A theory of spin waves for the spin structures found in the rare earth metals of h.c.p. crystal structure is described. The theory is developed for the conical spiral spin structure which contains the planar spiral, the nonplanar ferromagnet, and the planar ferromagnet as special cases. Included in the Hamiltonian are isotropic and anisotropic exchange interactions, single-ion crystal field terms, and magnetoelastic terms, both of the single-ion type (linear in the strains and up to fourth order in the spin operators) and of the two-ion type (linear in the strains and second order in the spin operators). The magnetoelastic effects are discussed in considerable detail, both in the "frozen lattice approximation" and in the opposite limit in which the strains closely follow the motion of the (uniformly) processing spins. Equations of motion for the spin operators are linearized with the help of the random phase approximation which makes it possible to express some spin-wave interaction effects in terms of powers of the reduced magnetization. Expressions for the spin-wave energies are given for the planar spiral, the nonplanar ferromagnet, and the planar ferromagnet, taking advantage of the simplifying features in each case.
Nanometric skyrmion lattice from anisotropic exchange interactions in a centrosymmetric host
