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- W1978108508 abstract "A method for determining the frequency distribution of crystal lattices on the basis of Born—von Kárman theory is proposed. This is a full account of the previous work, and a generalization to the case of three-dimensional crystal lattices is described. The procedure for obtaining the frequency distribution involves the following stages: (1) One must solve the secular equation for determining the squared circular frequency ω2 along the edges of an appropriate part of the first Brillouin zone in the reciprocal lattice in the same manner as did Houston. (2) One must find, in addition, a few important contours of constant frequency on the surface of the zone as accurately as possible. (3) On the basis of (1) and (2), one can calculate or trace the desired contours of constant frequency on the surfaces by the proposed interpolation scheme [Eq. (1) or Eq. (2)]. (4) The contours of constant frequency in the inner planes can be calculated or traced by the proposed interpolation scheme described in Sec. IV. (5) The volume V(ω2) enclosed by the surface of constant frequency means the number of modes of vibrations whose frequencies are less than ω2, and its magnitude may be computed by the proposed numerical or mechanical mean of integration described in Sec. V. (6) V(ω2) thus calculated is fitted in an analytical form, and at this time the results of van Hove's analysis for singularities may be taken into account to obtain the actual shape of frequency distribution in the vicinity of singularities. A differentiation of V(ω2) with respect to ω2 yields the frequency distribution. Applications to the two-dimensional and three-dimensional simple cubic lattices are made, and an excellent agreement with the exact frequency distributions calculated by Montroll and Rosenstock is obtained. The present method is thought to be far easier to handle than the conventional ones." @default.
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- W1978108508 date "1962-10-15" @default.
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- W1978108508 title "Frequency Counting in Lattice Dynamics" @default.
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- W1978108508 doi "https://doi.org/10.1063/1.1733355" @default.
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