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. 4.2, pp. 221-229

Section 4.2.4.2.2. Theoretical Rayleigh scattering data: σR

D. C. Creaghb and J. H. Hubbelld

4.2.4.2.2. Theoretical Rayleigh scattering data: σR

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If each of the atoms gives rise to scattering in which momentum but not energy changes occur, and if each of the atoms can be considered to scatter as if it were an isolated atom, the cross section may be written as [\sigma _R = \pi r ^2_e \textstyle\int\limits^1_{-1} (1 + \cos ^2 \varphi) \,f^2(q,Z) \,{\rm d} (\cos \varphi), \eqno (4.2.4.6)]where

  • [r_e] is the classical radius of the electron;

  • [\varphi] is the angle of scattering ([=2 \theta] if [\theta] is the Bragg angle);

  • [2 \pi \,{\rm d} (\cos \varphi)] is the solid angle between cones with angles [\varphi] and [\varphi + {\rm d} \varphi];

  • [f(q,Z)] is the atomic scattering factor as defined by Cromer & Waber (1974[link]);

  • q is [ [\sin (\varphi /2) / \lambda],] the momentum transfer parameter. Here λ is expressed in ångströms.

Reliable tables of f(q, Z) exist and have been reviewed recently by Kane, Kissel, Pratt & Roy (1986[link]). The most recent schematic tabulations of f(q, Z) are those of Hubbell & Øverbø (1979[link]) and Schaupp et al. (1983[link]). The data used in these tables have been derived from the tabulation for q = 0.02 to 109 Å−1, for all Z's from 1 to 100 by Hubbell & Øverbø (1979[link]) based on the exact formula of Pirenne (1946[link]) for H, and relativistic calculations by Doyle & Turner (1968[link]), Cromer & Waber (1974[link]), Øverbø (1977[link], 1978[link]), and high-q extensions using the Bethe–Levinger expression in Levinger (1952[link]).

As mentioned in Creagh & Hubbell (1987[link]), the atoms in highly ordered single crystals do not scatter as though they are isolated atoms. Rather, cooperative effects become important. In this case, the Rayleigh scattering cross section must be replaced by two cross sections:

  • the Laue–Bragg cross section [\sigma _{\rm LB}],

  • and the thermal diffuse scattering cross section [\sigma _{\rm TD}].

That is, [\sigma _R] is replaced by [\sigma _{\rm LB} + \sigma _{\rm TD}].

These effects are discussed elsewhere (Subsection 4.2.3.2[link]). Briefly, [\sigma _{\rm LB} = (r^2_e \lambda ^2 / 2 NV_c) \textstyle\sum\limits _H [C_p md | F | {}^2 \exp (-2M)]_H. \eqno (4.2.4.7)]In equation (4.2.4.7)[link], which is due to De Marco & Suortti (1971[link]),

  • [C_p= \textstyle {1 \over 2}(1+ \cos ^2 \varphi)];

  • [d_H] is the spacing of the (hkl) planes in the crystal;

  • [m_H] is the multiplicity of the hkl Bragg reflection;

  • [F_H] is the geometrical structure factor for the crystal structure that contains N atoms in a cell of volume [V_c];

  • [\exp (-2M)_H] is the Debye–Waller temperature factor.

It is assumed that the total thermal diffuse scattering is equal to the scattering lost from Laue–Bragg scattering because of thermal vibrations. [\sigma _{\rm TD} = (r^2_e \lambda ^2/2NV_c) \textstyle\sum\limits _H \{ C_p md | F | {}^2 [1- \exp (-2 M)] \} _H. \eqno (4.2.4.8)]This equation is not in a convenient form for computation and the alternative formalism presented by Sano, Ohtaka & Ohtsuki (1969[link]) is often used in calculations. In this formalism, [\sigma _{\rm TD}= 2 \pi r ^2 _e \textstyle\int \limits^1 _{-1} C _p\,f^2 (q,Z) \{ 1 - \exp [-2M (q)] \} \,{\rm d} (\cos \varphi). \eqno (4.2.4.9)]

The values of f(q, Z) are those of Cromer & Waber (1974[link]).

Cross sections calculated using equation (4.2.4.8)[link] tend to oscillate at low energy and this corresponds to the inclusion of Bragg peaks in the summation or integration. Eventually, these oscillations abate and [\sigma _{\rm TD}] becomes a smoothly varying function of energy.

Creagh & Hubbell (1987[link]) and Creagh (1987a[link]) have stressed that, before cross sections are calculated for a given ensemble of atoms, care should be taken to ascertain whether single-atom or single-crystal scattering is appropriate for that ensemble.

References

First citation Creagh, D. C. (1987a). The resolution of discrepancies in tables of photon attenuation coefficients. Nucl. Instrum. Methods, A255, 1–16.Google Scholar
 First citation Creagh, D. C. & Hubbell, J. H. (1987). Problems associated with the measurement of X-ray attenuation coefficients. I. Silicon. Acta Cryst. A43, 102–112.Google Scholar
 First citation Cromer, D. T. & Waber, J. T. (1974). Atomic scattering factors for X-rays. International tables for X-ray crystallography, Vol. IV, edited by J. A. Ibers & W. C. Hamilton, Chap. 2.2, pp. 71–147. Birmingham: Kynoch Press.Google Scholar
 First citation De Marco, J. J. & Suortti, P. (1971). Effect of scattering on the attenuation of X-rays. Phys. Rev. B, 4, 1028–1033.Google Scholar
 First citation Doyle, P. A. & Turner, P. S. (1968). Relativistic Hartree–Fock X-ray and electron scattering factors. Acta Cryst. A24, 390–397.Google Scholar
 First citation Hubbell, J. H. & Øverbø, I. (1979). Relativistic atomic form factors and photon coherent scattering cross sections. J. Phys. Chem. Ref. Data, 8, 69–105.Google Scholar
 First citation Kane, P. P., Kissel, L., Pratt, R. H. & Roy, S. C. (1986). Elastic scattering of γ-rays and X-rays by atoms. Phys. Rep. 150, 75–159.Google Scholar
 First citation Levinger, J. S. (1952). Small angle coherent scattering of gammas by bound electrons. Phys. Rev. 87, 656–662.Google Scholar
 First citation Øverbø, I. (1977). The Coulomb correction to electron pair production by intermediate-energy photons. Phys. Lett. B, 71, 412–414.Google Scholar
 First citation Øverbø, I. (1978). Large-q form factors for light atoms. Phys. Scr. 17, 547–549.Google Scholar
 First citation Pirenne, M. H. (1946). The diffraction of X-rays and electrons by free molecules. Cambridge University Press.Google Scholar
 First citation Sano, H., Ohtaka, K. & Ohtsuki, Y. H. (1969). Normal and abnormal absorption coefficients of X-rays. J. Phys. Soc. Jpn, 27, 1254–1261.Google Scholar
 First citation Schaupp, D., Schumacher, M., Smend, F., Rullhusen, P. & Hubbell, J. H. (1983). Small-angle Rayleigh scattering of photons at high energies: tabulations of relativistic HFS modified atomic form factors. J. Phys. Chem. Ref. Data, 12, 467–512.Google Scholar








































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