HOME    |    BROWSE    |    SEARCH    |    SUBSCRIPTION   |    SUBMISSION    |    REGISTRATION

Quick Search:
Author: Title/Abstract: Vol./No: Page:
PTP Online Top > Volume 28 (1962) > Number 1
| Next Article >

Prog. Theor. Phys. Vol. 28 No. 1 (1962) pp. 33-79

[ Full Text PDF : FREE ACCESS (3287K) ]

Energy Spectrum of Lattices with Defects. II

— Frequency Spectrum of Lattice Vibration —

Shozo Takeno

Department of General Education, Osaka University of Engineering, Osaka

(Received January 17, 1962)

Abstract:

A systematic investigation of the vibration of a lattice containing isotopic impurities of a single kind is made in terms of the Green's function, and a general expression for the squared-frequency spectrum is obtained from a solution of difference equations, which can be applied to any number of dimensions. For a fixed spacial configuration of impurities, the results yield the spectrum of the main continuum and of the localized modes. When the number of isotopes is few, the frequencies of the localized modes, the criterion for their appearance, and the distribution of in-band frequencies are studied in detail, and the results are applied to a simple cubic lattice. The localized modes due to a substitutional impurity atom are also studied. In the case of a random distribution of isotopes, the spectrum is obtained in a closed form with use of a graphical method for the perturbation calculations. It is shown that (1) the perturbation due to lighter isotopes causes an impurity band or a tail of the spectrum into the forbidden region, and (2) when heavier isotopes are present, the spectrum in the low frequency part is increased, while in the high frequency region it adds a tail. The criteria for the appearance of the impurity band and for its merging into the main continuum are obtained. For a simple cubic lattice, the construction of the spectrum is performed.


URL : http://ptp.ipap.jp/link?PTP/28/33/
DOI : 10.1143/PTP.28.33

[ Full Text PDF : FREE ACCESS (3287K) ] Citation:

References:

  1. E. W. Montroll and R. B. Potts, Phys. Rev. 100 (1955), 525[APS]; ibid. 102 (1956), 12[APS].
  2. P. Mazur, E. W. Montroll, and R. B. Potts, J. Wash. Acad. Sci. 46 (1956), 2.
  3. A. A. Maradudin, P. Mazur, E. W. Montroll, and G. H. Weiss, Rev. Mod. Phys. 30 (1958), 175[APS].
  4. R. E. Wallis and A. A. Maradudin, Prog. Theor. Phys. 24 (1960), 1055[PTP].
  5. I. M. Lifshitz, Nuovo Cim., Supple. Al, 3 (1956), 716.
  6. J. Hori and T. Asahi, Prog. Theor. Phys. 17 (1957), 523[PTP].
  7. E. Teramoto and S. Takeno, Prog. Theor. Phys. 24 (1960), 1349[PTP].
  8. F. J. Dyson, Phys. Rev. 92 (1953), 1331[APS].
  9. H. Schmidt, Phys. Rev. 97 (1955), 425[APS].
  10. C. Domb, A. A. Maradudin, and G. H. Weiss, Phys. Rev. 115 (1959), 18[APS]; ibid. 115 (1959), 24[APS].
  11. J. Hori, Prog. Theor. Phys. 23 (1960), 475[PTP].
  12. Y. Fukuda, private communication.
  13. P. Dean, Proc. Roy. Soc. A 254 (1960), 507; ibid. 260 (1961), 261.
  14. P. A. Flinn, A. A. Maradudin, and G. H. Weiss, Sci. Paper Westinghouse Research Lab. 024-G000-P10.
  15. J. S. Faulkner and J. Korringa, Phys. Rev. 122 (1961), 390[APS].
  16. J. S. Langer, J. Math. Phys. 2 (1961), 584[CrossRef].
  17. P. Dean and J. L. Martin, Proc. Roy. Soc. A 259 (1961), 409.
  18. I. M. Lifshitz and G. I. Stepanova, J. E. T. P. 30 (1956), 938.
  19. S. Takeno, Prog. Theor. Phys. 25 (1961), 102[PTP].
  20. T. Matsubara and Y. Toyozawa, private communication, to be published in this issue.
  21. G. F. Koster and J. C. Slater, Phys. Rev. 95 (1954), 1167[APS]; ibid. 96 (1954), 1206[APS].
  22. G. F. Koster, Phys. Rev. 95 (1954), 1436[APS].
  23. E. W. Montroll, J. Chem. Phys. 10 (1942), 218[CrossRef]; ibid. 11 (1943), 481[CrossRef].
  24. E. W. Montroll, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley, 1955), p. 209.
  25. A. Erdelyi, W. Magnus, F. Oberhettinger and F. Tricomi, Table of Integral Transforms I (McGraw-Hill, New York, 1954).

Citing Article(s) :

  1. Journal of the Physical Society of Japan 58 (1989) pp. 3899-3909 :
    Exact Anharmonic-Localized-Mode Solutions to the d-Dimentional Discrete Nonlinear Klein-Gordon Equation
    Shozo Takeno
  2. Progress of Theoretical Physics Vol. 28 No. 4 (1962) pp. 631-662 :
    Energy Spectrum of Lattices with Defects. III
    Shozo Takeno
  3. Progress of Theoretical Physics Vol. 29 No. 2 (1963) pp. 191-205 :
    Resonance Scattering of Lattice Waves by Isotopes
    Shozo Takeno
  4. Progress of Theoretical Physics Vol. 29 No. 2 (1963) pp. 328-330 :
    Frequency Spectrum of a Lattice with Heavy Isotopes
    Shozo Takeno
  5. Progress of Theoretical Physics Vol. 30 No. 1 (1963) pp. 1-16 :
    Surface Vibration of a Diatomic Simple Cubic Lattice
    Shozo Takeno
  6. Progress of Theoretical Physics Vol. 30 No. 4 (1963) pp. 565-566 :
    Impurity States in a Heisenberg Ferromagnet
    Shozo Takeno
  7. Progress of Theoretical Physics Vol. 30 No. 6 (1963) pp. 731-742 :
    Spin-Wave Impurity Levels in a Heisenberg Ferromagnet
    Shozo Takeno
  8. Progress of Theoretical Physics Vol. 33 No. 3 (1965) pp. 363-379 :
    A High-Frequency Resonant Mode in Lattice Vibrations
    Shozo Takeno
  9. Progress of Theoretical Physics Vol. 33 No. 4 (1965) pp. 614-623 :
    Localized Modes of the Extended Defect
    Itaru Kobori
  10. Progress of Theoretical Physics Vol. 34 No. 4 (1965) pp. 534-538 :
    On the Asymptotic Green's Function in an n-Dimensional Lattice Model
    P. M. Mathews and M. S. Seshadri
  11. Progress of Theoretical Physics Vol. 38 No. 5 (1967) pp. 995-1022 :
    Theory of Impurity-Induced Infrared Absorption in Alkali Halide Crystals
    Shozo Takeno
  12. Progress of Theoretical Physics Vol. 40 No. 5 (1968) pp. 942-957 :
    A Solution of a Dyson Equation in Many-Impurity Problems in Solids
    Shozo Takeno

[PTP HOME] [BROWSE] [SEARCH] [SUBSCRIPTION] [SUBMISSION] [REGISTRATION]

Copyright © Progress of Theoretical Physics 2013 All rights reserved.