## ABSTRACT

Spinwave dispersions of magnetically ordered materials have been calculated by various authors. The reader is referred to seminal publications by Holstein-Primakoff, Dyson, Keffer, Akhiezer, etc., on the subject matter. Spinwave excitations in magnetic materials as well as superconductivity materials involve the collective excitations or interactions of 1022 particles in a macroscopic body. Second quantization calculational methods have been formulated to approximate the interaction energy of 1022 particles. These theoretical formulations are beyond the scope of this book. We have adopted a much simpler approach whereby interactions of particles at a local site are represented by a molecular ﬁeld, and it is of the same form from site to site. Our calculational method utilizes a semiclassical approach where the mag-

netic moment of an ordered magnetic material is represented by a classical vector ~m rather than by a quantum spin operator. The magnetic moment is assumed to be uniform in amplitude over a small region of the material. The object of our calculations is to determine the spatial variation of ~m, dynamic magnetization vector, under the inﬂuence of a molecular exchange ﬁeld. We assume that within the crystal unit cell ~m is uniform, although there may be many magnetic sublattices within the crystal unit cell. For example, in garnet ferrite materials there may be as many as three magnetic sublattices in the crystal unit cell. Clearly, spinwave excitations whose wavelengths are greater than the lattice constant of a unit cell are considered semiclassical for the dispersion of spinwaves, since ~m is uniform over one unit cell. For wavelengths smaller than the lattice constant, one may no longer assume ~m to be constant over the unit cell, especially if it contains magnetic sublattices. For these cases, spinwave excitations occur at much higher frequency, usually in the optical frequency regime. For the former case, where ~m may be assumed to be uniform over the crystal unit cell, the spinwave dispersion is often referred to as the acoustic magnon dispersion, since it is the lowest frequency excitation. It is well known that in the semiclassical approach there are two uniform dynamic ﬁelds besides static internal ﬁelds: the dynamic exchange and the volume demagnetizing ﬁelds. The exchange dynamic ﬁeld is a molecular ﬁeld representing exchange coupling between spins. The volume demagnetizing ﬁeld can be approximated to be constant over one-half wavelength. Clearly, ~m changes sign over that distance. This means that the dynamic magnetization vector, ~m*, ‘‘sees’’ a uniform demagnetizing ﬁeld in

its motion, since ~m is uniform over the unit cell, which is much less than the wavelength of the spinwave. As such, it is meaningful to treat the motion of ~m within the unit cell in a classical way, where all the ﬁelds acting on ~m are uniform. The exchange ﬁeld is deﬁned as follows (see Chapter 4):

~hexc: ¼ 2AM2r 2~M,

where A exchange stiffness constant (erg=cm) 106 erg=cm. Let’s assume that ~M¼ ~M0þ~m, where M

0 is the static magnetization and is constant throughout the magnetic sample. Then,

~hexc: ¼ 2AM2 k 2~m, ~m / ej k

It is too cubersome to carry the subscript ‘‘0’’ on M in the denominator of above equation. There may be internal demagnetizing ﬁelds as a result of spinwave

excitations. For example, assume the following spinwave conﬁguration (Figure 8.1a).