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, p. 659

Section 7.4.3.2.3. Exact calculations

N. G. Alexandropoulosa and M. J. Cooperb

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
Li0.8790.8780.877
B0.8790.8780.877
O0.8780.8770.876
Ne0.8750.8750.875
Mg0.8630.8630.872
Si0.8510.8500.868
Ar0.8430.8260.877
V0.6630.7160.875
Cr0.5680.6360.875

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

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 Bloch, F. (1934). Contribution to the theory of the Compton line shift. Phys. Rev. 46, 674–687.Google Scholar
First citation Eisenberger, P. &Platzman, P. M. (1970). Compton scattering of X-rays from bound electrons. Phys. Rev. A, 2, 415–423.Google Scholar
First citation 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.Google Scholar
First citation Slater, J. C. (1937). Wavefunctions in a periodic crystal. Phys. Rev. 51, 846–851.Google Scholar








































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