# Cromer–Mann coefficients

### From Online Dictionary of Crystallography

## 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. and Turner, P. S. [(1968). *Acta Cryst.* A**24**, 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 *A relativistic Hartree–Fock atomic field calculation*], and in 1968 by Cromer, D. T. and Waber, J. T. using the unpublished wavefunctions of J. B. Mann [*International Tables for X-ray Crystallography* (1974), Vol.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.* A**45**, 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, especially Table 6.1.1.4.