Talukdar, B

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Indian J. Phys. 73B (5), 8 2 1 -825 (1999)U P  B—  an in ternationa l jo u rn a lA to m ic  C o m p to n  p r o fileB Taluk darD epartm en t o f  Physics, V isva-B harati. Santim ketan-731 23S, West Bengal, IndiaA b s tra c t  A tom ic K  and /.-shell C om p to n -p ro tiles  for M oK (i rad ia tion  are s tud iedw ith in  the fram ew ork  of a foim  factor approxim ation  by using screened hydrogenie w ave functions N um erical results presented foi the lack o f sym m etry in the profile values about </ = 0 (</ is the usual Com pton param eter! for atoms from ,Hcto 1KAi are used to judge the quality  of approxim ation methods for correcting the impulse double differential cross section fo r  C o m p to n  sc a tte rin gK eyw ords : C om pton p io files , im pulse approxim ation , cross section  PACS No. * 32 80 CvWhen a radiation is Compton scattered, the emerging beam is Doppler broadened because of the motion of the target electron. The analysis of the broadened lineshape or the so-called Compton-profile provides detailed information about the electron momentum distribution in the target. The line profile J (P j  is determined by the probability that the scattering electron has a component momentum p  , that is=  1,, x*lp)xlp'>‘lpjp>* ( 1)where X(p)  is the electron wave f unction in the momentum representation obtained by Fourier transformation of the real space wave function t//(r). Eq. (1) gives the expression in the impulse approximation which implies that the interaction between the photon and electron takes place in such a short time that the scattering interaction is over belore the electron had had a chance to move in the potential well and change its potential energy. Naturally, the outgoing electron is represented here by a plane wave. The initial state of the electron issupposed to have a momentum p , and energy not the electron binding energy - E B. The energy momentum conservation relation is expressed accordingly. However lor a typical element, the above picture tends to break down at intermediate and small momentum transfers. One then takes course to an exact hydrogenic {EH) calculation 11 ], which consists in using more realistic final state wave functions than plane waves and introducing the electron binding energy to express the energy momentum conservation. CO I999IACS822 B TalukdarThe appropriate theory in respect of this was developed as early as 1934 by Felix Bloch |2]. But the topic did not receive much attention for quite a long time although there is no single identifiable reason to account for the sudden inactivity. In recent years there has been resurgence of interest in the exact Compton -profile calculation, which on the one hand, can give improved results over the corresponding impulse-model values and, on the other hand, can be used to examine the domain of validity of the impulse-model [3]. In an exact calculation [4] the Compton peak may move to slightly higher or lower energies when compared with the impulse profile and the lineshape may change. The inconsistency of the experimentally observed profile maximum with the prediction of the IA gave further stimulus for theoretical works. In the present paper, we shall try to clarify some of the conceptual aspects of the profile asymmetries.In extending the theory of Wentzel [5], Bloch [2] first studied the scattering of photons from any initial hydrogcnic bound state with the wave functions expressed in parabolic coordinate system. The exact hydrogenic results were, however, left in terms of residue integrals and their derivatives. These were not evaluated because of their complexities. After forty years, Bloch and Mendelsohn [6] re-examined and re-dcrived the theory of exact hydrogcnic Compton profile in response to the needs of the contemporary experimental observations. They pointed out certain mathematical errors in the seminal work of Bloch [2] and also worked out the results for the residue integrals leading to closed form expression for the EH Compton profile.t HThe exact Compton-profile J,t) (q) for scattering of photon by an orbital electron (nl) of a hydrogenic system is given byJnf  (<7)=| —  127.212* Z/„f (<?)/«v2 .(2 )The quantity k is related to the photon scattering vector and can be written ask2 = io ~ VOil 1ZZ(27.2I2)(nf + M'2 -  2W| w'2 cos &j. (3)In eq. (3), if energies w, and w2 of the incident and scattered photon are expressed in eV, k comes out in atomic units. Obviously, 0 represents the photon scattering angle. In this unit, the projection q of initial electron momentum on the scattering vector is given bynw kZ 27.212 kZ 2/i (4)with the energy transfer w = Wj -  vcr  The expression for the 1 s profile is given by/ eh _ 256 k ,v 3 Z1 + 3 k 2+p2£(1 +k2 - p2)1 +4p2J-exp -2 -i—  tan 2 P1 + k 2- p 2(5)wherey2 -  2 [ — 3 —27.212 Z2 2 (6)Atomic Compton profile 823The expressions for the 2s and 2p profile are also equally simple but somewhat lengthy. For a given incident photon energy w{ and scattering angle 0 the energy \\\ of the outgoing photon can be determined for various values q by using the result2a +(}2q2+Pq(4a + P2q2y  <7)withandor =  1 +2u’j sin2  ^511000 (8)„ 2sinvP  — -— -■ , (9)137.036 wThe above analysis completes the formalism necessary for computing values of (q) as a function of q. For clarity of presentation, we have written eqs. (2), (4) and (6) in terms of atomic number Z. Since we are working within the framework of a screened hydrogenic model, the quantity Z should, in fact, be replaced by the effective nuclear charge (Z ) appropriate to the particular subshell under consideration. Also the (Z^ ) of the continuum electron (c) will enter into the expression for the exact Compton-profile. For the bound state electron (b), we shall use the values of Z ^  chosen in such a way that the impulse hydrogenic profiles for these states match the respective Hartcc Fock profiles at the center, q = 0. The problem of obtaining z;„ is a far more difficult problem. In their discussion of the photoeffcct, Bethc and Salpetcr [71 (while recognising that Z ^  —» 0 as r —»<») point out that since the greatest contribution to the matrix element occurs for distances of the order of the bound electron’s Bohr radius, the most preferable choice is to set Z ^  = Z ^The impulse profile J ln[(q) as a function of q is symmetric about q = 0 and has a maximum there. As opposed to this, the maximum of J ( q )  computed from (2) occurs at q values not equal to zero. In other words, the maxima shift slightly (Sq) either towards let! (-Sq)  or towards right (+ Sq). The quantity Sq is often termed as the Compton detect. We arc interested in studying the K and L-shell Compton defects of MoKa radiation (\Cj =• 17.374 eV) scattered through an angle 0 =  155°. Using the chosen effective nuclear charges, we have computed the number for J„iH(q) as a function of q and noted down the values of Sq where the exact profile exhibits a maximum. The result for Sq and J ln)H (q) lor various Z arc shown in Table 1.The specific results presented include Is, 2s and 2p^av) profile values. The 2p impulse profile seems to average the 2 and 2p ^  exact results. Since there arc four electrons in the 2p(±,) state and two electrons in the 2p ^  state, we have computed the 2p(m) result by using the formula/ EH2/2 J EHZtJipU) (q) + J™BI(q)\ (10)824 B TalukdarTabic I. Values o f Sq and J tt) ( S q )  in au.sub-shellE le m e n tIsSq JuiStO2sSq ,£ / /  c ,J2s (<>q) Sq- pHe ~ 0 .02 0 .5 3 3 0Li 0 06 0 3 2 5 0 0 01 1 3 8 5 2Be -  0 11 0 2 3 2 0 • 0 02 1 .3 3 5 2B - 0 17 0 1794 -  0  03 0 9821C - 0 23 0  1450 -  0 05 0 7 9 3 5 0 21 0  4 8 2 2N -  0 .29 0 1203 -  0 07 0 .6 6 1 2 0 .2 4 0 .4 2 5 2O -  0 31 0 1018 -  0 10 0 5 6 3 2 0 27 0 3768F -  0 .33 0 0 8 6 8 -  0 13 0 4 8 9 9 0 .2 9 0  3367Ne -  0 16 0 4 3 4 0 0 31 0 3 0 3 0N a -  0 21 0 3 6 8 9 0 .3 5 0 .2 5 2 7M g -  0 26 0 3 2 3 2 0 .3 7 0 2 1 8 4A1 -  0 31 0 .2 8 1 7 0 37 0 1928Si -  0 37 0 2521 0 38 0 .1 7 1 0P -  0 42 0 2 2 9 0 0 39 0 1529S -  0 48 0 .2 0 8 1 0 43 0 1370Cl -  0 54 0 1872 0 S 1 0 1222A r - 0 60 0 1732 0 67 0 1076and have noted the corresponding position of the maxima accordingly. As expected, the values of Sq for Is and 2s subshcll are negative while those for 2pUtv) arc positive confirming the expectation of Mukhopadhyaya et at [4] and of Bell 18]. Both for s and p electrons I Sq I increases as Z increases. We have found that our numbers for Sq arc in exact agreement with those obtained by the approximation method of Holm and Ribberfors (9|. This implies that the so-called first correction to the impulse approximation is good enough at the MoKo energy. The correction introduced by Holm and Ribberfors is essentially a high energy approximation of the exact profile. For example, if we assume that the energy and momentum transfer involved are large relative to the characteristic energy and momentum of the bound-state, then the wave vector p of the final state electron is a large quantity and we have the following results2 ( \1 2pexp ---- tanP [l + k-  -/>-)_In- —ta n '1i P ( 1 1 )withv = k - pJ(1 + k 2 - p 1 Y +4p2j = 64p6(l + v2)'1 (12)[ '- ]H (13)Atomic Compton profile 825and1 + 3k 2 + p^ = 4 p^V (14)In view of (11 -14) the high energy result of j ( f ( q )  comes out in the form8 V 3 r3flZ(l + v2) ' _p.1 + — tan 1 v +> 2 P(15)Within a normalisation factor, this high energy result has essentially been used by Holm and Ribberfors [9] for estimating the first order corrections to the impulse profile. However, at low energies an exact calculation like that of ours may turn out to be unavoidable. The values of for the Kv orbital are given tor atoms, He (Z= 2) toF(Z=9), because forZ>9thcconservation oi energy leads to such a cut off on the negative wings of the Compton profiles, that does not permit one to investigate the maxima there. With Bloch and Mendelsohn [6| we have explicitly included this point in our calculation. The q cutoff has been determined by using the formula<7 =X - aP yfx (16)whereThe expressions for J%H(q) given by Bell [8] are very simple and appear to be quite convenient to study profile asymmetries and Compton defects with particular emphasis on relative deviation (R) of the exact profile from the corresponding impulse result. We have verified that the mathematical expressions of Bell can be obtained from those in [6] quoted in this work by means of straightforward algebraic simplification.R e fe re n c e s[ ! ]  L B  M endelsohn and B J Bloch P h vs R ev  A12 551 (1975)[21 F  Bloch P h ys  R ev  46  6 7 4 (1 9 5 4 )13 J P E  E isenberger and P M P lat/m an  P h ys R ev  A2 415 (1970)[4 ] S M ukhopadhyaya. S N Roy and B Talukdar J. Chem Phys. 76 2484 (1982)[5) C  W entzel Z  P h ys  58 348 (1929)161 B J B loch and L B M endelsohn Phys. R ev  A9 129 (1974)[71 H A  B ethc  and  E E  S alpeter Q uantum  M c th a n u s  of O ne a n d  Two E le ttro n  A tom s  (N ew  York A cad e m ic )  (19 5 7 )|8 ]  F  Bell J  C hem  P h ys. 85  303 (1986)[9 ]  P H olm  and R R ibberfors P h ys. Rev. A 40 6251 (1989)

Atomic Compton Profile

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Atomic Compton Profile

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