Neutron scattering by phonons in quasi-crystals

Abstract

Using the spectral moments method, the authors studied the inelastic scattering by phonons in very long Fibonacci chains. The results show that the pseudo-acoustic dispersion curves can be associated with Bragg peaks. The intensity of the acoustic phonon lines is proportional to the intensity of the corresponding Bragg peak. A study of disordered Fibonacci chains shows that the intensity of the phonon lines decreases strongly with increasing disorder. These results could explain the difficulties encountered in measurements of the acoustic modes in quasi-crystals.

Spectral gaps and mode localization in Fibonacci chains of metal nanoparticles

“…These spectral features, including the appearance of more Miller indices than the dimensionality of the physical system, induce fascinating physical properties in Fibonacci structures, which have been widely investigated in different physical contexts [3,4]. In particular, It has been shown that in Fibonacci structures the transport of wave-like excitations (such as electronic [5], optical [6], mechanical [7], spin waves [8]) as well as "mixed-waves" (polariton waves [8]) exhibits unique properties described by multi-fractal energy spectra [9], critical wavefunctions with power-law scaling [10] and anomalous diffusion [11].…”

Computation of electromagnetic waves diffraction by spectral moments method

“…with (17) and (18) The space discretization of (17) and (18), using the finitedifference scheme, leads to the following motion equation: (19) which can be written in matrix form (20) The vector is given by (21) where if (22) and if (23) here represents a current density source. is equivalent to a vector of the canonical base.…”

“…Indeed, in 3-D space, there is one matrix for every bound between site and the six neighboring sites . These matrices can be expressed schematically as with (53) where and arise, respectively, from the discretization of (17) and (18). Spatial derivatives in take into account the heterogeneity of the media.…”

“…In the dynamics of condensed matter, the exact evaluation of the response was developed by Benoit [10], [11] and applied to the study of fractals as the Sierpinski's gasket by Benoit [12], percolating networks by Royer [13], [14], silica aerogels by Rahmani [15], [16], dynamical properties of polymers by Poussigue [17], quasi-crystals by Benoit [18] and Poussigue [19], and to damped systems by Benoit [20]. In solid-state physics, the moments method has been developed to study electronic properties by Gaspard and Cyrot-Lackmann [21], Lambin and Gaspard [22], Turchi [23] and Jurczek [24].…”

Quasicrystals

“…An interesting question is to ask whether there are propagating modes of well-defined wave vectors (plane-wave excitations) in the neighbourhood of these 6-functions. This question has been studied for the case of vibrational [309,310,3141, electronic [311 to 3131, and magnetic [277] two-dimensional Penrose lattices by examining the dynamic structure factor. It has been found that the profile of the dynamic structure factor (frequency of the maximum versus the wave vector) yields a dispersion-like pattern.…”