---

Prog. Theor. Phys. Vol. 48 No. 3 (1972) pp. 709-723

Dynamical Theory of Quantum Solids

— Phonon Dispersion Relation —

Hiroshi Namaizawa

Institute of Physics, College of General Education, University of Tokyo, Megro, Tokyo

(Received March 3, 1972)

Abstract:

A unified dynamical theory of quantum solids is formulated in the self-consistent reaction matrix approach. The phonon dispersion relation is derived in a determinantal form by linearizing the coupled equation of motion for operators bilinear in creation and annihilation operators of localized single-particle states. Comparison with the traditional lattice dynamics is given.

URL : http://ptp.ipap.jp/link?PTP/48/709/
DOI : 10.1143/PTP.48.709

[ Full Text PDF : FREE ACCESS (1296K) ] Citation: Plain Text BiBTeX EndNote(RIS) RSS

---

References:

1. M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, London and New York, 1954).
2. R. A. Guyer, in Solid State Physics, edited by F. Seitz, D. Turnbull and H. Ehrenreich (Academic Press, New York, 1969), Vol. 23, p. 413.
3. T. Nagamiya, Proc. Phys. Math. Soc. Jpn. 22 (1940), 492.
   M. Toda, Proc. Phys. Math. Soc. Jpn. 22 (1940), 503.
   L. H. Nosanow and G. L. Shaw, Phys. Rev. 128 (1962), 546[APS].
4. L. H. Nosanow, Phys. Rev. 146 (1966), 120[APS].
   J. H. Hetherington, W. J. Mullin and L. H. Nosanow, Phys. Rev. 154 (1967), 175[APS].
   W. J. Mullin, L. H. Nosanow and P. M. Steinback, Phys. Rev. 188 (1969), 410[APS].
5. K. A. Brueckner and J. Frohberg, Prog. Theor. Phys. Suppl. Extra Number (1965), 383[PTP].
6. R. Jastrow, Phys. Rev. 98 (1955), 1479[APS].
   F. Iwamoto and M. Yamada, Prog. Theor. Phys. 17 (1957), 543[PTP]; ibid. 18 (1957), 345[PTP].
   F. Wu and E. Feenberg, Phys. Rev. 122 (1961), 739[APS].
7. F. Iwamoto and H. Namaizawa, Prog. Theor. Phys. Suppl. Nos. 37 & 38 (1966), 234[PTP]; Prog. Theor. Phys. 45 (1971), 682[PTP].
   Hereafter referred to as I and II, respectively.
8. R. A. Guyer, Solid State Commun. 7 (1969), 315[CrossRef].
   H. Horner, Phys. Rev. A 1 (1970), 1712[APS]; ibid. 1 (1970), 1722[APS].
9. C. Ebner and C. C. Sung, Phys. Rev. A 4 (1971), 269[APS].
10. K. A. Brueckner and C. A. Levinson, Phys. Rev. 97 (1955), 1344[APS].
11. As a general review of the theory of phonons in quantum solieds, see N. R. Werthamer, Am. J. Phys. 37 (1969), 763.
12. W. Brenig, Z. Phys. 171 (1963), 60.
13. D. R. Fredkin and N. R. Werthamer, Phys. Rev. 138 (1965), A1527[APS].
14. D. J. Hooton, Phil. Mag. 3 (1958), 49.
    N. Boccara and G. Sarma, Physics 1 (1965), 219.
    T. R. Koehler, Phys. Rev. Lett. 17 (1966), 89[APS]; ibid. 18 (1967), 654[APS]; Phys. Rev. 165 (1968), 942[APS].
15. J. Ranninger, Phys. Rev. 140 (1965), A2031[APS].
    P. F. Chouquard, The Anharmonic Crystal (W. A. Benjamin, Inc., New York, 1967).
    N. R. Werthamer, Phys. Rev. B 1 (1970), 572[APS].
    S. Takeno, Prog. Theor. Phys. Suppl. No. 45 (1970), 137[PTP].
16. There are diagramatic approaches also. See for example, H. Horner, Z. Phys. 205 (1967), 72; ibid. 242 (1971), 432.
17. See for example, N. S. Gillis, N. R. Werthamer and T. R. Koehler, Phys. Rev. 165 (1968), 951[APS].
    N. S. Gillis, T. R. Koehler and N. R. Werthamer, Phys. Rev. 175 (1968), 1110[APS].
    V. V. Goldman, G. K. Horton and M. L. Klein, Phys. Rev. Lett. 24 (1970), 1424[APS].
    H. R. Glyde and R. A. Cowley, Solid State Commun. 8 (1970), 923[CrossRef].
    H. Horner, Phys. Rev. Lett. 25 (1970), 147[APS]. Numerical computations are done also in the single-particle approach. See for example, F. W. de Wette, L. H. Nosanow and N. R. Werthamer, Phys. Rev. 162 (1967), 824[APS].
18. See Ref. 4), 5), 11) and 17).
19. Summary Talk by N. R. Werthamer at Banff Conference on Quantum Crystals, Banff, 1971.
20. J. Goldstone, Nuovo Cim. 19 (1961), 154.
21. K. Sawada, Phys. Rev. 106 (1957), 372[APS].
22. A similar sum rule in the case of interatomic potential without hard-core is first derived by Fredkin and Werthamer. See Ref. 13).

Citing Article(s) :

1. Progress of Theoretical Physics Vol. 49 No. 5 (1973) pp. 1779-1781 :

   Ground-State Energy and Phonon Spectra of bcc Solid Helium

   Shigeo Homma, Katsuhiko Nagai and Hiroshi Namaizawa

2. Progress of Theoretical Physics Vol. 51 No. 4 (1974) pp. 959-972 :

   Theory of Quantum Crystals

   Yasunari Kurihara, Yoshihiro Kuroda and Norikazu Ishimura

3. Progress of Theoretical Physics Vol. 61 No. 5 (1979) pp. 1273-1286 :

   Theory of Collective Modes in Quantum Crystals of Spinless Fermions

   Hisao Jichu and Yosuke Nagaoka

---

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

Copyright © Progress of Theoretical Physics 2013 All rights reserved.