Cromer–Mann coefficients

Definition

The set of nine coefficients [math]a_i, b_i, c (i=1,\dots, 4)[/math] in a parameterization of the scattering factors for neutral atoms as a function of [math](\sin \theta)/\lambda[/math]:

[math]f(\sin\theta/\lambda) = \sum_{i=1}^4 a_i \exp[-b_i(\sin\theta/\lambda)^2] + c[/math]

for [math]0\lt(\sin\theta)/\lambda\lt2.0\,\mathrm{\AA}^{-1}[/math].

History

Atomic scattering factors for non-hydrogen atoms were calculated from relativistic Hartree–Fock wavefunctions by Doyle, P. A. & Turner, P. S. [(1968). Acta Cryst. A24, 390–397. Relativistic Hartree–Fock and electron scattering factors] using the wavefunctions of Coulthard, M. A. [(1967). Proc. Phys. Soc. 91, 44–49. A relativistic Hartree–Fock atomic field calculation, and in 1968 by Cromer, D. T. & Waber, J. T. using the unpublished wavefunctions of J. B. Mann (1968) [International Tables for X-ray Crystallography (1974), Vol. IV, p. 71. Birmingham: Kynoch Press]. The latter are based on a more exact treatment of potential that allows for the finite size of the nucleus, but the effect on the scattering factors is small. The calculations of Cromer & Waber (1968[link]) were originally made for [0\lt(\sin\theta)/\lambda\lt2.0\,{\rm \AA}^{-1}], but these have been extended to 6 Å−1 by Fox, O'Keefe & Tabbernor (1989[link]); this has been done because there are increasing numbers of applications for high-angle scattering factors.

See also

Intensity of diffracted intensities. P. J. Brown, A. G. Fox, E. N. Maslen, M. A. O'Keefe and B. T. M. Willis. International Tables for Crystallography (2006). Vol. C, ch. 6.1, pp. 554-595 doi:10.1107/97809553602060000600, especially Table 6.1.1.4.