JOURNAL O F RESEARCIrl of the National Bureau of Standards - A. Phys ics and Chemistry Val. 78A, No.4, July-August 1974

Theoretical and Experimental Compton Scattering Cross Sections at 1. 12 MeV in the Case of Strongly

Bound K-Shell Electrons *

P. N. Baba Prasad** and P. P. Kane**

(March 4, 1974)

Measure me nt s a re re port ed for the diffe re ntial c ross sec tions for Co mpt on sca tt e ring of 1.12 Me V gamma rays by th e K-s he ll e lec tro ns of tin , tantalum, gold, and th orium . A few discrepa nc ies be tween approxim ate th eo reti ca l ca lc u lations and th e ex perime ntal res ult s for diffe re nt e ne rgies are point ed ou t. The need for an e xact re lati vis ti c ca lc ul a ti on is indi ca ted.

Key words : Co mpton sca tt e ring; diffe re nti a l c ross section ; e lectron bindin g; ga mma rays; K-she ll ; photons .

Th e Compton scattering of gamma rays by s trongly bound atomic K-shell electron s has been studi ed experimentally a t 0.279 MeV [1],1 at 0.320 MeV [2] , ve ry intens ively at 0.662 MeV [3 . . . 11] and at an average energy of 1.002 MeV [12]. W e have give n preliminary reports on similar measurements made with a zinc-65 so urce at an energy of 1.12 MeV [13J.

Motz and Missoni [3J sugges ted that the res ults for large momentum transfers can be unders tood within the fram ework of an impulse approximation. Recentl y, a jus tifi cation [14] of thi s approximation within the framework of nonrelativi sti c quantum mechanics has been given. If the target elec tron momentum p makes angles a and a' with the mome nta TI and TI' of the in­cident and the scattered photons respectively, the e nergy k of the photon scattered through an angle () is given by eq (1).

k 1- f3 cos a ko koE - I(l - cos (}) + (1- f3 cos a')

(1)

where ko is the incide nt photon energy, f3 is the ratio of the initial electron velocity to the veloci ty of light and E is the sum of the res t e nergy and the kinetic e nergy T of the targe t electron. In view of the virial theorem , a convenient measure of T is taken 2 to be the bindin g energy of the elec tron. Unfortu nately , in the work of lauch and Rohrl; ~ h [15], there was a

*Wnrk s upported in pari by a gra nt from tli e Na tiona l Bureau of S ta ndards. W as hin gton . D.C. undf!rt he I'L- 480 p r-u ~ rallllll e.

** Prescnt address: Ph ysics Department. Indian Instit ute ufTechnulogy, Powai. Bomhay-76. India .

1 Figures in bracke ts ind ica te the lit era ture referen ces at th e e nd of this paper. t At:curdin g tu the virial theorem , the a verage ki ne ti c c ncrgy of an e lec t run bou nd by

t he attrac ti ve Coulomb pot e nti al in an a tom turns ou t to be equa l 10 the negative of half the average p(Iten ti a l ('nergy. The rrfo rc. the average kine tic e nergy is the nega t ive uf the su m of the po te ntial and the kine ti c ene rgies. Thus. the kine tic c nc '·gy ma y be put equal to the binding energy of 1he e lectron.

mi s prinl SO that a appeared in th e de nominator in­stead of a'. The final result for the cross section

(duK) is obtained in te rm s of an average , as in eq. dO tmp .

(2).

(du K) dO tmp.

J+ 1 J+ 1 (du) _ I _ I do d (cos Ct) d (cos cf»

(2)

J+I J+I _ I _ 1 d (cos a) d (cos cf»

where cf> is the angle bet wee n the planes formed by the pairs of vectors (TI , p ) and (TI ', p ) a nd the relativi s ti c expression for du/ dO in th e case of a free but fast electron has been give n by l auch and Rohrli ch.

In the calculation of Motz and Misso ni for 0.662 MeV , the misprinted formula with Ct in ste'ld of a' was used and so the average over cf> was not evaluated. On the basis of the correct eq (2) , Shimizu et al. [7] proceeded to derive an extremely complicated ex­pression in closed form. A graphical integration of (2) was performed by Ramalin ga Reddy et al. [2 , 12] for 0.320 MeV and 1.002 MeV ga mma rays . However , on the basis of a repeti tion of their calculations, it appears to us th at they might have used the mis printed form ula. Pingo t [9] evaluated th e averages for 0.662 MeV by a num erical integration procedure. His th eoreti cal values for gold were in di sagree me nt with th e expe rime ntal data and with the first two calculation s.

In a few recent re ports [1, 9, 10], an attempt has bee n made to compare the experimental results for du K/ du KN at all angles with the calculated values of the incoherent scattering function 5 K for K-shell electrons. Here, du K/ du KN is the ratio R of the scatter-

461

ing cross section of an electron bound in the K·shell to the Klein·Nishina prediction for an electron initially free and at rest. 3 On the basis of a nonrelativistic treatment and the incoherent scattering function approximation, Shimizu et aL [7] have given an analyti· cal expression for SK, which should be really valid only for small momentum transfers. But the same expression was used in the above· mentioned reports even at large scattering angles. The resulting S K

values for gold at 0.279 MeV and 0.662 Me V were in approximate agreement with the measured values of the ratio R. However, on the other hand, the data for lead at 0.662 Me V do not agree well with the S K values computed in this way or according to an earlier relativistic treatment [41-

We have repeated the numerical evaluation associ· ated with eq 2 and with the well·known Klein·Nishina

dCTKN formula for df},' for five gamma energies including

1.12 MeV. The resulting estimates for the ratio R1mp.

are less than unity at all angles and are in disagree· ment with several experimental values of R at 0.279 MeV, 0.320 MeV, 0.662 MeV, and 1.002 MeV. Some of the new experimental results for R at 1.12 MeV, outlined briefly in the following paragraph, are also in disagreement with the theoretical estimates of R1mp.• Our estimates for a gold scatterer and 0.662 MeV gamma rays agree extremely well with those reo ported by Pingot and thus confirm the correctness of the estimation procedure.

The experimental results at 1.12 MeV were ob· tained in a conventional coincidence experiment with 14.9 mgfcm2 thorium , 12.9 mgfcm2 gold, 22.1 mgfcm2

tantalum and 18.8 mgfcm2 tin scatterers. The bias level in the gamma channel was chosen to correspond to about one third of the photon energy calculated for the Compton scattering from a free and stationary electron. However, an additional consideration deter· mined the final choice. It was desirable to keep the bias substantially above the target atom K, x·ray energy in order to minimize the spurious coincidences arising from K x·rays registering in the gamma de· tector and the scattered gamma rays simultaneously registering in the thin x-ray detector. The final bias values were 0.300 MeV , 0.165 MeV , 0.165 MeV, and 0.100 MeV in the case of measurements made at scattering angles of 25, 60, 90, and 120° respectively except that for thorium at 120°, the bias was 0.165 MeV. Measurements of random coincidences were made as usuaL In addition, counts due to the nat­ural radioactivity of the thorium foil were deter­mined in an auxiliary experiment. The experimental results obtained by us are summarized in the third column of table 1. The values estimated according to

:) The two symbols R and 5 11 are introduced, for the purposes of this paper, to distinguish between (1) an ope rational d efinition of the ral io R of bound-elec tron incoherent scatt e ring 10 free-electron (Klein-N ishin a) sca tt ering obtained e ither from (a) ex periment or (b) a theoreti ca l mode l not rest ri cted to small scatt ering angles, and (2) S(q,Z) (here SK for K-she ll elec trons onl y), the same ratio, but de fin ed as th e probabilit y of atomic exc it ation or ioniza tion res ulting from an y impulsive ac tion imparting a recoil momentum q to an atom ic electron (see. C.g., Crodstein [16], Evan s [17]. Davisson [181. Hubbell [19] , or Storm and Israel [20D, calculat ed lip to now from models cla iming va lidit y onl y for appli cation s to small angles.

TABLE 1. Experimental and theoretical values of R = dd<TK for <TK N

1.12 MeV gamma rays from a zinc-65 source

Element 8 R ex pt. R,mp.a SK(Shimizu)b

25° 0.39 ± 0.13 0.896 0.420 60° .96 ± 0.13 .918 1.149

Thorium 90° 1.35 ± 0.13 .934 1.489 14.9 mg/cm2 120° 0.92± 0.18 .869 1.655

25° .67 ± 0.09 .925 0.560 60° .82 ± 0.12 .937 1.575

Gold 90° 1.39 ± 0.12 .952 2.046 12.9 mg/cm 2 120° 1.07 ± 0.16 .898 2.277

Tantalum 120° O.96 ± 0.26 .914 2.735 22.1 mg/cm 2

25° .78 ± O.07 .967 1.491 Tin 60° .97 ± O.08 .975 4.406 18.8 mg/cm2 90° 1.09 ± O.09 .985 5.765

a The values of RIm". are estimated for K·shell electrons on the basis of eqs (1) and (2) in the text , and the Klein-Nishina formula.

b SK is computed from an equation given by Shimizu et al (refe r­ence [7]). In their eq uation, b and not b2 should be put equal to

Z!'!:... ao

impulse approximation mentioned above are listed in the fourth column. SK values calculated according to the formula of Shimizu et al. are tabulated in the last column. It should be noted that several SK(Shimizu)

values are larger than unity and much larger than the

2.4

2 .0

1.6

~I~ " ~1.2 l-

II

GOLD 1.12 MeV

I I

I I

I I

I I

I

I

r--IX ..- -f 0.8 l- I

F I

I

0.4 r "

I I

I

, , , , ,

/

" SK (SHIMIZU) /

f !

(IMPULSE)

-

I I

O. 0 0~-L..--::;4~0:--"---;8;!-;0;;---'---;1*2"0-L--;':;:!6"0--'--'

ANGLE OF SCATTER I NG (DEGRE ES)

F 1 A I d' 'b' J h .. R d<TK IGU RE . ngu ar Lstn utwn 0.1 t e cross sectwn ratw = d<TK N'

rjJ-Present experimental results at 1.12 MeV for a go ld scatt erer. .... .... SK cal culat ed according to the formula of Shimizu et al. --= The theoretical values of R1mp. calculated accord ing to eqs (1) and (2), and the Klein-Nishina formula.

As point ed out in the text , the impulse approximation is expected to be valid for large transfers; whereas the SK (Shlmh.U) formula has been de rived in a nonrelativistic trea tment and for small momentum transfers.

462

corresponding experimental values of R. In figure 1, the experimental results for a gold scatterer are com· pared with the theoretical values co mputed according to the above mentioned approximations.

The detailed spectral di stribution s of 1.12 MeV photons Compton scattered by the K-shell electrons of different atoms are bein g de termined at present. The results of thi s stud y will be repo rted in a later communi cation.

For 0.662 MeV gamm a rays, a relativi sti c calculation [16] employing Dirac e igenfunc tions for the initial and final elec tron states and the bound electron propagator for intermediate states has been done. It is clearly necessary to extend thi s type of calculation to other gamma ray energies. A similar suggestion has been made recently by other workers [17].

References

[1] Pingot , 0 ., Nuc l. Phys. A 133,334 (1969). [21 Rarna linga Reddy, A. , Laks hrnina raya na, V., a nd Jn anananda,

S. , Proc. Phys. Soc. (Lond.) 91, 71 (1967). [3] Motz , j. W. , and Missoni , G., Phys. Re v. 124,1458 (1961). [4] Sujkowski , Z. , and Nagel, B. , Arkiv Fys. 20 ,323 (1961 ). [5] Varrna, j. , and Eswaran, M. A., Phys. Re v. 127,1197 (1962). [6] Missoni , G. , and Di Lazzaro , M. A. , Istituto S lIperiore Di Sanita

Re ports, ISS- 63/7 , 63/8 , 66/6, 66/7 , and 66/8 (1963) and (1966).

[7] Shimizu, S., Nakayama, Y., a nd Mukoyarna , T. , Ph ys. Rev. 140A,806 (1965).

[8] Rarnalinga Reddy , A., Lakshrninarayana , Y. , and Jna nananda, S., Ind. J . Pure and Appi. Phys. 4, 371 (1966).

[9] Pingot, 0., N ucl. Phys. A1l9, 667 (l968). [10] Krishna Reddy, D. V., Naras irnhacharyulll E., and Murthy ,

D. S. R. , Proc. of the Nuclear and Solid S tate Phys ics Syrn · posium, Roorkee (1969) 84 and ibid. Madurai (l970), p. 177.

[11] East, L. V., and Lewis, E. R. , Physica 44 ,595 (1969). [12] Ramalinga Reddy, A., Parthasarad hi , K., Lakshminarayana, V. ,

and J nanananda, S. , Ind. J. Pure a nd A ppi. Phys. 6, 255 (1968).

[13] Baba Prasad, P. N. , and Kane, P. P., Proc. of the Nuclear and Solid State Physics Syrnposi um , Roorkee , (1969) 81 and ibid. Madurai (1970) , p. 187.

[14] Eisenberger, P. , and Platzman , P. M. , Phys. Re v. A2, (1970) 415. [151 Jau ch , J . M., a nd Rohrlich , F. , The Theory of Photons and

E lectrons (A ddi so n Wes ley Publ ishin g Co.; 1955) p. 230, eq 11-8.

[161 Grodstein , G. W., Nat. Bur. S tand. (U.S.), Circ. 583, 54 pages (1957).

[171 Evan s, R. D. , Encyc l. Phys. (Springer,V erlag , Berlin), S . Flii gge, Ed. , 34 (1958) , p. 218.

[18] Davisson , C. M. , in Alph a· , Beta· ,and Gam ma-Ray Spectros­co py (North·Holl and , K. S iegbahn , Ed.; 1965), p. 57.

[19] Hubbell , J . H. , Na t. S tand. Ref. Data Ser., Na t. Bur. S tand. (U.S.), 29, 85 pages (A ug. 1969).

[20] S torm , E., a nd Israel, H. J. , Nucl. Data T ables A7 , 565 (1970). [21J Whittingham , LB. , J . Phys. A. (Gen. Phys.) 4,21 (1971). [22] Tseng, H. K_ , Gavrila, M., a nd Pratt , R. H. , Report No. 2

(1973), unpublished.

(Paper 78A4-826)

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