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. 657659

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(E_{1}, E_{2}, 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 highenergy limit of , S(E_{1}, E_{2}, K, Z) Z but as Table 7.4.3.1 shows this condition does not hold in the Xray regime.
The evaluation of the matrix elements in equation (7.4.3.4) was simplified by Waller & Hartree (1929) who (i) set E_{2} = E_{1} 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 manyelectron atom.
According to Currat, DeCicco & Weiss (1971), equation (7.4.3.5) can be improved by inserting the prefactor (E_{2}/E_{1})^{2}, where E_{2} 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 nonrelativistic Hartree–Fock selfconsistentfield wavefunctions. Table 7.4.3.2 is taken from the compilation by Hubbell, Veigele, Briggs, Brown, Cromer & Howerton (1975).

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 oneelectron wavefunctions in manyelectron 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 Lshell photoelectric cross sections (Pratt & Tseng, 1972). The method has been applied in a limited number of cases to Kshell (Eisenberger & Platzman, 1970) and Lshell (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 Lshell 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.
S_{exact} is the incoherent scattering factor calculated analytically from a hydrogenic atomic model. S_{imp} is the incoherent scattering factor calculated by taking the Compton profile derived in the impulse approximation and truncating it for ΔE < E_{B}. S_{W–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.
References
Bewilogua, L. (1931). Incoherent scattering of Xrays. Phys. Z. 32, 740–744.Bloch, B. J. & Mendelsohn, L. B. (1974). Atomic Lshell 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). Crosssections for coherent/incoherent Xray scattering. Reports D21251361 and 1251371. 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 Xrays 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 crosssections. 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 Xray scattering cross sections: final report. Report KN379673(R). Kaman Sciences Corp.
Waller, I. & Hartree, D. R. (1929). Intensity of total scattering of Xrays. Proc. R. Soc. London Ser. A, 124, 119–142.