4. Recommendations
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# 4. Recommendations

1. The term anisotropic displacement parameters (abbreviated ADPs) should be used in referring to the individual atomic coefficients in the exponent of the factor that describes the effects of atomic motion and static displacement.
2. The elements of the tensors and should always be superscripted when the refinement is referred to a crystal system rather than to a Cartesian system. This definition follows from the definition of the elements of and as contravariant tensor components (see Section 2.1, ). The frequent use of subscripts for the ADPs, and specifically for those not referred to Cartesian systems, is inconsistent with their tensorial properties.
3. With the common Gaussian approximation, use either the quantities , which have dimension (length)2 , defined in eq. (2.1.25), or the dimensionless , defined in eq. (2.1.22).
4. When the Gaussian approximation to the probability density function is not deemed valid, the use of the Gram-Charlier expansion of eq. (3.1.62) is recommended, although other formalisms may sometimes be advantageous for special problems.
5. Standard uncertainties of ADPs obtained from a full-matrix refinement are valid within the system in which the refinement is made. If ADPs are transformed to any other axial system, Cartesian or not, then the uncertainties may also be calculated by transforming the original variance-covariance matrix to this new axial system and taking the square roots of its diagonal elements, i.e., the variances. The required variance-covariance matrix is usually not available for ADPs taken from the literature. Hence, although ADPs can still be transformed, their uncertainties cannot be. Calculations involving published ADPs and their (published) uncertainties should therefore be referred to the same system of coordinates as the original refinement in order to retain the significance of the published uncertainties.
6. Avoid using the term ``temperature factor", both because the phenomenon represented may not be due entirely to thermal motion and because that phrase has in the past been used in several quite distinct senses (see Section 1.5).
7. Avoid using the Gaussian anisotropic parameters that are now usually symbolized as and are defined in eq. (2.1.26). These quantities are directly proportional to the recommended , the ratio being 8 .
8. Avoid using ADPs that do not represent matrix elements. In some early references and computer programs, cross-terms were sometimes doubled in magnitude, being represented, for example, as instead of , for programming convenience. This was possible because the matrix representing the ADP is symmetric, with only six independent terms. This practice is not found in modern crystallographic software.
9. Published values of should always be accompanied by their standard uncertainties. The ratio of the minimum to the maximum eigenvalues of the corresponding anisotropic displacement tensors should also be published, either in the primary publication itself or in the secondary (deposition) publication.

Authors of crystallographic software and crystallographers who maintain their own software are encouraged to introduce the minor modifications that are required for the implementation of these Recommendations.

Next: References Up: ATOMIC DISPLACEMENT PARAMETER NOMENCLATURE Previous: 3. Beyond the Gaussian Approximation