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3. Beyond the Gaussian Approximation next up previous
Next: 4. Recommendations Up: ATOMIC DISPLACEMENT PARAMETER NOMENCLATURE Previous: 2.3.2 Mean-square displacement surface

3. Beyond the Gaussian Approximation

The situation is less straightforward if the distribution function is not Gaussian. A large variety of different approximation formalisms, as well as different nomenclature for similar formulations, is found in the literature. Summaries have been given by Johnson & Levy (1974), Zucker & Schulz (1982), Coppens (1993), and Kuhs (1992). By virtue of eq. (1.4.8), one may express either pdf(u) or tex2html_wrap_inline2578 as a series expansion and obtain the other quantity by Fourier transformation.

The most widespread approaches are based on formalisms developed in statistics to describe non-Gaussian distributions (Johnson, 1969). They use a differential expansion of the Gausssian pdf. Two formulations are found in frequently used refinement programs, the cumulant or Edgeworth expansion gif


and the quasi-moment or Gram-Charlier expansion


with tex2html_wrap_inline2796 the Gaussian Debye-Waller factor (see sections 1.4 and 2.1) and tex2html_wrap_inline2798 , tex2html_wrap_inline2800 , ... the third, fourth, ... order (anharmonic) tensorial coefficients. There are in general ten cubic, fifteen quartic, ... terms that enter into the treatment. In statistics they are called cumulants and quasi-moments, respectively. They constitute the parameters of the refinement. Various symbols for these coefficients are scattered through the literature. Greek letters are chosen here to comply with the tex2html_wrap_inline2802 's of the Gaussian case, which may thus be considered as second-order coefficients. For the same reason the factors tex2html_wrap_inline2804 , with N the order of the tensor, are included, also to follow standard physical notation, which uses tex2html_wrap_inline2808 as the scattering vector. The factors tex2html_wrap_inline2810 and/or the factors 1/N! (e.g., Kuhs, 1992) are sometimes omitted in the literature. For comparability of future results, it is therefore proposed that only coefficients defined as in (3.1.61) and (3.1.62) be published, and that subscripts be used to indicate the type of expansion employed.

The tex2html_wrap_inline2798 , tex2html_wrap_inline2800 , ... are dimensionless quantities. As proposed by Kuhs (1992), they may be transformed to quantities of dimension (length)N by


with tex2html_wrap_inline2636 to be replaced by tex2html_wrap_inline2638 , tex2html_wrap_inline2824 ... Note that this is a generalization of eq. (2.1.38). It must be stressed, however, that the tex2html_wrap_inline2798 , tex2html_wrap_inline2800 , ... are simple expansion coefficients and (in general) have no direct physical meaning. The transformation (3.1.63) thus has no such merits as in the Gaussian case, and some real-space illustrations should always be given to permit the results to be appreciated. The best way is certainly to plot the corresponding pdf, obtained by inversion of (3.1.62) or (1.4.8). Only programs that produce sections of the pdf's seem to be currently available, although a three-dimensional visualisation similar to ORTEP would be highly desirable. Another way of presenting the results is by tensor contraction (Kuhs, 1992). For even-order terms, full contraction yields an invariant scalar,


For the Gram-Charlier series, this quantity indicates flatness (for negative values) or peakedness (positive values) of the pdf. The tex2html_wrap_inline2830 are the components of the real-space metric tensor. Note that tex2html_wrap_inline2832 , i.e., (3.1.64) is an extension of eq. (2.2.53). Similarly, vector invariants may be calculated for odd-order terms,


giving the direction of maximal skewness. Partial contraction of even-order terms,


reveals the directions of flatness and peakedness.

Various discussions in the literature (see, e.g., Kuhs, 1992, and references therein) indicate that the Gram-Charlier formalism is the best choice in routine crystallographic work. In particular, it has the advantage that the reverse Fourier transformation (1.4.8) can be carried out analytically,


with tex2html_wrap_inline2834 Hermite polynomials, and tex2html_wrap_inline2836 the harmonic part of the pdf. These polynomials are tabulated by Johnson & Levy (1974) up to the fourth order and by Zucker & Schulz (1982) up to the sixth order [see also Coppens (1993)]. The use of the Gram-Charlier expansion (3.1.62) is therefore recommended, although other formalisms may sometimes be advantageous for special problems. In any case, the results should always be carefully checked, especially if higher order terms are used merely to improve the agreement of the fit. Strong and extended negative regions in the pdf indicate inadequacy of the results. One also has to remember that, with anharmonic refinements, the positions and tex2html_wrap_inline2802 obtained are not necessarily faithful representations of the mean and variance of the pdf, respectively. This must be borne in mind if bond distances and Gaussian displacement ellipsoids are to be derived from the refined parameters. In some situations, it may be better to use only the Gaussian approximation, even though the resulting R-factors may be higher.

Another possibility is the expansion of the so called one-particle-potential (OPP) tex2html_wrap_inline2840 , which in the classical limit [ tex2html_wrap_inline2842 ] is related to the pdf by Boltzmann statistics


with Z the partition function. The second equality is obtained by setting tex2html_wrap_inline2846 .

The latter approach was formulated by Dawson & Willis (1967) and Willis (1969) for cubic point groups and later generalized for any symmetry by Tanaka & Marumo (1983). The OPP is written as


with tex2html_wrap_inline2848 the harmonic (quadratic) OPP and tex2html_wrap_inline2850 and tex2html_wrap_inline2852 the third and fourth order coefficients, respectively, which are defined in a Cartesian system. Since application of eqs. (3.1.68) and (1.4.8) does not lead to an analytical expression for tex2html_wrap_inline2578 , the anharmonic part tex2html_wrap_inline2856 is approximated in (3.1.68) by


The final expressions for tex2html_wrap_inline2858 are rather lengthy and may be found in Tanaka & Marumo (1983). Refinable parameters are the tex2html_wrap_inline2850 and tex2html_wrap_inline2852 . Other formulations with simpler expressions for tex2html_wrap_inline2858 have been introduced by Coppens (1978), Kurki-Suonio, Merisalo & Peltonen (1979) and Scheringer (1985). None of these approaches seems to have been used much in crystallographic studies, and final recommendations must await further developments in this field. It should also be noted that the OPP approach treats each atom as an individual (Einstein-)oscillator, which is a poor approximation for tightly bound atoms in molecules.

The OPP approach is physically meaningful only for purely dynamic displacive disorder (giving, e.g., the directions of weak and strong bonds), and is limited to rather small anharmonicities through the approximation (3.1.70). Occasionally special expansions (e.g., symmetry-adapted spherical harmonics) of pdf(u) or tex2html_wrap_inline2578 have been used for special problems (e.g., curvilinear motion, molecular disorder); see Johnson & Levy (1974), Press & Hüller (1973) and Prandl (1981). Again, these expansions do not seem yet to have entered routine crystallographic work. It should be remembered that the classical limit tex2html_wrap_inline2842 , which is assumed in eq. (3.1.68), may be far from the actual situation even at room temperature.

gif In eqs. (3.1.61), ( 3.1.62), and many of the remaining equations in this section, the summation convention has been used. It is assumed that summation occurs over indices that are repeated, such as j, k, l and m in the terms on the right-hand side of ( 3.1.61) and (3.1.62). gif
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Next: 4. Recommendations Up: ATOMIC DISPLACEMENT PARAMETER NOMENCLATURE Previous: 2.3.2 Mean-square displacement surface

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