Introduction to Quantum Mechanics · Problem 5.33

Identical Particles — David J. Griffiths

Physics — Quantum Mechanics · pauli_paramagnetism

Problem

Pauli paramagnetism. If the free electron gas (Section 5.3.1) is placed in a uniform magnetic field

\mathbf{B} = B\hat{k}

, the energies of the spin-up and spin-down states will be different:

E^{\pm}_{n_x n_y n_z} = \frac{\hbar^2 \pi^2}{2m} \left( \frac{n_x^2}{l_x^2} + \frac{n_y^2}{l_y^2} + \frac{n_z^2}{l_z^2} \right) \pm \mu_B B.

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/\mu_B \ll \rho

, find the magnetization that minimizes the total energy. Hint: Use the result of Problem 5.32(b). (b) The magnetic susceptibility is

\chi = \mu_0 \frac{dM}{dB}.

Calculate the magnetic susceptibility for aluminum (

\rho = 18.1 \times 10^{22}

^{-3}

) and compare the experimental value of

22 \times 10^{-6}

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