Section 1.9.2.1. Tensorial properties of (quasi)moments and cumulants

By separating the powers of Q and u, one obtains in equations (1.9.2.4), (1.9.2.5) and (1.9.2.7) the higher-order displacement tensors in the form of moments, cumulants or quasimoments, which we shall denote in a general way as ; note that b^ij is identical to β^ij. They transform on a change of the direct-lattice base according to

The higher-order displacement tensors are fully symmetric with respect to the interchange of any of their indices; in the nomenclature of Jahn (1949), their tensor symmetry thus is [b^N]. The number of independent tensor coefficients depends on the site symmetry of the atom and is tabulated in Sirotin (1960) as well as in Tables 1.9.3.1–1.9.3.6. For triclinic site symmetry, the numbers of independent tensor coefficients are 1, 3, 6, 10, 15, 21 and 28 for the zeroth to sixth order. Symmetry may further reduce the number of independent coefficients, as discussed in Section 1.9.3.

In many least-squares programs for structure refinement, the atomic displacement parameters are used in a dimensionless form [as given in (1.9.1.4) for the harmonic case]. These dimensionless quantities may be transformed according to(no summation) into quantities of units Å^N (or pm^N); aⁱ etc. are reciprocal-lattice vectors. Nowadays, the published structural results usually quote U^ij for the second-order terms; it would be good practice to publish only dimensioned atomic displacements for the higher-order terms as well.