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 7.4.3.2.1. Semi-classical radiation theory

N. G. Alexandropoulosa and M. J. Cooperb

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−1)
0.100.200.300.400.500.600.700.800.901.001.502.00
1 H0.3430.7690.9370.9830.9950.9980.9940.9991.0001.0001.0001.000
2 He0.2960.8811.3621.6571.8171.9021.9471.9701.9831.9901.9992.000
3 Li1.0331.4181.7952.1432.4172.6132.7462.8342.8912.9282.9892.998
4 Be1.1702.1212.4712.7443.0053.2373.4293.5793.6933.7773.9543.989
5 B1.1472.5313.1903.4993.7323.9484.1464.3204.4694.5904.8954.973
6 C1.0392.6043.6434.1844.4784.6904.8785.0515.2085.3485.7815.930
7 N1.082.8584.0974.7925.1825.4375.6355.8095.9686.1136.6306.860
8 O0.9772.7994.2935.2575.8286.1756.4116.5966.7556.9017.4627.764
9 F0.8802.6914.3475.5526.3396.8327.1517.3767.5527.7038.2888.648
10 Ne0.8122.5474.2695.6446.6407.3207.7748.0858.3128.4909.1139.517
11 Na1.5032.8914.4315.8046.9037.7248.3138.7299.0289.2529.93910.376
12 Mg2.0663.4444.7716.0647.1818.0868.7849.3049.6899.97510.76611.229
13 Al2.2644.0475.2506.4357.5238.4599.2259.83010.29610.65211.59212.083
14 Si2.2934.5205.8086.9037.9378.8679.66710.33010.86411.28612.40812.937
15 P2.2064.7326.3127.4358.4199.32310.13110.82711.41111.88813.20913.790
16 S2.1514.9606.7958.0028.9609.82910.62611.33611.95212.47213.99014.641
17 Cl2.0655.0747.1828.5539.53910.38211.15811.86712.49913.05014.75015.487
18 Ar1.9565.0337.3778.99810.10610.96711.72612.42413.06113.62915.48916.324
19 K2.5005.3017.6529.40510.65011.56812.32913.01413.64514.22016.21217.152
20 Ca3.1055.6907.9819.79011.15712.16312.95313.63514.25614.83016.92117.970
21 Sc3.1365.8018.16910.07111.56112.64813.54514.25614.88515.46017.63018.782
22 Ti3.1145.8608.31210.30411.90113.14014.09314.85615.50916.09518.33419.585
23 V3.0675.8588.37510.45412.15613.51414.57415.41316.11116.72119.03220.379
24 Cr2.6095.5778.20610.41512.26413.77014.96015.90216.67017.32319.73021.168
25 Mn2.9495.7918.38010.60412.48614.06215.34616.37617.21117.91020.41121.938
26 Fe2.8915.7818.43210.73312.68714.34315.71616.83117.73718.48821.09722.704
27 Co2.8325.7648.46910.84412.86714.59616.05017.24918.22919.03921.77723.462
28 Ni2.7725.7268.46110.89412.98014.78016.31717.60218.66419.54322.44524.211
29 Cu2.3485.4558.31010.77812.94214.84716.49417.88519.04320.00223.10724.957
30 Zn2.6545.6318.38810.90113.09415.02016.70918.16319.39520.42723.74525.683
31 Ga2.7915.9398.59911.08213.29015.23316.94718.44519.73420.83124.37026.400
32 Ge2.8396.2298.91211.33813.53615.48617.21518.74120.07421.22424.98327.109
33 As2.7936.3659.23611.65813.82815.77517.51119.05620.42021.61225.58327.810
34 Se2.7996.5899.60112.03314.16816.09817.83519.39120.77822.00326.17128.504
35 Br2.7716.7489.94012.44014.55216.45618.18519.74721.14922.39926.74729.190
36 Kr2.7036.76010.15712.82814.96916.84918.56220.12321.53522.80427.31329.870
37 Rb3.2257.06210.43113.20615.41017.28218.97420.52621.94023.22127.87130.543
38 Sr3.8317.46410.74613.57615.86017.74519.42020.95622.36723.65428.42331.210
39 Y3.9997.70011.01013.89916.27918.21519.89121.41622.82024.11028.97031.870
40 Zr4.0647.87911.23614.17616.65818.67220.37321.89523.29424.58329.51732.522
41 Nb3.6727.68411.21314.31716.94919.08120.84422.38623.78725.07730.06733.167
42 Mo3.6257.69011.26014.44417.19619.45521.30022.87724.28825.58130.62033.808
43 Tc3.9877.98411.51214.65317.45619.81621.74823.37024.79726.09331.17334.447
44 Ru3.5597.85711.53114.78217.68520.15022.17223.85525.31226.62131.74035.081
45 Rh3.4997.86311.59114.88317.85820.42822.55724.31825.81927.14832.30935.715
46 Pd3.1037.72511.44114.82417.94326.65322.90424.75626.31627.67732.88836.349
47 Ag3.3627.78511.59814.96918.08220.85823.21225.16226.79228.19533.46536.983
48 Cd3.7007.98011.81215.18518.26321.06423.50125.54627.25228.70534.04637.618
49 In3.8528.29712.08315.44418.48921.28823.77925.90627.69129.20334.63438.255
50 Sn3.9178.61512.41515.74618.76021.54124.05926.25228.11329.68735.22638.894
51 Sb3.8718.81112.77716.08819.06721.82324.34926.59028.51830.15735.82239.536
52 Te3.0979.07613.17116.46619.40722.13425.65526.92728.91230.61336.42240.181
53 I3.9039.28713.56416.87619.22722.47124.98027.26929.29831.05637.02440.827
54 Xe3.8419.34013.89217.30720.17522.83325.32427.61929.68031.48837.62841.477
55 Cs4.3209.61514.21717.75320.61223.22825.69127.98130.06431.91438.23242.129

References

First citation Bloch, B. J. & Mendelsohn, L. B. (1974). Atomic L-shell Compton profiles and incoherent scattering factors: theory. Phys. Rev. A, 9, 129–155.Google Scholar
First citation Cromer, D. T. (1969). Compton scattering factors for aspherical free atoms. J. Chem. Phys. 50, 4857–4859.Google Scholar
First citation Cromer, D. T. & Mann, J. B. (1967). Compton scattering factors for spherically symmetric free atoms. J. Chem. Phys. 47, 1892–1893.Google Scholar
First citation Currat, R., DeCicco, P. D. & Weiss, R. J. (1971). Impulse approximation in Compton scattering. Phys. Rev. B, 4, 4256–4261.Google Scholar
First citation 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.Google Scholar
First citation Waller, I. & Hartree, D. R. (1929). Intensity of total scattering of X-rays. Proc. R. Soc. London Ser. A, 124, 119–142.Google Scholar








































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