The vibrational density of states of three-dimensional quasicrystals

Abstract

We have computed the vibrational densities of states (DOS) both of several ideal approximants of icosahedral quasicrystals, up to 13/8, and of an actual Al70Pd21Mn9 icosahedral alloy, in the framework of the harmonic approximation. The dynamic matrix has been constructed for a spring potential in the cluster approximation. In order to compute the DOS we have used the continued-fraction expansion of the Green's function and the coefficients of this expansion have been obtained with the recursive technique. The DOS of the 13/8 approximant is in agreement with the one already obtained in the literature, by using the commensurate approximation and the direct diagonalization. This strongly suggests that, provided the size of the cluster is large enough, the two methods are equivalent. On the other hand, recursive technique algorithms are much more convenient from the point of view of computing time. The DOS computed for the actual case of AlPdMn shows a satisfactory qualitative agreement with the neutron-weighted DOS, measured for the same compound. This is an interesting result, since it indicates that, in spite of the simple assumption adopted for the potential, the model describes quite well the main features of the vibrational spectrum of this quasicrystal. Further improvements in this direction can then be made.