Abstract

The standard approach applies the Gaussian distribution function to estimate atomic displacements due to thermal vibrations in periodic and aperiodic systems, which is used in a form of the Debye–Waller factor during the structure refinement. Acoustic phonons provide the largest contribution to the Gaussian correction although the character of other phonon modes remains relatively unclear. In this paper, we provide an alternative description of localized and dispersionless phonons based on an assumption of the harmonic displacement distribution function, which was recently proposed for model quasicrystals, and apply this approach for a decagonal Al-Cu-Rh quasicrystal that was previously studied by Kuczera et al. in 2012. We used the same X-ray diffraction data and the statistical method of structural analysis of the aperiodic systems. The correction function for phonons takes the form of a Bessel function instead of a conventional (Gaussian) Debye–Waller factor. This allowed us to achieve R-factor of 7.2% compared to 7.9% reported in the original paper. A significant improvement of the calculated atomic composition towards experimentally obtained and minor positional changes is also reported compared to the original paper. The results show the usefulness of investigating different corrective terms for diffraction data during a structure refinement.

Highlights

  • Decagonal (d-) Al-Cu-Rh quasicrystals were originally investigated by Kuczera et al and a comparative structural study with d-Al-Cu-Ir and d-Al-Cu-Co systems was performed [1]

  • Cell (AUC) concept [3,4,5], which is a method that involves the modeling of the aperiodic systems, which is the main alternative to the multidimensional one

  • We focus on the novel approach to phonons, which is accessible within the statistical method

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Summary

Introduction

Decagonal (d-) Al-Cu-Rh quasicrystals were originally investigated by Kuczera et al and a comparative structural study with d-Al-Cu-Ir and d-Al-Cu-Co systems was performed [1]. Cell (AUC) concept [3,4,5], which is a method that involves the modeling of the aperiodic systems (called the statistical method in the parts of the paper), which is the main alternative to the multidimensional one. The same method is used in this paper. We focus on the novel approach to phonons, which is accessible within the statistical method. In reference [6], we theoretically speculated about possible benefits from replacing the standard Debye–Waller (D–W) factor with different functions, such as Bessel functions (ordinary or spherical). In references [6,8], we showed the usefulness of Bessel-like functions (ordinary and spherical) as good correction functions for phonons, which suggests the crucial role of weak reflections in the refinement of quasicrystals.

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