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Prog. Theor. Phys. Vol. 17 No. 4 (1957) pp. 523-542

[ Full Text PDF : FREE ACCESS (1396K) ]

On the Vibration of Disordered Linear Lattice

Jun-ichi Hori and Takashi Asahi

Department of Physics, Hokkaido University, Sapporo

(Received November 26, 1956)

Abstract:

The method of transfer matrix which was recently developed by Kerner to treat the problem of electronic band structure of mixed linear was applied to the problem of vibration of atomic linear chain. It was shown that this method affords to the problem of calculating the eigenfrequency-distribution of several kinds o lattices, including disordered ones.
The well-known results for regular monatomic and diatomic lattices, and the results which were obtained by Montroll and Potts for linear lattices containing a few impurity atoms, were rederived by the matrix method in a much simpler way. The eigenvalue equations for the lattices containing several impurities at arbitrarily given positions were then derived. In cases where the force-constant is not altered by impurities the formulas become so simple that we can obtain the eigenfrequencies without laborious calculations. It is also possible to derive in such cases a simple approximate eigenvalue equation for lattices containing impurities distributed completely at random. This equation indicates that the mode of change in the distribution of eigenfrequencies is qualitatively the same as in the case of one or two impurities.
Monatomic lattice with a single hole and diatomic lattice with a single were treated as simple examples. An attempt to extend the matrix method to the case in which the next-nearest interaction between atoms is taken into account was also mentioned.


URL : http://ptp.ipap.jp/link?PTP/17/523/
DOI : 10.1143/PTP.17.523

[ Full Text PDF : FREE ACCESS (1396K) ] Citation:

References:

  1. E. H. Kerner, Phys. Rev. 95 (1954), 687[APS]; Proc. Phys. Soc. London A 69 (1956), 234.
  2. E. W. Montroll and R. B. Potts, Phys. Rev. 100 (1955), 525[APS].
  3. F. J. Dyson, Phys. Rev. 92 (1953), 1331[APS].
  4. R. Bellman, Phys. Rev. 101 (1956), 19[APS].
  5. M. Lifschitz, Nuovo Cim. Suppl. 3 (1956), 716.

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 18 No. 4 (1957) pp. 367-374 :
    On the Vibration of Disordered Linear Lattice, II
    Jun-ichi Hori
  2. Progress of Theoretical Physics Vol. 20 No. 3 (1958) pp. 327-355 :
    Effects of Defects on Lattice Vibrations
    Kaoru Yamahuzi and Tomoyasu Tanaka
  3. Progress of Theoretical Physics Vol. 23 No. 3 (1960) pp. 475-489 :
    On the Vibration of Disordered Linear Lattice. III
    Jun-ichi Hori
  4. Progress of Theoretical Physics Vol. 24 No. 6 (1960) pp. 1296-1306 :
    Lattice Vibration and Random Walk Problems
    Ei Teramoto
  5. Progress of Theoretical Physics Vol. 24 No. 6 (1960) pp. 1349-1368 :
    Time Dependent Problems of the Localized Lattice Vibration
    Ei Teramoto and Shozo Takeno
  6. Progress of Theoretical Physics Vol. 27 No. 3 (1962) pp. 571-588 :
    Statistical Dynamical Behaviors of a One-Dimensional Lattice with an Isotopic Impurity
    Shohei Kashiwamura
  7. Progress of Theoretical Physics Vol. 27 No. 4 (1962) pp. 811-836 :
    A New Approach to Green's Function of a Particle in One Dimension
    Hirotsugu Matsuda
  8. Progress of Theoretical Physics Vol. 28 No. 1 (1962) pp. 33-79 :
    Energy Spectrum of Lattices with Defects. II
    Shozo Takeno
  9. Progress of Theoretical Physics Vol. 32 No. 2 (1964) pp. 183-189 :
    Structure of the Spectra of Disordered Systems. I
    Jun-ichi Hori and Hirotsugu Matsuda
  10. Progress of Theoretical Physics Vol. 33 No. 4 (1965) pp. 624-637 :
    Scattering Matrix and Surface Modes. II
    Masahisa Fukushima
  11. Progress of Theoretical Physics Vol. 36 No. 3 (1966) pp. 488-503 :
    A Method of Calculating the Energy Spectrum of an Electron in the Field of Many Potentials
    Tōru Hirota
  12. Progress of Theoretical Physics Vol. 48 No. 3 (1972) pp. 808-825 :
    Application of the Transfer Matrix Method to the Hückel Molecular Orbital Analysis of Complex Conjugated Molecules
    Yuji Jido, Takashi Inagaki and Hideo Fukutome
  13. Progress of Theoretical Physics Supplement No.53 (1973) pp. 139-201 :
    Density of States in Aperiodic Solids
    Hiroshi Hasegawa

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