PHY 4523 Spring 2000 – Homework 8
Due by 5:00 p.m. on Wednesday, April 26.
Answer all questions. To receive full credit, you should explain your reasoning and show all
working. Please write neatly and remember to include your name.
1. Heat capacity contributed by ferromagnetic spin waves: Reif Problem 10.1.
This problem asks you to modify the Debye theory of lattice vibrations to describe the
low-energy normal modes of a three-dimensional ferromagnet: “magnons,” or in the
classical limit, “spin waves.” Magnons, like phonons, are bosons. However, there is
only one branch of magnons (not three), and the long-wavelength modes (|q|a 1)
have a quadratic (rather than linear) dispersion: ω(q) = A|q|2 .
2. Magnetization of the Ising model.
Consider the mean-field solution of the ferromagnetic Ising model in zero external
magnetic field, for which the magnetization is
M = N gµB SBS (Sη) =
N gµB kB T
η.
2zJ
Here N is the number of spins of size S interacting with exchange coupling J, z is the
lattice coordination number, T is the temperature, BS is the Brillouin function, and η
is the mean-field parameter defined by the second equality.
Just below the Curie temperature, kB Tc = 23 zJS(S + 1), all solutions for η are small
and we can expand BS (x) ≈ (S +1) x/(3S) − Ax3 .
Based on the above, calculate the dependence of the magnetization on Tc − T in the
ordered phase of the Ising model for Tc − T Tc .
3. Heat capacity of the Ising model.
Again consider the mean-field solution of the ferromagnetic Ising model in zero external
magnetic field as discussed in the previous question.
Calculate Cv , the heat capacity at constant volume, in each of three cases: (a) at very
low temperatures, 0 < T Tc ; (b) just below the Curie temperature, 0 < Tc −T Tc ;
(c) above the Curie temperature, T > Tc .