Free Electron Fermi Gas

Yu. M. Suhov

John B. Ketterson

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.

K. Yamada

D. C. Johnston

A free-electron Fermi gas is a gas of electrons with spin https://www.w3.org/1998/Math/MathML" display="inline"> S = 1 / 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003606857/19698a40-dbd9-4f9a-9cb7-890a1d39adce/content/math8_1.tif "/> . A fermion is a particle with half-integer spin S such as the electron. A boson is a particle with integer spin https://www.w3.org/1998/Math/MathML" display="inline"> S ≥ 0 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003606857/19698a40-dbd9-4f9a-9cb7-890a1d39adce/content/math8_2.tif "/> . By a gas is meant that there are no interactions between the particles making up the gas. Thus a Fermi gas can exhibit no phase transitions by itself. On the other hand, a free boson gas can exhibit an interesting quantum phase transition into a superfluid state, qualitatively similar to that of the superconducting state of a metal, called Bose-Einstein condensation . In contrast, an electron Fermi liquid would be a system containing electrons that do interact with each other. We will see in the last two chapters of this book that when such interactions occur, magnetic or superconducting transitions associated with the electrons can happen.

Hui Liu

The properties of an ideal Fermi gas in finite gravitational field are studied,based on the semiclassical(Thomas-Fermi) approximation.The general analytical expressions of the total energy,Fermi energy,chemical potential and heat capacity are derived.Moreover,the analytical expressions of the chemical potential and heat capacity in the high-and low-temperature approximations are given,and then,the example of electron gas is used to calculate exactly.All these results are compared with those of free Fermi gas and Fermi gas in infinite gravitational field.The effects of finite gravitational field on properties of Fermi gas are discussed.