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. 6.1, p. 590

Section 6.1.1.6.6. The quasi-Gaussian approximation for curvilinear motion

E. N. Maslen,e‡ A. G. Foxb and M. A. O'Keefec

6.1.1.6.6. The quasi-Gaussian approximation for curvilinear motion

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The p.d.f.'s defined by (6.1.1.77)[link], (6.1.1.78)[link], (6.1.1.86)[link] and (6.1.1.87)[link], and their Fourier transforms given in §6.1.1.6.5[link] may be considered `inverted series' since zero-order terms describe uniform distributions. The inverted series converge slowly if the density is concentrated near the mode. If [\sigma^2] in (6.1.1.76)[link] is sufficiently small, the cyclic overlap on the circle becomes unimportant and the summation for [n\ne0] can be neglected. In this limiting case, the p.d.f. assumes the same form as a one-dimensional rectilinear Gaussian density function except that the variable is the angle [\varphi]. A similar relation must exist between the p.d.f. on the sphere and the two-dimensional Gaussian function. This `quasi-Gaussian' approximation is the basis for a number of structure-factor equations for atoms with relatively small amplitude components of curvilinear motion (Dawson, 1970[link]; Kay & Behrendt, 1963[link]; Kendall & Stuart, 1963[link]; Maslen, 1968[link]; Pawley & Willis, 1970[link]).

References

First citation Dawson, B. (1970). Neutron studies of nuclear charge distributions in barium fluoride and hexamethylenetetramine. Thermal neutron diffraction, edited by B. T. M. Willis, pp. 101–123. Oxford University Press.Google Scholar
 First citation Kay, M. I. & Behrendt, D. R. (1963). The structure factor for a harmonic quasi-torsional oscillator. Acta Cryst. 16, 157–162.Google Scholar
 First citation Kendall, M. G. & Stuart, A. (1963). The advanced theory of statistics, Vol. 1, Chaps 2, 3 and 6. London: Griffin.Google Scholar
 First citation Maslen, E. N. (1968). An expression for the temperature factor of a librating atom. Acta Cryst. A24, 434–437.Google Scholar
 First citation Pawley, G. S. & Willis, B. T. M. (1970). Neutron diffraction study of the atomic and molecular motion in hexamethylenetetramine. Acta Cryst. A26, 263–271.Google Scholar








































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