Question

A magnon is a spin wave found in ferromagnetic solids. The properties of a magnon include: (1) Magnons are bosons with only one possible polarization. (2) The number of magnons is not conserved as temperature increases. (3) There is a finite number (N) of magnon states in a solid. (4) The magnon energy is proportional to its angular frequency: E -hca (5) The magnon dispersion relation is quadratic: i.e., a) Ak" (where A is a constant) Throughout this question, consider the properties of magnons in a three-dimensional ferromagnetic solid of volume, V. 1/2 (a) Show that the density of magnon states is: 2 3/2

Answer 1

Number of states with in the thick ness dk of spherical shell is dN = 4 pi k^2b dk/(2 pi/L)^3 = L^3 4 pi k^2 dk/(2 pi)^3 = v k^2 dk/2 pi^2 rightarrow dN/dk = v k^2 /2 pi^2 rightarrow dN/dw = vk^2/2 pi^2/d w/d k = v k^2/2 pi^2 /2 A k = V k/4 A pi^2 = v (w/A)^1/2/4 A pi^2 = V w^1/2/4 A^3/2 pi^2 [where we used w = A k^2 rightarrow k = (w/A)^1/2]