Anand, S

Singh, M

Sood, B S

English

IACS Institutional Repository

ON THE ESTIMATION OF ELASTIC SCATTERING CROSS SECTIONS OF GAMMA RAYS FROM DIFFERENT ELEMENTSM SINGH, S. ANAND and J3 S SOOl)P h y s i c s  D e i 't t . ,  P c n t j a u i  U js iiv j!m M iT \ , P v r i A j . \{Reraivcd October 27, 1905)ABSTRACT A mclhoii for Uio oHtimi\tjon oi olastu cuj ictioiiH ul luw‘t'y;y {J[rtramn lays from difloront clomojitK a1, dillufc'iii Mijilliainp lr.s( i il)(^ d Itof tlid exact ( lalculalions of Urow ii aiul Alnyc'rs lot meienry and Uie oNpeiimenl ul dul.i on Z  (U'jiondoneo 'Plio o,stimatod vaJuos show a f?ood af?ii<ein(‘iit with (he avadahlo i'V|u)rimental datar N ' V R O D U (J T 1 O ]S]Itayloigh sojiitenng frt)in atomic olocirons ami Tliomsoii KcaUcriiig from mi- (l(‘ai eliargc' an  ^ tJu' only piocitssos which make any sigmlicant c.ontrihiiljoiiK to tli(‘ clastic scattering o f  low cm^rgy gamma rays Whih^ Thomson scattering from till'll'rent clcmt^rils lian lie accurately calculated, tli(‘, exact calculations of llayliiigh s( attcring are availahle. only for /f-idcctrons o f mercury. HoM^cver, various ajijiroxi- male form factor (.alcnlations have- heoii proposed by many workers to esti­mate Rayleigli scattering from different elements. Unlortiinati^ly, these form laitoi i-alculations liave been found to be in largi’* errors. In a rei:ent commimi- eat-ioii, we suggested tliat our experimental studies of Z  dependence o f elastic statti'riug o f  gamma rays can be combined with thi‘ I'clined calculations ol'Brown mid Mayers (1957) to give reliable estimates of elastic scattering cross sei.tions of gamma rays from different elements at various scattering aiigl(‘,s and gamma ray (Miergiiis (Aimnd at al., 1904) in  the following sections w^ e outline the method o f estimation and compare the results so obtaintid with the available experimental dataM y: r  h o d  o r s t  i m  a  t  i o nThe 1 dined calculations o f Browm and Mayeis (1957) for llaylc.igh scattering are given in terms o f fi'-shell scattering amplitudes for nu^nmry ivith and ivithout [lolarizatioii change The cross section for Raykugli scattering is given in the loim :{d(Tld^l)j, =  l “ l ■■■ ( ')^^hero subscripts \k and 2k denote the 7f-shell amplitudes with no polarization change and with polarization oliange respectively, and is the classical electron38367radius. Those amplitudes are complex quantities; the contributions o f the imagi- nary part are particularly small at low values cif momentum transfer ( q. 'e \ 'sin j . Neglecting the imaginary parts, Bernstein and Mann (1958) have shown by plotting the real jiarts o f the scattering amplitudes that these may be writteni368 M. Singhy S, Anand and B. S. Sood1 + cos  6="-^2z(Q')1 —cos 0Fn  and are smooth functions o f q above q =  0.6 and for all values of q res- Xiectively. Furtluu', F i^ is very nearly equal in magnitude to the Beiijie’s -shell form faetfw (Bethe 1952), and consistently large than Bernstein and Mann(1958) have calculated the contributions o f the L  shell electrons to the elastic scattering by assuming that the non-spin flip and sjun flip amplitudes for L  shell contribute in the same ratio as the corresponding K  shell amjilitudes. However, Brown and Mayers (1957) suggest that this assumption may not bo correct at largo values o f q and exiierimental data suggest that it is .better to uetgleet the contribution o f iiou-siun flqj anqiliLude for L  shell electrons for large values of q So, in our work Ave have altogether neglected the contributions o f non-sjiin flip amplitudes foi Ij shell electrons at values o f q larger than 1. The L  shell anq)li- tudes were taken from the work o f Bernstein (1958).The contributions o f nuclear Thomson scattering, which interferes construc­tively with Tlayloigh scattering, are calculated from the well known classical rela­tionCombining the various contributions, Rayleigh plus nuclear Thomson stiattcr- ing cross section for mercury at a scattering angle 0 is given by{d(rldn)pau, + Thom.^r^^m{{A^+B^){i^ cos ^)2+(6’2_^2)2)(l-cos ^)2] ... (3)whore,A-ViB F,^{q)+F^j^{q)Jra ... (4)C -\ - i D  =  F ^ ,,{q ) -\ -F .; .jX q )^ aand,aHero m is the electronic mass, M  the nuclear mass and Z  the atomic number nf the scattorer. F^ jJ^ q) and F, j^.{q) contain both real and imaginary parts as given in the papers o f Brown and Mayers (1957), andB iM  =- (a ix .+ iM /(l +  cos 0)12 ... (5)=  (a2i+ii>2(fc)/(l—cos 6)12On the Estimation of Elastic Scattering, etc. 309Tlio values o f taken for aifft^-cni scatUiimg anglosfrom the data o f  Brown and Mayors (1957) for gamma ray onorgias of 1 28 and 2 and acjoordmgly and F^ n calculated as functions of moniontumtransfer. Tlio values o f F^ j^  and F.^  ^ wei-o taken from the work of Bernstein (1948) for different values olr/ The values of ii-and C'“ (-1^  ^for mercury, so obtained, are plotted as functions ol‘ momentum transfer m Fig. (1). ByFig (1) PJots of unci T)'^  againsf mnmenlum fi'unHfer for raorcury.For definitions roforenfi' may lu’ mado lo textmeans o f these (uirves, it is possible to estimate the elastic scattering cross sec­tions for mercury at various gamma ray eneigii^s and scattering anglesr0 0 0 4 . f  1 . 2 1 6  2 0  2 4 2 8  n . 2 3 . 0  4 0Fig. (2) A Plot of index ‘ n’ of Z  dopondenco agmiiat momentum transfer for elastio scattering of ganoma rays370 M. Singh, S, Anand and B. S, SoodTlie oxpiTiinental Z dopftndcncc ciirvo wliicli shows tho variation, of index: n of Z depomlenc-o is given in Fig (2) for various values of moTnenturn transfer (Anand H aJ., 19H4) This eurve has heen obtained from a study ot tho variation ol tho eJastic scattiTing cross secitioiis vvitli atomic number ol tho scatterer for 0.280, 0.41] andO 002 MeV gamma rays scattered from Ag, 8n, W  and Pb at different scattering angles.In ordt'r to (\stimate tin? scatte-ring (to.ss section from the above data for any deslr(^d element (Z  47) at paiticulai vahuis of gamma ray energy and acattining angle, the eorre^spondmg cross section for meremy was first (‘valuatiMl|froni Fig (I), and o(pi (2) To conviM't this cross section for meicnvy to tlie desired (dcmuMit oJ atonii(‘ number Z, the following relation was used, \the index n was obtained from the (‘Xperiinental curve in Fig (2) corViisponding to the partieuhir value o f q.It 7^  S U L T 8 A N  I) C O N C U S I n N SThe estnnatiMl cross sections are compared with the available cxpornmuital results 111 Figs (.‘1-0) Figs, (.‘la. 3b) show a comparison of I'stimated cross siwtmns for Pb, Pt, Ta and Sn loi' 0.002 MeV gamma rays at diHcrcift angles v\ i'tli tlu^  ox- perimeiita] data of Narasimbamurty et al . (1004) Also shown in thcsf‘. Figs are0 lO 20 30 40 TjO GO 70 80 90 100Scattering angle. Scattering angle,pig. (3a, 3b) CompuriHon of tho ‘ostimatod’ and cxpenmontal elastic scattering cross soctimiH for 0.GG2 MeV gamma rays for Pb, Pb, Ta and Sn.071 the, EstimcbtioTi of Elastic ScaitcTiyig^ 371ioi* I 12 JMrV prtmruu ivs fm J’l). I’ l, Ta and Snl\ >1 Ju........\  '  ^\\\ ' \ X0 20 40 00 SO 100 120 Enorgy (MeV)l'’jK. (C) CompHrison of tlio‘efitimatod’ and axprii- Fi^ (0) Comymi’iMon of ‘oslntiafcd’rnt*nial oJastic* Hcatieririg oroBS HHctinriH for 1.33 MpV anti oxptirnTinnUd oliiHf-ic scat-tonnggamma rays for U. Pb, and Sn rrobs HorlionH for Pb, Sn, In and Cdal. 90" for difloront onorgioa.t'lio extrapolaiod Brown et al, values of scatteM’iiig cioss soctions as reported by Narasiniliamurty et al (1964). The agroeniemt is better with oiu estimates. In l^ igs (4a) and (4b) a comparison of the estimated cross sections tor Pb, Pt Ta372 M, Singh, S. Anand and B. S. Soodand Sn at 1 .12  MoV is made with the experimental results o f Hara et al. (1958), Cmdro and Ilakovac (1958) and Narasimhamiirty at al. (19G4), and extrapolated theoretical values o f calculation o f Brinvn and Mayers (Narasimhamurty et al., 1964). Again the agreement is quite good witli the results o f Hara et al. and Narasimhamurty et al. while the results o f Cindro and Ilakovac are somewhat lower. The extrapolated values are also lower. Fig (5) comjiarcs the estmates at 1.33 MeV for V, Pb and Sn with exporiiiKuitalvaliKiS o f Standing and Jovanovich (1962), Goldzahl and Eborhard (1957), Bernstein and Mann (1958) and Hara et al. (1958). The ri^sults o f Hara et al. and Standing and .lovanovuil^ agree well with the estimated values wliile those o f Goldzahl and Eherliard and Bernstein and Mann arc appreciably liigher. This discrepancy is resolved if, as pointed out by Standing and Jovanovich, the experimental results o f Bornsteiii and Mann are assumed to include some contribution of incoherent scattering. In Fig. (6) wo have compared our estimates for Pb, Sn, In and Cd at 90° at different gamma ray energies with the experimental i-esults o f Burkhardt (1955) and Hara el al. (1958) Here, we find that the experimental data o f Burkhardt are f;onsistoiitly higher than the latter more reliable data (Hara et al , 1958, Standing and Jovanovich1962) which are quite scanty This may be duo to the presence o f appreciable amounts o f incoherent scattering in the measuromonts o f Burkhardt (1955).Thus there exists an over all agreement between tlu‘ estimated and reliabh^ experimental values o f the differential scattering cross sections for different ele­ments at various sc:attoring angles and gamma ray energies. Moreover, the esti­mates o f the scattering cross sections made by Narasimhamurty et al (1964) from extrapolation o f calculations of Brown et al are lower than our estimates and the experimental values. This is particularly noticeable for tin where any deficiency in the method o f extrapolation should bo clearly exposedThe cross sections o f non-resonant tJastic scattering estimated by tlie method described above can be used witli confidentre to determine the cross section of resonant scattering o f gamma rays by c;omparisoii V'ith non-resonant scattering cross section, especially when the cxiienmcntal arrangement needed to restore resonance condition is complicated and it is not possible to calculate the solid angles from the geometrical set up (see for example Moon 1951).R E F E R E N C E SAnand, S ; Singh, M. and Sood, B S., 1964, Curr Sr,i 33, .333.Bernstein, A and Mann, A K  , 19.^8, Phya. Rev. 110, 805Bemstoin, A.. 1958, P h D . Thvyii>, Univ of PeHnsylvaniuHotho, H  A., 1952, Private, Commvmcation to Levinger, J. »S’. 1952. Phys. Rev. 87, 656.Brown, 0  E. and Mayer, D. F., 1957. Proc. Roy Soc. .4242, 89.Burkhardt, .T. L 1955, P?i.ys. Rev. 100. 192.Cmdro, N. and Rakovan, K ., 1958, Nucl. Phys 5, 647Gold/ahl, L. and Eberhard, P. 1957, J. Phys. Ro(hum, 18, 33.Hara, E. et al 1958. J. Phys rad%um, 19, 668.Moon, P. B., ]951, Proc. Phys Soc 1064, A04, 76.Narasimhamurty, V. A. et al Proc. Nvcl. Phys. Symj). Chandtgarh, Pago 312, long darli 1965, Nucl. Phys. 62, 296Standmg, K , G. and Jovanovich, J. V., 1962, Canad. J. Phys. 40, 622.

On the Estimation of Elastic Scattering Cross Sections of Gamma Rays from Different Elements

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On the Estimation of Elastic Scattering Cross Sections of Gamma Rays from Different Elements

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