ACOUSTIC PHONONS IN HEISENBERG PARAMAGNETS

HAL Id: jpa-00213998

https://hal.archives-ouvertes.fr/jpa-00213998

Submitted on 1 Jan 1971

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ACOUSTIC PHONONS IN HEISENBERG PARAMAGNETS

H. Bennett

To cite this version:

H. Bennett. ACOUSTIC PHONONS IN HEISENBERG PARAMAGNETS. Journal de Physique

Colloques, 1971, 32 (C1), pp.C1-526-C1-527. �10.1051/jphyscol:19711176�. �jpa-00213998�

JOURNAL DE PHYSIQUE Colloque C 1, supplément au n° 2-3, Tome 32, Février-Mars 1971, page C l - 526

ACOUSTIC PHONONS IN HEISENBERG PARAMAGNETS

H. S. BENNETT

National Bureau of Standards Washington, D. C. 20234

Résumé. — La propagation d'ondes sonores dans des isolants ferro- et antiferromagnétiques est étudiée en tenant compte d'interactions de magnétostriction de volume. Des expressions sont obtenues donnant les coefficients d'atténuation ultrasonique et les glissements de fréquence des phonons en fonction des susceptibilités statiques, des coefficients de diffu- sion des spins et des chaleurs spécifiques. Les résultats sont en accord avec une loi d'échelle et qualitativement avec l'expé- rience.

Abstract. — The propagation of sound waves in ferromagnetic and antiferromagnetic insulators is examined in terms of a volume magnetostrictive interaction. Expressions for the ultrasonic attenuation coefficients and for the phonon fre- quency shifts near the transition temperature are given as functions of the static susceptibilities, the spin diffusion coeffi- cients, and the specific heats. The results agree with scaling law calculations and qualitatively with experiment.

Introduction. — The attenuation of a sound wave propagating in a magnetic insulator increases as the temperature approaches the transition temperature of the magnetic system. In addition, the frequency of a sound wave decreases under similar circumstances [1,2] because the phonons are renormalized. Any study of ultrasonic attenuation a and changes in the phonon frequency Aco requires a knowledge about bulk proper- ties such as specific heats (C

), static susceptibilities (x), and spin diffusion (D for ferromagnets and A for antiferromagnets).

The interaction between the magnetic system and a sound wave propagating in the magnetic medium is referred to as the phonon-magnon interaction. The effective forces between the ions comprising the lattice determine how sound waves propagate. At tempera- tures sufficiently far from the critical temperature r

, the effective lattice forces vary only slightly within the medium, if at all. However, as T

is approached, the critical fluctuations in the spin system produce large changes in the effective forces between the lattice sites. The force changes may extend near T

over distances of the order of several lattice spacings. They lead to spatial inhomogeneities in the manner in which ions respond to the sound wave. When these inhomogeneities become comparable in size to the wavelength of the sound wave and when they exist for times comparable to the period of the sound wave, then greater attenuation and frequency shifts of the sound wave occur. This is the case for which the effective size of the scatterer, the correlation length £, changes relative to the fixed wavelength X of the sound wave. The effective spatial extent of the scatterer (critical fluctuations) varies with temperature. The inhomogeneities arise because blocks of cooperatively moving spins with an average size £, exist for long times near T

and are coupled to the lattice.

Theoretical Concepts. — There have been a number of theories developed for ultrasonic attenuation at magnetic phase transitions. They all assume a coupling

mechanism between the spins and the phonons [3-7].

The spin-phonon interaction responsible for the cri- tical effects are thought to arise in many cases by means of the strain modulation of the exchange inter-

action (volume magnetostrictive coupling). Other cou- pling forms such as the single ion coupling and the energy density coupling have been studied also. Only a model based upon the volume magnetostrictive coupling is discussed here. The Hamiltonian for this model has the form,

3& = H

+ H

+ £ { U(a, f) - U(b, 0 } x

x V

J ( a - b ) x S(a, f)-S(b, t ) . The three dimensional, isotropic Heisenberg Hamilto- nian and the non-interacting phonon Hamiltonian are denoted respectively by H

and H

. The phonon operator U(a, t) determines the displacement of the ion at site a from its equilibrium position. The spin operator is S(a, t). The spatial gradient of the exchange integral /(a — b) is denoted by V

/(a — b).

By subtracting the number of phonons with wave vector q emitted from those absorbed one finds that the attenuation coefficient and the frequency shift are proportional to the space-time Fourier convolutions of a four spin correlation function with kernels contain- ing the exchange interaction. In order to evaluate the convolution integrals four major assumptions are introduced.

1. Factorizations of the four spin correlation func- tions give approximations valid for the paramagnetic region in terms of products of two spin correlation functions.

2. The use of hydrodynamic forms for the two spin correlation functions introduce the static susceptibility X and decay time of the characteristic spin fluctuations.

The inverse of the decay time gives the spin diffusion coefficient (D(q) for ferromagnets and yl(q) for anti- ferromagnets). The hydrodynamic forms are valid only for regions in which q% -4 1.

3. The unperturbed phonon frequency co is much smaller than the inverse of the spin decay time.

4. An heuristic attempt is made to account for the fact that the spin factorization overestimates the critical fluctuations and predicts that the specific heat C

varies as x'

- When %

appears in expressions for the attenuation a and the frequency shift Aco, it is replaced by the specific heat C

.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711176

ACOUSTIC PHONONS. I N HEISENBERG PARAMAGNETS C 1 - ⁵²⁷

Results and Conclusions. - With these assumptions, a and Aw become explicit functions of CB, X, D, and A as given in Table I for T R T,. It is not possible to solve within the present formalism for the tempe- rature behavior of these physical quantities. It is necessary therefore to resort to some other theories.

Using the scaling law results for CB, X, and D in Table I predicts that a -

for the ferromagnet.

The quantity

is the reduced temperature

ments to determine the four spin correlations [6].

Laramore and Kadanoff [5] use the mode-mode coupl- ing theory of Kadanoff and Swift [7] to estimate the four spin correlation functions. These theories predict that the attenuation is proportional to

The critical attenuation exponent q, is (513) for isotropic ferromagnets and is 1 for isotropic antiferromagnets.

T , - ' I T - T , I . Attenuation coeficien fs a and frequency shifts Molecular field theory gives A -

Inserting this into ratios (holm) predicted by conventional theories.

Table I for the antiferromagnet gives a - ^{w Z}

The quantities a and (Aw/w) are proportional These results agree qualitatively with experiments to the factors listed in this Table.

[I, 21. F AF

Other theories retain modifications of assumptions ^-

(2) and (3) and circumvent the factorization of the a o2 CB xD-I w2 cTA-'

four spin correlations. Kawaski uses scaling law argu- (Aw/w) - CB - CB References

[I] GOLDING (B.), Phys. Rev. Letters, 1968, 20, 5. [5] LARAMORE (G. E.) et al., Phys. Rev., 1969, 187, 619.

[2] POLLINA (R. J.) et al., Phys. Rev., 1969, 177, 841. [6] KAWASAKI (K.), Solid State Comm., 1969, 6, 57.

[3] BENNETT (H. S.) et al., Phys. Rev., 1967, 155, 553. [7] KADANOFF (I,. P.) et al., Phys. Rev., 1968, 166, 89.

[4] TANI (K.) et al., Phys. Letters, 1966, 19, 627.