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. 657661

In many diffraction studies, it is necessary to correct the intensities of the Bragg peaks for a variety of inelastic scattering processes. Compton scattering is only one of the incoherent processes although the term is often used loosely to include plasmon, Raman, and resonant Raman scattering, all of which may occur in addition to the more familiar fluorescence radiation and thermal diffuse scattering. The various interactions are summarized schematically in Fig. 7.4.3.1, where the dominance of each interaction is characterized by the energy and momentum transfer and the relevant binding energy.
With the exception of thermal diffuse scattering, which is known to peak at the reciprocallattice points, the incoherent background varies smoothly through reciprocal space. It can be removed with a linear interpolation under the sharp Bragg peaks and without any energy analysis. On the other hand, in noncrystalline material, the elastic scattering is also diffused throughout reciprocal space; the pointbypoint correction is consequently larger and without energy analysis it cannot be made empirically; it must be calculated. These calculations are imprecise except in the situations where Compton scattering is the dominant process. For this to be the case, there must be an encounter, conserving energy and momentum, between the incoming photon and an individual target electron. This in turn will occur if the energy lost by the photon, ΔE = E_{1} − E_{1}, clearly exceeds the oneelectron binding energy, , of the target electron. Eisenberger & Platzman (1970) have shown that this binary encounter model – alternatively known as the impulse approximation – fails as (E_{B}/ΔE)^{2}.
The likelihood of this failure can be predicted from the Compton shift formula, which for scattering through an angle can be written.
This energy transfer is given as a function of the scattering angle in Table 7.4.3.1 for a set of characteristic Xray energies; it ranges from a few eV for Cr Kα Xradiation at small angles, up to ∼2 keV for backscattered Ag Kα Xradiation. Clearly, in the majority of typical experiments Compton scattering will be inhibited from all but the valence electrons.

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.
The Compton effect is a relativistic phenomenon and it is accordingly more satisfactory to start from this basis, i.e. the Klein & Nishina (1929) theory and the Dirac equation (see Jauch & Rohrlich, 1976). In secondorder relativistic perturbation theory, there is no overt separation of and terms. The inclusion of electron spin produces additional terms in the Compton cross section that depend upon the polarization (Lipps & Tolhoek, 1954a,b); they are generally small at Xray energies. They are of increasing interest in synchrotronbased experiments where the brightness of the source and its polarization characteristics compensate for the small cross section (Blume & Gibbs, 1988).
Somewhat surprisingly, it is the spectral distribution, d^{2}σ/dΩ dE_{2}, rather than the total intensity, dσ/dΩ, which is the better understood. This is a consequence of the exploitation of the Compton scattering technique to determine electron momentum density distributions through the Doppler broadening of the scattered radiation [see Cooper (1985) and Williams (1977) for reviews of the technique]. Manninen, Paakkari & Kajantie (1976) and Ribberfors (1975a,b) have shown that the Compton profile – the projection of the electron momentum density distribution onto the Xray scattering vector – can be isolated from the relativistic differential scattering cross section within the impulse approximation. Several experimental and theoretical investigations have been concerned with understanding the changes in the spectral distribution when electron binding energies cannot be discounted. It has been found (e.g. Pattison & Schneider, 1979; Bloch & Mendelsohn, 1974) that, to a high degree of accuracy, the spectral distribution is merely truncated at energy transfers .
This has led to the suggestion that the incoherent intensity can be obtained by integrating the spectral distributions, i.e. from
Unfortunately, this requires the Compton profile of each electron shell as input [Compton line shapes have been tabulated by Biggs, Mendelsohn & Mann (1975)] for all elements.
Ribberfors (1983) and Ribberfors & Berggren (1982) have shown that this calculation can be dramatically simplified, without loss of accuracy, by crudely approximating the Compton line shape. Fig. 7.4.3.2 shows the incoherent scattering from aluminium, modelled in this way, and compared with experiment, Waller–Hartree theory, and an exact integral of the truncated impulse Compton profile.
In typical Xray experiments, as is evident from Table 7.4.3.1, the energy transfer may be so low that Compton scattering will be inhibited from all but the most loosely bound electrons. Indeed, in the situation in metals where K, the momentum transfer, is less than k_{F} (the Fermi momentum), Compton scattering from the conduction electrons may be restricted by exclusion because of the lack of unoccupied final states [see Bushuev & Kuz'min (1977)]. Fortunately, in these uncertain circumstances, the incoherent intensities are low. In this regime, the electron gas may be excited into collective motion. For almost all solids, the plasmon excitation energy is 20–30 eV and, in the random phase approximation, the incoherent scattering factor becomes S(ΔE, K) ∝ (K^{2}/w_{p})δ(ΔE − hω_{p}), where ω_{p} is the plasma frequency.
At slightly higher energies , Compton scattering and Raman scattering can coexist, though the Raman component is only evident at low momentum transfer (Bushuev & Kuz'min, 1977). The resultant spectrum is often referred to as the Compton–Raman band. In semiclassical radiation theory, Raman scattering is usually differentiated from Compton scattering by dropping the requirement for momentum conservation between the photon and the individual target electron, the recoil being absorbed by the atom. The Raman band corresponds to transitions into the lowest unoccupied levels and these can be calculated within the dipole approximation as long as Ka , where K is the momentum transfer and a the orbital radius of the core electron undergoing the transition. The transition probability in equation (7.4.3.4) becomes which implies that the nearedge structure is similar to the photoelectric absorption spectrum.
Whereas plasmon and Raman scattering are unlikely to make dramatic contributions to the total incoherent intensity, resonant Raman scattering (RRS) may, when . The excitation involves a virtual Kshell vacancy in the intermediate state and a vacancy in the L (or M or N) shell and an electron in the continuum in the final state. It has now been observed in a variety of materials [see, for example, Sparks (1974), Eisenberger, Platzman & Winick (1976), Schaupp et al. (1984)]. It was predicted by Gavrila & Tugulea (1975) and the theory has been treated comprehensively by Åberg & Tulkki (1985). The effect is the exact counterpart, in the inelastic spectrum, of anomalous scattering in the elastic spectrum. It is important because, as the resonance condition is approached, the intensity will exceed that due to Compton scattering and therefore play havoc with any corrections to total intensities based solely on the latter.
Although systematic tabulations of resonance Raman scattering do not exist, Fig. 7.4.3.3, which is based on the calculations of Bannett & Freund (1975), shows how the intensity of RRS clearly exceeds that of the Compton scattering for incident energies just below the absorption edge. However, since the problems posed by anomalous scattering and Xray fluorescence are generally appreciated, the energy range is wisely avoided by crystallographers intent upon absolute intensity measurements.
Finally, and for completeness, it should be noted that the intensity of Compton scattering from a magnetic material with a net spin moment will, in principle, differ from that from a nonmagnetic material. For unpolarized radiation, the effects are only discernible at photon energies greatly in excess of the electron rest mass energy, mc^{2} = 511 keV, but for circularly polarized radiation effects at the 1% level can be found in Compton scattering experiments carried out at on ferromagnets such as iron. See Lipps & Tolhoek (1954a,b) for a comprehensive description of polarization phenomena in magnetic scattering and Lovesey (1993) for an account of the scattering theory.
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