InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 1.2, p. 23
## Section 1.2.12.3. The one-particle potential (OPP) model |

When an atom is considered as an independent oscillator vibrating in a potential well , its distribution may be described by Boltzmann statistics. with *N*, the normalization constant, defined by . The classical expression (1.2.12.10) is valid in the high-temperature limit for which .

Following Dawson (1967) and Willis (1969), the potential function may be expanded in terms of increasing order of products of the contravariant displacement coordinates: The equilibrium condition gives . Substitution into (1.2.12.10) leads to an expression which may be simplified by the assumption that the leading term is the harmonic component represented by : in which *etc.* and the normalization factor *N* depends on the level of truncation.

The probability distribution is related to the spherical harmonic expansion. The ten products of the displacement parameters , for example, are linear combinations of the seven octapoles and three dipoles (Coppens, 1980). The thermal probability distribution and the aspherical atom description can be separated only because the latter is essentially confined to the valence shell, while the former applies to all electrons which follow the nuclear motion in the atomic scattering model.

The Fourier transform of the OPP distribution, in a general coordinate system, is (Johnson, 1970*a*; Scheringer, 1985*a*) where is the harmonic temperature factor and *G* represents the Hermite polynomials in reciprocal space.

If the OPP temperature factor is expanded in the coordinate system which diagonalizes , simpler expressions are obtained in which the Hermite polynomials are replaced by products of the displacement coordinates (Dawson *et al.*, 1967; Coppens, 1980; Tanaka & Marumo, 1983).

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