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## Spin wave theory of complex ferrimagnets : The Saga on YIG

A review of magnon properties of yttrium-iron garnet (YIG), a classical object for experimental studies in magnetism, is presented. Both experimental and theoretical results concerned with thermodynamics and kinetics of YIG are described. The main purposes of the review are to introduce a new method of approximate calculation of the magnon spectra in magnets with large unit cell and to obtain by means of this method some basic properties of YIG. In particular, it is shown that the problem of calculating the frequencies of all the 20 magnon branches over the entire Brillouin zone contains two small parameters. First, because of the large number of magnetic atoms in the unit cell the distance between the nearest interacting magnetic atoms is small in comparison with the lattice constant and, accordingly, with the wavelength of a spin wave. An effective long-wavelength character thus arises in the problem. Second, there are a large number of wave-vector directions along which many elements of the Hamiltonian matrix vanish by symmetry in the basis which diagonalizes this matrix for *k* = 0. These matrix elements thus have an additional, angular smallness for arbitrary directions of *k*. These matrix elements can be taken into account using perturbation theory. As a result, the large elements of the Hamiltonian matrix are few in number, and they can be eliminated by several two-dimensional rotations. Approximate expressions, differing from the computer calculations by ⪅ 10%, are thus obtained for the frequencies.

Neutron scattering data are used to find the values of the exchange integrals in YIG and to obtain the magnon spectra. It is shown that in the energy range *T* ⪅ 260 K only magnons of the lower branch are excited; the spectrum of these “ferromagnons” is quadratic in the wave vector only up to 40 K and becomes linear in the region ω*k* ⪆ 40 K. For temperatures up to 400 K the temperature dependence of the magnetization is calculated in the spin-wave approximation and good agreement with experimental data is found.

A brief review of experimental data on magnon relaxation in YIG is presented. The magnon-magnon interactions which cause the magnon relaxation are described. The amplitude of the four-magnon exchange interaction is determined, and the temperature correction to the frequency is evaluated. This temperature correction is positive, in contrast to the case of simple cubic ferromagnet with nearest-neighbour interaction. The exchange relaxation rate is calculated for normal and umklapp processes. It is shown that the magnetic dipole interaction is important only for the ferromagnons; the amplitude of this interaction and the corresponding relaxation rate are determined. Three-magnon scattering processes are allowed only for wave vectors larger than a certain *k*/; at *k* = *k*/ there is a discontinuity in the wave-vector dependence of the damping. A calculation is given for the nonvanishing contribution to the relaxation at **k** = 0 on account of scattering processes involving optical magnons; this contribution is due to the local uniaxial anisotropy. The relative role of each of the investigated relaxation mechanisms is discussed, and the correspondence of the present results with the experimental data is examined.