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, pp. 584-585

Section 6.1.1.5. The temperature factor

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

6.1.1.5. The temperature factor

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The atoms in a solid vibrate about their equilibrium positions, with an amplitude that increases with temperature. As a result of this vibration, the amplitude for coherent scattering is modulated by the Fourier transform of the probability distribution for the vibrating atom, known as the temperature factor. The reduction in the intensity of the coherent scattering is accompanied by thermal diffuse scattering, for which the phase relationship between the incident and diffracted beams is altered by the thermal wave, or phonon.

The first term in an expansion of the probability density [\rho({\bf u})] for displacement u about an equilibrium position at the origin is [\rho_o({\bf u})={{\rm det\,{\bf \boldsigma_u}^{-1/2}}\over8\pi^3}\exp(-\textstyle{1\over2}{\bf u}^T\cdot{\bf \boldsigma_u}^{-1}\cdot{\bf u}), \eqno (6.1.1.30)]where [{\bf\boldsigma_u}] is the dispersion matrix describing the second moments of the displacements about the mean position. The corresponding expression for the temperature factor is [T_o({\bf S})=\exp(-\textstyle{1\over2}{\bf S}^T\cdot{\boldsigma}_{\bf u}\cdot{\bf S}),\eqno (6.1.1.31)]which is the Fourier transform of [\rho_o({\bf u})].

The mean-square displacement of the atom from its mean position in the direction of the vector v is given by [\langle{\bf u}^2\rangle_{\bf v}={\bf v}^T{\bf g}^T{\boldsigma}_{\bf u}{\bf g}{\bf v}/({\bf v}^T{\bf g}{\bf v}),\eqno (6.1.1.32)]where [g_{ij}] is the covariant metric tensor with the scalar products of the unit-cell vectors [{\bf a}_i\cdot{\bf a}_j] as components.

The thermal motion for atoms in crystals is often displayed as surfaces of constant probability density. The surface for the thermal displacement u is defined by [{\bf u}^T{\boldsigma}^{-1}_{\bf u}{\bf u}=C^2.\eqno (6.1.1.33)]The square of the distance from the origin to the equiprobability surface in the direction v is [C^2{\bf v}^T{\bf g}{\bf v}/({\bf v}^T{ \boldsigma}_{\bf u}^{-1}{\bf v}).\eqno (6.1.1.34)]This is equal to (6.1.1.32)[link] for C unity only if v coincides with a principal axis of the vibration ellipsoid.

The probability that a displacement falls within the ellipsoid defined by C is [(2/\pi){}^{1/2}\textstyle\int\limits^C_0q^2\exp(-q^2/2)\,{\rm d}q.\eqno (6.1.1.35)]








































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