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New exact results for the defective square lattice

Published online by Cambridge University Press:  24 October 2008

G. Ronca
G. Ronca
Affiliation:
Istituto di Chimica, Politecnico di Milano, 20133 Milano
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Extract

Since the publication of the fundamental papers by Lifshitz (1, 2) and Montroll and Potts (3, 4) many authors have investigated the effect of an isotopic impurity on the lattice vibrations of a harmonic crystal at zero temperature. A fairly broad knowledge is now available on scattering amplitudes, localized modes and resonance modes (6, 7). Nevertheless, as pointed out by Maradudin and Montroll (see (7), p. 430), a closed form solution to the problem has been found only for the one-dimensional crystal, the work done on two and three-dimensional crystals being predominantly numerical. Unfortunately the one-dimensional crystal, as an approximation for a real crystal is an oversimplified model, incapable as it is of exhibiting resonance modes. To the author's knowledge the most significant exact result concerning the classical behaviour at zero temperature of crystals having a dimensionality higher than one is the connexion, calculated by Mahanty et al. (5) between localized mode frequency and impurity mass for the case of a square lattice undergoing planar vibrations.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 80 , Issue 2 , September 1976 , pp. 365 - 381
Copyright
Copyright © Cambridge Philosophical Society 1976

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References

REFERENCES

(1)Lifshitz, I. M.J. Phys. U.S.S.R. 7 (1943), 215.Google Scholar
(2)Lifshitz, I. M.J. Phys. U.S.S.R. 8 (1944), 89.Google Scholar
(3)Montroll, E. W. and Potts, R. B.Phys. Rev. 100 (1955), 525.CrossRefGoogle Scholar
(4)Montroll, E. W. and Potts, R. B.Phys. Rev. 102 (1956), 72.CrossRefGoogle Scholar
(5)Mahanty, J., Maradudin, A. A. and Weiss, G. H.Progr. Theor. Phys. 24 (1960), 648.CrossRefGoogle Scholar
(6)Takeno, S.Progr. Theor. Phys. 29 (1962), 191.Google Scholar
(7)Maradudin, A. A. et al. Theory of Lattice Dynamics in the Harmonic Approximation. Solid State Physics, Suppl. no. 3, chap. 8 (New York, Academic Press, 1971).Google Scholar
(8)Ronca, G. and Allegra, G.Contraction effects in ideal networks of flexible chains. J. Chem. Phys. 63 (1975), 4104.CrossRefGoogle Scholar
(9)Kittel, C.Introduction to Solid State Physics, pp. 158–159 (New York, Wiley, 1966).Google Scholar
(10)Abramowitz, M. and Stegun, I. A.Handbook of Mathematical Functions, pp. 600–601 (New York, Dover, 1966).Google Scholar
(11)Carrier, G. F., Krook, M. and Pearson, C. E.Functions of a Complex Variable, pp. 272–275 (New York, McGraw-Hill, 1966).Google Scholar