Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 1.2, p. 23   | 1 | 2 |

Section The one-particle potential (OPP) model

P. Coppensa*

aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260-3000, USA
Correspondence e-mail: The one-particle potential (OPP) model

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When an atom is considered as an independent oscillator vibrating in a potential well [V({\bf u})], its distribution may be described by Boltzmann statistics. [P({\bf u}) = N \exp \{- V({\bf u})/kT\}, \eqno(] with N, the normalization constant, defined by [{\textstyle\int} P({\bf u})\ \hbox{d}{\bf u} = 1]. The classical expression ([link] is valid in the high-temperature limit for which [kT \gg {\it V}({\bf u})].

Following Dawson (1967)[link] and Willis (1969)[link], the potential function may be expanded in terms of increasing order of products of the contravariant displacement coordinates: [{V = V_{0} + \alpha_{j}u\hskip 2pt^{j} + \beta_{jk}u\hskip 2pt^{j}u^{k} + \gamma_{jkl}u\hskip 2pt^{j}u^{k}u^{l} + \delta_{jklm}u\hskip 2pt^{j}u^{k}u^{l}u^{m} + \ldots .} \eqno(] The equilibrium condition gives [\alpha_{j} = 0]. Substitution into ([link] leads to an expression which may be simplified by the assumption that the leading term is the harmonic component represented by [\beta_{jk}]: [\eqalignno{P({\bf u}) &= N \exp \{- \beta^{'}_{jk}u\hskip 2pt^{j}u^{k}\}\cr &\quad \times \{1 - \gamma^{'}_{jkl}u\hskip 2pt^{j}u^{k}u^{l} - \delta^{'}_{jklm}u\hskip 2pt^{j}u^{k}u^{l}u^{m} - \ldots \}, &(}] in which [\beta^{'} = \beta / kT] 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 [u\hskip 2pt^{j}u^{k}u^{l}], for example, are linear combinations of the seven octapoles [(l = 3)] and three dipoles [(l = 1)] (Coppens, 1980[link]). 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, 1970a[link]; Scheringer, 1985a[link]) [\eqalignno{T({\bf H}) &= T_{0} ({\bf H}) \left[1 - {4 \over 3} \pi^{3} i \gamma^{'}_{jkl} G\hskip 2pt^{jkl} ({\bf H}) + {2 \over 3} \pi^{4} \delta^{'}_{jklm} G\hskip 2pt^{jklm} ({\bf H})\right.\cr &\quad\left. + {4 \over 15} \pi^{5} i \varepsilon^{'}_{jklmn} G\hskip 2pt^{jklmn} ({\bf H}) - {4 \over 45} \pi^{6} i \varphi^{'}_{jklmnp} G\hskip 2pt^{jklmnp} ({\bf H}) \ldots\right],\cr& &(}] where [T_{0}] 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 [\beta_{jk}], simpler expressions are obtained in which the Hermite polynomials are replaced by products of the displacement coordinates [u\hskip 2pt^{j}] (Dawson et al., 1967[link]; Coppens, 1980[link]; Tanaka & Marumo, 1983[link]).


First citationCoppens, P. (1980). Thermal smearing and chemical bonding. In Electron and magnetization densities in molecules and solids, edited by P. J. Becker, pp. 521–544. New York: Plenum.Google Scholar
First citationDawson, B. (1967). A general structure factor formalism for interpreting accurate X-ray and neutron diffraction data. Proc. R. Soc. London Ser. A, 248, 235–288.Google Scholar
First citationDawson, B., Hurley, A. C. & Maslen, V. W. (1967). Anharmonic vibration in fluorite-structures. Proc. R. Soc. London Ser. A, 298, 289–306.Google Scholar
First citationJohnson, C. K. (1970a). Series expansion models for thermal motion. ACA Program and Abstracts, 1970 Winter Meeting, Tulane University, p. 60.Google Scholar
First citationScheringer, C. (1985a). A general expression for the anharmonic temperature factor in the isolated-atom-potential approach. Acta Cryst. A41, 73–79.Google Scholar
First citationTanaka, K. & Marumo, F. (1983). Willis formalism of anharmonic temperature factors for a general potential and its application in the least-squares method. Acta Cryst. A39, 631–641.Google Scholar
First citationWillis, B. T. M. (1969). Lattice vibrations and the accurate determination of structure factors for the elastic scattering of X-rays and neutrons. Acta Cryst. A25, 277–300.Google Scholar

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