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, pp. 659-660

Section 7.4.3.3. Relativistic treatment of incoherent scattering

N. G. Alexandropoulosa and M. J. Cooperb

7.4.3.3. Relativistic treatment of incoherent scattering

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The Compton effect is a relativistic phenomenon and it is accordingly more satisfactory to start from this basis, i.e. the Klein & Nishina (1929[link]) theory and the Dirac equation (see Jauch & Rohrlich, 1976[link]). In second-order relativistic perturbation theory, there is no overt separation of [{\bf p} \cdot {\bf A}] and [{\bf A} \cdot {\bf A}] terms. The inclusion of electron spin produces additional terms in the Compton cross section that depend upon the polarization (Lipps & Tolhoek, 1954a[link],b[link]); they are generally small at X-ray energies. They are of increasing interest in synchrotron-based experiments where the brightness of the source and its polarization characteristics compensate for the small cross section (Blume & Gibbs, 1988[link]).

Somewhat surprisingly, it is the spectral distribution, d2σ/dΩ dE2, 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[link]) and Williams (1977[link]) for reviews of the technique]. Manninen, Paakkari & Kajantie (1976[link]) and Ribberfors (1975a[link],b[link]) have shown that the Compton profile – the projection of the electron momentum density distribution onto the X-ray 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[link]; Bloch & Mendelsohn, 1974[link]) that, to a high degree of accuracy, the spectral distribution is merely truncated at energy transfers [E\le E_B].

This has led to the suggestion that the incoherent intensity can be obtained by integrating the spectral distributions, i.e. from [{{\rm d}\sigma \over {\rm d}\Omega}=\int\limits^\infty_{E_1-E_B}{{\rm d}^2\sigma \over {\rm d}\Omega\,{\rm d}E_2}\,{\rm d}E_2. \eqno (7.4.3.6)]

Unfortunately, this requires the Compton profile of each electron shell as input [Compton line shapes have been tabulated by Biggs, Mendelsohn & Mann (1975[link])] for all elements.

Ribberfors (1983[link]) and Ribberfors & Berggren (1982[link]) 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[link] 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.

[Figure 7.4.3.2]

Figure 7.4.3.2| top | pdf |

The incoherent scattering function, S(x, Z)/Z, per electron for aluminium shown as a function of x = (sin θ)/λ. The Waller–Hartree theory (dotted line) is compared with the truncated impulse approximation in the tabulated Compton profiles (Biggs, Mendelsohn & Mann, 1975[link]) cut off at E < EB for each electron group (solid line). The third curve (dashed line) shows the simplification introduced by Ribberfors (1983[link]) and Ribberfors & Berggren (1982[link]). The predictions are indistinguishable to within experimental error except at low [(\sin\theta/\lambda)]. Reference to the measurements can be found in Ribberfors & Berggren (1982[link]).

References

First citation Biggs, F., Mendelsohn, L. B. & Mann, J. B. (1975). Hartree Fock Compton profiles for the elements. At. Data Nucl. Data Tables, 16, 201–309.Google Scholar
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 Blume, M. & Gibbs, D. (1988). Polarisation dependence of magnetic X-ray scattering. Phys. Rev. B, 37, 1779–1789.Google Scholar
First citation Cooper, M. J. (1985). Compton scattering and electron momentum determination. Rep. Prog. Phys. 48, 415–481.Google Scholar
First citation Jauch, J. M. & Rohrlich, F. (1976). The theory of photons and electrons. Berlin: Springer-Verlag.Google Scholar
First citation Klein, O. & Nishina, Y. (1929). Uber die Streuung von Strahlung durch freie Electronen nach der neuen relativistichers Quantendynamik von Dirac. Z. Phys. 52, 853–868.Google Scholar
First citation Lipps, F. W. & Tolhoek, H. A. (1954a). Polarization phenomena of electrons and photons I. Physica (Utrecht), 20, 85–98.Google Scholar
First citation Lipps, F. W. & Tolhoek, H. A. (1954b). Polarization phenomena of electrons and photons II. Physica (Utrecht), 20, 395–405.Google Scholar
First citation Manninen, S., Paakkari, T. & Kajantie, K. (1976). Gamma ray Compton profile of aluminium. Philos. Mag. 29, 167–178.Google Scholar
First citation Pattison, P. & Schneider, J. R. (1979). Test of the relativistic 1s wavefunction in Au and Pb using experimental Compton profiles. J. Phys. B, 12, 4013–4019.Google Scholar
First citation Ribberfors, R. (1975a). Relationship of the relativistic Compton cross-section to the momentum distribution of bound electron states. Phys. Rev. B, 12, 2067–2074.Google Scholar
First citation Ribberfors, R. (1975b). Relationship of the relativistic Compton cross-section to the momentum distribution of bound electron states. II. Effects of anisotropy and polarisation. Phys. Rev. B, 12, 3136–3141.Google Scholar
First citation Ribberfors, R. (1983). X-ray incoherent scattering total cross-sections and energy absorption cross-sections by means of simple calculation routines. Phys. Rev. A, 27, 3061–3070.Google Scholar
First citation Ribberfors, R. & Berggren, K.-F. (1982). Incoherent X-ray scattering functions and cross sections by means of a pocket calculator. Phys. Rev. A, 26, 3325–3333.Google Scholar
First citation Williams, B. G. (1977). Compton scattering. New York: McGraw-Hill.Google Scholar








































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