International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 7.4, pp. 657-659

Section 7.4.3.2. Non-relativistic calculations of the incoherent scattering cross section

N. G. Alexandropoulosa and M. J. Cooperb

7.4.3.2. Non-relativistic calculations of the incoherent scattering cross section

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7.4.3.2.1. Semi-classical radiation theory

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For weak scattering, treated within the Born approximation, the incoherent scattering cross section, (dσ/dΩ)inc, can be factorized as follows: [\bigg ({{\rm d}\sigma \over {\rm d} \Omega}\bigg)_{\rm inc}=\bigg ({{\rm d}\sigma \over {\rm d} \Omega}\bigg)_{0}S(E_1,E_2,{\bf K},Z), \eqno (7.4.3.2)]where (dσ/dΩ)0 is the cross section characterizing the interaction, in this case it is the Thomson cross section, [(e^2/mc^2)^2\boldvarepsilon_1\cdot\boldvarepsilon_2]; [\boldvarepsilon_1] and [\boldvarepsilon_2] being the initial and final state photon polarization vectors. The dynamics of the target are contained in the incoherent scattering factor S(E1, E2, K, Z), which is usually a function of the energy transfer [\Delta E=E_1-E_2], the momentum transfer K, and the atomic number Z.

The electromagnetic wave perturbs the electronic system through the vector potential A in the Hamiltonian [H={e\over me}{\bf p}\cdot {\bf A}+{e^2\over 2me^2}{\bf A}\cdot {\bf A}. \eqno (7.4.3.3)]

It produces photoelectric absorption through the [{\bf p}\cdot{\bf A}] term taken in first order, Compton and Raman scattering through the [{\bf A}\cdot{\bf A}] term and resonant Raman scattering through the [{\bf p}\cdot{\bf A}] terms in second order.

If resonant scattering is neglected for the moment, the expression for the incoherent scattering cross section becomes [S=\textstyle\sum\limits_f (E_2/E_1)^2\bigg| \langle\psi_f| \textstyle\sum\limits_j\exp (i{\bf K}\cdot {\bf r}_j)|\psi_i\rangle\bigg|^2\delta(E_f-E_i-\Delta E),\eqno (7.4.3.4)]where the Born operator is summed over the j target electrons and the matrix element is summed over all final states accessible through energy conservation. In the high-energy limit of [\Delta E\gg E_B], S(E1, E2, K, Z) [\rightarrow] Z but as Table 7.4.3.1[link] shows this condition does not hold in the X-ray regime.

The evaluation of the matrix elements in equation (7.4.3.4)[link] was simplified by Waller & Hartree (1929[link]) who (i) set E2 = E1 and (ii) summed over all final states irrespective of energy conservation. Closure relationships were then invoked to reduce the incoherent scattering factor to an expression in terms of form factors [f_{jk}]: [S=\textstyle\sum\limits_j[1-|\,f_j({\bf K})|^2]-\textstyle\sum\limits_j^{\phantom{X}j}\textstyle\sum\limits^{\ne \,\,k\phantom{XX}}_k|\, f_{jk}({\bf K})|^2,\eqno (7.4.3.5)]where [f_j({\bf K})=\langle\psi_j|\!\exp (i{\bf K}\cdot {\bf r}_j)|\psi_j\rangle]and [f_{jk}=\langle\psi_k|\!\exp [i{\bf K}\cdot ({\bf r}_k-{\bf r}_j)]|\psi_j\rangle,]the latter term arising from exchange in the many-electron atom.

According to Currat, DeCicco & Weiss (1971[link]), equation (7.4.3.5)[link] can be improved by inserting the prefactor (E2/E1)2, where E2 is calculated from equation (7.4.3.1)[link]; the factor is an average for the factors inside the summation sign of equation (7.4.3.4)[link] that were neglected by Waller & Hartree. This term has been included in a few calculations of incoherent intensities [see, for example, Bloch & Mendelsohn (1974[link])]. The Waller–Hartree method remains the chosen basis for the most extensive compilations of incoherent scattering factors, including those tabulated here, which were calculated by Cromer & Mann (1967[link]) and Cromer (1969[link]) from non-relativistic Hartree–Fock self-consistent-field wavefunctions. Table 7.4.3.2[link] is taken from the compilation by Hubbell, Veigele, Briggs, Brown, Cromer & Howerton (1975[link]).

Table 7.4.3.2| top | pdf |
The incoherent scattering function for elements up to Z = 55

Element(sin θ)/λ (Å−1)
0.100.200.300.400.500.600.700.800.901.001.502.00
1 H 0.343 0.769 0.937 0.983 0.995 0.998 0.994 0.999 1.000 1.000 1.000 1.000
2 He 0.296 0.881 1.362 1.657 1.817 1.902 1.947 1.970 1.983 1.990 1.999 2.000
3 Li 1.033 1.418 1.795 2.143 2.417 2.613 2.746 2.834 2.891 2.928 2.989 2.998
4 Be 1.170 2.121 2.471 2.744 3.005 3.237 3.429 3.579 3.693 3.777 3.954 3.989
5 B 1.147 2.531 3.190 3.499 3.732 3.948 4.146 4.320 4.469 4.590 4.895 4.973
6 C 1.039 2.604 3.643 4.184 4.478 4.690 4.878 5.051 5.208 5.348 5.781 5.930
7 N 1.08 2.858 4.097 4.792 5.182 5.437 5.635 5.809 5.968 6.113 6.630 6.860
8 O 0.977 2.799 4.293 5.257 5.828 6.175 6.411 6.596 6.755 6.901 7.462 7.764
9 F 0.880 2.691 4.347 5.552 6.339 6.832 7.151 7.376 7.552 7.703 8.288 8.648
10 Ne 0.812 2.547 4.269 5.644 6.640 7.320 7.774 8.085 8.312 8.490 9.113 9.517
11 Na 1.503 2.891 4.431 5.804 6.903 7.724 8.313 8.729 9.028 9.252 9.939 10.376
12 Mg 2.066 3.444 4.771 6.064 7.181 8.086 8.784 9.304 9.689 9.975 10.766 11.229
13 Al 2.264 4.047 5.250 6.435 7.523 8.459 9.225 9.830 10.296 10.652 11.592 12.083
14 Si 2.293 4.520 5.808 6.903 7.937 8.867 9.667 10.330 10.864 11.286 12.408 12.937
15 P 2.206 4.732 6.312 7.435 8.419 9.323 10.131 10.827 11.411 11.888 13.209 13.790
16 S 2.151 4.960 6.795 8.002 8.960 9.829 10.626 11.336 11.952 12.472 13.990 14.641
17 Cl 2.065 5.074 7.182 8.553 9.539 10.382 11.158 11.867 12.499 13.050 14.750 15.487
18 Ar 1.956 5.033 7.377 8.998 10.106 10.967 11.726 12.424 13.061 13.629 15.489 16.324
19 K 2.500 5.301 7.652 9.405 10.650 11.568 12.329 13.014 13.645 14.220 16.212 17.152
20 Ca 3.105 5.690 7.981 9.790 11.157 12.163 12.953 13.635 14.256 14.830 16.921 17.970
21 Sc 3.136 5.801 8.169 10.071 11.561 12.648 13.545 14.256 14.885 15.460 17.630 18.782
22 Ti 3.114 5.860 8.312 10.304 11.901 13.140 14.093 14.856 15.509 16.095 18.334 19.585
23 V 3.067 5.858 8.375 10.454 12.156 13.514 14.574 15.413 16.111 16.721 19.032 20.379
24 Cr 2.609 5.577 8.206 10.415 12.264 13.770 14.960 15.902 16.670 17.323 19.730 21.168
25 Mn 2.949 5.791 8.380 10.604 12.486 14.062 15.346 16.376 17.211 17.910 20.411 21.938
26 Fe 2.891 5.781 8.432 10.733 12.687 14.343 15.716 16.831 17.737 18.488 21.097 22.704
27 Co 2.832 5.764 8.469 10.844 12.867 14.596 16.050 17.249 18.229 19.039 21.777 23.462
28 Ni 2.772 5.726 8.461 10.894 12.980 14.780 16.317 17.602 18.664 19.543 22.445 24.211
29 Cu 2.348 5.455 8.310 10.778 12.942 14.847 16.494 17.885 19.043 20.002 23.107 24.957
30 Zn 2.654 5.631 8.388 10.901 13.094 15.020 16.709 18.163 19.395 20.427 23.745 25.683
31 Ga 2.791 5.939 8.599 11.082 13.290 15.233 16.947 18.445 19.734 20.831 24.370 26.400
32 Ge 2.839 6.229 8.912 11.338 13.536 15.486 17.215 18.741 20.074 21.224 24.983 27.109
33 As 2.793 6.365 9.236 11.658 13.828 15.775 17.511 19.056 20.420 21.612 25.583 27.810
34 Se 2.799 6.589 9.601 12.033 14.168 16.098 17.835 19.391 20.778 22.003 26.171 28.504
35 Br 2.771 6.748 9.940 12.440 14.552 16.456 18.185 19.747 21.149 22.399 26.747 29.190
36 Kr 2.703 6.760 10.157 12.828 14.969 16.849 18.562 20.123 21.535 22.804 27.313 29.870
37 Rb 3.225 7.062 10.431 13.206 15.410 17.282 18.974 20.526 21.940 23.221 27.871 30.543
38 Sr 3.831 7.464 10.746 13.576 15.860 17.745 19.420 20.956 22.367 23.654 28.423 31.210
39 Y 3.999 7.700 11.010 13.899 16.279 18.215 19.891 21.416 22.820 24.110 28.970 31.870
40 Zr 4.064 7.879 11.236 14.176 16.658 18.672 20.373 21.895 23.294 24.583 29.517 32.522
41 Nb 3.672 7.684 11.213 14.317 16.949 19.081 20.844 22.386 23.787 25.077 30.067 33.167
42 Mo 3.625 7.690 11.260 14.444 17.196 19.455 21.300 22.877 24.288 25.581 30.620 33.808
43 Tc 3.987 7.984 11.512 14.653 17.456 19.816 21.748 23.370 24.797 26.093 31.173 34.447
44 Ru 3.559 7.857 11.531 14.782 17.685 20.150 22.172 23.855 25.312 26.621 31.740 35.081
45 Rh 3.499 7.863 11.591 14.883 17.858 20.428 22.557 24.318 25.819 27.148 32.309 35.715
46 Pd 3.103 7.725 11.441 14.824 17.943 26.653 22.904 24.756 26.316 27.677 32.888 36.349
47 Ag 3.362 7.785 11.598 14.969 18.082 20.858 23.212 25.162 26.792 28.195 33.465 36.983
48 Cd 3.700 7.980 11.812 15.185 18.263 21.064 23.501 25.546 27.252 28.705 34.046 37.618
49 In 3.852 8.297 12.083 15.444 18.489 21.288 23.779 25.906 27.691 29.203 34.634 38.255
50 Sn 3.917 8.615 12.415 15.746 18.760 21.541 24.059 26.252 28.113 29.687 35.226 38.894
51 Sb 3.871 8.811 12.777 16.088 19.067 21.823 24.349 26.590 28.518 30.157 35.822 39.536
52 Te 3.097 9.076 13.171 16.466 19.407 22.134 25.655 26.927 28.912 30.613 36.422 40.181
53 I 3.903 9.287 13.564 16.876 19.227 22.471 24.980 27.269 29.298 31.056 37.024 40.827
54 Xe 3.841 9.340 13.892 17.307 20.175 22.833 25.324 27.619 29.680 31.488 37.628 41.477
55 Cs 4.320 9.615 14.217 17.753 20.612 23.228 25.691 27.981 30.064 31.914 38.232 42.129

7.4.3.2.2. Thomas–Fermi model

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This statistical model of the atomic charge density (Thomas, 1927[link]; Fermi, 1928[link]) considerably simplifies the calculation of coherent and incoherent scattering factors since both can be written as universal functions of K and Z. Numerical values were first calculated by Bewilogua (1931[link]); more recent calculations have been made by Brown (1966[link]) and Veigele (1967[link]). The method is less accurate than Waller–Hartree theory, but it is a much simpler computation.

7.4.3.2.3. Exact calculations

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The matrix elements of (7.4.3.4)[link] can be evaluated exactly for the hydrogen atom. If one-electron wavefunctions in many-electron atoms are modelled by hydrogenic orbitals [with a suitable choice of the orbital exponent; see, for example, Slater (1937[link])], an analytical approach can be used, as was originally proposed by Bloch (1934[link]).

Hydrogenic calculations have been shown to predict accurate K- and L-shell photoelectric cross sections (Pratt & Tseng, 1972[link]). The method has been applied in a limited number of cases to K-shell (Eisenberger & Platzman, 1970[link]) and L-shell (Bloch & Mendelsohn, 1974[link]) incoherent scattering factors, where it has served to highlight the deficiencies of the Waller–Hartree approach. In chromium, for example, at an incident energy of ∼17 keV and a Bragg angle of 85°, the L-shell Waller–Hartree cross section is higher than the `exact' calculation by ∼50%. A comparison of Waller–Hartree and exact results for 2s electrons, taken from Bloch & Mendelsohn (1974[link]), is given in Table 7.4.3.3[link] for illustration. The discrepancy is much reduced when all electrons are considered.

Table 7.4.3.3| top | pdf |
Compton scattering of Mo Kα X-radiation through 170° from 2s electrons

ElementSexactSimpSW–H
Li 0.879 0.878 0.877
B 0.879 0.878 0.877
O 0.878 0.877 0.876
Ne 0.875 0.875 0.875
Mg 0.863 0.863 0.872
Si 0.851 0.850 0.868
Ar 0.843 0.826 0.877
V 0.663 0.716 0.875
Cr 0.568 0.636 0.875

Sexact is the incoherent scattering factor calculated analytically from a hydrogenic atomic model. Simp is the incoherent scattering factor calculated by taking the Compton profile derived in the impulse approximation and truncating it for ΔE < EB. SW–H is the Waller–Hartree incoherent scattering factor. Data taken from Bloch & Mendelsohn (1974[link]).

In those instances where the exact method has been used as a yardstick, the comparison favours the `relativistic integrated impulse approximation' outlined below, rather than the Waller–Hartree method.

References

Bewilogua, L. (1931). Incoherent scattering of X-rays. Phys. Z. 32, 740–744.
Bloch, B. J. & Mendelsohn, L. B. (1974). Atomic L-shell Compton profiles and incoherent scattering factors: theory. Phys. Rev. A, 9, 129–155.
Bloch, F. (1934). Contribution to the theory of the Compton line shift. Phys. Rev. 46, 674–687.
Brown, W. D. (1966). Cross-sections for coherent/incoherent X-ray scattering. Reports D2-125136-1 and 125137-1. Boeing Co.
Cromer, D. T. (1969). Compton scattering factors for aspherical free atoms. J. Chem. Phys. 50, 4857–4859.
Cromer, D. T. & Mann, J. B. (1967). Compton scattering factors for spherically symmetric free atoms. J. Chem. Phys. 47, 1892–1893.
Currat, R., DeCicco, P. D. & Weiss, R. J. (1971). Impulse approximation in Compton scattering. Phys. Rev. B, 4, 4256–4261.
Eisenberger, P. &Platzman, P. M. (1970). Compton scattering of X-rays from bound electrons. Phys. Rev. A, 2, 415–423.
Fermi, E. (1928). Statistical methods of investigating electrons in atoms. Z. Phys. 48, 73–79.
Hubbell, J. H., Veigele, W. J., Briggs, E. A., Brown, R. T., Cromer, D. T. & Howerton, R. J. (1975). Atomic form factors, incoherent scattering functions and photon scattering cross-sections. J. Phys. Chem. Ref. Data, 4, 471–538.
Pratt, R. H. & Tseng, H. K. (1972). Behaviour of electron wavefunctions near the atomic nucleus and normalisation screening theory in the atomic photoeffect. Phys. Rev. A, 5, 1063–1072.
Slater, J. C. (1937). Wavefunctions in a periodic crystal. Phys. Rev. 51, 846–851.
Thomas, L. H. (1927). Calculation of atomic fields. Proc. Cambridge Philos. Soc. 33, 542–548.
Veigele, W. J. (1967). Research of X-ray scattering cross sections: final report. Report KN-379-67-3(R). Kaman Sciences Corp.
Waller, I. & Hartree, D. R. (1929). Intensity of total scattering of X-rays. Proc. R. Soc. London Ser. A, 124, 119–142.








































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