Nuclear Spin-Lattice Relaxation in Magnetic Insulators

D. Beeman; P. Pincus

doi:10.1103/PhysRev.166.359

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Nuclear Spin-Lattice Relaxation in Magnetic Insulators

D. Beeman^* and P. Pincus^†

Department of Physics, University of California, Los Angeles, California

^*Present address: Theoretical Division, Atomic Energy Research Establishment, Harwell, Didcot, Berkshire, England.
^†Alfred P. Sloan Foundation Fellow.

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Phys. Rev. 166, 359 – Published 10 February, 1968

DOI: https://doi.org/10.1103/PhysRev.166.359

Abstract

First-order nuclear spin-lattice relaxation arises in magnetic insulators when a nuclear spin directly interacts with one or more spin waves via the hyperfine interaction. The direct process, in which a single magnon is emitted, is not ordinarily allowed on the basis of energy conservation, since the energy of a nuclear spin flip is considerably lower than the minimum energy of a spin wave. When the hyperfine interaction is isotropic and the axes of quantization of the nuclear and electronic spins are collinear, the conservation of the $z$ component of spin angular momentum forbids the Raman process, in which a thermal magnon is scattered to a state of different wave vector, accompanied by a nuclear spin flip. The three-magnon process is usually allowed. We consider here some second-order two- and three-magnon processes which arise when a virtual magnon, emitted in the direct process, is scattered by thermal magnons via the exchange interaction or the magnetic dipole-dipole interaction. This is possible, since (1) the magnon spectrum is lifetime broadened to overlap the nuclear spin resonance frequency, and (2) the dipole interaction does not conserve spin angular momentum. The second-order three-magnon process, arising from the four-magnon exchange interaction, enhances the three-magnon process relaxation rate by a factor of 8 in ferromagnets, and results in a temperature-dependent enhancement of about one order of magnitude in antiferromagnets. Three-magnon terms in the dipole-dipole interaction may induce a two-magnon process in both ferromagnets and antiferromagnets, which is of significance when the first-order Raman process is forbidden. We also calculate the relaxation rate due to second-order exchange-scattering-induced two-magnon process which is often more important than the first-order process in canted antiferromagnets of both the "easy-axis" and the "hard-axis" anisotropy type.

References (27)

N. Kaplan, R. Loudon, V. Jaccarino, H. J. Guggenheim, D. Beeman, and P. A. Pincus, Phys. Rev. Letters 17, 357 (1966)
A. Narath and A. T. Fromhold, Jr., Phys. Rev. Letters 17, 354 (1966)
L. B. Welsh, Phys. Rev. 156, 370 (1967)
T. Moriya, Progr. Theoret. Phys. (Kyoto) 16, 23 (1956) ibid.16, 641 (1956)
J. Van Kranendonk and M. Bloom, Physica 22, 545 (1956)
G. E. G. Hardeman, N. J. Poulis, and W. van der Lugt, Physica 22, 48 (1956)
P. Pincus, Phys. Rev. Letters 16, 398 (1966)
T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940)
F. Keffer, in Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1966), Vol. 18, pt. 2
V. Jaccarino, in Magnetism, edited by G. T. Rado and H. Suhl (Academic Press Inc., New York, 1965), Vol. 2A, Chap. 5
N. Bloembergen, E. M. Purcell, and R. V. Pound, Phys. Rev. 73, 679 (1948)
A. Mitchel, J. Chem. Phys. 27, 17 (1957)
Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegen (U.S. Government Printing Office, Washington, D.C., 1964), p. 998
T. Oguchi and F. Keffer, J. Phys. Chem. Solids 25, 405 (1964)
S. J. Allen, R. Loudon, and P. L. Richards, Phys. Rev. Letters 16, 463 (1966)
A. Okazaki, K. C. Turberfield, and R. W. H. Stevenson, Phys. Letters 8, 9 (1964)
A. J. Freeman and R. E. Watson, in Magnetism, edited by G. T. Rado and H. Suhl (Academic Press Inc., New York, 1965), Vol. 2A
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T. Holstein (private communication)
F. J. Dyson, Phys. Rev. 102, 1217 (1956)
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C. Kittel, Quantum Theory of Solids (John Wiley & Sons, Inc., New York, 1963), Chap. 4
D. Beeman, J. Appl. Phys. 38, 1276 (1967)
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R. Loudon and P. Pincus, Phys. Rev. 132, 673 (1963)

[c1] N. Kaplan, R. Loudon, V. Jaccarino, H. J. Guggenheim, D. Beeman, and P. A. Pincus, Phys. Rev. Letters 17, 357 (1966)

[c2] A. Narath and A. T. Fromhold, Jr., Phys. Rev. Letters 17, 354 (1966)

[c3] L. B. Welsh, Phys. Rev. 156, 370 (1967)

[c4] T. Moriya, Progr. Theoret. Phys. (Kyoto) 16, 23 (1956) ibid.16, 641 (1956)

[c5] J. Van Kranendonk and M. Bloom, Physica 22, 545 (1956)

[c6] G. E. G. Hardeman, N. J. Poulis, and W. van der Lugt, Physica 22, 48 (1956)

[c7] P. Pincus, Phys. Rev. Letters 16, 398 (1966)

[c8] T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940)

[c9] F. Keffer, in Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1966), Vol. 18, pt. 2

[c10] V. Jaccarino, in Magnetism, edited by G. T. Rado and H. Suhl (Academic Press Inc., New York, 1965), Vol. 2A, Chap. 5

[c11] N. Bloembergen, E. M. Purcell, and R. V. Pound, Phys. Rev. 73, 679 (1948)

[c12] A. Mitchel, J. Chem. Phys. 27, 17 (1957)

[c13] Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegen (U.S. Government Printing Office, Washington, D.C., 1964), p. 998

[c14] T. Oguchi and F. Keffer, J. Phys. Chem. Solids 25, 405 (1964)

[c15] S. J. Allen, R. Loudon, and P. L. Richards, Phys. Rev. Letters 16, 463 (1966)

[c16] A. Okazaki, K. C. Turberfield, and R. W. H. Stevenson, Phys. Letters 8, 9 (1964)

[c17] A. J. Freeman and R. E. Watson, in Magnetism, edited by G. T. Rado and H. Suhl (Academic Press Inc., New York, 1965), Vol. 2A

[c18] V. Jaccarino and R. G. Shulman, Phys. Rev. 107, 1196 (1957)

[c19] T. Holstein (private communication)

[c20] F. J. Dyson, Phys. Rev. 102, 1217 (1956)

[c21] T. Moriya, in Magnetism, edited by G. T. Rado and H. Suhl (Academic Press Inc., New York, 1963), Vol. 1

[c22] K. Lee, A. M. Portis, and G. L. Witt, Phys. Rev. 132, 144 (1963)

[c23] R. J. Mahler, A. C. Daniel, and P. T. Parrish, Phys. Rev. Letters 19, 85 (1967)

[c24] C. Kittel, Quantum Theory of Solids (John Wiley & Sons, Inc., New York, 1963), Chap. 4

[c25] D. Beeman, J. Appl. Phys. 38, 1276 (1967)

[c26] M. Sparks, R. Loudon, and C. Kittel, Phys. Rev. 122, 791 (1961)

[c27] R. Loudon and P. Pincus, Phys. Rev. 132, 673 (1963)

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Nuclear Spin-Lattice Relaxation in Magnetic Insulators

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