## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

12-15-1963

#### Publication Source

Physical Review

#### Volume

132

#### Issue

6

#### Start Page

2398

#### Last Page

2409

#### DOI

10.1103/PhysRev.132.2398

#### Abstract

The problem of deducing the values of the exchange integrals in yttrium and gadolinium iron garnet from measurements of the magnetization and the magnetic contribution to the specific heat at low temperatures is considered. For these garnets the spin-wave normal modes can be found by solving the semiclassical equations of motion which give rise to a set of *n* simultaneous linear equations, where *n* is the number of magnetically inequivalent ions in the unit cell. Expressions for the thermodynamic functions at low temperatures in terms of the frequencies of the normal modes are given assuming the validity of the spin-wave approximation. It is argued that the temperature variation of the frequency of these normal modes on the macroscopic properties can be completely accounted for without considering the zero point energy explicitly. Due to the size of the unit cell, the equations for the frequencies of the normal modes can only be solved numerically for general values of **k**. Such solutions are obtained for **k** lying along a [111] direction for various values of the exchange integrals, and the thermodynamic functions corresponding to these choices of parameters are calculated. In the case of yttrium iron garnet, the value of D, the coefficient of a^{2}k^{2} in the acoustic dispersion law, is reliably known and fixes one linear combination of J_{aa}, J_{ad}, and J_{dd}. By comparing our calculations with the magnetization data of Solt, it was established that J_{aa}/J_{ad}=0.2, but since the magnetization was not very sensitive to variations of the ratio J_{dd}/J_{ad} its value could not be estimated precisely. Taking J_{dd}/J_{ad}=0.2 gives J_{aa}=J_{dd}=6.35 cm^{−1} and J_{ad}=31.8 cm^{−1}. For GdIG the specific heat data below 20°K is not very much influenced by the exact values of the iron-iron exchange integrals which were taken to be those quoted above for yttrium iron garnet. Again one combination of J_{ac} and J_{dc} is known from the calorimetric determination of the single ion splitting. By comparing the specific heat data below 5°K with calculations for various values of J_{ac}/J_{dc} it was possible to determine J_{ac} and J_{dc} separately: J_{dc}=7.00 cm^{−1} and J_{ac}=1.75 cm^{−1}. These values are about 25% larger than what one would expect using the Weiss molecular field approximation.

#### Recommended Citation

Harris, A.
(1963).
Spin-Wave Spectra of Yttrium and Gadolinium Iron Garnet.
*Physical Review,*
*132*
(6),
2398-2409.
http://dx.doi.org/10.1103/PhysRev.132.2398

**Date Posted:** 12 August 2015

This document has been peer reviewed.

## Comments

At the time of publication, author A. Brooks Harris was affiliated with Duke University. Currently, he is a faculty member in the Department of Physics at the University of Pennsylvania.