Physics 573 (Solid State Physics)

Physics 573: Problem set 6

What are the orbital, spin, and total angular momentum values for ^3dV? What is the Landé g factor? What is the magnetic dipole moment?
A certain monatomic material has spin-split levels separated by Δ. Calculate the partiction function, Z and the internal energy E at temperature T. Now calculate the heat capacity C = (∂E/∂T). Plot this function to see how behaves near where Δ = k_BT. The name for these bumps in response functions is Schottky anomaly. If Δ is induced by an external magnetic field H discuss the changes in C(H). If the splitting is small and arises from an ionic or crystal field (such as hyperfine interactions) how does C depend on temperature?
An even richer version of the above problem obtains when the values of Δ are distributed randomly throughout the crystal as happens in glasses. A tractable version of this problem might have a flat distribution of Δ between 0 and Δ_max. Start with states at energies +Δ and -Δ, and recalculate E and C. Now show that C(T) ∝ T for very low temperature T << Δ_max/k_B
A simple model for an antiferromagnet is a long row of spins where next nearest neighbors are aligned parallel and neighbors are antiparallel. That is there are two interleaved ferromagnetic sublattices. The field on sublattice a is B_a = B_ext - μ₁M_a - μ₂M_b and the effective field on sublattice b is B_b = B_ext - μ₁M_b - μ₂M_a. Show that the Néel temperature is [(μ₂ - μ₁)/ (μ₁ + μ₂)]θ, where θ is the effective temperature offset for the Curie-Weiss "law."
A superconducting plate is aligned perpendicular to the z axis. There is a constant magnetic field in the space near the plate that lies parallel to the plate (e.g. along the x axis). Obtain B(z) near the surface inside the superconductor. Obtain the magnetization. Describe the magnetization if the plate's thickness d is small compared to the penetration depth. This is slightly less annoying if you set the middle of the plate at z = 0. The magnetization in the plate contribute extra free energy even at T = 0. Average the magnetization across the thickness of the plate and estimate the critical field. You should find that the critical field for a very thin plate can be very different from the bulk value.