the photocathodes. Light striking the photocathode is converted with an efficiency which can be as high as 10% t o photoelectrons, and the electrical pulse thus pro- duced, amplified by the photomultiplier tube itself, and followed by external amplification has the characteristics of amplitude, shape, and time of occurrence. These characteristics are used to study the primary physical event under consideration.

1.4.1.10.2. ENERGY Loss IN LIQUID SCINTILLATORS. The energy deposited in the scintillator in the first place depends on the particle involved, its energy, and of course on the size of the detector. The scintillation efficiency is also a function of the particle involved and in some instances, its energy as well.

Discussions of 'the energy loss of various particles, i.e., gamma rays

* See Section 2.2.4 of this volume. :Io F. Reines, C . L. Cowan, F. B. Harrison, and 1). 8. Carter, Rev. Sci. Znstr. 26,

41 €3. C. Diven, H. C. Martin, R. F. Taschek, and J. Terrell, Phys. Rev. 101, 101'2

32 E. C . Anderson, I R E Trans. on Nuclear Sci. NS-3, 93 (1956).

1061 (1954).

(1956).

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144 1. PARTICLE DETECTION

electrons, protons, 1nesoiis,33~3* arid iieutroiisS6 caii be fouiid in lnaiiy places.* We note here the fact that in the hydrocarbons comprising liquid scintillators, the energy loss of gamma rays is, for eiiergies less thaii 10 Mev, primarily by mealis of the Comptoii eEect and helice takes place over ail extended region. Figure 10 shows the total mean free path

70

60

- I 50 u I I-

: 40 W w LL

-

a

0 I 2 3 4 5 6 7 E 9 10

GAMMA ENERGY (MEV)

FIG. 10. Total gamma-ray lnean free path versus energy in toluene (C,H8, p = 0.87 gm/cm3).

x(cm) versus energy for a gamma ray of energy E (MeV), in toluene (C,H,, p = 0.87 gm/cm3)). Since the iiieaii free path is inversely propor- tional to density this curve can be simply scaled to other organic solvents. Because the Comptoii effect results only in a partial energy loss, the gamma-ray absorption process requires several collisions. The absorption

* Refer to Section 1.1 2 of this volume. 3 3 R. Latter and H. Kahn, (;anma ray absorption coefficients. Rand Report &I70

(1949); see the article by H. A. Bethe and .J. Ashkin: Passage of radiation through niatter, i n “Experiinentnl Xurlear I’hysics” (E. SegrP, ed.), Vol. I, pp. 166-357. Wiley, Nrw York, 1953.

3 4 M. Itich and R. Mady, Range Energy Tal~les. UCItL-2301 (1954). 3 5 Bec the article by B. Feld: The neutron, i n "Experimental Nuclcni P ~ ~ R I v ~ ’ ’

(E. SegrB, ed.), Vol. 11, pp. 208-586. Wiley, New York, 1953.

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length may be roughly ehtimated by random walk considerations modified by the correlation in direction of the incident arid scattered gamma ray: e.g. for a 1-Mcv ganima t>hc Compton mean free path is 17 cm implying absorption lengths a h i t 35 cm. The Comptmi recoil electrons are for these energies readily :~l~sorJwci by the scintillator with an energy loss

I 1 I I I I I I I 1

30

a e

aQ 10

I I-

5

0 I 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4

NEUTRON ENERGY (MEW

1710. 11. Neutron, proton cwlhsion incan free path in toluene.

of about 1.6 Mev/cm so that the energy deposition of a gamma ray is given essentially by the Comptori process.

Neutrons give up their energy to the scintillator largely by elastic c~ollisions with protons. The process of nentroii slowing down arid diffusion has been extensively and it is known that the distance travelled by a fast neutron prior to its thermalieation is of the order of the mean free path for the first collision. Figure 11 gives the mean free path for n,p collisions in toluene. The slowing down process is quite rapid ( -2 X

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146 1. PARTICLE DETECTION

sec) so that the sequence of proton recoils involved occurs within the resolving time, 2 X lo-’ sec, of more or less conventional electronics. Under these circumstances the neutron slowing down pulses “pile up” giving one pulse. Because of the nonlinearity of the scintillator response to protons and the various combinations of recoil energy loss possible for a neutron, the sum pulse varies. Consequently, there is no unique scintillation response for a given neutron energy.36 The role of the other major scintillator constituent, carbon, in slowing down the neutron is small because of the relatively great mass of the carbon nucleus and also because the neutron collision cross section presented by the scintillator protons is somewhat larger than that of scintillator carbon. Nevertheless, the neutron loses an average of 14% of its energy in each collision with a carbon nucleus and since the carbon recoil nuclei are so inefficiently signaled by the scintillator, this fact introduces additional nonlinearity into the response to neutron energy.

The light output of a liquid scintillator (5 gm/liter terphenyl in toluene) for electrons and protons as determined by Harrison87 is shown in Fig. 12.

As an example of the photon yields to be expected from liquid scintilla- tors we quote the absolute photon yield determined by Post38 for ter- phenyl (8 gm/liter) in toluene, 150 ev/photon.

1.4.1.10.3. DESIGN CONSIDERATIONS. As outlined in Section 1.4.1.10.1, the response of the detector to a primary event is a consequence of several factors :

1. the energy deposited in the scintillator and the fraction of this

2. the transparency of the liquid to its own scintillation; 3. the reflectivity of the container walls and the fraction of the wall

area covered by photocathodes; 4. the photoelectric efficiency of the photocathodes and the electrical

characteristics of the photomultipliers, photomultiplier ganging circuits, and amplifiers.

This lack of uniqueness could be eliminated in principle by the use of a system fast enough to observe the individual proton recoils. Indeed, since one is here con- cerned with correcting for a nonlinearity, it need not be made with precision. Thus far such corrections have not been attempted although the multiplicity of recoiLs has been observed by D. W. Mueller’s group a t Los Alamos (private communication), and was actually used to discriminate between neutrons and gammas by F. D. Brooks, in “Liquid Scintillation Counting” (C. G. Bell and F. N. Hayes, eds.), pp. 268-269. Pergamon, New York, 1958.

energy which appears as scintillation light,

F. B. Harrison, Nucleonice 10(16), 40 (1952). 88 R. F. Post, Phys. Rev. 79, 735 (1950), as interpreted by S. C. Curran in hie book

entitled “Luminescence and the Scintillation Counter.” Academic Press, New York, 1963.

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1.4. SCINTILLATION COUNTERS 147

The fraction F of scintillation light which eventually reaches a photo- cathode can be related to the mean transmission over the light path between reflections t of the scintillation light by the liquid, the reflectivity

I 2

10

- W _I

0

* U

a

‘ I , s

a a t m a 6

P

R - %

$ 4

W I

_I 2 a

I I I I I I I I I

ELECTRONS

-.

0 1 2 3 4 5 6 7 8 ’ 9 1 0

PARTICLE ENERGY (MEV)

FIG. 12. Scintillation light output for electrons and protons versus particle energy in a 5 gm/liter terphenyl toluene solution (Harrison37).

of the wall r , and the fraction f of the wall surface uniformly covered by photocathodes by the simple formula:*

F = tf/[l - tr(1 - f)]. (1.4.10)

The mean transmission t is in general a complicated function of the mean free path for scattering A, and absorption A. of the light by the liquid and the detector size and shape. For t,he simple case of a spherical detector and a liquid which has a scattering mean free path much greater than the detector diameter, t is given by the relationship

(1.4.11)

* A more precise formula would replace t in the numerator by to, where fa is the light transmission, averaged over points uniformly distributed throughout the scin- tillator, to the wall.

A, d

t = - (1 - e-”Aa).

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148 1. PARTICLE DETECTION

As an example, suppose d = 1 meter, A, = 2 meters, r = 0.9, f = 0.05. Then t = 0.787 and the fraction of scintillation light collected F = 0.121. The enhancement of light collection caused by the reflectivity of the container walls is seen in this example, to be a factor of 2.4. The reflec- tivity of a given wall coating, scintillator combination can be measured by using a light source centrally placed in a sphere which is so small that t = 1.0. A uniform distribution of holes around the sphere would enable the light to emerge and appropriate filters can be used around the light sensing element, to match scintillation light. Small optical mockups have been used to measure the uniformity of light collection in this limiting, t = 1, case.39

The total mean free path for scattering plus absorption can be measured for the scintillating liquid by means of a Beckman spectrophotometer, provided a standard is available for comparison. As a standard we have used reagent grade toluene measured in a six foot long box with a modu- lated light source at one end to provide an ac signal, a photoelectric detector a t a variable distance from the source, and light baffles between for collimation. The total mean free At path was measured as the depar- ture from inverse quar re,^" i.e.,

I(r,At) = Ioe-r/X1/4~r2. (1.4.12)

With this arrangement we have found reagent grade toluene to have a total mean free path of 5 meters. Thus far no measurements have been made which enable a direct and accurate determination of the absorption mean free path A, for a given scintillator. In principle, given the total mean free path A,' (= A;' + A;') and a light, source which mocks up the scintillation light, an experiment the result of which depends on the ratio X,/At can be done. For example, a centrally placed light source sur- rounded by a spherical volume of liquid with photoelectric detectors on a spherical black wall represents a calculable system. A drawback is the rather large radius (-A,) which is required in order to make a useful measurement.

Because of the complicated dependence of the various light collection factors on the geometry, the eventual optimum design and resultant, characteristics for any given application is probably better determined from studies of complete detectors than from an a priori synthesis.

1.4.1.10.3.1. Scintillator Transparency and ReJlecting Paint. The prob- lem of scintillator transparency4I has been met, by standard chemical

3a H. W. Kruse and F. Reines, unpublished. *O C. L. Cowan, Jr., F. B. Harrison, A. D. McGuire, and F. Reines (unpublished). 4 * For more details see A. R. Ronzio, C. L. Cowan, Jr., and F. Reines, Rev. Sci.

Instr. 29, 146 (1958).

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1.4. SCINTILLATION COUNTERS 149

means in the case of triethylbenzene (TEB, C2Hy, p = 0.87 gm/cm3). Reagent grade toluene with a total mean free path for s5attering and absorption of about 5 meters at a wavelength of 4200 A is generally acceptable without further treatment. Purified triethylbenzene has a mean free path of from 5 to 10 meters. Purification of the crude TEB is accomplished by digestion wit,h sodium methylate (about one pound per hundred gallons of TEB) for four hours followed by fractional distil- lation in which a narrow cut is t,aken at. the constant boiling point range. The liquid is best st,ored in glass or metal cont,ainers painted wit8h Epon base enamel because of it,s tendency, especially in t,he case of toluene, to leach impurities from t,he container walls. Storage under an inert atmosphere such as argon is advisable in order to avoid the lowering of scintillation efficiency due to the act.ion of dissolved oxygen.

An essential ingredient in any large scint’illator is the so called wave- length shifter which absorbs the primary scintillation light and re-emits it a t a lower frequency which is: (a) less readily absorbed by the scintil- lat,ing solution; (b) more easily reflected by available inert,, adhesive container coatings;42 (c) more efficient,ly detected by the photomultiplier tubes. A popular shifter is 1’0P0P43 which moves the maximum in the emitted spectrum from the region of 3800 to 4200 A. The 5-meter mean free path quot,ed above is in purified solvent and does not apply to t,he case of an actual scintillator in which e.g. 3 gm/liter of terphenyl and 0.3 gm/liter of POPOP have been added t,o the solvent. For a scintillating solution the total mean free path drops to about one meter, presumably due to fluorescent or reradiative scattering. Judging from the actual performance of large detectors, itJ seems that only a small part, of the mean free path reduction is due to increased absorption by t,he terphenyl and POPOP.

1.4.1.10.3.2. Uniformity of Light Collection. If it is desired to employ the detector in any manner which requires energy resolution then it is clearly necessary to be able to make the amount of light collected by the photocathodes dependent only on the number of photons emitted. In view of the fact that the optical t#rarismission of the liquids and the reflectivity of the container are imperfect, this aim is accomplished in an approximate sense by surrounding the liquid with a multitude of photo- multiplier tubes. The fraction of the liquid surface area covered by photo- cathode is + in the largest detectors (-1.5 meterss) constructed at Los Alamos. Numbers from 45 2-in. h b e s to 110 5-in. tubes have been

42 Plasite paint with TiO:! pigment has been found to be suitable coating material.

F. N. Hayes, D. G. Ott, and V. N. Kerr, Nucleonics 14, 42 (1956). See also Ohtained from Wisconsin Protective Coat.ing Go., Green Bay, Wisconsin.

Table I, Section 1.4.1.3.1 of this volnmc.

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150 1. PARTICLE DETECTION

used to help obtain uniformity of light collection. Figure 13 shows three arrangements designed to this end. All the tubes are usually connected in parallel although it is on occasion useful to divide them into two interleaved banks which are connected in prompt coincidence to dis-

I

X X X X X X

SCINTILLATING LlOUlD

\ 1

SCINTILLATING

\

X X X X X X

ISOLATION LlQUlO TO /OBTAIN OPTICAL MATCH

x- -- x- - -

.TRANSPARENT WINDOW

SCINTILLATING LlOUlO PHOTOMULTIPLIER TUBES

DISTRIBUTED AROUND CYLINDRICAL WALL

r c ) RECTANGULAR PARALLELOPIPED

FIQ. 13. Photomultiplier tube arrangements designed for collection.

uniformity of light

criminate against tube noise. In arrangement (a) light from events which occur over an appreciable distance compared with the photomultiplier tube dimensions is collected in a uniform manner. In (b) the photo- multipliers are isolated from the scintillating liquid by a nonscintillating, optically matching so that short-range particles cannot deposit their energy in the near vicinity of a photomultiplier cathode. Arrange-

'* Cerenkov radiation will of course reRult in some light from energetic particles passing through this region.

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1.4. SCINTILLATION COUNTERS 151

ment (c) is a variant of (b) which has been found useful in work where a large, unobstructed area was necessary. The light collection characteristic of a uniformly distributed source was measured in an optical mockupa9

of a rectangular detector 9 X 4+ X 2 ft to have a half-width a t half- maximum of 7%. The over-all light collection was estimated as %. Given this figure, 1 Mev deposited in the scintillator, a conversion to photons of 1 photon/l50 ev deposited, and 1 photoelectron emitted by the photo- cathode per 10 incident photons, we conclude that 170 photoelectrons are produced per Mev absorbed by the liquid. This figure implies a statistical uncertainty of k 4 1 7 0 or ?8% due to fluctuations in the number of photoelectrons. The over-all uncertainty for 1 Mev deposited in this example is therefore conservatively taken as k 15%.

1.4.1.10.3.3. Photomultiplier Selection and Circuitry. * Very large detectors built thus far have employed 2-in. RCA and Dumont and 5-in. Dumont (K-l198), photomultiplier tubes. Sixteen-inch tubes which are just becoming available in quantity have a wide variation of response across the photocathode and hence, assuming only a few are used, intro- duce nonuniformity in the light collection of the system. As mentioned, photomul4iplier tubes have been used in gangs with as many as 110 5-in. tubes connected in parallel. Under these circumstances, uniformity of light collection requires that all the tubes have the same gain. Since tubes differ from each other in gain for a fixed voltage distribution along the dynodes, it is necessary to adjust the voltage in each case so as to equalize the gains of all tubes in the gang. This is done by observing the response to a source such as Csla7 placed in a reproducible fashion near a NaI crystal which is viewed by the photomultiplier tube under adjustment. The tubes are similar enough to require only the selection of an appro- priate load resistor. Figure 14 shows a resistance network on one tube. Figures 15 indicate how a number of tubes can be connected in parallel. Also shown in Fig. 15(a) is a simple switch which enables a given tube to be thrown out of the gang. This switch is useful in checking the performance of individual tubes in situ, especially in the location of tubes which have become noisy, In addition to equalizing gains it is also of importance to select tubes of acceptable noise level because despite the large capacitance of the ganging network even one noisy tube can result in variable and unacceptable backgrounds. Occasionally problems arise in which even the best collection of tubes results in unacceptable levels due to tube noise. In such cases it has been found that tube noise can be greatly reduced by the expedient of dividing the tubes into two inter- leaved bunks axid requiring a coincidence between the two banks. The price one pays for this reduction in background is in a smaller light collec-

*See also Vol. 2, Section 11.1.3.

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152 1. PAltTICLE DETECTION

2 MEG

I

I

I

GAIN BALANCE RESISTOR

H .V. -

1 SCREEN (13) G- CATHODE (14)

SWITCH

- -

FIG. 14. Voltage divider network used on a 5-in. 1)umont K-1198 photoInultiplier tube.

t,iori and hence in decreased energy resolution per bank. In c,ases where the energy resolution required in making a coiiicidence is not unduly restrictive it is possible t,o add the signal from the two banks through iso1ztt)ion networks (i.e., separate preamplifiers) and regain the resolution lost in t,he division.

A problem occasionally eucountered is in the electrical oscillations which sometimes result., in the ganging “yoke.” Parasitic resistors c:m be added t,o suppress such ringing as show11 in Icig. 15(b). The variatiou of

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1.4. SC'INTILLATION ( 'OUNTElIS

FIG, 15(:1). Ganging yokc for a largc number of 5-in. photomultiplier

93"COAXIAL SIGNAL LEAD ( T O PREAMPLIFIER)

COAXIAL HIGH VOLTAGE LEAD

t rrhes.

SIGNAL YOKE

I53

FIG. 15(b). Schematic drawing of ganging yokc for a large number of 5-in. photo- inultiplier tubes.

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154 1. PARTICLE DETECTION

photomultiplier response due to ambient magnetic fields can also be a cause for concern. In general the use of mu metal shields is cumbersome, expensive, and in many cases impractical because of the restriction imposed on light collection. One solution to this problem is to build the

18

17

16

15

14

13 t 3 12

a a 1 1

- z

>

E g 10

G 8

U - 9 W

&

0 7 z F 2 6 3

8 5

0 20 30 40 50 60 70 80

PULSE HEIGHT (VOLTS)

FIG. 16. Peaked spectrum due to cosmic rays which pass through the giant slab detector of Fig. 5. The liquid depth, 56 cm, corresponds to a peak energy of -90 MeV. These data were taken under about 200 f t of rock at an altitude of 7300 f t and cor- respond to a muon rate through the detector of 13/sec. (Cowan and Reines, unpub- ished, 1957.)

detector of steel which will shield the tubes, another is to recalibrate the system a t each position of use simply foregoing the loss of resolution due to the magnetic field effects.

1.4.1.10.4. CALIBRATION AND USES. Two general calibration schemes

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1.4. SCINTILLATION COUNTERS 155

are available for use with large detectors: t,he first makes use of cosmic radiation, the second of radioactive sources. Minimum ionizing cosmic- ray particles, for example, deposit an amount of energy which is propor- tional to the track length in the scintillator. In consequence a pulse-height spectrum due to cosmic rays penetrating the detector has a peak which may be associated with the mean energy loss of minimum ionizing cosmic rays. A typical “through peak” is shown in Fig. 16. Once the through peak is located, the detector can be replaced by a standardized pulser which is then adjusted to give an output voltage equal to the through

B I I I 1 I I I 1

PULSE HEIGHT ( M E W

Fro. 17(a). Mu-meson decay electron spectrum seen with 75-crn cylindrical detector. Data were taken at 7300 ft above sca level and 40,000 counts were recorded in 36 hours. (Reines et al.so)

peak value. Energy gates can then be set using the pulser. An independent energy calibration can also be obtained, along with a check of the system used to study events in delayed coincidence, by employing the phenome- non of muon decay,

p*-+ 8’ + Y- + v+. (1.4.13)

In this instance, a delayed coincidence is required electronically with a fist pulse of, say 20-40 Mev energy followed within 10 psec by a second pulse of energy > 15 MeV. The energy spectrum of these second pulses is due to the decay electron. Fortunately, backgrounds are small, and despite considerable distortion of the decay electron spectrum due to bremmstrahlung losses and edge effects, the end point (53 MeV) is sharp. Figure 17(a) shows such a decay spectrum measured with the 75-cm cylin-

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156 1. I’AHTICLE DETECTION

drical detector. Figure 17(b) shows the associated time interval or decay time spectrum. These calibration schemes give the energy response for a distributed source: the variation of response across the detector can be investigated by means of aperture detectors used in coincidence to gate the detector under study. The distributed or localized response can also be determined by means of radioactive sources such as the posit,ron

10

v)

I- z 3 0 0

10 z

IC

I I I I I

I I I I I, 10 \ 4 2 6 8 12

MICROSECONDS

FIG. 17(b). Half-life measurement of the 8-decay of cosmic fi mesons stopping in the detector used as a check on the time calibration of the apparatus. The entry of the meson yielded a ‘ I first pulse ” and the decay electron the “second pulse.’’ This measure- ment was made in conjunction with the energy calibration using the decay electron spectrum end-point. (Reines et ~ 1 1 . 3 ~ )

emitter Cu‘j4 which can be dissolved as a salt into the scintillator and the resultant spectrum observed. If the Cu64 is encapsulated, only the anni- hilation radiation enters the detector, giving gamma rays of unique energy for measurement.

These calibrat,ioii techniques incidentally indicat,e some of the uses which can be made of large volume liquid scintillation detectors. Another degree of freedom which can be incorporated into the detector is a sensi- tivity to neutrons which are fast or are produced in association with a

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1.4. SCINTILLATION COUNTEItS 157

charged particle. In this case the delayed coinviderice technique men- tioned above in connection with muon decay can be employed, the first pulse being due to the associated rharged particle or a recoil proton, the second to the capture of the moderated neutron by the scintillator solu- tion. A good neutron capturer is cadmium which has a large capture cross section (5300 barns at thermal energies for the natural isotopic mixture) and on the average, four capture gamma rays with energy totalling 9 MeV. The use of energetic gamma ray.; helps discriminate

> K

K a

t m a 4

w

r

0

a

a a

0 20

0.16

0 12

0 08

0 04

1.0 c

W

I - z

0 W t

0 5 2

u a a

0 0

CAPTURE TIME t ( p SEC)

Fro. 18. Keutron capture versus time spectrum seen with 75-cm cylindrical detector CY = 0.003%. These measurements mere made using cosmic-ray neutrons: 9 X lo4 neutrons were rounted in 20 hours. The solid lines represent theoretical values oh- tained by means of a Monte Carlo calculation. (Iteines et ~ 1 . ~ ~ )

against background. Cadmium-bearing compounds such as Cd pro- pionate and more recently Cd octoate have been used with some success.41 Figure 18 shows a typicd neutron capture versus time curve for the 75-cm cylindrical detector with a Cd to H at,omic ratio, a = 0.0032. In this case the neutrons were fast, arising from various cosmic-ray events such as p- cnpt'ure and stars in 90 kgm of Pb placed on top of the detector. 4 5

The kinds of electronic circuitry employed with large detectors are indicated schematically in Fig. 19. Shown are the positive high-voltage supply (h.v.) (required t o maintain the photocathode a t ground poten- tial so as to prevent degeneration due to high-voltage gradients at the photocathode), preamplifiers, amplifiers, coincidence circuits, scalers, and

45 In some experiments this cosmic-ray neutron background can be quite trouble- some. A partial remedy is to construct the detector of light elements and avoid using heavy elements close to the detector as part of the shielding.

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158 1. .PARTICLE DETECTION

‘ I l -

SWITCH

FIG. 19. Schematic of electronics associated with a large liquid scintillation detector to measure delayed coincidences, e.g., muon decay.

it registers a delayed coincidence on the scaler and, a t the same time triggers the pulse-height analyzer gating circuit. The second pulse is then analyzed by the pulse-height analyzer. If it is desired to analyze the first pulse, the delay line can be used to store it pending the electronic decision that an appropriate delayed coincidence has occurred. In addi- tion to this sequence, the time interval between the two pulses which comprise the delayed coincidence is measured by a time delay analyzer triggered by the two pulses. The second pulse scaler reads the rate,at which single pulses occur in the energy range 15-60 MeV.

The example just given is only meant to be indicative of the kind of use to which such detectors can be put. The reader is referred to the literature for more details. a0--aa-46 In general, however, it should be noted that the field of large scintillation counters is still a new one and in many

*On these circuit elements consult also Vol. 2, Chapters 6.1, 6.2, Sections 7.2.2, 9.1.1, and Chapter 9.6.

44 F. Reines, in “Liquid Scintillation Counting” (C. G. Bell and F. N. Hayes, eds.), pp. 246-257. Pergamon, New York, 1958.

pulse-height analyzer. * Photographic records of oscilloscope traces triggered by appropriate pulses from coincidence circuits are sometimes employed to assist in the identification of the signals and the elimination of noise and background events.

As an example of how this circuitry is used, consider the case of muon decay as measured with a large liquid scintillation detector. A pulse corresponding to an entering muon of 20-40 Mev passes through circuit I and registers as a pulse on the scalar. In addition a pulse is sent to coincidence unit I1 making it sensitive for, say, 10 psec. If a pulse in the energy range 15-60 Mev passes through circuit I1 during this 1 0 ~ s e c

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1.4. SCINTILLATION COUNTEItS 159

cases the answers to questions associated with specific uses must be found by t,he experiment,alist. *

1.4.2. Solid Luminescent Chambers

Almost as soon as the nature and usefulness of scintillation counters became apparent, the possibility of “seeing” the path of a charged par-

FIG. 20. (a) Sample scintillator filaments. (b) A filament scintillation counter. (c) A comparison solid scintillation counter.

ticle in the scintillator was discussed. Early reports, later borne out by publication^,^^ of Russian work date back to 1954. This work involved the photography of tracks of protons a t two times minimum ionization in CsI crystals, with the aid of an image intensifier tube.48 In this ar- rangement, involving as it does a solid CsI crystal, the depth of focus

* The author acknowledges collaboration with Dr. C. L. Cowan, Jr. on the problems associated with the development of large volume liquid scintillation detectors. 47E. K. Zavoiakii, M. M. Butslov, A. G. Plakhov, and G. E. Smolkin, J . Nuclear

Energy 4, 340 (1957). B. R. Linden and P. A. Snell, Proc. IRE (Insl. Radio Engrs.) 46, 513 (1957).

Page 160: n

problem is severe in the optical link coupling the crystal with the image intensifier. This is particularly true since, as will he discussed later, the light output of a scintillator is very low compared with photographable intensities.

Recently a technique has been d c t v e l ~ p e d ~ ~ ~ ~ ~ for the preparation of plastic scintillators in the form of long, thin filaments as shown in Fig. 20. Such filaments are now available commercially. 61 The filaments are

F I G . 21. Crossed filament array with simulated stereo views light piped from rear to front faces.

arranged in rows, stacked alteriiatively a t right angles as shown in Vig. 2 1, to furnish the stereoviewing necessary for three-dimensional recon- struction of the particles’ path. Each of the two orthogonal sets of fila- ments is viewed separately by image intensifiers. The major advantage of such a system is that since the filaments act as light pipes for the scintillation output, only those filaments traversed by the particle actually put out light, and this light is piped to the end of the respective filaments, so that the optics problems are restricted to a plane source. Thus the coupling to the image tubes can be made directly. Filament arrays with individual diameters from 0.5 to 1.0 mm have been prepared.

48 G. T. Reynolds and P. E. Condon, Rev. Sci. Instr. 28, 1098 (1957).

51 Available froin Pilot Cherniral Corp., Watertown, Connrcticrit, and Xuclear G. T. Reynolds, N?ccleonies 16 (6), 60 (1958).

Enterprises, Edinburgh, Srotland.

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1.4. SCINTILLATION COUNTERS 161

It has been s h o ~ 1 1 ~ ~ ~ that for a 1-mm diameter filament traversed by a minimum ionizing particle, approximately 16,000 photons/cm’ result. This shows that recording on fast film requires an image intensifier with a gain of lo5. Figure 22 shows a track of a minimum ionizing p meson

FIG. 22. A minimum ionizing p-meyon track 1 in. long obtained with a scintillation chamber niadc up of &5-1nm diameter fi1ament.s 1 i in . long.

1 in. long obtained in n chamber made up of ~.5-111m diameter filaments. Six stages of image intensification prereded an image orthicon tube and the track was photographed on the face of 3, kinescope. The Russian reports imply n tube of that gain, and there is every evidence that similar tubes will he available commercially to Western scieiitists i l l t.he near future.

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162 1. PARTICLE DETECTION

A solid scintillator detector composed of plastic scintillator filaments has the advantage of simultaneous fast timing (approximately 3 X sec) and good space resolution, allowing the detection of relatively rare events in fluxes 1000 to 10,000 times those possible with bubble chambers. With the image tube? gating techniques available, the scintillation chamber can be triggered after the event, similar to cloud-chamber operations. There are no moving parts; the nuclear composition is simple (carbon and hydrogen) and loading with selected 2 material can be easily accomplished by placing thin sheets between the rows of filaments.

Following a suggestion of Kalibjian,62 Jones and Per16a have applied the idea of regenerative feedback to the problem of viewing a CsI crystal. Although lack of precise registry of successive images in practice prevents simple application of the regenerative idea, forced registration, or alter- natively, alternate cycling of two image tubes in the regenerative chain, offer promising approaches for this general idea.

Several commercial laboratories are currently engaged in the develop- ment of channeled image intensifiers in which secondary electron cascade (and possible subsequent photon internal regeneration) paths are re- stricted to small cross-section channels.

1.5. cerenkov Counters*

1.5.1. Introduction

cerenkov counters have recently been playing an increasingly im- portant role in the detection of high-energy particles, especially in experi- ments performed in particle accelerators in the multi-Bev range. Not only do these Cerenkov detectors prove to be extremely useful in many counter experiments but they can also be employed in conjunction with bubble and scintillation chambers to select and identify high-energy particles as desired. $

t See also Vol. 2, Section 11.2.3. 6* R. Kalibjian, UCRL 4 i 3 2 (1956). 65 L. W. Jones aud M. L. Perl, Rev. Sci. Instr. 29, 441 (1958). $ Regarding the Cerenkov effect see also Vol. 4, A, Section 1.5.3.

*Chapter 1.5 is by S. J. Lindenbaum and Luke C. 1. Yuan. -

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1.5. EERENKOV COUNTERS 163

A cerenkov counter can be constructed from any relatively trans- parent optical medium which possesses an index of refraction sufficiently greater than 1 in the region of the visible spectrum and its neighborhood. When a charged particle of velocity v(cm/sec) travels in a medium of index of refraction n such that v > (c/n)-i.e., when tJhe particle velocity exceeds the velocity of light in the medium Cerenkov radiation (first observed by Cerenkov) is 'The Cerenkov photons are radi- ated with uniform probability along the elements of conical surfaces of angle 6 relative to the direction of motion of the particle, where 0 is given by2nS

( I .5.1)

and p = the ratio of the particle velocity to the velocity of light in vacuum. n(v) = the optical index of refraction of the medium at the frequency v of the emitted photon.

The instantaneous apex of the cone passes through the position (macro- scopic) of the particle. The Cerenkov radiation is polarized such that the electric vector lies in the plane formed by the photon direction and the direction of motion of the particle. The intensity of cerenkov radiation per unit length per unit frequency interval is then given by

sin2 e = 2Tz2 - sin2 0 (1.5.2) d2N dx dv 137c -=-

where d2N/dx dv is the number of photons emitted per cm of path per unit frequency interval,

v is the frequency of emitted photons, e is the electron charge, Z is the ratio of the magnitude of the charge of the moving particle

c is the velocity of light in vacuum in cm/sec, and h is Planck's constant.

to the electronic charge,

Figure 1 summarizes the relevant features of the Cerenkov radiation.

1P. A. cerenkov, Compt. rend. acad. sci. U.R.S.S. 2, 451 (1954); Phys. Rev. 62,

* I. Frank and I. Tamrn, Compt. rend. aead. sci. U.R.S.S. 14, 109 (1937). 3 G. B. Collins and V. G. Reiling, Phys. Rev. 64, 499 (1938). 4 H. 0. Wyckoff and J. E. Henderson, Phys. Rat. 64, 1 (1938). 6 J. Marshall, Ann. Rev. N d e a r Sci. 4, 141 (1954); CERN Sumposium, ffenevo,

378 (1937).

Proc. 2, 62 (1956).

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164 1. PARTICLE DETECTION

Figure 2 depicts the relationship between index of refraction and velocity for a series of Cereiikov angles.

Figure 3 depicts the variation of Cerenkov angle with p for various fixed indices of refraction corresponding to some of the commonly avail- able values. For most practical cases the index of refraction is relatively constant over the visible spectrum which is contained in a frequency

CHARGED

CTnRY

FIG. 1 . Relevant features of Cerenkov radiation. (Instant = Instantaneous.)

1.28 1.26 1.24 1.22

I \ I \ , , I I

1.20 - 1.18 - 1.16 -

n 1.14 - 1.12 - 1.10 - 1.08 - 1.06 - 1.04 - 1.02 I.00. 1

-

B O O ,825 ,050 .075 ,900 .925 ,950 ,975 1.00

B

FIG. 2. Index of refraction a8 a function of velocity for a series of Cerenkov angles.

interval -3.5 X 10'" cycles/sec. Furthermore, practical Cerenkov counters use photons only in the visible and near ultraviolet regions of the spectrum.

For a particle with 2 = 1 traveling in a medium of coilstant, index of refraction, the number of photons in the visible spectrum generated per em of path length (dN/dz ) is found by evaluating (1.5.2) to be

I x 500 sin2 8 (1.5.3)

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1.5. EERENKOV COUNTERS 165

where I = number of photons generated in the visible spectrum per cm of path.

Commercial photomultipliers in general use a t present, have equiva lent photocathode efficiencies of -0.05 to 0.10 electron per photon over a frequency interval approximately equal to the visible frequency interval. Therefore if all the generated Cerenkov light is collected without absorption or other loss and a conservative average photocathode effi- ciency of -0.05 is assumed for the photomultipliers, one obtains as the

50”-

40’-

0 60 0 70 0 80 0 90 10

B FIG. 3. Variation of Cerenkov angle with B for various fixed indices of refraction.

resultant electrical signal, S(photoelectrons/cm) , generated at the photo- multiplier cathode :

(photo td t rons = 25 sin2 8. (1.5.4)

The maxipum Cerenkov signal will obviously be obtained for any medium when the particle velocity approaches c, i.e., when 0 3 1. This will be the case, for example, for a relativistic electron 2 10 MeV.

If the particle traverses wat>er (n = 1.33) one can evaluate (1.5.4) and one finds S - 10 photoelectrons/cm. For Lucite, another commonly used medium, ( r ~ = 1.5) arid 8 - 14 photoelect,rons/t:m. For Pb loaded glass TL = 1.7 and S - 16 photoelectrons/cm.

On the other hand a minimum ionizing particle in plastic scintillator would lose about, 1.5 Mev/cm and generate -6300 photons/Mev. *

* Other organic phosphors such as anthracene, stilbene, and diphenyl acetylene are either generally equal or superior to plastic scintillator in photon yield. The numerical evaluation is based on data listed in “Handhuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. XLV, p. 145 (Nuclear Instrumcntation 11). Springer, Berlin, 1959.

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166 1. PARTICLE DETECTION

This would yield -475 photoelectronsjcm if all the light were collected. Hence for a relativistic ionizing particle of @ -+ 1 the ratio of scintillator photoelectrons/cerenkov photoelectrons n. 33 for Lucite and = 44 for water. In many practical cases the ionization loss in plastic scintillator would be several times minimum with a corresponding proportional* increase in photoelectrons/cm generated in the scintillator, whereas also in many practical cases @ < 1 which reduces the Cerenkov signal.

Hence one can say that generally speaking the ratio of photoelectrons generated by charged particles in plastic scintillator/cm to that generated by Cerenkov radiation in optical media/cm is greater than -30 to 50, i.e., S(scintillator)/S(Cerenkov) > 30 to 50.

From the foregoing it is obvious that Cerenkov counters would only be employed when one wishes to make use of the special characteristics of this radiation.

The main use of Cerenkov counters is to restrict the velocity range of the particles counted. This can be done in the following ways.

1. Detection of cerenkov light sets a threshold for /3, i.e., @ > l/n. 2. Measurement of the angular range of the Cerenkov cone deter-

mines the generating particle velocity to lie in a range < @ < @2

where 81 and @ z are determined by the index of refraction of the medium and the details of the measuring system.

One might remark at this point that with the best of modern techniques signals of -5 to 10 photoelectrons on the average can be utilized to count with an efficiency approaching 100%. Average signals of even 2-3 photoelectrons have been used to count with maderate efficiency.6sC Hence although the Cerenkov signal levels are much lower than scintillation signal levels it has proven quite feasible to construct many types of useful Cerenkov counters.

There are two general categories of Cerenkov counters: focusing or angular selection counters and nonfocusing count,ers. In the focusing type, a system for focusing the photons in the Cerenkov cone on the detector photomultiplier or photomultipliers is included. This type allows a selection of a range of angles O1 < 8 < e2 of the Cerenkov light, The nonfocusing type most commonly used merely att$empts to collect as

* Saturation effects for very high specific ionization cases are neglected in the above treatment.

S. J. Lindenbaum and A. Pevsner, Rev. Sci. Znstr. 26, 285 (1954). 6a Using modern high gain photomultipliers such as the RCA-6810A, the authors

have found it quite easy even without further amplification to attain efficiencies which correspond to counting all cases where one or more photoelectrons are generated i.e. eff. = 1 - e-< where 7t is the mean number of photoelectrons generated.

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1.5. ~ E R E N K O V COUNTERS 167

many cerenkov photons as possible onto the photomultipliers more or less independent of their cone angle. However, there is a special class of nonfocusing counters which select an angular range of cerenkov light by making use of the properties of internal reflection and the appropriate use of absorbing coatings of black paint. Both types and combinations of them have been extensively employed.

Probably one of the first working Cerenkov counters was built by D i ~ k e . ~ It mas the focusing counter schematically depicted in Fig. 4. A design proposal for the type of counter Dicke used was previously made by Getting.s Dicke employed the 20-Rlev electrons from a betatron to test his counters. A 20-Mev electron t,raveling parallel to the axis generates Cerenkov radiation (as shown) which is internally reflected

INCIDENT PARTICLE- - - TRAJECTORY

EERENKOV RAY

PHOTO M ULT I PL I E R

FIG. 4. Cerenkov counter designed by Dicke.

by the rod and cone until it leaves the base of the cone and is focused by the lens as shown on the 1P28 photocathode. A fast particle of different velocity such that 0 differed sufficiently would not be focused a t the photocathode and hence could be discriminated against. Although Dicke probably detected Cereiikov light he was not able at the time to rule out all other possibilities.

Jelleyg later achieved success with a nonfocusing water cerenkov counter shown in Fig. 5. The cone of Cerenkov light generated in the water is relected by the silver-coated glass cylindrical container, and the light enters the photocathode of the photomultiplier at the bottom di- rectly, and also from the back side of the photocathode by reflection from the MgO cone. The black paint on the outside of the tube was removed to allow this.

Jelley showed quite clearly that the cosmic-ray counts were due to cerenkov light since he painted black the end wall of the glass container opposite the photomultiplier (i.e., the top end in the diagram) and by a

R. H. Dicke, Phys. Rev. 71, 737 (1947). I. A. Getting, Phys. Rev. 71, 123 (1947). J. V. Jelley, PTOC. Phys. SOC. (London) AM, 82 (1951).

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I G8 1. PARTICLE DETE("PI0N

roincidencc method selected part,icles moving downward. He then showetl that rotating the counter by 180" so that the photomultiplier was on top, caused the counts in the photomultiplier to be reduced to a small frac- tion of their former value. This demonstrated conclusively that most of thc counts were due to light which was dirrcted forward, and cerenkov radiation is the only known possibility.

GLASS END PLATE WITH BLACK PAPER

DISTILLED WATER

CONTAINER SILVERE D ON THE OUTSIDE

LIGHT TIGHT ENVELOPE

PHOTOMULTIPLIER ' LIGHT GATHERING CONE

AMPLIFIER

FIG. 5. Nonfocusing water Cerenkov

COATED WITH M g 0

coiinter designed hy Jelley.

Several general reviews6, l0--lla have been written on Cerenkov counters. In the present article representative types of the most generally useful types of counters will he discussed without necessarily including all reported counters.

1.5.2. Focusing Cerenkov Counters

Basically a focusing Cerenkov counter consists of three elements: a radiator, a focusing system including in some cases an output coupler, and a photomultiplier or series of photomultipliers.

1.5.2.1. Radiator. This is the opticd medium in xvhich the Cerenkov radiation is generated. The radiator is generally designed so as to main-

"

10 Cerenkov and other fast countcr terhniqurs. P E R N Symposium, Geneva, Proc 2,

11 J . V. Jelley, "Cerenkov Radiation." Pergnmon, Kew York, 1958 1 1 ~ D. Blanc, " DPtecteurs de particulrs," pp. 170-187. Masson, Paris, 1950

61-103 (1956).

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1.5. 6ERENKOV COUNTERS 169

tail1 a defiiiite relatioil between the cone angle of Cerenkov light a i d the direction of the particle in order to allow a measurement of this angle. A common type of radiat,or consists basically of a cylinder of solid optical medium such as Lucite or glass. The bases and cylindrical surface are optically polished. The charged particles are incident on one base of the cylinder in a direction parallel to the axis as shown in Fig. 6. If a fast charged particle does not scatter or interact, or appreciably slow down in such a radiator, the unique angle 0 of the cerenkov photons relative to the cone axis is maintained regardless of the number of reflections from the cylindrical surfare.

FIG. 6 . Cercmkov rays in a radiator.

This can easily be seen since the photon momentum p originally has a component pll = p cos 8 along the cone axis and a (*omponelit p , = p sin 8 perpendicular to the cone axis. The perpendicular component can be further broken down into pI = ps + p. where p+ is tangent to the cylindri- cal surface a t the point of contact hut perpendicular to the axis, and p. is perpendicular to the surface and the axis. lcor a specular reflection a t the cylindrical surfwe pi1 and ps are obviously unaffected, p, is reversed iii direction from an outward going normal to the surface to an inward going normal to the surfave without change of magnitude. Hence i t is obvious that the angle 8 relative to the axis is maintained and also that the minimum distance of appiwch to the axis for a skew ray is maintained. If @ < I , the critical angle i h exceeded a t the cylindrical surface and all light is internally reflected.

When the photon reaches the exit base surface of the cylinder and enters the air, its angle to the axih is changed due to refraction from 0 tn 8, where

( I . 5 .5 )

If sin 8 < 1 the ray is transmitted (at least partially) to the air. How- ever, if sin F) > I the ray is interirally reflecte,! and is trapped.

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170 1. PARTICLE DETECTION

Since P 5 1; it follows that: sin 0 5 -\/nz - 1. Hence, if n 5 -\/2 there is always some transmission out of the cylinder. However, for even common optical media such as glass or Lucite, if ---t 1 entrapmeiit re- sults. Hence in these cases an output coupling section is necessary to allow escape of the light from the radiator.

In the Getting-Dicke counter (Fig. 4) the flared cone of the radiator cylinder serves this purpose.

Another type of radiator for use with liquid optical media is a cylindri- cal container with specularly reflecting walls which is filled with a trans- parent liquid and is fitted with a thin glass window at the exit end to allow escape of the Cerenkov radiation. The principle of operation is similar to the Lucite or glass rod described above, except for the fact that specular reflection a t the boundary replaces the internal reflection. If a thin polished glass cylinder is used as the cylindrical container internal reflection can still be employed.

One should remark that for p = 1 the internal reflection will even, in the case of the solid cylindrical radiator (Lucite), be complete only for particles exactly parallel to the axis, and hence a reflecting coating or a slight outward taper toward the front face of the cylinder to insure internal reflection may be in order if the beam divergence is appreciable.

In many cases the base of the radiator where the charged particles enter is coated with black paint to absorb Cerenkov radiation of particles proceeding in the wrong direction.

Gas radiators have also been employed although not to the extent of liquid or solid radiators. Gas is used for very fast particles where /3 -+ 1 and it is desired to employ a low index of refraction medium to set a high p threshold or to improve the velocity revolution de/d,B. A major advantage in a gas Cerenkov counter is the feasibility of varying the index of refrac- tion n by simply varying the pressure of the gas and, to a lesser extent, by varying the temperature. Thus charged particles of a desired velocity or momentum can be easily selected during the course of an experiment by providing the appropriate pressure of the gas in the counter.

1.5.2.2. Focusing System. The focusing systems generally make use of n series of cylindrically or spherically symmetrical surfaces around the axis parallel to the designed-for direction of motion of the incident particle.

Following the work of Jelley, Marshall6 employed the focusing counter system shown in Fig. 7. The Lucite radiator is joined to a Lucite henii- spherical lens which has a focal length equal to twice its radius of curva- ture, Therefore there is a sharp focus a t 3 radii for rays froin the radiator coplanar with the axis of the system. A cylindrical mirror constructed of glass tubing which is aluminized on the inside is inserted such that the rays strike its surface at a distance -1.5 radii from t,he radiator end so

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h

D

171

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172 1. PARTICLE: DETECTION

that a sharp focus is made a t the axis of the lens. The hemispherical lens serves as a coupler to efficiently remove the photons from the radiator and avoid the total internal reflection for very fast particles (p + I ) .

For rays skew to the axis the focusing does not work due to conserva- tion of angular momentum, and has been shown5 to lead to an image diameter;

sin e D > _ n d - - - sin 8’

(1.5.6)

where D is the diameter of the image and the equality holds under ideal conditions; n is the index of refraction; d is the diameter of the radiator; fl is the C‘erenkov cone angle and 8’ is the half cone angle of the rays which form the image.

CYLINDRICAL MIRROR

BAFFLE PHOTOMULTIPLIER > Lll!LL- FIG. 8. Schematic of the most commonly applied t,ype of focusing counter.

At the position of the image a photomultiplier is used as a detector which converts the incident photon into an electronic pulse. A dia- phragm can be used a t the image to limit the acceptance circle. Dia- phragms can also be used in other parts of the system.

Marshall has described5 variations of his counter in one of which the hemispherical lens is split by a light shield, and two plane mirrors are placed colinear with the axis to form two images on two photomultipliers such that stray eerenkov light? produced in the lens cannot lead to a roincideiice but the desired light from tjhe radiator does. Also a two photomultiplier coincidence eliminates the counts due toa particle directly striking the photomultiplier and greatly reduces phototube noise counts.

A schematic of the most commonly applied type of focusing COUII-

ter6~12J3 is shown in Fig. 8. It uses the cylindrical radiator and cylindrical mirror but not the

hemispherical coupler. The radiator can be a solid polished cylinder of an optical medium such as Lucite, glass or quartz, or a polished glass

’* S. J. Liudenliaiim and I,. C. I,. Yuan, (’ERN Symposic~m, OPrwiw, Z’io(. 2, (23 (1956).

l 3 0 Chamhrrlain and C Wicgand, (‘ERN Syi:iposzum, Geneva, Proc 2, 6 3 (IR5ti)

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1.5. ~ E R E N K O V COUNTERS 173

cylindrical shell container filled with a liquid which acts as the radiator. One can also use as a radiator a metal cylindrical shell which is filled with liquid or gas and contains a polished aluminum cylindrical inner wall, or a separate polished aluminum cylindrical mirror and an exit window a t one end with plane surfaces perpendicular to the axis to allow the photons to escape.

The cerenkov light will escape from the end of the radiator only if eo is small enough. Let us denote the index of refraction of the generating medium by no, that of the glass by n,, and that of the air by n,.

Also denote the angle of the Cerenkov light to the axis by 80, OQ, and 8,. Then due to the relations no sin eo = n, sin $# = n, sin 8, and the geome-

tries used, the angle to the axis ea of the cerenkov light escaping into the air is given by

sin 8, = 2 sin eo c-. no sin eo

and, provided 0, is real, is independent of the index of refraction of the glass exit window.

na

0, is real provided

Since n, = 1 this reduces to (no2 - l/p2) 5 1 < pno. Thus if Cerenkov light is generated in the radiator these inequalities are always satisfied for no I z/% For no > 4 3 , P must not be too large to satisfy these inequalities.

The cerenkov photons of cone angle 8, are then reflected from the cylindrical mirror as shown in Fig. 8. The position of the image along the axis is determined by the Cerenkov cone angle 8. Provided that the diam- eter of the cylindrical mirror D, is 2 3 times the diameter of the radiator d, the image is not much affected by the optical aberrations of the focusing system. The magnification of the system is approximately one and the image is a circle of diameter equal to the effective radiator diameter. However, it can be shown that the effective angular un- certainty of acceptance of such a system is proportional to d/D,. Hence high-resolution counters require a large D,/d.

In many practical cases, two or more photomultipliers, generally off the axis, are employed in coincidence to effectively eliminate counts due to tube noise coincidences, direct excitations of the photomultipliers by a charged particle, and stray light coming through a part of the baffle system due to a particle of wrong velocity proceeding through the radiator in the wrong direction. Such a particle may be due to a scattering or inelastic interaction of the incident particle, or due to background

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174 1. PARTICLE DETECTION

/

/ - /

/ /

/

/ /

t ZERENKOV RADIATOR -

particles. Possible ways of splitting the light are indicated in Figs. 9 and 10.

One should remark here that in the case of liquid or gas radiators, cerenkov light will be generated in the solid transparent exit window.

PHOTOMULTIPLIER

\ \ \

/ \ \ '

PLANE \ MIRROR \ \ I--.---- \ '

I

I--/ ' ----

BLACKEN ED BAFFLE

u 0 2 4 6 8

INCH

HALF TRANSPARENT MIR

LINED WITH ALUMINIZED POLYSTYRENE

FIG. 9. High-velocity resolution gas counter designed by Lindenbaum and Yuan.

FIG. 10. Velocity selecting counter designed by Chamberlain, SegrB, Wiegand, and Ypsilantes.

For p + 1, if the exit window, as is generally the case, is made of glass, quartz, or other transparent optical media of n > 4 2 , the light generated by a particle parallel to the axis will be trapped in the window and can be absorbed by blackening the outer boundary of the window.

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1.5. EERENKOV COUNTERS 175

However, a sufficiently slow particle will produce light in the window of the same cone angle as the desired signal and hence will get through the optical system. But the number of photons involved will be smaller than those of the desired signal by approximately the ratio of the window thickness to the radiator length. Hence by designing a large ratio of radiator length to window thickness, they can be discriminated against.

Particles proceeding along some directions in the window may also generate an appreciable signal some of which gets through the baffle system. However, in general this can also be discriminated against by pulse height and light splitting with a double coincidence requirement. Furthermore, a directional requirement can be made by requiring one more coincidence after the particle passes through the focusing counter.

1.5.2.3. Resolution. The velocity resolving power of a focusing cerenkov counter can be expressed in terms of the partial derivative ae/ap which from Eq. (1.5.1) is

1 - ae - - ap p2n sin 0'

(1.5.7)

For counters designed for high energy resolution of relativistic particles p -+ 1 and n = 1. Hence, since the variation of p with energy is very slow, large values of ae/ap are required to obtain good energy resolution. But

a6 1 ap sin 6 - N -.

Hence small values of 6 are required for high resolution.

(1.5.8)

However, intensity - const sin2 6. Therefore :

ae 1 const W s1n dintensity - W v N

Hence an index of refraction must be used such

(1.5.9)

that 6 - 0 in order to obtain a good resolution. However, it is also obvious from Eq. (1.5.9) that 6 must be sufficiently larger than zero to allow an adequate number of photons to be generated. Gases under variable pressure and tempera- ture are the practical sources of these low index optical media. Figure 11 shows the index of refraction of one of the commonly used gases as a function of pressure a t several temperatures.

If it is desired to cover a wide range of index of refraction from near unity to -1.2 to 1.3, then a gas with a critical temperature near or above room temperature is desirable. This allows liquefaction and the resultant high values of index of refractions to be obtained with moderate pressures, at temperatures which are not excessively high or low.

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176

- - co2

25.05% 321)80 40°C 49.71'

4 x

1. PARTICLE DETECTION

1.20

1.18

1.16

1.14

1.1 2 c

2 1.10

5 1.08

n

1.06

1.04

1.02

1.00

PRESSURE -(ATOM.)

FIG. 11. Index of refraction of COZ gas as a function of pressure at several tem- peratures.

1.5.2.4. Practical limitations. Let us now consider the various practical limits to the resolution of a focusing cerenkov counter. These fall into three categories :

(a) the finite width (AO) of the cerenkov angular cone radiated relative to the instantaneous position of the particle;

(b) the deviations of the particle trajectory tangent from the optical system axis direction, and various geometrical and optical factors con- tained in the resolution;

(c) characteristics of the incident beam of particles and the effects resulting from their interaction with the cerenkov counter itself.

In category (a) we have the following effects. 1. Difraction-The Cerenkov cone angle has a width A6 due to dif-

fraction which essentially depends on the length of path in the radiator over which the coherence conditions are unchanged. Although i t has been ~ h o w n ' ~ - ~ ~ that the emission of individual photons do not affect this coherence, large enough Coulomb scattering and nuclear shadow scatter- ing do.

Lil43lS and Dedrickle have shown that the characteristic distance which determines the diffraction width of a Cerenkov cone is much greater than

l 4 Yin Yuan Li, Phys. Rev. 80, 104 (1950). 16 Yin Yuan Li, Phys. Rev. 82, 891 (1952). l6 K. G . Dedrick, Phys. Rev. 87, 891 (1952).

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1.5. ~ E R E N K O V COUNTERS 177

the mean distance between emission of successive photons but much smaller than the total path length in the radiator material. In practical cases the diffraction width is generally negligible.

2. Dispersion-Since the index of refraction n is a function of frequency Y , it follows from Eq. (1.5.1) that there will be a dispersion width A0, introduced.

The dispersion width Ad, can be estimated as follows:

ae An an pn2 sin 0'

AB, = - A n (visible) = (1.5.10)

For a relativistic particle in Lucite A0, = 0.8", in fused quartz A0, = 0.6" in water AO, = 0.5'. It is obvious from Eq. (1.5.10) that At?, can become large a t small angles.

In this connection it is interesting to compare the behavior of the ratio of the dispersion width AO, to the velocity resolving power a0/ap as a function of angle 8.

Using the foregoing we find

that the above ratio is independent of angle. Hence, the increasing dispersion width a t small angles is accompanied

by a proportional increase in velocity resolving power a t small angles. Therefore, it is generally desirable to go to small angles for increased resolution, since beam angular divergence, and Coulomb and shadow scattering widths are more or less independent of angle and generally larger than dispersion widths.

In category (b) we have the following effects. 1. Scattering-Even for a particle originally parallel to the optic axis

of the system, both Coulomb and nuclear shadow scattering change the direction and position of the trajectory in the counter. Since the Cerenkov photons are radiated at a polar angle 0 to the trajectory direction] the scattering of the trajectory leads to a distribution in 0 when all the photons radiated from the various parts of the trajectory are considered together. Light media of low atomic number such as water or Lucite or especially gas minimize both optical and Coulomb scattering.

2. Optical resolution-Any practical optical system has a characteristic angular resolution due to the finite size of the object, the inherent resolu- tion limits of the optical system and optical aberrations.

For the optical system shown in Fig. 8, the angular resolution can be defined as l /A0, where A 0 is the width of the range of cone angles leaving the exit window of the radiator which are transmitted with greater than

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178 1. PARTICLE DETECTION

half the peak intensity by the optical system to the photomultipliers comprising the detectors.

For a fixed angle 8, the optical resolution l / A e - D,/d. Hence it is obvious that a large enough mirror to radiator diameter ratio is required for high optical resolution.

Practical counters used by the authors and others use a D,/d ratio -3 to 10. For these values of D,/d spherical and other optical aberrations of the system have little additional effect on the resolution.

In category (c) (beam characteristics and effects of interactions) we have the following effects.

1. Ionization loss in the radiator-The ionization loss leads to a system- atic decrease in p as the particle progresses through the radiator. This of course leads to a decreasing C'erenkov cone angle which gives an energy loss width term A8dr,dz to the cone angular spread. This effect is only important for thick radiators and lower energy particles. Marshall6 has shown how the use of tapered outward toward the front radiators can correct for this efiect.

Another indirect effect of the ionization loss on the Cerenkov radiation is the production of 6 rays of sufficient velocity to themselves produce Cerenkov radiation. This latter effect, of course, leads to a type of general background light. However, this effect is generally not a serious back- ground limitation in practical Cerenkov counter applications.

2. Nuclear interactionsThe special cases of Coulomb and nuclear shadow scattering which lead to small changes of particle direction have been previously considered. In addition, one can also have inelastic nuclear interactions which change the direction, energy, and type of particle as well as adding new particles. These interactions generally terminate the Cerenkov radiation pattern of the original particle a t the point of interaction, but also in many cases supply new Cerenkov light emitted by the products of the interaction which acts as a background. Actually background-producing particles can enter the counter from any point of its surface and both directly emit C'erenkov light and also in- directly via products of nuclear interactions which they induce. It is again desirable to use light media as radiators to reduce the number of nuclear interactions.

3. Beam characteristicsA practical beam of particles even if momen- tum analyzed so as to be nearly monochromatic in energy has both an energy spread, and an angular spread. These two effects obviously lead to a spread AObeam in the cerenkov cone radiated which limits the practical velocity resolution of the counter. 4. Magnetic J ie ldsThe presence of strong stray magnetic fields can

cause a curvature of the radiating charged particles path and hence impart a distribution to the Cerenkov cone angle relative to the axis of

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1.5. EERENKOV COUNTERS 179

the system. This effect is not important in most practical cases. Stray electric fields can also in principle modify the cerenkov cone angle but the fields usually encountered are too weak.

1.5.2.5. Photomultipliers.* The photomultiplier characteristics most useful for application to Cerenkov counters are the following.

1. End window semitransparent photocathode type of large enough cathode area to efficiently cover the image in a focusing type counter and collect as many photons as possible. For nonfocusing counters the large area end window type are also desirable for highest detection efficiency.

2. A high efficiency for converting photons to photoelectrons over as wide as possible a section of the visible and ultraviolet spectrum as is transmitted by the radiator. A peak of conversion efficiency in the blue or ultraviolet is in general desirable.

3. As high a gain as feasible to reduce the need of electronic amplifica- tion of the small Cerenkov pulses. 4. A good signal-to-noise ratio. 5. Preferably a small spread of transit time from photocathode to first

dynode structure) and a small time spread in the photomultiplier struc- ture, in order to take advantage of the very short time spread in the cerenkov pulses.

The authors) experience has been that the fourteen-stage 56AVP (Philips) and the 6810A and 7264(RCA) represent a reasonable compro- mise with the above requirements for general purpose use.

There are many other phototubes manufactured by Dumont, EMI, RCA and others which are more suitable for particular cases. Some of these are described in the various references given for individual counters.

In particular the 5-in. and 16%. diameter RCA and Dumont photo- tubes are useful for large counters.

1.5.2.6. Some Practical Focusing Counters. Figures 9 and 10 show various practical focusing counters of the type depicted schematically in Fig. 8.

Figure 9 shows a type of high-velocity resolution counter with both a liquid and a gaseous radiator which was first constructed in 1952 by the authors and tested at the Brookhaven Cosmotron.lEs

The gas generally employed is COz which can be varied continuously in index of refraction over the range 1.004 to 1.21 by varying the pressure over the range 0 to 200 atmospheres and the temperature over the range 25" to 50°C (see Fig. 11).

*See also Vol. 2, Section 11.1.3. 16* Recent modifications include an anti-coincidence channel to improve background

rejection when K mesons are detected in the presence of a large r-meson background and these changes are not shown. S. Ozaki a.nd J. Russel have collaborated with the authors in these modifications.

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180 1. PARTICLE DETECTION

At 25"C, a pressure of approximately 75 atmospheres liquefies the CO, which then has an index of refraction -1.2. Hence when the counter is set for a particular angle of detection, the index of refraction and hence the velocity interval accepted can be changed a t will.

For the counter shown, the mean value of P accepted can be varied from = 0.83 to /3 = 1.00 with a velocity resolution AD = ,0.005.

Since the cerenkov cone angular range is not changed, the photon intensity and geometrical resolution properties are constant as the index of refraction is changed. Hence also the efficiency of detection is approxi- mately constant. When used a t detection angles of -10" the efficiency is 290%.

The liquid radiator can be used with Minnesota Mining & Mfg. Co. fluorochemical 0-75 with a variable temperature to cover the range n = 1.26 to 1.31. Water, sugar water, and then various standard liquids can be used to cover the range n = 1.33 to 1.7. A list of the index of refraction of some solid and liquid substances is shown in Table I.

TABLE I. Index of Refraction of Solid and Liquid ~ ~~~ ~~~

Index of refraction Reciprocal dispersion Material n d V = (na - l) /(w - nJ

Solid (at 18°C) Fused quartz 1.458 65

Quarts 1.5443 70 Polystyrene 1.592 30 Glass (ordinary Crown) 1.517 60

(light fiint) 1.580 42

Polymethyl methacrylate (Lucite) 1.489 - 1.493 49

(dense flint) 1.655 34

Liquid (at -20°C) Fluorochemical FC-75 Water Paraldehyde Carbon tetrachloride Toluene Benzene Chlorobenzene Carbon-di-sulfide

1.276 1.333 1.405 1.46 1.497 1.501 1.525 1.630

56

49

30 31

The small gaps that exist are easily filled by using the movement of the photographic bellows to change the angle of detection. With a par- ticular liquid the mean value of fi accepted can be varied over a consider- able range by changing the distance between the light splitter and the radiator via the bellows. Although the efficiency can in principle change

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1.5. EERENKOV COUNTERS 181

as the angle of detection is changed this effect is small since theefficiency can be made close to 100%. The velocity resolution however does change somewhat in a calculable way with angle. The major use of such a counter is as a mass spectrometer wherein small changes of resolution are not too important.

A velocity selecting counter of this general type was employed by Chamberlain et ~ 1 . ' ~ in their discovery of the antiproton at Berkeley. Figure 10 shows a diagram of their counter.

STAINLESS CYLINDER 4 " 1.D x 8" LONG

I" QUARTZ WINDOW

FRONT SURFACE

TWO ELEMENT LUCITE LENS

RING APERTURE

PHOTOMULTIPLIER

C7170 (RCA)

FIG. 12. Focusing counter using gaseous or liquid fluorochemical 0-75 designed by Baldwin et al.

Using a fused quartz solid radiator and a cylindrical mirror arrange- ment as shown in Fig. 10 they were able to attain a velocity resolution such that when AD - 0.03 the counting rate dropped to -3% of the peak value. This counter was used as an element in a counter telescope to select antiprotons from negative pions in a momentum analyzed beam.

A gas focusing counter using gaseous or liquid fluorochemical 0-75 (normally liquid) at elevated temperatures and high pressures has been designed and used by an M.I.T. group, Baldwin et ul.18 A diagram of this counter is shown in Fig. 12. By varying the temperature up to -255°C

l7 0. Chamberlain, E. SegrB, C. Wiegand, and T. Ypsilantis, Phys. Rev. 100, 947 (1955). .

E. Baldwin, D. Caldwell, S. Hamilton, L. Osborne, and D. Ritson, Scintillation Counter Conference, Washington, D.C., January, 1958; D. Hill, private com- munication.

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182 1. PARTICLE DETECTION

and the pressure over the range 1-20 atmospheres the index of refraction can be varied from near unity to -1.28.

The optical system employed essentially consists of a lens system with Schmitt correction which focuses light of a particular Cerenkov polar cone angle relative to the lens axis into a narrow ring located approxi- mately one focal length behind the lens. For small angles the ring diam- eter is proportional to the Cerenkov cone angle and hence a small annular disc opening before the photomultiplier restricts the photons accepted to a small Cerenkov cone angular interval.

FIG. 13. Focusing liquid counter employing a spherical mirror, designed by Huq and Hutchinson.

One advantage of this system is that for obtaining high-velocity resolution with large diameter radiators the necessity for a large cylindri- cal mirror can be avoided. However the alignment of the lens system must be rather carefully performed.

This counter has been used in the detection of K+ mesons in positive analyzed momentum beams a t the Bevatron.

The velocity resolution is such that the counting rate dropped by a factor of 20 when Afl - 0.006.

Another variation of the ring type of focusing counter employing a spherical mirrorIg instead of a series of spherical lenses is shown in Fig. 13. This counter is particularly suited for liquid radiators.

The liquid radiator fills the space between a spherical mirror through which the beam enters and a plane mirror face through which the beam leaves. Both mirrors are front silvered. After reflection from the plane

1s M. Huq and G. W. Hutchinson, Nuclear Instr. and Methods 4, 30 (1959).

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1.5. EERENKOV COUNTERS 183

mirror the Cerenkov cone is focused into a ring defined by an annular stop of inner radius equal to 4.25 cm and external radius of 4.5 cm.

The focused ring diameter is an increasing function of Cerenkov angle. The counter described was used in a 900-Mev proton beam and exhibited an over-all angular resolution of *+' for a cone angle -35'. This cor- responded to an energy resolution -54%.

A counter similar in principle using ethylene gas as the radiator has been designed by von Dardel and his co-workers*O at CERN to be used for experiments with the 25-Bev alternate gradient synchrotron beam. Oper- ating a t a gas pressure of up to 70 atmospheres (at room temperature) they were able to separate antiprotons from k- and T- up to 16 Bevlc momentum. A drawing of this counter is shown in Fig. 14a.

A counter designed by the authors and their co-workers for particle separation at the Brookhaven 32 Bev Alternate Gradient Synchrotron is shown in Fig. 14b. The major differences from the CERN counter is the use of COS gas and the addition of an anti-coincidence channel which collects the n-meson light when K-meson or anti-proton light is tuned into the signal channel. This technique greatly reduces the background level.

One should remark a t this point that cerenkov radiators and optical systems of the focusing type with film or other integrating detectors2I have been employed to measure the velocity distribution of particles in a beam. Since we are concerned here with cerenkov counters we shall not describe these.

There is a special class of velocity interval selecting counters which do not use focusing. The lower velocity limit is set by the threshold and the upper velocity limit by the internal reflection and subsequent absorp- tion of light with a cone angle greater than ec where ec is that angle for which total internal reflection occurs. A counter of this type designed by Fitch and Motleyzz is shown in Fig. 15. The velocity range selected is 0.65 < p < 0.78.

From the discussion in Section 1.5.2.2 it is clear that n L .\/z is required in order for total internal reflection to occur at the exit face. This is the major limitation of this kind of velocity interval selector. One is restricted to a low threshold velocity and also one has a much poorer resolution than obtainable in the focusing type of counter. How- ever the simplicity of the device is of course an advantage for those cases where it can be used.

Another variation of velocity interval selection can be obtained by

20 G. von Dardel, private communication (1960). 21 R. L. Mather, Phys. Rev. 84, 181 (1951); J. V. Jelley, AERE NP/R 1770 Atomic

22 V. Fitch and R. Motley, Phys. Rev. 101, 496 (1956). Energy Research Establishment, Harwell (1955), unpublished.

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FIG. 14a. Focusing gas counter designed by von Dardel et al.

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signal light; (F), special collecting mirror for Cerenkov anti-coincidence light; (G) , reflecting mirror for directing light into photomultipliers; (a), RCA type 6810A or Amperex 56UVP photomultiplier; (I), 4 RCA type 6810A or Amperex 56UVP photo- multiplier connected in parallel.

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186 1. PARTICLE DETECTION

using black paint or other arrangements to eliminate the largest cerenkov cone angles corresponding to the highest velocities and of course a thresh- old velocity is set by the index of refraction.

A counter of this type has been designed by Hughes, Palevsky, and co-workers23 for use in detecting high-energy neutrons. The counter consists of a high-pressure COZ cylinder which allows a variable index of refraction to set a variable threshold velocity; the cylinder is painted with black paint so that only small angle light will escape.

BLACK PAINT

FIQ. 15. Velocity interval selecting counter designed by Fitch.

1.5.3. Nonfocusing Counters

The counter described in Fig. 5 constructed by Jelley represents one general type of nonfocusing counter, namely what can be. called the “end on type.” This ‘counter is mainly suitable for use as a last element in a telescope, as the thickness of material in the counter and the presence of the phototube in the beam do not make it convenient as in one of several counter elements in a telescope.

A relatively thin transmission type nonfocusing cerenkov counter has been designed by Lindenbaum and Pevsner.s In this counter two

23 D. Hughes, H. Palevsky, and co-workers, private communication.

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1.5. EERENKOV COUNTERS 187

5819 RCA photomultiplier tubes faced the sides of an aluminum liquid container of 3 in. X 3 in. cross section and 14 in. thick with 31-mil walls. Liquids of various indices of refraction were used as cerenkov radiators. The ends of the phototubes were immersed directly in the liquids so that the semitransparent photocathode was covered by liquid. O-ring seals around the tubes were used to make a tight seal. Aluminum foil lined the inside of the container except for the ends of the phototubes.

This counter was used as an element in a counter telescope in an 87-Mev negative pion beam which had been momentum analyzed. Differential range curves taken in this beam with turpentine (n = 1.475), ethylene glycol (n = 1.427), and water (n = 1.33) all exhibited the usual T- and pmeson peaks appropriately shifted to correspond to a velocity threshold a t that cerenkov angle which provided -2 photoelectrons at the photomultipliers.

The only observable background was small and due to accidental coincidences. The absolute counting efficiency obtained was greater than 90% relative to filling the counter with a scintillator. There was no evidence for any background counts due to scintillation or other non- cerenkov counts. It has been the general experience with Cerenkov counters that, except for substances that tend to scintillate, the cerenkov effect is large compared to general light background due to other sources even for nonfocusing counters.

A number of improved versions of this type of counter using 6810A 14-stage RCA photomultipliers have been designed and employed by the authors at the Brookhaven Cosmotron for the past few years. A typical example is shown in Fig. 16. Counting efficiencies of -95-98 % have been attained for relativistic particles. The counter output was amplified and limited with one distributed 140 mc amplifier with 18 db gain. The output of the amplifier was fed to one grid of a 6BN6 dual grid tube coincidence circuit. Another counter element or series of elements were put in coin- cidence with the Cerenkov counter via the other grid.

The inside of the counter is coated with white reflecting paint to diffuse the light or in some cases coated with aluminum foil.

A gas counter of this type designed to use CO:, mainly is shown in Fig. 17. The mirrors increase the efficiency of light collection at very small angles.

The pressure can be varied from one to 200 atmospheres. The tempera- ture can be regulated from 0" to 100°C. This allows the index of refraction to be varied continuously from 1.004 to 1.21.

The liquid version of this counter (Fig. 16) allows the index of refrac- tion to vary with suitable liquids over the range 1.33 to 1.7. The index of refraction of fluorochemical o-75 can be varied over the range 1.26 to

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188 1. PARTICLE DETECTION

- - - -

1.32 by varying the temperature. Hence except for some small gaps most of the index of refraction range 1.004 to 1.7 can be attained by these counters.

A series of these transmission type counters were employed by the authors24 a t the Brookhaven Cosmotron in an investigation of positive

SCALE 1:2

- - - - - -

LIT€ WINDOW

DIRECTION OF BEAM

FIG. 17. Nonfocusing gas counter designed by Lindenhaum and Yuan.

K-meson production in positive proton collisions. After momentum selec- tion via magnetic deflection, a velocity interval is selected by requiring a coincidence in a counter with a threshold p1 and an anticoincidence in a counter with a threshold pz where pz > pl. Hence the velocity range selected is represented by: p1 < p < p2.

2 4 S. J. Lindenbaum and L. C. L. Yuan, Phys. Rev. 106, 1931 (1957).

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1.5. EERENKOV COUNTERS 189

Such a coincidence anticoincidence pair then acts as a mass spectrom- eter in a momentum analyzed beam, and can be employed to detect only K+ or another mass component in the beam. To insure a high efficiency in the anticoincidence, 2 or 3 counters are used.

As a matter of fact the major practical use of the focusing type of Cerenkov counter is also to act as an element which by selecting a velocity interval in a momentum analyzed beam a t a high energy accel- erator selects a particular mass of particles.

Another major advantage of Cerenkov counters is that they do not detect background due to low velocity particles (i.e., below their thresh- old). Hence the accidental counting rates are reduced and jamming is avoided.

Heiberg and Marshallz6 and also Porter26 have reported using a fluores- cent material additive to a water Cerenkov counter so that the violet and ultraviolet components of the radiation can be transformed into a nondirectional light of more usable wavelength for the photomultiplier. Gains of less than a factor of two have been realized in certain cases with this technique. Atkinson and Perez-Mendez have reportedz7 a gas Cerenkov threshold device for discriminating against inelastically scattered pions in a negative pion momentum analyzed beam.

1.5.4. Total Shower Absorption eerenkov Counters for Photons and

Kantz and Hofstadter first suggested28 the principle of using a total absorption cerenkov counter to measure the energy of a photon or electron of energy >, 100 MeV. The basic idea is that a block of a relatively clear optical medium of short radiation length such as lead loaded glass, with dimensions equal to many radiation lengths is used as a shower producing medium for a photon or electron entering near its center. If the block is large enough, the mean total path length of electrons and photons is approximately linearly related to the energy of the incident photon or electron. Since for electrons of energy 2 several Mev the mean number of Cerenkov photons emitted per unit path length is independent of the energy, the mean number of Cerenkov photons emitted in the counter is also a linear function of the incident energy. If the side walls of the block are reflecting for the shower photons, and the front end is

Electrons*

* Refer to Section 2.2.3.7. 26 E. Heiberg and J. Marshall, Rev. Sci. Znslr. 27, 618 (1956). ** N. Porter, Nuowo cimento [lo] 6, 526 (1957). 2’J. H. Atkinson and V. Perez-Mendez, Rev. Sci. Znstr. SO, 864 (1959). 28 A. Kantz and R. Hofstadter, Nucleonics 12, (3), 36 (1954); R. Hofstadter, CERN

Symposium, Geneva, Proc. 2, 75 (1956).

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190 1. PARTICLE DETECTION

optically coupled directly to one or a series of large photocathode photo- multipliers, a sizeable fraction of the photons generated will strike the photocathodes. A fraction of the light emitted will be reabsorbed before reaching the photocathodes. Both this reabsorption and the fraction of photons collected on the photocathode will be only slightly dependent on the incident energy over the energy range for which a well-designed counter is useful. Hence the mean number of photoelectrons generated at the photomultiplier photocathodes will be approximately a linear function of the energy of the incident photon or electron. Known energy electron beams can be used to calibrate the counter.

Obviously there will be appreciable fluctuations from the mean num- ber of photoelectrons for individual showers caused by the same energy electron. These fluctuations arise from the stochastic nature of the shower, the partial loss of electrons from the counter even in large counters, the fluctuations in Cerenkov photon emission, photon collection, and photo- electron production.

The over-a11 effect of the fluctuations can be expressed in terms of the per cent energy resolution. Although different definitions have been employed, a convenient one is the full width at half-maximum of the counter response to monoenergetic electrons divided by the energy.

The most commonly used type of total shower absorption Cerenkov counters employ Pb-loaded glasses or heavy crystals which are nearly colorless. A typical glass of this type is manufactured by the Corning Glass Co. It has a radiation length of 1-in., a density of 3.9, an index of refraction of 1.65, and a critical energy of 16 MeV. The critical energy is that energy at which ionization loss equals radiation loss.

The Schott glass works in Germany makes two varieties of Pb-loaded glass suitable for total absorption counters. The lighter one, type SF-1, is clear white, contains 62% PbO, has a density of 4.44, and a radiation length of 2.0 cm. Counters of this type have been designed and utilized by Kantz and Hofstadter12* C a s ~ e l s , ~ ~ Brabant et u Z . , ~ ~ Swartz13* Filosofo and Y a m a g a t ~ , ~ ~ Koller and S a c h ~ , ~ ~ and others.

Jester30 employed a 12-in. diameter Corning glass cylinder 14 in. long (2 optically coupled 7-in. cylinders). Four 5-in. diameter Dumont 6364 photomultipliers were placed with their cathodes in optical contact with

29 J. M. Camels, CERN Symposium, Geneva, Proc. 2, 74 (1956). 30 M. H. L. Jester, Univ. of California Radiation Laboratory, Report No. 2990

91 J. M. Brabant, B. J. Moyer, and R. Wallace, Rev. Sci. Znstr. 28, 421 (1957). C. Swartz, I R E Trans. on Nuclear Sci. NS-8, 65 (1956).

33 I. Filosofo and T. Yamagata, CERN Symposium, Geneva, Proc. 2, 85 (1956). 34 E. L. Koller and A. M. Sachs, Phys. Rev. 116, 760 (1959).

(1957).

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1.5. ~ E R E N K O V COUNTERS 191

one end of the cylinders using Dow-Corning Silicone No. 200 as the optical coupling. He has reported obtaining a linear response from 50 to 200 Mev with a resolution of -45%. An improved version of Jester’s counter was developed by Brabant et and its linear response until -1.5 Bev was demonstrated. A typical counter of this type designed by Swartz a t Brookhaven is shown in Fig. 18. The observed resolution was better than 30% for 400-Mev electrons.

FIG. 18. Typical shower detector designed by Swartz.

Another obvious way to obtain a linear relation with electron or photon energy is to use a total shower absorption scintillation counter. Various versions of this type have been reported10*11.31 utilizing NaI at low energies ( 5 100 Mev), ordinary and heavy liquid scintillators, and combinations of liquid or plastic scintillators sandwiched between Pb plates to reduce the radiation length.

In general the cerenkov type appears more likely to provide a compact, high resolution, low background, higher absolute energy accuracy, counter for the several hundred Mev to several Bev region.

1.5.5. Other Applications

discussed are the following. Several useful applications of cerenkov counters which have not been

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192 1. PARTICLE DETECTION

1. As a directional device-All focusing counters are highly directional. Furthermore front to back discrimination is easily attained with non- focusing counters by painting the back end of a cylindrical radiator black. Wincklerss used this method to measure albedo in the atmosphere. Various other methods of obtaining directional sensitivity are also possible.

2. Since the number of Cerenkov photons is proportional to Z2 for a known velocity particle, various charged particles can be separated by pulse height. There is some advantage over scintillators in that the broad Landau ionization loss distribution does not exist. Of course the statistical fluctuations of the number of photons and photoelectrons are larger in the Cerenkov case due to the smaller numbers. Nevertheless one can probably do better in many cases with a cerenkov counter than a scintillator.

3. As a source of very fast light pulses for very short resolution time of flight work-A particle traveling along the axis of a radiating cylinder like that in Fig. 1 will produce a light pulse a t the front face of the cylinder of width in time equal to:

At8o,, = 1 (- 1 - 1) v p C O S ~ e (1.5.10)

where 1 is the cylinder length in cm, v is the particle velocity in cm/sec, and 0 is the Cerenkov cone angle.

The cause of this time width is due to the fact that photons emitted in the interior of the radiator arrive later a t the exit than those emitted at the exit face. This is due to two reasons:

(a) the velocity of light is less than the particle velocity; (b) the particle travels along the axis while the photons travel a t

angle 8 to the axis.

For a relativistic particle p + 1

(1.5.11)

For a Lucite radiator of 2411. length

At - sec.

For a gaseous cerenkov radiator of 6411. length operated at a 10" Cerenkov angle with p 4 1,

At - 2 X sec. J. R. Winckler, Phys. Rev. 86, 1034 (1952).

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1.5. EERENKOV COUNTERS 193

An optical system of the type shown in Fig. 8 will in principle approxi- mately preserve the value of At for a photocathode placed in the image plane.

In this respect one should note that an additional path length of 1.2 in. in air gives a delay of -1O-lO second and an additional path length of 0.12 in. gives a delay of -10-l1 second.

to 10-l1. Of course there is no existing production type photomultiplier of end type photocathode of sizeable area which will allow one to trans- form these short time width light pulses into electrical pulses without appreciable lengthening.

The best photomultipliers presently available would lead to a width of one to several millimicroseconds sec) for an instantaneous light pulse. Although developmental types may provide widths of -1O-lO sec in the near future.

Techniques employing rf gating of the first dynode for short intervals to reduce timing errors to -10-10 sec have also been considered.

In this connection one should note that the other general and older method for measuring velocity is by electronically measuring the time of flight between two counters. Presently available photomultipliers and ordinary coincidence and chronotron techniques allow a time of flight measurement of an accuracy close to 10-lo second, when one demands counting each individual particle with a moderate efficiency.

Therefore for a typical relativistic particle ( p 4 1) timed over say 10 f t , which is a typical telescope distance, we would have A@ - 0.1. If the distance were increased to -50 f t we would have Ap - 0.01. With a 50-ft distance and an accuracy of timing of -1O-lO sec we would have A 0 - 0.001. Of course short lifetime particles cannot be effectively counted over such large distances.

In order to generate pulses with sharp enough timing to maintain 10-10 sec coincidence resolution, entails a series of severe problems. Also changes of signal path length of -1 in. would bring the counters out of coincidence.

Gas cerenkov counters of the focussing type can in the future be expected with proper design and precautions to reach velocity resolutions of AD < 0.001. This would allow one, for example, in a 20 Bev/c beam a t the Brookhaven AGS or CERN strong focusing 25-Bev proton accelerators, to separate antiprotons from all other known particles. *

From the foregoing one can conclude that cerenkov counters appear most promising in providing highest velocity resolution. Furthermore they can be used with existing photomultipliers and electronic techniques

Hence great care must be taken to maintain a pulse width of

* Refer to Section 2.2.1.3.

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194 1. PARTICLE DETECTION

most conveniently, and do not require large time of flight distances. Finally their direct velocity selectivity makes them extremely useful in reducing background and pileup problems.

1.6. Cloud Chambers and Bubble Chambers*

Cloud chambers and bubble chambers are used to make visible the paths of high-speed charged particles. In cloud chambers, the track of the particle is formed when a supersaturated vapor condenses preferen- tially on the ions formed by the charged particle as it passes through a gas. Droplets formed on the ions grow large enough so that, with the proper illumination, they are visible and can be photographed. Bubble chambers operate quite differently. The path of the particle is delineated by the bubbles formed when a charged particle passes through a super- heated liquid. Energy deposited along the track by the ionizing particle creates locally heated centers around which bubbles of vapor start to grow. When these bubbles reach a suitable size they, too, may be illumi- nated and photographed.

The most important basic difference to be noticed between cloud chambers and bubble chambers is that the former operate with gases, and the latter with liquids. There are many other differences, aside from technical operating problems, and they will be discussed in a section concerned with the advantages and limitations of each method. We will now treat only the fundamental principles involved in the operation of these devices.

1.6.1. Cloud Chambers

obtained in two ways: The supersaturation of vapor needed to form droplets on ions may be

(1) by the rapid expansion of a volume of gas containing the vapor (expansion cloud chamber) ; and,

(2) by the diffusion of vapor from a warm region where it is not supersaturated to a cold region where it is supersaturated (diffusion cloud chamber).

The technical problems of construction and operation of these two types of cloud chambers are quite different. However, once supersatura- tion is obtained, by whatever method, the formation of the droplets proceeds according to well-known thermodynamic principles.

*Chapter 1.6 is by W. B. Fretter.

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1.6. CLOUD CHAMBERS AND BUBBLE CHAMBERS 195

The theory of the formation of liquid droplets from a supersaturated vapor, treated by J. J. Thomson’ in 1888, and developed by various other physicists is summarized by J. G. Wilson in his excellent treatise on cloud chambers,2 and by Das Gupta and Ghosh3 in their review article. More recent developments in cloud-chamber technique are described in a report4 of a conference on cloud chambers.

Liquid droplets may form in a supersaturated vapor on nuclei present in the gas or, if the supersaturation is sufficiently high, spontaneously on microscopic fluctuations in density in the vapor. The latter process determines the upper limit of supersaturation desirable to attain in an expansion type cloud chamber, but is not of use in track formation. The nuclei present in cloud chambers, upon which droplets form at lower supersaturations include ions, both those of the track and others formed in the chamber, foreign suspended particles such as dust, chemical com- pounds which may act as condensation nuclei, and re-evaporation nuclei. The latter are produced by the evaporation of large droplets to a point where further evaporation ceases.

Of all these condensation nuclei, the only ones wanted to exist in the chamber a t the time of production of supersaturation and subsequent photography are the first, those ions produced by the passage of the charged particles. All others must be cleared from the chamber in various ways.

1. Unwanted ions are removed by an electrostatic clearing field. 2. Dust particles and re-evaporation nuclei are removed by production

of supersaturation successively, in the case of expansion chambers, and continuously, in the case of diffusion chambers, until the nuclei are carried to the bottom of the chamber, where they adhere to the wall.

The condensation of vapor on ions involves in the first approximation the dielectric constant of the liquid and the external medium, the surface tension of the liquid, the molecular weight of the vapor, and the degree of supersaturation of the vapor. The theory is given by Wilson2 and only the broad outlines will be given here. The effect of the charge is to modify the surface energy of an incipient droplet in such a way as to permit it to grow by the condensation of molecules out of the vapor. If the super- saturation, that is the ratio of the vapor pressure existing ( p ) to the

1 J. J. Thomson, “Applications of Dynamics in Physics and Chemistry.’’ Macmillan, London, 1888.

2 J. G. Wilson, “The Principles of Cloud-Chamber Technique.” Cambridge Univ. Press, London and New York, 1951.

a N. N. Das Gupta and S. K. Ghosh, Revs. Modern Phys. 18, 225 (1946). 4 ‘‘Report of the Conference on Recent Developments in Cloud-Chamber and Asso-

ciated Techniques, March, 1955’’ (N. Morris and M. J. B. Duff, eds.), University College, London, 1956.

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196 1. PARTICLE DETECTION

equilibrium vapor pressure (PO) at the temperature after the expansion is sufficiently high, charged droplets will grow and continue to grow until other limitations occur.

The value of the supersaturation necessary for drop formation on ions depends on the nature of the vapor and the sign of the ion. The latter fact indicates that, in the initial stages of formation of the drop, the polar nature of some of the vapors used plays an important role. For example, water vapor condenses preferentially on negative ions, and higher super- saturation is needed for condensation on positive ions, while for ethyl

EXPANSION RATIO

FIQ. 1. Positive and negative ion thresholds, 70% ethanol and 30% water, in a cloud chamber filled with oxygen. Curve at left is for positive ions; curve at right is for negative ions.

alcohol, the opposite effect occurs, as is shown6 in Fig. 1. The value of the supersaturation required2 varies from p / p o = 4.14 in water to 1.94 in ethyl alcohol, to name two commonly used vapors. Mixtures of alcohol and water are also sometimes used, in which case the value of p / p ~ may be as low as 1.62.

The rate of growth of drops from a supersaturated vapor determines the length of time required for a drop to reach visible (or photographable) size. Rapid growth makes possible short photographic delay times, thus minimizing distortion effects due to motion of the gas, and it also is desirable in producing large droplets which fall quickly to the bottom of the chamber, leaving no residue of re-evaporation nuclei. The rate of drop growth has been studied by Hazen" and by Barrett and Germain.'

C. E. Nielsen, Ph.D. Thesis, University of California, Berkeley, California, 1941. * W. E. Hazen, Rev. Sci. Znstr. 13, 247 (1942). ' 0. E. Barrett and L. S. Germain, Reu. Sci. Instr. 18, 84 (1947).

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1.6. CLOUD CHAMBERS AND BUBBLE CHAMBERS 197

Theoretically, the drop growth is determined by the diffusion of vapor toward the growing drop and by the conduction of heat away from the growing drop. Practically, it has been founda that the heat conductivity of the gas is the predominant factor in ordinary operation of cloud chambers near atmospheric pressure. The rate of growth of droplets in xenon, which has a very low heat conductivity, is very small, but if an equal pressure of helium, with high heat conductivity, is added to the xenon, the drop growth is nearly as rapid as i t is in pure helium. Kepler et al. also found that the rate of drop growth is not very dependent on the gas pressure in the range 0.2 atmos to 1.4 atmos, indicating that diffusion processes are not limiting the growth.

In practice, the time required for a drop to reach visible size is between 50 and 250 milliseconds, depending on the observational conditions. The diameter of the drops in a cloud chamber, a t the instant of photog- raphy, is of the order of 10W cm.

1.6.1.1. Expansion Chamber. Supersaturation is produced by a rapid, nearly adiabatic expansion of the mixture of gas and vapor. The drop in temperature during such an expansion is given by T1/T2 = ( V Z / Z J I ) Y - ~

where y = c,/c,, the ratio of specific heats a t constant pressure and constant volume of the gas mixtures, or Tl/T2 = (pl/pz)(~-l)’u, depend- ing on whether the expansion is volume-defined or pressure-defined. Here Tl is the initial (absolute) temperature, T 2 the final temperature, v 1 the initial volume, and v 2 the final volume, with p l and pz the corre- sponding pressures. The change in temperature is clearly dependent on y, and the large value of y for monatomic gases makes them desirable for use in cloud chambers. Most cloud chambers in current use are volume-defined, generally by mechanical means, but pressure-defined cloud chambers, with only a thin rubber diaphragm separating the pres- sure vessels, are sometimes used, especially in cloud chambers containing metal pIates.

There are .many factors to consider in the design of an expansion cloud chamber, aside from the purely mechanical ones, and those connected directly with the experimental setup. Some of these factors are discussed briefly as follows.

1.6.1.1.1. PRESSURE AND TYPE OF GAS. Cloud chambers may be operated a t widely varying pressures, from a fraction of an atmosphere, to 50 atmospheres. At low pressure, the vapor becomes an appreciable fraction of the total amount of gas present, changes the value of y, and provides an appreciable amount of ionization. In the range of pressures from 0.1 atmos to 2 atmos, the operation of a cloud chamber is not

8 R. G. Kepler, C. A. d’Andlau, W. B. Fretter, and L. F. Hansen, Nuovo cimento 1101 7, 71 (1958).

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198 1. PARTICLE DETECTION

difficult. At higher pressures, the chamber becomes more difficult to clear of old droplets, and the time necessary to wait between expansions increases as the operating pressure increases. Scattering of the passing particle by high-density gas reduces the accuracy of curvature measure- ments if the chamber is placed in a magnetic field. On the other hand, the sensitive time of high-pressure chambers is long compared with chambers operated near atmospheric pressure. A good discussion of the properties of high-pressure chambers has been given by Burh0p.O When the type of gas is not specified by the experiment to be done, noble gases are pre- ferred because of the lower expansion ratio required. Argon is most commonly used, or a mixture of half argon and half helium which gives easily visible tracks and rapidly growing droplets.

1.6.1.1.2. USE WITH MAGNETIC FIELD. Very often information on the momentum of the tracks passing through a cloud chamber is required, and the cloud chamber must be placed in a magnetic field. One factor to be considered here is the design of the chamber and its expansion mechanism to make the best use of the field. See Fig. 2. It is desirable to make the expansion mechanism occupy the least space possible, and often most of it can be placed outside the magnet, with a rod or tube leading through the iron to compress or expand the chamber. The mag- netic field should be as uniform as possible to avoid large corrections in the momentum, and accurate fiducial marks should be provided in the chamber to serve as points of reference. Generally the larger the magnetic field, the better, since the spurious curvatures produced by scattering and by motion of the gas are relatively less import,ant if the field is large. Wilson2 summarizes the relative importance of these two types of error for various lengths of track and magnetic field values. The term “maxi- mum detectable momentum” is often used as an index of performance of a chamber. This is the particle momentum for which the true curvature is equal to the uncertainty in curvature, and for a chamber where tracks of length about 40 cm can be measured in a field of lo4 gauss, the maxi- mum detectable momentum under very good conditions may be as high as 50 Bev/c.

1.6.1.1.3. USE WITH PLATES. Since the stopping power of gas is so low, and the probability of nuclear interaction in a typical cloud chamber is very small, it sometimes is desirable to place sheets of heavier material in the chamber, leaving gas spaces between, in which the tracks may be seen. The minimum distance between such plates is about 3 in., and they must be coated with reflecting materials to increase the light scattered by the droplets. Such multiplate chambers have proved valu- able in investigations of nuclear reactions, and the short-lived unstable

E. H. S. Burhop, Nuowo dmento [9] 11, Suppl. No. 2 (1954).

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1.6. CLOUD CHAMBERS AND BUBBLE CHAMBERS 199

\ i? FRONT WINDOW

,//Y ,///; ./// ,/.v //+ FRONT

B

particles produced in these reactions. The range of particles can be measured if the particle stops in one of the plates, and scattering in the plates can also be determined. If y rays traverse the chamber, the prob- ability of pair production may be increased if plates are introduced, and

A

APPROX SCALE IN INCHES

6 1 2 3 4

REAR I

C

FIG. 2. Cloud chamber designed for use in a magnetic field. The back plate of the cloud chamber moves to produce the expansion. (A) Vertical section parallel to front. (B) Vertical section parallel to side. (C) Horizontal section.

details of nuclear reactions can be observed. Multiplate chambers do not operate well in regions of high background unless sufficient shielding is available. A multiplate chamber has been operated near the Bevatron'O with a strong pulsed electric field between the plates, which reduced the ion background to the point where counter control may be used.

1oR. W. Birge, H. W. J. Courant, R. E. Lanou, and M. N. Whitehead, Univ. of California Radiation Laboratory Report UCRL-3890 (1957).

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200 1. PARTICLE DETECTION

1.6.1.1.4. COUNTER CONTROL. Expansion cloud chambers may be operated a t random, with a repetitive cycle accelerator, or with counter control. Normally counter control is used for cosmic-ray experiments. Here a particle passes through the chamber and associated counters, giving a signal which triggers the expansion of the chambers.* Since the speed of expansion is of the order of 10 milliseconds, counter controlled tracks are broadened by diffusion to widths of about one or two milli- meters. If the chamber is triggered before the particles pass through, as with an accelerator, the tracks are much sharper, easier to photograph, and to measure. If unusual events are to be observed, however, it is sometimes advantageous to use counter control even a t an accelerator.

1.6.1.1.5. TEMPERATURE CONTROL. No cloud chamber operates con- sistently without adequate temperature control, and if accurate momen- tum measurements are required, extreme precautions must be taken to avoid certain types of temperature gradients. The order of magnitude of temperature control required may be 10.1"C for most applications, and +O.Ol"C for accurate momentum measurements. Normally a tem- perature gradient of about O.Ol"C/cm is maintained from top to bottom of an expansion cloud chamber to provide stability of the gas. It is also good practice to measure the temperature and the temperature differences around a cloud chamber as routine operating procedure.

1.6.1.1.6. SPEED OF EXPANSION. The speed of expansion of a cloud chamber is usually an important factor only in the case of counter con- trol, when diffusion of the ions before they are immobilized by the form- ing drops may make the track too wide for accurate measurement. Cloud chambers can be made to complete their expansion in as little as 0.004 set," but expansion times of 10 to 20 milliseconds are more com- monly used. The width of a track is given by2

X = 4.68(D7)'I2

where X is the width which contains 90% of the drop images, D is the diffusion coefficient in cm2 sec-1, and 7 is the expansion time. Tracks about 1 mm wide are obtained with expansion times of 14 milliseconds in air a t NTP. The speed of expansion is increased if the moving parts are of low mass; however, if the speed is too great and gas must move at speeds near the speed of sound, undesirable shock wave effects occur which often can completely spoil the operation of the chamber. In cloud cham- bers expanded by a moving piston, provision should always be made to catch and damp the motion of the piston at the end of its stroke.

* See also Vol. 2, Part 8. 11 R. V. Adams, C. D. Anderson, P. E. Lloyd, R. R. Rau, and R. C. Saxena, Revs.

Modern Phys. 20, 334 (1948).

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1.6. CLOUD CHAMBERS AND BUBBLE CHAMBERS 20 1

1.6.1.1.7. RECYCLING TIME. A conventional expansion cloud chamber requires a t least one minute to prepare for each expansion. This time is occupied by slow, clearing expansions, waiting for the motion of the gas to cease, and the vapor to diffuse back through the gas. Such chambers cannot operate a t the repetition frequency of a pulsed accelerator, and in this respect are far inferior to bubble chambers, which can recycle every few seconds. Various attempts have been made12 to shorten the resetting time by quick recompression, overcompression, etc., but it seem difficult to operate on much less than a one minute cycle. Under these circumstances the operating characteristics are quite different from those at longer times, and the chamber must be adjusted to take these differences into account.

1.6.1.1.8. PHOTOGRAPHY AND ILLUMINATION. Although the illumination problems of diffusion and expansion cloud chambers have some common features, the illumination of an expansion chamber is sometimes easier because it may be possible to illuminate from the rear and use the large amount of forward-scattered light. Illumination a t right angles to the direction of viewing gives about 100 times less light than illumination from the rear but in many cases, because of mechanical reasons, right angle illumination is necessary. The design of the illumination system depends on the degree of detail required in the tracks. If individual drops must be photographed, the requirements are stringent. Flash-tube light sources are now universally used. A brief discussion of recent techniques of photography and illumination is given in the paper by flretter;I2 and Wilson2 discusses fully the photographic problems involved in drop photography.

1.6.1.2. Diffusion Chambers. Supersaturation in a diffusion cloud chamber is produced by the diffusion of a vapor from a warm region where supersaturation does not exist into a cold region where supersaturation occurs. The diffusion cloud chamber is a continuouslg sensitive instrument. The region of supersaturation is necessarily horizontal because it is maintained by temperature gradients in a gas, where stability in the gravitational field occurs when the thermal gradient is vertical. The thickness of the sensitive region depends on the thermal conditions, but cannot be made more than two or three inches.

A review article on diffusion cloud chambers, covering the theory of operation and the techniques of experimental use was published by Snowden,l3 and more recent developments were reviewed by Fretter. l2

In several ways, the design factors for diffusion cloud chambers are similar to those for expansion cloud chambers. A magnetic field is often

I* W. B. Fretter, Ann. Reu. Nuclear Sci. 6, 145 (1955). 18 M . Snowden, Prop. in Nuclear Phys. 3, 1 (1953).

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202 1. PARTICLE DETECTION

required, and the chamber, together with the cooling arrangement must then be designed to fit in a magnet. The illumination and photography present similar problems, but the photographic background in a diffusion chamber may consist of a black-dyed liquid which gives excellent contrast with the brightly illuminated track. It is considerably more difficult to utilize plates of heavy material, as in a multiplate expansion chamber because of thermal problems involved, and although some attempts have been made along these lines, the use of plates in a diffusion chamber has not become an important device. There are, however, certain unique design factors in diffusion cloud chambers.

1.6.1.2.1. PRESSURE AND TYPE OF GAS. Diffusion cloud chambers operate satisfactorily with air and argon at atmospheric pressure. Methyl or ethyl alcohol are commonly used as vapor. Dry ice (solid CO,) usually provides the cooling for the bottom of the chamber, either directly by contact, or by circulation of a liquid such as acetone over dry ice and through cooling tubes on the bottom of the chamber. A low-pressure chamber has been constructed for use with helium by Choyke and Nie1~en.l~ In this chamber the bottom was cooled with liquid air and the chamber operated in the pressure range of 75 cm to 15 cm Hg. The temperature of the top had to be maintained a t less than -20°C to provide mass stability. Such a chamber might be used, for example, for the observation of low-energy electrons whose range would be small in an atmospheric pressure chamber.

The most' useful diffusion cloud chamber for high-energy nuclear research has been the high-pressure hydrogen chamber. Shutt16 has shown that the light gases such as hydrogen, deuterium, and helium are un- suitable for use in diffusion chambers near atmospheric pressure, but work well a t pressures of the order of 25 atmos. Thus a desirable increase in density is obtained along with proper operation. Such chambers have been widely used in connection with accelerators and until the advent of the hydrogen bubble chamber provided the only means of observing directly interactions of fast particles with protons and deuterons. Al- though the technical problems of operating a t 25 atmos pressure are substantial, the high-pressure hydrogen diffusion chamber is an important instrument in nuclear physics.

1.6.1.2.2. USE WITH ACCELERATORS. The diffusion cloud chamber, being continuously sensitive, is adaptable to the rapid cycling of pulsed accelerators. For such operation, the recharging of the condenser bank supplying energy to the flash tubes illuminating the chamber must be done at a rapid rate. More basic problems are those of background

W. J. Choyke and C. E. Nielsen, Rev. Sci. Insts. 2S, 207 (1952). 1sR. P. Shutt, Rev. Sci. Instr. 22, 730 (1951).

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1.6. CLOUD CHAMBERS AND BUBBLE CHAMBERS 203

produced in the chamber during the acceleration cycle, and the ion load supportable by the chamber during the pulse. The first of these is handled by proper shielding of the chamber and insertion of the target late in the acceleration cycle. The ion load allowable in the chamber is limited by the diffusion rate of the vapor into the region depleted by formation of tracks during the previous exposure. If the cycling time is not less than five seconds, the chamber will usually recover adequately.

1.6.2. Bubble Chambers

It has long been known that liquids may be heated above the boiling point, without actually boiling. Such superheated liquids are unstable and erupt into boiling after short periods of time. Boiling may start, that is bubbles may form, at surfaces or a t nucleation centers within the liquid. D. A. Glaser was the first to conceive the idea that nucleation centers within the liquid might be created by deposit of energy by passing charged particles, and to see that such a process could be used to detect fast-moving charged particles. The bubble chamber can be thought of as the inverse of a cloud chamber, with a gas bubble forming in a superheated liquid instead of a liquid drop forming in a supersaturated gas.

The first bubble chambers were constructed so that the only nucleation centers were provided by the ionizing particle. They were made entirely of glass and were thus limited in size. Later experiments showed that gasketed chambers could operate satisfactorily if the expansion conditions were properly controlled. Development of this technique has been rapid and bubble chambers of large size are in operation or under construction. Many different types of liquids have been used, for example, liquid helium, liquid hydrogen, organic liquids, liquid xenon, and certain other inorganic liquids.

Although the general principles of the operation of a bubble cloud chamber are known, there is as yet no satisfactory theory which predicts, for example, the degree of superheat required or the number of bubbles formed as a function of energy loss. It is found experimentally that bubble chambers operate with pressure'8 about two-thirds of the critical pressure and the temperature about two-thirds of the way from the normal boiling temperature to the critical temperature. Some examples of pressure and temperature are given in Table I.

The liquid in a bubble chamber is superheated by a sudden reduction of pressure. After the track forms and the photograph is taken, the pressure is increased to the initial value, the bubbles collapse and the chamber is ready for another expansion. The great advantage of the bubble chamber over the expansion cloud chamber is that all this can take place in a few

16 D. A. Glaaer and D. C. Rahm, Phys. Rev. 97, 474 (1955).

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204 1. PARTICLE DETECTION

TABLE I. Operating Conditions of Typical Bubble Chamber Materials For the methyl iodide-propane chamber the ratiation length is 7 cm; for liquid

xenon it is 3.1 cm.

Operating Operating Density Substance pressure (psi) temperature gm/cm3

Hydrogen 70 Heliums 15 Xenonb 300 Propane 315

Isopentane 350 Methyl iodide-propane 450 WFs*' 426

28°K 4°K

- 19°C 58°C

157°C 125°C 149°C

--- 0.07 0 . 0 7 2 . 3 0 . 4

(0.078 gm/cm* of H) 0 . 5 1 . 3 2 .42

~ ~

0 W. M. Fairbank, E. M. Harth, M. E. Blevins, and G. G. Slaughter, Phys. Rev.

* J. L. Brown, D. A. Glaser, and M. L. Perl, Phys. Rev. 102, 586 (1956). 100, 971 (1955).

seconds, making the bubble chamber match a pulsed accelerator in its duty cycle. A chamber described by Glaser and Rahm16 filled with iso- pentane, became fully sensitive to minimum ionizing particles 3.5 milli- seconds after the expansion was initiated, and remained sensitive for about 10 milliseconds. Photographs must be taken within this interval. In liquid hydrogen bubble chambers" the bubbles grow much more slowly, and delay times of the order of 50 milliseconds are required.

Although the exact process of nucleation of bubbles is still not under- stood, the rate of growth of the bubbles can be e ~ p l a i n e d ' * ~ ~ ~ in terms of the heat flow in the liquid. In liquids of high thermal conductivity the rate of growth of bubbles is expected to be large, and the experimental results give close agreement with the theory.

The nucleation process itself has been discussed16,20.21 in connection with measurements on bubble density. Deposit of a substantial amount of energy, such as might occur when a delta ray is made, seems to be neces- sary. That the bubble density varies with velocity in the same way delta rays do supports this idea. Another theoretical approach to this problem has been made by Askar'ian,22 who finds an expression giving the specific number of bubbles.

17 D. Parmentier, Jr. and A. J. Schwemin, Rev. Sci. Znstr. 26, 954 (1955); also D. E. Nagle, R. H. Hildebrand, and R. J. Plano, ibid. 27, 203 (1956).

18 H. K. Forster and N. Zuber, J . Appl. Phys. 26, 474 (1954). 19 M. S. Plesset and S. A. Zwick, J . Appl. Phys. 26, 493 (1954). 20 D. A. Glaser, D. C. Rahm, and C. Dodd, Phys. Rev. 102, 1653 (1956). 11 G. A. Blinov, I. S. Krestnikov, and M. F. Lomanov, Soviet Phys. J E T P 4, 661

22 G. A. Askar'ian, Soviet Phys. J E T P 4, 761 (1957). (1957).

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1.6. CLOUD CHAMBERS AND BUBBLE CHAMBERS 205

1.6.2.1. Bubble Chambers. Design Factors. 1.6.2.1.1. TYPE OF LIQUID. The choice of liquid depends primarily on the nature of the experiment to be done. Liquid hydrogen has the advantage of presenting a purely pro- tonic target to the incident particle, but its low density makes it necessary to construct rather large chambers to have appreciable probability for interaction, and the low temperature involved creates cryogenic problems. A source of liquid hydrogen must be a t hand. Organic liquids such as pentane or propane present a mixture of nuclei as targets, complicating the analysis of the pictures, but their density is much greater. The density of free protons is about the same in liquid hydrogen as in propane. Organic liquids must be heated above room temperature to produce the required superheat. Neither liquid hydrogen nor organic liquids are very efficient in materializing photons. Liquid xenon, liquid SnC14, and liquid WFs23 may be used as h i g h 4 materials with good detection efficiency for pho- tons, but the cost of sufficient xenon to fill a reasonable size bubble chamber is very high. Another type of bubble chamber24 is that in which a gas is dissolved in a liquid under pressure. When the pressure is released the gas forms bubbles. The liquids used are usually organic liquids; thus the matter of choice of liquid is the same as for an ordinary organic liquid bubble chamber.

1.6.2.1.2. CONTROL OF TEMPERATURE. In order to ensure reproducible conditions, particularly for bubble counting, the temperature of a bubble chamber must be controlled to about 0.1"C. This implies accurate thermo- statting and consideration of temperature gradients. Rapid recycling causes heating of the liquid, and compensation for this must be provided. Generally the production and maintenance of the proper temperature conditions presents a major problem in bubble chamber design.

1.6.2.1.3. MAGNETIC FIELD. It is often desirable to immerse the bubble chamber in a magnetic field so that measurements of momentum may be made. See Fig. 3. For liquid hydrogen the multiple Coulomb scattering is not important compared to the deflection in the magnetic field a t 10 kilogauss or higher. Scattering is more serious in organic and other heavier liquids. Since single scattering can often be detected by visual inspection of the track with fields of 20 kilogauss measurements of momentum accurate to 10% may be made in bubble chamber filled with organic liquids. In most cases it is desirable to design chamber and magnet together because of the many interlocking mechanical and thermal problems.

Am. Phys. Soe. 2, 175 (1957).

B. Hahn and J. Fischer, Rev. Sci. Instr. 28, 656 (1957).

23 J. H. Mullins, E. D. Alyea., L. R. Gallagher, J. K. Chang, and J. M. Teem, Bull.

2 4 P. E. Arzan and A. Gigli, Nuovo czmento [lo] 3, 1171 (1956); 4 ,953 (1956); see also

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206 1. PARTICLE DETECTION

1.6.2.1.4. ILLUMINATION. The first photographs of tracks in a bubble chamber were taken with bright-field illumination. I n this system a bright diffuse source of light is placed back of the chamber and light is scattered out of the beam by the bubbles. Thus the bubbles appear dark against a bright background. The scattering angle can be quite small, of the order

FIG. 3. Large liquid hydrogen bubble chamber and associated magnet.

of one degree. Dark field illumination has also been developed, similar to the " straight-through" illumination in cloud chambers. This may be done by a series of plastic strips placed a t an angle, illuminated by a flash tube.

1.6.2.1.5. PHOTOGRAPHY AND REPROJECTION. Photography of a small bubble chamber presents no serious problems, and reprojection of the photographs is similar to reprojection of cloud-chamber photographs. For a large bubble chamber, however, the optical problems may be formidable.

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1.6. CLOUD CHAMBERS AND BUBBLE CHAMBERS 207

The index of refraction of the liquid is not negligible and the glass through which the photograph is taken may be quite thick. Thus the images are displaced, and for a large chamber the displacements may not be linear with distance off axis. The analysis of the pictures is very complicated unless suitable optical elements are introduced.

1.6.2.1.6. SENSITIVE TIME, COUNTER CONTROL. Measurements of sensitive time in a pentane chamber have been made by Glaser, who found that the chamber was sensitive for 10 milliseconds. Further experiments on the nucleation centers in this chamber showed that the lifetime of such centers was never more than 1 millisecond and was usually less. Since several milliseconds are required to perform the expansion, counter controlled expansions seem to be impossible, at least with present techniques. Thus for cosmic ray studies where countercontrolled expan- sion are often required, bubble chambers have not been very useful. Some attemptsz5 have been made to recycle bubble chambers a t a very high rate and have therefore a large fraction of the time during which the chamber was sensitive.'Photographs of the chamber would be taken only when an interesting event is detected by a system of counters, analyzed for events occuring during the sensitive time. However, it seems likely that the principal use of bubble chambers will be with accelerators, to which they are ideally adapted.

1.6.2.1.7. SAFETY. The dangers inherent in operation of bubble cham- bers are so great that all possible precautions against accident must be taken. Hernandez et aLZ6 described the safety measures taken in the construction of hydrogen bubble chambers, and tests on explosions of hydrogen gas. The safety precautions to be taken with bubble chambers containing organic liquids must be also carefully considered when the chamber is large, since such liquids are necessarily hot and at high pres- sure, and an explosion would be disastrous.

Phys. JETP 6, 773 (1957). 26 E. V. Kurnetsov, M. F. Lomanov, G. A. Blinov, and Chuan Chen-Niang, Soviet

26 H. P. Hernandez, J. W. Mark, and R. D. Watt, Rev. Sci. Instr. 28, 528 (1957).

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1.7. Photographic Emulsions*

1.7.1. Introduction

in the field of nuclear physics dates as far back as the discovery of radioactivity, the latter being first observed by photographic methods. A new era in the use of photographic emulsions was initiated when Kinoshita8 and Reinganumg were able to identify a trajectories-rows of developed silver grains-marking the passage of an a particle through emulsions. After Rutherford’s discovery of the disintegration of light elements by a particles, there arose a definite need for sensitive tools to detect and measure protons emitted in these dis- integrations. Since only a few sensitive instruments were available a t this time, experiments with photographic emulsions were initiated. The trajectories of slow protons were first detected in 1925;‘O in the following years the tracks of faster protons-up to about 50 MeV-were observed, due to the subsequent improvements of the quality of emulsions, process- ing techniques, and thickness of emulsion layers. The grain density in proton tracks was smaller than in a tracks of equal velocity, and it was soon definitely established that the grain density in tracks is a function of the specific ionization loss which a particle suffers in the penetration of matter.

The earliest experiments were concerned with particles emitted in the disintegration of nuclei by a particles of radioactive origin. Attempts were made to determine the yields and angular and energy distribution of disintegration products in these reactions. The low intensity of radiation, available from radioactive sources seriously limited the accuracy of the

The use of

1 M. M. Shapiro, Revs. Modern Phys. 13, 240 (1941). a P. Demers, Can. J . Research A26, 223 (1947).

H. Yagoda, “Radioactive Measurements with Nuclear Emulsions.” Wiley, New

J. Rotblat, Prop . in Nuclear Phys. 1, 37 (1950). A. Beiser, Revs. Modern Phys. 24, 273 (1952).

OL. J. Vigneron, J . phys. radium 14, 121 (1953). Y. Goldschmidt-Clermont, Ann. Rev. Nuclear Sci. 3, 141 (1953). L. Voyvodic, in “Progress in Cosmic Ray Physics” (J. G. Wilson, ed.), Vol. 11,

Ib M. M. Shapiro, in “Handbuch der Physik-Encyclopedia of Physics” (S. Flugge,

?OC. F. Powell, P. H. Fowler, and D. H. Perkins, “The Study of Elementary

York, 1949.

p. 219. North Holland Publ., Amsterdam, 1953.

ed.), Vol. 45, p. 342. Springer, Berlin, 1958.

Particles by the Photographic Method,” Pergamon Press, London, 1959. S. Kinoshita, Proc. Roy. SOC. A83, 432 (1910).

9.M. Reinganum, Phys. 2. 12, 1076 (1911). lo M. Blau, J . Phys. 34, 285 (1925).

* Chapter 1.7 is by M. Blau. 208

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1.7. PHOTOGRAPHIC EMULSIONS 209

measurements. Emulsions were exposed to cosmic radiation on high mountains and in balloon flights, leading to the discovery of fast neutrons in cosmic radiation simultaneously with cloud chamber experiments. In these exposures multiple disintegration of emulsion nuclei by cosmic radiation-stars-were observed for the first time. l1

With the availability of collimated proton-deuteron- and a-particle beams from accelerators, it became possible to correlate the residual range of particles in emulsions and the grain density in tracks with mass, charge, and energy of the incident particles; these calibration tracks were then used for the energy determination of particle tracks emitted in di~integrations.l~-'~ The investigations were greatly enhanced by the availability of new emulsion types, containing higher concentration of silver halides, with which much denser and therefore better defined tracks could be obtained. These emulsions were manufactured first by Ilford and later also by Kodak and Eastman-Kodak. However, the photographic method up to 1948 was limited to the detection of particles with velocities p 5 0.4; the then available emulsions were not sensitive enough to record particles of higher velocity and therefore smaller specific ionization. In 1948 Kodak, Ltd., in England, and soon afterward Eastman-Kodak and Ilford were successful in manufacturing the so-called electron sensitive or minimum ionization emulsions with which all charged particles, regardless of velocity, can be recorded.

Another shortcoming of the older emulsion techniques was overcome by a new processing technique16-the so-called temperature method- with which plates with up to 1 mm emulsion thickness can be developed. Before the invention of this procedure, the maximum thickness which could be developed within a reasonable length of time was 200 microns. The development of plates with emulsions thicker than 1 mm, although possible, is very lengthy; i t is also difficult to obtain uniform development and to avoid distortion. However, since many experiments in the high- energy range require thicker emulsion layers, the manufacture of stripped emulsions or pellicles must be considered a very great improvement in emulsion techniques. Tightly compressed stacks of these emulsion sheets are exposed to the radiation and later developed separately. Various marking systems have been devised which make i t possible to follow the particle trajectories through adjacent sheets. This increases the measura-

l 1 M. Blau and H. Wambacher, Sitzber. Akad. Wiss. Wien, Math.-naturw. Kl. Abt. Zla 146, 623 (1937).

l2 W. Heitler, C. F. Powell, and C. E. F. Fertel, Nature 144, 283 (1939). I3T. R. Wilkins, J . Appl. Phys. 11, 35 (1940). l 4 J. Chadwick, A. N. May, C. F. Powell, and T. C. Pickavance, Proc. Roy. SOC.

l6 C. C. Dilworth, C. P. S. Occhialini, and R. M. Payne, Nature 162, 102 (1948). A183, 1 (1944).

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210 1. PARTICLE DETECTION

ble path length of high-energy particles and therefore allows a greater number of measurements on a single track to be made. Consequently the statistical error in the measurement is diminished. The greater observable path length is of special importance in the investigation of interaction and decay events because a greater number of events becomes observable.

Another milestone in the development of the photographic method is the introduction of a measuring technique with which multiple scattering in particle trajectories can be determined.16~~7 This technique is indis- pensable for mass measurements of particles in the relativistic energy range; for lower energy particles the results of scattering measurements supply a valuable complement to ionization and range measurements.

Perhaps the greatest triumph of the photographic method is the dis- covery of unstable particles. In 1947 Perkins18 discovered the negative ?r meson and shortly afterward Lattes et ~ 1 . ~ 9 found the positive counter- part. Since then a great number of unstable particles-heavy mesons and hyperons-have been detected and their properties investigated through work in nuclear emulsions. The first heavy meson, unambiguously defined by its decay, was the 7 meson, discovered by Brown et a1.2Q in nuclear emulsions. The contribution of nuclear emulsion work in the field of strange particles could be adequately described only together with the development of particle physics. The recent improvements in mass and energy measurements are primarily the result of these experiments.

So far the above discussions have mainly dealt with field of high-energy particles. The method has also been successfully applied in the field of slow neutrons, or photo-disintegrations, and in problems connected with fission. In most of these experiments the emulsions are loaded (impreg- nated) with small amounts of the element under investigation. There is, furthermore, a large field of application for nuclear emulsions in problems of biochemistry, biophysics, medicine, and mineralogy.

1.7.2. Sensitivity of Nuclear Emulsions

The process of latent image formation for particles in nuclear emulsions is essentially the same as the case of light in ordinary emulsions. The fact that one observes rows of silver grains in the former case, and not in light exposures, may be explained by the larger energy of the particles and by the different mechanism of energy dissipation. The general theory of

P. H. Fowler, Phil. Mag. 171 41, 169 (1950). Y. Goldschmidt-Clermont, Nuovo cimento [91 7, 331 (1950). D. H. Perkins, Ndure 169, 126 (1947).

l9 C. M. G. Lattes, G. P. S. Occhialini, and C. F. Powell, Nature 160, 486 (1947). ao R. H. Brown, U. Camerini, P. H. Fowler, H. Muirhead, C. F. Powell, and D. M.

Ritson, Nature 163, 82 (1949).

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1.7. PHOTOGRAPHIC EMULSIONS 21 1

latent image formation is a solid state problem and will not be discussed in detail here.

In a recent articlez1 the latest experimental and theoretical data were reviewed on which the theory of the latent image formation is based. Electrons and positive holes are liberated through irradiation and move independently through the crystal. The authors consider as a first step in the latent image formation the creation of a “pre-image speck,” which is a combination of an electron and a silver ion, absorbed near a disloca- tion site. The pre-image speck is unstable and decays in a fraction of seconds, if not, another electron and silver ion is deposited at the same dislocation site. The pre-image is now converted into the “ sub-image,” a neutral complex Agz. The sub-image has a longer lifetime and can be developed, but only with strong developers or large induction periods.

The next step then is the absorption of another silver ion and the subsequent neutralization by a photoelectron. Experimental investiga- tions make it plausible that the neutral aggregate Ag, can be considered as the origin of the stable latent image. Since silver in contact with silver halide acquires positive charge, i t is likely that Ag3 will combine with a silver ion to form a stable tetrahedral combination Ag,. Ag, due to its positive charge, is now easily reduced by the developer. It is believed that in exposures of extreme short duration predominantly sub-images are formed, since recombination phenomena prevent the formation of stable Ag, complexes. Fast particles traverse a silver halide grain in about 10-14 sec, a time interval which is exceedingly short in comparison to the small mobility of ions. Therefore most of the ionization energy of fast particles will be spent in the formation of sub-images, which are less effective than stable latent images in the subsequent developing pro- cedures. That explains why, in spite of the relatively high ionization power of fast charged particles, special types of emulsions are necessary for the detection of particles.

Problems connected with the sensitivity of nuclear emulsions, i.e. , the maximum particle energy which can be detected and the maximum grain density in tracks of particles of given properties, have been well studied.1~22~23 A few a ~ t h o r s ~ * ~ 4 , ~ ~ attempted to describe the relation between a grain density and specific ionization loss by semiempirical mathematical expressions. However, in later experiments with particles of higher energy and in emulsions of higher sensitivity it was found that

21 T. W. Mitchell and N. F. Mott, Phil. Mug. [8] 2, 1149 (1957). ** C. M. G. Lattes, P. H. Fowler, and P. Cuer, PTOC. Phys. Soc. (London) 69,883 (1947). 25 J. H. Webb, Phgs. Rev. 74, 511 (1948). a 4 M. Blau, Phys. Rev. 76, 279 (1948). a5L. van Rossum, J . phys. radium 10, 402 (1949).

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212 1. PARTICLE DETECTION

these relations are valid only within a limited energy range. More detailed information about the phase of the theoretical and experimental investiga- tions prior t o the use of electron sensitive emulsions, may be found in the l i terat~re.26~-~~ * These articles also contain discussions of the first experiments in electron-sensitive emulsions and the earlier investigations on stopping power and the range energy relation in emulsions.

Electron-sensitive emulsions are able to detect all charged particles, no matter what their energy is, thereby opening a completely new area of nuclear physics to emulsion research. It became necessary to re-examine and to revise the methods of earlier investigations in order to adapt the techniques to the new problems. In particular, the study of grain density as a function of specific ionization or energy loss has been resumed during the last years. As a result i t has been found necessary to introduce certain changes due to theoretical considerations and because of practical reasons connected with the new measuring techniques.

At very high energies the grain density decreases slowly approximately proportional to the ionization loss until a minimum value is reached a t energies of about three times the rest mass of the particle. The grain density starts then to rise again slowly for still higher energies (relativistic increase)a1 up to the so-called plateau value, which is about 10% higher than the minimum grain density.

Grain density is not only a function of energy loss, but depends also upon emulsion sensitivity and development conditions. However, it has been found that the ratio g/gmin or g/gpl is nearly independent of develop- ment and changes in emulsion sensitivity; where g is the grain density in the track element under investigation and gmin and gpl are the grain densi- ties a t minimum ionization and plateau value. For slower or multiply charged particles the relationship between specific energy loss and grain density becomes more complicated. With increasing ionization loss the grain density in particle tracks tends to reach a saturation value which is due to the limited number of grains per unit length. The saturation value depends strongly on development (size of grains) and upon the emulsion sensitivity.

* See also Vol. 4, A, Section 2.1.7. 258 J. W. Mitchell, ed., “Fundamental Mechanism of Photographic Sensitivity.”

26 P. H. Fowler and D. H. Perkins, in reference 25a, p. 340. R. Morand and L. van Rossum, in reference 25a, p. 317.

** R. W. Berriman, in reference 25a, p. 272. ISL. Vigneron and M. Boggardt, in reference 25a, p. 265.

a1 E. Pickup and L. Voyvodic, Phys. Rev. SO, 98, 251 (1950). Also refer to Section

Butterworths, London, 1951.

J. Rotblat and C. T. Tay, in reference 25a, p. 331.

1.7.6 of this volume.

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1.7. PHOTOGRAPHIC EMULSIONS 213

In a later chapter we will discuss theoretical and semiempirical equa- tions, governing the correlation between specific energy loss and grain density or related parameters. These relations are extremely important in problems of particle identification.

It can be considered as a general statement that quantitative results in nuclear emulsions can be obtained only if appropriate calibration methods are employed. Therefore, it is understandable that, for emulsion experi- ments, as for any other measuring technique, based on calibration methods, technical details and reproducibility considerations will play an important role.

We will return to this topic later; here, only the more technical aspects of emulsion properties, sensitivity requirements, and processing conditions will be treated.

The chief purposes of emulsion experiments are: (a) the identification of particles by mass and charge measurements; (b) the determination of particle energies; (c) the investigation of lifetime and decay characteristics of unstable particles; (d) the study of scattering, interaction, and produc- tion cross sections; and (e) the detailed study of the nature and energy as well as angular distribution of the particles emitted in these events. In many problems the emulsion serves only as the detector of particles from an external source, while in others it is utilized as a reaction chamber in which the particles interact with nuclei of the emulsion itself, or with additional nuclei introduced into the emulsion for the specific purpose of the experiment.

Such experiments can be carried through successfully only if: (1) the emulsion sensitivity is sufficient for the detection of particles in the energy interval under consideration; (2) the discrimination among trajec- tories of particles with different properties is satisfactory, i.e., the dif- ference in grain density is appreciable; (3) the emulsion thickness is large enough to observe, on the average, appreciable segments of the trajectory; and (4) the geometrical relations prevailing a t exposure are well repro- duced in the developed emulsion. The latter depends on the processing conditions. Item (3) depends upon the size of the emulsion as well as the thickness of the emulsion layer, if glass-backed plates are used, or on the number of sheets within the emulsion stack; (1) and (2) depend mainly on emulsion properties but can be changed slightly through development conditions. The simultaneous attainment of the highest sensitivity and the best discrimination properties is not always possible. In emulsions of high sensitivity the number of developed grains tends to reach the satura- tion value for particles of relatively high kinetic energy (about 30 MeV for protons). Thus, a further increase in ionization energy does not essentially change the grain density in the trajectory. Therefore i t is very fortunate

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214 1. PARTICLE DETECTION

that various types of emulsions are available, permitting a choice ap- propriate to the problem at hand.

The various types of emulsions and their specific properties have been discussed in great detai1;1-7b,32 in the following discussion the various types of emulsions are merely enumerated.

Ilford, Kodak (England), and Eastman Kodak (Rochester) manu- facture various types of emulsions of different sensitivity. The most sensitive emulsions, which are selected for work with fast electrons and in the field of high energy are: G6 by Ilford, NTd by Kodak and NTB-, by Eastman-Kodak. The emulsion in widest use is the G-5 emulsion which is available in various sizes and thicknesses either with glass backing or as free sheets called pellicles. The emulsions next in sensitivity are the CZ, NTZa, and NTB emulsions from Ilford, Kodak, and Eastman Kodak respectively in which protons up to 5G70 Mev and electrons up to 30-100 kev can be detected. In El, NTA emulsions, where protons up to l(f20 Mev can he detected, the discrimination between proton and (Y particle tracks is very good. Ilford’s D and Eastman Kodak’s NTC emulsions detect only slow particles and no protons. They are designed for the detection of fission products which can easily be distinguished from par- ticles in these emulsions. Ilford manufactures another type of emulsion, the Go, with a sensitivity between El and CZ emulsions; it can be de- veloped in the same way as G6 emulsions and is very useful if sandwiched between GS pellicles for the detection of multiple charged particles of high energy (e.g., heavy primaries in cosmic radiation). Ilford manufactures G5 emulsions in gel-form, with which one can prepare fresh emulsion layers and avoid background tracks, due to cosmic radiations. This technique is used whenever the intensity of the radiation under investiga- tion is very low as in cosmic-ray experiments at great depths below the earth’s surface or in measurements of the radiation of pure isotopes. It is furthermore useful in geological and biological problems where the emul- sion can be directly poured over the substances whose radioactivity is to be measured.

Within the last few years, Ilford Ltd., had introduced new types of small grain emulsions, sensitive to particles of minimum ionization, with essentially the same constitution as G-5, which has a mean crystal diam- eter of 0.27 j~ (microns). The new emulsions, K-5 and L-4, with mean crystal sizes of 0.20 p and 0.15 p diameter respectively, thus exhibit developed grains of smaller diameter than does G-5. The fine grain emulsions prove especially useful in the identification of dense tracks, in the analysis of large stars, and in general provide sharper resolution in the measurement of small distances. However, these emulsions must be

32 C. Waller, J . Phot. S&. 1, 41 (1953).

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1.7. PHOTOGRAPHIC EMULSIONS 215

processed very soon after exposure, because the fading of the latent image is more rapid in emulsions of smaller crystal size.

Among other types are the so-called diluted emulsions which have a smaller silver halide content than other nuclear emulsions. These emul- sions are used usually together with ordinary emulsions in experiments in which interactions of particles with light and heavy elements are com- pared; the former being more frequent in diluted emulsions. A similar purpose is served by plates with alternating emulsion and thin gelatin layers where the trajectories of particles emitted in the interactions with the light elements of the gelatin can be followed into both adjacent emulsion layers.

Furthermore, all companies supply emulsions loaded with boron, lithium, and bismuth for experiments in which certain reactions with these elements are studied. Unfortunately, the amount of foreign element which can be introduced is small.

Experiments connected with loaded and sandwiched emulsions are described in references 1-7b, mentioned above.

Most nuclear emulsions have very similar chemical composition, with the exception of diluted emulsions and Eastman-Kodak NTC emulsions which contain smaller amounts of silver halides.

TABLE I. Composition of Dry Ilford Gg Emulsions.

Element Weight in gm/cm3

Silver Bromine Iodine Carbon Hydrogen Oxygen Sulfur Nitrogen

2.025 1.496 0.026 0.30 0.049 0 .20 0.011 0.073

a Ilford Nuclear Research Emulsions (Ilford Research Lab., Ilford, London, England, 1949).

The composition of the Gs emulsion is given in Table I. The density of the dry emulsion is 4.18. However, emulsion density changes if the emul- sion is brought into surroundings of higher relative humidity; because the changes take place very slowly, an appreciable length of time will elapse before equilibrium is reached. These problems were recently investigated with great care by Barkas and co-workers and are discussed below. An exact knowledge of the emulsion composition, which of course includes the water content, is very important in investigations of the range-energy

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216 1. PARTICLE DETECTION

relation in emulsions and for cross section experiments. Since the increased water content changes the volume of the emulsion layer, the spatial relationship between particle trajectories can only be evaluated if the actual volume a t exposure is known.

During processing, the emulsion layers experience several large density changes, the most important occurring during fixing, when most of the original silver halide is dissolved. After drying, the thickness of the emul- sion layer is considerably decreased with respect to the original value. The ratio of emulsion thickness before and after processing (provided that the emulsion was mounted on glass during the processing), the so- called shrinkage factor, determines the relationship between geometrical conditions a t exposure and in the processed emulsions. However, inasmuch as the processed emulsion is also hygroscopic, emulsion work should be done in humidity controlled laboratories and the emulsion thickness should be checked frequently through repeated emulsion thickness measurements. The magnitude of the shrinkage factor depends slightly on the processing conditions and is greatly influenced by the concentration of the glycerin, or other plastisizer solutions, which is used in the last bath to which the emulsions are subjected. Details on shrinkage factor measure- ments may be found in references 1-7b.

The large thickness of emulsion layers required the development of new processing methods in order to achieve uniform development throughout the emulsion and to avoid distortion. The latter is very important not only for the true reproduction of the geometrical condition prevailing a t exposure, but even more so on account of multiple scattering measure- ments which represent one of the most important measuring techniques in nuclear emulsions.

Emulsions, if not developed immediately, should be kept in deep and narrow wells both before and after exposure in order to minimize exposure to cosmic radiation. The latent image tends to fade under the influence of humidity and high temperature, necessitating special care in the storage of exposed emulsions. The fading effect is discussed in references 1-7b and is treated exhaustively in a more recent paper by Demers et

1.7.3. Processing of Nuclear Emulsions

1.7.3.1. Processing Techniques. The development of various types of emulsion of thicknesses up to 2 0 0 p has been discussed in detail in references, 3, 4, and 5.

The larger the emulsion thickness, the more difficult it is to obtain uniform development because of the time needed for thorough penetration of the developer. This difficulty was removed by the so-called temperature

3 3 P. Demers, T. Lapalme, and T. Thonvenin, Can. J . Phys. 31, 295 (1953).

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1.7. PHOTOGRAPHIC EMULSIONS 217

development method.34-36 The principle of this technique is based on the fact that a developer is chemically inactive at very low temperatures (about 4%). Amidol has been generally adopted as developing agent. The plates are soaked in cold developer until its diffusion in the emulsions is completed. Then the plates are removed from the solution and warmed to the required temperature in stainless steel containers. After the development at the higher temperature is concluded, the developing action is stopped by a weak solution of acetic acid. The emulsion is then washed and finally readied for the fixing solution. The temperature of the dry-development stage determines the degree of development, i.e., the size of the developed grains and the number of minimum ionizing tracks.

The fixing procedures for thick emulsion layers have also been com- pletely changed just as in the development stage. While fixing a t room temperature would be rapid, it has been found to lead to grave distortion in the emulsion; hence the fixing solution is also kept a t very low tempera- tures. In order to minimize distortion, the fixing solution is never changed abruptly during the fixing process which lasts several days, but is slowly and carefully replenished by fresh solution. When the fixing process is concluded, the solution is slowly removed and replenished by cold water.

The plates are then soaked in a plasticizing solution, placed between guard rings (old emulsion sheets), and dried under controlled temperature and humidity conditions. The plasticizer is introduced in order to avoid stripping of the emulsion from the gIass and to restore part of the original thickness; the latter has been considerably decreased by the dissolution of all the silver halide which was not activated during the exposure. The dry emulsions are often covered with a thin plastic coat in order to protect the emulsion and to avoid excessive fluctuations in the water content of the processed pellicle, even if the ambient relative humidity should change abruptly. Some authors introduce a clearing solution after fix- ing, especially if plates thicker than 600 p are used.

The purpose of all these involved procedures is t o ensure the most homogeneous development possible and to minimize distortion. The latter is accomplished by avoiding abrupt changes in the pH of all solutions brought into contact with the emulsion, and by avoiding sharp tempera- ture changes. To this end, i t has been proposed that the temperature of the hot stage be lowered, or that only cold developer be used, consequently increasing the development time. However, i t has been found that such

34 C. C. DiIworth, C. P. S. Occhialini, and L. Vermassen, Bull. centre phys. nucltaire,

35 A. Bonetti, C. C. Dilworth and C. P. S. Occhialini, R711L. centye phys. nucldaire, univ. libre Bruselles No. 13a (1950).

univ. libre Bruxelles No. 13c (1951).

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218 1. PARTICLE DETECTION

emulsions show a smaller ratio of grain density to background density.3e In the processing t e ~ h n i q ~ e s ~ ~ , ~ ~ ~ ~ ~ - ~ ~ for thick emulsions (Ilford, G-5),

Amidol is used as the basic developing agent. However, the Brussels group use boric acid Amidol while the Bristol group uses a combination of Amidol and bisulfite.

TABLE 11. Temperature Development for 600 p Plates Used in Belgian Laboratories. Boric Acid Amidol Developera

Amidol 4 . 5 gm Sodium sulfite (anhydrous) 18 gm Potassium bromide (10% solution) 8 cma Boric acid 35 gm Distilled water 1000 ml PH 6 . 4

a From Dilworth, Occhialini, and V e r m a ~ s e n . ~ ~

TABLE 111. Temperature Development for GOO p Plates Used in Belgian Laboratories"

Operation Bath Temperature Time

Preliminary soaking Distilled water Cold stage Boric acid Amidol Warm stage slow Dry (after wiping the

heating plate surface with a soft tissue)

Development Dry Slow cooling Dry Stop bath Acetic acid, 0.2% Silver deposit cleaning Washing Running water Fixing Hypo 40% (sodium sul-

fate u p to 10% is added if swelling is excessive)

sulfate) Slow dilution Water (adding sodium

Glycerin bath Glycerin, 2%

Slow drping with guard rings

Cooling down to 5°C

5°C to 28°C 5°C

28°C 28°C to 5°C 5°C to 14°C

14°C 14°C cooling to 5°C

5°C

5°C to room tem- perature

20°C

120 min 120 min

5 min

60 rnin 5 min

120 min

120 min Until clear

100 hr

120 min

7 days

a From Y. Goldschmidt-Clermont, Photographic emulsions. Ann Rev. Nuclear

*OA. J. Her2 and M. Edgar, Proc. Phys. SOC. (London) A66, 115 (1953). 87 A. D. Dainton, A. R. Gattiker, and W. 0. Lock. Phil. Mag. 171 42, 396 (1951). 38A. J. Herz, J . Sci. Znstr. 29, 60 (1952). 39 B. Stiller, M. M. Shapiro, and F. W. O'Dell, Rev. Sci. Instr. 26, 340 (1954).

Sci. 3, 149 (1953).

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1.7. PHOTOGRAPHIC EMULSIONS 219

Tables 11, 111, IV, and V describe processing solutions and processing procedures used in Brussels and in the Naval Research Laboratory in Washington.

TABLE IVa. Developer for 600 p Emulsions in Development Procedures by Stiller, Sha.piro, and O'Dell of the Naval Research Laboratory88

Amidol Sodium suIfite (anhydrous) Potassium bromide (10% solution) Boric water Distilled water PH

4 . 5 gm 18 gm 8 cm

35 gm 1000 ml

6 . 6

TABLE IVb. Fixing Solution (pH 5.3)

Distilled water Sodium thiosulfate cp Sodium bisulfite Ammonium chloride

~~ ~

1000 om3 400 gm

7 gm 7 gm

TABLE IVc. Clearing Solution (pH 5.2)

Distilled water Ammonium acetate Citric acid Thiurea

500 cm2 15 gm 5 gm 5 gm

TABLE V. Processing Procedure

Temperature Time

Presoaking in distilled water Penetration of cold developer Warm dry development Dry cooling Acid stop (0.5%) Fixing Clearing solution Washing Plasticizing solution (Flexoglass) Coating and drying

Room temp. to 5°C 5°C

18°C 18"C-5"C

5°C 5°C 5°C 5°C 5°C

Room temp.

150 rnin 150 rnin 180 min

5 min 150 rnin

Clearing time + 50% 24 hr 36 hr 1 hr 5 days

After each exposure a sample emulsion should be carefully examined for the degree of development by grain density measurements on the tracks of fast electrons. The uniformity of development may be investi- gated by determining the variation of grain density of fast tracks which

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220 1. PARTICLE DETECTION

traverse the thickness of the emulsion, or by measuring the grain density in electron tracks near the top, center, and bottom of the pellicle. Finally, one has to determine the degree of distortion in the processed emulsion; this may be accomplished through a technique described by Cosyns and Vanderhaeghe.40 First, several steeply dipping tracks are chosen at various positions in the emulsion; the distortion is then determined by measuring the angle, fi - a, between the tangent and the chord of the curved line representing the projection of the distorted track. The quantity

Distorted trock

FIG. 1. Perspective drawing of a distorted track inclined to the emulsion surface.

d sin(@ - a) as shown in Fig. 1, where d is the chord length of the track, is then the component of the distortion vector perpendicular to the pro- jection of the track on the emulsion surface. Here AC would be the path of the particle if no distortion were present; AB is the chord length of the track; and ARB is the actual distortion path of the particle.

The problem of distortion measurements and the various types of distortion configurations are treated in greater detail in various papers on multiple scattering, where methods for distortion diminution in scattering measurements are also described (see Section 1.7.5).

In some laboratories, emulsions are subjected to erradication processes

4 0 M. G. E. Cosyns and C . Vanderhaeghe, Bull. centre phys. nuclbaire, univ. libre Bruxelles No. 16 (1951).

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1.7. PHOTOGRAPHIC EMULSIONS 221

before expoeure. The latter is based on the destruction of the latent image by water v a p ~ r . ~ ~ , ~ ~

The wide use of large stacks of emulsions necessitates a description of the assembly and processing techniques peculiar to stacks. Pellicles were first used by D e m e r ~ , ~ ~ and the first commercial pellicles were manu- factured by Eastman-Kodak-NTB emulsion sheets, 250 p thick. The pellicles most commonly used a t present are Ilford G-5 emulsions which are available in thicknesses ranging from 200 to 1000 p. Inasmuch as the pellicles suffer considerable lateral expansion and contraction during processing, it is necessary to mount them on glass plates prior to develop- ment. 43,44 The mounting techniques in various laboratories differ slightly from each other, although the essential point is the uniform wetting of both the emulsion surface and the glass plate in order to ensure freedom from air bubbles. The pellicle and glass are generally soaked in cold water and are then pressed to each other either with a rubber roller or by passing the pellicle and glass plate through a mangle in which the pressure can be adjusted. Again, care must be taken that the pressure is uniform over the entire emulsion sheet. After mounting, the pellicle is developed and fixed in exactly the same way as are ordinary plates.

For exposure the emulsion sheets are tightly pressed together by blocks of Bakelite to ensure close contact of emulsion surfaces. Holes are some- times punched in the pellicles to aid in the packing and subsequent align- ment of the stack. Prior to development the plates are sometimes exposed to narrow beams of X-rays passing near the edges of the stack in order to facilitate, after development, the aligning of emulsion sheets relative to each other and thereby aid in tracing tracks from one plate to another. Other more elaborate have also been developed for successful alignment. One method which has gained wide acceptance is the printing of labeled grids on the surface of each pellicle. Before printing each plate is carefully adjusted in a jig with respect to small holes which were previously punched in the stack. Another method is the placement of brass tabs at the corners of the emulsion which then allow each emulsion to be mounted a t the same relative position of the microscope stage. The align- ment of the emulsions by this method is generally within 50 microns.

4 l M. Wiener and H. Yagoda, Rev. Sci. Znstr. 21, 39 (1950). 42 P. Demers, Can. J. Research A24 628 (1950). 43 B. Stiller, M. M. Shapiro, and F. W. O’Dell, Phys. Rev. 86, 712 (A) (1952). 4 4 C. W. F. Powell, Phil. Mag. [i] 44, 219 (1953). 46 R. W. Burge, L. T. Kerth, C. Richman, and D. H. Stork, U. C. R. L. 2690 (1954). 46 G. Goldhaber, S. J. Goldsack, J. E. Lannuti, and H. L. Whetstone, U. C. R. L.

47 E. Silverstein and W. Slater, J . Sci . Instr. 33, 381 (1953). 2928 (1955).

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222 1. PARTICLE DETECTION

In a novel a p p l i c a t i ~ n ~ ~ of the stripped emulsion technique the two outside emulsion sheets move relative to a fixed stack, providing a time record for particles entering or leaving the stack. This method is especially suited for investigation of the heavy primary component in cosmic radiation.

1.7.3.2. Water Content of Emulsions. The emulsion is composed of a mixture of silver halides, gelatin, and glycerin; the last two constituents are hygroscopic and will normally contain a certain amount of adsorbed wat,er. A precise knowledge of the water content of emulsion is of great importance in cross section and range measurements and for a n exact determination of angular relations in scattering, disintegration, and decay events.

The influences of the water content of emulsion on shrinkage factor and range measurements have been treated e ~ t e n s i v e l y . ~ ~ ~ ~ ~ - ~ ~

It is assumed that the absorption of water by gelatin and the con- sequent swelling of the emulsion occur without strong chemical inter- action. On the strength of this assumption, one obtains that the absorp- tion of w gram of water, of density one, by 1 cm3 of emulsion, density p ,

should result in a final volume, (1 - w) cm3, of material of density d. The latter is then related to the emulsion densit,y p through the expression d = [ ( p + to) / ( l + w)] gm cm-3. However, detailed experiment^^^-^^ have shown that Av/Aw, the volume change in cm3 per mass change in grams, due to absorption or evaporation of water, is a quantity smaller than unity. The deviation of this ratio from unity is particularly evident if the time interval between the accrual or removal of water from the emulsion and the actual measurement is short. After long time intervals-several days -Av/Aw reaches an equilibrium value, which is 0.875, 0.84, and 0.94 for G5 emulsions, according to determinations by Batty, Ilford Lab., and Barkas respectively. This effect, which Barkas attributes to the porosity of the emulsion, requires that both the mass and volume of the emulsion be obtained when precision measurements of particle ranges are needed.

The slow diffusion of water vapor into and out of emulsions was measured carefully by Oliver and B a r k a ~ . ~ ~ , ~ ~ Table VI gives the loss of

48 J. J. Lord and M. Schein, Phys. Rev. 80, 304 (1950). 49 F. K. Goward and T. T. Wilkins, Proc. Phys. Soc. (London) A63,662, 1171 (1950). 6o H. Bradner and A. S. Bishop, Phys. Rev. 77, 462 (1950). 61 J. Rotblat, Nature 166, 387 (1950). 62 J. J. Wilkins, A. E. R. E., Harwell c/r 664 (1951). 6 8 A. J. Oliver, Rev. Sci. In&. 25, 326 (1954). 6 4 W. H. Barkas, Rev. Sci. Instr. 25, 329, (1954). 66 C. J. 'Batty, Nuclear Znstr. 1, 138 (1951). 6 e W. H. Barkas, P. H. Barrett, P. Cuer, H. H. Heckman, F. M. Smith, and H. K .

Ticho, Nuovo cimento [lo] 8, 185 (1958).

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1.7. PHOTOGRAPHIC EMULSIONS 223

weight due to the ambient humidity of a 1000 fi G5 emulsion kept a t 50% r. h. (relative humidity) after unpacking. Only after 30 days in a vacuum do 1 0 0 0 ~ plates lose their water content entirely, exhibiting then a density of 4.03 gm/cm3.

TABLE VIm

Weight loss in grn per gm of emulsion Days at 50% r. h.

1 4 5

11 15 18

5.02 X 10-3 8.07 8.36 9.07 9.'16 9.22

"From A. J. Oliver, Rev. Sci. Instr. 26, 327 (1954).

Table VII gives the ratio of thicknesses of plates, kept a t various values of relative humidity, to the thickness at 50% r. h.; the thicknesses represent equilibrium values. Thickness rather than volume determina- tions are permissible if changes in length and width of the emulsion are negligible.

TABLE VIIa

% r. h. T / T a o r . h .

10 0.9657-0.0035 20 0.9720-0.0011 50 1 .ooo 60 1.0202-0.0016 70 1.0466-0.004 81 1.1090-0.0035

5From A. J. Oliver, Rev. Sci. Instr. 26, 326 (1954).

If the physical conditions a t the time of exposure are known precisely, the shrinkage factor of the processed emulsion may be determined by measuring the thicknesses of the processed and unprocessed emulsions. Because this factor depends strongly on the humidity of the surroundings, measurements must be made repeatedly if the work is not done in humidity controlled rooms. Only if the shrinkage factor is known exactly can one relate the measurements in the processed plate to the situation prevailing at the time of exposure. The shrinkage factor also depends strongly on the concentration of plasticizer used in the last step of processing, on the concentration of hardener used in the fixing solution, and according to Oliver, on the time of fixing and washing.

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224 1. PARTICLE DETECTION

One of the most accurate measurements of the shrinkage factor employs a narrow beam of particles (e.g., a particles from a radioactive source) inclined at a small angle to the emulsion surface. The ratio of the tangent of this angle to the tangent of the angle observed in the processed emulsion gives the shrinkage factor directly.

The density of emulsion may be made by comparing residual ranges of particles from accelerators or in decay processes (e.g., p mesons from r - p

decay) with the ranges found in standard emulsion.

1.7.4. Optical Equipment and Microscopes

The contents of this section do not purport to be a complete description of microscope procedures for nuclear emulsions, but rather to highlight certain aspects which differ from ordinary microscopic work.

Because many hours are spent in searching the emulsions and in measuring events, the use of a binocular microscope is necessary for the viewer’s comfort. The total magnification depends on the specific problem at hand, and may vary between 100 and 2000 times. The eyepiece lenses should be of the best possible quality; the magnifications which are commonly used may go from 6 to 20X. Dry objectives, lox, 20X, or 25X can be used when low magnification is desired, while oil immersion objectives, 45-70X and 90-100X, are commonly used for higher magnifi- cations. The aperture of these objectives should be as high as is possible on account of depth measurements and in order to ensure optimum working conditions. Emulsions which are thicker than 400 p (or emulsions whose shrinkage factors are reduced by special processing) require objec- tives with long working distances, such as those now manufactured by Cooke, Throughton, and Sims, Leitz, and Koristka. The relatively low numerical aperture (n. a.) is a drawback which must be tolerated in order to allow observation of the entire emulsion thickness.

Table VIII describes the characteristics of various objectives with long working distance.

TABLE VII I

Manufacturer Magnification n. a. Working distance

Cooke,

Leitz KS Objective Leitr KS Objective Leitr KS Objective Koristka Koristka Koristka

Throughton, and Sims 45 x 0.95 1.50mm 22 x 0.65 2.30 m m 53 x 0.95 1.00 mm 100 x 0.95 0.370 mm 30 X 1.05 3.00 mm 55 x 0.95 1.35 mm 100 x 1.25 0.530 mm

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1.7. PHOTOGRAPHIC EMULSIONS 225

The low magnification oil immersion objectives are useful because they permit better visibility and because their use avoids changing from dry to immersion lenses when switching magnifications.

Dry objectives with large working distances such as the Newton and reflecting objectives which are manufactured by Beck (England) have the advantage of enabling one to place one emulsion atop another, and then to trace directly a track which passes from one pellicle to the other; this procedure is often convenient when one is examining a stack of emulsions. However, the magnification is so small that it is difficult to follow mini- mum tracks; furthermore, the visibility is impaired by the emulsion-glass- emulsion sandwich, and the degree of optical alignment which is neces- sary to permit easy tracing of tracks is quite critical.

Measurements of length are performed with an eyepiece micrometer which has been calibrated against a stage micrometer. The other eyepiece may contain a reticle in which a line, or two parallel lines, are engraved; this line is used as a fiducial line in making angular measurements in the plane of the emulsion. The rotational movement needed to superimpose this line on the track under consideration may be determined by a pro- tractor device connected to the eyepiece. Angular measurements to within fractions of a minute may be made with precise eyepiece goniometers. For very accurate length measurements, so-called filar eyepiece micrometers are available; these devices contain one or two hairlines which are moved normal to a calibrated scale by a micrometer screw. Depth measurements are performed with the fine vertical adjustment screw, the emulsion being viewed with an oil immersion objective of highest magnification and n. a., in order to minimize the depth of focus. Another method of depth deter- mination utilizes depth gages for measuring the vertical motion of the objective.

Microscopy with nuclear emulsions differs markedly from ordinary applications in its requirements for precise and extended stage move- ments. The microscopes which are most widely used in emulsion work are the Cooke, Throughton, and Simms, type M4000, and the Leitz Ortholux; both instruments possess sufficiently smooth movements along the two stage axes. The former microscope has the advantage of a micrometer movement, thereby enabling one to read the stage position to within 2 p , while the rigid structure of the Ortholux is a desirable feature. A serious drawback in the use of both microscopes is the fact that neither can accommodate plates which are larger than 3 in. X 4 in. These disadvan- tages are especially evident in high-energy work, where large emulsion sizes are now widely used.

In order to overcome these difficulties, nuclear emulsion workers have themselves designed and built completely new stages, or modified the

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226 1. PARTICLE DETECTION

original stages, to allow them to accommodate the large plate sizes. The simplest kind of modification consists of merely extending the linear dimensions of the stage itself; the chief disadvantages of this method are the fact that the very largest plates cannot be used and that the relatively short movements must be tolerated. A further step towards versatility is the construction of a superstage which is mounted atop the original stage on runners, allowing movement of the plates in addition to that of the stage. Many workers have retained only the frames and the optics of their microscopes, and have constructed entirely new stages, according to their specifications. Others have gone even further and have utilized only the optical components from commercial microscopes and have, in effect, manufactured their own microscopes. Such an instrument has been built by ZornS7 from a universal table and drill press stand; it can accommodate the largest emulsionfi used and its movement is adequate for use as a scattering microscope. Another complete microscope has been constructed by Schein at the University of Chicago. In all of these endeavors the requirements of precision and care are understandably high.

The above-mentioned microscopes are not always satisfactory for scattering measurements, where the movement must be accurately linear over a range of several inches. One must check each microscope for the linearity of its movement, which is found to vary with the individual instrument.

1.7.5. Range of Particles in Nuclear Emulsions

1.7.5.1. Measurement of the Residual Range of Particles in Nuclear Emulsions. The length of a track in the emulsion is determined by first measuring the projection of the track in the focal plane ( 2 , ~ ) and then finding the angle of inclination (dip angle) to this plane. The former measurement is executed with a carefully calibrated eyepiece micrometer, while the latter is found by measuring the z coordinates, of two more grains in a track, on the fine adjustment depth screw of the microscope. If the direction of the trajectory changes, separate determinations of these quantities must be made for each segment of track with a different dip.

The actual length of a track is sometimes given by

R = (P + S2z2)1/2 = Z(l + S2 tan2 a)lI2 (1.7.1)

where 2 = AX)^ + ( A Y ) ~ . The shrinkage factor S is defined as the ratio of the original emulsion thickness to to the thickness after develop- ment t d , while a is the angle of inclination as measured in the processed emulsion.

0’ G, Zorn, Rev. Sci. Instr. 27, 628 (1955).

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1.7. PHOTOGRAPHIC EMULSIONS 227

Inasmuch as the track length depends on the magnitude of the shrink- age factor, the latter must be determined with great accuracy. The shrinkage is a function of the water content of both the undeveloped and shrunken emulsions. Furthermore, the slow diffusion of water into and out of the emulsion63 requires that before exposure the plates be kept in surroundings of constant humidity for an extended period of time (several days). The humidity content of the developed emulsion should also be kept constant during search and measurement.

Methods of shrinkage factor measurements are described by many a~thors.4~~8-6~ Dip angle measurements must be made under the highest magnification (smallest depth of hcus) and with oil immersion objectives. The index of refraction varies slightly with the water content of emulsions; for G5 emulsions, it has the values 1.539, 1.533, and 1.521 for 31, 51, and 75% r. h., respectively. The projected length, dip angle, and shrinkage factor must be determined separately for each emulsion sheet when a track passes through several sheets in a stack. These measurements presuppose a knowledge of the thickness of each plate a t exposure, the value of which may vary as much as 5% throughout the stack. As a consequence of this fluctuation in thickness, when accurate range determinations are required, it is not adequate merely to measure the total thickness of the stack and then to divide by the number of sheets to obtain a mean value.

Barkas et uZ.66,62a describe the measuring techniques and all the neces- sary precautions in obtaining accurate values of the density and of water content of emulsions. A precise knowledge of the emulsion composition is a prerequisite for range measurements, the latter being meaningful only in a well-defined medium. The residual range Ad, of a particle in a dry emulsion (it may be dried in a vacuum or over Hzso4) of density do is related to its range in an emulsion of density d and water content w through the Eq. (1.7.2) :

’ (1.7.2)

Here, Ado and Ad are the ranges in emulsions of densities do and d, respec- tively. A, is the range of the particle in water, and T [ = ( A v l A w ) 5 I] is the ratio of the increase of volume to that of weight of an emulsion after

- rd - 1 r(da - d ) Ado - Ad rdo - 1 -l- rdo - 1 L‘

6* L. Vigneron, J . phys. radium 10, 309 (1949). 6 8 J. RotbIat, Nature 167, 550 (1951). E o V. L. Telegdi and W. Zunti, Helv. Phys. Acta 23, 754 (1950). 61 F. A. Roads, in “Fundamental Mechanism of Photographic Sensitivity” (J. W.

82 M. Gailloud, C. H. Heanny, and R. Weill, Helv. Phys. Acta 27, 337 (1954). 8% W. H. Barkas, F. M. Smith, and W. Birnbaum, Phys. Rev. 98, 605 (1955).

Mitchell, ed.), pp. 327-330. Academic Press, New York, 1951.

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228 1. PARTICLE DETECTION

the absorption of a certain amount of moisture. In the case of dry emul- sions, the authors find r = 0.94 and 4.004.03 gm/cma for the density, while Ilford Laboratories obtain 0.84 and 4.033 gm/cm3 for the same quantities.

Another factor which, if not properly taken into account, may lead to inaccuracies in range determination is emulsion distortion. Cosyns and Vanderhaeghe40 were the first workers to treat the distortion problem mathematically. Distortion calculations are based on simple geometrical considerations, which describe the coordinate displacement of an origi- nally straight track, due t o the action of stresses present in the emulsion before solidification or induced d\ring the processing and drying of the emulsion. The maximum displacement AO will occur a t the air surface of the emulsion, since the other surface is firmly attached to the glass and will remain fixed. The displacement A of any other depth h in the emulsion will be a function of the ratio h : t , where t is the thickness of the emulsion sheet. The authors introduced a unit distortion vector, called the “Covan” and defined by C = Ao/ t2 , where the surface displacement A0

is measured in microns and the emulsion thickness t in mm. The relation of distortion to range measurements is discussed by Barkas et aE.6s and its connection with angular and scattering measurements by La1 et u Z . ~ ~ The apparent range of a particle in distorted portions of the emulsion differs from the value in the undisturbed part. If A0 = Ct2 is the maximum displacement occurring on the emulsion surface and if the second deriva- tive of the distortion vector with respect to the depth coordinate in the emulsion is constant (C-shaped curvature, which is the most widely observed), then the change p in the position of a point in the xy plane is given by p = Ao(1 - h2/t2) where h is the z coordinate of the point, measured from the emulsion surface. The range variance arising from small local distortions is often called “microscopic distortion straggling,” which is related to cavities and irregularities in the emulsion that are caused by the fixing process. The variance due to this type of straggling depends on the grain diameter, and lies between 0.02fi and 0.03E where R is the mean particle range.

The finite grain size and separation of the grains introduces another uncertainty into range measurements in that the actual range may be larger than the measured range. However, the effect is generally small, except in the case of trajectories of very small residual range. Finally, the observer error must also be taken into account. While this contribution is small for experienced workers, it may be considerable for steeply dipping or strongly scattered tracks.

The resultant of all errors and uncertainties responsible for range 68 D. Lal, Y. Pal, and B. Peters, Proc. Indian Acad. Sn’. A38, 398 (1953).

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1.7. PHOTOGRAPHIC EMULSIONS 229

straggling, which have been mentioned thus far, are smaller than the effect of “Bohr straggling.” The latter is inherent to the process of energy loss and will be discussed below in the section on range-energy theory.

1.7.5.2. Range-Energy Relation in Nuclear Emulsions.* The residual range of an ionizing particle is a function of its velocity, charge and mass.

R = F(v,Z,mi). (1.7.3)

The range-energy relation in emulsions is derived partly from theoretical considerations and partly from actual range measurements on particles whose energies have been accurately determined. Because the stopping power of nuclear emulsions is relatively high (it has more than 1000 times the stopping power of air) even fast particles, available from high-energy machines, can be brought to rest in the emulsion, provided that thick layers or stacks of emulsions are employed. This feature is one of the greatest assets of the photographic method in that it permits the observa- tion of the entire trajectory of a particle and its investigation by a variety of experimental methods. Although the relation is derived from protons, the ranges of other singly charged particles with velocities equal to that of protons of range R , can be obtained immediately from the following equation:

where mi and mp are the masses of the particle and proton respectively. This relationship follows from the energy loss equation (1.7.5), which asserts that the energy loss is independent of particle mass. The range of a particle, R(miz), with a velocity equal to that of a proton, but with different mass mi and charge z is given by the expression

(1.7.5)

The quantity f(z) represents a range correction due to electron capture, and will be discussed later in connection with the ranges of multiply charged particles.

The range-energy relation for e l e c t r ~ n s ~ ~ - ~ ~ differs somewhat from the case of heavier particles; but will not be discussed here, inasmuch as range

* Refer to Section 1.7.6. 64 H. Ross and B. Zajac, Nature 162, 923 (1946). 6 6 R. H. Hertz, Phys. Rev. 76, 478 (1949). 6 6 J. Blum, Compt. rend. 228, 918 (1949).

68 J. P. Lonchamp and C. GBgauff, J . phys. radium 17, 132 (1956). B. Gauthe and J. Blum, Compt. rend. 234, 2189 (1952).

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230 1. PARTICLE DETECTION

measurements of electrons are rarely performed in emulsions. The trajectories of slow electrons are not straight, but, due to scattering phenomena, curved in a complicated way. The rectified length of electron tracks or the number of grains rendered developable in electron tracks of certain energy is of importance in problems of &ray emission and will be discussed in the section on particle charge (2.1.1.3).

The residual range of particles of given mass, charge, and energy can be calculated if the emulsion composition and the differential stopping power of the constit,uent emulsion elements are known. This follows from the fact that the stopping power of a homogeneous mixture is equal to the sum of the contributions from each element. In earlier emulsion experi- ments the differential stopping power relative to air was used because the residual ranges-in air-of a particles from radioactive elements were well-known quantities.

The range-energy relation in emulsion was first calculated by Webbz3 for Eastman-Kodak emulsions. Similar calculations were later performed by Wilkins,62 based on the experimental datasg-72 for the computation of the irlntegral stopping power of emulsions. These data were adjusted to provide a smooth curve of stopping power versus particle velocity. Wilkins calculated range-energy curves for protons and a particles with velocities up to p = 0.31; the proton values, with the exception of the very low energy region, are in excellent agreement with the latest and most carefully determined data of Barkas. 7a Wilkins’ calculations are extremely useful in that they allow evaluation of the residual range of particles in emulsions of various compositions-in the so-called “loaded emulsions,” and in emulsions which contain a higher percentage of water as a result of humidity conditions.

A more direct approach to the calculation of the residual range is the evahation of the integral of the reciprocal of the energy loss per unit length which an article suffers along its trajectory through matter.

(1.7.6)

The energy loss, * dE/dx is a function of the mean excitation potential I, of the elements of the stopping material and of the charge z , and velocity B = v / c of the ionizing particle. The following equation shows that the

* See Eq. (1.7.13) Section 1.7.6. 88 M. S. Livingston and H. A. Bethe, Revs. Modem Phys. 9, 245 (1937). T o J. D. Hirschfelder and J. L. Magee, Phys. Rev. 73, 207 (1948). ’1 R. Warshaw, Phys. Rev. 76, 1759 (1949). 7 1 E. L. Kelly, Phye. Rev. 76, 1006 (1949). 73 W. H. Barkas, Nuovo cimento [lo] 8, 201 (1958).

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1.7. PHOTOGRAPHIC EMULSIONS 231

energy loss does not depend on the mass mi of the particle, provided that P/mc < mi/m, where p is its monentum and m the electron mass.

Here N is the number of stopping atoms per unit volume of atomic num- ber Z and Ci is a correction term for non-participating electrons in the i th shell of the atoms. The Bethe-Bloch equation (1.7.7) is valid for heavy particles with velocities greater than approximately 1.5 X lo9 cm/sec. The factor ZCi becomes negligible if the particle velocity is well above the orbital velocity of K electrons. At still higher velocities a different correc- tion term must be introduced into the Bethe-Bloch equation which repre- sents the reduction of energy loss arising from the polarization of the medium. This polarization effect or the “density effect,”* so called be- cause of its evidence in dense media, was first treated mathematically by Fermi.74 The density correction term (6), which must be added to the energy loss equation, depends both on the particle velocity and the mean excitation potential of the medium. This effect will be discussed in greater detail in the section on grain density and energy loss.

The energy loss, and hence the residual rangeof the particle [Ey. (1.7.6)], can be determined in the region of validity of Eq. (1.7.7) if I and Ci are known. However, until recently, because these quantities were not known directly from experiments, appreciable uncertainties existed in the range- energy relation. It is generally quite difficult t o perform very accurate ionization potential measurements. The magnitude of ionization poten- tials I, may be obtained from energy loss and range measurements on particles with known momentum; however, in both cases In I , and not I, enters into the equation. Therefore, extremely accurate measurements are required in order to determine the value of I and to decide about the velocity dependence of the ionization potential. Another difficulty arises from the dependence of dE/dx on the electron density of the medium, and therefore on the density and composition of the emulsion.

A number of a ~ t h o r s 7 ~ - 7 ~ have attempted to represent the range- energy relation empirically by power law equations which, however, are valid only in a restricted energy region. A semiempirical relation,

* Refer to Section 1.7.6. 74 E. Fermi, Phys. Rev. 67, 485 (1940). 7b U. Camerini and C. M. G. Lattes, Ilford technical data (Ilford Research Lab.,

76H. Bradner, F. M. Smith, W. H. Barkas, and A. S. Bishop, Phys. Rev. 77, 462

77 W. M. Gibson, D. J. Prowse, and J. Rotblat, Nature 173, 1180 (1954). 78 H. Fay, K. Gottstein, and K. Hain, Nuovo cimento [91 11, Suppl. No. 2, 234 (1954).

London, 1948).

(1950).

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232 1. PARTICLE DETECTION

R E 2 = K . Ral * RPb, for high-energy particles has been proposed.79 This equation compares the range in emulsions, RE, with ranges in aluminum and lead, as determined from absorption experiments where K is a slowly varying constant.

The first theoretical range-energy curve was calculated by VignerongQ for particles with velocities p 5 0.3 in dry emulsions of density 3.815 gm/cm3, employing a constant value of I = 332 ev. The curve is based on the most reliable experimental data known a t that time, and extends down to proton energies of 0.1 MeV. The low-energy portion of the curve is based on measurements by Mano.81

Baroni et aLsz have calculated the range-energy relation for energies up to several Bev, taking into account the density effect by utilizing Stern- heimer’se3 equation. The emulsion density used in deriving this curve is taken to be 3.92 gm/cm3.

Barkas and Youngs4 have also extended Vigneron’s curve to higher energies by calculating the ranges from the Bethe-Bloch equation, in which they used an ionization potential of 331 ev and an electron density corresponding to an emulsion density of 3.815 gm/cm3, the same value used by Vigneron. The calculated values for high energy protons are based on Sternheimer’s work.

In a more recent paper, B a r k a ~ ~ ~ refines the approach to this problem by performing much more detailed calculations. The correction factor C, in the Bethe-Bloch equation is evaluated for the K and L shells of all emulsion nuclei, except hydrogen, by using Walske’sss calculations. Stern- heimer’s expression for the density effect correction is used at very high energies; the mean excitation potential, which enters this correction, is determined experimentally.

The value of the mean ionization potential can, in principle, be obtained from a single accurate measurement of the range in an emulsion of known composition (humidity content) and shrinkage factor on a particle of well- defined momentum. The energy of the particle must lie below the value where the density effect becomes noticeable, in order to utilize the simplest form of the Bethe-Bloch equation. Once I has been determined, however, the exact ranges or higher energy particles may be employed for the direct evaluation of the density effect correction.

7 0 R. R. Daniel, G. G. George, and B. Peters, Proc. Indian Acud. Sci. A41,45 (1055). 8oL. Vigneron, J. phys. radium 14, 145 (1953); Compt. rend. 232, 1199 (1951). M. G. Mano, Compt. rend. 197, 1759 (1933); Ann. phys. [ l l ] 1, 407 (1934).

8 2 G. Baroni, C. Castagnoli, G. Cortini, C. Franrinetti, and A. Manfredini, Bureau

81 R. M. Sternheimer, Phgs. Rev. 103, 511 (1956). 81 W. H. Barkas and D. M. Young, U. C . R. L. 2579 (1954). 8b M. C. Walske, Phys. Rev. 88, 1283 (1952), 101, 940 (1956).

of Standards, Bull. No. 9, CERN, Geneva (1956).

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1.7, PHOTOGRAPHIC EMULSIONS 233

A mean value of the excitation potential of 331 zk 6 ev has been estab- lished from several very exact range determinations. If one uses the Bloch relations6 I = kz, and substitutes for Z the mean atomic number of the emulsion elements, one obtains the value 12.25 -t 0.22 ev for k. The latt,er value is in satisfactory agreement with the measurement of Bichsel et aZ.,s7 who find that k varies between 12.5 and 13.1 in absorption measurements of low-energy protons in various elements. The above value of k is definitely greater than that of 9.1 which was found from the experiments of Mather and Segr&8s

The excellent agreement between calculated and measured values presents a strong argument in favor of adopting a mean excitation potential near 331 ev; furthermore, it seems to strengthen Caldwell’s*9 assumption concerning the constaiicy of I . The calculated values of I have not only been compared with the work of B a r k a ~ ~ ~ but with experimental data from other laboratories as well.

2.2)p, of the p meson in the T-P decay is in excellent agreement with the Berkeley results and with measurements on the G stack.g0 The ranges of mesons from the decay of K,, mesons agree with Barkas-Birge tables91 to within an experimental error in the deter- mination of the K meson mass. The measurements of Heinzg2 on 342 Mev protons are about 1.5% lower, those of Friedlander el aL93 on protons of 87, 118, and 146 Mev are lower by about 1%, and of De Carvalho and Friedman94 on 208 Mev protons are in good agreement with Barkas’ values.

Figure 2 gives the residual range of protons as a function of kinetic energy in the emulsion; the curves are drawn according to the tabulated range data in Barkas’ paper.73 For energies below 1 MeV, the ranges were calculated from the E3’2 relation of Geiger and Bohr. The range-energy relation due to Baroni et aZ.S2 deviates from Barkas’ curve only at proton energies greater than 1 Rev, while the curve from Fay et aZ.,78 which is based on a power law, exhibits considerable deviation a t proton energies as low as 150 MeV.

Figure 3, taken from Barkas’ paper,73 gives the percentage increase in

8 6 F. Bloch, 2. Physik 81, 363 (1933).

88 R. L. Mather and E. SegrB, Phys. Rev. 84, 191 (1951). *9 D. 0. Caldwell, Phys. Rev. 100, 291 (1955).

$1 R. W. Birge, D. H. Perkins, J. R. Peterson, D. H. Stork, and M. N. Whitehead,

*z 0. Heinr, Phys. Rev. 94, 1726 (1954). *3 M. W. Friedlander, D. Keefe, and M. G. X. Menon,Nuovocimento [lo] 6,461 (1957). *4 H. G. De Carvalho and J. I. Friedman, Rev. Sci . Instr. 26, 261(1955).

The calculated range, (602

H. Bichsel, R. F. Morley, and W. A. Aron, Phys. Rev. 106, 1788 (1957).

G. Stack Collaboration, Nuovo cimento 1101 2, 1063 (1955).

Nuovo cimento [lo] 4, 834 (1956).

Page 234: n

234 1. PARTICLE DETECTION

2 -

FIG. 2. Range-energy relation for protons in emulsions. The energy of protons in Mev is plotted versus the range in cm. (According to tabulated data by Barka~'~.)

- - -

I I I I I I I 1 I

FIG. 3. The percentage increase of ionization potential causing a 1% increase in emulsion range is plotted versus particle velocity (after Barkas73).

Page 235: n

1.7. PHOTOGRAPHIC EMULSIONS 235

0.9

0.8

0.7

0.6

0.5

0.4

the mean excitation potential which would cause a one per cent increase in emulsion range, as a function of particle velocity. Similarly, Fig. 4, also from Barkas’ work,66 gives the relative decrease in range resulting from a one per cent increase in emulsion density, as a function of P ; these curves are drawn assuming that the ratio of water volume decrease to water

-

-

-

-

-

-

I .o I I I I I I I l l I I 1 I I I l l

0.3 I I I I I I I l l I I I I I I l l

0.01 0.1 I B

FIG. 4. The percentage range decrease for one per cent increase in emulsion density is plotted versus particle velocity. The curves are calculated for 3 different assumptions about the ratio of the water volume to water weight decrease in emulsions (after Barkas et al.66).

weight decrease is A * - * equal to unity, B * * equal to 0.94, and C - - * equal to 0.84. It may be noted how critically the residual range is affected by density variations, especially a t high particle velocities.

Fowler and Scharffg6 propose a simple range-energy formula which is believed to be accurate to within 5% for ranges lying between 0.1 and 3000 gm/cm2:

E = +(R)[l . 1R + 25 z / R - 21. (1.7.8) 96 P. H. Fowler and M. Scharff, cited by Friedlander et al., see reference 93.

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236 1. PARTICLE DETECTION

Here R is measured in gm/cm2 and the factor 9 is close to unity but varies somewhat with emulsion density. Another range-energy relation which is valid within a rather wide energy interval and which was used extensively in the early fifties, may be written in the form:

E = aZzn&fl--nRn. (1.7.9)

1.7.5.3. Range Straggling.* The effect of range straggling on the range-energy relation, due to fluctuations in the rate of energy loss, must be considered if experimental range data is* to be interpreted correctly. Range straggling in homogeneous matter was first studied by B0hr.~6 The problem was treated relativistically by Lindhard and Scharff , 97 who showed that the exact straggling may be obtained from the equation:

where n is the electron density; E the kinetic energy, dE/dR, the mean rate of energy loss, R the mean range, and 0, the mean value of R2. Lewisgs has applied several corrections to the nonrelativistic form of Bohr’s equation without appreciably altering the magnitude of the Bohr straggling effect in emulsion. Furthermore, the difference between the mean range and the most probable range found by Lewis to be caused by a slight skewness of the range distribution, is of negligible magnitude.

Barkas et aLg9 have tabulated the percentage range straggling for protons in emulsions. Inasmuch as both quantities u = Z2ZB/mi1/2 and r = Z2E/mi, where u is a measure of the range straggling and r is a measure of the residual range, depend only on the velocity of the particle, the percentage straggling 100a/r is also a function, albeit slowly varying, of the velocity alone. The quantity (100a/r) = (100mi1’2Z~/R), and hence the percentage straggling, does not depend on the charge of the particle, but varies inversely with the square root of its mass, since varies directly with the mass. According to Barkas, the percentage straggling is greater than 2% for very slow protons and about 1 % for very fast protons. How- ever, the actual range straggling in emulsion is greater than the Bohr effect alone, as a result of straggling due to distortion and inhomogeneity

* Refer to Section 1.7.5.1. 96 N. Bohr, Phil. Mag. [6] SO, 581 (1915). 91 J. Lindhard and M. Scharff, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 27,

98 H. W. Lewis, Phgs. Rev. 86, 20 (1952). 99 W. H. Barkas, F. M. Smith, and W. Birnbaurn, Phys. Rev. 98, 605 (1955).

No. 15 (1953).

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of emulsion, reading errors, and end effects, as has been noted in previous sections.

1.7.5.4 Range-Energy Relation for Multiply Charged Particles.* Residual ranges of a particles of various energies were known even before those of protons, because the former were available a t well-defined ener- gies, emitted from radioactive sources. These experiments are described in references 1-7b. The problem of a-particle ranges was also extensively studied by Wilkins,62 Cuer and Lonchamp,'O' and Neuendorffer et c z L 1 0 2

Wilkins determined the ratio of ranges of a particle and proton of equal velocities by means of a comparison with the like ratio in air (Livingston and Bethesg), and by baking into account the stopping powers of both media. He found that the relation obeyed in emulsion by a particles is given by

(1.7.11)

where the range extension C = 1.5 N for CZ emulsions of density 3.92 gm/cm3.

The excess of the range of multiply charged particles over the range of protons of the same velocity is caused by the occasional capture of orbital electrons from atoms of the traversed medium. This effect becomes im- portant, when the ion velocity is equal or smaller than the orbital velocity of electrons. Consequently during a portion of its trajectory through the emulsion the effective charge of the a particle ZHe < 2. Thus the energy loss is not proportional to Z2, but is given by -dE/dR = Zf(P), wheref(P) depends only on the particle velocity.

The effect of range extension becomes very noticeable for ions of still higher charge, since the ion velocity a t which electron pickup sets in in- creases with higher ion charge. It is usually assumed that the cross section for electron capture and electron loss become comparable when the ion velocity approaches the velocity of the most loosely bound electron, and that the capture cross section increases rapidly and hence the effective ion charge decreases when the ion velocity drops below this value. Knipp and TellerlO3 have calculated the ratio ionic to nuclear charge for slow ions in gases, using range data, available at this time. The authors describe in detail the process of orbital electron capture, basing their cal- culations on the Thomas-Fermi charge distribution, and calculate the effective charge as a function of the ion velocity. I n these calculations

* Refer to Sections 1.1.3.2 and 1.1.3.3. 101 P. Cuer and J. P. Lonchemp, Compt. rend. 232, 1824 (1951). 102 T. A. Neuendorffer, D. R. Inglis, and S. S. Hanna, Phys. Rev. 82, 79 (1951). 103 J. Knipp and E. Teller, Phys. Rev. 69, 659 (1941).

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238 1. PARTICLE DETECTION

enters a parameter, y, assumed to be constant, which has to be determined by comparison with experimentally determined range energy curves.

A semitheoretical study of the range-energy relation of heavy ions in emulsions, based on the calculations of Knipp and Teller was made by WilkimS2 The range-energy relation for heavy ions is given by

(1.7.1 la)

where the range extension CZ(@ is a function of velocity: CZ@) is different for ions of different charge.

Wilkins determined C Z @ ) for various heavy ions in emulsions of density 3.92 gm/cm3 and compared the results with the experimental data of various authors.

Since then a great number of range measurements of heavy ions, Li, Be, B, C, and N in emulsions have been made. The heavy ions used for the purpose of analysis originated from disintegrations of light emulsion nuclei or from boron atoms (boron-loaded emulsions), caused by irradia- tion with particles of well-defined energy. In other experiments carbon or nitrogen ions, recoiling in elastic collisions with monoenergetic beam protons, and finally magnetically analyzed ions, produced in the target of accelerators were used. The latter method was used by Barkaslo4 who has measured the ranges of H and He isotopes and of Li8 and Be8-the last two being easily identified by their decay schemes (“hammer” track). The particles were emitted in the bombardment of a thin target by a! particles from the 184-inch cyclotron. The difference in range extension can be measured for particles of equal velocities. Barkas derives from Knipp and Teller’s work a general relation for the quantity BZ which is defined by the equation r = (Z2R/mi) - B z ; r + B Z is thus a function of velocity only. He assumes the validity of the relation Bz = aZ3 and determines a from experimental data to a = 1.2 X cm. The range extension CZ is then given by CZ = Bz(mi/Z2) . The value of CZ for hydrogen is just equal to BZ = 1.2 X cm and thus evidently a negligible quantity; CZ was determined for a particles Li7, Be9, and C12 ions to Cz = 0.8 p, Cs = 2.4 p, Cq = 4.2 p, and Ca = 8.5 p, where the Cz values of the heavier ions were found with reference to the range-energy relation of a! particles.

A very similar procedure was previously used by Lonchamp,105 while in his later paper106 range-energy relations of heavier ions are calculated with reference to protons. There is furthermore an essential difference in the

104 W. H. Barkas, Phys. Rev. 89, 1019 (1953). 106 J. P. Lonchamp, J. phys. radium 14, 89 (1953). lo8 J. P. Lonchamp, J. phvs. radium 18,239 (1957).

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1.7. PHOTOGRAPHIC EMULSIONS 239

treatment of the relationship of effective charge versus ion velocity. The ratio between effective charge and ion velocity, y, which in the work of Knipp and Teller,’Wilkins and Barkas was assumed to be constant, is now treated as a variable. This assumption is justified, since according to measurements of Reynolds et aZ.lo7 in gases and Reynolds and Zuckerlo8 in emulsions, y decreases with increasing velocity. The experiments were performed with nitrogen ions in the energy interval between 4-28 MeV.

The range-energy relation for various heavy ions in CZ emulsions were investigated by Lonchamplo6 and the corresponding values for range extension were derived; Lonchamp’s values are in general somewhat lower than Barkas’ and Wilkins’ values, with exception of Cz, which was found to be Cz = 1.6 p instead of 0.8 p , given by Barkas. The later paper of Lonchamplo6 is dedicated to the investigation of the range-energy relation of Li7 ions. The ions were emitted from the target of the 184-in. Berkeley cyclotron, and after being deviated by the cyclotron magnet, strike photographic emulsions, situated a t various distances from the target ; each position corresponds to a different radius of curvature and therefore different particle velocity. In a painstaking way the author compares the energy loss in trajectories of protons and Li7 ions, starting in each case from a point of known energy; the values of d E / d X were measured within very small energy intervals. Since the range energy relation for protons is known, one finds the energy loss of Li7 ions according to the relation

(1.7.12)

provided that the velocity of protons and Li7 ions is the same. 2, in the above equation is equal to 1, a t least down to energies larger than 0.32 Mev.lo9 In this way Lonchamp has determined the effective charge of Li7* ions for various energy values, from which in turn the range extension and thus the complete range-energy relation can be derived. Figure 5 gives the range-energy relation for Li7 ions, drawn according to the tabulated values in Lonchamp’s paper. From the data of Li7 the curves for Li8 and Lie ions can be obtained easily, considering the proportionality between mass and residual range for particles of equal charge and velocity. The author compares the calculated data with experimental values of various authors and especially with the more recent range measurements of Livesy,l10 which are in excellent agreement. Livesy discusses in detail the difficulties confronting any theoretical approach to the problem of

107 H. L. Reynolds, J. W. Scott, and A. Zucker, Phys. Rev. 96, 671 (1954). 108 H. L. Reynolds and A. Zucker, Phys. Rev. 96, 393 (1954). 109 T. Hall, Phys. Rev. 79, 504 (1950). 110 D. L. Livesy, Can. J . Phys. 34, 219 (1956).

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240 1. PARTICLE DETECTION

effective charge versus particle velocity. He proposes a semiempirical method in which the range-energy relation is approximated by a power law with coefficients determined from experimental data.

However up to now the range energy relation in emulsions is exactly known only for He and Li ions, while the question of range extension in heavy ions is still unsettled. It is evident that the knowledge of CZ values is especially important for slow ions, if energy determination from range measurements is needed. Problems of this kind arise in investigations of

I00

I

0.1

FIG. 5. Range-energy relation for Li7, Lie, and Lie particles. The energy E is given in Mev and the range in microns.

binding energies and studies of energy levels of disintegrating atoms and calculations of the total energy release in stars caused by the capture and subsequent decay of unstable particles.

Fortunately several heavy ion accelerators were recently built and it is to hope that in the near future a great number of experimental data will become available, so that the problem of velocity dependence of effective ion charges can be solved in a general way.

1.7.6. Ionization Measurements in Emulsions

The problem of energy loss in emulsions as a function of particle charge and velocity has been discussed in a previous section in connection with the range-energy relation in emulsions. The theory is baaed on the Bethe- Bloch relation and modified for high-energy particles to include the

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1.7. PHOTOGRAPHIC EMULSIONS 24 1

relativistic deiisi t y effect . S ternheimer 83, l s l l2 performed detailed calcula- tions of the density effect in various materials, including nuclear emulsions.

Inasmuch as the theory is fully treated in Section 1.1, here we will dis- cuss only Sternheimer’s equation for fast particles in connection with the corresponding ionization parameters in the emulsion. This relation is :

g , = - 1 dE - 2me2W0 PZ ( ) = $, [In Iz + In ~ - 6 1 . (1.7.13) P r i b wo 1 - 02

In this equation, which is valid for singly charged particles heavier than electrons, the energy loss per gram cm-2 is represented as a function of particle velocity, p, and depends on the value of the mean excitation potential I , the polarization effect 6 , and WO. The latter represents the maximum energy transfer, which contributes to the local ionization, limited to the silver halide crystals within the path of the particles; g, does not include the energy spent in the production of fast knock-on electrons (6 rays), whose range in silver bromide exceeds the mean grain diameter. The choice of W O is somewhat arbitrary and estimated values between 2 and 30 kev can be found in various publications; m is the mass of the electron, and A is a constant defined by A = 4?me4/mc2p, where n is the number of electrons per cm3 in the substance; thus A = 0.0698 MeV/ gm cm-2 for Ilford emulsions, and A = 0.0671 Mev/gm cm-2 for AgBr.

The importance of the mean ionization potential I for energy loss cal- culations was already mentioned in the section dealing with the range- energy relations in emulsions. Sternheimer has chosen C a l d ~ e l l ’ s ~ ~ value, I Ei‘ 132. The value of 6 in AgBr is given by

6 = 4.606 loglo(py) - 5.95 + 0.0235[4 - logl~(pr)]4.03 (1.7.14a)

for all values of loglo(py)

and 0.30 < logio(P7) < 4

6 = 4.606 loglo(py) - 5.95 * * 10glo(Pr) < 4. (1.7.14b)

The constants in these equations were calculated by Sternheimer, taking into consideration mean values of excitation potentials of Ag and Br.

According to Eqs. (1.7.13), (1.7.14a), and (1.7.14b) the ionization loss decreases slowly for high-energy particles until a minimum is reached a t 0 = 0.95. For still higher velocities, the second term in Eq. (1.7.13) rapidly increases so that a continuous increase in energy loss would be expected if it were not for the 6 term, which a t very high energies increases proportionally to loglo(0y).

111 R. M. Sternheimer, Phys. Rev. 88, 851 (1952). 111 R. M. Sternheimer, Phys. Rev. 91, 256 (1953).

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242 1. PARTICLE DETECTION

It has been found that the grain density closely follows, within a wide energy range, the energy loss versus velocity curve. The grain density curve goes through a broad minimum, reaching the lowest point a t y = 4 and then increases slowly for still higher energies. The relativistic increase was first observed by Pickup and V o y ~ o d i c ; ~ ~ 3 these authors found that the plateau value, which is reached for y = 50, is about 14% higher than the minimum grain density. Other authors110-118 have more recently confirmed the realtivistic rise in grain density. However, there is still dis- agreement among the various authors about the ratio gPl,t,,u/g,i, as well as the rate of increase of grain density with y beyond the value y = 4, qplateau being the maximum value the grain density reaches for values of y > 4. The experimental results of Stiller and Shapiro"' and Fleming and LordlZ1 seem to be in good agreement with Sternheimer's calculations.

Alexander and Johnston122 have determined the rate of plateau to minimum grain density to be 1.133; the error in these measurements is estimated to be less than 1 %. Actually this investigation was not based on grain densities but on blob densities; the relation between these two parameters will be discussed in the next paragraph.

The authors, furthermore, determine from the ratio of blob densities Bpiateau/Bmio the parameters I and Wo where Bplatellu is the maximum blob density beyond y = 4 and I and Wo have the same meaning as in Eq. (1.7.13). This can be done because the energy loss at the plateau value does not depend on Eo/I but only on Wo

= A[ln 2mc2W0 - 21n(hvp)] (1.7.15)

where v p is the plasma frequency of the mean emulsion nucleus,l13 m the mass of the electron, and A the constant defined in Eq. (1.7.13). On the other hand In(2mc2Wo/12) can be calculated from (1.7.16)

(1.7.16)

118 E. Pickup and L. Voyvodic, Phys. Rev. 80, 89 (1950). 114 A. H. Morrish, Phil. Mag. 171 43, 555 (1952). us A. H. Morrish, Phys. Rev. 91, 425 (1953). llE M. M. Shapiro and B. Stiller, Phys. Rev. 87, 682 (1952). 117 B. Stiller and M. M. Shapiro, Phys. Rev. 171 92, 735 (1953). 11* R. R. Daniel, J. H. Davies, J. H. Mulvey, and D. H. Perkins, Phil. Mag. [7] 43,

'19 M. Danysz, W. 0. Lock, and G. Yekutieli, Nature 169, 364 (1952). 120 R. P. Michaelis and C. E . Violet, Phys. Rev. 90, 723 (1953). 121 J. R. Fleming and J. J. Lord, Phys. Rev. 92, 511 (1953). 122 G. Alexander and R. H. W. Johnston, Nuovo n'mento [lo] 6, 363 (1957).

753 (1952).

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1.7. PHOTOGRAPHIC EMULSIONS 243

where j3 is the particle velocity (determined from the residual range) for which minimum ionization is observed. In these investigations ?r and p

mesons from K,, and K,, decays were used. The authors find for W o the value (2.9 & 0.5) X lo4 ev and for the mean excitation potential I = 12.92; the latter value is in good agreement with Caldwell’s value but differs somewhat from the value found by B a r k a ~ ’ ~ for particles of lower energy. The above value of W O seems high, considering that an electron ejected with this energy produces an easily visible &ray track in the emulsion.

Before analyzing the theoretical basis of various ionization parameters, it is well to discuss experimental det,ails and methods. The measurement of grain density requires oil immersion objectives of 90-1OOx magnifica- tion and eyepieces with carefully calibrated micrometer scales. One counts the number of grains lying within given scale intervals, being care- ful to choose track segments which lie near the center of the field of view. When the trajectory is inclined to the horizontal plane, the true grain density is found by multiplying the measured value by cos a, where cr is the dip angle of the track in the emulsion before development.

It is generally assumed that there exists a simple relationship between the probability of activating an emulsion crystal and the ionization loss of the particle traversing the emulsion.

This relationship can be represented, assuming a (Poissonian) dis- tribution law,

P = 1 - exp( -p~) = 1 - exp( - y) (1.7.17)

where P = n,/nt is the probability of rendering developable nu crystals out of a total number nt crystals. In Eq. (1.7.17) y is given by y = qv2 where q is a parameter which depends only on development conditions, 2 is the mean path length of the ionizing particle through the crystal, and v is the number of ionization acts per crystal; the latter must be identified with the restricted ionization loss, since, otherwise (including 6 rays), would depend on the velocity of the particle activating the grain. Equa- tion (1.7.17) assumes that all crystals have equal size and that their centers are aligned.

If one assumes with Demers2 that the crystals are spheres of equal size, distributed at random about the trajectory of the particle then

] exp (:) - - . (1.7.18)

Fowler and PerkinsIza propose a more general approach accounting for fluctuations in crystal sizes; they assume random distribution of crystals

1 z S P . H. Fowler and D. H. Perkins, Phil. Mag. [7] 46, 587 (1955).

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244 1. PARTICLE DETECTION

and a distribution of crystal sizes in which all diameters between 0 and 22 are equally probable. P is then given by

] (1.7.19) 1 + (I + y)exp(-zy) .

Because of the wide distribution chosen (the distribution of crystal sizes is sharper) Eq. (1.7.18) can accommodate variations in the parameter q, by expressing y = f[(QZ)v].

FIQ. 6. A plot of the relations between mean development probability and restricted ionization loss. The curves refer to Eqs. (1.7.17), (1.7.18), and (1.7.19) in the text. The experimental points are: o experiment A; + experiment B; 0 experimental values which have been corrected for the apparent loss of grains in clusters; X track with dip angle 40", A tracks with dip angle of 30", calculated according to Eq. (1.7.43a).

In Fig. 6, the values of defined by Eqs. (1.7.17), (1.7.18), and (1.7.19) are plotted as functions of y. (The experimental points in this figure will be discussed later.) The three curves are nearly identical €or small values of y, but differ considerably for larger values. From grain density measurements in tracks of lightly ionizing particles it is known that values of normalized grain density (g* = g/go) as a function of ( S / g o ) ? can be easily fitted to the initial part of the 3 curves. However, thus far, it has not been possible to decide which one of the 3 curves, or if in- deed any one of them, does truly represent g* = f[(g/go),] for g* 2 6. The reason for this failure is due to difficulties of performing grain density measurements in dense tracks.

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Grain counting is a simple procedure in tracks of very fast particles, especially if the development is light. In this case the grains are well separated from each other and clogging of grains occurs only occasionally. However, in denser tracks cluster formation is quite frequent and it be- comes difficult to resolve the clusters into single grains. The number of grains per cluster is either estimated by the observer, or determined by the application of length criteria based on the mean grain diameter of developed grains, or finally by investigating the diffraction patterns of grain clusters. However, all these methods are subjective and tedious, leading to the belief that the grain counting method should be replaced by other more objective procedures.

1.7.7. Ionization Parameters: Blob Density, Gap Density, Mean Gap length, and Total Gap Length

At present the general practice is to measure “blob” density instead of grain density, a blob being a single grain as well as a cluster of grains. An advantage of the blob counting method is that i t leads directly to another ionization parameter, the gap ” density, where a gap is the blank space between blobs. However, i t is necessary to emphasize that the grain density, or the number of silver crystals per unit length, made develop- able by the traversing particle, and not the blob density, is directly related to the ionization loss. The blob density depends upon the spacing of crystals in the emulsion and upon the size and configuration of the developed grains, these being influenced by the processing conditions.

As in the case of grain density, one introduces, instead of blob density 3, the normalized value 3*, which refers t o the ratio of actual blob densities to the respective value at minimum ionization, or more often to blob densities a t the plateau value. It has been found that for low ionization densities-singly charged particles of high energy-the normalized blob densities are reasonably independent of development. Therefore, by exposing emulsions to particles in this energy region one can obtain calibration values of B”, which are valid also in emulsions of a different batch or which were processed under slightly different conditions. However, for higher ionization densities the independence of B values of development conditions ceases to be valid, and blob density, therefore, loses its value as an ionization parameter. There is still another reason why blob density measurements are not meaningful in the region of higher ionization density. The blob density increases with increasing ionization loss up to a certain maximum value and then decreases slowly for still higher ionization values; thus in a certain energy interval, blob density is a bivalent function of energy loss.

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246 1. PARTICLE DETECTION

H o d g s ~ n ~ ~ ~ recommended gap length, the total blank space between blobs per unit length, as a useful parameter in ionization measurements in tracks with blob densities larger than 2 times plateau value. Shortly afterward Renardier and AvignonlZ6 modified this technique by counting, instead of measuring, the total number of gaps per unit length. Since that time gap measurements in a variety of forms have played an important role in nuclear emulsion problems. Gap measurements are now generally used instead of, or in combination with, blob density measurements if the ionization exceeds 2 times the minimum value or when light but steeply dipping tracks have to be measured.

The following parameters are used in ionization measurements :

(a) blob density B, the number of blobs per unit length; (b) H , the number of gaps per unit length; H is, of course, equal to B,

(c) L H , gap length or the total width of gaps per unit length; (d) A, mean gap length, defined as the mean distance between the

the number of blobs per unit length;

inside edges of neighbor grains.

The last parameter was proposed by O'Ceallaigh. lZ6

Ritson127 has devised a very convenient method for gap measurements : The track is aligned in the microscope with one of the two stage move- ments which in turn is driven by a motor a t a low constant speed. The observer is provided with two counters, driven by a single pulser. One of the counters runs continuously with the stage movement, while the other one is activated only when the observer presses a button which is done whenever the hairline in the eyepiece lies over a region of track unoccupied by grains. The ratio of the two counter readings gives directly the total gap length in the track under investigation or the gap length LH per unit length if counter readings are made at certain predetermined time intervals. Baroni and CastagnolilZ8 describe a similar arrangement, using a motor-driven stage moving with constant velocity. The authors add several improvements, the most important being an RC circuit connected with a counting device which enumerates only those gaps which are longer than a certain predetermined minimum value. Therefore, one can separately record, with this apparatus, the total number of gaps, the number of gaps longer than a minimum value I, the total width of gaps

I z 4 P. E. Hodgson, Phil. Mag. [7] 41, 725 (1950). ' 2 6 M. Renardier and Y. Avignon, Compt. rend. 233, 393 (1951). IaSC. O'Ceallaigh, Proc. Intern. Union Pure and Appl. Phys. Conf. on Cosmic

Radiation, BagnBres, France, 1953 (unpublished). l27 D. M. Ritson, Phys. Rev. 91, 1572 (1953). l28 G. Baroni and C. Castagnoli, Nuovo cimento [9] 12, Suppl. No. 2, p. 364 (1954).

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per unit length and the total width of gaps surpassing a minimum length. The introduction of a minimum gap length is connected with the findings of various authors that the inclusion of gaps below a certain minimum length in gap measurements makes the measurements unnecessarily tedious and dependent on observational errors and optical conditions (resolution). Therefore, only those gaps are included whose widths exceed a certain minimum value 1 which should be large enough for accurate and rapid measurements, but small enough to prevent loss of information, especially in tracks of high ionization density. The choice of 1 depends upon the problem, the length, density, and dip angle of the track. The corre- sponding parameters referring to data above a minimum value 1 will be denoted in the following expressions: H(1), X(I), and A H @ ) . For tracks in- clined to the emulsion surface the optimum value of gap length 1 is given by 1 = I’ sec a, where (Y is the dip angle and 1‘ the minimum value corre- sponding to a track with equal gap density, but negligible dip angle.

In the following paragraphs we will discuss the relationships which exist among blob density, total gap length, mean gap length, and grain density. Grain density, as pointed out earlier, is a direct measure of the energy loss of the traversing particle; i t depends, of course, also upon the processing conditions; however, the dependence is of such a nature that the nor- malized value g* becomes independent of development conditions and so represents, for a given emulsion type, the true ionization parameter.

The normalized blob density B* is, as stated before, also reasonably independent of development in tracks of low ionization density; this is due t o the fact that only a small percentage of grains will coalesce to greater complexes. However, for higher ionization densities, the process of cluster formation becomes more and more important and the degree of coalescence depends on the strength of development which determines the final size to which the crystals grow during processing. The relationships between ionization loss and blob density or any other parameter con- nected with gap measurement, are much more complicated than in the case of grain density. An analytic expression for the former relationship must be based on a theory of track formation in the emulsion. Various a~thors123~12~-1a7 have worked on this problem and several models describ-

139 C. O’Ceallaigh, Bureau of Standards Document No. 11, CERN, Geneva (1954). 130 M. G. K. Menon and C. O’Ceallaigh, Proc. Roy. Soc. A221,292 (1954). 131 R. H. W. Johnston and C. O’Ceallaigh, Phil. Mag. [7] 46, 424 (1954). 132 M. Della Corte, M. Ramat, and L. Ronchi, Nuovo cimento [9] 10, 509, 958 (1954). 133 M. Della Corte, Nuovo cimento [9] 12, 28 (1954). 134 M. W. Happ, T. E. Hull, and A. H. Morrish, Can. J . Phys. SO, 669 (1952). 1 3 5 A. J. Hem and G. Davis, Australian J . Phys. 8, 129 (1955). 136 J. M. Blatt, Australian J . Phys. 8, 248 (1955). 137 C. Castagnoli, G. Cortini, and A. Manfredini, Nucwo cimento [lo] 2, 301 (1955).

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248 1. PARTICLE DETECTION

ing this process were proposed. The essential difference between these models lies in the treatment of spatial distribution of silver crystals.

O’Ceallaigh’s work126~12p-1s1 is based on the assumption, shared by Happ et al.,la4 that there exists no correlation between the position of silver crystals in the emulsion, while Herz and Davis136 visualize the emulsion as a lattice with silver crystals spaced a t regular intervals; other authors use somewhat modified models supporting their assumptions by comparison with experimental data. BlattL3‘j and Fowler and PerkinslZ3 discuss in great detail the theory of track formation and the importance of the theoretical assumptions for the practical use of emulsions in ionization measurements. In comparing their theories with experimental data the authors arrive at different conclusions, Blatt supporting a modified con- stant spacing model, while according to Fowler and Perkins’ findings, agreement between experiment and theory can be obtained with a variable spacing model; the authors introduce certain refinements in the earlier theory which are connected with fluctuations occurring in the distribution of crystal positions, size, and developability.

For low ionization densities all theories give satisfactory agreement with experiments, due to the fact that the blob separation in lightly ionized tracks is not greatly influenced by any type of fluctuation, and exceeds greatly such distance8 as required in the constant spacing model.

We will discuss here first O’Ceallaigh’s work based on the simplest model. O’Cealla.igh126~12v and Menon and O’Ceallaigh130 have shown that the distribution in the length of gaps along tracks is exponential and given by

(1.7.20)

where h is the mean gap length. The connection between mean gap length A, grain density g, and total gap length LH, is given by the equation

(1.7.21)

where (Y is the mean diameter of the silver halide crystal.’30 During development the grain increases in size and reaches finally a diameter which is about twice the size of the original value. Therefore, the free space between grains becomes smaller, and some free spaces may dis- appear completely while the grains coalesce and form larger complexes. However, because of the exponential distribution, the homogeneous growth of all grains does not change the mean gap length A; the latter is, therefore, a parameter which is largely independent of the diameter of the developed grains and hence of developing conditions. The model is based

1 9 - - a! = x and LH = 1 - ag

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on the assumption of a random distribution of crystal centers along the track.

The blob density B which is equal to the number of gaps H ( 0 ) can then be written

B = gexp(--) (1.7.22)

where K is a constant, depending strongly on developing conditions, must be determined for each stack and i t is related to the crystal diameter, a, by the relation K = d - a + e, where d is the diameter of the developed grain and e is the smallest distance between grains which can be resolved microscopically. K can be found by the simultaneous measurement of B and X in track segments of constant ionization. The expression exp( - K/X) represents the conversion factor for grains into blobs, which for constant energy loss depends only upon developing conditions.

Alexander and JohnstonlZ2 develop a relationship which allows the conversion of normalized blob densities from one stack to another (different development); however, this relation is limited to tracks of low densities for reasons previously mentioned.

In this case one can replace B* = (B/BO) by

B* = g* exp[-K(g - go)] = g* exp[-K(B - Bo)] (1.7.23)

where go and Bo refer to the values of grain and bIob density in the plateau or minimum ionization region. If now K1 and K z are the developing con- stants for two different stacks, it can be easily seen that the two blob densities B1* and Bz* are related by

B1* = Bz* exp([Kz(Bz - B231 - [K1(B1 - BIO)]). (1.7.24)

The procedure of finding the mean gap length can be greatly simplified by using the method proposed by Johnston and O’Cealleaigh.’31 The authors propose to count the number of gaps H(I1) and H(E2), exceeding two predetermined lengths l1 and l z ; these data are related to each other and the mean gap length by

(1.7.25)

With a set of 4 t o 5 different values of I , the mean gap length X can be quite accurately determined. Such measurements, performed with con- ventional methods (microscope and eyepiece scale), ar’e lengthy and time consuming. This difficulty can be removed by the use of “gap analyzers” ;

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250 1. PARTICLE DETECTION

recently a number of such devices have been c o n ~ t r u c t e d . ~ ~ * - ' ~ ~ The principle of such gap analyzers is a system of counter circuits, which are actuated by the closing of a microswitch; the microswitch is closed by a number of equidistant contacts on a motor-driven disk; if the system is actuated by photoelectric means, then the contacts are replaced by holes in the disk. In this way the gap lengths are chopped into increments which are determined by the constants of the quantizer and the gaps are meas- ured in subunits of time, in a time dependent circuit, or in subunits of length if the stage and disk are driven by synchro motors.

Another method for the measurement of X was proposed by Fowler and perk in^,'^^ in which blob measurements are combined with the measure- ments of gaps exceeding a predetermined length I

H ( I ) = B exp (- $ a (1.7.26)

This method has the advantage, that i t can be performed with conven- tional methods, fixed eyepiece scale and manually driven stage, especially if a conveniently large value of I is chosen; however, one is likely to lose information if E is too large since, especially in denser tracks, the number of small gaps is large and thus important for the measurement. The authors have calculated the optimum gap length A, which, of course, depends on the track density. They found a relatively broad optimum region between 1.5X < 1 < 2.5X. A great advantage of mean gap length measurements by (1.7.25) or (1.7.26) is, that these methods can be easily adapted for dipping tracks. Mean gap length measurements, instead of blob measurements alone, should be made, even in very light tracks, if the dip angle exceeds 15" (developed e m ~ l s i o n ) . * ~ ~ - ' ~ ~ The measured distance I in (1.7.25) or (1.7.26) is the projected length and has to be replaced by I t which is related to I and dip angle a by

It = I sec a + d(sec a - 1) (1.7.27)

where d is grain diameter and d(sec a - 1) accounts for the obscuration due to the apparent increase of blob sizes.

The "smooth model" by O'Cealleigh neglects any type of correlation, which may exist among the positions of silver crystals in the emulsion. In

la* J. E. Hopper and M. Scharff, Bureau of Standards Document No. 12, CERN, Geneva (1954).

la8 K. Enstein, Electronic Eng. 29, 277 (1957). 140 A. DeMarco, R. Sanna, and G . Tomasini, Nuovo cimento [8] 9, 524 (1928). 141 S. C. Bloch, Rev. Sci. Instr. 29, 789 (1958). 144 M. Della Corte, Nuovo cimento [lo] 12, 28 (1959). 148 H. Winzeler, Nuovo cimento [lo] 4, Suppl., p. 259 (1956). 14( R. C. Kumar, Nuovo cimento [lo] 6, 757 (1957).

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1.7. PHOTOGRAPHIC EMULSIONS 251

fact, such a correlation may be of no consequence in light tracks with widely separated grains; however, it will become important in dense tracks.

Fowler and PerkinsIz3 assume that the gaps, found in very dense (saturated) tracks, can be identified with the gelatine gaps existing be- tween neighboring crystals. Furthermore, they assume (based on experi- ments) that the gap length distribution is exponential, but cuts off sharply a t a distance-lco-equal to the mean crystal diameter. The distri- bution function of gaps between developed grains is found in the following way. If u is the distance between the grain centers of two grains, bounding a gap of n undeveloped crystals, then the frequency with which a gap can appear will depend on the product a(n)F,(u) du. ~ ( n ) = p2(1 - p ) n is the probability that one grain is developed and followed by an undeveloped and finally another developed grain. The function Fn(u) is defined by

Wn Fn(u) = - exp(-W)

n! (1.7.28)

where W = (u/ko) - (u + 1)2 is the path length through gelatine, ex- pressed in units of lco.

The total differential distribution, or the number of gaps per unit length between u and u + du, is obtained by summation over all values of n, and is given by

The calculation of the integral distribution is quite complicated, be- cause the sum appearing in (1.7.29) cannot be simplified. Numerical evaluation shows for small grain separations a certain degree of roughness in the distribution curve; which strongly depends on the choice of the numerical values for LO and 2 . However, for grain separations (center to center) greater than about 0.8 p and reasonable assumption for the values of Lo and 2, the curves appear smooth. Fowler and Perkins assumed LO = 2 = 0.2 p. In general i t will be possible to describe the total integral gap length by

(1.7.30)

which is an asymptotic approximation of the distribution function for u -+ a. In this equation X is defined by

(1 - $)exp(- :) = 1 - F. (1.7.31)

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252 1. PARTICLE DETECTION

In one assumes that the asymptotic distribution form is also valid for the calculation of blob densities and if u’ is the distance between neigh- boring crystal centers, then B is given by

1 l l k o 5k03 B = +(u’P) = -exp ( - :) - [ 1 + 4 8 ~ + m3 * * * (1.7.32) X

where the series in the squared parenthesis converges rapidly and reaches a maximum value for ( k o / X ) = 1. Thus far we have neglected to take into account fluctuations in the size of the developed grains in the distribution function. The existence of such fluctuations can be easily observed; Fowler and Perkins showed that the distribution function is Gaussian and determined the standard deviation t o be 0.14 p or 20%, with a mean grain diameter of 0.7 p. They then calculated a correction factor C = [ l f (p2/4X2)] by which both blob and gap densities have t o be multiplied in order to account for fluctuations in the size of the developed grain. It is easily seen that this correction factor is negligible for light tracks.

Fowler and perk in^'^^ have plotted 1 / X as a function of residual range for proton and pion tracks in two sets of emulsions which were developed in different ways. The curves for both emulsions coincide over a wide range of energy, down to about 100 Mev (protons) ; but from there on the curves diverge considerably and hence A * in this region cannot be con- sidered as a parameter which is independent of development conditions; A * = X/Xu, where Xo is the mean gap length found in tracks of high-energy electrons; X* was determined from blob density in the region of small energy loss and in the denser region from blob and gap measurements [Eq. (7.1.26)].

The constant spacing model of Herz and Davis (H-D) is based on the following assumptions: the crystals in the emulsion are arranged in a definite lattice, so that each silver crystal can be assumed to lie within a cell of certain small dimensions. Therefore, there exists a strong correla- tion between crystal positions, which will affect the results in the dense region of the track; however, in light tracks where grain spacing is usually many times the length of the small unit cells, the correlation will cease to be effective and the (H-D) model leads to the same results as the two previously discussed models. For very high densities the gap length will not reduce to zero, but will assume a constant value which is deter- mined by the original lattice spacing and the dimensions of the developed grain. The grain size is assumed to have a constant value (grain size fluctuations are neglected) and the model has been calculated in “linear approximation” (the crystal centers are assumed to be aligned along the path of the particle). The intrinsic difference in the calculations of the

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various parameters of the (H-D) model in comparison with other models stems from the assumption of a discrete gap length distribution.

In the following we call b the length of the unit cell containing one crystal and d the diameter of the developed grain; the zero gap length is then given by Go = b(I’ + 1) - d, where r is the largest integer less than d/b , and it is the integral part of a factor determining the growth of the crystal during development. The size of the next gap (first-order gap) will be G = b(I’ + 2) - d and the probability of finding an n th order gap is given by ~ ( n ) = P z ( l - P)(r+n-l). One can now calculate the integral gap length distribution and the various track parameters exactly as in the case of other models. Although the problem was calculated rigorously by Blatt,136 we indicate here only the results in the simplified form used by Castagnoli et ~ 1 . l ~ ’ The following equations (1.7.33-1.7.35) refer to blob density B = H(O) , the number of gaps larger than 1 per unit length H(Z), and, the mean gap length X :

where rl is the integral part of (d + Z)/b and

(1.7.33)

(1.7.34)

( 1.7.35)

The expression for X in (32) becomes identical with O’Cealleigh’s value for P << 1 and X >> b. The total gap length LH per unit length is equal t o the product B X X.

Blatt has compared the O’Ceallaigh and (H-D) models and various other slightly modified models with experimental data, and has found good agreement with the predictions of the (H-D) model. Castagnoli et al. and more recently O’Brien, 146 based their investigation of the (H-D) model on a large number of experimental data. Castagnoli et al. measured about 100 particle tracks in an energy region from 600 Mev (protons) down to very small energies using the gap counting machine, previously mentioned.

Equations (1.7.33)-(1.7.35) show that the various parameters are related to each other through the probability P . In order t o calculate P one has to know the mean grain diameter d, which can be found by direct measurements and b and r. The unit cell is not strictly defined; it is at least the size of the mean crystal diameter and can probably be identified with 8 = ko + X where 8 is the projected mean distance between centers of successive crystals as defined by Fowler and Perkins. 12* Castagnoli

146 B. T. O’Brien, Nuovo cimento [lo] 7, 147 (1958).

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254 1. PARTICLE DETECTION

et aZ. define b by b = 1/0.455($a), where a is the mean crystal diameter and 0.455 the fraction of emulsion volume occupied by silver bromide. Finally r, which by definition is an integer number, can for experimental reasons assume only values between 1 and 3. Although neither b nor I’ are exactly known, one can determine these magnitudes from the following considerations. Only certain pairs of values of b and r are compatible with the observed grain diameter and the crystal diameter known to be be- tween 0.2-0.3 p. The best combination of b and I? can be found by compar- ing calculated and observed values at the maximum of the B versus P curve, since B,, = ( l / b ) [ l / ( l + r)][l - 1/(1 + I?)]’ depends only on b and y. The value of r, of course, is development dependent, and must be determined for each set of emulsions. With the values of r and b found at the blob density maximum, B versus X curves were calculated and com- pared with experimental data. The authors found good agreement between experimental and calculated curves within a wide region of ionization, starting with near minimum densities down to values of about 8 times minimum value.

O’Brien145 tested the (H-D) model for relativistic heavy primaries in G-5 emulsions which were very lightly developed; he found good agree- ment for light and medium tracks, but disagreement for very dense tracks (relativistic Fe nuclei).

Figure 7 (Castagnoli et ~ 1 . ~ ~ 7 ) is a plot of the measuring parameters B, A, LH, and H(Z) per 200 track length versus R / m = r, where m is the electron mass and R the residual range measured in microns; the range of a particle with a mass mi times the electron mass is then given by r X mi. It can be seen that for dense tracks (small values of r ) the slope of the LH curve is steep (contrary to B and X curves) and therefore, in this region, L H

is sensitive to changes in ionization loss. Furthermore, it has been found, that LH in the dense region is less subject to personal errors than other parameters, since the occasional omission of small gaps in the determina- tion of the total gap length does not seriously affect the results. In the light region the slope of the Lcr curve decreases and LH ceases to be sensi- tive to changes in ionization loss; in this region the best parameters are R and A, the latter having the additional advantage of being independent of development conditions. The role of the parameter H(1) (the curve refers to 1 = 0.8 p ) has been discussed before.

It must be emphasized that in the (H-D) model the parameter X is not, as in the (F-P) model, derived from the slope of the gap distribution curve. but is found from the ratio L H / B . The experimental values should be identical in both cases if the gap length distribution were truly exponential However, experiments show that this is only approximately true for light tracks, but certainly not for medium and heavy tracks. The discrepancy

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2.0 0.5 1.0

0.2

between the two X values is quite serious, because Fowler and Perkins (and subsequently many other authors) express the ionization loss in tracks by the parameter l / X * . In general, the value 1 / X derived from the slope of the gap distribution curve is larger than the value found by L H / B , and consequently the values of the probability derived from Eqs. (1.7.31) and (1.7.35) will be quite different when tracks of medium or large densities are considered.

H(l)/200p

- _/_------- - - k(microns)

-

200 1 I

100 1 100 - 50 -

0/200p

20 - 10

5.0

The situation is rather complicated for the following reasons. The ionization parameters proposed thus far are applicable only within certain density intervals, and furthermore, their mutual interdependence as well as their relationship to are based on assumptions (models) and emulsion parameters [Eqs. (1.7.31) and (1.7.35)] which cannot be verified in a simple experimental way. While the exact values of these emulsion parameters are of no consequence in light tracks, they are of great im- portance for tracks of medium and large density. Thus the problem of

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256 1. PARTICLE DETECTION

ionization measurements in the latter regions has not yet been solved satisfactorily, even though this is an important region for the identifica- tion of slow singly charged, and multiply charged particles of all velocities.

100 90 00 70 60

50

40

30

20 - 8 u - - - I

10 9 8 7 6

5

4

3

2

I \ , 0 1 2 3 4 5 6

GAP LENGTH (IN MICRONS)

FIQ. 8. Gap length distribution curves (normalized). Experiment A: pion, 1.3 Bev/c. Experiment B: proton, R = 9300 p.

The discrepancy between the X values, measured by different methods can be partly understood from observations recently published by Cortini et ~ 1 . ' ~ ~ The authors found that the gap length distribution deviates from an exponential curve for small gap lengths, an observation which was confirmed by other authors.

Figure 8 gives the semilogarithmic plot for a pion with momentum I46 G. Cortini, G. Luzzatto, G . Tomasini, and A. Manfredini, Nuovo cimento [lo] 9,

706 (1958).

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1.3 Bev/c and a proton of 9 mm residual range and it can be easily seen that the curves deviate from a straight line for gap lengths 1 5 0.5 p. If h is calculated from the straight line (F-P), the result will be smaller than the value found from h = LH/B and consequently the parameter 1/X in the (F-P) model becomes too large. This effect will be more pronounced, the denser the track, since the fraction of small gaps (which are neglected in the slope measurements) will increase with ionization loss. Cortini et ul.l4* explain their observations as the result of a development effect and propose a modification of Eqs. (1.7.26) to (1.7.36)

1 - F ( Z ) H ( Z ) = B exp [ - (1.7.36)

where 6 is a function, which is zero at 1 = 0 and increases to a constant value a t the gap length, where (1.7.36) becomes exponential in 1.

If h is defined by Eq. (1.7.36), then evidently Eq. (1.7.31) which relates h and P is no longer valid, but must be replaced by another relation which now contains in addition to ko and x still another parameter 6. The exact value of 6 is difficult to determine, since it involves distances, which are close to the limit of optical resolution.

Because of the ambiguities contained in the use of the parameter A, another more empirical approach to the ionization problem has been attempted, based on the use of a parameter which is more directly related to grain density.ld7 The method is based on the blob length distri- bution; the importance of the latter has already been pointed out by Della Corte132-133 who investigated the differential blob length distribution in tracks of various density. The differential distribution curves consist of a distinct Gaussian part and a tail, which later becomes more important with increasing ionization.

The integral blob length distribution curves for tracks of various densities are plotted in Fig. 9. (The curves refer to measurements in two different experiments A and B.) The number of blobs B(Z) as a function of blob length decreases exponentially with increasing blob length, except for the initial part of the curve. The curved part in the integral distribu- tion reflects the Gaussian part of the differential distribution. It is reason- able to assume that all blobs falling within the Gaussian part are single grains, while all blobs outside the Gaussian part are clusters, composed of at least two grains. In the following the density of single grains will be denoted by N , and the cluster density by N,. The dividing point between grains and clusters can be determined from the shape of differential and

147 M. Blau, S. C. Bloch, C. F. Carter, and A. Perlmutter, Rev. Sci. Instr. 31, 289 (1960).

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258 1. PARTICLE DETECTION

integral distribution curve. If S is the abscissa of the dividing point, then all blobs with 1 5 S are considered to be single grains while all blobs with 1 > S are clusters with a mean cluster length of w = A + S, if A is the reciprocal slope of the exponential curve. The percentage a of blobs, being single grains ( S 5 I) decreases with increasing ionization loss, while A and therefore, w increases (see Figs. 10a, 10b and 1Oc). Since the

FIG. 9. Integrated blob length distribution curve; the blob length is given in microns. The curves represent measurements on trajectories of the following particles. Experi- ment A: A 1 - proton, Reff = 1890 p; A B - proton, Reff = 1.5 cm; A3 - proton, R,rr = 3.66 cm; A , - pion with equivalent proton range, R = 17.7 cm. Experiment B: I?, - antiproton, Ref{ = 1.01 cm; Bf - antiproton, R,ff = 1.45 cm; B3 - 680 Mev/c pion. The curves were fitted to the experimental points by the method of least squares. The errors shown are statistical.

exponential behavior of the curves is probably related to the randomness of the agglomeration process, one can assume that w is a function of the number of grains contained in a cluster.

In order to evaluate the blob length distribution curves certain assump- tions have to be made:

The first assumption is connected with the number of grains per cluster in minimum tracks and it was assumed that the mean cluster length w o corresponds to just two grains per cluster; w o is smaller than 2d, where d

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is the mean grain diameter found from the Gaussian distribution. I n the minimum case the size of the second grain is defined by d, = wo - d, which is less than d, due to geometrical conditions or reasons connected with development conditions.

It is furthermore assumed that the number of grains per cluster, g,, in denser tracks with w > W O is given by

w - d ge = 1 + -*

w o - d (1.7.37)

This assumption seems to be reasonable, as a first approximation, at least in tracks of medium density. Linearity of the value of g, - 1 as a function of w - d up to P = 0.6 follows for instance from the (H-D) model, using parameters /3 = 0.35, I' = 2, and d = 0.7 p . However, i t is clear that the linearity relation will break down for large densities for a variety of reasons connected with the spatial distribution of silver crystals as well as with phenomena appearing in the saturation region. Therefore, relation (1.7.37) is no longer valid a t high ionization densities and a certain value of w might correspond to a larger number of grains than expected from the proportionality relation; the number of apparent grains per cluster is too small, because a certain number of grains are not efficiently observed, while others, due to saturation processes, are not efficiently produced. This statement is equivalent to the assumption that the grain diameter in clusters d, decreases with increasing ionization and this effect was considered tentatively by writing

d, = (w, - d) exp - - ( 3 (1.7.38)

The ratio in the exponent is the ratio of the mean blob length to the free space in a measuring cell. The procedure of evaluating the data is as follows. First one determines in minimum tracks the number of blobs B and the total blob length LB per unit cell, while the mean blob length A in minimum tracks is obtained from the integral blob length distribution, and the mean grain diameter from the differential distribution. (The use of a blob length analyzer increases considerably the measuring speed.) From the measurements one obtains readily the following equations :

and LB, = Bo[aod + (1 - ao)wo]

no, - - Bo[ao + (1 - ad21

( 1.7.39)

( 1.7.40)

based on the assumption that clusters in minimum tracks contain just two grains.

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260 1. PARTICLE DETECTION

RESTRICTED IONIZATION LOSS

FIG. 10(a). Experimental values of function a, determining the percentage of blobs which are considered to be grains.

I I 2 4 6 0 10 20

RESTRICTED IONIZATION LOSS

.I 4

Fro. lO(b). Experimental values of the reciprocal slope A of the exponential part of the integral blob length distribution. The dimensions of A are microns.

For tracks heavier than minimum ionization one uses the equation

LB = B[ad + (1 - a).] (1.7.41)

and because of

(1.7.42)

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one finds the total number of grains as given by

(1.7.43)

Finally, in very dense tracks, the number of grains per cluster gc has to be corrected in accordance with Eq. (1.7.38).

Equations (1.7.41) and (1.7.43) have to be modified, if tracks with dip angle >20° are analyzed, because all the lengths measured by the analyzer are projected lengths, which have to be converted in true lengths.

i oJ I 1 2 4 6 8 10 20

RESTRICTED IONIZATION LOSS

FIG. 10(c). Experimental values of the total blob length L g per unit length in microns per 100 microns. The abcissas for all three curves are restricted ionization loss. The curves were drawn as best fits to the experimental points. The symbol 0 refers t o experiment A, + to experiment B.

This can be accomplished by multiplying the number of blobs by cos 0 and the blob lengths I' by sec 8. However if the integrated blob length distribution is plotted versus 1' sec 8, one would measure each blob too long by an amount of d(sec e - 1); (see Della Corte133). Therefore the abscissa of the integrated distribution, from which the inverse slope is determined should be chosen to be 1' sec e - d(sec e - 1). Applying this correction Eq. (1.7.43) becomes

w - d

The triangles and crosses in Fig. 6 represent dippingItracks of 40" and

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262 1. PARTICLE DETECTION

30" dip angle calculated in this way. The formula probably gives a value which is somewhat too high because of the possibility of including some single grains within the clusters.

There are several reasons why the parameter n, is preferable to others, mentioned before. no is a sensitive parameter in the entire range of ionization. This can be seen from Figs. 10a, b, and c, representing the plot of the parameters: -a-, -A-, and L g as functions of the restricted ionization loss (cut off value 5 kev). For near minimum tracks the parameter -B- (not plotted here) increases rapidly with ionization loss, and, since -a- is near unity, while A and therefore w is small, the value of n, is essentially blob density which here is nearly equal to grain density. With increasing ionization loss -a- decreases rapidly while -A- increases, so that the second term becomes more important. Even in the dense region some of the parameters, used in (1.7.41) and (1.7.43) are still sensitive to changes in ionization, so that n, is a sensitive parameter in the entire range. In the dense region subjective errors in blob density measurements might be considerable, but this fact does not affect the results greatly because the mean blob length, measured simultaneously by blob length analyzer compensat.es the error, as can be seen from Eqs. (1.7.41) and (1.7.43); since in dense tracks the value of -a- is small, the compensation will not suffer through fluctuations in the value of -a-.

If the parameter no, defined by Eq. (1.7.43), is really identical with the true grain density, then no must be equal to n, = Pn,; where n1 is the total number of developable crystals per unit length and P is the prob- ability of development defined by Eqs. (1.7.17) to (1.7.19). Therefore, n,, calculated from Eq. (1.7.43), must be a function of y = QZV, where u,

the number of ionization acts per unit length, can be identified with the restricted ionization loss 4,. The experimental points in Fig. 6 represent experimental values found in two different experiments. Since the abso- lute value of y = q f u is not known, the value of y for the minimum track was found by normalization to the theoretical curve, while all other points were plotted by using the experimental values P = n,/nr and for y the value of the minimum track multiplied by g/go, (ratio of re- stricted ionization values; cut off at 5 kev).

The points (0 . . . experiment A, + + * * experiment B) lie fairly close to the curve within the error limits, with exception of the points referring to dipping tracks (A * . 40" dip angle, X . * - 30" dip angle) which have been calculated according to Eq. (1.7.43a) and the very dense region ( 0 ) which were corrected for the apparent loss of grains according to Eq. (1.7.38). The fact that measurements from two different experiments show agreement with a single theoretical curve indicates that n, is a parameter which is independent of development.

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1.7. PHOTOGRAPHIC EMULSIONS 263

The problem of finding a suitable model is by no means trivial; there is an urgent need for the standardization of ionization parameters in the medium and dense region. The present situation is quite involved, because the results of various authors do depend not only on the method of measurements, but also on the model used for the evaluation of experi- mental data. This dependence extends still farther to the problem of error evaluation. For instance, in O’Ceallaigh’s model, which assumes random distribution of crystals along the track, the standard deviation is simply given by the square root of the numbers counted, provided that sys- tematic errors and irregularities in the emulsion constants can be neg- lected. For other models, however, the problem is more involved, because then, one has to account not only for the statistical fluctuations of the parameters but also for the correlation between the fluctuations of each of the parameters involved.

The error problem in emulsion work was studied in great detail by various authors. 1 2 3 . 1 3 6 v 1 4 6 , 1 4 8 Blatt introduces fluctuation parameters for the evaluation of errors; the parameters are defined as the ratio of stand- ard deviation to the square root of the mean value of the respective parameters measured. Blatt’s fluctuation theory was experimentally verified by O’Brien for the case of the (H-D) model.

In connection with the problem of heavily ionizing particles, which includes particles near the end of their range, Alvial et ~ 1 . ~ 4 ~ have recently proposed another method. This method is based on the measurement of track profiles; the tracks are projected on a screen and the distances of both borders of the track from a fiducial line are measured. The authors found in a preliminary investigation that the thickness of tracks for slow particles depends on the velocity of the particle and reaches a maximum near p = 0.1. The increase in thickness and the maximum is explained as being caused by short 6 rays, ionizing crystals in the immediate neigh- borhood of the tracks; for p 7 0.1 the track width diminishes, because the range of the emitted 6 rays is too short to reach crystals of the main track. If further measurements should confirm that the position of the maximum can be easily established, the method would be very valuable for mass determination of slow particles, since the position of the maxi- mum varies with particle mass. Unfortunately it seems that the observa- bility of the maximum depends on developing conditions and on the dip of the track; therefore, the measurement of each individual case affords a great amount of precalibration work.

148 G. Lovera, Nuovo cimento [lo] 8, 1476 (1956). 149 G. Alvial, A. Bonetti, C. C. Dilworth, M. Ladu, J. Morgan, and G. Occhialini,

Nuovo cimento [lo] Suppl. No. 4, 244 (1956).

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264 1. PARTICLE DETECTION

1.7.8. Photoelectric Method

In photoelectric methods the observer’s eye is replaced by photoelectric cells. In all measurements photomultiplier cells are used, and the image of the track is projected through a slit, in the image plane of the eyepiece of the microscope, onto the sensitive screen of the photocell. In most photoelectric devices the stage is motor driven and the output of the photocell is recorded, so that the function of the observer is limited to the task of keeping the track image in focus and in the center of the slit. The width of the slit is usually 2-3 p ; it should not be wider, because otherwise the photocell sees and records in addition to the track segment a considerable amount of the background. However, the slit should not be too narrow, because then the adjustment of the track within the slit becomes difficult; furthermore the slit must be wide enough to accommo- date also tracks of slow particles, which frequently suffer deviations from the initial direction. The length of the slit is determined by the purpose of the measurements. With short slits and high magnification the photo- cell acts as a blob and gap counter; in this case the measurements are relatively independent from background conditions and the depth posi- tion of the track. For long slits the photocell acts more as a densitometer, covering, but not resolving larger segments of the track; it is clear that this type of measurement depends greatly on the background and position of the track and it becomes necessary to perform accurate background measurements in order to evaluate the net photoelectric effect arising from the track alone. The length of the slit is somewhat limited by the dip of the track, because during the measurement the whole exposed track segment must be kept in focus. There is still another type of photoelectric measurement in which the profile of a track is measured by sweeping out the cross section of a track with oscillating prism or mirrors.

A great number of authors have performed photoelectric measurements and details of measurements and devices used are given in the following papers.lS0-l68 We will come back to this section in the chapter on mass measurements and will describe in greater detail the work of the Lund school, where the method has been applied with great success and has been constantly improved.

The author warmly thanks Dr. A. Perlmutter for reading the manu- script and suggesting many style improvements.

l l r0 M. Blau, R. Rudin, and M. Lindenbaum, Rev. Sci. Znstr. 21, 978 (1950). 161 S. yon Friesen and K. Kristiansson, Arkiv Fysik 4, 505 (1951). 152 M. Ceccarelli and G. T. Zorn, Phil. Mag. [7] 43, 356 (1952). 153 C. Kayas and 0. Morellet, Comp. rend. 234, 1359 (1952). 154 P. Demers and R. Mathieu, Can. J . Phys. 31, 97 (1953). 15.5 L. Van Rossum, Comp. rend. 236, 2234 (1953). 156 M. Della Corte and M. Ramat. Nuovo cirnento [9] 9, 605 (1952). 157 M. Della Corte, Nuovo cimento [lo] 4, 1565 (1956). 168 P. C. Bizzeti and M. Della Corte, Nuouo cimento [lo] 7 , 231 (1958).

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1.8. Special Detectors

1 A.1. The Semiconductor Detector*

1.8.1.1. Introduction. The junction region of a reversed-biased p-n diode is essentially a solid state version of the conventional gaseous ionization chamber. For the purpose of analogy, it is well to recall the essential features of a gas-filled ion chamber which are shown schematic- ally in Fig. 1. The region between the plates of a charged parallel-plate capacitor is filled with a gas such as argon a t a pressure near one atmos- phere. The externally applied voltage V establishes an electric field E = V/d, where d is the interelectrode spacing. The field E is of sufficient magnitude to prevent recombination of the positive ions and electrons, but not large enough to permit gas multiplication. If a single a-particle,

i f PULSE OUT

FIG. 1. Essential features of a gas-filled ionization chamber.

say, passes through the chamber it will lose energy by elastic and inelastic collisions with the argon atoms. The net effect of these interactions is the formation of a number of positive ion-electron pairs which are swept apart by the electric field. Frequent collisions with gas molecules preventj both the ions and electrons from obtaining enough energy from the field between collisions to produce secondary ion-electron pairs. As the elec- trons and ions drift apart, the collector electrode potential rises from zero to ne/C, where n is the number of ion pairs formed, e is the electronic charge, and C is the chamber capacitance. The rise-time and shape of the leading edge of the output pulse is dependent upon the interelectrode distance d, the mobilities p e and pi of the electrons and ions in the chamber gas, and the applied potential V. The decay time is determined by the RC time constant where R is an external resistor (see Fig. 1) and C is the chamber capacitance.

A discussion of the principles of operation of a semiconductor detector

* Section 1.8.1 is by S. S. Friedland and F. P. Ziemba. 265

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266 1. PARTICLE DETECTION

is aided by referring to Fig. 2. A p-n junction’ is formed close to one surface of a slab of high-resistivity p-type (boron-doped) silicon by a shallow diffusion of phosphorous. A reverse bias applied to the junction establishes a depletion region (space-charge region) on both sides of the junction. The thin phosphorous doped (n-type) region near the junction has a positive space-charge due to ionized donors, whereas the boron doped (p-type) region has a negative space charge due to ionized acceptors. This distributed dipole layer resembles the charged parallel- plate capacitor of the conventional gaseous ionization chamber. If an a particle passes through the space-charge region, electron-hole pairs are produced by inelastic collisions with the silicon atoms. These carriers are swept apart by the electric field set up by the dipole layer, giving rise to an electrical pulse similar to that obtained in a gas chamber.

Another variation of the semiconductor detector, often referred to as a “surface barrier counter,” has also been developed. These counters are made by evaporating a thin layer of gold (100-2OOOw) onto high- resistivity n-type silicon (or germanium). A distributed p-type layer is formed by surface states at the interface between the metal and the semi- conductor. 11,12 Positively charged ionized donors in the n-type material along with the p-type states form a dipole layer. The region in the semi- conductor which is nearly stripped of conduction electrons is called a surface barrier. The charge distribution, potential gradient, barrier capacitance, and barrier depth can be calculated from Poisson’s equation and the Fermi-Dirac distribution of charge carriers. The results are

1 I. Van der Ziel, “Solid State Physical Electronics.” Prentice-Hall, Englewood Cliffs, New Jersey, 1957.

J. W. Mayer and B. R. Gossik, Rev. Sci. Instr. 27, 407 (1956). J. W. Mayer, J. Appl. Phys. 30, 1937 (1959).

a F. J. Walter, J. W. T. Dabbs, L. D. Roberts, and H. W. Wright, Oak Ridge National Laboratory, CF 58-11-99 (1958).

F. J. Walter, J. W. T. Dabbs, and L. D. Roberts, Oak Ridge National Laboratory, ORNL-2877 (1960); Bull. Am. Phys. Soc. [11] 3,181 (1958); 3,304 (1958); 6,38 (1960); 6, 22 (1960).

F. J. Walter, J. W. T. Dabbs, and L. D. Roberts, Rev. Sci. In&. S1, 756 (1960). J. M. McKenzie and D. A. Bromley, Phys. Rev. Letters 2 , 303 (1959); Bull. Am.

Phys. Soe. [11] 4, 457 (1959). 7 E. Nordberg, Bull. Am. Phys. Soc. [11] 4, 457 (1959).

M. L. Halbert, J. L. Blankenship, and C. J. Borkowski, Bull. Am. Phys. Soc. [II]

J. L. Blankenship and C. J. Borkowski, Bull. Am. Phys. Soc. [11] 6, 38 (1960). 6, 38 (1960).

lo M. L. Halbert and J. L. Blankenship, Nuclear Instr. and Methods 8, 106 (1960). 11 W. Schottky, Z . Physik 118, 539 (1942). l2 R. H. Kingston, ed., “Semiconductor Surface Physics.” Univ. of Pennsylvania

Press, Philadelphia, 1957.

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1.8. SPECIAL DETECTORS 267

similar to those to be described for the p-n junction. The nuclear char- acteristics of both the p--n diffused junction detector and the surface barrier counter are the same, since they are dependent upon the nature of the interaction of radiation with the semiconductor material.

On the average, 3.5 ev of incident particle energy is required to produce one electron-hole pair in silicon (as opposed to 32 ev for a typical gas), To dat,e, all of the experimental evidence indicates that this value is independent of the particle type. Thus any particle losing energy E (electron volts) in the space-charge region produces n = &/3.5 electron- hole pairs (as compared to &/32 ion-electron pairs for the gas chamber).

SURFACE CONTACT T- . r P TYPE SILICON

BASE CONTACT

D = DIFFUSION DEPTH Xp= WIDTH OF DEPLETION

REGION IN P MATERIAL X,,= WIDTH O F DEPLETION

REGION IN N MATERIAL d = X, + X, = TOTAL WIDTH

x = 0 O F DEPLETION REGION

L N TYPE SIL ICON

FIG. 2. Scheniatic diagram of a p-n junction under reverse bias.

Statistical arguments show that the fundamental device resolution limit is set by the characteristic fluctuation. One, therefore, theoretically expects and experimentally obtains a significant improvement in the energy resolution of the solid state chamber over that obtained with a gaseous chamber. The high carrier mobilities and drift velocities in silicon combined with the small width of the depletion region result in pulses with millimicrosecond rise times. Since the range of energetic particles in silicon is measured in microns as compared to centimeters in a gas, the physical size of the detector is several orders of magnitude smaller.

For the p-n junctions under consideration the thickness of the deple- tion layer in microns is given approximately by d = (pV)l'2/3 where p

is the resistivity of the p region in ohm centimeters and V is the applied reverse bias voltage. It is necessary that the incident particle lose all of its energy within the depletion region in order that there be a linear relationship between the pulse height and particle energy. It is obvious from the above result that the product pV be made as large as possible when the detectors are to be used as spectrometers for penetrating par- ticles such as protons and electrons. The results of several workers using

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268 1. PARTICLE DETECTION

detectors with p = 10,000 ohm-cm and V = 400 volts are shown quali- tatively in Fig. 3. The relative pulse height is linearly related to the elec- tron energy to about 1 MeV, for protons to 10 MeV, a's to 40 MeV, heavy ions and fission fragments > 100 MeV. These results are consistent with the known range-energy relationships of these various particles in silicon. The typical noise level of amplifiers is shown to be near 20 kev. Resolu- tions of 0.3% for 5-Mev a particles have been obtained. It should be pointed out that if one wishes merely to count individual events (not

I I I 1 1 1 1 1 ~ I ll l l l l l~ I 1 1 1 1 1 1 1 ~ I 1 1 1 1 1 1 1 ~ I 1 1 1 1 ~

HEAVY IONS /

ELECTRONS

a

I l f l l l l ' I I 111111' I I 1 1111J I I I 111d I ' ' I .01 0.1 1.0 10 100 1000

PARTICLE ENERGY IN MEV

FIG. 3. Relative pulse height versus energy of a semiconductor detector for various nuclear particles.

measuring the particle energy) then even minimum ionizing particles may be detected inasmuch as they will expend energy at approximately 0.35 kev/micron in silicon.

Early attempts to use solid state devices as particle detectors date back to the work of Jaffe13 and Schiller14 who observed small changes in the conduction current of crystals irradiated with cr particles. Measurable pulses produced by individual & particles penetrating a AgCl crystal were first obtained by Van Heerden.16 Extending that work, McKayl8 studied conductivity changes in diamond under electron bombardment, and Ahearn17 tested a large number of crystals for conduction pulses pro-

13 G. Jaffe, 2. Physik 33, 393 (1932). l4 H. Schiller, Ann. Physik [4] 81, 32 (1926). 16P. J. Van Heerden, "The Crystal Counter, A New Instrument in Nuclear

Physics." North Holland Publ., Amsterdam, 1945. l6K. G. McKay, Phys. Rev. 74, 1606 (1948). 1 7 A. J. Ahearn, Phys. Rev. 76, 1966 (1949).

i

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1.8. SPECIAL DETECTORS 269

duced by a particles. It was generally found that the crystal counter was not useful as a spectrometer because of poor resolution and polariza- tion effects. Review articles by Chynowethls and Hofstadterlg summarize much of the work with crystal counters.

McKay20 reported the response of point contact germanium rectifiers to a particles and suggested the use of p-n junctions. Orman et McKay,22 and Airapetiants et ~ 1 . ~ ~ ~ 9 ~ ~ measured the voltage pulses pro- duced by germanium p n junctions struck by a particles.

In 1955 Mayer and GossikL8 used gold-germanium surface barrier counters as a-particle spectrometers. They found that the pulse height from such counters was proportional to the a-particle energy up to about 8 Mev and obtained good resolution from units with 0.8 to 16.0 mm2 active surface area. The Ge-Au surface barrier counter was developed further by Walter et C L Z . , ~ ~ ~ , ~ McKenzie and Bromley16 and Dearnaley and Whitehead.25 For good resolution, Ge-Au counters have had to be oper- ated a t low temperatures. Room temperature operation was obtained with the introduction of silicon which has a larger energy gap, and, therefore, a much lower (reverse) saturation current. Si-Au surface barrier counters were developed by McKenzie and Bromley16 Nordberg17 Halbert et d.,* Blankenship and Borkowski,Y and Halbert and Blanken- ship. lo Such detectors have excellent properties a t room temperature, but until recently the fabrication has been most difficult.

Use of a p-n junction diffused in silicon which resulted in an operational room temperature spectrometer was developed by Friedland et aLZ6 Further properties of the p-n diffused junction detector have been reported by a number of ~ o r k e r s . ~ ~ - ~ l

A. G. Chynoweth, Am. J . Phys. 20, 213 (1952). l9 R. Hofstadter, Proc. IRE (Znst. Radio Engrs.) 38, 726 (1950). ,OK. G. McKay, Phys. Rev. 70, 1537 (1949). 2 1 C. Orman, H. Y. Fan, G. T. Goldsmith, and K. Lark-Horowitc, Phys. Rev. 78,

z* K. G. McKay, Phys. Rev. 84, 829 (1951). 28 A. V. Airapetiants and S. M. Ryvkin, Zhur. Tekh, Fiz. 2 7 , l l (1955); Soviet Phys.,

Tech. Phys. (Eng. Trans.) 2, 79 (1958). z 4 A. V. Airapetiants, A. V. Logan, N. M. Reinov, S. M. Ryvkin, and I. A. Sokolov,

Zhur. Tekh. Fiz. 27, 1599 (1957); Soviet Phys., Tech. Phys. (Eng. Trans.) 2, 1482 (1957).

zs G. Dearnaley and A. B. Whitehead, Atomic Energy Research Establishment, Harwell, Berkshire, United Kingdom, AERE-R 3278 (1960).

26 8. S. Friedland, J. W. Mayer, J. M. Denney, and F. Keywell, Rev. sci. Instr. 31, 74 (1960).

27 Report on the Seventh Scintillation Counter Symposium, Washington, D.C. February 25-26, 1960. Nucleonics 18 (5), 98 (1960).

28 Complete Proceedings of the Seventh Scintillation Counter Symposium, Wash-

646 (1950).

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270 1. PARTICLE DETECTION

1.8.1.2. Junction Capacitance and Junction Width. The potential dis- tribution due to the distributed dipole layer in the space-charge region of the plc junction may be calculated by solving Poisson’s equation with the appropriate boundary conditions. For sufficiently large-area diodes, edge effects may be ignored and the resulting one-dimensional problem is readily handled. The result so obtained allows for the determination of the capacitance and width of the space-charge region. The details of the calculation are omitted and may be found elsewhere.’ For a “Schottky-type potential barrier” where there is a sudden step from p type to n type, the thickness of the transition is d = xp + x,; xp is the distance from the junction into the p-type material and xn is the distance from the junction into the n-type material. The thickness d is readily found to be given by

d = [2eeo(Vd + V)(N,, $- Nd)/eNoNd]112. (1 A.1. I)

The acceptors and donors per unit volume are N , and Nd;

ED = 8.85 X 10-lafarad/m

and e is the relative dielectric constant; Va is the diffusion potential (built in voltage) and V is the applied bias. For the devices under con- sideration V >> Vd, Nd >> N,, and we obtain

d [2ee0V/eNd]’/~. (1.8.1.2)

Since the total charge must be zero for over-all charge neutrality in

ington, D.C., February 25-26, 1960. IRE Trans. on Nuclear Sci. NS-7, 2-3 (1960).

tory-Brookhaven National Laboratory Report BNL 4662 (1960). *9 G. L. Miller, W. L. Brown, P. F. Donovan, and I. M. Mackintosh, Bell Labora-

3o G. F. Gordon, Univ. of California Radiation Lab. Rept. 9083 (1960).

a* H. Mann, Argonne National Laboratory, private communication. J. Beneveniste, Univ. of California Radiation Lab., private communication.

E. L. Zimmerman, “Comments on the Use of Solid State Detectors for Neutron

34T. A. Love and R. B. Murray, Oak Ridge National Laboratory, CF 60-5-121

36 C. T. Raymo, J. W. Mayer, J. S. Wiggins, and 5. S. Friedland, Bdl. Am. Phys.

ae S. 6. Friedland, J. W. Mayer, and J. S. Wiggins, Nucleonics 18, 54 (1960).

38 R. L. Williams, Bull. Am. Phys. SOC. [11] 6, 354 (1960).

Detection.” Solid State Radiations, Inc., Culver City, California, unpublished.

(1960).

SOC. [I11 6, 354 (1960).

J. D. Van Putten and J. C. Vander Velde, Bull. Am. Phys. SOC. [11] 6, 197 (1960).

J. W. Mayer, R. J. Grainger, J. W. Oliver, J. 5. Wiggins, and S. S. Friedland, BuU.

‘O P. F. Donovan, G. L. Miller, and B. M. Foreman, Bull. Am. Phys. SOC. [I11 6,355

41 J. M. McKenaie, Bull. Am. Phys. SOC. [11] 6, 355 (1960).

Am. Phys. SOC. [11] 6, 355 (1960).

(1960).

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1.8. SPECIAL DETECTORS 27 1

the junction we must have N d X n = N o X p . ( 1.8.1.3)

Typical values for zn and rp are 0.1 micron and 0.5 mm respectively. The transition region capacitance per unit area is given by

c = [ccoeN,N,&(V,j + V)(Na + N d ) ] ' / ' = t c o / d . (1.8.1.4)

The space-charge layer thus acts as a parallel plate capacitor with plate separation d. For a sudden step junction, one obtains the approximate expression

C = [ ee0eN~/2V] l /~ . ( 1.8.1.5)

In terms of resistivity and mobilities, Eqs. (1.8.1.2) and (1.8.1.4) take on the more convenient forms

d = [2eeoVpspo]1'2 = (+)(PV)~/ ' microns c ZZ [Eeo/2VphP,p,]"' C (+)(pV)-"' 10' ppf/Cm2 (1.8.1.6)

where p. and ph are the mobilities of electrons and holes respectively with V expressed in volts and p in ohm-em. The above relationships have been combined by Blankenship27 into a very useful nomograph.

1.8.1.3. Properties of the Semiconductor Detector. The response of a semiconductor detector is linearly related to the energy of the incident particle energy provided the range of the particle (in silicon) is less than the width of the depletion region. Equation (1.8.1.6) demonstrates that the depletion width is proportional to the product (pV) l/'. Detectors with resistivities up to 13,000 ohm-cm have been described and e~aluated. '~ Operating such a device with the not unreasonable bias of 750 v would lead to a depletion depth of approximately 1 mm which corresponds to the range of about a 15-Mev proton in silicon.

Equation (1.8.1.6) shows that the capacitance is inversely proportional to the product ( P V ) ~ ' ~ . Increasing the quantity pV therefore reduces the capacitance, increases the output voltage V o = ne/C, and increases the signal energy E. = (ne) '/2C. The signal-to-noise ratio, however, does not in practice increase with bias voltage. The detector reverse leakage current increases monotonically with the applied voltage. At large voltages this current gives rise to a diode noise energy which increases more rapidly with voltage than the signal energy. The frequency dis- tribution of the noise, the effects of surface leakage, and other matters relating to signal-to-noise ratios have not been studied extensively a t this time. At relatively low bias voltages, the detector noise level is usually below typical amplifier noise levels. This state of affairs has initiated a considerable effort to develop low noise preamplifiers for use

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272 1. PARTICLE DETECTION

with solid state detectors.27 Preamplifiers with noise levels as low as 7.5 kev have been reported.

As discussed above, the capacity of an abrupt junction should vary with the voltage according to the relation W 2 C = constant. In a “graded” junction such as that obtained in a grown junction, the acceptor and donor densities N , and Na are slowly varying functions of position in the region of the junction. For a “graded” junction one expects the relation

I 10 100

REVERSE BIAS IN VOLTS

FIG. 4. Approximate high-frequency equivalent circuit of a semiconductor detector. The junction capacitance Cj and resistance Rj determine the pulse rise-time. The resistance Rb is due to the bulk silicon outside of the space charge region.

to be V I W = constant.2 Experimental measurements indicate that VfC = constant with 4 < f < $; typical data for capacitance versus reverse bias for typical detectors are shown in Fig. 4. The temperature dependence of the capacitance is negligible.

An approximate equivalent circuit of a solid state detector is shown in Fig. 5. The radiation source is replaced by a charge generator charging the junction capacitance Cj through a resistance Rj which may be esti- mated by setting the time constant RjCj equal to the transit time of the carriers through the space charge region. Rise times in the millimicro- second region are commonplace in contrast to the microsecond rise

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1.8. SPECIAL DETECTORS 273

times obtained with gaseous ion chambers. The development of low- noise wide-band amplifiers appears to be quite desirable. The resistor Ra is due to the bulk silicon outside of the space charge region and the ohmic contact with the p-type material. It may be reduced by: (i) having the wafer thickness comparable to the depletion depth; and (ii) a proper doping at the ohmic contact. The latter should also have the effect of producing a more uniform field within the space charge region, whence a more uniform collection efficiency and resolution in large-area detectors.

Surface leakage current, space-charge-generated current, and diffusion current all contribute to the reverse current of a semiconductor de- t e ~ t o r . ~ ~ ~ ~ ~ It is difficult to determine the relative contributions in a par-

-

CJ

CHARGE I GENERATOR I

FIQ. 5. The dependence of the junction capacity upon applied bias voltage.

ticular device; however, the absence of a well-defined breakdown in many units indicates that surface leakage is usually the most important of the three sources. The temperature and voltage dependence of the reverse current are at present under considerable investigation. Surface effects will have to be minimized before any definite conclusions are drawn.

1.8.1.4. Experimental Results. The response of a semiconductor detec- tor to various types of nuclear radiations is shown in Figs. 6 to 11. The energy range over which the device responds linearly to the different types of radiation and the resolution for each of the particle types are discussed and illustrated with experimental data.

1.8.1.4.1. HEAVY IONS AND FISSION FRAGMENTS. Fission fragments have relatively short ranges in silicon and there is no difficulty in obtain- ing depletion depths which are wide enough to ensure linearity with energy. The same remark applies to heavy ions such as C12. Figure 6 demonstrates that the device is linear for CI2 ions with energies to 120 M ~ v . ~ O The kinetic energy spectrum of fragments from the spon-

42 J. H. Shive, “Semiconductor Devices.” Van Nostrand, Princeton, New Jersey, 1959.

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274 1. PARTICLE DETECTION

taneous fission of Cf262 observed29 with a p-n detector is in agreement with the accepted time-of-flight measurement^.^^ The detector appears to be “windowless” for fission fragments provided the phosphorous surface layer (n-type) is less than -0.1 p . The semiconductor detector does not exhibit the “ionization defect ’m7 characteristic of gaseous chambers and negligible columnar recombination appears to exist

ENERGY IN MEV

FIG-. 6. The relative pulse-height versus energy of a semiconductor detector for Cl* ions with energies from 30 to 120 Mev.

along the tracks of fission fragments even though carrier densities are -lozo ~ m - ~ .

1.8.1.4.2. PROTONS AND a PARTICLES. The proton response31 of a 5 mm X 5 mm area detector made from 10,000 ohm-cm silicon, and operating a t a reverse bias of 400 v, is shown in Fig. 7. The device is linear for protons to about 10 MeV. Energy-range relationships for protons in silicon show that the range of a 10-Mev proton is about 700 1.1. This agrees well with the calculated width of the depletion region.

The resolution versus bias of a typical detector for 8.78-Mev a particles from Pb2I2 is shown in Fig. 8. The poor resolution a t low bias voltages

4 a J. C. D. Milton and J. S. Fraser, Phys. Rev. 111, 877 (1958).

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1.8. SPECIAL DETECTORS 275

FIQ. 7. The relative pulse-height versus energy of a semiconductor detector, 6,000 ohm cm, 400-v bias, for protons with energies from 2 to 12 MeV.

0 I I I I 0 0 50 100 150 200 250

REVERSE BIAS (VOLTS)

FIQ. 8. The energy-resolution for 8.78-Mev a particles and reverse current versus bias voltage of a semiconductor detector, 6000 ohm cm, 6 mm X 5 mm.

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276 1. PARTICLE DETECTION

is probably due to a combination of a poorer signal-to-noise ratio and a nonuniformity in collection efficiency over the sensitive area. The absence of a well-defined avalanche breakdown in the current-voltage character- istic would indicate considerable surface leakage. A maximum resolution of 0.3% for 5-Mev a! particles which has been obtaineds2 is currently limited by the noise level of the amplifiers.

1.8.1.4.3. ELECTRONS. Figure 9 shows that the detector response is linear for electrons to nearly 1 MeV. Using a calibrated charge-sensitive amplifier, Mann32 shows that about 3.7 ev of incident electron energy

ENERGY IN KEV

FIG. 9. The relative pulse-height versus energy of a lo4 ohm cm semiconductor detec- tor for electrons with energies from 50 to 800 kev. Bias, 360 v; X, Pml47; 0, Agllom.

is required to produce one electron-hole pair in silicon. The result is in reasonable agreement with the results obtained for protons and a’s in silicon. The data are shown in Fig. 10. Internal conversion lines in CsI3’ are shown in Fig. 11. The K and L lines are distinctively resolved. The data were kindly supplied by C. S. Wu.

1.8.1.4.4. N E u T R o N S . ~ ~ Since the semiconductor detector is an excellent device for observing heavy charged particles, it is obvious that its use- fulness may be extended to include neutron detection by applying coat- ings which react with neutrons to produce heavy charged particles. Efficient thermal neutron detectors can be realized by BIO, Lis, and UZ35 coatings. Such devices are not directly useful as neutron energy spec-

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1.8. SPECIAL DETECTORS 277

trometers since the reaction energies are large compared to the incident neutron energy. A combination of bare and cadmium-covered detectors will, however, give some indication of the thermal neutron distribution.

Threshold detectors based upon the Np239 (n,f) and U238 (n,f) reactions will be useful for high-energy neutrons. Neutron energy spectrometers based upon “proton recoil techniques,”* Lie (n,a) HS, and SiZ8 (n,p) AlZ8

FIQ. 10. The number of charges collected versus energy of a semiconductor detector for electrons with energies from 50 to 800 kev.

reactions hold considerable promise. I t is interesting to note that no coating is needed for the SiZ8 (n,p) A128 detector. Preliminary results of Love and on a Lie ( n , ~ ) H3 neutron spectrometer are most encouraging. The promising ranges of usefulness of the variously coated semiconductor neutron detectors are illustrated in Fig. 12. The height of each curve is a rough indication of the degree of utility of each of the possible arrangements.

1.8.1.4.5. PHOTONS. The only experimental data available on the * Refer to Section 2.2.2.1.

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278 1. PARTICLE DETBCTION

photon response of semiconductor detectors is some preliminary work with high-energy CosO gammas.36 The device is relatively insensitive to gamma radiation in this energy range (Compton effect) due t o the low-absorption cross section of silicon. The p-n detector is quite sensitive

2000

1500

w i

3 1000 0 z I-

0 u

500 - BIASED SO THAT ZERO CHANNEL AT - 5 6

K - LINE 624 KEV

100 150 200 CHANNEL NO

FIQ. 11. The internal conversion lines in Cs1*7 aa measured with a semiconductor detector.

to photons with energies comparable to the gap energy in silicon (1.1 ev) and should not be exposed to light when used as a nuclear particle detector.

1 A.1.4.6. HIGH-ENERGY PARTICLES. Recently, several laboratories have made investigations to determine whether high-energy particles in

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1.8. SPECIAL DETECTORS 279

the minimum ionizing region can be measured with semiconductor de- tectors. The results obtained so far are quite en~ouraging .~~

Figure 13 shows the energy spectrum of a positive pion and proton beam with a momentum of 750 Mev/c from the Brookhaven Cosmotron obtained in a silicon junction detector. The resistivity of the silicon junction detector is 10,000Q-cm and it was operated at a reverse bias

si 28cn.p) AL~~SPECTROMETER

!/--- ( n, cx INTEGRAL Li’ In. ~)H31NTEGRAL

I Li6 (h.oC)H3 SPECTROMETER

N$”(n,f) THRESHOLD

Cn.p) THRESHOLD

P - RECOIL

.dl d.i i!o Ib Ib2 1’0. I3 I2 I’O’ I 1 0 7 I b e ilop ;do NEUTRON ENERGY-EV

4.5 MEV

0.26 I.!& MEV

i \ / Li6 (h.oC)H3 SPECTROMETER

NEUTRON ENERGY-EV 4.5 MEV

0.26 I.!& MEV

FIQ. 12. The relative utility of variously neutron sensitized semiconductor detectors versus the neutron energy.

of 100 v. At this momentum the pions are close to minimum ionization whereas the protons are twice minimum. The ionization losses for the pions and protons are found to be 110 kev and 200 kev, respectively, indicating a linear response, and the pion and protons are very clearly separated.

1.8.1.5. Conclusion. The small size of the semiconductor detector makes it possible to arrange a linear array of detectors in the focal plane of a spectrometer or a t several angles inside of a reaction chamber. Two

44 L. C. L. Yuan, Application of solid state devices for high energy particle detec- tion. Intern. Conf. on Instrumentation for High Energy Physics, Berkeley, California, September, 1960.

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280 1. PARTICLE DETECTION

dimensional arrays can be assembled to obtain large-area a-survey instru- ments, for example, with low power requirements. A three-dimensional array along with an appropriate data handling system would obtain the equivalent of a “solid state cloud chamber.”46

The effects of radiation on semiconductor devices have been summar- ized in many report^.^^.^^ Radiation damage studies in semiconductor

302

2oc

IOC

-

7T+ AND PROTON BEAM, 750 y, IOK COUNTER, 1.4 CM DIAMETER, g- 100 VOLT BIAS. 5

LL

s! a- J

0

>- w Y

z

> U w- z W

1 I I I I I I ) 20 30 40 50 60 m 80 90

I 100

300

5 5 0

?oo

I 50

100

50

- CHANNEL NUMBER

FIG. 13. 750 Mev/c positive pion and proton spectrum obtained in a Brookhaven Cosmotron beam. Number of counts per pulse-height analyzer channel is plotted versus the channel number which is a measure of the particle energy.

detectors is not available at the present. The data available a t present indicate that no changes in detector operation are observed after exposure to 10l2 14-Mev protons.

The physical properties, low power requirement, wide range of linear relationship of pulse height versus particle energy, and high speed sug- gests that the semiconductor detector will soon become one of the basic operational devices in the field of radiation detection.

46 S. S. Friedland, “The Solid State Cloud Chamber.” To be published. *6 G. J. Dienes, Radiation effects in solids. Ann. Rev. Nuclear Sci. 2, 187 (1953). 41 F. J. Reid, The effect of nuclear radiation on semiconductor devices. REIC

Report No. 10, Battelle Memorial Institute, Columbus, Ohio (1960).

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1.8. SPECIAL DETECTORS 281

1.8.2. Spark Chambers*

Frequently it is desirable to have good spatial resolution of high-energy particle interactions because so many modes of interaction are possible. Generally, the interesting processes also have a small cross section so that good time resolution is desirable too. Bubble chambers have been very effective in experiments where it is reasonable to expand the chamber and then have approximately ten charged particles incident on the chamber. If the chamber is large enough and if there are enough interest- ing events, then this is a good technique. The electromagnetic spectrom- eters which have been used to select the desired mass of the incident charged particles have helped to extend the use of bubble chambers. However, these will be considerably less effective at higher energies and new techniques will be desirable.

Cloud chambers have been used in conjunction with scintillation counters to select interesting events and then the chamber is expanded to get tracks. Also diffusion chambers have been used in this manner, the lights being flashed to detect an interaction. The resolving time of such chambers is greater than 10 psec and background radiation is a rather severe limitation of these chambers.

The scintillation chambers are being developed which have both good spatial and time resolution. For some experiments these have many desirable features. The chief disadvantages are the still rather small size and the limited flexibility for a variety of experiments.

In some experiments' where only moderate spatial resolution is re- quired, such as differential cross section scattering of antiprotons and K mesons, it has been quite practical to use arrays of scintillation counters and Cerenkov counters. In these experiments the constraints due to two- particle interactions have simplified the analysis.

In many cases it is now necessary to investigate specific details with good statistics. This frequently requires that rare events be selected from a background of many other interactions. The spark chamber used in conjunction with scintillation counter and Cerenkov counter telescopes is a device that permits both good spatial and time resolution of high- energy charged particle interactions.

An early type of discharge chamber2 consisted of bundles of slightly

"2. A. Coombes, B. Cork, W. Galbraith, G. R. Lambertson, and W. A. Wenzel, Phys. Rev. 112, 1303 (1958).

M. Conversi and A. Gorrini, Nuovo cimento [lo] 2, 189 (1955);'M. Conversi, S. Focardi, C. Franzinetti, A. Gozzini, and P. Murtas, Nuovo cimento Suppl. [lo] 4, 234 (1956). __

* Section 1.8.2 is by Bruce Cork.

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282 1. PARTICLE DETECTION

conducting glass tubes filled with one-half atmosphere of neon and placed between parallel metal plates. When a high-energy charged particle was transmitted through the tubes and plates a high electric field pulse was applied between adjacent plates. The tubes that transmitted the particle would then give a glow discharge and the light from the ends of the tubes could be photographed. These devices had a long recovery time, approxi- mately one second, due to the electrostatic charges on the glass.

1 ji

i

I

4

-

FIQ. 1. A parallel plate spark chamber. The plates are made of +-in. thick aluminum separated by a gap of 4 in.

A similar device has been described by Cranshaw and DeBeer.3 How- ever, they omitted the glass tubes, immersed the metal plates in air at one atmosphere, and applied a 20-kv pulse to the plates when a charged particle was transmitted. The efficiency for minimum ionizing particles was 99%, for a 3-mm gap.

Then Fukui and Miyamotoe immersed the plates in an atmosphere of neon. They observed that minimum ionizing particles could be detected with nearly 100% efficiency and by applying a +psec pulse to the plates, the sensitive time was observed to be approximately 10 Fsec.

Several other groups have built similar spark chambers. The details

8 T. E. Crawhaw and J. F. DeBeer, Nuovo cimento [lo] 6, 1107 (1957). S. Fukui and S. Miyamoto, Nuovo eimento [lo] 11, 113 (1959).

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1.8. SPECIAL DETECTORS 283

of a chamber6 built by Beall, Cork, Murphy, and Wenzel are given below. The chamber consisted of seven parallel plates of g in . thick aluminum, separated by gaps of s i n . thickness, Fig. 1. The chamber has been filled with one atmosphere of argon or neon. This chamber has been tested in a beam of high-energy pions and protons a t the Bevatron.

- Coincidence circuit

Discriminator 8 gote circuits

T h u r n t v n m I

1--D

. '.>'"."",. I 2i----Jy Thyratron 2

FIG. 2. Block diagram of electronics, scintillators C , Cz, Cs and spark chamber.

The diagram, Fig. 2, shows a charged particle being detected by a scintil- lator coincidence telescope. The output of the coincidence circuit is used to operate a hydrogen thyratron which applies a 20-kv pulse, approxi- mately 0.2psec, to alternate plates of the spark chamber. A battery supplies a dc clearing field between the parallel plates so that electrons produced in the gap between the plates can be swept away, thus reducing the sensitive time of the chamber.

The efficiency of the chamber when filled with argon or neon is given by Fig. 3 for a clearing field of 0 v/cm and 270 mFsec delay after the traversal by a minimum ionizing charged particle. To determine the

6E. Beall, B. Cork, P. G. Murphy, and W. A. Wenzel, UCRL-9313 (1960).

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284 1. PARTICLE DETECTION

sensitive time of the chamber for various values of clearing field, varying time delays were inserted in the trigger line to the thyratron. The effi- ciency as a function of delay time is given by Fig. 4. It is noted that the sensitive time can be made less than 0.5 psec.

The recovery time of the chamber should be of the order of the deioniza- tion time of an inert gas at one atmosphere. The observed recovery time (Fig. 5) was long compared to the deionization time. This was measured by selecting a charged particle that was transmitted by the chamber, firing the chamber, and then selecting a second charged particle a t a pre- determined time and again firing the chamber. One reason for the appar-

I 1 I a 1

5 10 15 20 25

Pulse voltoge ( k v )

FIG. 3. Efficiency of a single +in. gap in 1 atmos of argon or neon as a function of pulsed voltage across the t i n . gap.

ent long recovery time may be due to impurities in the gas. This is not a practical limitation for proton synchrotrons where the beam time is of the order of 100 msec.

Typical photographs of the 6-gap chamber are given in Fig. 6. Mini- mum ionizing pions enter from the left. In Fig. 6(a) one pion interacted with the plate and a second pion entered during the sensitive time of the chamber. A second interaction is shown by Fig. 6(b) where one reaction product was scattered at an angle of 25 degrees, the second a t 36 degrees.

When a magnetic field of B equal to 13 kg was applied parallel to the plates, the efficiency was still nearly 100% per gap. If a clearing field E of 80 v/cm was applied and the time of applying the hv pulsed electric field was delayed for 1 psec, the electrons from the ion pairs were displaced an amount proportional to E X B, and the delay time. The photograph (Fig. 7) shows a displacement of the tracks for the above conditions of

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FIQ. 4. Efficiency of a single *in. gap in 1 atmos of argon as a function of delay in application of the high-voltage pulse. The zero of the delay axis is the time at which the particle passed through the chamber.

3 100 E c

'p 80 c 0 V

$ 60

I I

8 Argon

A Neon

5 10 15 20

Time between partieter (maec)

FIG. 5. Efficiency for a single gap to a spark on a second particle as a function of the time between particles. The clearing field was -40 v/cm.

285

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286 1. PARTICLE DETECTION

(b)

FIG. 6. Typical photographs of the 6-gap chamber. In Fig. 6(a) one pion interacted with the aluminum plate and a second pion entered during the sensitive time of the chamber. Figure 6(b) shows two large angle scatters.

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1.8. SPECIAL DETECTORS 287

FIG. 7. A magnetic field of 13 kg parallel to the plates and a clearing field of 80 v/cm cause a displacement of the sparks of ki in. if the high-voltage pulse is delayed for 1 usec.

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288 1. PARTICLE DETECTION

approximately 1 cm. Particles arriving “off time” could be detected by this means.

Besides the good spatial and time resolution of the spark chamber, the chamber can be arranged in a manner that is appropriate to the particular experiment. For example, the quantity of light from the spark is so great that stereographic photography is easy from an extensive assembly of chambers. The plates can be made of metal or graphite, for example, to preferentially scatter polarized protons. It should be possible to make the plates of scintillator or cerenkov material so that the spark chamber and counter telescope are integral. Large solid angles for detecting short-lived particles can be obtained by this method.


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