On the Effects of Thermal Velocities in Two-Dimensional and Axially Symmetric Beams*

PETER T.

Summary-The effects of thermal velocity in two-dimensional and axially symmetric beams are evaluated, assuming a Maxwellian distribution of velocities at the cathode. The linear treatment is similar to that of Cutler and Hines, but the approach is somewhat different. Analytic expressions and curves are given for the current density in terms of the error and modified Bessel functions; the arguments of these functions are the solutions of the usual paraxial equations, neglecting thermal effects. Our predictions agree with those of Cutler and Hmes for the case they considered.

An approximation is given for the nonlinear effects of the electro- magnetic fields; these nonlinear effects may be due to aberrations in the system, or to changes in the space-charge fields due to the linear effects of thermal velocities. As an example, the problem of a beam with a finite transverse temperature in a field-free drift space is considered. Analytic expressions are given for the current density in such a beam.

I. INTRODUCTION HE ESTIMATION of the effects of thermal veloci- ties in particle beams has been important since the early work on the cathode-ray tubes. In the applica-

tion of ion beams, where acceleration-deceleration systems are being considered, thermal effects become even more important, and cannot be considered independently of space-charge effects. Pierce [I] has given an excellent re- view of the work up to 1954. However, his treatment is very general, and does not give results or methods of approach for most of the problems encountered in practice. Cutler and Hines [2] first gave a systematic discussion of thermal velocities in electrostatic round beams with a constant current density at the cathode. Their method can be extended to a wide range of problems, namely to those problems where the field normal to the trajectory of the beam varies linearly with distance from the beam center. All beams for which the paraxial assumptions are valid [3]-[6] fall into this category. Thermal effects change the current distribution in the beam, and this affects the space-charge forces. Danielsolz, et aZ. [7] extended and formalized Cutler and Hines’ [a] treatment, and allowed for the change in space-charge forces by considering their effect on a particular ‘Ltypical” particle.

In this paper the treatments of Cutler and Hines [a] and Danielson, et aZ. [7] are extended in several directions. First, their method is related to the paraxial theories de- veloped in [3]-[6]. This method is valid even with trans- verse magnetic fields, but the effect of the spread in longitudinal velocities is neglected. Both axially symmetric and two-dimensional problems are treated. Expressions

* Received August 8, 1962. t High Voltage Engineering Corporation, Burlington, Mass. On leave from the European Organization for Nuclear Research, Geneva, Switzerland.

KIRSTEINt

are derived for the current density and total current as a function of the transverse position in the beam. These expressions are compared with those of Cutler and Hines [2], and are found to be identical for constant cathode current density. I n sheet beams with deflection focusing, or in crossed-field guns, the current density varies linearly at the cathode. For this reason the case of sheet beams with a linear variation of cathode current density is treated in detail. Analytic expressions and curves are presented.

A higher-order theory is stated which gives a correction for the effects of nonparaxial fields, which result from two sources. First, aberrations in the applied fields will pro- duce nonparaxial forces. Secondly, finite temperature effects will, even to the paraxial approximation, change the current distribution; this c,hange in current distribu- tion will lead to a nonlinear component of the space- charge field. The theory gives an approximation to the actual current density distribution. This is different to the approach of Danielson, et al. 171, who considered a specific “average” particle, and assumed that the corrections to this particle were a good approximation to the behavior of the beam as a whole. Usually our higher-order theory can be only applied numerically. However one particular analytic example is given to illustrate the method. This example is the computation of the current distribution in a two-dimensional beam in a field-free region, with initially a constant current density and a Maxwellian distribution of transverse velocities. The method may of course be applied to more complex paraxial conditions and nonlinear applied fields. Throughout this paper the mks system of units is used. Our results are valid through- out for relativistic beams.

11. PARAXIAL THEORY

A. Introduction and Notation I n this section we will develop a theory for the spatial

distribution of current density in axially symmetric and two-dimensional beams. The following assumptions are made:

The current density at the cathode is uniform in one direction in two-dimensional beams, and is axially symmetric in axially symmetric beams. The current density at the cathode has a Maxwellian distribution of velocity, and any direction of emission is equally possible. The effects of longitudinal velocity spread may be neglected.

70 IEEE TIZANSACTIONS ON ELECTRON DEVICES March

4) All electromagnetic transverse forces are linearly proportional to distance from some central tra- jectory C,,.

Assumption 4) is almost, but not quite, the assumption which leads to the paraxial theory. In the paraxial theory assumption 4) is made in the beam. In the absence of thermal effects this covers all paraxial particles. With finite transverse temperature, some particles will stray far from the central trajectory of the paraxial theory. We will assume that assumption 4) holds for all particles. Later, in the treatment of higher-order effects, the cor- rections which may be made for the error in assumption 4) are considered.

The coordinate system is illustrated in Fig. 1 €or round beams, and in Fig. 2 for sheet beams. The theory could equally well be applied to hollow beams with curvilinear central trajectories C,, but these will not be considered in this section. In the round beam case the central tra- jectory is the x axis, and r , 6 are the usual cylindrical polar coordinates. All physical parameters of the flow are assumed independent of 6. The cathode is then assumed to be z = constant. The theory would hold equally well ir, spherical coordinates (r, e, +). In this case if the sub- stitutions

x + r > r + r e , 0 - 4

are made, the results would apply t o spherical cathodes. In the sheet beam case of Fig. 2, the beam is infinite in the y direction. The central trajectory of the system is C,, and the cathode is the plane x = constant. This co- ordinate system is the same as those of [3], [4f. I n the rest of this section we will discuss the two-dimensional case, but the results hold for the round beam by replacing x by r.

From assumption 4) it is seen that the electromagnetic force has the form xh(x). The relation between h(x) and the variation of fields on the actual trajectory Cg has been discussed in [3]-[6] and does not concern us here. In this case the relativistic Lorentz Force Law can be written in the form

dp,/dt = eh(,z)x, dx/dt = p J m , (1)

where

p = (dx/dt)/c = v,/c, y = 1/(1 - pa)'/', m = m,y. (2)

In (11, p , is the momentum in the x direction, m, e , the mass and charge of the particle; in (2) , v, is the axial velocity, c the velocity of light, and m, the rest mass of the particle. Defining new parameters

s = p,/(m,c), g(z1 = (T2 - l)-l/z,

f ( 4 = eh(x)/(m3c2/3).

(3)

Eq. (1) becomes

dx/dx = g(z)s, ds/dz: = f(x)x. (4)

It is to be noted that in (3) g, f are stated to be functions only of x . This follows from assumptions 3) and 4), so

r I

/; Z

Fig. 1-The coordinate system for round beams.

Fig. 2-The coordinate system for curvilinear sheet beams.

that it is assumed that all particles at the same x have the same axial velocity. It is easily verified that if sl, x1 and sz, 5% are two independent solutions of (4), then the cross-products

sp dx, /d .~ - 82 dxl/dx = XI dsa/dZ - 5 2 ds,/dZ = 0 , (5)

so that

slxz - xlsz = constant. (6)

We may therefore define, in the same way as Sturrock [3] the two solutions

x = all(z), s = al,(x) and x = s = azz(z), (7)

where the pairs of x, s have the values, a t z = 0,

a,, = 1 = azz, a,, = 0 = azl. (8)

Any solution of (3) can then be written in the form

where X, S are the values of (x, s) at x = 0, and, from (6)-(S), the determinant of aii is unity.

We will not concern ourselves with detailed expressions for ai;. They result from the normal solution of the paraxial equation with the boundary conditions of (8). The initial plane x = 0, may be taken as the cathode, or any other plane where the initial x, s are known or assumed. The conclusions which will be drawn are independent of the actual expressions €or the aii.

B. Boundary Conditions at the Initial Plane

It is observed experimentally that from an emitting cathode or plasma sheath, particles are emitted with equal probability in all directions according to a Maxwellian distribution of velocities. This experimental fact, which was assumption 3), leads to the conclusion that the transverse velocity distribution is also Maxwellian. In particular in the two-dimensional beams of Fig. 2, the

1963 Kirstein: Thermal Velocities in Beams 71

probability of having an initial transverse momentum in the x direction between p , and p , + dpz , is P(p,) dz where

Ph,) = (27rmokT)-”” exp - b t / ( 2 m o k T ) ] , (IO)

where k is Boltzmann’s constant, and T is the tempera- ture in O K of the source. If we define the parameter X by the relation

x = [moc2/(2kT)]1’2, (1 1)

and evaluate the probability of having transverse CO-

ordinate S, as defined by ( 2 ) , between S and S 4- ds this probability is P(s) ds where

P(S) = (X/ 4;) exp (-X’S”). (12)

In an axially symmetric system, (12) holds identically with P(sz)P(s,) ds, ds,, replacing P(s) ds.

If the current density at the cathode varied linearly with x, then the current between x and x + dx is i c ( x / a ) dx where

ic(x/a> = IIc/(2a)l[1 + &/a)l, (13)

the boundaries of the two-dimensional beam are f a , I , is the total current per unit length in the beam, and q is the factor of proportionality. Eq. (12) is often satisfied in practice, and is always satisfied in paraxial beams from a planar cathode. If j(x, s) dx ds is the current density in 2, s phase space, then from (12) and (13) the value of j(x, s> at the cathode is j c ( x , s), where

j h , 4 = [Jc/(2a)l[l + ! ? ( 4 a ) l ( V 4 4 Sexp (-X’s’), -a 5 x 5 a . (14)

= o otherwise I = o otherwise 1

At the cathode of axially symmetric beams with current density variation iC(r/a) , the current density at the point (x, y, s,, s,) in four-dimensional phase space is given by

j o b , Y, 8, , s,) = i c (?/a> (X”3 -exp [-X2(s: + si)], 0 i r 5 a , (15)

where now s,, s, are the x and y components of s, a is the radius of the beam, and r, s are related to x , y and s,, s, by the formulas

r2 = x’ + y2, s” = s: + s:. (161

We will usually consider the case of constant current density, where i, is constant, but will derive expressions for the general case.

C . The Current Distribution in Round Solid Beams

We are now in a position to find the current density in (x, s> or (r, s ) space, and therefore in real space, in two-

dimensional and axially symmetric beams. Let US first consider round beams.

In Cartesian coordinates with axially symmetric fields, it may be verified that the x, y coordinates obey identical equations, namely ( I ) , so that their transformation laws are given by (9).l Hence the point (x, y, s,, s,) comes from ( X , Y , Ex, 8,) if, from (91,

a l l X + alz& = x , a l l y + a l zS , = Y. (17)

The line elements ds,, ds,, for constant x, LJ are related to dS, dS, by the relations

ds, = az, dX + aZ2 dS, = dS,/all , ds, = dS,/all , (18)

so that the current density i ( x , y, x ) at (x, y) is given by

‘(y - a l z S J / a l l , %, X,] dX, dX,. (19)

Clearly i(z, y, x) depends on x through the a i j . To obtain the current density in an axially symmetric beam, we may transform into axially symmetric coordinates (r, 0) in real space. With the transformation

x = r cos 8, y = T sin 8,

alzSz = r cos - r cos 4, alzS, = r sin 0 - {sin 4

the differential line elements become

Substituting (15), (20) and (21) into (19), and integrating over e to find the total current i l(r , z) dr between r and r + dr, we find that

i l ( r , x) = ri(x, y, x) d8 s,

fields, there is coupling between the x and y planes. Eq. (9) then, 1 This is only true for electrostatic fields. If there are magnetic

still holds if x is replaced by z = iy, and now aij may take complex values. It is then necessary to replace aij by aii in equations such ILS (23 ) or (27) (i.e., Reference [lo]).

72 JEEE TBANSACTIONX ON E'LECTRON DEVICES March

Normalized Distance from Axis +/x = r / (a l la)

Fig. 3-Curves showing the character of the current density variation in a round beam which has been dispersed by thermal velocities.

Defining now the convenient parameters

x = aa~,X/a~,, 1c. = rX/al2, 4 = Xt/a12, (23)

and using the relation [2]

s, exp (a cos u) du = 27rI,(U), 27r

(241

where I,(a) is the modified Bessel function of zero order, (22) becomes

i l ( r , 2) E (4.n~'Xz) Lx ic(t/x)t exp - ( t z + $*jIn(2t$) dt, (25)

where i, was defined in (15). If i, is constant, the i, of (25) is the same as that

obtained by Cutler and Wines [2] in (31) of their paper. To compare their results with ours, one must use the substitution

u = u12/( di), r , a,,?,

x = T . / ( d % T ) , $ = r / ( 4%). (26)

For the case i, constant, curves are given in j2]. We see that the effect of the current density variation at the cathode puts in the extra term i,(E/x) in the int,egral. The i l (r , x ) from (25) have been evaluated numerically

in [21 and [71, for constant i,. We present curves, however, for the constant current density case in a slightly different way from the curves of 121. In Fig. 3, the quantity i(r, x ) / i o , where i o ( T , x) is the value of i in the beam in the absence of thermal effects, is plotted vs $/x, which is (./a ulJ. For highly convergent beams, with x less than unity, these curves are difficult to use. Hence in Fig. 4 we plot i(r, x)/'(x2io) vs $. For hollow beam applications, the integral is taken from x1 t,o xa instead of from 0 to x where xl and x, are expressions analogous to x with the inner and outer cathode radii replacing a.

Finally it is important in Section I11 to have expres- sions for the total current I (r , x) inside a radius r. Using the definitions of (25) we see that

Eq. (27) has been integrated for the constant current density case in [2].

We will not evaluate any part'icular electromagnetic structure as an example of this theory. Suffice to say that it is valid, to the linear approximation, for any beam or gun for which the paraxial equation is valid. For the higher- order corrections, it is necessary to use the methods of

1963 Kirstein: Thermal Velocities in Beams 73

Normalized Dist,ance from Axis rl. Fig. &Curves showing the character of the current density variation in a round beam

which has been dispersed by thermal velocities. These curves are in more suitable u n i t s than those of Fig. 3 for highly convergent beams.

Section 111. The theory of this section can easily be applied to hollow beams with a linear variation of current density at the cathode. In this case the same parameters can be used, but the limits of integration, and the functions i(r, x ) will be slightly changed. It is still possible, however, to express these functions analytically in terms of the error functions.

D. The Current Density in Two-Dimensional Beams The methods of Section 11-G can be applied equally

well to the evaluation of the current density and total current in strip beams, infinite in one direction. Assuming a linear variation of current density according to (13) at the initial plane, which will usually be taken as the cathode, the current density in phase space at this plane is givGn by (14). Assuming a current density &(%/a) at x = 8, we may make a computation similar to that of (19)-(261, with x replacing r. The current density i(x, x ) at x may be derived by similar transformations, namely

i(x, x) = j(x, s) ds = (- j c (X , dX/all

= X / ( G a l l > J-1 i,[(x - a,,~>/(a,,a)l exp -(AS)' as

= [l/(V'&~)l 1-1 i d t / x ) exp -(t - 49' d t , (28)

1, d - m

where now a is the half-width of the beam at the cathode, and $, X are given by (23) with x replacing r . If we re- placed the exponential term by the Dirac Delta function, (28) would give the current density to be expected in the absence of thermal effects. For the particular case of a linear variation of current density at the cathode, i,(x/a) takes the form of (13), and the current density is given by the expression

where E(%) is the usual error function

(2,' &) [ exp (- x2) dx.

In the low temperature limit, I $ 1, [ x I 3 03 , the expo- nential terms tend to zero, while the bracket with the

IEEE TRANXACTIONS ON ELECTRON DEVICES March

Normalized Distance from Central Trajectory $./X = x/(a,,a)

h

0

Normalized Distance from Central Trajectory $/x = x/( alia) Fig. 5-Curves showing the character of the current density variation in a sheet beam, with linear current

density variation at the cathode, dispersed by thermal velocities. The ratio of maximum to minimum current

q = 1. density at the cathode is ( I f q ) / ( l - q). (a) Constant current density, q = 0. (b) Linear current density,

1963 Kirstein: Thermal Velocities in Beams 7 5

Fig. 6-Curves showing the character of the current density variation in a sheet beam dispersed by thermal velocities as in Fig. 5. These curves are in more suitable units for highly convergent beams. (a) Constant current density, q = 0. (b) Linear current density, q = 1.

error functions gives unity inside the original beam, where I I,5 I 5 x and zero outside it. This is the current density in the absence of thermal effects. The expression for i(z, x ) takes a particularly simple form for constant current density, where q = 0.

Curves are given in Figs. 5 and 6 for i(z, z ) / x . For highly convergent beams, Le., for small x less than unity, the curves of Fig. 6 are more useful. The different curves refer to values p = 0, p = 1 of the parameter p, which determines the initial current density variation at the cathode. Since i(x, x) is linear in q, the values of i ( z , x ) for different q may be found by linear interpolation be- tween the values on the q = 0 curve, and those on the q = 1 curve.

A quantity which is of value later in Section 111, is the total current I ( z , x), between the nominal center of the beam and the plane 2, at the axial position x . This current is given by the rela.tions

I ( x , x) = i(x, X ) dx = aa,, i(x, x) d$/x r o IS I+

= [1,/(2x)1[9(1 - dFd$ + x> - 9 0 + q)F2(I,5 - x) + @,(x) + t 4 / ( 2 x ) ) {PI(# + x) - Fl(# - x) - 2FI(X)J I, (30)

F,(x) = +(x" + ~ ( l c ) + [ X / ( ~ V % I exp (-x2> . (31)

~ , ( x ) = z ~ ( z ) + (l/v'G) exp (-z'>

In Fig. 7, curves are given for I(x, x j / I c as a function of t,b for different values of the parameters x, q. The function I ( z , x ) / I , , is the percentage of the total current con- tained between x = 0 and z = x a t the plane 2.

An illustration of the use of this theory is given at the end of Section 111.

where F , and F , are given by the relations

1 J

Normalized Distance from Central Trajectory $/X = z/ (al la)

March

Normalized Distance from Central Trajectory $/x = z / ( a ~ ~ a ) Fig. 7-Curves showing the per cent of the total beam current to be found between the

planes z = 0 and z = z in a sheet beam dispersed by thermal velocities as in Fig. 5. (a) Constant current density, q = 0. (b) Linear current density, q = 1.

111. HIGHER-ORDER CORRECTIONS space-charge distribution to the nonlinear fields. Finally we consider in Section IV-D a particular example, the spreading of a sheet beam in the absence of applied fields,

In Section I1 of the paper, we discussed the effects of but allowing for nonlinear space-charge effects due to an linear applied fields, i e . , paraxial fields, on particles which initially Maxwellian distribution of transverse velocities. originally had a Gaussian distribution in phase space. The effects discussed in Section I1 will be called linear B. The Formalism with Slightly Nonlinear Fields effects. In this part will be discussed the perturbation introduced by slightly nonlinear fields. These nonlinear In Section 11 the effects of linear fields were discussed.

fields may come from two factors. First, aberrations may In this section we discuss the perturbations which arise

cause the applied fields to become nonlinear. Secondly the in slightly nonlinear fields. Let it be assumed that there

h e a r thermal effects will redistribute the space-charge, in addition to the linear h(x). In the axially is a nonlinear transverse field &(x, x) in the x direction

causing nonlinear space-charge fields. In Section 111-B we develop the general nonlinear theory of the motion case the extra field would be e(?+, x ) in the r direction. In

of particles in phase space. In this theory, the linear the x, s coordinate system of (3 ) and (4), the equations behavior is assumed known. In Section 111-C the theory of motion become, for a two-dimensional beam,

is applied to round and strip beams to evaluate the dx/dx = g(x)s, ds/dz = f ( 2 ) ~ 9 F(z, z), (32)

A. Introduction

1963 Kirstein: Thermal Velocities in Beams 77

where f , g, s were defined in (3) and F is related to E

by the expression

F ( x , x) = eE(x, x)/(mopc2). (33)

If we write

x = x,+(, s = so+ b , (34)

where z0, so are the solution of (32) with F = 0, then substitution into (32) yields the relation

dE/dx = g(x)f, d{ /dz = f ( x ) E + F(X0 + E , 4. (35)

F ( Z 0 + E , x ) -h F ( X 0 , 4 + tF,(XO , 4 7 (36)

If the perturbation is small, then to first order in f we may write

I

(a>

where F,(xo, x ) is the derivative of F(xo , z) with respect to X. By successive iteration, we find that to second order the solution of (35) is

>. (37)

+ 1' f(u) du u o Iu fog(.) dv + l' t0F,(xo, u) du

In (37) to and f a are the original value of E and {. If we assume that the initial position of a particle in phase space is given by the linear theory of Section 11, SO that E,,, c0 are zero, then (37) takes the simple form

Eq. (38) shows that while the linear theory transforms the region of the beam from Fig. 8(a) to 8(b), the non- linear theory to this approximation transforms the region into that of Fig. 8(c). However, since the perturbation term is only dependent on the position of the particle in the absence of nonlinear effects zo, the actual current density at the point corresponding to zo is unaltered. However, the point corresponding to zo is shifted by the amount of (38). The total current between z and (z + dz) remains J i(xo, z) dzo, while t,he element of length is (dxo + d t ) . Therefore the current density i(z, 2) at the physical point zo is given by

i ( Z 0 , x ) = io(x0 - t , z)/(l + dEldx0)

= io(z0, x ) - d [i,(zo, x)El/dxo,

i o (xo , x) - i: di,/dxo - io dfldzo (39)

where io is the i of the linear theory. The formulas for I (%, x ) are unchanged, though it must be remembered theory.

75 IEBE TRANSACTIONS ON ELZCTRQN DEVICES March

that they refer to the current inside (xc + 0 . The param- eter $ of (23) is given by

$ = xox/a,z. (40)

I n axially symmetric systems all the results of this section arc applicable with rr ro replacing x, xo. The spread of transverse momenta is not changed a t x by the nonlinear perturbation, however, the s of each particle is given the extra increment j- of (38). Thus the nonlinear term is equivalent to a long lens of strength (Ji F(xO, u) du) at the point corresponding to xo. We note that the method adopted here is quite different from that used by Danielson, et a1.171. They considered the nonlinear forces experienced by an average particle, and estimated that these forces were.a good approximation to the forces experienced by the beam as a whole. We compute, to the second approxi- mation in x, the forces experienced at each point in space. Clearly at no part of this section is it material whether the problem is two dimensional or axially symmetric. Such considerations only enter into the Computation of F .

C. The Computation of the Space-Charge Forces

In Section III-B a theory was developed based on a nonlinear forcing term F(x , x). It was stated that this non- linear term was made of two terms; the first term is due to aberrations of the system, the second is due to space- charge. The first term can be computed by the methods developed by Sturrock [3] . This term will not be discussed in this paper, we will however give an estimate of the space-charge term F,.

1) Round Beam: Let us assume an axially symmetric beam, uniform in the x direction. It can be shown from Green’s functions considerations, that the electrostatic radial field E , due to a current distribution i(r, z) can be given by the expression

E&, X ) = i(r* x)/(2nre0v,) dr ,

where the axial velocity v, is assumed constant across the beam, i(r, x ) is assumed a slowly varying function of x, and eo is the dielectric constant of free space. I n the relativistic case the self-magnetic field reduces the force due to the field by y2 so that the space-charge force F 8 ( ~ , z ) is given using (25) and (33) by the expression

PB(r, x ) = (y2 - l)”zAI(rl x) /Ic , (42)

.l (41)

where A has the value

A = - I ) -”/z / (7r~) , (43)

and I ( r , x) given by (25). In (43), X is a constant which has the value in mks units from [9]

X = 3.82 X for electrons

= 2.08 X for protons J

In the nonrelativistic case, if the beam has voltage V , it is shown in [9] that

A = [X’/(na)] [ Ic / V3”] , (45)

where

X’ = 4.78 X lo4 for electrons

= 2.05 X lo6 for protons J

If the rate of change of axial velocity with distance is fast, as a t a space-charge-limited cathode, then (41) is not strictly valid.

2) Two-Dimensional Beam: If the beam is two-dimen- sional, and it is assumed again that the axial velocity does not vary too rapidly in the x direction, the expression analogous to (41) for the transverse electric field is

Using the definitions of I ( x , x ) and F(x , x) of (30) and (33), it can be shown that the space-charge force F,(x, z) is given by [9]

Fa(%, X ) = [2I(z1 x) - I( - , X)

where A is given by the expression

and X is given by (44), while I , is the current per unit width of beam. In the nonrelativistic case,

where X’ is given by (46). The higher-order correction due to space-charge may now be evaluated by the methods of Section III-B.

D. The Space-Charge Spreading of a Sheet Beam in a Drift Region

To illustrate the method, we will make a computation of the space-charge spreading of a sheet beam in a drift region.

This problem has been treated by many authors [2], [7], [9]. It will be assumed that the beam is in a region of zer0 applied field, and that originally the current density is uniform across the beam with a Maxwellian distribution of velocities. This means that the beam originally occupies the region of phase space shown in Fig. 6. Unlike previous authors, we will make a nonlinear computation which allows for the change of space-charge

1963 Kirstein: Thermal Velocities in Beams 79

forces due to the change in current distribution. The linear behavior assumes no space-charge spreading, the nonlinear term will include all the space-charge effects. It is possible to choose, for the linear model, the situation with constant space-charge. In order to solve the problem on a digital computer, this would be the best procedure. We intend to give an analytic solution; this is difficult to do if the a,,, al, of the matrix of (9) are complicated.

Let us first consider the linear behavior. The assumption is made that the beam originally has thickness 2a, tempera- ture T, current I , and ratio of energy t.o rest energy y. Since the equations of motion in the absence of any field are, from (3) and (4)

dx/dx = s/ d z , ds/dx = 0, (5 1)

the a,,, a,, of (7) and (8) are given by

a,, = 1, a12 = z / d q - 1. (52)

Hence according to the linear theory, using (19) #, x may be defined by the relations

x = aa/z , # = xoa/x, (53)

where x. is the position of the particle according to the linear theory and

a = x-. (54)

Using (52) and (53) for a,,, x , # the current density is obtained directly from (28). Since it is assumed that the initial current density is constant in real space, q = 0 in (28). The expression for &(xo, x ) then becomes

io(x , ,x) = [1,/(2a)][BE(# + x ) - %E(# - x)!. (55)

Using the expression for F,(zo, z ) of (48) and for I(%, x ) of (30) with q = 0, it is seen that F,(xo, x ) is given by the expression

Fs(xo, Z) = A d y 2 - 1 [ F Z ( # i- X) - PZ(# - X)]. (56)

where #, x are given by (53) and F 2 ( x ) from (31) by the expression

_-

F*(x) = xE(x) - e ? / G . (57)

Using the F , of (56), g(z) of (3 ) and #, x of (53) , we obtain the expression for E , from (38),

E = A J z du /’ (F2[a(xo f a>/vl

- Fz [a(xo - u>/v]) dv. (58)

The right-hand side of (58) can be integrated analytically in terms of t,he error function E ( z ) and the exponential integral Ei(z) defined by

Ei(x) = Trn eUz/z dx. (59) “ 2

In terms of these functions it may be verified that (58) yields the relation

where F3(x) has the form

and the positive sign holds for positive x, the other for negative. Clearly for general a, x, it would be necessary to evaluate (60), and this does not seem worth the trouble. However, for large x, the asymptotic expression may be used

F3(x) M =tx/2 f 1/(4 %‘%xz) exp (-x2), (62)

the positive sign refering to positive x. Using this ex- pansion, (60) becomes

although this approximation is not good enough near the edge of the beam if a(zo =k a) = z. The first term gives the usual linear behavior in the absence of thermal effects, since the bracket gives zo for x. 5 a and a for x. 2 a, while the second term gives the nonlinear term due to the thermal velocities changing the space-charge distribution. The current density i(xo, x ) may be found from (39), namely

In (64), io, 5 are given by (55) and (60). This problem is of insuBScient practical interest to justify calculating i exactly from (64). However, one may use the asymptotic expansions for of (63) for large a(xo f a ) / z and a simiIar one for i(xo, x ) using the asymptotic expansion

Using these expansions, the current density is given by

where F4(z1 z) is given by the expression

The approximation of (67) is not good for x ‘v a. The ia(zo, z) in (65) has already been plotted in Fig. 6, the

80 IEEE TRANXACTIONS ON ELECTRON DEVICES March

0%-her term is the nonlinear perturbation. It seems of !&le practical purpose to evaluate the [ and i of (63) and (66). This example illustrates the method, which can be used for far more complex problems. As even this simple example shows, the analytic application of the method will usually be impossible. However the method can be applied numerically without difficulty. It is to be noted that unlike Danielson’s [7] inclusion of nonlinear effects, this application is equally good both inside and outside the beam. Finally, more complicated electro- magnet’ic forces, due to aberrations in the system, can be used for F .

ACKNOWLEDGMENT

The author would like to acknowledge the help of the computer department of the Joint Institute for Nuclear Research Dubna, U. S. S. R., and in particular of P. A. Polubojarova who did the numerical computations of this paper.

REFEREKCES [I] J. R. Pierce, “Theory and Design of Electron Beams,” D. Van

[a] C. C. Cutler ,Fnd M. E. Hines, “Thermal velocity effects in Nostrand Company, Inc., New York, N. Y., 2nd. ed.; 1954.

[3J P. A. Sturrock, “Static and Dynamic Electron Optics,” Cam- electron guns, PROC. IRE, vol. 43, pp. 307-315; March, 1955.

bridge University Press, Cambridge, England; 1955. [4] P. T. Kirstein, “Paraxial formulation of the equations of electro-

static space-charge flow,” J. A p p l . Phys., vol. 30, pp. 967-975; July, 1959.

/5] P. T. Kirstein, “A paraxial formulation of the equations for

vol. 8, pp. 207-225; March, 1960. space-charge flow in a magnetic field,” J . Electronics and Control,

[6] W. E. Waters, “Rippling of thin electron ribbons,” J. A p p l .

[7] W. E. Danielson, J. L. Rosenfeld and J. A. Saloom, “A detailed Phys., vol. 29, pp. 100-104; January, 1958.

analysie of beam formation with electron guns of the Pierce type,” Bell Sys. Tech. J., vol. 35, p 375-420; March 1956.

[8] P. T. Erstein, “On the variation of t eam boundaries mi linear beam transport elements and the computation of space-charge effects,” Joint Institute for Nuclear Research, Dubna, U.S.S.R.,

Control. Dubna Preprint E-1023; To be published in J . Electronics and

[9] E. R. Harrison, “On the space-charge divergence of an axially symmetric beam,” J . Electronics and Control, vol. 4, pp. 193-200; March, 1960.

1101 P. T. Kirstein, “Thermal velocity effects in axially symmetric solid beams,” submitted to J . A p p l . Phys.

Parametric Interaction in Electron Beam Waves - A System Function Characterization*

CASPER W. BARKES-J-, SENIOR MEMBER, IRE

Summary-In this paper we develop a technique for characterizing parametric interaction in electron beam waves in terms of a system transfer function. We consider systems in which the parametric interaction is described by a matrix equation of the form (a /a f $- uoa/d,)a = - 1 9 3 a where a is a state vector of the system and H is the matrix characterizing the interaction. Solutions in the time domain can be expressed in the form a(z, f ) = M(z, t ) a(0, t - z/u0) where M (a matrix) is called the system function in the time domain. Solutions in the frequency domain can be expressed in the form 8(z, o) = J &(z, o - o‘)B(o, o’)do’ where S(z, o) is the Fourier transform of a(z, f) and &(z, o) is the transform of M(z, t ) .

Solutions for parametric interaction in transverse beam waves are worked out for two cases: In Case I we derive the well-known result that describes the behavior of a single pump wave, electron beam parametric amplifier. In Case I1 we consider a parametric interaction involving two pump waves in which an infinite number of fre- quencies are parametrically coupled.

I. INTRODUCTION HE PURPOSE of this paper is to develop a simple and direct technique for analyzing the b e h a ~ o r of parametrically coupled electron beam waves.

This technique permits us to characterize the behavior of these waves in terms of a general system transfer function which is obtained directly from the solution of the time-domain coupled-mode equations.

ber 23, 1962. This work was supported by the Aeronautical Systems * Received October 5, 1962; revised manuscript received Novem-

Div., Wright-Patterson Air Force Base, Ohio.

Menio Park, Calif. t Physical Electronics Laboratory, Stanford Research Institute,

In the conventional method of analyzing traveling-wave parametric systems,‘-’ one assumes the presence of waves a t a discrete set of frequencies*&?., signal frequency, idle frequency, and perhaps additional waves at higher parametric sidebands. This set of assumed waves is then substituted into the appropriate coupled-mode equations, and the resulting terms of equal frequency are equated separately. Although this procedure is valid provided one is careful to assume the correct set of frequencies, it does

in propagating circuits,” J. A p p l . Phys., vol. 29, pp. 1347-1357; 1 P. K. Tien, “Parametric amplification and frequency mixing

September, 1958. 2 W. H. Louisell, “A threefrequency electron beam parametric

amplifier and frequency converter,” J . Electronics and Control, vol. VI, pp. 1-25; January, 1959.

fication,” J . A p p l . Phys., vol. 31, pp. 338-345; February, 1960. 3 Curtiss C. Johnson, “Theory of fast-wave parametric ampli-

electron-beam parametric amplification,” J. A p p l . Phys., vol. 32, * R. W. Gould and C. C. Johnson, “Coupled mode theory of

pp. 248-258; February, 1961. R. W. Fredricks, “Traveling wave analysis of a class of para-

metric amplifiers based upon the hill equation,” J . A p p l . Phys., vol. 32, pp. 901-904; May, 1961.

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May, 1961. J. W. Kluver, “Parametric coupling between the transverse

waves on 0- and M-type beams,” J . A p p l . Phys., vol. 32, pp. 1111- 1114; June, 1961.

E. I. Gordon and A. Ashkin, “Energy interchange between

J . A p p l . Pkys., vol. 32, pp. 1137-1144; June, 1961. cyclotron and synchronous waves in quadrupolar pump fields,”

0 William H. Louisell, “Coupled Mode and Parametric Elec- tronics,” John Wiley and Sons, Inc., New York, N. Y.; 1960.