Inelastic scattering of X-rays and gamma rays by inner shell electrons

INELASTIC SCATTERING OF X-RAYS AND GAMMA RAYS BY INNER SHELL ELEC TRONS

P.P. KANE

Department of Physics, Indian Institute of Technology, Powai, Bombay 400 076, India

NORTH-HOLLAND

PHYSICS REPORTS (Review Section of Physics Letters) 218, No. 2(1992) 67—139. North-Holland

PHYSICS R E PORTS

Inelastic scattering of X-rays and gamma rays by inner shell electrons P. P. Kane Department of Physics, Indian Institute of Technology, Powai, Bombay 400 076, India Received January 1992; editor: J. Eichler

Contents: 1. Introduction 2. Theoretical treatment of photon inelastic scattering 2.1. Description of main features 2.2. Detailed theoretical calculations 2.3. Impulse approximation 2.4. Other relativistic calculations 2.5. Incoherent scattering function approximation 2.6. X-ray resonant inelastic scattering (resonant Raman and resonant Raman—Compton scattering) 2.7. Higher order effects and other recent studies (not covered in this review) 3. Experimental details 3.1. Sources 3.2. Targets 3.3. Detectors and scattered beam analysis 3.4. Electronics and data analysis 3.5. Target Z dependent false coincidences/counts 3.6. Determination of absolute values of cross sections 3.7. Scattering arrangements 3.8. Direct measurement of total Compton scattering cross sections of K and L shells 4. Experiments 4.1. A list of experimental studies of inelastic X-ray and

69 72 72 78 80 84 85 88 91 91 92 93 93 94 94 97 100 101 102

gamma ray scattering by inner shells of medium and high Z atoms 4.2. Presentation of some data 5. Discussion of results 5.1. Triple differential scattering cross sections 5.2. Double differential scattering cross sections determined in coincidence experiments 5.3. Double differential scattering cross sections determined in the singles mode 5.4. Resonant Raman and Raman—Compton scattering 5.5. Single differential cross sections for Compton scattering 5.6. Total cross sections for K shell Compton scattering 6. Conclusions and outlook Appendix A. Relation between photon energy after Compton scattering and initial electron momentum under the impulse approximation Appendix B. A summary of studies of non-resonant Raman and Compton scattering of low energy X-rays by inner shell electrons of low Z atoms References Note added in proof

102 102 114 114 115 119 120 123 127 128

129

131 132 139

Abstract: A survey is presented of theoretical and experimental studies of inelastic scattering of X-rays and gamma rays by electrons in inner atomic shells. The photon energy range from 2 keV to about 1 MeV is considered. The main focus is on both non-resonant and resonant inelastic scattering by strongly bound electrons. Applications of the results to studies of X-ray attenuation and transport in condensed matter, and of selected topics in solid state physics are also briefly pointed Out. 2 Theoretical treatments the different processes are described. Limitations simpler treatments of Compton scattering the A approximation, the impulseofapproximation and the incoherent scattering functionof approximation are explained in the context relying of the on relativistic second order S matrix approach. Additional aspects of inelastic processes such as double Compton scattering, electron magnetic moment dependent scattering, and scattering by plasmons and phonons are mentioned briefly but not reviewed. Comments are made regarding experimental details concerning sources, methods of monochromatisation, choices of targets, detectors, coincidence techniques and scattering arrangements, and estimation of false counts. Extensive tables listing different types of experiments are presented. The experimental data are discussed from the standpoint of the best available calculations. Possible directions of further progress are indicated.

0370-1573/92/$15.00

©

1992 Elsevier Science Publishers B.V. All rights reserved

1. Introduction The subject of X-ray and gamma ray scattering has attracted considerable interest in recent years. Experimental and theoretical advances have helped the resurgence of this interest. The advances in the field of elastic scattering have been covered in recent reviews [Gavrila 1982; Kissel and Pratt 1985; Kane et a!. 1986]. Detailed reviews [Williams1977; Bushuev and Kuzmin 1977; Williams and Thomas 1983; Cooper 1985] have described the study of Compton scattering with particular reference to weakly bound electrons in condensed matter. Inelastic photon scattering from inner shells of heavy atoms is not very sensitive to details of interatomic interactions in the initial state. On the one hand, such scattering provides an unavoidable contribution in the study of spectra of Compton scattered photons as pointed out in the above mentioned reviews and thus needs to be understood quantitatively even in the investigation of outer electron scattering. On the other hand, relativistic treatments become necessary as the electron binding energies increase, Further with increasing binding energies, the region of validity of the well known and frequently used impulse approximation has to be examined. In addition there is a general theoretical prediction that inelastic photon scattering cross sections should increase with decreasing energy of scattered photons in the low energy limit. This interesting feature called infra-red divergence (IRD) is also expected to be stronger in the case of inner shell electrons. See also section 2.2 and the end of section 2.1 in this context. The term inner shell electrons may be employed to designate in an approximate way electrons whose wave functions, energies etc. are affected very little by interatomic interactions in molecules and solids. When scattering leads to ionisation of a participating atom, the scattering process is called Compton scattering. Raman scattering involves the excitation of a struck electron to a higher energy unoccupied bound state. The same term has also been used occasionally in situations in which the struck electron is excited to vacant states within conduction bands in solids. Several approaches have been adopted for an experimental study of inelastic photon scattering from inner shells. For example, if the scattering occurs from K shell electrons of high Z atoms, K X-rays will be emitted a short time (1016 to 1017 s) later. Thus from an experimental standpoint, the X-ray emission may be considered to be in time coincidence with the scattered photons. Scattered gamma rays detected in coincidence with characteristic K X-rays lead to a selective study of K shell Compton scattering ([Motz and Missoni 1961; Sujkowski and Nagel 1961] and other work to be described in sections 3, 4 and 5). However, the coincidence requirement reduces substantially the signal rates and thus results in larger statistical errors than those in measurements made in the singles mode of whole atom Compton scattering. Ideally the scattered photon should be detected in coincidence with the emerging electron, with or without the detection of the associated characteristic X-ray. Although such experiments were initiated some time ago [Bothe and Geiger 1924, 1925; Burcham and Lewis 1936; Picard and Stahel 1936; Bothe and Maier-Leibnitz 1936; Hofstadter and McIntyre 1950; Cross and Ramsay 1950], it is only recently that thin enough 30 ~g/cm2) targets and semiconductor detectors have been used successfully for such studies in conjunction with synchroton X-ray beams [Rollason et a!. 1989b; Bell et al. 1990]. In the earlier studies, the poor energy resolution of detectors or multiple scattering of electrons within the targets was a likely source of error in the determination of emerging electron momentum vectors. (‘—‘

69

70

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

In the case of Compton scattering of photons from electrons assumed initially free and at rest, the scattered photon energy depends in a unique way on the scattering angle i~and the incident photon energy hi’, through the well known Compton relation

hv~=

1 + (1

(1.1)

2

—

cos i~)hi’1Imc

where m is the mass of an electron, c is the velocity of light, h is Planck’s constant and hv~is the “free” Compton peak energy (see also appendix A). When Compton scattering occurs from a bound electron, the final state consists of an ion, an electron and a scattered photon. The ion carries away momentum but negligible kinetic energy on account of its large mass. Thus the available energy (h v~ B) is still effectively shared between the emergent electron and the scattered photon, where B is the electron binding energy in the initial state. In the case of bound electron Compton scattering, the final electron momentum is not uniquely determined by the change in photon momentum. So the requirement of energy conservation leads to the result —

0
(1.2)

With larger values of B, the upper cut off indicated in inequality (1.2) occurs at lower values of hi’1. For example, if 662 keV photons are scattered by a lead target, the K shell and L shell cut offs occur at about 574 keV and 646 keV, respectively. (See fig. 3.1 for more details.) Note that the free Compton peak energy hi’~is about 197 keV in this case for a scattering angle of 145°.The widths of the scattered photon energy distributions are much smaller in the case of outer electron scattering than those for inner electrons. Thus the sharp steps in the distributions near the cut offs, though of a small absolute magnitude, can be studied at large scattering angles when efforts are made to reduce background counts. Such experiments have been performed in the single detector mode with silicon and germanium counters giving resolutions of about 0.2 to 0.7% at different energies [e.g. Schumacher 1971; Fukamachi and Hosoya 1972b; Pattison and Schneider 1979a and 1979b; Rullhusen and Schumacher 1976; Reineking et al. 19831. Although scattering from individual electron shells is not strictly isolated in these experiments, attention is focussed on inner shell scattering. The energy distribution of photons detected after Compton scattering is much broader than that of the incident photons on account of the spread of bound electron momenta and the finite angular acceptance in an experiment. The broad Compton distribution observed in low energy experiments has sometimes been called the Compton band [Das Gupta 1959, 1964; Faessler and Mühle 1966]. Intensity jumps arising from the onset of K and L shell Compton scattering have been observed also in low energy investigations. Inelastic scattering of photons of a few keY energy from very light elements has been studied with wavelength dispersive spectrometers based on crystal diffraction and giving resolutions of a few eV. In the case of low incident photon energies and over an appreciable range of scattering angles, the K shell binding energies BK are larger than the Compton shifts (hi~ hv~).For example, values of BK are 112 eV, 188 eV, and 284 eV, respectively, in the case of beryllium, boron and carbon, and the Compton shift in the case of copper K,,1 excitation (8.047 keV) is less than 63 eV for scattering angles smaller than 60°.Thus intensity jumps, expected on account of the above considerations, were seen for scattered radiation wavelengths close to and larger than AK, where the cut off wavelength AK is hc/(hv1 BK). A brief introduction to studies in this field of long standing interest is given in appendix B. When the incident photon energy is slightly less than the binding energy of an inner shell electron, —

—

P. P. Kane, Inelastic scattering of X.rays and gamma rays by inner shell electrons

71

for example BK in the case of a K shell electron, a large enhancement of the inelastic scattering cross section is seen [Sparks1974; Bannett and Freund 1975]. A theoretical framework for an understanding of these observations has been given [Costescuand Gavnla 1973; Gavrila 1974; Nozières and Abrahams 1974; Gavrila and Tugulea 1975]. Whether the final electron is in an excited bound state or in a continuum state, the term resonant Raman scattering (RRS) has usually been employed to designate the enhancement in both situations. In very high resolution studies of the first type of situation, a sharp and symmetric peak at an energy (hi’, Bex) is expected in the scattered photon energy distribution, where the excitation energy Be,, is a few eV less than the binding energy threshold BK. In the lower resolution experimental studies mentioned above, the intensity distribution of the scattered photons exhibited a broad and asymmetric peak extending downwards from the energy (hM BL), where BL is the average binding energy of a 2p electron. The observed peak corresponded to the second type of situation mentioned above. Since the incident photon energy in these experiments was significantly less than the K shell binding energy BK, the observed peak was explained on the basis of the virtual creation of a K shell vacancy and its filling by an L shell electron with the emergence of a photon and an electron in the final state. In these cases, the observed peaks represented K—L resonant inelastic scattering. When the incident photon energy is very close to a binding energy threshold, the resonant scattering cross section is so large (~.~~1022 cm2/sr) that accurate measurements are possible even with gas targets [Czerwinskiet al. 1985; Deslattes 1987]. A simultaneous observation of resonant features arising from the two different types of situations mentioned above was made for the first time in a very high resolution (better than ±0.5eV) experiment with monochromatised synchrotron radiation near the manganese K edge of a KMnO 4 target [Briandet al. 1981]. There is a strong atomic-like ls—4p line, the so-called white line, in the manganese K absorption spectrum in this case about 15 eV below the K edge. Resonant scattering associated with electron excitation from the is state to the bound atomic-type 4p state and leading to a symmetric sharp peak in the scattered photon spectrum was observed along with an asymmetric peak which is expected when the final electron state is in the continuum. The terms resonant Raman scattering and resonant Compton scattering, respectively, are appropriate in the case of the two observed intensity distributions. Briand et a!. [1989]designated the second case by the term resonant Raman—Compton scattering. A theoretical treatment covering both cases has been given [Tulkkiand Aberg 1982; Aberg and Tulkki 1985]. Ambiguity in the current usage of the term resonant Raman scattering has been commented on earlier [Abergand Tulkki 1985; Kane 1988]. However, in conformity with wide usage, the term resonant Raman scattering will generally be used in this report in the discussion of lower resolution experiments. In available tabulations of photon attenuation coefficients, the contribution of resonant scattering has not been included even at energies close to absorption edges. The angular distribution of Compton scattering from strongly bound electrons varies less rapidly than that in the case of free electrons as the angle of scattering i~increases from 0 to 90°.Further, unlike elastic and Compton scattering cross sections, the resonant Raman scattering cross sections are found to be nearly isotropic. Thus the results described in this report are relevant for accurate evaluation of attenuation and transport of photon beams in matter. As extensively discussed, for example, by Cooper [19851,Compton scattering studies provide a sensitive probe of momentum distributions, particularly of valence and conduction electrons in solids. The background in the spectra of Compton scattered photons is contributed by inner shell electrons. Therefore, an improvement in the understanding of inner shell Compton scattering is expected to result in an improvement in the accuracy of results obtained in the above mentioned studies. Precise experiments concerning the absorption of X-rays are difficult especially at low energies. So near K —

—

72

P.P. Kane, inelastic scattering of X-rays and gamma rays by inner shell electrons

edges of low Z elements and near L edges of medium Z elements, it is difficult to perform accurate investigations of X-ray absorption near-edge structure (XANES) and of extended X-ray absorption fine structure (EXAFS). A convenient complementary probe of the same underlying phenomena is provided by high resolution investigations of scattering of higher energy X-rays [Tohji and Udagawa 1989]. A new probe of unpaired electron spin distributions in magnetic materials has become available through work performed with circularly polarised photon beams (e.g. [Sakai et a!. 1991] and earlier references mentioned therein). Inverse Compton scattering of ultra relativistic electrons and positrons from low energy photons is also very interesting from several points of view [e.g. Dehning et a!. 1990; Bini et al. 1991]. However, the topics mentioned in this paragraph are outside the scope of this review.

The basic topics highlighted earlier in this section will be covered in some detail in the present report. The organisation of the rest of the report is as follows. The different theoretical approaches pertaining to inelastic scattering of X-rays and gamma rays by inner shell electrons are described in section 2. A few details of experimental techniques are outlined in section 3. The photon energy range from 2 keV to about 1 MeV is considered. A summary of experiments of different types and their results is presented in section 4. A few representative experiments are discussed in more detail in section 5 from the standpoint of the best available calculations. The final section indicates possible directions of further research. The frequently used relations between the photon energy after Compton scattering and the initial electron momentum are derived in appendix A. A brief summary of studies carried out over nearly six decades with low energy X-rays and low Z targets is provided in appendix B.

2. Theoretical treatment of photon inelastic scattering 2.1. Description of main features We will be concerned with an incident beam of photons of energy hM, unit polarisation vector e1 and wave vector k,, and a scattered beam of energy hi’1 (i.~ i’,), unit polarisation vector e~and wave vector kf (fig. 2.1). Linear polarisations are shown in the figure. Alternatively, a circular polarisation description is sometimes more convenient. Different representations of photon polarisation have been discussed, for example in a recent review of elastic scattering [Kane et at. 1986]. In the following discussion, photon energies are low enough (<1.2 MeV) so that probabilities of nuclear excitation can be neglected. The intensity of photon beams will be considered small enough to justify neglect of

non-linear photon density dependent effects. As pointed out in the introduction, we shall be interested in inelastic scattering from fully occupied inner electron shells. Therefore, electron spins can be considered to be randomly oriented. The final photon polarisation is usually not measured. Thus the cross sections are averaged over initial polarisation and summed over final polarisation states. Further, inelastic photon scattering involving energy transfer to elementary excitation modes such as phonons is not considered here, particularly since the meV resolution required for such studies is still not easily available (see section 2.7 for a recent reference). Purely formal aspects of such scattering phenomena have been explored in a general way [Van Hove 1955]. Cross sections revealing an increasing degree of sensitivity to details of treatment can be defined as follows; total scattering cross section o, do’1d4 or differential scattering cross section per unit solid angle without reference to the energy of scattered photons, dff/d(hv1) or differential 2o~/d12 cross section per unit range in energy with the scattered photon emerging in any direction, d 1 d(h i.~) or double

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

73

p

(i

ha)

1)

\

~,#_.:~~

“\~kj,hV8)

\

fl

__________

\ \

Fig. 2.1. Photons of energy hi1, wave vector k1 and unit polansation vector e are incident on an electron shown by a filled circle. After scattering through an angle ~I,the corresponding photon variables are hi’,, k, and e,, respectively. The symbols .1. and with e and e, designate two independent directions of the linear polarisation of the photon with reference to the scattering plane containing the vectors k and k,. The scattering vector K = k — = KIc, where k is a unit vector along K. Before the collision, the electron momentum is p, where p = mc-y~,m is the mass of an electron, fi is the ratio of the electron velocity v to the velocity c of light and y = (1 — f32)_~I2~ Note that the electron momentum before the collision has been indicated without a subscript i in order to ensure a direct comparison with the formulae found in many papers. The angles between the vectors p and k1 and between the vectors p and k, are x and x1~respectively. The angle between the scattering plane and the plane containing the vectors p and k1 is ~‘.

3oidt2f d(hi~)due differential cross section withsection respectwith to scattered direction and energy, d energy with the or triple differential cross respect photon to scattered photon directionandand electron emerging in a unit solid angle around a given direction with reference to the incident photon direction. It should be noted that, on account of infra-red divergence, the total scattering cross section and the differential scattering cross section do/d(1 1 are not defined unless a lower energy threshold is specified for scattered photons. The matrix element for the calculation of the photon scattering cross section can be obtained if the interaction with the radiation field is treated as a perturbation. As is well known, the non-relativistic form of the interaction operator in the case of an electron of charge e is given by the following equation except for magnetic moment contributions mentioned at the end of this subsection: Hiflt

— —

2A2 2mc2 e

e(AP+PA) 2mc

‘

(2.

)

where A is the vector potential of the radiation and P is the electron momentum operator. Since V•A = 0 in the Coulomb gauge, eq. (2.1) can be written as HIflt

e2A2eAP 2mc2 mc

(2.2)

The scattering process is characterised by the incident photon and at least one final photon. So in order to obtain results to the lowest non-vanishing order, the A2 dependent term is treated in first order perturbation theory and the A P term is treated to the second order. Then the triple differential cross section for Compton scattering by an electron is given by

74

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

d3o

r~~ M~2d~

=

1d(hv1) due,

(2.3)

2Imc2 and M is the matrix element for the process. The where is theofclassical electrona radius e ago [Kramersand Heisenberg 1925; Wailer 1929], resultingr0 form M was derived long time

M= e

• ~ f

(eu1~~r)

— Ii

m

~ (e~”~”e~ P)t~(ehhdre ,,

E~—(E~+hi’,)—i~ 1 P)~1 m ~,, (e~.re1 E~—E1+hv1 p)1~(e~.re1 —

(2.4)

where i, n and f indicate initial, intermediate and final electron states, r is the position vector of the electron, ~,, indicates summation—integration over accessible intermediate states and the small positive quantity ~ prevents the occurrence of a singularity in the absorption first matrix element. If the small recoil ion energy is neglected, the requirement of energy conservation leads to E1+hv1E1+hv~.

(2.5)

2 term in the interaction. The absorption and emission The first term in M inarising eq. (2.4) the A first matrix elements fromarises the Afrom P interaction term are indicated by the two summations in eq. (2.4). If the central self-consistent potential model is adopted for electrons in an atom, the scattering process involves a single electron transition and the appropriate matrix element for the initially bound electron is approximately given by eq. (2.4). As pointed out by Gavrila [1972],this result is approximate because orthogonality properties of the individual electron orbitals are needed in order to reduce the matrix element to the form of eq. (2.4). These properties require the presence of the same central potential for the atomic electrons in the initial, intermediate and final states, which is not really true in the case of a many electron atom. The consequences of invariance under gauge transformations of the vector potential have been studied in a basic paper by Sachs and Austern [1951].A deep connection between the A2 and the A P matrix elements for processes involving two photons, revealed through such studies, will be now described briefly. The total Hamiltonian of a quanta! system interacting with an external radiation field described by the vector potential A can be written in general as a sum of terms involving ascending powers of A, •

(2.6)

where H

0 is independent of A, and H1 {A} and H2 (A) are in general functionals involving first and second powers of A, respectively, and may be compared in a simplistic way with the second and the first

terms in eq. (2.2), respectively. Gauge transformations in the Coulomb gauge are effected by a space coordinate dependent gauge function G(r) such that V VG = 0. On account of gauge invariance, the transformed wave function ç!ig(r) and the Hamiltonian are given by çfrg(r)

=

exp(—g) 1/1(r),

(2.7)

H{A

—

grad G} = exp(—g) H{A) exp(g),

(2.8)

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

75

where g = (iI~1c)eG(r).

(2.9)

Expanding the left hand side of eq. (2.8) about A

=

0, the exponentials on the right hand side in powers

of g, and comparing corresponding terms on the two sides, we get H1 (grad G} = [g, H0],

H2{grad G} = [g, [g, H0]]

(2.10) (2.11)

,

where the square brackets [g, H0] indicate the commutator (gH0 H0g) and the double square brackets indicate a double commutator involving the same operators. Equations (2.10) and (2.11) constitute an instructive way of formulating gauge invariance without the need to specify explicitly the forms of H1 and H2. The above relations were used by Sachs and Austern show that, electric 2 matrixtoelements cancelina the subset of thedipole A P approximation to be described in section 2.2.1, the A matrix elements for elastic photon scattering. The same approach was used by Eichler [1974]for the case of emission of two photons of any electric multipolarities and can be extended to Compton scattering. However, Aberg and Tulkki [1985]have argued that the invariance of the transition amplitudes under a gauge transformation is not guaranteed in the formulation involving the usual form of non-linear terms in A and further considerations will be required. The velocity form of the interaction operator, elmc)A P, can be changed to so-called length form er E) in the dipole approximation (to be discussed in more detail in section 2.2.1) by a gauge transformation. In the Coulomb gauge mentioned above, 4i =0 and V A = 0. In the dipole approximation, A is only a time dependent function A(t) and the electric field E = (—1/c) aAIat. If the gauge function G is chosen as r• A(t), VG = A and we get —

(—

•

Ag=A~VG=0~

(2.12)

çb

9=~+(1Ic)oGIat=(1Ic)r~aA/t3t.

(2.13)

Therefore, the interaction term linear in A becomes ecbg = er~E. The equivalence of the corresponding matrix elements holds only if a complete set of intermediate states is included in eq. (2.4) [Fried 1973]. If a truncated set of intermediate states is chosen as in the treatment of resonant scattering to be described in section 2.6, the two forms do not give the same results and the velocity form is found to be more accurate. When the emerging electron is not recorded, we get the following equation after averaging over initial and summing over final photon polarisations: —

2 dtr

3 i~ dr dQ1d(hv1) ~2 ~ d111 d(hvf) due dQ~.

(2.14)

Detailed non-relativistic calculations of Compton scattering cross sections start from eq. (2.4). However, the sums over intermediate states in the A P matrix elements are difficult to evaluate. .

76

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

Therefore, an explicit evaluation of these matrix elements was frequently avoided by the recognition that the same are smaller in the Compton peak regime when hi’1 = B, where B is the electron 2 terms under such circumstances. binding energy in the initial state, and by the retention of only the A In the resulting A2 approximation [Wentzel 1927, 1929; Kaplan and Yudin 1975], one obtains ~‘

d~u

=

~r~(1 + cos2~)~ ~ (f~eht~T~i)j28(E 1

(

+

hi~ E1 —

—

h~).

(2.15)

p,f~i

i’~)

2i~)is the Thomson scattering cross section. Equation (2.15) is called the form Here, (r~I2)(1 cos factor of the same type also appears in the theoretical treatment of ionisation by factor result. A+form charged particles [e.g. Bethe 1930; Choi 1973; Sera et al. 1980a, 1980b]. The result of eq. (2.15) was utilised a long time ago [Schnaidt 1934] to derive the double differential cross section for Compton

scattering by a K shell electron of a hydrogenic atom. An independent derivation has been given for a K shell electron subject to a point Coulomb potential [Gummel and Lax 1957]. The use of nonrelativistic wave functions leads to d2o

—

dQ

2

1

+

2~

6 (hK)2(a2 + p~+ 3h2K2) 256a 2 + (p~+ hK)2]3[a2 + (p 2]3 ~ 3 [a 1 hK) ~~‘

1

1 d(hi’f)

—

2r0(

cos

—

2 + 112K2 (2 16 1—exp(—2~a/p 1a/(a 1) 1/137, the energies and momenta are expressed in units of mc2 and mc,

< exp{—(2a/pf) tan’[2p

2/hc)

-=

where a = Za, a =is (ethe momentum transfer respectively, IlK

—

=

111k

1 kf and pf is the asymptotic momentum of the 212 in units of mc2) and the energy electron in the final state. The initial bound electron energy E1 (= a conservation equation are used to determine pf in units of mc. Similar non-relativistic evaluations of eq. (2.15) have been performed for different electron shells by Bloch and Mendelsohn [1974], Schumacher —

I’

—

et at. [1975]and Belkié [1984].A simplification of these evaluations has been demonstrated by Mukhopadhyaya et a!. [1982] and Bell [1986].Relativistic wave functions have been used for the evaluation of the form factor in the case of s-state electrons by Randles [1957],Jamnik and Zupanèi~ [1957], Lambert et a!. [1966], Schumacher [1971],Pradoux et at. [19771and Whittingham [1981],and in the case of p-state electrons by Wenskus et al. [1985].The resulting expressions involving hypergeometric functions are quite complicated and are not reproduced here. On the basis of computations for samarium, tantalum, lead and uranium, Whittingham has stated the following conclusions regarding spectral distributions of photons scattered from K shell electrons. In the case of 279.2 keY gamma rays, the relativistic and non-relativistic spectral peaks are shifted upwards with respect to hz’~’.The peak shift with respect to hi’~,called the Compton defect, is larger for the relativistic case. In the case of 661.6 keY gamma rays, the relativistic peak shifts towards higher energies but the non-relativistic peak shifts towards lower energies. The latter behaviour is not observed in the non-relativistic computations of Pradoux et at. for germanium K shell electrons at the same incident energy. The available experimental information concerning Compton defects in K shell scattering is summarised in section 5.2. The calculations of Dumond [1929]and Bloch [1934]at lower incident energies in the case of low Z atoms lead to bound electron Compton peaks shifted towards energies higher than hv~.Several experiments concerning whole atom Compton scattering have confirmed this

trend [e.g. Nutting 1930; Dumond 1933; Ross and Kirkpatrick 1934; Weiss

Ct

al. 1977; Holt et al.

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

77

1979b]. Note that non-relativistic calculations of Mukhopadhyaya et al. [1982]for 2p electrons and of Bell [1986]for 2p and 3p electrons indicate a peak shift towards energies lower than hv~and thus suggest a dependence of the sign of the Compton defect on the parity of the initial electron state. An experimental verification of these conclusions has not been possible so far. In pure A2 coupling, the cross section for non-resonant Raman scattering involving the excitation of

a K shell electron of a hydrogenic atom to a final unoccupied bound state of principal quantum number n was derived in a similar non-relativistic treatment by Schnaidt [1934],

7 du\ \dQflRaman

=~r

2

2

512

i~I11K\21

/IIK\2

n2—1

0(1+cos~)——I———J131—I + 3n i.~\ a / L \ a / 2 + (11K/a)2)” 31{[(n + 1)/n]2 + (1lK/a)2)”~3 x {[(n 1)/n]

(2.17)

.

—

The dependence of the Raman cross section on hi’,, Z, IlK, ~ and n in this equation is quite complicated. Note that the energy of the scattered photon in this case is hi’, (E~ E 1), where E~and E1 are electron energies in states of principal quantum numbers n and 1, respectively, and excited state widths have been neglected. Further, in the case of an initial s-state, a dominant transition is to an excited p-state. In the case of atomic number Z larger than about 70, n has to be larger than or equal —

—

to 6.

The experimental observations of Raman scattering are mentioned in appendix B. The magnitude of the non-resonant Raman cross section for a given n is typically less than a few per cent of the Thomson scattering cross section. In the regime of small IlK, thecross angular variation of the with Raman cross section is 21~)sin2(i~/2).The section decreases increasing n. An approximately given by (1 + cos estimate of cross sections summed over n has also been obtained by a different method based on the incoherent scattering function approximation to be introduced in section 2.5 and has been found to be consistent with the above mentioned results [e.g. Sujkowski and Nagel 1961]. According to relativistic theory, the interaction Hamiltonian in Coulomb gauge is written as ea A), where a is the well known Dirac velocity operator in four dimensional spinor space. Now the interaction is linear in A and the intermediate states have both positive and negative total energies. The non-diagonal a operator connects positive energy states with those of negative energy and vice versa.

Akhiezer and Berestetskii [1965,pp. 487—489] have shown in detail how the contribution to the scattering matrix elements from the negative energy intermediate states in the relativistic theory approaches the A2 contribution in the Pauli or non-relativistic limit, when I E 2 ~ mc2, I I E~ I 1 mc mc2I ~ mc2 and binding in intermediate states is neglected. In this limit, the corresponding cross sections are again given by eq. (2.15) with the initial and final states being represented by relativistic —

—

wave functions. A complete treatment of the relativistic matrix elements is described briefly in section 2.2.2.

The consequences of both gauge and relativistic invariance have been explored in several papers [e.g. Low 1954; Gell-Mann and Goldberger 1954], and in text books on quantum field theory [e.g. Mandl 1959; Bjorken and Drell 19651. General arguments involving gauge invariance have also been used to show that a cross section for a radiative process such as bremsstrahlung by charged particles exhibits infra-red divergence, and in the soft photon limit can be determined exactly in terms of the cross section for the related non-radiative process [Low 1958; Burnett and Kroll 1968]. This result,

known as the low energy theorem, can be extended to other processes such as multi-photon

78

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

bremsstrahlung and Compton scattering from bound electrons ([Rosenberg 1991; Rosenberg and Zhou 1991] see also the latter part of section 2.2.2). The relativistic interaction has been shown in the non-relativistic limit to lead to an additional small term (—it V x A) or (—it H), where z is the magnetic dipole moment of an electron and H is the magnetic field of the radiation. This term is of crucial importance in the treatment of magnetic moment dependent scattering of polarised photons. Since such scattering is significant only in the case of unpaired outer electrons, it is not discussed here further. •

2.2. Detailed theoretical calculations 2.2.1. Non-relativistic treatment The evaluation of the complicated sums over intermediate states in eq. (2.4) has been performed

analytically in the case of a K shell electron of a hydrogenic atom by Gavrila [1972].The intermediate sums have been expressed in terms of momentum space integrals involving the Green’s function for the

Coulomb field. An integral representation of Schwinger was used for the Green’s function. The final result for the matrix element was given in terms of a linear combination of generalised hypergeometric functions of the Lauricella type. The complicated details of the derivation are not reproduced here. It was shown that the A P terms lead to an infrared divergence in the double differential cross section for small values of hz~,i.e., a cross section increasing with 1/hi’1. The divergence is connected with the first •

order radiative corrections to the photoeffect cross section [Botto and Gavrila 1982]. A simple classical explanation of the infrared divergence (IRD) was presented by Bannett et at. [1977].The IRD arises essentially because an initially bound electron is ejected promptly during a collision with a photon. Then a Fourier analysis of the emitted radiation leads to a flat power spectrum. Therefore, the probability for the emission of a photon of low energy hi’1 varies as 1/hi’1, i.e., exhibits IRD. The divergence effect becomes stronger with increasing Z. A numerical evaluation of the deduced expressions was performed by Gavrila in the electric dipole approximation, which is expected to be valid when kfrK ~ k.r~‘~ 1. Here, the radius rK of a K shell electron is a0IZ is the first scattering Bohr radius of hydrogen.theThe binding energyenergy BK of ahi’K shell electron 2Z2mc2. In and mosta0 Compton experiments, incident photon is ~a 1 is larger than 2a 2BK. Since k~= (2irhi’1Ihc), kIrK is typically larger than 2~ra 0(mcIh)Z,i.e., typically larger than (Z/ 140). Thus, the dipole approximation is a priori expected to be applicable in the case of K shell electrons only if (Z/140) is much smaller than unity. In this approximation, the final expression for the double differential cross section is given by 2o 2 2 d4d d(hi’ 1) = ~r0(C1+ C2 cos ~ (2.18)

where C1 and C2 are found to decrease monotonically with increasing h i.~. dimensionless 2R11c) and k 2R1lc), where RThe is the Rydberg andenergies Z2R11c k1 the and binding k2 are expressed = hi’,!electron (Z 2 = hi’~/(Z atom. is energy ofasa k1 K shell of a hydrogen-like Further, the initial and the final electron states correspond to different energies in the case of inelastic scattering and are therefore orthogonal. Thus the matrix element of the A2 term in this approximation, being proportional to (fIi), is zero. The results of the numerical evaluation were expressed in terms of an auxiliary variable where = k 2/(k1 1). The dimensionless energy k2 varies between 0 and (k1 1) so that ~varies from 0 to 1. ~,

—

—

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

79

Values of the coefficients C1 and C2 have been tabulated by Gavrila for different values of k1 from 1.05 to 20 and for different values of The increase appears to be faster when ~decreases below about 0.25, i.e., when2 k2and decreases below about 1). A P terms have also 0.25(k1 been considered by Froelich and Weyrich [1986].Noting that the ~.

—

.

Both A evaluations of Gavrila were available only in the dipole approximation and that the A2 numerical contribution vanishes in this approximation, Basavaraju et al. [1987b] and Lad et al. [1990]made a purely heuristic attempt to test the feasibility of an explanation of experimental K shell Compton scattering data in the case of high Z elements by combining the numerical evaluations and A2

calculations on the basis of a particular assumption about relative phases. As pointed out earlier, in any case a non-relativistic treatment based on the dipole approximation is not expected to be appropriate in the case of K shell electrons of high Z atoms. Therefore, a fresh analysis based on detailed theoretical calculations in different circumstances is desirable. The connection between the A P contribution in inelastic scattering and that in anomalous elastic scattering has also been explored [Suortti 1979b]. Most of the calculations pertaining to Compton scattering from inner shell electrons have been performed for isolated atoms. Only a few attempts have been made to treat the effects of condensed matter on the wave functions of final continuum electrons [Mizuno and Ohmura 1967; Ohmura and Suzuki 1984; Ohmura and Sato 1987a, b; Sato and Ohmura 1988]. Within the framework of the well known Orthogonalised Plane Wave Method (OPW), the final electron state was represented by a plane wave orthogonalised with respect to the initial localised core electron orbitals. Then the scattered photon spectrum exhibited two broad peaks, called R and C peaks by the authors, instead of only one peak as in the other calculations. However, as we shall see in section 5.2, there is no convincing

evidence from experiments for the existence of such a two peak structure. Further, in a recent calculation based on the one site approximation, Sato and Ohmura [19911have recovered essentially the single peak structure. 2.2.2. Relativistic second order S matrix treatment The relativistic theory of photon scattering from free electrons, developed by Klein and Nishina [1929], has been described in detail by Jauch and Rohrlich [1955],and by Akhiezer and Berestetskii [1965].As mentioned at the end of section 2.1, the relativistic photon—electron interaction operator can be written as ea A). The Furry picture is used in order to calculate photon scattering from electrons (—

bound in an atomic field. The Dirac wave functions in a point Coulomb potential have been used until recently to describe both the initial bound state and the final continuum state. Binding in intermediate states is taken into account by the use of the bound electron propagator. The relativistic theory has been applied to the calculation of Compton scattering by an electron in hydrogen [Henry 1969], and by K shell electrons of high Z elements [Wittwer 1972; Whittingham 1971, 1981]. The recent calculations of the Pittsburgh group will be described at the end of this subsection [Surié et al. 1991; Bergstrom and Pratt 1991]. Since a larger number of multipoles have to be considered in the case of the more spread out L shell orbitals, and since there are three L subshells, the numerical calculations based on the S matrix approach are more laborious in the case of the L shell. Fortunately, simpler calculations are usually found to be adequate for an explanation of Compton scattering data in the case of relatively less strongly bound electrons. A detailed formulation for the evaluation of the relativistic second order S matrix was developed by Brown and coworkers [1954,1956, 1957] in the case of elastic photon scattering from bound electrons,

8t)

P. P. Kane, inelastic scattering of X-rays and gamma rays by inner shell electrons

i.e. atomic Rayleigh scattering. The procedure has been summarised in a recent review [Kane et a!. 1986] and can be extended in principle to the calculation of Compton scattering. However, in Compton scattering the final electron is in continuum states of different energies rather than in a bound state of definite energy and angular momentum. Therefore, as seen from the work of Whittingham, the S matrix computations of Compton scattering are much more complicated. As a result, additional simplifications have been adopted in practice.

In the calculations of Henry, the emission first matrix elements were neglected and only single differential cross sections were calculated. The calculations of Wittwer neglected intermediate bound states and included only the dipole and the quadrupole contributions. These calculations were made for 145 keY and 320 keV gamma rays in the case of K shell electrons of tin and gold. In the calculations of Whittingham for 279.1 keY and 661.6 keY gamma rays and K shell electrons of samarium, tantalum, lead and uranium, the final electron partial waves were limited to four in order to keep the required computer time within manageable limits. This limitation is particularly serious at high energies and can lead to errors of the order of 20% at 661.6 keY. Further, at small values of hi’1, i.e., at larger values of the final electron energy, dominant contributions to the absorption first radial integrals come from large distances and thus require substantial screening corrections. Since an adequate treatment of potentials incorporating screening makes the computations even more laborious, a point Coulomb potential was used in the calculations of Wittwer and Whittingham. The reader is referred to the original reports for additional details. Calculations incorporating the effect of screening through the use of Dirac—Hartree—Slater potentials have recently become available [Surié et a!. 1991; Bergstrom and Pratt 1991). These calculations have been performed for incident photon energies up to 100 keV in the case of carbon, aluminium and copper, and for 145 keY, 279 keY and 320 keV in the case of tin, holmium, gold and lead. In order to verify the reliability of the extensive numerical procedures, several tests of the new S matrix code were made in regimes where simpler treatments are expected to be applicable. For example for low photon energies and low Z values, the new calculations agreed to -=0.1% with the non-relativistic A P calculations of Gavrila [1972] in the dipole approximation. Another successful test of the code was possible with the help of the low energy theorem mentioned in section 2.1, according to which the differential cross section for Compton scattering in the soft photon limit can be written in terms of the unpolarised cross section for the photoeffect. Further, under the assumption of a free propagator and a free outgoing electron, the amplitude calculated for high incident energies was found to agree with the relativistic amplitude evaluated analytically in the Born approximation. In some cases, the new calculations can be compared directly with the previous calculations. The new results for 279 keY gamma rays differ significantly from those of Whittingham in regimes of low and high values of hi’f. In particular, the new calculations reveal a much stronger infra-red divergence closer correspond2 calculations near the kinematicand limita (hi’ ence at large angles with the non-relativistic A 1 BK). On the other hand, particularly at scattering angles larger than about 60°,the recent results follow rather closely the trends of Wittwer’s calculations. An example of this situation can be seen in fig. 5.4. Simpler treatments based on various approximations are described in the following four subsections. .

—

2.3. Impulse approximation 2.3.1. Double differential cross sections and Compton profile in non-relativistic theory It is instructive to note the historical development of the impulse approximation (IA). A detailed formulation of the IA was presented in the pioneering work of Chew [19501and of Chew and

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

81

Goldberger [1952]in connection with inelastic neutron—deuteron scattering. It has been discussed, for example, by Goldberger and Watson [1964]and by Mott and Massey [1965],and reviewed by Peterkop and Yeldre [1966]and McCarthy and Weigold [1991]in the case of collisions with electrons, and by Coleman [1969] in the case of ion—atom and electron—atom collisions. A related approach, called the binary encounter approximation (BEA), has been reviewed by Yriens [1969] and by Garcia et al. [1973].An application of the IA to photon—atom collisions was developed by Eisenberger and Platzman [1970] and by Currat et al. [1971a,b]. In the case of photon—atom collisions, only the A2 interaction dependent contributions are considered in the IA. In any case, dominant A P interaction effects such as infra-red divergence and resonant scattering were not identified until later. As in the BEA, it is assumed that an incident wave interacts with a single atomic electron at a time and that its amplitude is not appreciably diminished when crossing the atom. A necessary condition for the validity of the IA is that the collision time r~ should be much smaller than the characteristic orbital period of a bound electron. Except for resonant scattering [Goldberger and Watson 19641, the collision time r,~may be taken to be the period of the

incident wave (1/i’

1 = hIhi’.~).A measure of the characteristic orbital period of an inner shell electron is h/B, where B is the electron binding energy. Thus the necessary condition is given by the inequality hi’, a~B. During the short collision time, the binding potential can be considered to be nearly constant. Under these circumstances, the scattering cross section can be calculated from the electron momentum distribution, which in turn is obtained through a Fourier transformation of the bound electron wave function. 2 treatment, a time dependent formulation and an expression for the ~ function in eq. Usingin an A of an integral over time, Eisenberger and Platzman [1970] deduced an additional (2.15) terms condition for obtaining a fairly good fit to the spectral shape in the Compton peak regime. The additional condition is given by the inequality (hv~ hi’ 1) B and is obviously stronger than the necessary condition mentioned earlier. In many calculations, a simple plane wave representation is used for the final state electron of momentum p,~= p + IlK, where p is the electron momentum in the initial state and K = k1 kf. On the basis of detailed calculations of the moments (hii, hi’1)’~” over the scattered photon spectrum for N = 0, 1, 2 and 3, according to the IA and eq. (2.15), Eisenberger and 2.However, unfortunately, Platzman showed that corrections to the IA are of order [BI(hi’, hi.~)] perhaps in view of the specific cases considered [i.e. (hi.’, hi’ 1) significantly larger than B], the ratio [BI(hi’, hi’1)] was described in the text as the ratio of binding energy to electron recoil energy. The same verbal description has been used by some of the later authors. In the neighbourhood of the kinematic cut off, hvf s (hi’, B) and the ratio is nearly unity, whereas the verbal description indicates values approaching infinity. In the hard photon regime near the kinematic cut off, the additional condition is obviously not satisfied in any2case. However,can in sometimes the neighbourhood the cutofoff, P contributions are relatively contributions lead to a of peaking the Adouble differential cross section. small and the A A “peaking” approximation formulated in the work of Chew and Goldberger [1952]is then useful for an understanding of the fair success of the IA even under these circumstances. An example is shown in fig. 5.5. Surié [1991]and Bergstrom et al. [1991]have compared their calculations based on the IA and on the relativistic second order S matrix over an extensive spectral range of the scattered photons and suggested a new criterion for the validity of the IA. However, this work will not be discussed here further, since it is yet to be published and some understanding of the successes and limitations of the IA —

~‘

—

—

—

—

—

—

.

is already available from the formulations mentioned earlier and also in section 2.3.2.

82

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

Following eq. (2.15) based on the A2 interaction term, the double differential cross section is given in the IA by the following equations:

dn

21~)~ 1d(h~) = ~r~(1 + cos 2

____

f

n(p) d3p 6(hv~ hi’

2K2I2m

—

1

—

—

11K p/rn),

(2.19)

Il

2

d12 1 d(hv1)

=

~r0(1 + cos

~

—

~

J(p~).

(2.20)

Here n( p) is the probability of an electron having momentum p in the initial state, ~ is the projection of the initial electron momentum anti-parallel to the direction of the scattering vector K according to the convention adopted, for example, by Eisenberger and Reed [1974]and the Compton profile J(p,) is expressed through J(p2)=JJn(p)dpxdpy.

(2.21)

P, P~

2 and mc, respectively. In the above equations, energies are au, expressed in units of in mcdiscussions of Compton Another unit of momentum, calledand themomenta atomic unit is frequently used profiles. Note that 1 au = (h/a 2Ime2). Note 0), where a0 is the first Bohr radius of hydrogen and a0 = (Il further that 1 au = a mc (mc1137). In the case of an isotropic distribution, J(p~)can be written in an alternative form. If N( p) is the density in momentum space of electrons having a scalar momentum p, we get J(p 2)=

f

Ip,I

2rrpN(p)dp=~

f

~-~dp,

(2.22)

1p51

where the number 1(p) dp of electrons having scalar momentum between p and p 2N(p). 1(p) = 4irp

+

dp is given by (2.23)

Yalues of J( p 5) for different shells in atoms of different Z have been tabulated by Biggs et a!. [1975].In the IA, the characteristic property J( p5) of a target can be determined directly from the double differential cross section at the appropriate final photon energy. The relation between p~and hi’1 is given by [Manninen et a!. 1986a; Pitkanen et a!. 1987] (see also appendix A). 2 224 p5z~—i’1+hiy~(1—cosi~)/rnc mc (p~+ v~—2i’ 1v1cos ~)1I2 —

.

(

)

Applications of the above equations for the determination of Compton profiles and from there of wave functions of valence and conduction electrons have been discussed in great detail by Cooper [1985]. 2.3.2. Relativistic approach and impulse approxilnation The impulse approximation as have outlined was derived in a non-relativistic 2— (v2)1c2 beenabove considered by Eisenberger and Reedtreatment. [1974]. InCorrecmany tions to order B/mc

P.P. Kane, Inelastic scattering of X.rays and gamma rays by inner shell electrons

83

Compton scattering studies, the photon energy change (hM hr~)is less than mc2 and the ratio of B to photon energy change is larger than the ratio BImc2. Therefore in such cases, it is possible for the IA to become inappropriate before relativistic effects become important. However, it is worthwhile to examine the often used concept of a Compton profile even in the relativistic context. A relativistic expression given by Jauch and Rohrlich [1955]for Compton scattering from moving electrons was used by Ribberfors [1975]and by Ribberfors and Berggren [1982]in such a way that it was possible to numerically obtain scattering cross sections from tabulations of wave functions in momentum space. The validity of the procedure has been further examined by Holm [19881and by Holm and Ribberfors [1989]. Their conclusion is that the expression for a cross section in terms of a profile gives reasonable predictions near the Compton peak energy and also on the high energy side of the peak, particularly at large angles. The expression gives poorer predictions at smaller angles of scattering and for scattered photon energies lower than the Compton peak energy. The last mentioned trend is different from that noticed by Eisenberger and Platzman [1970]in the non-relativistic case. The Ribberfors procedure will be mentioned in slightly greater detail (see also fig. 2.1). Let /3 be the ratio of the initial electron velocity to c. The energy E~of a moving electron is expressed through —

E, = (m2c4

+ p2c2)V2

=

(1

132)1/2

=

yrnc2.

(2.25)

It is possible to use the following equation in order to express hi’

1 in terms of J3 (see appendix A):

vf

lf3cosX1

226

1—/3cosX1+(1—cos1~)hMIE~~

(.

Here, ~, and are the angles made by the direction of the initial electron velocity with the directions of the incident and the scattered photons, respectively. If ~ is the angle between the plane formed by the vectors k. and p and the plane formed by the vectors k, and k1, we have the relation ~

cos

=

cos ,y, cos

i~+

sin

~

sin

i~cos

(2.27)

~.

Equation (2.26) is not directly used in the expression of the Compton profile in the relativistic theory. The treatment of Ribberfors starts with an equation of the following form [note that the choice of units in the work of Ribberfors is different from that adopted in this review]:

dQ1d(hv1)

=

~

~

J

3p n(p) d

X(K’, K”)6(E~+ hi’ 1

—

Ef — hi’1),

(2.28)

where K’

=

E~(hi.~) (1—13 cos x1), (mc)

K” = K’

—

(~~)(~~)

(1 cos ~

2c4 + p~c2)112, E1

=

(m

—

(2.29)

(2.30) (2.31)

84

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

P1=p+hK,

(2.32)

K’

K”

/1

1\

/1

~

1\2 .

(2.33)

The function X contains a dependence on x1~x1 and

i~in a complicated form. In general, one does not get an exact proportional relation between the Compton profile and the double differential cross section in the relativistic case. However, if the electron density is isotropic, it is possible to rewrite the

Ribberfors formula in the following approximate but useful form: d 2o~ dfl1d(hi.~)

22s,

i

AintkPmin 2r0 ~ IlK E(Pmin) j

2 ~‘1 ~

j

~

2 34

ø’~Pm,n

The expression for the additionally introduced function X~~,(p) is in general rather complicated and is not reproduced here. The minimum value of the initial electron momentum which can lead to a scattered photon of energy hi’1 at angle t~ is Pmin and the corresponding energy is E(pm,n). The relativistic relation between p~1,,and hi’1 is given by (see appendix A) —

mc

—

~

ii~)E(p~1,~) + hi’,i’1(1 —cos 2(i’~+ i’~—2i’~cos ~)1/2 mc

(2.35)

where E(p~j~) = (m2c4 +p~,t~c2)V2.

(2.36)

Different ways of writing eq. (2.35) have been presented by Ribberfors and HoIm. The function J 0 in eq. (2.34) is the first approximation to the Compton profile under the IA. Further corrections to 10 are obtained by an iterative calculation, which has been shown to be rapidly convergent, for example, in the case of scattering of 160 keY gamma rays through 90°and 150°by Z = 10 atoms. In the special case of 180°scattering, the corrections turn out to be negligible and the earlier results of Manninen et a!. [1974]are recovered. It should be borne in mind that, in consonance with the conditions mentioned in the beginning of section 2.3.1, only photon energies near the Compton peak and significantly away from kinematic cut offs were considered in the demonstration of the convergence.

2.4. Other relativistic calculations As mentioned in section 2.2.2, the relativistic second order S matrix calculations are available for only a few cases. In an attempt to simplify the calculations, various other approaches have been adopted in special cases. For example, Gorshkov et a!. [1969, 1973, 1974] have examined the situation for small values of Z and large momentum transfers, and pointed out the limitations of the simpler equation (2.4). Owen [1977, 1978] and Owen and Steinitz [1979] have examined the range of applicability of the semi-relativistic approach of Eisenberger and Reed [1974].Further, they have also tried to estimate the effect of accelerations of bound electrons on the results expected on the basis of the impulse approximation.

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

85

2.5. Incoherent scattering function approximation

The incoherent scattering function (1SF) approximation is used to calculate the single differential cross section (dcr/d121) for bound electron Compton scattering. It was developed by Heisenberg [1931] on the basis of a formulation by Waller [1928].The incoherent scattering function expresses the ratio of the cross section to the free electron cross section, which is represented by the well known Klein— Nishina expression in most treatments following the work of Grodstein [1957]. Thus the basic result in this approximation is written in the form

Z),

= ~KN5(x,

(2.37)

where the 1SF S(x, Z) is calculated in a non-relativistic treatment, the difference between the scattered photon energy and the incident photon energy is ignored, i.e., x = (K/4’ir) sin(i~/2)/A(A),with A the

wavelength of the incident radiation, and essentially a hybrid combination of non-relativistic and relativistic approaches been used for that convenience. Further, states the 1SFform approximation based the 2 interaction term andhasthe assumption the final electron a complete isset. Theonlatter Aassumption is not strictly correct on account of the Pauli exclusion principle and the requirement of energy conservation. The derivation of the form of S(x, Z) and its extension in the case of a K shell electron will be described below. The initial and the final atomic states will be indicated as 10) and Js’), respectively, where s’ is the energy difference between final and initial states and is greater than zero in the case of inelastic scattering. With the assumptions indicated above, eq. (2.15) can be used to express S(x, Z) in the following form: S(x, Z) =

~ FE,(x, Z)12,

(2.38)

‘>0

where FE.(x, Z)

= (~‘I~

e~’il0).

(2.39)

Since the closure property has been assumed, we get

~Is’)(s’I=i.

(2.40)

One can then rewrite eqs. (2.38) and (2.39) so as to express S(x, Z) in terms of only the ground state wave function, S(x, Z) =

(0I~

e~.n/~ 10) - (01 ~ e~.njI0)2

(2.41)

where the second term on the right hand side is the ground state form factor, which is useful in an approximate treatment of elastic scattering. A few important properties of S(x, Z) will be described below. When K or x approaches zero, the exponential factor in the above equations approaches unity.

86

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

Note that x—~0when hty—*0 or i~—~O. Since wave functions for different energies are orthogonal, we have S(x—~0,Z)= ~

As hi’1 —* ~, x we have

—~

I(~’I0)I~0.

(2.42)

~ for not too small values of i~and the ground state form factor approaches zero. Then

(0j~e~.”~I0) (0I(Z +

S(x~~,Z)

~ e~)I0).

(2.43)

ii
Therefore, as shown below, S(x—~,Z)--~Zor S(x->~,Z)IZ—~1.

(2.44)

Let ~ represent the ground state wave function of Z electrons in an atom. Then for given values of i and j, we get (01 elKfrl_rPIO)

=

=

J J

I~l~ e~,_n1)

3r 3r 3r d 1 d 2~ d 5 3r 3r d 1 d 1,

f(r1, r1) ~

(2.45)

where the probability density f(r1, r1) for two electrons i and j is a reasonably well behaved function of r~ and r.. The exponential in eq. (2.45) oscillates rapidly when K—I’ ~, and hence the integral approaches zero leading finally to eq. (2.44). In the Hartree—Fock treatment, 1/’~is given by

/1= 0

1 2 uu1(1) (Z!)” 1(Z)

u~(1) , u~(Z)

...

...

(2.46)

u~(j)= u5(r1)x,(o~),

(2.47)

where u5 (r1), x~ (o,) are space and spin dependent wave functions, respectively, for the jth electron in the sth state. Then combining eqs. (2.41), (2.46) and (2.47), we get 2 S(x, Z)

=

Z-

J

s=1 ~

f55(K)1

-

~

,~‘If,~(K)I2,

(2.48)

s-~1 s~’s’

f 3r. (2.49) In eq. 5~(K) (2.48),= theu~(r) double sum indthe last term runs only over orbitals with the same values of the spin. eIK.rus,(r) These exchange terms, whose importance was highlighted by Freeman [1959],arise from an application

P.P. Kane, Inelastic scattering of X.rays and gamma rays by inner shell electrons

87

of the Pauli principle, which forbids an electronic transition to an occupied state. Values of S(x, Z) were obtained on the basis of non-relativistic Hartree—Fock wave functions by Cromer and Mann [1967] and by Cromer [1969], and have been tabulated extensively by Hubbell et al. [1975]. In spite of the approximations underlying the above mentioned Waller—Hartree method of calculation, the tabulations of S(x, Z) have been found to be fairly successful in the explanation of experimental data concerning atomic Compton scattering cross sections [e.g. Dow et al. 1988; Kahane 1990].

The impulse approximation can also be used to obtain expressions for the cross section ratio described by S(x, Z). In such a treatment, the single differential scattering cross section is first obtained by an integration over the scattered photon spectrum of eq. (2.20) in the non-relativistic theory, or of eq. (2.34) or its corrected version in the relativistic approach. Note that the values obtained for S(x, Z) in the Waller—Hartree and impulse approximation treatments are significantly different in some cases, particularly in the case of small scattering angles. A similar approach can be used to express the incoherent scattering function SK(x, Z) for a K shell electron, which like S(x, Z) /Z is also found to increase from zero to unity with increasing x or momentum transfer. On the basis of non-relativistic wave functions, Sujkowski and Nage! [1961] calculated values of SK and expressed the same in terms of a function of only one variable, namely 11K/Zamc. In addition, values based on relativistic wave functions in the case of lead were also calculated by them for 661.6 keV gamma rays and were seen to be smaller than the corresponding non-relativistic values (see table 2.1). However, in a calculation based on relativistic wave functions and extending up to 30°scattering angle, Talukdar et al. [1971]obtained significantly larger values of SK, which is surprising. Considering an initial K shell electron of average momentum Po’ where p is equal to (2mBK)’12 and assuming the final state electron to be represented non-relativisticalty by a0plane wave, Shimizu et al. (1965) developed an explicit form for ~K’

sK

— —

32V~• 1372(E~~ E~) 3ir2Z3m312c3(1 + p~/b2)4~

( 2 50

—

Table 2.1 Calculated K shell cross section ratios for 661.6 keV gamma rays and a lead target at a few angles. The non-relativistic eq. (2.50) of Shimizu et al. [19651 for S,, has been used to obtain the values in the second column. The non-relativistic calculation of Sujkowski and Nagel [19611leads to a dependence of S~only on the ratio of the momentum transfer to Zamc. The corresponding relativistic calculation shows an additional dependence on Z and has been presented by them only for the case of lead. The relativistic calculations of Talukdar et at. [1971!for S~are available only for a~~ 300. Note that the calculations of the latter two groups of authors have been published only in graphical form. The values obtained from these graphs are given in columns 3, 4 and 5. The results of Whittingham [1981] based on a relativistic second order S matrix are shown in the last column. The different experimental values for lead are listed in table 5.3. The symbols (S), (SN) and (T) represent the calculations of Shimizu et al., Sujkowski and Nagel, and Talukdar et at., respectively. non-relativistic

relativistic

Angle

S,~(S)

S,~(SN)

S~(SN)

S~(T)

S matrix

30 50 70 120

0.30

0.51 0.83 0.93 0.97

0.40 0.74 0.86 0.93

0.58

0.42 0.83 1.17 0.91

0.56 0.79 1.10

88

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

where b is ZIl Ia0 and a0 is the first Bohr radius of hydrogen. Values of 5K calculated with this formula turn out to be reasonably satisfactory only in limited ranges of Z and momentum transfer [e.g. Baba Prasad and Kane 1974; Krishna Reddy et a]. 1974], presumably on account of the adopted simplifying assumptions. If T is the kinetic energy transferred to an electron initially free and at rest, we get T= hi’,

—

hi’~.

(2.51)

When electron motion in the initial state is considered, the maximum and the minimum values of the final electron kinetic energy, Emax and Em,n, respectively, correspond to the momentum transfer being parallel and anti-parallel to the initial electron momentum. Then we get T+

BK

+

2(TBK)~2,

(2.52)

Emin = T +

BK

—

2(TBK)”2.

(2.53)

Emax

An idea of the variation of the K shell cross section ratio with scattering angle, calculated for 661.6 keV gamma rays according to different theoretical treatments, can be obtained from table 2.1. 2.6. X-ray resonant inelastic scattering (resonant Raman and resonant Raman—Compton scattering)

Comments have been made in the introduction regarding the terminology used in this context. The relevant theory has been developed in considerable detail by Costescu and Gavrila [1973],Nozières and Abrahams [1974],Gavrila and Tugulea [1975],Tulkki and Aberg [1980,1982], Tulkki [1984a]and Aberg and Tulkki [1985].Resonant Raman scattering has been briefly described in section 1 and will also be discussed in section 5.4. Here the focus will be on resonant Raman—Compton scattering, usually called RRS as pointed out in the introduction. The concept underlying K—L RRS is illustrated in figs. 2.2a and 2.2b, the first referring to an isolated atom and the second to an atom in a solid. A K shell electron is virtually excited to a continuum state of kinetic energy ~e’ which is above the zero energy

J

~

(a)

(b)

Fig. 2.2. Schematic representation of K—L, resonant Raman—Compton scattering of X-rays of energy hr near a K shell threshold (a) for an isolated atom and (b) for an atom in a solid. See also section 1 for a discussion of terminology. The energy of the scattered photon is indicated by hi’,. The widths of the virtual K shell hole state and of the final L 3 subshell hole state are I’,~and f~,respectively. The binding energies of the K shell and the L3 subshell are labelled as B~and BL312, respectively. EF is the Fermi energy.

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

89

level in the case of an isolated atom and above the Fermi energy EF in the case of a solid. An L shell electron fills the K shell hole with the emission of a photon of energy hi’1. Since hM is assumed to be less

than the K shell electron binding energy BK, the K shell hole state in RRS is off the energy shell. Thus in the description of RRS on the basis of eq. (2.4), the second order matrix elements of the A P interaction term are essential. Note that the final state consists of a photon, a p-wave electron of kinetic energy ~e and a hole in an outer shell (L shell, M shell,. in the case of K—L, K—M,. RRS). The dipole selection rule implies a dominant transition from the K shell s-state to a p-state. The binding energy of the final state hole may be indicated as ~ where n and j are the principal quantum number and the total angular momentum quantum number, ~espectively.In order to simplify the notation, we may indicate the binding energies simply as BL or BM. The requirement of energy conservation in RRS leads to .

.

hi’, = hi’1

+

~

.

.

.

(2.54)

+

5e is EF in the case ofa solid target, the scattered photon energy cut offwill Since valuefor of example, at (hi’, BL EF) in the case of K—L RRS. When the incident be at the (hi’, minimum ~ EF), photon energy i~close to B 1, L3—M, L3—N,. , RRS will be expected in a similar manner. As will be seen in table 4.6, there are only a few studies of RRS with hi’, close to B1. Therefore, in the interest of clarity of the discussion, we will consider RRS mainly near a K shell threshold in the remaining part of —

—

—

—

.

.

this subsection.

When the incident photon energy hv~is close to E~ E~,the corresponding term in the first sum in eq. (2.4) is dominant and is responsible for a resonant enhancement of the cross section. When we consider only the dominant term for the case of K—L RRS, we get —

2oRRS d

di’11d(hi’1)

r~ i~ (fi e~~~’e 2 in

1Pln)(nl e1k~~TetPIi) (E~E1)—hv,iI’15/2

P 1

h

2

(

—

i’j

h

—

B

i’f

255

—

L

~),

(

.

)

where I~is the width of the intermediate K shell hole state and takes the place of 2~in eq. (2.4). A simple treatment of eq. (2.55) is possible in the dipole approximation, which will be seen in section 5.4 to be quite successful for quantitative explanations of experimental RRS cross sections. As discussed in section 2.2.1, the exponentials in eq. (2.55) are approximated by unity in this approximation. Using the energy conserving equation (2.54) and assuming a slow variation of matrix elements except near the threshold energy, we obtain the following equation in a more explicit notation: 2oj~~~l 1” d \d111 d(hv1) !K

r 0

2

i.’1 ‘~

~

15

m

—

256

2 + F~

(B ~

—

hPf)

(

5/4

.

)

where the state function I sep) indicates an electron in a continuum p state of energy e~and is normalised per unit energy interval, and np1) is a bound p state with principal quantum number n and

total angular momentum j.0pK Note that the iswith wavethefunction spherically symmetric that the K shell is connected square isofthe first-order matrix and element [Ha!!et al. photoeffect cross section 1979], as shown in the following equation, where TpK is evaluated at (B 15

+

se):

m(B +s)c~~(B+s)

1< ~epki

Pus)

2 =

is

e 2a/ic ~‘ 8ir

is

e

(2.57)

90

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

Let R Is or R5~be, the radial wave function of a is or npj electron, respectively, in the presence of an 1icy. Then we get [Cooper1962] n~,1or is vacar

l(lsle

2

1 Pjnp1)j

=

J

~m(B 15

—

B~~)2~

3dr~.

(2.58)

R15(r)R5~(r)r

Aberg and Tu!kki have shown that a small anisotropy and polarization dependence of RRS will

result from an interference between the resonant term and the non-resonant terms in the complete matrix element, and from electron correlation effects in wave functions. It has not become possible so far to verify these predictions experimentally. Theoretical investigations of anisotropy and polarisation dependence of RRS at incident photon energies near L 3 thresholds are needed, since the intermediate L3 vacancy state is not spherically

symmetric and since there is a possibility of an orientation effect in the case of an orbital with a I value greater than one half.

The double differential cross section for K—L RRS shows a sharp rise as hi~decreases through (hi’, BL EF) and sufficiently below this cut off decreases approximately as hi’f(BK BL when the term F~5I4in the denominator of eq. (2.56) can be neglected. The calculated asymmetric line shape below the cut off has been verified in several studies (see sections 4.2 and 5.4). Similar remarks will be applicable in the case of K—M or K—N RRS with BM or BN in the place of BL in the above expression. In the case of a solid target, the double differential cross section for K—L RRS has been written in the following form [Eisenbergeret a!. 1976a, b; Suortti 1979a]: 2ff~~~ ‘\ r~i’f(BK+ ~e)(~K BL)glS d 2~ 2 59 2 + F~ ~dQ1d(h~)1K-I i’j[(BK BL hi’1) 5I4]\ dE~!BK+ee’ ( ) 1 and (2p)’ hole states and where g152~ is the oscillator strength of the transition (is)removed from the K shell. (dg~/ds~) is the density of states at kinetic energy ~efor between an electron The single differential cross section for K—L RRS is obtained after integration of the expression in eq. (2.59) from 0 to the cut off (hi’, BL EF), —

—

—

(

~

—

—

—

.

—

(dcrRRS~ \ d11 1 1K-I

—

r~(BK+ ~

—

2

~(

(dgK’\

5)(8~ BL)

F~I2

260

\~

—

2i’ h 1I~,I2

~

tan

~BK

+ EF

—

hM!’

(

.

)

where ~eis the average energy of the final state electron. As the incident photon energy exceeds the K shell threshold, RRS goes over into K X-ray fluorescence. As shown by Eisenberger et al., the L shell hole state width can be easily incorporated in the formulation. Modifications of the above treatment due to quadrupole terms and target atom orientations have been indicated by Aberg and Tulkki. The infrared divergence accompanying resonant Raman—Compton scattering has been investigated experimentally by Bannett et a!. [1977]and by Simionovici et at. [1990](see also section 5.4). A resonant effect associated with Auger electron emission following excitation of xenon by X-rays of energies close to the L2 threshold was demonstrated by Brown et al. [1980].Such an effect was also studied by an indirect technique [Kodreet a!. 1986]. When the energy spread of the incident photon beam is sufficiently small and when hi~approaches

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

91

close enough to the binding energy threshold, high resolution studies of resonant scattering and of the related Auger channel have provided evidence for the single stage nature of the overall process, rather than for a two stage description involving inner shell vacancy creation and its subsequent filling by an outer shell electron. A narrowing of the linewidth of the resulting spectra under these circumstances is mentioned briefly in section 5.4. An electron emerging after photon scattering from an atom can get scattered from neighbouring

atoms and consequently EXAFS (Extended X-ray Absorption Fine Structure) type intensity oscillations should be expected in the scattered photon spectrum in the neighbourhood of the cut off energy. In RRS, the hole state is off the energy shell and so there is only a limited time for electron scattering from neighbouring atoms. Thus a significant reduction of the anticipated fluctuations is likely to occur, particularly as the incident photon energy is tuned away from threshold. Further, the energy resolution of the spectrometer must be good enough to enable the fluctuations to be measured (see section 5.4 for details). Investigations at photon energies close to but larger than a binding energy threshold, although strictly outside the purview of this report, will be mentioned very briefly in view of incisive and

somewhat complementary information obtainable from them about processes near a threshold. In these cases, an inner shell electron emerges with very small kinetic energy. The effect of the interaction of the core hole with the slow photoelectron, called the post collision interaction (PCI), and the effect of the final state interaction between a photoelectron and an Auger electron has been seen in recent studies of Auger electron spectra obtained with a xenon gas target [Borstand Schmidt 1986; Schmidt 1987; Tulkki et al. 1987; Armen et al. i987a, b; Armen 1988]. The underlying theory of these processes has also been presented [Niehaus 1977; Aberg 198i; Niehaus and Zwakhals 1983; Ogurtsov 1983; Russek and Mehlhorn 1986]. 2.7. Higher order effects and other recent studies (not covered in this review)

Low probability events arising from higher order processes such as Compton scattering with the emergence of two or more photons have been studied theoretically [Brownand Feynman 1952; Mandl and Skyrme 1952; Mork 1971; Ram and Wong 197i; Marchetti and Franck 1987a; Ohmura and Sato 1988] and experimentally with targets of very small Z [Cavanagh 1952; Bracci et al. 1955, 1956; Theus and Beach 1957; McGie et al. 1966, 1968; Sekhon et a!. 1988]. Thin targets of very small Z were chosen

in order to minimise complications arising from bremsstrahlung. The measurements are in fair agreement with theory. As expected, the higher order cross sections are typically less than 1% of the values calculated in the lowest non-vanishing order. The other recent studies refer to electron magnetic moment dependent scattering [e.g. Sakai et al. 1991] and scattering involving plasmons [Schülkeet al. 1989], phonons [Burkel et at. 1989], and standing wave fields [Schülke1989]. These topics, being not important in the study of inelastic scattering by inner shell electrons, are not covered in this review.

3. Experimental details In this section, a description will be given concerning the methodology involved in measurements of non-resonant and resonant inelastic scattering of X-rays and gamma rays from inner shell electrons.

92

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

Considerations pertaining to sources, targets, detectors and electronics will be described briefly in sections 3.1 to 3.4. The bremsstrahlung contribution in the scattered beam will be discussed in section 3.5. The methods adopted for the determination of absolute values of scattering cross sections and the commonly used scattering arrangements will be outlined in sections 3.6 and 3.7, respectively. A few available direct measurements of K and L shell total scattering cross sections will be mentioned in section 3.8.

3.1. Sources Only radioactive sources were used until recently for the study of Compton scattering from inner shell electrons by the coincidence technique (see tables 4. i to 4.4 in section 4). Americium-241, cobalt-57, cerium-i4i, mercury-203, chromium-Si, cesium-137, scandium-46 and zinc-65 sources were used to provide gamma rays of 59.54 keV, 122 keV, 145 keV, 279.2 keV, 320 keV, 66i.6 keV, an average energy of 1002 keV and 1116 keV, respectively. Cerium-14i, mercury-203, and chromium-Si sources have relatively short half lives of 32.5, 46.6, and 28.0 days, respectively. A mercury-203 source is accompanied by characteristic K X-rays of energies higher than 70 keV, which can affect the scattered beam spectrum in the lower energy regime. In the case of a chromium-Si source, only about 10% of the decays give rise to the desired 320 keV gamma rays. A strong gold-i98 source of high specific activity and providing 412 keV gamma rays has been shown to be convenient for the study of Compton profiles, although the half life of only 2.7 days necessitates repeated neutron activation in a nearby nuclear reactor. The large number of early X-ray scattering experiments mentioned in appendix B and several RRS experiments were performed with conventional (-~..-1 kW) or rotating anode high power (—10kW) X-ray generators. With monochromators of moderate resolution, Ka1 and Ka2 doublets cannot be separated. In order to achieve larger photon intensities, curved focussing monochromators have been used ([Suortti et at. 1986] and references mentioned therein). Several methods of obtaining high monochromaticity of synchrotron X-ray beams of small angular divergence (of the order of 0.1 mrad) have been summarised by Schfllke [1986].Most of these methods utilise narrow slits at a large distance from the emission point, and perfect double crystals or channel cut crystals of silicon or germanium. Energy spreads of less than an eV have been obtained in this way at energies less than 10 keV [Eisenbergeret a!. 1976a, b]. For achieving much smaller energy spreads, a pre-monochromator of the above mentioned type was used and the pre-monochromatised beam was allowed to undergo Bragg reflection in near back scattering geometry [Dorneret al. 1987], or repeated back reflections with additional double crystal arrangements [Faige!et a!. 1987]. Moderately monochromatised synchrotron X-ray beams have been utilised in some of the recent coincidence experiments [Marchettiand Franck i987b, c, 1989; Rollason et al. 1989b; Bell et al. 1990], although the pulsed nature of synchrotron radiation results in a larger percentage of chance coincidence counts than in the case of a continuous source of the same average intensity. The synchrotron radiation source is particularly well suited for RRS studies on account of its tunability and strong linear polansation in the plane of the electron orbit. When scattering of an X-ray beam through 90°is studied in the plane of the electron orbit, the elastic and Compton components are weak on account of the last mentioned characteristic and therefore measurements of polarisation insensitive RRS become easier.

PP. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

93

3.2. Targets

Ideally a target should have as small a thickness as possible in order to reduce secondary effects such as multiple photon scattering and bremsstrahlung. In many studies of Compton profiles, a different strategy was adopted. When bremsstrahlung contributions to the scattered photon spectra were not troublesome, very thin targets did not have to be used provided elaborate algorithms were developed in order to correct scattered beam spectra for the effects of multiple photon scattering [Pitkanenet al. 1987; Rollason et al. 1989a]. In studies of resonant scattering near, for example, K shell thresholds, attention is mainly focussed on scattered beam spectra near (hi’1 B1) or (hi’, BM), i.e., near photon energies comparable to appropriate photoelectron energies. Since bremsstrahlung intensity in the tip region is small, targets of —

—

fair thickness have been used in RRS experiments and in Compton scattering experiments near the kinematic limits (hi’, BK) or (hM BL) (see tables 4.6 and 4.5, respectively). However, if a study of infrared divergence effects is intended, it is essential to use extremely thin targets [Simionoviciet al. —

—

1990]. As shown by Schaupp et al. [1984],very pure targets are essential in resonant scattering experiments in order to ensure absence of characteristic X-rays of impurities of slightly lower Z than the target of interest.

3.3. Detectors and scattered beam analysis

High efficiency of detection and good energy resolution are the main desirable characteristics of detectors. Frequently, convenience and affordable cost are additional considerations affecting the choice of detectors. Thallium activated sodium iodide detectors have only a moderate resolution (e.g., photopeak full width at half maximum of about 50 keV at 662 keV) but satisfy the other criteria rather well. Further the energy distributions of photons after Compton scattering by inner shell electrons of medium and high Z atoms are expected to exhibit comparable or even larger widths. As a result, sodium iodide detectors have been used in many experiments to determine pulse height distributions and double differential cross sections for Compton scattering of high energy gamma rays by inner shell electrons (tables 4.2 and 4.4). In the case of several studies, pulse height distributions were not determined but pulses with amplitudes above an electronically selected bias level were counted and the counts so obtained were used to report single differential cross sections (table 4.3). Semiconductor silicon and germanium detectors have good resolution with typical FWHM values of about 170eV at 6keV, 400eV at 60 keY and i.4keV at 600 keV. Tables 4.1 and 4.5 list studies of non-resonant scattering performed with semiconductor detectors. In a study of Compton scattering by L shell electrons ofgold, Basavaraju et a!. [1982]utilised the good resolution of a Si(Li) detector to select coincident La, Lj3, or Li X-rays. With this technique, it was possible to change the contributions of L3 or (L1 + L2) subshells to the scattering signals. Double silicon crystal Bragg spectrometers with very high resolution (—0.8 eV) have been used with synchrotron X-ray beams by Eisenberger et al. [i976b], see also table 4.6. With a bent crystal spectrometer and a position sensitive counter on the focal circle, a resonantly scattered beam was simultaneously analysed over an energy interval of 80 eV at 0.3 eV resolution in an attempt to separate resonant Raman and Raman—Compton contributions [Briandet al. 1981]. Position sensitive detectors

94

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

of different types have been useful in high resolution studies of Compton profiles [Loupias and Petiau 1980; Frouin et a!. 1988; Shiotani et al. 1989; Itoh et a!. 1989]. 3.4. Electronics and data analysis

The objective of a scattering experiment is to determine the number n(hi’1) d(ht.~)of scattered photons in the energy interval d(hv1) at energy hi~.If N(V) dV is the measured number of pulses with amplitudes between V and V + dv, we can write (hPf)max

N(V) dV=

J

R(V, hi’f)n(hi’f) d(hi’f) dV,

(3.1)

where R(V, hi’1) is the response function of the detector, i.e., the probability that a detected photon of energy hi’1 generates a pulse of amplitude near V. The spectral distribution n(hi’f) of scattered photons can be determined from the measured pulse height distribution by a deconvolution of eq. (3.1), if the response function has been determined with calibrated monochromatic sources of gamma rays of

different energies. In order to develop confidence in the final answer, the integrity of the unfolding procedure has to be tested with previously known spectral distributions. In most coincidence experiments, the scattered photons were detected in coincidence with characteristic X-rays of the target. The rate of net coincidences N, obtained with a given target is expressed as T—N~h—Nf.

(3.2)

Nt=N

Here, NT is the measured rate of coincidences with the given target, N~,is the chance coincidence rate and N 1 is the target Z independent rate of false counts. Strictly, the rate N1~of false counts arising from target Z dependent secondary processes such as bremsstrah!ung and characteristic X-ray production by electrons should also be subtracted from the right right hand side of eq. (3.2). Since N1~cannot be measured accurately but has to be estimated by auxiliary measurements and calculations, it has become customary to express N5 in the above form and to correct it later for N1~in a manner described in section 3.5. Note that a single channel analyser is used at the output of a thin X-ray detector in order to select target characteristic X-rays within a narrow energy interval. In the case of experiments with incident gamma rays of energies larger than about 100 keY, the rate Nf was measured with a target of low Z and was found to be between about 10% and 40% of the total rate NT. On the other hand, in the case of lower energy experiments, the rate N1 was measured in the absence of any target. Resolving times between 20 and 120 ns have been used in reported studies of Compton scattering. Except in the case of experiments with synchrotron radiation, the rate N~varied typically between 10% and 40% of NT.

3.5. Target Z dependent false coincidencesIcounts

As mentioned in the discussion following eq. (3.2), the target Z independent false count rate Nf does not take into account all sources of false counts. Electrons released through the photoeffect and Compton scattering cause bremsstrahlung emission and further ionisation during the process of slowing down in the target. These processes are target Z dependent. Therefore, their contributions have to be

P.P. Kane, Inelastic scattering of X.rays and gamma rays by inner shell electrons

95

separately estimated and then subtracted from the net counts N5 in order to determine the corrected count (Nt)corr due only to scattering events of interest. A large number of possible secondary processes can in principle contribute false coincidence counts. Following earlier work [Basavarajuet a]. i987a], these are listed for the case of a high Z target such as gold in table 3.1 with a few modifications and comments. Note that only a few processes make significant contributions to false counts, bremsstrahlung from photoelectrons being dominant in the case of high Z and electron produced secondary ionisation being relatively more important in the case of medium Z. Table 3.1 A list of target Z dependent contributions to coincidence counts arising from photons of primary and secondary origin in K shell Compton scattering experiments. The relative importance of each contribution in the case of a high Z target such as gold is indicated in the third column. Nature of process giving rise thin X-ray detector with SCA

toevents in gamma counter

relative importance

Primary processes

(1) K X-ray following Compton scattering by K shell electrons

gammaray Compton scattered from K shell electrons

events of main interest

(2) gamma ray from double Compton process

gamma ray from double Compton process

negligible

(3) X-ray from two photon emission after K shell Compton scattering

gammaray Compton scattered from K shell electrons

negligible

(4) K X-ray following K shell photoeffect

bremsstrahlung photon from photoelectron

significant

(5) K X-ray following K shell photoeffect

K X-ray due to K shell ionisation caused by K shell photoelectron

weaker than process (4)

(6) K X-ray after K shell Compton scattering

bremsstrahlung photon from K shell Compton electron

much weaker than process (4)

(7) K X-ray following K shell Compton scattering

K X-ray due to K shell ionisation by K shell Cosupton electron

weaker than process (6)

(8) bremsstrahlung photon from K shell Compton electron

gammaray Compton scattered from K shell electrons

much weaker than process (4)

(9) K X-raydue to K shell ionisation by K shell Compton electron

gamma ray Compton scattered from K shell electrons

weaker than process (8)

(10) bremsstrahlung by Compton electrons except from K shell

gamma ray Compton scattered from other than K shell electrons

weaker than process (4)

(11) K X-ray due to K shell ionisation by Compton electrons except from K shell

gamma ray Compton scattered from other than Kshell electrons

weaker than process (10)

Secondary processes

96

P. P. Kane, inelastic scattering of X-rays and gamma rays by inner shell electrons

Let us consider coincidence experiments concerning Compton scattering by K shell electrons. It is possible for the gamma counter to detect K X-rays in coincidence with bremsstrahlung photons accepted by the single channel analyser window at the output of the X-ray detector. Secondary ionisation caused by photoelectrons can also lead to K X-ray emission. Since the positions and widths of K X-ray peaks in the gamma counter spectra are accurately known, their contribution can be subtracted in a straightforward manner. We will discuss in more detail false counts arising from the detection of K X-rays by the X-ray detector in coincidence with the detection of bremsstrahlung photons by the gamma counter. The bremsstrahlung contribution makes it very difficult to identify the infrared divergence signal in bound electron Compton scattering. The rate of net coincidence counts per unit energy interval in the gamma counter spectrum due to K shell scattering can be written in the form z~N A(hv 1)

2d2r

w (N0n~~

=

~

flKQKaKeKQfaV~eP1) du11

d(hi’5)

(3.3)

where N0 is the number of gamma rays incident on the target per unit time, n5~is the number of target atoms per unit area, WK is the K shell fluorescence yield, t~K is the single channel analyser window acceptance factor for target K X-rays, ~K’ a K and EK are the solid angle, the transmission and the detection efficiency for K X-rays, respectively, (1~,a,,~and r~ are the solid angle, the transmission and 2uKId~’1fd(hi.~) the gamma counter detection efficiency, respectively, for photons of energy hi’1, and 2 d is the double differential cross section of two K shell electrons for Compton scattering. Similarly, the Z dependent false coincidence count rate per unit energy interval due to bremsstrahlung can be written in the form w

I

= (N 0n5~

~.

~Kf1KaKeKQfaSfePf)aPK

d~u

J dfl~d(hv1) dn5~,

(3.4)

2ub/dflfd(hi.~)is the double where crPK is thesection K shellforphotoelectric cross sectionbyper d differential cross bremsstrahlung emission an target electronatom, of kinetic energy T and the integral is evaluated over the path of an electron slowing down within the target. The quantity I UPKJ

~

dQ

1d(hi.~)dn5~

may be called the “apparent” bremsstrahlung cross section. Note that the factors in parentheses of eqs. (3.3) and (3.4) are the same. An accurate estimate of the apparent bremsstrahlung cross section is very difficult on account of the combined effects arising from the angular distribution of photoelectrons, the energy loss and multiple scattering of electrons in a target and the anisotropy of bremsstrahlung emission. Detailed Monte Carlo calculations of the different processes will be necessary for a comprehensive treatment of the situation. In many experiments an attempt was made to assess the contribution of the bremsstrahlung intensity by making measurements with targets of different thicknesses. Contrary to expectations, in many cases the double differential cross sections deduced without consideration of electron multiple scattering did not show an appreciable thickness dependence within the typical error of about 20%. In the beginning,

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

97

this situation was taken to mean that the bremsstrah!ung contribution was not significant in those cases. However, the considerable increase in the effective track length of electrons due to multiple scattering has to be taken into account. Thus even when the normal target thickness t is of the order of only 10% of the electron range D calculated in the continuous slowing down approximation, the bremsstrahlung intensity approaches its limiting value. Hence the naively expected target thickness dependence of

bremsstrahlung was not observed in the cases referred to above. Reference may be made to the work of Marchetti and Franck [1989]and of Lad et a!. [1990]for further details. Even in the case of recent work in the singles mode with synchrotron radiation of 14 to 17.4 keY and with zirconium targets of very small thickness (20 to 200 p..g/ cm2 on mylar backing [Briand et a!. 1989; Simionovici et al. 1990]), the smallest tID value was about 0.05 and therefore effects of electron multiple scattering cannot be considered negligible without further detailed calculations [Kane and Basavaraju 1991]. A semi-empirical thick target formula for the estimation of bremsstrah!ung was developed a long time ago [Kramers 1923] and has often been used in the absence of more detailed estimates based on

elaborate Monte Carlo calculations of the complexity pointed out above. The formula may be expressed as 2 do~

J

d

dQ1 d(hi’1)

2kZ T ~sc

—

0—hi’1 4~. hi’1

35

where the initial photoelectron kinetic energy and the parameter k’ turns out to be of 1. Different valuesT0ofisk’(hM used BK) by different authors reflect the approximate the order of 0.5 x 10-6 keV nature of the semi-empirical formula. The experimental counts corrected for false events arising from characteristic X-rays and bremsstrahlung can be used to obtain double differential cross sections for K shell Compton scattering alone as shown in the next subsection. —

3.6. Determination of absolute values of cross sections 3.6.1. Coincidence experiments pertaining to K shell Compton scattering It is convenient to eliminate N 0 and f1~and to simultaneously express the transmission and detection

efficiency for scattered gamma rays in terms of suitable ratios. For this purpose, a low Z target such as one of aluminium is employed in the singles mode to determine the rate N5 of counts in the corresponding Compton scattering peak. The rate N5 is proportional to the product of the Klein— Nishina cross section per electron and the incoherent scattering function S(x, Z = 13), N5 = NOnAI

tI2

thr KN

S(x, Z = i3)a~s~

1,

(3.6)

where is the number of aluminium target atoms per unit area, a~is the transmission of gamma rays Compton scattered by the aluminium target and is the efficiency of the gamma detector for ~Aj

aluminium scattered photons. Note that the width of the pulse height of the Compton peak in the case of aluminium arises from the intrinsic detector width, the variation of the scattered beam energy due to finite angular acceptance and the momentum distribution of electrons in aluminium. Rewriting eq. (3.3) in terms of the corrected count rate and using eq. (3.6), we get

98

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons 2duK 2

~N

(

5

dfl5d(hi’1)

—

\

AIAI

nAte.., a~~ do~KN \~(hi’1,)!corrnSCN,eVa,wKr~KQKaKeK S(x, 13). -~-~—

(3.7)

A numerical integration of the double differential cross sections from a specified low energy up to the kinematic limit (hi’, BK) leads to the single differential cross section, whose value is thus sensitive to the chosen low energy. A few authors have used an integration over only the peak portion to determine —

the so-called quasi-Compton single differential cross section.

Values of the transmission are determined with the help of tabulated attenuation coefficients [Storm and Israel 1970; Veigele 1973; Hubbel 1982; Saloman et al. 1988; Weast 1989]. The values of the detection efficiencies are determined by interpolation between values measured with different sources 12K is of known and energy. The [Bambynek solid angles et areal.calculated numerically. The fluorescence yield and is taken fromstrength well known tabulations 1972; Krause 1979; Krause and Oliver 1979] known to better than 2% in the case of high Z elements. The window acceptance factor ij~ is determined by recording the X-ray detector spectrum in the se!f-gated mode, the single channel analyser output providing the gating signal for the pulse height analyser. The combined error in the factors other than ~~1Kand [~Nt/~(hi’t)]corr is typically about 8% and turns

out to be weakly dependent on hi’

1 except at low energies. Statistical errors in the corrected count rate per unit energy interval have been usually comparable or larger. In several reports, only the statistical errors have been given explicitly. 3.6.2. Coincidence experiments pertaining to L shell Compton scattering2o-K/d4 d(hi.~)in eq. (3.7) will In the casereplaced of L shell scattering the expression WKflK ~2d have to be by Compton R4d2uL/df1fd(hz.~)],V,where [d2tT 1/dl2td(hVt)]av is the L shell electron

scattering cross section, the subscript av indicates a weighting of the three subshells depending upon the window setting of the single channel analyser at the output of the L X-ray detector and R~,expresses the probability of the resulting L X-ray energy to lie within the chosen acceptance window of the single channel analyser. Details of this procedure including the possibility of changing subshell contributions have been described earlier [Basavaraju et a!. 1982]. The three other quantities in eq. (3.7) with the subscript K will have to be relabelled appropriately to represent L X-ray detection in the present case.

As shown in the last mentioned work, a reduction of the final error in the scattering cross section can be achieved by means of a comparison with counts representing coincident detection of K and L X-rays by the two counters in the same experiment. Note that magnetic substate alignment is possible in the case of the L3 subshell with a total angular momentum of 3/2. A discussion of this effect with reference to L shell scattering has been given by Bell and Böckl [1985]. 3.6.3. Experiments in the singles mode with high energy gamma rays for the study of intensity discontinuities near kinematic limits associated with K and L shell binding energies Typical data obtained by Reineking et al. [1983]with 661.6 keV rays and targets of copper, tin and lead are shown in fig. 3.1. Calculations based on the relativistic impulse approximation are shown by

solid lines. The expected discontinuities are certainly seen. However, the data for lead in the neighbourhood of the discontinuities deviate systematically from theoretical calculations, indicating a need for further work.

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

I

I

A

-

/

99

I

I

ht~ e 1=145° 661.6keV

\~

~~24

‘II,’

-

yII\\

N

N

~

E E

-26 10

—

U

I I

b

-

‘0

~10

0

V

I 10

—28

4 V C

100 —31

10

-

—32

1013

200 I

300 400 I 500 I 600 PHOTON ENERGY(keV)

Fig. 3.1. Double differential atomic cross sections for Compton scattering of 661.6 keV gamma rays through 145°in the case of (a) lead, (b) tin and (c) copper, determined by Reineking et al. [1983].In the interest of clarity, the data for tin and copper are shifted downwards by factors of 10 and 100, respectively. Discontinuities in the data near the energies (hs~— BK) and (hs1 — BL) can be seen. The solid curves show the calculations of Reineking et al. on the basis of the relativistic impulse approximation. Note the deviations between experimental results and calculations, particularly in the case of the high Z lead target.

3.6.4. Resonant scattering cross sections

In resonant scattering studies the incident photon energy is varied in the neighbourhood of the target atom binding energy. So a different method can be used for the determination of absolute values of cross sections [Manninen et al. 1986b; Simionovici et al. 1990]. The incident photon energy is varied in the range close to but above a binding energy threshold, say BK in the case of K—L, K—M,. resonant scattering, andt~1K theand intensity fluorescent K X-ray measured a function If the Kofshe!! photoeffect crossemission sectionisare known,asthe rate of of K hi’,. X-ray the fluorescence emission at valuesyield of hi’~just above the K shell threshold can be used to determine the incident beam intensity. The relative variation ofthe beam intensity in a narrow energy interval from above to below .

threshold was inferred from the current variation in an ionisation chamber. Note that an energy dependent correction for absorption in the chamber gas can be applied reliably in the narrow energy interval of interest, which is much higher than any binding energy thresholds of the chosen gas. Thus absolute values of resonant scattering cross sections can be determined without having to measure

100

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

directly the absolute value of the incident beam intensity. As a by-product of this approach, the usually

neglected contribution of resonant scattering to the X-ray attenuation coefficient was also determined at energies slightly lower than the respective thresholds. 3.7. Scattering arrangements 3.7.1. Studies utilising coincidence techniques Some of the target Z independent false coincidences N1 arise from scattering of gamma rays by one detector into the other. In an attempt to reduce such false coincidences, the X-ray detector direction

with reference to the target was chosen in some experiments to be normal to the scattering plane [Spitaleand Bloom 1977; Singh et al. 1984, 1985; Marchetti and Franck 1987b, 1988, 1989]. If such a choice is made, it is necessary to tilt the plane of the target away from the X-ray direction in order not to reduce unduly the transmission of target characteristic X-rays. If characteristic X-rays from a shielding material such as lead cannot be resolved by the X-ray detector from the characteristic X-rays of a high Z target, it is advisable to use graded shielding of high Z—intermediate Z—low Z metals next to all surfaces facing the target and the detectors. In order to maximise the transmission of characteristic X-rays, a target is frequently mounted in reflection geometry with respect to the X-ray detector. Air scattering is eliminated by ensuring evacuated paths

for the different beams. An arrangement for a study of scattering at a large angle is shown in fig. 3.2. 3.7.2. High energy photon scattering studies in the singles mode The scattered intensity in the neighbourhood of a cut off energy such as (hi’1

—

BK) is typically only

of the order of 0.1% of the intensity at the main Compton peak. Therefore, a great deal of attention has to be devoted to the reduction of background in these studies. The experiments designed to determine spectral shapes in the neighbourhood of discontinuities at (hv BK) etc. are usually —

LEAD GRADED SHIELD

Fig. 3.2. A typical geometrical arrangement for the study of K shell Compton scattering by the coincidence technique. S: source, T: target, D,: X-ray detector, D,: gamma ray counter, 7,: angle between the incident photon direction and the normal to the target, 72: angle between the scattered photon direction and the normal to the target, 72.: angle between the X-ray direction and the target normal. A shielding arrangement is also shown.

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

101

performed at large angles of scattering at which the scattered photon energy spread caused by a given angular acceptance is relatively small and the validity of the impulse approximation is better. If a source such as gold-198 has a short half life (2.7 days in this case), it is possible to make a direct measurement of the elastic line shape without a long delay after the conclusion of the experiment. The latter shape is needed particularly in the analysis of discontinuities in the intensity caused by L or higher shells. 3.7.3. Study of resonant scattering

Although initially and in the case of high Z materials, X-ray tubes and radioactive sources were used for the study of resonant scattering, tunable synchrotron radiation of small angular divergence is now the usual choice. Further the polarisation of the synchrotron X-ray beam in the plane of the electron orbit is helpful in achieving a substantial reduction of Compton and elastic scattering components. 3.8. Direct measurement of total Compton scattering cross sections of K and L shells

The experimental method relies on a careful measurement of K X-ray intensities resulting from K shell ionisation produced by scattering as well as the photoeffect and a subsequent subtraction of the calculated contribution of the photoeffect. Initial experiments were attempted in the case of gold and lead targets with gamma rays of 279.2 keY, 661.6 keV and 1252 keY (average energy of a coba!t-60 source). Since the K shell Compton scattering cross sections are only about 1% of the K shell photoeffect cross sections in the case of high Z materials at 279.2 keY, the attempt was not successful at

this energy (Ghumman and Sood 1967]. In later work from the same laboratory, targets of lower Z and a comparison method were used [Allawadhiet al. 1978; Verma et a!. 1980]. Errors were still of the order of 15 to 30%. The comparison consists of measurements of target Compton scattering cross sections for gamma rays relative to photoeffect cross sections for X-rays associated with internal conversion accompanying the gamma rays. For example, cesium-137 sources of 661.6 keY and mercury-203 sources of 279.2 keY gamma rays are

accompanied by barium and thallium K X-rays, respectively. Iron absorbers of different thicknesses were placed in turn in the incident beam, and target K X-ray intensities arising from interactions of gamma rays as well as conversion X-rays were determined with each absorber thickness. The attenuation coefficients are substantially larger for conversion X-rays, which have much lower energy than the respective gamma rays. It was possible to express the probability for target K shell ionisation due to gamma ray interactions in terms of that due to conversion X-ray interactions. The latter, being dominated by the photoeffect, could be calculated with fair precision from accurately known attenuation coefficients. As in the initial attempts, a subtraction of the contribution arising from the gamma ray photoeffect was made in the last step. The 279.2 keV experiment was performed with iron, nickel, copper, zinc, arsenic, selenium and bromine targets of 10 to 40 mg/cm2 thickness and the 661.6 keY

experiment was carried out with targets of similar thickness of yttrium, zirconium, molybdenum, silver, cadmium and tin. In the analysis, the production of photons in the energy range of target K X-rays due to secondary processes was assumed to be negligible. However, the discussion in section 3.5 shows that this assumption cannot be considered reliable without a detailed examination.

In further work [Verma et al. 1981], errors of about 10% have been reported and the measurements

have been extended to targets from arsenic to iodine at 145 keV, from iron to tungsten at 279 keY, from yttrium to uranium at 662 keY and from niobium to uranium at 1250 keV. Similar work has been reported for scattering of 662 keY gamma rays by L shell electrons of atoms with Z between 74 and 92

102

P.P. Kane, inelastic scattering of X-rays and gamma rays by inner shell electrons

[Verma et al. 1983]. The reported values of total K and L shell Compton scattering cross sections were consistent with those calculated on the basis of the Klein—Nishina formula for free electrons. A further brief discussion of these results will be presented in section 5.6.

4. Experiments 4.1. A list of experimental studies of inelastic X-ray and gamma ray scattering by inner shells of medium and high Z atoms The relevant information is presented in the form of tables. Brief comments regarding the nature of

the results reported in each study are given in the tables in order to assist the reader in the assessment of the overall situation. Table 4.1 mentions coincidence experiments employing at least one semiconductor detector. Table 4.2 lists studies of pulse height spectra and double differential scattering cross sections with sodium iodide detectors in the coincidence gated mode. Table 4.3 also refers to K shell Compton scattering studies carried out with coincidence techniques but directed primarily towards measurements of single differential cross sections. Table 4.4 refers to L shell Compton scattering. Note that absolute values of double differential scattering cross sections are available from only some of the experiments listed in tables 4.1 and 4.2. Experiments performed with semiconductor detectors and directed towards a study of discontinuities in the pulse height spectra of scattered photons near the high energy end are indicated in table 4.5. The resonant scattering investigations are listed in table 4.6. 4.2. Presentation of some data It is appropriate at this stage to present some data for illustrative purposes. Other data will be presented in section 5 and will be discussed in relation to theoretical calculations. Coincidence mode data obtained by Marchetti and Franck [1989]with semiconductor detectors, concerning scattering of 70 keY synchrotron radiation through 90°by copper K shell electrons, are shown in fig. 4.1. The triple differential scattering cross sections obtained at energies lower than about 35 keV were shown to be mainly due to bremsstrahlung effects and are not shown in the figure. The rise of the cross section near the kinematic limit of about 61 keV should be noted. Results of two sets of calculations are also shown in the figure. The data will be discussed further in section 5.2. Data obtained in the coincidence mode with sodium iodide detectors in the case of 279.2 keY gamma

radiation incident on a tin target [Basavarajuet al. 1987b] are presented in fig. 4.2. The FWHM width of the Compton peak obtained in the singles mode with an aluminium target was about 25 keV. As expected, the FWHM width of the peak shown in fig. 4.2, corresponding to K shell Compton scattering, is significantly larger. Resonant scattering spectra, obtained with a Si(Li) detector in the case of scattering of mono-

chromatised synchrotron radiation of different energies through 90°by a copper target, are exhibited in fig. 4.3. Note that the incident beam was linearly polarised in the electron orbital plane, which was also chosen to be the scattering plane. Then at 90° scattering angle, e, e5—0 (see fig. 2.1), and so the contributions of elastic and Compton scattering to the different spectra are relatively small. Note also the changes in the vertical scale for different portions of the figure. Shapes of pulse height spectra of Ku and K13 X-rays can be seen in the topmost portion of the figure for 8.985 keY incident energy, which is larger than the K shell binding energy. Weak Ku and K~3components indicated in the lowest portion of

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons 4L

103

P~

1~70ktV

Cu 90° In

V x 10

ENERGY (key) Fig. 4.1. Triple differential cross sections perunit solid angle per unit energy range of scattered photons per unit solid angle for fluorescent K X-ray calculation of Marchetti and Franck on theofbasis of the Herman—Skillman potential.90°by The broken emission denoted as FX, determined by Marchetti2and Franck [1989]for Compton scattering 70 keY synchrotron radiation through copper K shellrepresents curve electrons.aThe calculation solid curve according represents to the theSchnaidt A formula for a point Coulomb potential. Results at energies lower than about 35 keV are not shown.

hV~= 279.2 keV e =115° TIN In > V

12 hY~

..__(~

E

hY(-BK

8

‘0

~ø 0 0 ______________________________________ 80 160 240 FINAL PHOTON ENERGY (key)

320

Fig. 4.2. Experimental results of Basavaraju et al. [1987b]for double differential cross sections in the case of Compton scattering of 279.2 keV gamma rays through 115°by K shell electrons of tin. The “free” Compton energy hv~and the cut off energy (hv~— BK) are indicated in the figure.

104

P. p. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

Table 4.1 K and L shell Compton scattering studies utilising coincidence techniques and at least one semiconductor detector (in order of decreasing h~). source and h~(keY) K shell scattering °7Cs 661.6

targets and thickness 2) (mg/cm

photon scattering angles (deg)

detectors

reference, source strength

nature of reported results

Au 5

60,90

1 mm thick lithium drifted Si detectorfor electrons (about 20 keY resolution) and two NaI(Tl) for hvf and K X-rays

a) 5 Ci

(i) Triple differential cross sections determined. (ii) Experimental cross sections smaller than calculated ones by factor between 2 and 3; deviations attributed to multiple Coulomb scattering of electrons in the target.

Ta

15, 20, 30, 40,

0.5cm thick Ge(Li) for KX-rays and hv 1

b)

(i) First experiment with two

100 Ci

semiconductor detectors. Chance coincidence rate 60 to 95% of total. (ii) Pulse height distributions obtained but not energy distributions.

Ge(Li) as scatterer and recoil electron energy detector and two NaI(Tl) for hi~and KX-rays

c), d) 86 mCi

(i) Triple coincidence studied among recoil electron energy detector and two NaI(T1) detectors for K X-rays and hv1. (ii) Relativistic calculations performed but without A P type terms.

11 39.5

Pt 11.3 Au 9.64

50,70

Ge 0.6mm

90, 135

Pb 6.8

15, 30 55, 70, 80, 90, 100,110, 130, 150

Ge (Li) for hPf and HpGe for KX-rays

e) 1 Ci

(i) Target Z dependent false counts determined using X-ray detector SCA window slightly above target K X-ray energies. (ii) K shell cross section ratio larger than unity around 90°and falling off for 0>90°. (iii) Pulse height distributions not reported.

Sn 5.1 Pb 6.8

15,40, 60, 75, 150 100, 125,

Ge(Li) for hvf and HpGe for KX-rays

f) 1 Ci

(i) False counts also estimated with targets 2ofsilver andplatinum upto 108 thickness. (ii) K shell cross section ratios similar for mg/cm Pb and Sn.

51Cr 320

Al —0.5 to 5 mg/cm2

106

Ge for hvf and PIPS for electron

g) 160 Ci

(i) Triple differential cross sections measured with relatively thin targets.

synchrotron radiation (monochromatised to —1 keY)

Al —0.032

141 for 151 keV 142 for 144 keV

HpGe for hi’f and PIPS for electron

h), i)

(i) Triple differential cross sections for scattering from electrons in aluminium determined and found to agree with impulse approximation (see fig. 5.1).

synchrotron radiation 148 keV (monochromatised to—i keY)

Cu —0.07

140

As in h) and i)

i’)

(i) Absolute values oftriple differential cross sections (with errors of about 40%) in fair agreement with impulse approximation.

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

105

Table 4.1 (cont.) source and hj~(keY)

targets and thickness 2) (mg/cm

synchrotron

Cu

photon scattering angles (deg) 49, 70,

detectors

reference, source strength

HpGe for hv 5 and Si(Li) for KX-rays

j), k), I)

proportional counter for X-rays and Si(Li) for hvf

m) 15 mCi

50

Ge(Li) for hvf and NaI(Tl) for K X-rays

n) 100 mCi

135

HpGe for hi’~and K X-rays

o) 100 mCi

radiation —6.9 (monochromatised) 70 and 62

88, 90, 118

241Am 59.54

Fe, Ni Cu (thick)

140

Cu, Mo (thick)

Fe 79 Cu 45

p)

q)

L shell scattering 203Hg 279.2

Cu —15

132

HpGe for hv~and K X-rays

r) 900 mCi

Cu —15 Zr —19.5

125 128

HpGe for hvf and NaI(Tl) for K X-rays

s)

Au 9.2

54,90, 115

Si(Li) for L X-rays andNaIforhi.~

t), u), v)

800 mCi °~

d)

Ljubiëiè et al. [19671. Pradoux et al. [1977].

~ Rollason et al. [1989a]. Bell et al. [1991]. ° Marchetti and Franck [1989]. °~ Namikawa and Hosoya [1984]. ° Manninen et al. [1987]. °~ Basavaraju et al. [1982].

nature of reported results

b) °~

East and Lewis [1969]. Wolff et al. [1989].

~ Rollason et al. [1989b]. Marchetti and Franck [1987b]. ‘°~ Fukamachi and Hosoya [i972a]. ° Manninen [1986]. ‘~ Manninen et al. [1990a]. v) Kane [1984].

(i) First experiment with SR. (ii) Detailed analysis of target Z dependent false events showed importance of bremsstrahlung. (iii) Double differential cross sections presented. (iv) Ratio of K~to K~X-rays associated with K-shell scattering determined (v) Chance coincidence rate 96%. but (i) Pulse height spectra measured not double differential cross sections. (ii) Data not in agreement with calculations. (i) Double differential cross sections near (hM — BK) in 5 keV steps shown. (ii) Calculations on the basis of relativistic wave functions and A2 interaction term. (i) Three peaks seen; one ascribed to double Thomson and one to Compton scattering. (ii) The Compton peak showed a defect of about 10 keV toward lower energies. Comment on the work of ref. o) and highlight on detector to detector scattering as origin offalse coincidences. A reply to above comment. (i) Spectral shapes believed not to agree with impulse approximation based calculations. (i) Spectral shapes agreed with IA calculations, although ratio of binding energy to recoil energy is larger than 1. (See section 2.3) (i) X-ray detector SCA window covered ~ permitting partial separation of subshell contributions. (ii) An improved ratio method. (iii) Pulseheight distributions presented. c) Pradoux et al. [1973]. ~ Wolff et al. [1991]. ° ~° n) q)

Bell et al. [1990]. Marchetti and Franck [i987c1. Pradoux et al. [19771. Namikawa and Hosoya [1986]. Basavaraju et al. [1981].

106

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

Table 4.2 Experimental work presenting double differential cross sections and/or pulse height distributions of gamma rays after Compton scattering by K shell electrons, determined with the help of coincidence techniques and sodium iodide detectors (in order of decreasing hv~l. source and hr’ (keV) °5Zn 1116

targets and thickness 2) (mg/cm Au 12.85 Pb 30.3 Th 14.9

‘37Cs 661.6

Pb 37, 46, 82, 182, 550 Pb 28.6, 58.8, 93.3 Zn Au 7.27 to 176 U Sn Sm Ta Pb

10.91 51.29 93 14.8

Pt 11.64 19.74, 55.06 128.7, 144.3

51Cr 320

reference and source strength

60,90, 100

a) —1 Ci

(i) Negligible Compton defect within ±15keY. (ii) Energy distribution FWHM about 160 keY for Au at 100° and l80keVforPbat60°.

28, 68, 132.5

b) 0.3 Ci

(i) Detailed pulse height distribution shown. (ii) Incoherent scattering functions for Kshell electrons calculated. (iii) Targetthickness effects studied.

c) 0.04 Ci

(i) Normalised double differential scattering cross sections shown. (ii) Cross section ratio was as large as 1.78 ±0.16 at 124°.

12, 20, 26, 40, 60, 75, 110

d), e)

(i) Compton defect zero for Au at 75°but 35 keV downwards for U, and zero at 26°. (ii) Marked infrared divergence at low hv~is seen in the case of gold. (iii) Relativistic calculations performed.

30,45, 60, 90, 130

f) 2.5 Ci

(i) A pulse height distribution in the case of thick Ta target shown. (ii) A quasi-Compton peak shape was deduced and used to determine single differential cross sections.

g) 6 Ci

(i) Kshell Compton scattering pulse height distribution is surprisingly narrower than that of aluminium in the singlesmode.

60, 124

30

10 Ci

nature of reported results

Sn 26.6, 135,192 Au 17, 108, 262 Au—Cu alloy

20, 30, 45,60, 90, 120, 137, 142

h)

(i) Double differential cross sections shown forAuat45°andl2O°. (ii) Negligible Compton defect within ±20keV. (iii) Pulse height distribution width smaller than that in several other studies. (iv) Infrared divergence seen.

Fe 15.8 Sn 19.56 Ho 21.9 Au 26.6, 135, 192 (in alloy form)

20, 30, 45,60, 90, 120, 136, 137

h)

(i) Low energy cut off at 45,50,80 and 30 keY for Fe, Sn, Ho and Au, respectively. (ii) IRDeffectlargerforHothanforAu. (iii) Quasi-Compton peak integrated cross section used for obtaining single differential cross sections. (iv) Cross section ratio for 0 90°is larger with Ho than with Sn or Au.

Zr Ag

40,70, 100

i) 0.5 Ci

(i) Pulse height data presented for Zr at 40°. (ii) Single differential cross sections reported for the cases studied.

j)

(i) Low energy cut off about 30 key. (ii) Detailed estimates of bremsstrahlung made; IRD effects shown to besignificantly obscured by bremsstrahlung.

64.2 105

Ho 30 Au 9.17,40.5 203Hg 279.2

angles (deg)

W Er

12.74, 197 25.5

Ta 20.75, 41.5,62.25,83

45, 115

0.8 Ci 30, 50, 70,100, 125, 130

k) 0.5 Ci

(i) Pulseheight distribution for W at 125°shown from 65 keY up to kinematic limit of210 keV. (ii) K shell Compton scattering pulse height distribution is surprisingly narrower than that ofaluminium in the singles mode.

70

I) 0.4 Ci

(i) Pulse height distribution shown todepend on target thickness. (ii) K shell Compton scattering pulse height distribution significantly wider than the singles mode aluminium distribution.

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

107

Table 4.2 (cont.) source and hv 1 (key)

targets

reference

2) and thickness (mg/cm Sn 18.8 Au 12.85,40.5

(deg) angles

‘°1Ce 145

Fe 15.8

20,30, 40, 50, 90, 120, 137

h)

57Co 122

TI t = 9.58, 2t, 3t, 4t

45,90, 135

o) 0.01 Ci

°~

d)

44, 54, 90, 115

Baba Prasad, et al. [1977]. Lazzaro and Missoni L1963].

Krishna Reddy et al. [1970]. ~ Lad et al. [1990]. Basavaraju et al. [1987a]. g)

strength and source

nature of reported results

m), n) 0.7 Ci

(i) Pulse height and energy distribution presented. (ii) Observation of a large IRD in the case of Au claimed; However, see ref. j). (iii) Fair agreement with Whittingham’s [1981]calculations for lead. (i) FWHMof the K shell scattering peak decreases from about 26 keY at 90°to lower values for larger scattering angles.

(i) Pulse height distribution presented. (ii) Here the target K X-ray energy ofabout 74 keY is higher than the cut off hp — BK of about 36.5 keV. (iii) Cross section ratios are nearly unity although binding effects should be expectedto be substantial.

5) °~

Sujkowski and Nagel [1961]. Lazzaro and Missoni [1966].

Spitale and Bloom [1977]. Murty et al. [1973]. °~ Basavaraju et al. [1987b]. 0

k)

c) t)

Yarma and Easwaran [1962]. Ramalinga Reddy et al. [1966].

Ramana Reddy et al. [1985]. B. Singh et al. [1984]. °~ Dowe [1965].

108

p. p. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

Table 4.3 Studies mainly of single differential scattering cross sections of K shell electrons by coincidence techniques with sodium iodide detectors. Studies mentioned in tables 4.1 and 4.2 are not listed here again. The ratio of the single differential cross section to the Klein—Nishina predictions is called the cross section ratio. source and hi~(keV) °5Zn 1116

targets and thickness 2) (mg/cm Sn 18.8 Ta 22.8 Au 12.85 Pb 14.01 Th 14.88

angles (deg)

reference and source strength

nature of reported results

25,60, 90,100, 120

a), b) 1 Ci

(i) For Au and Th, cross section ratios at 120°werelower than those at 100°. (ii) Calculations based on incoherent scattering function of Shimizu et al. [1965]much larger than experimental cross section ratios.

46Sc 889, 1112 average = 1002

Sn 10.9 Sm 31.3 Ta 9.12 Pb 14.85

40,60, 90, 128

c) 1.9 Ci

(i) With each target, cross section ratios larger than unity at 90°and 128°.

‘37Cs 661.6

Pb 92.8

10, 17.5, 25,35, 50, 65, 85

d) 10 Ci

(i) False coincidences not measured. (ii) Cross section ratios varied from 2.8 to about 14.9, the errorsbeing about ±20%

Sn 7.5 Au 6.7 also Au up to 40

20, 30, 40, 60, 110

e) 10 Ci

(i) Need for thintargets established. (ii) Cross section ratios larger than unity obtained at 60°and 110°. (iii) Angle integrated total cross section found to beequal to the Klein—Nishina cross section integrated over angles.

Sn 6.74 Ta 10.54 Pb 17.16

20, 35, 50, 65, 100

f) 1.4 Ci

(i) Fair agreement between these data and the data of Motz and Missoni [1961]. (ii) A formulafor incoherent scattering function derived.

Au 20

20,40, 60,90, 120, 140, 165

g), h) 2 Ci

(i) A large angular range studied in a single experiment. (ii) Calculations based on IA not in agreement with data.

Pt 11.64 Bi 3.7 Th 11.59

30, 50, 70, 105, 125

i), j) 6 Ci

(i) Cross section ratios increasing with increasing angle ofscattering. (ii) Cross section ratio as large as 1.8 at back angles.

Zr Ag

40,60, 80, 100

k) 2 Ci

(i) Cross section ratios aslarge as 1.48 measured at 100°with medium Z targets.

15,30, 50, 70, 105, 125, 150, 168

1) 0.5 Ci

(i) Cross section ratio increases monotonically from about 0.6 to about 1.5, with increasing angle of scattering (ii) Cross section ratio as large as 1.48 at 168° measured with a relatively thin target.

45,60 90,110

m) 5.5 Ci

(i) Cross section ratios increasing with angle but Iessthan unity. (ii) Cross section ratios at 45°and 60°much larger than calculated values.

64.26 105.4

Mo 8.5

5tCr 320

Sm 54.14 Ta 9.32 Pb 14.85

p. p. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons Table 4.3 (cont.) source and hv 1(keY) 203Hg 279.2

141Ce 145

°~

d)

targets and thickness 2) (mg/cm Au 15.64 Ag 52 Sm 19 Ta 41

reference and source strength

angles (deg)

nature ofreported results

20,55, 90, 125, 160

n),o) 5 Ci

(i) Crosssectionratiosoforder0.Sinthe backward hemisphere in the case of Sm, Ta and Au, and about unity in the case of Ag underthe same conditions.

30,50 70, 105, 125, 150

p) 0.75 Ci q) 0.75 Ci r) 0.5 Ci

(i) Cross section ratios increasing with increasing angle. (ii) Cross section ratios smallerthan unity except at 150°.

30,50, 70,90, 110,130 150

s) 0.4 Ci

(i) Cross section ratios smallerthan unity. (ii) Cross section ratios increasing with increasing angle.

Ag 37.8 Mo 20 to 50 Sn 20 to 50

30,50, 70,90, 110, 130, 150

u), v), w) 0.4 Ci

(i) Cross section ratios smaller than unity. (ii) Cross section ratios larger than values calculated with incoherent scattering function of Shimizuet al. [1965].

Y 134 Ag 106 Au 161

40,70, 100

x) 0.4 Ci

(i) Results for Ag not in agreement with those in ref. v). (ii) Relatively thick targets used.

Bi

3.87, 17

Th

4.89, 11.89

Pt

11.64

Pb W

30 40

Baba Prasad and Kane [1974]. Brini et al. [1960].

~ Pingot [1968]. ° Krishna Reddy et al. [1974]. ‘°~ Ramalinga Reddy et al. [1967]. i0 Nageswara Rao et al. [1977]. ° Singh et al. [1985]. °> Acharya et al. [1981].

t)

0.4Ci

b) °~

Kane and Baba Prasad [1977]. Motz and Missoni [1961].

Pingot [1971]. Ramana Reddy et al. [1983]. °~ Pingot [1969]. ‘~ Murty et al. [1977]. ° Singh et al. [1986]. w) Ghumman et al. [1981].

c)

°

Ramalinga Reddy et al. [1968]. Shimizu et al. [1965].

Murty et al. [1971]. Nageswara Rao and Murty [1989]. °~ Pingot [1972]. ° Swamy and Murty [1978a]. u) Acharya and Ghumman [1980]. ° Raghava Rao et al. [1982].

0

°

k)

°

109

110

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

Table 4.4 Studies of differential scattering cross sections of L shell electrons with the help of coincidence techniques and sodium iodide detectors. Studies mentioned at the end of table 4.1 are not listed here again.

source and hr’ (keY) ‘37Cs

targets and thickness 2) (mg/cm Au ito 20

661.6

U

203Hg 279.2

angles (deg)

reference and source strength

nature of reported results

20, 35, 50,60, 75,90, 120

Lazzaro and Missoni [1963] lOCi

(i) Cross section ratio close to unity in the case of gold. (ii) Pulse height distributions presented.

Th 4.87

30, 50, 70, 105, 125, 150

Swamy and Murty [1978b]

(i) L shell cross section ratios increasing with angle from about 0.7 to 1.2.

Pb 3.86 Th 4.87

30, 70, 105, 125, 150

Swamy and Murty [1975,1976]

(i) L shell cross section ratiosincreasing with angle from about 0.6 to about 0.95 with an error ofabout ±0.1.

3 to9

Table 4.5 Non-resonant Compton scattering studied in the singles mode with a semiconductor detector with special attention to outgoing photon energies near the cutoffs (hr’, — BK) and (hv, — BL). Low energy experiments of a similar nature with low Z targets and high resolution are not listed here but are mentioned in appendix B. Source strength is stated when known from the reference. source strength and energy hr’ (keY)

targets and thickness (mg/cm2)

angles (deg)

ref.

nature of reported results

‘37Cs

Pb

62, 85,

a)

(i) Discontinuous jumps in hr’

661.6

21to464

135

Pb Sn Cu Au Pb

145 165

b) c), d)

Pb 80 to 260 Sn 70 to 570 Cu 80 to 1270

135

e)

Synchrotron radiation 141 keY

Ag 250 to500

120, 140

1)

(i) K shell Compton profiles deduced from data and calculations for outer shells. (ii) Deduced K shell profiles in agreement with calculations.

241Am 59.54 keY

Al, Si (thick)

20,40, 60,90, 120, 150

g)

(i) For 0>90°,the discontinuous jumps can be seen clearly (ii) For smallerangles, the spectrum is similar to that studied by Suzuki [1967]with lower Z targets. Pattison and Schneider [1979a]. Manninen et al. [1990b].

98Au —200 Ci ‘ 411.8 203Hg —0.5 Ci 279.2

d)

57 to 5200 36 to 7200 85 to 10500 4400 900,4300

Schumacher [1971]. Pattison and Schneider [1979b].

~ Fukamachi and Hosoya [1972b].

1 at 574keYand646keY. (ii) Form factor calculations for electrons.

~ Reinekin~et al. [1983]. ° Rullhusen and Schumacher [1976].

(i) Deviations between experiment and calculations near the cut off energies. (i) Fair agreement between experiment and calculated Compton profiles right up to cutoff energies. (i) Form factor calculations lower than experimental data at high scattered photon energies.

“

p. p. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

111

Table 4.6 Experimental studies (listed in order of increasing hM) of resonant Raman and Raman—Compton scattering with semiconductor detectors or with higher resolution spectrometers near K shell or L 3 subshell thresholds. SR: synchrotron radiation (monochromatised) source hr’1 (keY)

targets and threshold (keY)

angles (deg)

ref.

nature of reported results

Si Ka12 produced in Si target by protons —1.73 MeV

Si 1.84

scattered internally

a)

(i) Curved crystal analyser used (ii) Theory of internal RRS given. (iii) Interpretation of other features of hr’1 spectrum discussed.

SR 2.4 to 3.2 (focussed)

Sin SF6 2.472 chlorine 2.822 Ar 3.202

—90

b), c)

(i) is—4p and is—5p RRS studied with —0.5 eY resolution. (ii) With hr’ tuned to a value between ls—4p and is—5p thresholds, superposition of blue and red shifted RRS lines seen.

SR 3.7to4 I eY monochromatised

Uin compounds BM4 = 3.728

—90

d)

(i) 3d31, to Sf512 RRS seen shifted towards red end.

X-ray tube 42 eV to 940 eV below K shellfocussed beam thresholds, cylindrically

Cr Mn Fe Ni Cu

—20

e), f)

SR 6.53

Mn in

—90

g)

5.989 6.540 7.112 8.333 8.980

(i) RRS yields measured and compared 0K with fluorescence (ii) K—M yield ~RRS deduced from asymmetry of intensity distribution (iii) Connection ofRRS and anomalous elastic scattering amplitudes studied. (i) K—L 3 and K—L2 RRS separated with —0.3 eY resolution. (ii) Atomic-like Mn 4p state 15 eY below threshold involved in RRS; resonant Compton component also studied. (iii) Red and blue shift of RRS lines with respect to Kct X-rays seen.

KMnO4 6.540

X-ray tube, CuKa1 8.047

Ni 8.333 Cu 8.980 Zn 9.659 Ge 11.104 Ta BLS = 9.881

10 to 140

h)

(i) First striking observation ofRRS. (ii) Near isotropy of RRS found. (iii) First measurement near L3 subshell threshold.

X-ray tube Cu Ka monochromatised 8.04

Ni Cu Zn Ga Ge

90

i), j), k)

(i) Suggested the explanation of RRS. (ii) Deduced constructive interference between RRS and infrared divergence amplitudes. (iii) Except for nickel,measured intensities larger than those calculated.

8.333 8.980 9.659 10.367 11.104

112

PP. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons Table 4.6 (cont.) targets and threshold

angles

(key)

(deg)

ref.

nature of reported results

SR —8.3

Ni 8.333

87

I)

X-ray tube Cu Ka1 monochromatised —8.047

Ni 8.333

<90

m), n)

(i) RRS yield studied as a function of hr’. (ii) Ratio of K—M to K—L RRS related to ratio of K~3to Ko intensity. (i) Scattered photons analysed with —10 eV resolution. (ii) EXAFS-like modulations seen in the spectrum of hr’1.

SR

Cu 8.980

90

o), p)

source hr’ 1 (keY)

SR

—8.98

7.5 to 10

Cu

8.980

90

q)

Zu 9.659 Ho BL3 = 8.071

(i) Scattered photons analysed with 0.8 eV resolution. (ii) Linearity of RRS peak shift with hr’ established. (iii) Narrowing of RRS line width just below threshold seen. (iv) Theory of RRS discussed. (i) K shell hole state widths determined. (ii) holmium L3 hole state width determined and found to be in fair agreement with theory but somewhat larger than 5eV.

SR 8.7 to 10.3

Yb Ta

BLS = 8.948 BLS = 9.88

90

r)

X-ray tube monochromatised by LiF (spread —20 eV)

Kr

14.323

angle integrated effect

s)

(i) L3 hole state widths found to be —0.7 eV larger than calculated ones. (i) Relative total intensities of RRS and corresponding Auger yields determined indirectly. (ii) Fluorescence yield found constant within ±10%above and below K edge.

X-ray tube causing Sr Ka, TIL~andKrK5

Kr

angle integrated effect

t)

(i) K shell hole state width determined to be (2.15±0.10)eV.

SR 14 to 17.4

Zr 17.998

90

u), v)

(i) Thin targets of thickness 2 between 380 sg/cm used and 20 lowand energy spectrum also studied. (ii) At low values of hr’ 1, evidence for IRD deduced.

P. p. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons Table 4.6 (cont.) source hr’ (keY)

targets and threshold (keY)

angles (deg)

ref.

nature of reported results

X-raytube Mo K~ 17.4 Mo K~ 19.6

Mo 19.999

60,90, 120

w)

(i) K—MRRSmeasured. (ii) RRS found to be nearly isotropic. (iii) ~ — c”/(BK — hr’,) + terms in ln(BK — hi~),where 2 keY/atom c”—5 x l0~cm for Mo.

SR 33.94 to 34.54

Xe 34.566

60,90 120

x)

(i) Monatomicgas target used. (ii) RRS found tobe nearly isotropic and independent of

SR 43.01to43.35

Nd 43.57

8OtollO

y)

24’Am 59.54

Yb 61.332

60, 171

z)

polarisation state of incident beam. (i) K—NRRSmeasuredfor the first time; K—L 2 and K—L3 RRS separated. (ii) Agreement ofcross sections with theory based on the dipole approximation. (i) K—M RRS line shape studied and found to agree surprisingly well with predictions based on the dipole approximation.

241Am

—

l4Oto 160

aa)

‘°9Cd 88.03

Yb 61.332 Lu 63.314 Bi 90.527

(i) K—MRRSwithhighZ targets seen. (ii) K—L 2 and K—L3 RRS also noted. (iii) Cylindricallysymmetric arrangement enabled measurements even with weaksources.

‘°°Cd 88.03

Bi 90.527

125

ab), ac)

(i) K—M and K—L2 RRS observed. (ii) Cylindrically symmetric arrangement (measurements withweak sources). (iii) Measurements also of anomalous elastic scattering.

° 1)

59.54

Hall et al. [1979]. Ice et al. [1990].

Briand et al. [1981]. Bannett et al. [1976]. m) Suortti et al. [1987]. ~° Eisenberger et al. [1976b]. ° Kodre et al. [19861. ° Simionovici et al. [1990]. ~° Schaupp et al. [1984]. ~b) Kane et al. [1987a].

b)

°

Deslattes [1987]. Suortti [1979a].

Sparks [1974]. Bannett et al. [1977]. °~ Eteläniemi et al. [1988]. ‘~ Hämhllinen et at. [1989]. ° Kodre et al. [1980]. w) Kodre and Shafroth [1979]. ° Manninen et al. [1985]. °°~Kane et al. [198Th].

g)

5)

°

6)

Lindle et al. [1991]. ~ Suortti [1979b]. c)

Bannett and Freund [1975]. Manninen et al. [1986b]. °~ Eisenberger et al. [1976a]. ° Hlmäläinen et at. (1990). ‘° Briand et at. [19891. ° Czerwinski et al. [1985]. °° MacKenzie and Stone [1985].

°

113

114

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

(a)

hV

1r8.985 keV

1:/5 I

hV~.8.970keV

Itoo

~

hV~~8.600ke~

‘0

5:789

LINE

SCATTERED PHOTON ENERGY (key)

Fig. 4.3. Resonant Raman scattering spectra obtained with a copper target and with different energies of monochromatised X-rays at the Daresbury synchrotron radiation source [Etellniemi et at. 1988]. The K shell threshold of copper is at 8.980 keV. Part (a) of the figure shows a large intensity of fluorescent Ka and K13 X-rays of copper. Note the scales of I~,10/5, I~/100,and I~/400for parts (a), (b), (c), and (d), respectively. The shift in the RRS peak towards lower energies with decreasing hr’ and a composite peak in part (d) near the elastic line should be noted. When hr’ is lower than 8.980 keY, Kct and K13 X-ray contributions can still arise due to harmonics passed by the monochromator in the incident beam.

the figure are due to small intensities of harmonics passed by the monochromator in the incident beam. The changes in the shape of the resonant scattering contributions with decreasing hi~are discussed in section 5.4.

5. Discussion of results 5.1. Triple differential scattering cross sections Several early photon—electron coincidence experiments, mentioned in the introduction, were performed with detectors of only moderate resolution. Further the true angular distribution of the emerging electrons was likely to be by multiple of a thickness larger 2 were used. So modified these experiments doscattering, not lead tosince firm targets conclusions and will not be than aboutfurther 5 mg/cm discussed in this report. In a recent coincidence experiment of Rollason et al. [1989a]with a very strong source of 320 keV gamma rays, the angular divergence of the incident beam was 1.8°,and the angular acceptances of the semiconductor detectors for the scattered gamma ray and the recoil electron were 1.2°and 3.1°, respectively. The thickness of the aluminium targets varied from 1.8 p.m (—0.5 mg/cm2) to 20 p.m (—5.5 mg/cm2). Since the electron mean free path was estimated to be about 0.1 p.m under the conditions of the experiment, the necessary Monte Carlo calculations of electron trajectories were carried out. Although the coincidence counting rate was very low, it was possible to identify electron—photon coincidence signals arising from Compton scattering.

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

140~

4

115

4

1~°(hs)j-hP~) (key)

—~

Fig. 5.1. Doppler broadened coincident photon spectrum obtained by Bell et al. [1990]with a 0.12 im self-supporting aluminium foil and synchrotron radiation of 151 keY. The photon and electron detectors were placed in the scattering plane at angles of 141°and 15°,respectively, with respect to the direction of the incident beam. The “free” Compton energy hr’~is about 99 keV in this case. The solid curve shows the calculation of Bell et al. [1990]on the basis of the impulse approximation. The arrows at the top of the figure cover the energy range of photons after Compton scattering by valence electrons.

In a subsequent experiment with synchrotron radiation of 144 keV [Rollason et al. 1989b], the photon detector was placed at a scattering angle of 142°and an angular scan of the coincident electron emission was obtained around a mean angle of 15°.The distribution of electron momenta in the plane perpendicular to the scattering vector K was determined from these measurements. Unambiguous evidence was found for the influence of the initial electron momentum distribution on the emerging electron angular distribution. In the complementary study of Bell et al. [1990],the scattering of 151 keV synchrotron radiation by an aluminium target of only 0.12 p.m thickness was studied with the photon and electron detectors fixed at 141°and 15°,respectively, with respect to the incident beam direction. The spectrum of scattered photons is related to the initial electron momentum density expressed as a function of the momentum component parallel to the scattering vector K. The photon spectrum determined in this work is shown in fig. 5.1 along with a calculated one. Within the statistical errors, there is fair agreement between experimental data and the solid curve calculated on the basis of the impulse approximation. Note also the appreciable shift of the peak toward photon energies lower than hp~.With the use of smaller angular acceptances and other improvements, absolute values of triple differential scattering cross sections have been reported recently in the case of a copper target of 0.08 p.m thickness [Bell et al. 1991]. It should be borne in mind that Bell et al. have extended the IA treatment outlined in section 2.3 to their studies involving the coincident detection of recoil electrons. Measurements of small cross sections in the tails of the Compton peak may also become possible with the availability of much brighter beams from new synchrotron facilities. 5.2. Double differential scattering cross sections determined in coincidence experiments With 661.6 keV gamma rays, Pradoux Ct al. [1977]used a selected germanium sample as a scatterer and as a detector of recoil electron energy. The data obtained by them for 90°scattering are shown in

116

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

fig. 5.2. A second order S matrix calculation is not yet available for this case. Justifying omission of A P terms on the basis of order of magnitude estimates, Pradoux et al. calculated the relevant cross sections with the help of relativistic wave functions and found good agreement with experimental data in the peak regime. However, calculations performed with non-relativistic wave functions were not in agreement with experimental data at outgoing photon energies larger than the free Compton peak energy of about 288 keV in this case. The data of Spitale and Bloom [1977]at 120°scattering angle, obtained with 661.6 keV gamma rays and a gold target, are scaled in a manner mentioned below and are shown in fig. 5.3. The experimental values deduced from a figure in their paper have been reduced by a factor of six in order to facilitate a comparison with S matrix calculations of Whittingham [1981]for lead. The small difference between the Z values of gold and lead is not expected to be very significant at energies far away from the cut off energy (hz~ BK). The experimental distribution is seen to be much narrower than that deduced theoretically. Keeping in mind the role of the employed scaling factor, we can see that the experimental values of the cross sections near the peak regime are much larger than those calculated. Thus in spite of a substantial disagreement between experiment and theory as regards double differential cross sections, the experimental value of the single differential cross section will not be in disagreement with the calculated one. The recent data of Lad et at. [1990],obtained at 115°with 320 keV gamma rays and a gold target, are shown in fig. 5.4. Unpublished relativistic calculations of Wittwer [1972]for 120°and of the Pittsburgh group for 115°are also shown in the figure. As mentioned in section 2.2.2, the results of the two .

—

hV=6s2k2V ~

Z=32

e

30

=900

:~ NE

60

U 0 17

.

o

—2Oo

‘

-

-1

-

N

c

—

C’1’bo

~-3O

0

I

-

.._

III

200

00

KN

II

I

I~I 300

III

II

I

—-

II

400

IL

10 ~

0

PHOTON ENERGY (keV)

Fig. 5.2. Energy distribution of photons after Compton scattering of 662 keV gamma rays through 90°by germanium K shell electrons, determined by Pradoux et al. [1977].The “free” Compton energy hr’~is indicated by the arrow marked KN. The solid and the broken curves represent 2 interaction term. relativistic and non-relativistic calculations, respectively, of Pradoux et at. based on the A

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

117

hV

10662 keV

1.0

e

=1200

-

o’~ll 50

200

350

500

650

SCATTERED PHOTON ENERGY (key)

Fig. 5.3. Energy distribution of photons after Compton scattering of 662 keY gamma rays through 120°by K shell electrons. The solid curve is based on the relativistic second order S matrix calculations of Whittingham [1981]for lead. The filled circles represent the data of Spitale and Bloom [1977] obtained with a gold target, after reduction by a scaling factor of 6.

hV1 = 320 keV GOLD e=115°

2

6-

-x

2.ft ~

F

hV~

4.

In N

~

I

~ 140

180

220

SCATTERED PHOTON ENERGY (key)

Fig. 5.4. Double differential cross sections for Compton scattering of 320 keV gamma rays by K shell electrons of gold. The filled circles represent the experimental data at 115°obtained by Lad et al. [1990].The “free” Compton energy hv~and the cut off energy (hv1 — BK) are shown. The broken curve labelled W represents an unpublished relativistic calculation of Wittwer [1972]for scattering through 120°.The crosses indicate the results of the Pittsburgh group for scattering through 115°[Bergstromand Pratt 1991], shown before publication by permission from Professor R.H. Pratt. The dotted curve is drawn through the crosses to guide the eye.

118

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

independent calculations are seen to be not very different here. There are, however, noticeable differences between experiment and theory. In the experiment of Marchetti and Franck [19891concerning the scattering of 70 keV synchrotron radiation by copper K shell electrons, the linear polarisation of the incident radiation was perpendicular to the scattering plane. The data were presented in the form of triple differential cross sections per unit solid angle per unit energy range of scattered photons per unit solid angle for fluorescent K X-ray emission designated as FX in fig. 4.1. The double differential cross sections for K shell Compton scattering can be obtained by multiplying the exhibited values by the factor 4171WK, where the K shell fluorescence yield WK for copper is 0.44. The solid curve in fig. 4.1 shows the non-relativistic A2 calculation of Marchetti and Franck, which was based on the Herman—Skillman atomic potential (a self-consistent Hartree—Fock—Slater potential with a Latter tail). The broken curve shows a calculation based on the Schnaidt formula, which was derived for a point Coulomb potential. Note that the calculations have been performed for linearly polarised radiation and not for unpolarised radiation as in the case of radioactive sources. The role of the chosen potential in determining cross sections for Compton scattering has been investigated in greater detail by Issolah et al. [1988]in the case of a low Z atom such as carbon. The single differential cross section for a K shell electron estimated from either the solid curve or the broken curve is consistent with the value of the incoherent scattering function 5K deduced from the non-relativistic calculations of Sujkowski and Nagel [1961]. The experimental value of the double differential cross section near the kinematic limit is seen to be larger than the calculated ones. Otherwise, there is fair agreement between experiment and theory. In the work of Manninen et al. [1987,1990a], the dependences on scattered photon energy of double differential cross sections for the Compton scattering of 59.54 keV gamma rays by K shell electrons of copper and zirconium were determined but not their absolute magnitudes. As shown in fig. 5.5, Manninen et al. were able to obtain good agreement between their data and calculations based on the

.2

Photon Energy (key)

U

1000 ~ 800 (b)

1280

~j 600 400 2oo~y. 14~.f~cfl

I I

$...

TTI~

I

Photon Energy (key) Fig. 5.5. Double differential cross sections in arbitrary units for Compton scattering of 59.54 keV gamma rays by K shell electrons, (a) at 125°with copper and (b) at 128°with zirconium [Manninen et al. 1990a]. The histograms represent the calculations of Manninen et al. on the basis of the impulse approximation. The broken curves represent calculations based on the Schnaidt [1934]formula and normalised to the data Just below the respective kinematic limits.

p. p. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

119

impulse approximation. According to the analysis of Eisenberger and Platzman [1970], the impulse approximation gives reliable predictions near the Compton peak when the ratio of electron binding energy to the photon energy change is much smaller than unity. Near the kinematic limit, the value of this ratio is close to unity. The somewhat unexpected agreement in cases of this type has been briefly discussed in section 2.3.1. As shown in fig. 5.5, the A2 calculations based on the Schnaidt formula are also able to account for the data. The overall agreement regarding shapes of scattered photon spectra indicates the importance of A2 interaction dependent amplitudes in the energy range near the kinematic limit. The observed rise in the spectral intensity with increasing photon energy near the kinematic limits follows also from an application of the Schnaidt formula in the cases under consideration corresponding to KrK s 1 (see also section 2.2.1 for the definition of rK). As pointed out in tables 4.1 and 4.4, there is only limited information available concerning double differential cross sections in the case of L shell Compton scattering. Further, the available data have not revealed any new features which are essentially different from those seen in more extensive K shell studies. Hence these data are not presented here. The importance of target Z dependent false events particularly at low values of h v 1 was pointed out in section 3.5. The reported values of double differential cross sections for low values of hi.’~were believed to indicate a qualitative verification of the predicted infrared divergence effect [Spitale and Bloom 1977; Basavaraju et al. 1987a, b]. However, recent work [Marchetti and Franck 1989; Lad et al. 1990] has shown that bremsstrahlung contributions are significant and obscure the infrared divergence signals in the experiments performed so far. Thus there is no clear experimental evidence for infrared divergence effects in K shell Compton scattering. Experimental information concerning Compton defects in Compton scattering of high energy gamma rays by K shell electrons is somewhat confusing. With 279.2 keV gamma rays, defects towards lower energies were of order 30 keV at 90° and 115°when a gold target was used but were not measurable within the experimental error of ±15keY when a tin target was used [Basavaraju et al. 1987b]. No defect was observed within the error of about ±20keV in the case of high Z elements with incident gamma rays of 320 keV [Spitale and Bloom 1977; Lad et al. 1990] and of 1116 keV [Baba Prasad et al. 1977]. With 661.6 keV gamma rays, defects were about l5keV and 2OkeV towards higher energies in the case of germanium at 90°and 135°,respectively [Pradoux et al. 1977]; at 75°they were negligible in the case of gold but about 35 keV towards lower energies in the case of uranium [Lazzaro and Missoni 1966]. 5.3. Double differential scattering cross sections determined in the singles mode The scattered photon spectrum obtained with 661.6 keV gamma rays at a large scattering angle in the case of a lead target [Reineking et al. 1983] has been shown in fig. 3.1. There is a deviation of as much as 50% between theory and experiment near the cut off energies (hi.’ BK) and (hv~ BL). On the other hand, in a similar work performed with 412 keV gamma rays and gold and lead targets, Pattison and Schneider [1979a]obtained good agreement between experimental data and calculations based on the impulse approximation. Of course, absolute values of small cross sections are difficult to determine accurately on account of errors associated with background subtraction, and corrections for detector efficiency, detector resolution related broadening of pulse height spectra and possibilities of multiple scattering. Further, the target thickness used in the 412 keV work was significantly larger than that used in the later 661.6 keV work. However, it is difficult to identify precise reasons for the differences in conclusions noted above. —

—

120

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

A similar experiment was performed with 141 keV synchrotron radiation and a silver target [Manninen et al. 1990b]. The contributions of electrons other than the strongly bound K shell electrons were estimated from their Compton profiles and then subtracted from the measured counts in order to determine K shell Compton scattering alone. The Compton profile of silver K shell electrons, deduced in the range of 8 and 20 atomic units of momentum (1 au mc1137), is in fair agreement with calculations based on the impulse approximation except near the lower portion of the above mentioned range. Note that the reported experimental errors are larger than those in the above mentioned higher energy experiments and that the profile is not reported for momenta lower than 8 atomic units. —

5.4. Resonant Raman and Raman—Compton scattering We shall first discuss resonant scattering leading to a final state electron of variable kinetic energy. As pointed out in the introduction, this process should be called resonant Compton or resonant Raman—Compton scattering, although in most reports it has been designated as resonant Raman scattering (RRS). A summary of the relevant experimental work has been presented in table 4.6. Figure 4.3 shows striking changes in shape and intensity of the scattered photon spectra with reduction of incident photon energy from slightly above the copper K shell threshold (8.980 keV) to a value 0.78 keV below the threshold. In the top portion of the figure, the K X-ray fluorescence spectrum of copper is seen. At 10 eV below threshold, the peaks become weaker and shift downwards in energy by about 10 eV. As the incident photon energy is reduced further, the scattered intensities decrease, and the downward shifting peaks show an asymmetry and also larger widths. The decrease of intensity with decreasing hi~is in conformity with eq. (2.60). Similarly, the peak shapes are in accordance with expectations based on eq. (2.59). On the scale of the lowest part of the figure, the elastic line, the K—M RRS contribution and the Ka X-rays caused by weak harmonics passed by the monochromator result in a composite unresolved peak. A penetrating analysis of the main features of RRS was presented some time ago by Eisenberger et al. [1976a,b]. In the initial work with a semiconductor detector, the peak position shifted linearly with hz.’~when (BK ht.’~)was larger than about 40 eV, but the apparent change in peak position was faster as hi~approached BK closely from below. Note that the scattered photon spectrum rises sharply as hi.’~ decreases through (ht~ BL EF) and then falls off slowly below that energy. The observed variations in peak position were shown to arise from the combined effects of insufficient detector resolution and the changing shape of the RRS spectrum. However, in the subsequent high resolution experiment, the RRS peak position was throughout seen to shift linearly with hi~as expected. Further, the proportionality of the RRS integrated intensity to —

—

tan’{FK/2(BK

+

EF

—

—

hi~)}

or to ir/2 —tan’{2(BK

+

EF

—

ht.~)/FK}

was checked by Eisenberger et al. with a TK value of 1.48 eV for copper known from other studies (note that the symbol TK of Eisenberger et a!. indicates the half-width of the K shell hole state and is therefore put equal to 0.74 eV in their paper).

P.p. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

121

An interesting result of the high resolution experiment was that the linewidth of the RRS spectrum was shown to become smaller than the lifetime limited width of the K shell fluorescence as h v~ approached close to (BK + EF). Since the K shell hole state is involved only virtually in RRS, the corresponding linewidth is determined by the incident beam energy spread and the final state width, whereas in fluorescence the K shell hole state width is also effective. A detailed theory of this linewidth narrowing in RRS spectra, observed with extremely well monochromatised incident beams, has been presented by Aberg and Tulkki [1985].A linewidth narrowing effect of the same type has also been observed in Auger electron spectra measured near the L2 threshold of xenon [Brown et al. 1980] and has been commented on in section 2.6. An analysis based on eq. (2.60) and an approach similar to that mentioned above have recently been used by Hãmäläinen et a!. [1989,1990] to determine the widths of K shell hole states in copper and zinc, and of L3 hole states in holmium, ytterbium and tantalum. Their analysis also showed that the threshold energy (BK + EF) could be determined from the data to an error of ±0.5eV. Since the total angular momentum of the L3 orbital is 3/2, i.e. greater than 1/2, a magnetic substate alignment effect can influence to some extent the angular distribution of resonant scattering when hx.~is close to an L3 threshold. Actually, it will be interesting to study the possible anisotropy of the angular distribution both experimentally and theoretically. The K—M RRS was measured for the first time by Kodre and Shafroth [1979] in the case of a molybdenum target with molybdenum K13 radiation from an X-ray generator and was shown to be isotropic within experimental error. With 59.54 keV gamma rays of an americium-241 source, the line shape of K—M RRS observed with an ytterbium target (BK = 61.332 keY) was shown to be consistent 2dependence expected from eq. (2.59) [Manninen et al. with 1985].the approximate hv~(BK BM hi~) The data obtained by Schaupp et a!. [1984] in a study of scattering of 43.35 keV synchrotron radiation from a neodymium target (BK = 43.57 keY) are shown in figs. 5.6a and 5.6b after a subtraction of the Compton scattering contribution. For the first time, K—N RRS was clearly seen. Further, K—L 2 and K—L3 RRS contributions were resolved. Since Kc~2,Ka1 and K~3X-rays from a 0.0015% impurity of praseodymium (of Z = 59 rather than 60 for the target under study) are clearly noticeable in the figure, the importance of extreme target purity in such experiments is evident. The solid lines indicate the double differential cross sections calculated in the non-relativistic dipole approximation but with relativistic expressions for the K shell photoeffect cross section, and for the np112 and np312 electron wave functions [see eqs. (2.56) to (2.58)]. A xenon gas target was used in another experiment [Czerwinski et al. 1985]. The scattering of 33.94 keY to 34.54 keY synchrotron radiation was investigated close to the 34.566 keY binding energy of the xenon K shell. The scattering plane was alternately chosen parallel and perpendicular to the polarisation of the incident beam. Within the experimental error of 5%, the RRS cross section with this monatomic gas of spherically symmetric atoms was found to be independent of the polarisation of the incident beam and of the scattering angle, in agreement with theoretical expectations. Angular correlations between the resonantly scattered photons and the recoil electrons have not been studied so far but are likely to provide a more incisive probe of the adequacy of the dipole approximation. Quantitative agreement between experiment and theory in the case of both K—L and K—M RRS intensities implies that the Ka to K13 branching ratio known from K X-ray fluorescence is also applicable for the relative intensities K—Ltheand K—M RRS. Similarly, an approximate 0K both below and of above threshold energy BK was demonstrated by constancy Manninen of et the radiative yield ~ al. [1986]in the case of nickel, and by Kodre et al. [1986]in the caseof krypton. The studies mentioned —

—

122

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons 4000

(a)

1

h~?43.35kev 0=800

~

3000-

L2TI

2000

-~°

;

1000

.

M

.

0

45

39~/~. h~f[key]

4 OOC (b)

h~.4335~~y ~

~

In

‘~

3000

L

3

e=iio°

L21(

2000

1000

:~-~

M ~/~R

hYf [key] Fig. 5.6. Double differential cross sections of neodymium atoms for resonant Raman scattering of 43.35 keV synchrotron radiation, (a) through 80° and (b) through 110°,obtained by Schaupp etal. [1984]after subtraction of the usual Compton contribution. The binding energy BK of the K shell is 43.57 keY in this case. The symbol R indicates elastic scattering. Peaks marked a1, a2, 13~ represent charactenstic K X-rays of a minute praseodymium impurity. The solid curve shows the calculation of Schaupp et at. in the dipole approximation.

above provide an experimental confirmation of the assumptions underlying the simple theory outlined in section 2.6. The measurements of Bannett et al. [1977]with monochromatised copper Ka radiation and thick targets of nickel, copper, zinc, gallium and germanium did not extend to values of hv~below about 2.5 keY. However, the relatively flat observed spectrum between 2.5 keY and 5 keY was consistent with the assumption of an infrared divergent contribution at energies lower than 2.5 keY and the applicability of Gavrila-type theory to many electron atomic systems. The energy of the incident beam was only about 0.3 keY below the binding energy of the nickel K shell and much below the values of BK in the case of the other targets. Although the large measured cross sections in the case of the nickel target were only slightly larger than the calculated ones, the significantly smaller measured values in the case of the other targets were up to about twice the corresponding calculations, indicating a possible non-negligible contribution from secondary processes such as bremsstrahlung. In recent work with linearly polarised synchrotron radiation of 14 keY to 17.4 keY and thin zirconium

P.p. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

123

targets of 20 p.g/cm2 and 200 p.g/cm2 thickness on thin mylar backings (of less than 1 p.m thickness), the measurements extended downwards to about 1.5 keY and showed the characteristic signature of infrared divergence, namely, cross sections increasing with decreasing hv~[Simionovici et a!. 1990]. However, it has been argued that photoelectron generated bremsstrahlung might have been strong enough in these experiments to prevent a clear identification of infrared divergence [Marchetti and Franck 1990]. A refutation of this argument was also presented [Briand et al. 1990]. Although the thinner target thickness was only about 5% of the electron range calculated in the continuous slowing down approximation (CSDA), the electron track length within the target and the backing must have been substantially larger on account of multiple scattering and so, according to the discussion in section 3.5, a significant contribution of bremsstrahlung cannot be ruled out [Kane and Basavaraju 1991]. Thus an unambiguous experimental demonstration of infrared divergence is still awaited in such experiments. Monochromatised copper Ku 1 radiation (—8.04 keY) from a sealed X-ray tube and a focussing crystal spectrometer of about 10 eY resolution were used to study double differential cross sections with a nickel target in more detail [Suortti et a!. 1987; Eteläniemi et al. 1988]. The scattered photon spectrum showed small but noticeable modulations up to about 150 eV below the cut off energy (hv~ BL EF), similar to those usually seen in studies of Extended X-ray Absorption Fine Structure (EXAFS). Since L2 and L3 binding energies of nickel differ by only about 17 eV, the modulations are diluted in this case due to a mixing of K—L2 and K—L3 RRS contributions. In the resonant scattering work of Briand et a!. [1981]with a KMnO4 target, the scattered beam was analysed with about 0.3 eY resolution over an interval of about 80 eV. The very high resolution enabled a separation to be made of the hitherto discussed resonant Raman—Compton scattering from resonant Raman scattering involving the excitation of a manganese K shell electron to a sharp atomic-like bound final state. As expected, the dependence of the Raman cross section on incident photon energy was much faster than that of the Raman—Compton cross section. The scattered photon spectrum revealed two sharp and symmetric Raman peaks, analogous to Ku1 and Ku2 X-ray lines in the fluorescence spectrum, in addition to the asymmetric and broader Raman—Compton peaks. With respect to the fluorescence X-ray lines, the Raman peaks showed a “red”/”blue” shift when the incident photon energy was lower/higher than that corresponding to the white line in absorption. Excitation of a K shell electron to bound atomic 4p and Sp states has also been invoked to explain similar very high resolution measurements of resonant Raman scattering in the case of argon [Deslattes 1987]. Further, when the incident photon energy was tuned to a value halfway between those corresponding to is—4p and ls—Sp resonances, a complicated spectrum was observed, which was explained as a superposition of a blue shifted ls—4p Raman spin doublet and a red shifted is—5p Raman spin doublet. Similar work has also been reported near the K shell threshold of sulphur (—2.47 keY) in SF6 molecules [Deslattes 1987], of chlorine [Lindle et a!. 1991] and near the M4 threshold of uranium in compounds [Ice et a!. 1990]. Intense and highly monochromatised beams (~~.~1012 s’), and high resolution spectrometers were essential for the success of these promising studies. —

—

5.5. Single differential cross sections for Compton scattering 5.5.1. K shell scattering The ratio of the single differential cross section to the corresponding K!ein—Nishina prediction is called the cross section ratio. Experimental data obtained with 279.2 keY and 661.6 keY gamma rays by different workers are compared with the best published calculations [Whittingham 1981] in tables 5.1 to

124

p. p. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

Table 5.1 Cross section ratios for K shell Compton scattering in the case of 279.2 keV gamma rays and high Z scatterers. Experimental values in columns 2 to 5 are from the references mentioned. The values in the sixth column are obtained by dividing Whittingham’s [1981]calculations for lead by the Klein—Nishina predictions for two free electrons. Note that some of the values were read from graphs. Angle (deg)

Pt

20 30

Au

Pb

Bi

Theory

0.206±0.021°>

0.157

0.25 ±

002b)

40 44 50 55 60 70 90

0.37 0.45

0.28

±0.05°>

0.40 ±0.07’”

0.183 0.224

0.52

±0.05°>

0.41

±0.04’”

0.288

0.34

±0.05°>

0.56 ±0.04”

0.378 0.486 0.681

0.35

±0.05°>

±0.04°>

±0.l0t~)

0.317±0.029°> 0.62

o.o~’>

±

0.378 ±0.028~ 0.57

100 105 110 115 120 125 130 150

±0.06°>

0.723 0.78

0.75

±

±0.04’”

0.40 ±0.06°

0.723

0.96±0.08°> 0.699 0.88

±0.06w

1.16 ±0.l2~ ° d)

Pingot [1969]. Nageswara Rao et a!. [1977].

0.426 ±0.030°

0.92 ±0.04’” 0.57 0.54

0.449 ±0.024° b)

°

±0.07° ±0.04°>

1.03 ±0.04’”

Swamy and Murty [1978a]. Basavaraju et al. [1987b].

0.662 0.588

Singh et al. [19851.

Table 5.2 Cross section ratios for K shell Compton scattering in the case of 279.2 keV gammarays and lower Z scatterers. Experimental values in columns 2, 4 and 5 are from the references mentioned. The values in columns 3 and 6 are obtained by dividing Whittingham’s [1981]calculations for Sm and Ta, respectively, by the Klein—Nishina predictions for two free electrons. Note that some of the values were read from graphs. Angle (deg) 20 30

Sm

Theory

Ta

0.161 ±0.058”

0.149 0.201

0.189±0.038°

50 0.294 ±0.079°

90 100 105 110 120 125 130

0.550±0.082°

°

1.125 1.169

0.566±0.112>

O.689±O.O4~ 0.57 ±

0.358±0.057”

0.152 0.186 0.340

0.69 ±O.06i°

0.469 0.618 0.869 0.916

0819--0042° 0.37±0.075~

1.146 1.09

0.909 0.871

0.487 ±0.053° 1.019

Pingot [1972]. Singh et a!. [1984].

Theory

0.290 ±0.042° 0.613 0.815

0.740 ±0.042” ‘>

0.267±0.036~ 0.27 ±0.04”> 0.456 ±0.062~ 0.46±0.06

0.431

55 60 70

150 160

W

0.899 0.863 °

Murty et al. [1973].

0.940±0.080° 0.575 ±O.0651~ 0.64 ±0.07510 0.502 ±0.065° 10

Singh et at. [1986].

0.820 0.737 0.712

p. p. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

125

Table 5.3 Cross section ratios for K shell Compton scattering in the case of661.6 keYgamma rays and high Z scatterers. Experimental values in columns 2 to 5 are from the references mentioned. The values in the sixth column are obtained by dividing Whittingham’s [1981)calculations for lead by the Klein—Nishina predictions for two free electrons. Note that some of the values were read from graphs. Angle (deg) 10 15 20

30

Pt

Au

Pb

0.31 ±0.28°>

0.21 ±0.42° 0.47 ±0.09° 0.335 ±0.050°> 0.31 ±0.12° 0.61 ±0.08°

0.21 0.40

0.73 ±0.11° 0.522 ±Q0478)

±0.115’”

0.292

0.29 ±0.13” 0.515 ±

0.418

0055h)

0.79±0.10° 0.561 ±0.044° 0.79 ±0.09°

0.47

±0.11>

o.~oo

063~007b)

0.715±0.07610 1.14±0.08° 0.857 ±0.045°

1.04±0.10°

1.07 ±0.13° 0.743 ±0.040°

0.97±0.09° 0.85 ±0.03’° 0.863 ±0.07210

0.829

1.036

112~022d>

1.16±0.098°

1.10 ±0.20°

80 100

105 110 120 124/125

±0.0310

0.50 ±0.05’° 0.83±0.101>

55 60 65 68 70

Theory 0.226

35 40

45 50

Bi

0.87 ±0.08’> 1.07 ±0.11w 1.12 ±0.11”> 1.19 ±015b) 1.45 ±020d) 1.02 ±0.10’ 0.92 ±0.14”>

1.14±0.193’>

1.20 1.09

1.380±0.169’>

1.46±0.32° 1.38±0.15°> 1.154±0.035°>

1.660

±0.115°

1.263 ±0.036°> 1.335 ±0.039°> East and Lewis [1969]. Motz and Missoni [1961]. ° Sujkowski and Nagel [1961]. ° Krishna Reddy et al. [1974].

0.995 0.912

087~018b)

1.78 ±0.16° 0.89 ±0.09”> 1.04±0.50” 1.542 ±0.166”>

l3Oto 132.5

150 140 160

1.17

°

b)

°

d) g>

1.725 ±0.14° 0.835

0.79 ±017b) 0.74 0.86 012”) 0.1210

Wolff et a!. [1989]. Shimizu et al [1965]. Krishna Reddy et al. [1970]. Yarma and Easwaran [1962].

0.733 0.772 0.698 Wolff et al. [1991]. Pingot [1968]. “~ Ramalinga Reddy et at. [1966]. b)

°>

126

p. p. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons Table 5.4 Cross section ratios for K shell Compton scattering in the case of 661.6 keV gamma rays and lower Z scatterers. Experimental values in columns 2 and 4 are from the references mentioned. The values in columns 3 and 5 are obtained by dividing Whittingham’s [1981]calculations for Sm and Ta, respectively, by the K!ein—Nishina predictions for two free electrons. Note that some of the values were read from graphs. Angle (deg)

Sm

Theory

10 15 20 30 35 40 45 50 60 65 70 90 100 130

Ta

Theory

0.180

0.205 0.21 ±0.15° 0.42±0.15’” 0.47 ±0.15’” 0.485 ±0.052° 0.71 ±0.09° 0.83±0.17’” 1.09±0.11°> 0.727 ±0.077° 1.28 ±0.10° 0.90 ±0.18’” 0.921 ±0.098° 1.19±0.21’” 1.33±0.04° 1.132±0.120° 1.23±0.22’” 1.358 ±0.1441~

0.284 0.617 ±0.092°

0.469

0.697 0.755

±0.113°

0.940 0.915 ±0.137°

1.118 1.20 1.14 1.05 0.745

1.160±0.174° 1.375 ±0.206° East and Lewis [1969].

d>

Shimizu et al. [1965].

h)

0.290 0.444

0.655 0.901 1.04 1.22 1.18 1.09 0.81

Ramalinga Reddy

Ct

al. [1966].

5.4. Note that a summary of the relevant experiments has already been presented in tables 4.1 to 4.3. The discussion in section 3.5 has emphasized the importance of using targets with thickness very much smaller than the photoelectron ranges. Some of the experimental papers report only statistica! errors, whereas some others include systematic errors as well in the reported error values. The energy ranges over which integrations have been performed in order to obtain values of do~K/dQfare not the same in different experiments or in theoretical calculations. Further, the resolutions of detectors are different in different experiments. Therefore, a certain amount of caution is necessary in the interpretation of the tables. The recent data of Wolff et a!. [1989,1991] obtained with germanium detectors and a 6.8 mg/cm2 lead target are seen to be in good agreement with the calculations for 661.6 keY gamma rays over the angular range from 15° to 150°.The theoretically predicted decrease in the cross section ratio for angles larger than about 100° has been confirmed in this work. However, in the case of larger angles, most of the scattered intensity occurs for smaller photon energies, i.e. for larger recoil electron energies. So the convergence of the sum of terms over electron wave number Kf is slower. As pointed out in section 2.2.2, screening corrections are also likely to become more significant. Whittingham has urged caution in reaching firm conclusions regarding the decrease of calculated cross section ratios at large angles, particularly since simple arguments suggest an approach of the cross section ratio towards unity with increasing momentum transfer. More calculations and experimental data are certainly needed to settle the variation of the cross section ratio in the regime of high momentum transfer. The single differential cross sections are obtained by Whittingham by an integration of the double differential cross sections over an energy interval from (hM BK)!S to (hv BK), i.e., for example, —

—

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

127

upwards from about 45 keV in the case of 279.2 keY gamma rays and scatterers of samarium and tantalum. The experimental bias values, not always stated, have been typically around 100 keV. This circumstance may perhaps be responsible for the fact that the experimental cross section ratios for the cases under consideration are usually smaller than the calculated values (table 5.2). Further, there is substantial scatter between experimental data of different investigators, the scatter being by as much as a factor of two at large angles. The precise reasons for the deviations are not clear. As pointed out by various workers [e.g., Krishna Reddy et a!. 1974; Baba Prasad and Kane 1974], values of the incoherent scattering function SK calculated according to the formula of Shimizu et al. [1965] do not satisfactorily explain the data concerning cross section ratios except in limited ranges of the parameters hi.’,, Z and i~. At small angles of scattering, the impulse approximation is unsuitable and predicts too large cross sections. The absolute values of the cross section ratios are known from only a few experiments for hi.’1 below 145 keV. The general trends of variations of the cross section ratios with Z and i~at other energies such as 145 keV, 320 keY, 1002 keV and 1116 keV are similar to those evident in tables 5.1 to 5.4. As already shown in table 2.1, results based on approximate methods of calculation differ significantly in many cases from more detailed calculations. 5.5.2. L shell scattering As pointed out in section 4, coincidence work concerning L she!! Compton scattering is not extensive. In the relevant experiments, scattered gamma rays were detected with moderate energy resolution by sodium iodide detectors. Single differential cross sections have been graphically presented in the case of 279.2 keV gamma rays for a lead target [Swamyand Murty 1975] and for a thorium target [Swamy and Murty 1976]. The cross section ratio averaged over the three L subshells of lead increased monotonically from about 0.62 ±0.05 at 30° to about 0.9 ±0.1 at 150°. The corresponding ratio for thorium increased from about 0.56 ±0.06 at 30°to about 1.0 ±0.1 at 150°.Similar variations have been obtained in the case of 661.6 keV gamma rays (see table 5.4). The good resolution of a Si(Li) detector was utilised to resolve Lu, L13, and L1 X-rays of a gold target and thus to vary the contributions of the different subshells to the scattering signal in the coincidence mode [Basavaraju et al. 1982; Kane 1984]. In the case of 279.2 keY gamma rays and a gold target, the cross sectiOn ratio at 54°was 0.75 ±0.10 with the LuJ3 window method, and 0.56 ±0.10 with the Lf3 window method. Since nearly 75% of the L13 X-ray intensity arises from L1 and L2 subshell vacancies, the above mentioned values suggest the possibility of a subshel! dependence of the cross section ratio. Note that the L shell cross section ratios are usually larger/smaller than the corresponding K shell ratios for scattering angles smaller/larger than 90°. 5.6. Total cross sections for K shell Compton scattering An examination of tables 5.1 and 5.2 presenting data for 279.2 keY gamma rays in the case of Z 62 shows that the total cross section for these cases will be significantly smaller than that obtained by an angular integration of the Klein—Nishina expression. A similar conclusion holds in the case of 145 keY and 320 keV data, which have not been tabu!ated. On account of considerable scatter among the different data for 661.6 keY presented in tables 5.3 and 5.4, it is not possible to state a firm conclusion at this energy. The direct measurements of K she!! total Compton scattering cross sections in the case of 279.2 keY and 661.6 keY gamma rays are consistent with the values expected for free and stationary

128

p. p. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

electrons (see also section 3.8) and thus appear to be somewhat at variance with the trend mentioned in the beginning in the case of 279.2 keY gamma rays and higher Z values. Singh [1970]has examined the avai!ab!e data and has deduced the trends of variation of total K shell scattering cross sections with Z and incident photon energy.

6. Conclusions and outlook We shall first comment on studies of K shell Compton scattering. As expected, the energy distribution of photons after K shell Compton scattering is significantly broader than that calculated for free electrons at rest. The data for Compton defects do not reveal systematic trends as regards sign and magnitude. In the case of incident photon energies higher than 100 keV, there are very few measurements of energy distributions with semiconductor detectors. At lower energies, there are very few determinations of absolute values of Compton scattering cross sections. The interesting prediction of infrared divergence in scattering is yet to be unambiguously confirmed in spite of several experimental attempts. Further, as pointed out in section 5, the absolute values of cross sections determined by different investigators in comparable situations show large deviations in some cases. A careful repetition of experiments in such cases with modern instrumentation and thin targets is desirable. Relativistic second order S matrix calculations were published until recently only at 279.2 keY and 661.6 keY for four Z values larger than 61. Further, a systematic treatment of electron screening was needed. As pointed out in section 2.2.2, these deficiencies are being remedied by the Pittsburgh group. When the energy transfer (hv hi.~)is comparable to the electron binding energy, the effect of the solid state environment on the final state electron wave functions may also have to be taken into account. As pointed out in section 2.3.1, a thorough investigation of the theoretical basis of the often successful impulse approximation is desirable. Only a few studies of L shell Compton scattering with the help of the coincidence technique have been performed. Relativistic second order S matrix calculations for L shell Compton scattering in different ranges of Z are yet to be published. The dipole approximation has been very successful in a quantitative explanation of the main features of resonant Raman—Compton scattering cross sections. Intense, tunable, polarised and monochromatised radiation from synchrotron facilities has been recently used in the investigation of subtle features in resonant Raman scattering. It will be interesting to probe specific solid state environment effects in these phenomena with very high resolution spectrometers. There is no information so far concerning the angular distribution of resonant Raman—Compton scattering of photons of energies near L3 subshel! —

thresholds. Experiments involving coincidences between resonantly scattered photons and the recoil electrons should become possible with the new synchrotron facilities being set up. Such experiments will provide an incisive probe of the range of validity of the dipole approximation. Thus considerable progress should be expected during the next few years in the field of inelastic scattering of X-rays and gamma rays. Acknowledgements This work was supported in part by Grant No. INT-9102053 made by the National Science Foundation of the USA under the special, foreign currency programme.

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

129

At the outset, I wish to apologise for inadvertent omissions and possible misinterpretations of the work of any investigators, and to thank those whose results have been discussed in this report. It is a pleasure to acknowledge the hospitality of the group led by Malcolm J. Cooper of the University of Warwick and the award of a Fellowship by the Science and Engineering Research Council of the UK for a year from May 1987 to April 1988, which enabled me to participate in experiments with synchrotron radiation from the SRS source at Daresbury, UK. The participation of a colleague, Goti Basavaraju, and former students P.N. Baba Prasad, Suju M. George and Saharsha M. Lad in the research work at lIT, Bombay is acknowledged with thanks. The present report was written on the basis of experience gained from research work funded by grants made by the Department of Science and Technology of the Government of India, and by the National Bureau of Standards and the National Science Foundation of the USA under the special foreign currency programme. At various times during the preceding decade and a half, discussions with Richard H. Pratt of the University of Pittsburgh and with John H. Hubbell of the National Institute of Standards and Technology in Washington, DC have been very useful. I appreciated a communication from Mihai Gavrila of FOM Instituut, Amsterdam. Appendix A. Relation between photon energy after Compton scattering and initial electron momentum under the impulse approximation (i) First, the relativistic eq. (2.26) is derived. Refer to fig. 2.1. The initial electron momentum and energy are designated as p and E,, respectively. The corresponding quantities for the incident photon are Ilk1 and hi.’~, respectively. The fina! electron momentum and energy are p~and Ef, where 2c4 + p~c2)”2,and the scattered photon momentum and energy are Ilkf and hi.~,respectively. Ef = (menergy and momentum conservation, we get Using hzi~+ E,

=

hzi~+ (m2c4 +p~c2)”2,

(A.1)

P+hk

(A.2)

1—Pf+Ilkf.

Therefore, we have 2 m2c4 + c2(hk (hi.’, Further,

—

hii~+ E~)

(h~~)2 = h2c2k~,

5lk~+ p)2. 1

—

(h~~)2 = Il2c2k~2,

p=mcyfl=Efi/c,

jJ=vlc,

E~= (m2c4 + p2c2)t/2 = mc2y

(A.3)

i

/12c2 k~ k~= h2M i.~cos ~

(A.4)

,

y=(l—f32)”2,

(A.5) (A.6)

.

With the help of eqs. (A.3), (A.4), (A.5) and (A.6), we get —h2~,i.~(1 cos —

t~)

+

2(lc~ kf)• p = hc2 K• p = E.(hvfl cos

E~(hM hv —

—

1)

=

11c

x,

—

hz.’j3 cos Xt), (A.7)

130

P. P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

where ~ and Xf are the angles made by the direction of p with the directions of k, and kf, respectively. Then we get l—/3cosX1

hi~

hv>

—

1—/3cosXf+(1—cos8)h~/E.

(A.8)

2 and the well known In the caseequation of initially freefollow and stationary Compton (1.1) from eq. electrons, (A.8). p = 0 /3, the relation E1 = mc (ii) Now eq. (2.24) or eq. (2) of Pitkanen et a!. [1987]is derived, Assume that the initial electron momentum p is small in comparison with mc, pc ~ mc2

and E

2. 1 mc Then, instead of eq. (A.7), we get

(A.9)

—

—hi~i.~(1 cos —

i~)

+

mc2(hi~’ hi.~)= hc2 K~p, —

(A.10)

where hcK=(h2+h2v~_2h21fcosi9)H2

(A.11)

-

If the Z axis is chosen along (—K) as done, for example, by Bloch and Mendelsohn [1974] and by Eisenberger and Reed [1974],we get p~ x.’>.— ~~+hv,i.~(1—cosi~)Imc2 mc (i.. + 2r~i.~cos

.

(A.12)

—

Note that this choice corresponds to positive p,~values for hi~larger than the “free” Compton energy ni)

F

1. (iii) Now the relativistic approach of Ribberfors [1975]is followed in order to derive eq. (2.35). It is useful to consider the minimum momentum Pmin(Pz) which can2c4 lead+ p~ to a scattered photon of energy 2)”2. Then, instead of eq. hv~.at angle The energy E(pmin) corresponding to Pmin is (m 1~c (A.7), the following equation is obtained:

~.

—h2v i.~(1

—

cos ~) + (m2c4

+

p~>~c2)”2(hi~ hi.~)= hc2 K p. —

(A.13)

Therefore, Pm>,,

mc

—

~

P)E(Pmjn)+hLY’i(1~COSi~)

mc22 (~1

~

2 —2ty.~cos + t~

1/2

-

(A.1~

Note that this equation is eq. (39) of Ribberfors except for a different choice of units and the sense of the positive Z direction.

P.P. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

131

Appendix B. A summary of studies of non-resonant Raman and Compton scattering of low energy X-rays by inner shell electrons of low Z atoms As pointed out in section 1, K shell electrons can contribute to Compton scattering only when the energy transfer exceeds the K shell binding energy BK. Therefore, the scattered radiation spectrum is expected to show an intensity jump signalling the onset of K she!! Compton scattering when the wavelength after scattering exceeds the cut off wavelength AK, where AK is hc/(hv BK). If measurements are also to be made of sharp peaks in scattered intensity at wavelengths slightly smaller than AK and corresponding to excitation of K shell electrons to unoccupied bound states, a very high spectral resolution is necessary. Although the desirable resolution of better than 1 eV was not achieved in X-ray scattering studies until a few years ago, the possibility of observation of sharp peaks at wavelengths slightly smaller than AK or AL (where AL = hc/(h~> BL)) involving the excitation of K or L shell electrons, respectively, to bound states close to the continuum has been considered and debated for a long time [e.g. Krishnan 1928; Raman 1928; Davis and Mitchell 1929; Ehrenberg 1929; Bearden 1930; Dumond 1933; Sommerfeld 1936; Das Gupta 1950, 1959, 1962, 1964; Weiss 1965; Faessler and Mühle 1966; Cooper and Leake 1966, 1967; Suzuki 1966, 1967; Schülke and Berg 1967; Alexandropoulos and Cohen 1969; Alexandropoulos 1970, 1971; Suzuki et al. 1970; Doniach et al. 1971; Alexandropoulos et a!. 1971; Cohen et a!. 1973; Mühle 1973; Platzman and Eisenberger 1974; Alexandropoulos et a!. 1974; Eisenberger et a!. 1975; Alexandropoulos and Cohen 1975; Suzuki and Nagasawa 1975; Pattison et at. 1981]. The characteristic wavelength AK or AL depends on the energy of the incident radiation and on the atomic number of the target, and not on the scattering angle, unlike the wavelength A~in the case of Compton scattering from free electrons. On account of this fact and a formal similarity to scattering phenomena observed in the optical and ultraviolet regimes involving the excitation of rotational and vibrational modes, intensity enhancements both above and below the characteristic wavelengths were frequently designated as X-ray Raman scattering. Actually, the excitation of K or L shell electrons of light elements leading to discrete lines in the spectra due to non-resonant scattering of X-rays of energy much larger than BK or BL was shown in only a few of the above mentioned studies [e.g. Das Gupta 1962; Alexandropoulos and Cohen 1969; Faess!er and Mflh!e 1966; Mühle 1973; Alexandropoulos and Cohen 1975; Pattison et al. 19811. The term Raman band was used by some authors to indicate the intensity distribution near the characteristic wavelengths [Mizuno and Ohmura 1967; Suzuki et a!. 1970; Alexandropoulos et a!. 1974; Suzuki and Nagasawa 1975]. However, as pointed out above, a clear distinction should be made between phenomena associated with Raman and Compton scattering of X-rays. It should be noted that observations of the onset of K shell Compton scattering in singles experiments became easier, since the double differential cross section peaks close to AK though asymmetrically on the long wavelength side. This behaviour can be seen to follow, for example, from an application of the Schnaidt equation (2.16). Most of the above mentioned studies have been reviewed earlier [Bushuev and Kuzmin 1977] and so are not discussed here further. Recent observations of intensity enhancements for wavelengths larger than AK have been reported in the case of scattering of copper K13 radiation (—8.9 keV) from boron [Zdetsis and Papademitriou 1988a]. In this experiment employing a wavelength dispersive spectrometer, excitation of boron K shell electrons to states just above the Fermi energy involving the necessary energy transfer of about 190 eV was certainly seen as expected. In addition, intensity enhancements were also seen at still longer —

—

132

p. p. Kane, Inelastic scattering of X-rays and gamma rays by inner shell electrons

wavelengths corresponding to an energy transfer of about 275 eY and thus to possible excitation of K shell e!ectrons to states nearly 85 eY higher up in the conduction band. This interpretation has been questioned by Manninen et al. [1988],who used monochromatised copper Ku1 incident radiation and did not observe any feature in the intensity distribution of the scattered beam corresponding to an excitation to states so deep in the conduction band. They also conjectured that perhaps Zdetsis and Papademitriou had observed scattering of Zn Ku radiation produced in a zinc contaminant of the collimator materials, or zinc Ku fluorescence of a zinc contaminant in the target. Note that ionisation of a zinc K shell can in principle occur in the experiment of Zdetsis and Papademitriou on account of the possibility of a weak component of higher energy in the unmonochromatised incident beam. Note further that zinc Ku2 and Ku1 X-ray energies are lower than copper K13 energies by about 275 eY and 254 eY, respectively, i.e., by approximately the required value of 275 eV. The publication of this conjecture led to further debate [Zdetsis and Papademitriou 1988b]. Highly monochromatised synchrotron radiation and a very high resolution (—0.8 eY) spectrometer have been used recently to study excitation of K shell electrons in lithium and highly oriented pyrolytic graphite to valence and conduction bands by inelastic X-ray scattering [Schülke 1986; Schülke et al. 1988; Nagasawa 1989]. Other weak band-like structures have been seen earlier in the spectra of emergent photons when, for examp!e, targets of magnesium, aluminium, silicon and sulphur were used with incident chromium Ku radiation [Aberg and Utriainen 1969]. Some of the observed structures were attributed to processes such as the radiative K—LL Auger effect, which has been called the atomic inner Compton effect. Recently in a study of non-resonant scattering of X-rays by diamond and graphite with about 2 eY resolution EXAFS (Extended X-ray Absorption Fine Structure)-!ike modulations of the scattered beam intensity have also been seen [Tohji and Udagawa 1989]. More such experiments with very high resolution are expected with the new synchrotron facilities. References Aberg, T., 1975, Two photon emission, the radiative Auger effect and the double Auger process, in: Atomic Inner Shell Processes, ed. B. Crasemann (Academic Press, New York) Vol. 1, p. 353. Aberg, T., 1980, Phys. Scr. 21, 495. Aberg, T., 1981, The creation and decay of inner shell vacancies as a single scattering process, in: Inner Shell and X Ray Physics of Atoms and Solids, eds. Di. Fabian, H. Kleinpoppen and L.M. Watson (Plenum, New York) p. 251. Aberg, T. and i. Tulkki, 1985, in: Atomic Inner Shell Physics, ed. B. Crasemann (Plenum, New York) Ch. 10. Aberg, T. and J. Utriainen, 1969, Phys. Rev. Lett. 22, 1346. Acharya, yB. and B.S. Ghumman, 1980, md. i. Phys. 54B, 242. Acharya, yB., B. Singh and B.S. Ghumman, 1981, Phys. Scr. 23, 21. Akhiezer, Al. and YB. Berestetskii, 1965, Quantum Electrodynamics (Wiley, New York). Alexandropoulos, N.G., 1970, Phys. Rev. B 1, 4115. Alexandropoulos, N.G., 1971, Phys. Rev. B 3, 2670. Alexandropoulos, N.G. and G.G. Cohen, 1969, Phys. Rev. 187, 455. Alexandropoulos, N.G. and G.G. Cohen, 1975, Phys. Rev. Lett. 35, 1182. Alexandropoulos, N.G., S.H. Parks and M. Kuriyama, 1971, Phys. Lett. A 35, 369. Alexandropoulos, N.G., G.G. Cohen and M. Kuriyama, 1974, Phys. Rev. Lett. 33, 699. Allawadhi, K.L., S.L. Verma, B.S. Ghumman and B.S. Sood, 1978, Phys. Rev. A 17, 1058. Arimitsu, N., Y. Kobayashi and Y. Mizuno, 1987, J. Phys. Soc. Jpn. 56, 2940. Armen, G.B., 1988, Phys. Rev. A 37, 995. Armen, GB., S.L. Sorensen, SB. Whitfield, G.E. Ice, iC. Levin, G.S. Brown and B. Crasemann, 1987a, Phys. Rev. A 35, 3966. Armen, GB., I. Tulkki, T. Aberg and B. Crasemann, 1987b, Phys. Rev. A 36, 5606. Baba Prasad, P.N. and PP. Kane, 1974, 1. Res. Nat!. Bur. Stand. A 78, 461. Baba Prasad, P.N., G. Basavaraju and PP. Kane, 1977, Phys. Rev. A 15, 1984.

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