# Most spin-1/2 fermions, including electrons and helium-3

## Problem 36P Chapter 7

An Introduction to Thermal Physics | 1st Edition

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Problem 36P

Most spin-1/2 fermions, including electrons and helium-3 atoms, have nonzero magnetic moments. A gas of such particles is therefore paramagnetic. Consider, for example, a gas of free electrons, confined inside a three-dimensional box. The z component of the magnetic moment of each electron is ±µB. In the presence of a magnetic field B pointing in the z direction, each “up” state acquires an additional energy of −µB B. while each “down” state acquires an additional energy of + µB B.

(a) Explain why you would expect the magnetization of a degenerate electron gas to be substantially less than that of the electronic paramagnets studied in Chapters 3 and 6, for a given number of particles at a given field strength.

(b) Write down a formula for the density of states of this system in the presence of a magnetic field B, and interpret your formula graphically.

(c) The magnetization of this system is µB(N↑ − N↓) Where N↑ and N↓ are the numbers of electrons with up and down magnetic moments, respectively. Find a formula for the magnetization of this system at T  = 0, in terms of N µB, B and the Fermi energy.

(d) Find the first temperature-dependent correction to your answer to part (c), in the limit T ≪ TF. You may assume that μB B ≪ KT; this implies that the presence of the magnetic field has negligible effect on the chemical potential µ, (To avoid confusing µB with µ, I suggest using an abbreviation such as δ for the quantity µB B.)

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