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Manuscript Title: Phonon focussing patterns: Monte Carlo simulation of the motion of ballistic phonon beams in cubic crystals.
Authors: W.M. Gancza, T. Paszkiewicz
Program title: MCFOC
Catalogue identifier: ADAQ_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 85(1995)423
Programming language: Pascal, Assembler.
Computer: IBM PC/AT (286/386/486).
Operating system: MS-DOS 5.0.
RAM: 0.7 K words
Word size: 16
Keywords: Solid state physics, Crystal structure, Heat pulses, Ballistic transport, Phonon focussing, Cubic crystals, Monte Carlo computer experiments.
Classification: 7.8.

Nature of problem:
During the last decade the spectroscopy of phonon images of crystalline solids has been developed. This spectroscopy provides information about phonon surfaces of constant energy which is not available from the familiar acoustic and neutron scattering measurements. From these phonon images one can infer information about the local geometrical properties of surfaces of constant energy of phonons [1] or knowledge to assist in the testing of the existing models of vibrational dynamics of crystalline lattices (cf. [2]). Phonon images of crystals are highly inhomogeneous and as a rule they exhibit regions ("lines") of very high (theoretically infinite but integrable) density of energy and quasimomentum. These lines are called the caustic lines, and spatial directions corresponding to them, are termed the caustics. The underlying physical phenomenon of strong spatial anistropy of phonon beams is called phonon focussing. On the other hand studies of many properties of crystalline solids are performed with beams of phonons. This makes very important the detailed knowledge of the anisotropy of densities of these beams, the focussing directions and the phonon patterns. Experimenters working with beams of phonons need a fast and reliable tool for obtaining phonon patterns. It is highly desirable to supplement such programs with measuring facilities which are very useful in experiment planning. Our program contains such a measurement module. One can derive closed analytical formulae for the densities of fluxes of energy and of the components of quasimomentum [1]. However, the mentioned singularities make calculations based on these formulae a rather complicated task [3,4].

Solution method:
The simplest way to overcome this difficulty is to work with a finite number of phonons and with a detector of finite area. Both of these requirements are fulfilled in real and computer experiments. In this program we use a Monte Carlo simulation method to generate a homogeneous angular distribution of directions of the wave vectors of long wave-length acoustic phonons with an isotropic and finite density in reciprocal space. In this way integrals (defining the densities of energy and quasimomentum currents emanating from an extended source [1,3,4]) of locally infinite functions change to the summation over a finite set of finite numbers.

[1] Cz. Jasiukiewicz, D. Lehmann, T. Paszkiewicz, Z. Phys. B-Condensed Matter 84(1991)73.
[2] S.E. Hebboul, J.P. Wolfe, Z. Phys. B-Condensed Matter 74(1989)35.
[3] Cz. Jasiukiewicz, T. Paszkiewicz, Acta Phys. Polon. 84(1993)459.
[4] Cz. Jasiukiewicz, T. Paszkiewicz, D. Lehmann, Phonon focussing patterns: calculation of the response of finite area detectors to pulsed ballistic beams of dispersive and dispersionless phonons, Z. Phys. B Condensed Matter, in print.