COMBINED STANDARD UNCERTAINTIES FOR INTERATOMIC DISTANCES IN ONE-DIMENSIONALLY MODULATED CRYSTALS

A method is proposed that allows a variance estimation for interatomic distances in the case of a one-dimensional incommensurate (or commens urate) displacively modulated structure to be derived. This method tak es into account the covariance terms in the variance-covariance matrix Sigma of the atomic pair, resulting from true linear relations betwee n some positional parameters. These linear relations, arising in the c ase of symmetry-related atoms or in the case of special positions in c rystals of high symmetry (trigonal, tetragonal or hexagonal systems), are carefully scrutinized. The results of these linear relations on th e Sigma matrix are specified for any order of the Fourier expansion of the modulated displacements and some examples of matrices are given. A general formula written in matrix form is then derived for the bond- length variance. This method is then exemplified by different typical examples, showing the usefulness of such a calculation for the compari son of different results or for the discussion of the significance of some interatomic distance variations.