International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 7.4, pp. 657-659
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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: where (dσ/dΩ)0 is the cross section characterizing the interaction, in this case it is the Thomson cross section,
;
and
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
, the momentum transfer K, and the atomic number Z.
The electromagnetic wave perturbs the electronic system through the vector potential A in the Hamiltonian
It produces photoelectric absorption through the term taken in first order, Compton and Raman scattering through the
term and resonant Raman scattering through the
terms in second order.
If resonant scattering is neglected for the moment, the expression for the incoherent scattering cross section becomes 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
, S(E1, E2, K, Z)
Z but as Table 7.4.3.1
shows this condition does not hold in the X-ray regime.
The evaluation of the matrix elements in equation (7.4.3.4) was simplified by Waller & Hartree (1929
) 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
:
where
and
the latter term arising from exchange in the many-electron atom.
According to Currat, DeCicco & Weiss (1971), equation (7.4.3.5)
can be improved by inserting the prefactor (E2/E1)2, where E2 is calculated from equation (7.4.3.1)
; the factor is an average for the factors inside the summation sign of equation (7.4.3.4)
that were neglected by Waller & Hartree. This term has been included in a few calculations of incoherent intensities [see, for example, Bloch & Mendelsohn (1974
)]. 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
) and Cromer (1969
) from non-relativistic Hartree–Fock self-consistent-field wavefunctions. Table 7.4.3.2
is taken from the compilation by Hubbell, Veigele, Briggs, Brown, Cromer & Howerton (1975
).
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This statistical model of the atomic charge density (Thomas, 1927; Fermi, 1928
) 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
); more recent calculations have been made by Brown (1966
) and Veigele (1967
). The method is less accurate than Waller–Hartree theory, but it is a much simpler computation.
The matrix elements of (7.4.3.4) 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
)], an analytical approach can be used, as was originally proposed by Bloch (1934
).
Hydrogenic calculations have been shown to predict accurate K- and L-shell photoelectric cross sections (Pratt & Tseng, 1972). The method has been applied in a limited number of cases to K-shell (Eisenberger & Platzman, 1970
) and L-shell (Bloch & Mendelsohn, 1974
) 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
), is given in Table 7.4.3.3
for illustration. The discrepancy is much reduced when all electrons are considered.
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 ![]() |
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.
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