Pauli paramagnetism. If the free electron gas (Section 5.3.1) is placed in a uniform magnetic field B = Bk̂, the energies of the spin-up and spin-down states will be different:

There will be more spin-down states occupied than spin-up states (since they are lower in energy), and consequently the system will acquire a magnetization (see Problem 5.32).
(a) In the approximation that M/μ_B << ρ, find the magnetization that minimizes the total energy. Use the result of Problem 5.32(b).
(b) The magnetic susceptibility is

Calculate the magnetic susceptibility for aluminum (ρ = 18.1 x 10²² cm^-3) and compare the experimental value of 22 x 10^-6.

Problem 5.32(b)

Show that for M/μ_B << ρ Ξ (N₊ + N_-)/V (which is to say, |N₊ - N_-| << (N₊ + N_-)), the energy density is

The energy is a minimum for M = 0, so the ground state will have zero magnetization. However, if the gas is placed in a magnetic field (or in the presence of interactions between the particles) it may be energetically favorable for the gas to magnetize. This is explored in Problems 5.33 and 5.34.

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= 12 2m + + ±μBB.