Spin Waves

  • A. R. Mackintosh
  • H. Bjerrum Møller


In this chapter, we will review the knowledge which has been accumulated over the last few years on spin waves in the rare earth metals, and describe the way in which this information may be interpreted to elucidate the magnetic interactions which give rise to their characteristic magnetically ordered structures. Most of the discussion will be concerned with the hcp heavy rare earths, since the magnetic excitations in the light rare earths, which are complicated by strong crystal field effects, have not yet been extensively studied. Recent measurements on praseodymium will, however, be briefly described. The great majority of the information which has been obtained about the dispersion relations for the spin wave quanta, or magnons, has resulted from inelastic neutron scattering experiments since these allow, in principle and often even in practice, the determination of the magnon energy at any point in the Brillouin zone, as well as providing much valuable additional information about magnon interactions. An important supplementary technique is provided by the absorption of electromagnetic waves, which, however, only determines the magnon energies at certain high symmetry points within the zone. In addition, some information may be obtained from the low-temperature thermodynamic and transport properties, although the number of explicit features of the magnon dispersion relations which can be deduced from them is extremely limited. However, they do provide a useful check on the consistency of the measured or calculated magnon spectrum.


Dispersion Relation Crystal Field Spin Wave Heavy Rare Earth Indirect Exchange 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Liu, S. H., Phys. Rev., 121, 451 (1961).CrossRefGoogle Scholar
  2. 2.
    Ruderman, M. A. and Kittel, C., Phys. Rev., 96, 99 (1954)CrossRefGoogle Scholar
  3. Kasuya, T., Prog. T.eor. Phys. (Kyoto), 16, 45 (1956)CrossRefGoogle Scholar
  4. Yosida, K., Phys. Rev., 106, 893 (1957).CrossRefGoogle Scholar
  5. 3.
    Kaplan, T. A. and Lyons, D. H., Phys. Rev., 129, 2072 (1963)CrossRefGoogle Scholar
  6. Kasuya, T. and Lyons, D. H., J. Phys. Soc. Japan, 21, 287 (1965).Google Scholar
  7. 4.
    Levy, P. M., Solid State Commun., 7, 1813 (1969)CrossRefGoogle Scholar
  8. 5.
    See Chapters 1 and 2.Google Scholar
  9. 6.
    Turov, E. A. and Shavrov, V. G., Fiz. Tverd. Tela, 7, 217 (1965).Google Scholar
  10. English trans.: Soviet Phys. Solid State, 7, 166 (1965).Google Scholar
  11. 7.
    Lindgârd, P. A. J. de Physique, 32, CI-238 (1971)Google Scholar
  12. 8.
    Rhyne, J. J. and Legvold, S., Phys. Rev., 138, A507 (1965).CrossRefGoogle Scholar
  13. 9.
    Koehler, W. C., Child, H. R., Nicklow, R. M., Smith, H. G., Moon, R. M., and Cable, J. W., Phys. Rev. Letters, 24, 16 (1970).CrossRefGoogle Scholar
  14. 10.
    Holstein, T. D. and Primakoff, H., Phys. Rev., 58, 1098 (1940);CrossRefGoogle Scholar
  15. Brooks, M. S. S., Phys. Rev., BI, 2257 (1970)Google Scholar
  16. Goodings, D. A. and Southern, B. W., Can. J. Phys. 49, 1157 (1971).Google Scholar
  17. 11.
    Niira, K., Phys. Rev., 117, 129 (1960).CrossRefGoogle Scholar
  18. 12.
    Brinkman, W. F., J. Appl. Phys., 38, 939 (1967).CrossRefGoogle Scholar
  19. 13.
    Cracknell, A. P., J. Phys. C, 3, S175 (1970).CrossRefGoogle Scholar
  20. 14.
    See e.g. Lomer, W. M. and Low, G. G., in Thermal Neutron Scattering ( P. A. Egelstaff, Ed.). ( Academic Press, London, 1965 ), p. 2.Google Scholar
  21. 15.
    Brooks, M. S. S., Goodings, D. A., and Ralph, H. I., J. Phys. C, 1, 132 (1968).CrossRefGoogle Scholar
  22. 16.
    Trammell, G. T., Phys. Rev., 92, 1387 (1953); see also Lovesey, S. W. and Rimmer, D. E., Rep. Frog. Phys., 32, 333 (1969).Google Scholar
  23. 17.
    Steinsvoll, O., Shirane, G., Nathans, R., Blume, M., Alperin, H. A., and Pickart, S. J., Phys. Rev., 161, 499 (1967).CrossRefGoogle Scholar
  24. 18.
    Lindgârd, P. A., Kowalska, A., and Laut, P., J. Phys. Chem. Solids, 28, 1357 (1967).CrossRefGoogle Scholar
  25. 19.
    Moller, H. Bjerrum and Houmann, J. C. G., Phys. Rev. Letters, 16, 737 (1966)CrossRefGoogle Scholar
  26. Moller, H. Bjerrum, Houmann, J. C. G., and Mackintosh, A. R., J. Appl. Phys., 39, 807 (1968)CrossRefGoogle Scholar
  27. Moller, H. Bjerrum, in Neutron Inelastic Scattering ( IAEA, Vienna, 1968 )Google Scholar
  28. Nielsen, M., Moller, H. Bjerrum, and Mackintosh, A. R., J. Appl. Phys., 41, 1174 (1970).CrossRefGoogle Scholar
  29. 20.
    Moller, H. Bjerrum, Nielsen, M., and Mackintosh, A. R., in Les Eléments des Terres Rares ( CNRS, Paris, 1970 ), Vol. II, p. 277.Google Scholar
  30. 21.
    Nicklow, R. M., J. Appl. Phys., 42, 1672 (1971)CrossRefGoogle Scholar
  31. Nicklow, R. M., Wakabayashi, N., Wilkinson, M. K., and Reed, R. E., Phys. Rev. Letters, 26, 140 (1971).CrossRefGoogle Scholar
  32. 22.
    Moller, H. Bjerrum, Houmann, J. C. G., Nielsen, M., and Mackintosh, A. R., (to be published).Google Scholar
  33. 23.
    Lindgârd, P. A., (private communication).Google Scholar
  34. 24.
    Stringfellow, M. W. and Windsor, C. G., Proc. Phys. Soc., 92, 408 (1967).CrossRefGoogle Scholar
  35. 25.
    Houmann, J. C. G., Solid State Commun., 6, 479 (1968).CrossRefGoogle Scholar
  36. 26.
    Cooper, B. R. Phys. Rev., 169, 281 (1968); Lindgârd, P. A. J. de Physique, 32, CI-238 (1971).Google Scholar
  37. 27.
    Nielsen, M., Moller, H. Bjerrum, Lindgârd, P. A., and Mackintosh, A. R., Phys. Rev. Letters, 25, 1451 (1970).CrossRefGoogle Scholar
  38. 28.
    Mackintosh, A. R., Phys. Letters, 4, 140 (1963).CrossRefGoogle Scholar
  39. 29.
    Cooper, B. R., Elliott, R. J., Nettel, S. J., and Suhl, H., Phys. Rev., 127, 57 (1962).CrossRefGoogle Scholar
  40. 30.
    Rossol, F. C. and Jones, R. V., J. Appl. Phys., 37, 1227 (1966)CrossRefGoogle Scholar
  41. Bagguley, D. M. S. and Liesegang, J., Proc. Roy. Soc., A300, 497 (1967).CrossRefGoogle Scholar
  42. 31.
    Marsh, H. S. and Sievers, A. J., J. Appl. Phys., 40, 1563 (1969)CrossRefGoogle Scholar
  43. Sievers, A. J., J. Appl. Phys., 41, 980 (1970).Google Scholar
  44. 32.
    Wagner, T. K. and Stanford. J. L., Phys. Rev., 184, 505 (1969).CrossRefGoogle Scholar
  45. 33.
    Wagner, T. K. and Stanford, J. L., Phys. Rev., Bl, 4488 (1970).Google Scholar
  46. 34.
    Feron, J. L., Hug, G., and Pauthenet, R., in Les Eléments des Terres Rares (CNRS, Paris, 1970), Vol. II, p. 19; see also Chapter 4.Google Scholar
  47. 35.
    Callen, H. B. and Callen, E., J. Phys. Chem. So.ids, 27, 1271 (1966).CrossRefGoogle Scholar
  48. 36.
    Brooks, M. S. S., Phys. Rev., Bl, 2257 (1970)Google Scholar
  49. 37.
    Kasuya, T., in Magnetism (G. T. Rado and H. Suhl, eds.). (Academic Press, New York 1966), Vol. IIB, p. 215.Google Scholar
  50. 38.
    Moller, H. Bjerrum, Thesis, University of Copenhagen (1968).Google Scholar
  51. 39.
    Yosida, K. and Miwa, H., J. Appl. Phys., 32, 8S (1961).CrossRefGoogle Scholar
  52. 40.
    Baryakhtar, V. G. and Maleev, S. V., Fiz. Tverd. Tela, 5, 1175 (1963).Google Scholar
  53. 41.
    Moller, H. Bjerrum, Houmann, J. C. G., and Mackintosh, A. R., Phys. Rev. Letters, 19, 312 (1967).CrossRefGoogle Scholar
  54. 42.
    Nicklow, R. M., Mook, H. A., Smith, H. G., Reed, R. E., and Wilkinson, M. K., J. Appl. Phys., 40, 1452 (1969).CrossRefGoogle Scholar
  55. 43.
    Nicklow, R. M., Houmann, J. C. G., and Mook, H. A., (to be published).Google Scholar
  56. 44.
    Stringfellow, M. W., Holden, T. M., Powell, B. M., and Woods, A. D. B., J. Phys. C, 2, S189 (1970).CrossRefGoogle Scholar
  57. 45.
    Elliott, R. J. and Lange, R. V., Phys. Rev., 152, 235 (1966).CrossRefGoogle Scholar
  58. 46.
    Cooper, B. R., Phys. Rev. Letters, 19, 900 (1967).CrossRefGoogle Scholar
  59. 47.
    Sherrington, D., (to be published).Google Scholar
  60. 48.
    Elliott, R. J., (private communication).Google Scholar
  61. 49.
    Woods, A. D. B., Holden, T. M., and Powell, B. M. Phys. Rev. Letters, 19, 908 (1967)CrossRefGoogle Scholar
  62. Nicklow, R. M., Wakabayashi, N., Wilkinson, M. K., and Reed, R. E. Phys. Rev. Letters (1971)Google Scholar
  63. Holden, T. M., Powell, B. M., Stringfellow, M. W., and Woods, A. D. B. J. Appl. Phys., 41, 1176 (1970).CrossRefGoogle Scholar
  64. 50.
    Nagamiya, T., Nagata, K., and Kitano, Y., Frog. Theor. Phys. (Kyoto), 31, 1 (1964).CrossRefGoogle Scholar
  65. 51.
    Cooper, B. R. and Elliott, R. J., Phys. Rev., 131, 1043 (1963)CrossRefGoogle Scholar
  66. Cooper, B. R., in Solid State Physics ( F. Seitz, D. Turnbull, and H. Ehrenreich, eds). Academic Press, New York (1968), Vol. 21, p. 393.Google Scholar
  67. 52.
    Rossol, F. C., Thesis, Harvard University ( 1966 ); Liesegang, J., Thesis, Oxford University (1966).Google Scholar
  68. 53.
    Brooks, M. S. S., Goodings, D. A., and Ralph, H. I., J. Phys. C, 2, 1596 (1968).CrossRefGoogle Scholar
  69. 54.
    Hegland, D. E., Legvold, S., and Spedding, F. H., Phys. Rev., 131, 158 (1963)CrossRefGoogle Scholar
  70. 55.
    Cooper, B. R., Proc. Phys. Soc., 80, 1225 (1962).CrossRefGoogle Scholar
  71. 56.
    Lounasmaa, O. V. and Sundström, L. J., Phys. Rev., 150, 399 (1966).CrossRefGoogle Scholar
  72. 57.
    Houmann, J. C. G. and Nicklow, R. M. Phys. Rev., Bl, 3943 (1970).Google Scholar
  73. 58.
    Elliott, R. J. and Stern, H., in Inelastic Scattering of Neutrons ( IAEA, Vienna, 1961 ), p. 61.Google Scholar
  74. 59.
    Dyson, F. J., Phys. Rev., 102, 1217 (1956)CrossRefGoogle Scholar
  75. Bloch, M., Phys. Rev. Letters, 9, 286 (1962).CrossRefGoogle Scholar
  76. 60.
    Tyablikov, S. V., Ukr. Math. Zh., 11, 287 (1959);CrossRefGoogle Scholar
  77. Callen, H. B., Phys. Rev., 130, 890 (1963).CrossRefGoogle Scholar
  78. 61.
    Kascheev, V. N. and Krivoglaz, M. A., Fiz. Tverd. Tela, 3, 3167 (1961).Google Scholar
  79. 62.
    Nielsen, M. and Moller, H. Bjerrum, Acta Cryst., A25, 547 (1969).CrossRefGoogle Scholar
  80. 63.
    Jensen, J. Intern. J. Magnetism, 1, 271 (1971).Google Scholar
  81. 64.
    Mason, W. P., Phys. Rev., 96, 302 (1954).CrossRefGoogle Scholar
  82. 65.
    Wolfram, T. and Callaway, J., Phys. Rev., 130, 2207 (1963).Google Scholar
  83. 66.
    Blackman, J. A., J. Phys. Chem. Solids, 31, 1573 (1970).CrossRefGoogle Scholar
  84. 67.
    Kasuya, T., Prog. T.eor. Phys. (Kyoto), 16, 45 (1956).Google Scholar
  85. 68.
    Kasuya, T., Prog. T.eor. Phys. (Kyoto), 22, 227 (1959).Google Scholar
  86. 69.
    Mannari, I., Prog. Theor. Phys. (Kyoto), 22, 335 (1959).Google Scholar
  87. 70.
    Brun, T. O., (1965-unpublished).Google Scholar
  88. 71.
    Sze, N. H., Rao, K. V., and Meaden, G. T., J. Low Temp. Phys, 1, 563 (1969).CrossRefGoogle Scholar
  89. 72.
    Mackintosh, A. R. and Spanel, L. E., Solid State Commun., 2, 383 (1964).CrossRefGoogle Scholar
  90. 73.
    Koehler, W. C., Cable, J. W., Wilkinson, M. K., and Wollan, E. O., Phys. Rev., 151, 414 (1966).CrossRefGoogle Scholar
  91. 74.
    See Mackintosh, A. R. J. de Physique, 32, C1–482 (1971).CrossRefGoogle Scholar
  92. 75.
    Bleaney, B., Proc. Roy. Soc., A276, 39 (1963).CrossRefGoogle Scholar
  93. 76.
    Parkinson, D. A., Simon, F. E., and Spedding, F. H., Proc. Roy. Soc., A207, 137 (1951).CrossRefGoogle Scholar
  94. 77.
    Johansson, T., Lebech, B., Nielsen, M., Bjerrum Moller, H. and Mackintosh, A. R. Phys. Rev. Letters, 25, 524 (1970); Johansson, T., J. de Physique, 32, C1–372 (1971).CrossRefGoogle Scholar
  95. 78.
    Trammell, G. T., J. Appl. Phys., 31, 362S (1960).CrossRefGoogle Scholar
  96. 79.
    Rainford, B. D. and Houmann, J. C. G. Phys. Rev. Letters, 26, 1254 (1971).CrossRefGoogle Scholar
  97. 80.
    Bjerrum Moller, H., Houmann, J. C. G., Jensen, J. and Mackintosh, A. R. in Neutron Inelastic Scattering (IAEA, Vienna, to be published) have studied the anisotropy of the exchange in terbium by measuring the magnetic field dependence of the magnon energy, as a function of q in the c-direction. They find that, if the exchange is written as in (5.45), with a different parameter multiplying each Cartesian component, its isotropic and anisotropic components have the same order of magnitude.Google Scholar

Copyright information

© Springer Science+Business Media New York 1972

Authors and Affiliations

  • A. R. Mackintosh
    • 1
  • H. Bjerrum Møller
    • 2
  1. 1.H.C. Ørsted InstituteUniversity of CopenhagenDenmark
  2. 2.Atomic Energy Commission Research EstablishmentRisø, RoskildeDenmark

Personalised recommendations