The Non-interacting Fermi Gas

Abstract This chapter first discusses the quantum mechanics of non-interacting electrons in a box. The simplest model of a metal is to assume it consists of free electrons in a box, with their mutual Coulomb repulsion ‘switched off’. The atoms in these materials (Li, Na, K, Rb, and Cs) have low ionization energies, only weakly binding their single valence s electron to the nucleus. It can then be imagined that when assembled as a crystal, each atom contributes one valence electron to the formation of a kind of electron gas or quantum plasma having a number density equal to that of the atoms. On average, the nuclear charge will compensate the free or mobile electronic charge. The remainder of the chapter covers Fermi–Dirac statistics; evaluation of integrals involving the Fermi distribution function; the temperature dependence of the chemical potential; energy of an ideal Fermi gas as a function of temperature; the paramagnetic susceptibility of a Fermi gas; and a qualitative discussion of the behaviour of a Fermi gas. Sample problems are also provided at the end of the chapter.