Influence of phason dynamics on atomic Debye-Waller factors of incommensurate modulated structures and quasicrystals

Abstract

We analyze the influence of phason dynamics on atomic Debye-Waller factors in incommensurate structures. It is stressed that in the harmonic approximation a rigorous derivation leads to atomic Debye-Waller factors of the usual form exp[−( 1 2 )Σ Q α Q β<δ u αδ u β>] , where both the scattering vector Q and the atomic displacements δ u are defined in the real three-dimensional space. Due to the lack of periodicity of the structure, this factor is also in principle aperiodic, if its value for different atoms is considered. Phason thermal factors of the type exp(−( 1 2 ) n 2<δφ 2>) , proposed for incommensurate modulated structures, or its analogous for quasicrystals, exp(−BQ ⊥ 2), are shown to be incompatible with the previous general result. In the case of an incommensurately modulated structure with a one-dimensional modulation, the correct form of the contributions of both amplitudon and phason modes to atomic thermal parameters is derived. Within the sinusoidal regime, the phasons cause the build-up of a second-harmonic modulation of atomic thermal factors, which can be directly related with the static structural distortion. The introduction of these particular modulated atomic thermal tensors in a quantitative structure analysis would imply a minimum increase in the number of refinement parameters. Existing experimental studies concerning Debye-Waller factors in incommensurate systems are shown to be inconclusive.