Section 1.9.2.2. Contraction, expansion and invariants of atomic displacement tensors

Anisotropic or higher-order atomic displacement tensors may contain a wealth of information. However, this information content is not always worth publishing in full, either because the physical meaning is not of importance or the significance is only marginal. Quantities of higher significance or better clarity are obtained by an operation known as tensor contraction. Likewise, lower-order terms may be expanded to higher order to impose certain (chemically implied) symmetries on the displacement tensors or to provide initial parameters for least-squares refinements. A contraction is obtained by multiplying the contravariant tensor components (referring to the real-space basis vectors) with the covariant components of the real-space metric tensor g_ij; for further details on tensor contraction, see Section 1.1.3.3.3 . In the general case of atomic displacement tensors of (even) rank N, one obtains is called the trace of a tensor of rank N and is a scalar invariant; it is given in units of length^N and provides an easily interpretable quantity: In the case of , a positive sign indicates that the corresponding (real-space) p.d.f. is peaked, a negative sign indicates flatness of the p.d.f. The larger , the stronger the deviation from a Gaussian p.d.f. provoked by the atomic displacements of order N. The frequently quoted isotropic equivalent U value U_eq is also obtained by this contraction process. Noting that U^ij may be expressed in terms of b^ij (= β^ij) according to (1.9.2.9) and that the trace of the matrix U is given as , one obtainsNote that in all non-orthogonal bases, . In older literature, the isotropic equivalent displacement parameter is often quoted as B_eq, which is related to U_eq through the identity . The use of B_eq is now discouraged (Trueblood et al., 1996). Higher atomic displacement tensors of odd rank N may be reduced to simple vectors v by the following contraction:where v¹ is the 23 trace etc. ^Nvⁱ is sometimes called a vector invariant, as it can be uniquely assigned to the tensor in question (Pach & Frey, 1964) and its units are length^{N − 1}. The vector v is oriented along the line of maximum projected asymmetry for a given atom and vanishes for atoms with positional parameters fixed by symmetry; Johnson (1970) has named a vector closely related to ³v the vector of skew divergence. The calculation of v is useful as it gives the direction of the largest antisymmetric displacements contained in odd-rank higher-order thermal-motion tensors.

Atomic displacement tensors may also be partially contracted or expanded; rules for these operations are found in Kuhs (1992).