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 non-dispersive part of the atomic scattering factor for neutral atoms as a function of [math](\sin \theta)/\lambda[/math]:

[math]f^0(\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 \lt 2.0\,\mathrm{\AA}^{-1}[/math].

This expression is convenient for calculation in crystal structure software suites.

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 A relativistic Hartree–Fock atomic field calculation], and in 1968 by Cromer, D. T. & Waber, J. T. using the unpublished wavefunctions of J. B. Mann [International Tables for X-ray Crystallography (1974), Vol.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. Subsequent calculations [Fox, A. G., O'Keefe, M. A. & Tabbernor, M. A. (1989). Acta Cryst. A45, 786–793. Relativistic Hartree–Fock X-ray and electron atomic scattering factors at high angles] extended the useful range to 6 Å⁻¹ to accommodate the 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.