Synchrotron Radiation as an Infrared Source

James R. Stevenson, H. Ellis, and Roger Bartlett

The increasing availability of synchrotron radiation sources in a number of geographical regions of theworld has motivated an evaluation of the radiation from electron accelerators and storage rings as a pos-sible source for ir spectroscopy. As synchrotron radiation can be analytically described, a direct compar-ison is made with blackbody radiation for typical solid state spectroscopy. Both existing and proposedsynchrotron radiation sources are found to be attractive in the ir. The ultrahigh vacuum environment ofthe source is compatible with clean surface investigations, the analytical description of the radiation isappropriate for calibration studies, and the continuous nature is suitable for Fourier spectroscopy orwhenever a white light source is desirable.

I. IntroductionSynchrotron radiation is primarily regarded as a

soft x-ray and VUV source, and its applications inthese areas are well documented.1-5 Until the pres-ent the ir utilization of existing sources has been ne-glected; however, the increasing availability of syn-chrotron radiation as a spectroscopic source suggeststhe desirability of a reexamination of its propertiesin the ir. In 1966, prior to the recent growth in syn-chrotron radiation spectroscopy, a Solid State Panelof the National Research Council6 came to the fol-lowing conclusion concerning the building of a syn-chrotron source for ir applications:

It will be seen that radiation obtainable at wave-length of the order of 100,g from a synchrotron de-signed as an ultraviolet source is useful-particularlyin that it is sharply confined near a plane, but hard-ly justifies the development of an expensive machinespecially for this purpose. It also does not appear tobe justifiable to build a machine of this kind special-ly for the far infrared unless currents considerablylarger than those contemplated at present can be ob-tained in the future. One might be forced to modifythis conclusion if a workable scheme for utilizing alarger fraction of the radiation pattern can be de-vised.

The general conclusion is still valid, and onewould not justify building a synchrotron radiationsource on the basis of ir applications. However, thespecific statement is also true that the existing syn-chrotron radiation sources may provide useful ir ra-diation.

Current work at synchrotron radiation facilities isThe authors are with the School of Physics, Georgia Institute of

Technology, Atlanta, Georgia 30332.Received 28 March 1973.

providing a rather strange marriage between the eVspectroscopists in solid state physics, atomic physics,molecular chemistry, and molecular biology on onehand with the GeV particle physicists on the otherhand. As funding for major scientific facilities forbasic research becomes more difficult throughout theworld, the presence of this marriage of disciplines viaa major facility becomes more desirable. Currentconstruction of electron-positron storage rings atDESY in Hamburg, W. Germany exemplifies thedesirability of advance planning with synchrotronradiation users being considered in the initial design.Experimental facilities for both biology and physicsare being provided in the initial construction. Therings at DESY are designed to operate at 6 A at 1.75GeV or 300 mA at 3.5 GeV. An estimated 1.5 MWof synchrotron radiation should result at maximumpower.7 Although the storage rings are primarilydesigned for electron-position collision experiments,the proposed power levels will provide a brilliantsource from the x ray through the far ir. In contrastto the DESY facility, this paper will be primarilyconcerned with the existing UWPSL 240-MeV elec-tron storage ring located at the Physical ScienceLaboratory of the University of Wisconsin.8 TheWisconsin storage ring has typically operated in the2-10-mA range and is dedicated to synchrotron ra-diation users. A new proposed ring, Tantalus II,would considerably increase the energy and intensi-ty.

The analytical description of synchrotron radiationallows for an easy extrapolation between different fa-cilities.

11. Synchrotron Radiation-Basic Characteristics

Before a meaningful description of the ir character

2884 APPLIED OPTICS / Vol. 12, No. 12 / December 1973

of synchrotron radiation can be given, some back-ground material must be presented to describe thebasic nature and define the characteristic parame-ters. Many excellent treatments of the general de-scription of synchrotron radiation are available inthe literature. Theoretical background for this pre-sentation was primarily obtained from Schwinger9and Sokolov and Ternov.' 0

An expression for the power radiated by a singleelectron per unit range in angular frequency and in-tegrated over all angular dependence has been de-rived by Schwinger9 and is given below as Eq. (1):

3 3/2e2 / E \ ~ C

P(W) = 34wR (Mc 2) Wt2 KJ51 3(7)d7rAw. (1)

In the above expression, e is the electronic charge,mO is the rest mass of the electron, R is the radius ofthe orbit, E is the energy of the electron, and Aw isthe angular frequency range. The parameter o isdefined as the ratio of the velocity of light to the ra-dius of the orbit wo = cR. The characteristic angu-lar frequency wc is given by the expression

sion for the integral given in Eqs. (2) and (3) isshown along with the computer evaluated integral asthe upper curve. On the scale used the two curvesare not distinguishable, and the difference betweenthe two expressions is plotted as the lower curve inthe same figure. At a value of q = 0.01 the asymp-totic expression differs by 4% from the actual value.As seen from Fig. 2, the agreement is considerablybetter for smaller values of Xq or longer wavelengths.

Although the functions shown in Figs. 1 and 2 givea theoretical characterization to synchrotron radia-tion as Planck's radiation law gives for blackbody ra-diation, a useful comparison of sources for experi-mental purposes must also consider the collimationand available solid angle of the source, polarization,and the desired resolution. Frequently the type ofdetector, quantum or thermal, dictates the choicebetween G(w) and G(X) for describing synchrotronradiation. As these expressions do not give the com-plete picture, some additional characteristics of syn-chrotron radiation must be discussed.

= (3/2)wo(E/mc2)3.

The integrand K5/ 3(7) can be expressed" in terms offractional Bessel functions with imaginary argu-ments as

K,,(77) = (7r/2) {[ 51 /3(G7) - 1/ 3 (i7)]/sin(5r/3)j

or in integral form as

K5 /3(77) = exp(-77 cosht)cosh 5- dt.

As can be seen from the expressions for the integrandin Eq. (1), the calculation of the energy radiated bya single electron requires numerical integration ex-cept when asymptotic expressions can be used forthe Bessel functions. Asymptotic expressions areavailable whenever c < or c »> w. These ap-proximations as well as computer generated tables ofthe values of functions useful in synchrotron radia-tion calculations are available in a special reportfrom the authors. The spectral distribution of theenergy is determined by the functions

G(w) = f fK 51 3(X)dX

and

G(X) = i73f K5 3(X)dX,

lo,

10°

lo-'

10-2

10 -3

10-4

10-5

10-7 I0 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00

1/i = WAVELENGTH / CRITICAL WAVELENGTH10.00

Fig. 1. The function G(X) as a function of the wavelength rela-tive to the critical wavelength, calculated from Eq. (3).

1a-5(2)

-9,( 3 ) 1 o'a

(3) A,' 10-7

respectively. These analytical expressions for syn-chrotron radiation have their analogs in blackbodyradiation and enable a direct comparison betweenthese sources. A computer generated plot for G(X)is in Fig. 1. The maximum in G(X) appears at 7 =2.35, while the maximum in G(w) appears at =0.29. Figure 1 accentuates the weak wavelength de-pendence for wavelengths longer than the position ofthe maximum compared to wavelengths shorter thanthe maximum.

In Fig. 2 the long wavelength asymptotic expres-

10-9 0100 200 300 400 500 600 700 800 900 1000

l/ = WAVELENGTH / CRITICAL WAVELENGTH

Fig. 2. Asymptotic behavior of G(X) calculated for wavelengthslong compared to the critical wavelength is shown in the uppercurve. The lower curve is the absolute difference between G(X)

and the approximation given in Ref. 10.

December 1973 / Vol. 12, No. 12 / APPLIED OPTICS 2886

. I . 0

.1-9

EiV.

I-'r

10-6

6

10-8

The geometrical character of the ra(and the polarization are important to mtions in spectroscopy. For synchrotron raxis of the radiation cone is tangent to t]orbit and predominantly in the forwardmotion of the electron.' The intensityin the orbital plane away from the tangei

.stantaneous velocity is not critical to thisince the motion of the electrons in the c

8.0

6.0

C: 4.0

-2

2.0

0 0 4.0 8.0 12.0 16.0 20.0

VERTICAL ANGLE (MRAD)

Fig. 3. The distribution of intensity (in arbitrartwo linear polarization components of radiation fr(AC = 262 A) at 10 in the angle measured from thbital plane. The component polarized parallel

plane is the middle curve.

12.0TOTAL ENERGY

9.0

-I

3.0

3.0

liation field will sweep a distribution of tangents into an entrancelost applica- slit of finite width. The distribution of intensity inradiation the an angle 0 measured from the orbital plane dependshe electron's on frequency.direction of For wavelengths below or near the critical wave-distribution length, almost all the radiation is emitted within 2.nt to this in- mrad or 3 mrad of the orbital plane and is polarizedis discussion parallel to this plane. For wavelengths > , therbital plane angular spread of intensity becomes significant, and

the polarization is strongly dependent on the angle 0.The two components of polarization have been plot-ted for parameters appropriate to the UWPSL stor-

1, age ring ( = 262 A) at a wavelength of 10 in Fig.3. Figure 3 does show the possibility of a problemfor some ir spectroscopy not present for shorterwavelengths. For wavelengths longer than XA, astriation of the polarization appears and is seen inFig. 3 as one departs from the orbital plane. For ex-ample, at = 10 , the radiation between 0 mradand 4 mrad from the vertical or the central portionof the beam is approximately 90% polarized parallel

240 280 to the plane, while between 12 mrad and 16 mrad*4.0 28.0 the components of linear polarization parallel andperpendicular to the plane become almost equal.

y units) of the Thus a sample radiated with an image of the sourceom eleTantalus I would not be radiated with a uniform polarization.eetron or-a However, the use of selective apertures could resultto the orbital in a choice of polarizations.

The sum of the two components is plotted for sev-eral wavelengths in Fig. 4 and shows that as one goesfurther into the ir a larger vertical angle must be in-tercepted. Fortunately the angles are relativelysmall, and the use of mirrors can still intercept mostof the radiation. For comparative purposes a usefulparameter is the rms angle of the radiation as intro-duced by Hartman and Smith.6 Making use of theexpressions for XC given by Schwinger, 9

A = 5.59{R(meters)/[E(GeV)]3 A,

00 4.0 8.0 12.0 16.0 20.0 24.0 28.0

VERTICAL ANGLE (MRAD)

Fig. 4. The distribution of total power per unit frequency (inarbitrary units) for = 0.026 as radiated into the angle mea-

sured from the electron's orbital plane for several wavelengths.

we obtain

(02)1/2 = [m.C 2c/(5.59) 1 1 3 AlX1 /3 /R1/3 (4)

This value of rms angle is 15.5 mrad for the UWPSLstorage ring at 10 and includes about 80% of theradiated power, as seen from Fig. 4. Using this pa-rameter to characterize the angular spread, Table I

Table I. Angular Spread of Synchrotron Radiation for Some Storage Rings

(02)1/2, Tantalus IIc(0)1/2, Tantalus Ia (02)'/2 DESY b E = 1.76 GeV, y = 3500,

X in units of E-240 MeV, -y = 471, R =0.65 in E = 3 GeV, y = 5880, R = .0m R = 4.5 mX, 2.1 mrad, X = 2623A 0.17 mrad, X 10. 5A 0. 28 mrad, X = 4.6 A102Xc 9.8 mrad, X = 2 .6 2 a 0.79 mrad, x = ioso3A 1.32 mrad, X = 460 A103XA 21 mrad, X = 26.2 A 1.7 mrad, X = 1.05 A 2.8 mrad, X = 4600 A104%, 45.7 mrad, x = 262,u 3.66 mrad, X = 10.5 - 6.14 mrad, X =4.6 p

a Tantalus I is in operation at the University of Wisconsin and dedicated to synchrotron radiation users.b DESY is under construction in Hamburg, West Germany.

Tantalus II was proposed by the University of Wisconsin as an electron storage ring dedicated for synchrotron radiation users.Sinco the time this paper was originally written, the proposal to build Tantalus II as been declined. owever, the parameters providea useful comparison between Tantalus I and the higher energy machines at DESY and SLAC.

2886 APPLIED OPTICS / Vol. 12, No. 12 / December 1973

compares the UWPSL machine, Tantalus I, with thedesign parameters of the new DESY facility and theproposed new storage ring at UWPSL known asTantalus II. As seen in Table I the angular diver-gence of the radiation is quite small and confinedclose to the orbital plane. The collimation factor isimportant in the consideration of geometrical factorsfor spectroscopy applications.

III. Snchrotron Radiation-ExperimentalConsiderations

As seen from Table I, the higher energy machineprovides less divergence, but this advantage is some-what offset by the longer distance between the sourceof synchrotron radiation and the intercepting mirroror other entrance aperture of the experimental ar-rangement. If individual charged particles are con-sidered, a time average must be taken over the lengthof the orbit in which the synchrotron radiation can becollected by the entrance aperture. However, in deal-ing with the electrical current present in the storagering or synchrotron, the time averaging is implied.Other experimental considerations include the fi-nite size of the electron beam as well as the desiredresolution.

As a starting point of our discussion, Gahwiller etal.

3 have derived a meaningful expression for thenumber of photons radiated per second and Ang-strom bandwidth into 1 mrad of horizontal angle.This expression is given as

I(X)-NP()(X/AX),where J.(X) is the wavelength dependent analog ofEq. (1), N is the number of electrons present at agiven time in the curved sections of the storage ringso that N = 2r(RJ/ec), where J is the time averagecurrent. When the expressions for N and P(X) aresubstituted along with numerical values for the con-stants, the resulting expression is

1AX = 4.94 X 1015G () J(in mA)

(E in GeV)[R(meter)] 2 X (in A)AX. (5)

Before a reasonable comparison can be made be-tween synchrotron radiation and other sources, boththe resolution factor and geometrical factor must befolded into Eq. (5).

For most solid state spectroscopy, a resolution of r= (AX/X) 10-3 is satisfactory. However, ratherthan assume a numerical value, a constant value of

Irl = IAXI/IXI = Av|iv = API/IvI

will be used in the formulation of the expression forI(X). Ilaking the substitution AX = rX changes themeaning of I(X) from per AX bandwidth to a band-width of rX and will be denoted as I'(X). The primewill be used with other intensity and power quan-tities to show a fixed resolution.

Geometrical considerations are somewhat moredifficult to evaluate without resorting to specific ex-

perimental arrangements. However, some geometri-cal factors can be considered within limits imposedby typical solid state spectroscopy. Madden' et al.as well as Gahwiller3 et al. have considered the pri-mary geometrical factors associated with the appli-.cation of synchrotron radiation to solid state spec-troscopy, and the reader is referred to their treat-ment for detailed considerations.

The modifications to Eq. (5) now result in thesubstitution of AX = rX to take account of a fixedresolution and the multiplication of the equation bythe appropriate geometrical factor which we will as-sume to be F2 as defined by Madden et al.' Thenew equation is given by

I' = 4.94 X 1015 G(XA/X)J[(E)7 /(R)2]X2 F2 r, (6)

whereF2 = (dG/ 27rD2)(S/B)R,

and d& is the illuminated width of the grating, D2 isthe distance from the entrance slit to the grating, Sis the entrance slit width, and B is the width of theimage of the electron beam at the entrance slit. Theuse of the prime in Eq. (6) denotes a fixed resolution.Typical values that can be assumed for purposes ofcalculations in the ir are r 10-3, dG/2rD2 5 X10-3; S/B and Rx will be close to unity in the ir as awider entrance slit can be used for the constant reso-lution factor, and a high reflectivity is easilyachieved. Thus for the VUV rF2 - 2.5 x 10-8 whilein the ir rF2 5 10-6 would be more typical.Further considerations in this paper will refer to ra-diation collected in a horizontal angle, Ai = d /D2and will be usually expressed in milliradans. Unlessotherwise stated, we assume no significant loss in ra-diation due to a geometrical stop in the vertical di-rection.

The power radiated by a synchrotron into Aso mil-liradians of horizontal angle and a resolution of r =AX/X is

P' = 1.57 X 10-3(E7/ R2)G(Xc/x)x(A)J(mA)rA W.

If this is focused at the entrance slit of a spectrome-ter into an image of area a the intensity is

Y'= (P'/ a) W /m2 .

For example, if Xc 100 A, in the ir region, X >>Xc, and we may approximate G (XC/X) by

G(X,/X) = 22/3 r(2/3)(X\/X)7 3 .

as X = 5.59 {R(m)/[E(GeV)] 3 } A, we obtain

Y' = 0.186rA0kJ(mA)[1/X(A)] 4 1/3[R(m)]/

3(/a)W/m 2 (7)

Our analysis of the factors needed in calculatingapproximate intensities to be obtained from synchro-tron radiation in the ir is now complete. However, acomparison must be made with the intensity avail-able from other ir sources. Although ir lasers arenow available at a number of wavelengths, they stilldo not replace a continuous source for much of ir

December 1973 / Vol. 12, No. 12 / APPLIED OPTICS 2887

spectroscopy. The two sources most commonly usedare the blackbody in the NIR and the mercury arcfor wavelengths greater than approximately 100 .For analytical comparisons the blackbody is theeasier to compare with synchrotron radiation. Oncethis comparison is made, most ir spectroscopists willbe in a position to evaluate for themselves a compar-ison with other ir sources.

IV. Blackbody Source

The power emitted per unit area by a blackbodyat temperature T into a wavelength interval dX i12

Fl\dX = (27rhc2/X5)IdX/ [exp(hc/XkT)-1]} W/m 2. (8)

The function F dX has a maximum at a value Xm =(2898/T) A. For this value,

F,,,,dXn, = 1.286 X 10-"T5 dXm W/m2.

At a wavelength X Xm, the intensity is reducedfrom the maximum in the ratio

FcdX fX,\5 142.0 dXFIAmdX,n Y )[exp (4.965 X/X)- 1 dA

In a conventional ir spectrometer, the actualpower delivered to the entrance slit depends upondetailed geometrical considerations. An area B of ablackbody radiates into 27r sr, of which a fraction f iscollected by the first focusing mirror. If the mirrorhas an area Am and is located a distance D from thesource,

/ = Am/27D2 .

The radiation is focused at the spectrometer en-trance slit into an area Ar. The intensity of the ra-diation is, then,

Yb'(X)dX = FdXb(f/Af)W/m2.

If we choose a resolution AX/X = r to be constantover the wavelength interval to be studied, we canreplace d Xm = Xmr and dX = Xr. The power acceptedthrough the entrance slit becomes

Yb A 8,

where A, is the area of the entrance slit. The inten-sity at the entrance slit is given by

Yb(X) Y'(Xm) (Xm/X)4 {142.0/ [exp (4.965 Xm/X)

-1]1W/m2, (9)

where

Yb'(Xm) = 286 X 10' 1 T 5rXm(bf/Af) W/M 2 .

For parameters appropriate to a conventional spec-trometer used in solid state spectroscopy, the geo-metrical factor is G bf/Af = 7.9 x 10-4. Assum-ing a resolution of r 10-3 and a blackbody sourceat T = 1200 K, the resulting value of

Yb'(Am)

is

Yb'(Xm) = 0.05 W / M2 .

Thus,

Yb'(X) = 7.1(Xm / X)4 1/ [exp (4.965 Xm/X) - 1 W/ r 2 . (10)

For wavelengths much longer than Xm the exponen-tial results in the asymptotic expression

(11)

A comparison of Eq. (11) for blackbody radiationwith Eq. (7) for synchrotron radiation and with rea-sonable approximations for geometrical and resolu-tion considerations shows that the decrease in ir sig-nal at the entrance slit of the dispersing instrumentis considerably less dependent on wavelength forsynchrotron radiation than for a blackbody source.

V. Comparison of the Synchrotron and theBlackbody as an Infrared Source

A direct comparison can now be made of the in-tensity produced at the entrance slit of a monochro-mator by synchrotron and blackbody sources usingEqs. (7) and (9). The intensity distribution for allwavelengths is determined, for each source, if a sin-gle point is known. For our initial discussion we as-sume that near the blackbody peak Xm the black-body is the more intense source. Then, due to thewavelength dependence of the two distributions,there will be two wavelengths at which the twosources produce equal intensities: X < Xm and X >Xm. At wavelengths X > Xi and X < X, the synchro-tron is the more intense source.

As a specific example of the comparison of the longwavelength dependence of the two sources, we con-sidered a blackbody source to be at 1200 K (Xm =2.4 g) and have required that the synchrotron sourcematch the intensity of the blackbody at 10 , (i.e., X,= 10 g). Figure 5 shows the over-all comparison of

102

lo, r

I-0

-C 10--

10i-3

10i-4 -BAC BODY

1 i- 5 -

~ 0-6__

10-7 _

i a-8

0 20 40 60 80 100 120 140 160 180 200 220 240WAVELENGTH IN MICRONS

Fig. 5. Comparison as a function of wavelength of the intensityproduced at a spectrometer entrance slit by a blackbody at 1200K with that produced by a synchrotron whose maximum intensityis emitted in the uv, but which matches the blackbody intensity

at 10 u.

2888 APPLIED OPTICS / Vol. 12, No. 12 / December 1973

YAM - (X./X)'.

Table 11. Values of Circulating Current in MilliampsNecessary to Match 1200 K Blackbody at Various

Wavelengths XI Expressed in Microns

Tantalus I Tantalus II DESYXzinuA R = 0.65m R = 4.5m R = 50m

10 71 mA 37.4 mA 16.7 mA30 17.8 9.4 4.240 11.7 6.2 2.850 8.5 4.5 2.0

the two sources as a function of wavelength. At awavelength of X = 240 u, the synchrotron is about100 times as intense a source as the 1200 K black-body. Using parameters appropriate to several exist-ing and proposed synchrotrons, we have calculatedthe necessary electron current to match intensitiesproduced by the 1200 K blackbody at several wave-lengths X1. The results are in Table II. The pro-posed maximum circulating current for Tantulus IIis 100 mA, which is apparently capable of matchingthe blackbody near Xm and would provide an evenmore dramatic comparison than the result shown inFig. 5.

Although the theoretical comparison seems rea-sonable, a direct experimental comparison is desir-able to provide confidence in the geometrical factors.This experimental comparison was made on Tanta-lus I at UWPSL. At the time of the comparison thestorage ring was operating with a circulating currentof about 1 mA. Even with this low current we wereable to make the comparison and obtain agreementbetween the measured and predicted values to with-in a factor of 2.

A CaF2 window was used on an exit port of thestorage ring. A Perkin-Elmer model 99 monochro-mator with a NaCl prism was used as a dispersinginstrument. The monochromator was adjusted sothat the light made only two passes through theprism rather than the usual four passes. A singlespherical mirror was used at close to normal inci-dence to intercept 30 mrad of horizontal angle and15 mrad of vertical on either side of the orbitalplane. The focus on the entrance slit was approxi-mately 0.8 mm X 3.9 mm.

For comparative purposes a globar was used as ablackbody source. The temperature was determinedfrom the location of the maximum of the blackbodyradiation. Light from the globar was focused on theentrance slit by mirrors in a standard Perkin-Elmerreflectance attachment to simulate the geometricalfactors considered in Eq. (9).

Two considerations made matching of intensitiesin the visible region X < X more desirable. Thestorage ring was operating at a low current, and thesynchrotron radiation has a sufficiently small diver-gence that the thermocouple detector in the irblocked a significant fraction of the synchrotron ra-diation. Using a photomultiplier detector the mea-sured ratio of the synchrotron signal to that of theglobar was 4:1, while a predicted value of 7:1 was

calculated by using appropriate parameters in Eqs.(7) and (9).

Although the visible gave the best SNR because ofthe available detectors and the problem with the op-tical path mentioned above, the thermocouple detec-tor was used, and synchrotron radiation was record-ed through the 4.2-,u atmospheric absorption band.The measurements in the ir were consistent withthe predictions of Eqs. (7) and (9), but the signal tonoise was not as good as in the visible.

VI. Summary and Conclusions

An analysis of synchrotron radiation has shownthat the radiation from existing or planned sourceshas a significant intensity in the ir to be experimen-tally attractive. Facilities that are dedicated to syn-chrotron radiation users should plan for ir use in ad-dition to short wavelength use. In particular theutilization of such sources in the far ir seems promis-ing. A direct experimental verification of the calcu-lations was made using the electron storage ring atUWPSL.

One author, JRS, acknowledges the hospitality af-forded by the F-41 group at DESY during the sum-mer of 1972 when this study was initiated. The au-thors gratefully acknowledge the assistance ofUWPSL personnel as well as the assistance of Mr.Gholamenzhad and Mr. Zivitz. This research wassponsored by the Air Force Office of Scientific Re-search under AFOSR grant AFOSR-70-1892. Theresearch at the storage ring at UWPSL was supportedin part by AFOSR contract F44620-70-C-0029.

References

1. R. P. Madden, D. L. Ederer, and K. Codling, Appl. Opt. 6, 31(1967).

2. R. Haensel and C. Kunz, Z. Agnew. Phys. 23, 276 (1967).3. C. Gahwiller, F. C. Brown, and H. Fujita, Rev. Sci. Instrum.

41, 1275 (1970).4. R. P. Godwin, Springer Tracts Mod. Phys. 51, 1 (1969).5. D. H. Tomboulian and P. L. Hartman, Phys. Rev. 102, 1423

(1956).6. F. C. Brown, P. L. Hartman, P. G. Kruger, B. Lax, R. A.

Smith, and G. H. Vineyard, "Synchrotron Radiation as aSource for the Spectroscopy of Solids," NRC Solid StatePanel Subcommittee Rep. (March, 1966).

7. Private Communication with R. Haensel and C. Kunz.8. E. M. Rowe, R. A. Otte, C. H. Pruett, and J. D. Stebben,

IEEE Trans. Nucl. Sci. NS-16, 159 (1969).9. J. Schwinger, Phys. Rev. 75, 1912 (1949).

10. A. A. Sokolov and I. M. Ternov, Synchrotron Radiation En-glish Translation: (Akademie Verlag, Berlin, 1968).

11. G. N. Watson, A Treatsie on the Theory of Bessel Functions(Cambridge U. P., Canibridge, 1958).

12. R. A. Smith, F. E. Jones, and R. P. Chasmar, The Detectionand Measurement of Infrared Radiation (Oxford U. P., Lon-don, 1957).

December 1973 / Vol. 12, No. 12 / APPLIED OPTICS 2889