DOI: https://doi.org/10.30970/jps.01.106
INVESTIGATION OF LATTICE VIBRATION BY MEANS OF CLUSTERS WITH CYCLIC BOUNDARY CONDITIONS
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Journal of Physical Studies 1(1), 106–109 (1996)
DOI: https://doi.org/10.30970/jps.01.106 INVESTIGATION OF LATTICE VIBRATION BY MEANS OF CLUSTERS WITH CYCLIC BOUNDARY CONDITIONS |
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I. I. Talyansky
Ivan Franko Lviv State University, Chair of Theoretical Physics
12 Drahomanov Str., Lviv UA-290005, Ukraine
The cluster method is often applied for the calculation of atoms vibrations in crystals. However, this method has a weak point which consists in the distortion of vibration frequencies caused by the boundaries of the cluster.
In this paper the cyclic boundary conditions are applied for the elimination of the abovementioned difficulty. They consist in locking the atoms which are at the boundaries of the cluster at opposite sides. In this way it is possible to eliminate the influence of cluster boundaries.
One–dimensional clusters consisting of two and four atoms have been investigated. They are considered for two cases: atoms of the same kind (a model of covalent crystal) and atoms of two different kinds with the masses $m$ and $M$ (a model of ionic crystal). The nearest neighbors approximation has been used in both cases. The quasi–elastic force coefficients $k_{1}$ and $k_2$ were expected to differ for the atoms in one and two neighboring cells. Limit values of the frequencies of acoustic and optical bands obtained coincided exactly with the well known results for a chain of atoms of infinite size. This makes it possible to obtain the correct value of frequency of local oscillation of the impurity atom present in the crystal relatively at the edge of the band. It coincides with the corresponding value for the infinite chain with the accuracy to $(m_{1}/M)^{2}$ where $m_{1}$ is the mass of impurity.
The cyclic boundary conditions cluster method has been applied also to a three–dimensional simple cubic lattice. In this case the interaction with three coordination groups was taken into account. The limit frequencies for the obtained longitudinal and transverse acoustic waves coincide exactly with the corresponding value for the infinite crystal.