Cromer–Mann coefficients

Definition

The set of nine coefficients $a_i, b_i, c\, (i=1,\dots, 4)$ in a parameterization of the non-dispersive part of the atomic scattering factor for neutral atoms as a function of (sinθ) / λ:

$f^0(\sin\theta/\lambda) = \sum_{i=1}^4 a_i \exp[-b_i(\sin\theta/\lambda)^2] + c$

for $0 < (\sin\theta)/\lambda < 2.0\,\mathrm{\AA}^{-1}$.

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. and 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 A relativistic Hartree–Fock atomic field calculation], and in 1968 by Cromer, D. T. and Waber, J. T. using the unpublished wavefunctions of J. B. Mann [International Tables for X-ray Crystallography (1974), Vol.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. and 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.