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
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The p.d.f.'s defined by (6.1.1.77), (6.1.1.78), (6.1.1.86) and (6.1.1.87), and their Fourier transforms given in §6.1.1.6.5 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 in (6.1.1.76) is sufficiently small, the cyclic overlap on the circle becomes unimportant and the summation for 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 . 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; Kay & Behrendt, 1963; Kendall & Stuart, 1963; Maslen, 1968; Pawley & Willis, 1970).
References
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Kendall, M. G. & Stuart, A. (1963). The advanced theory of statistics, Vol. 1, Chaps 2, 3 and 6. London: Griffin.Google Scholar
Maslen, E. N. (1968). An expression for the temperature factor of a librating atom. Acta Cryst. A24, 434–437.Google Scholar
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