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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: ![[\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)]](/teximages/cbch7o4/cbch7o4fd33.svg)
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]](/teximages/cbch7o4/cbch7o4fi47.svg)
; ![[\boldvarepsilon_1]](/teximages/cbch7o4/cbch7o4fi48.svg)
and ![[\boldvarepsilon_2]](/teximages/cbch7o4/cbch7o4fi49.svg)
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]](/teximages/cbch7o4/cbch7o4fi50.svg)
, 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)]](/teximages/cbch7o4/cbch7o4fd34.svg)
It produces photoelectric absorption through the ![[{\bf p}\cdot{\bf A}]](/teximages/cbch7o4/cbch7o4fi51.svg)
term taken in first order, Compton and Raman scattering through the ![[{\bf A}\cdot{\bf A}]](/teximages/cbch7o4/cbch7o4fi52.svg)
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 ![[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)]](/teximages/cbch7o4/cbch7o4fd35.svg)
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]](/teximages/cbch7o4/cbch7o4fi54.svg)
, S(E1, E2, K, Z) ![[\rightarrow]](/teximages/abch3o1/abch3o1fi527.svg)
Z but as Table 7.4.3.1![[link]](/graphics/greenarr.gif)
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 ![[f_{jk}]](/teximages/cbch7o4/cbch7o4fi56.svg)
: ![[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)]](/teximages/cbch7o4/cbch7o4fd36.svg)
where ![[f_j({\bf K})=\langle\psi_j|\!\exp (i{\bf K}\cdot {\bf r}_j)|\psi_j\rangle]](/teximages/cbch7o4/cbch7o4fd37.svg)
and ![[f_{jk}=\langle\psi_k|\!\exp [i{\bf K}\cdot ({\bf r}_k-{\bf r}_j)]|\psi_j\rangle,]](/teximages/cbch7o4/cbch7o4fd38.svg)
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 ).
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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
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