A free-electron Fermi gas as defined here is a collection of a large number N of nonrelativistic noninteracting electrons with spin S = 1/2 in zero external potential. Thus the interaction of the electrons with a crystal lattice of positive ions that would give rise to energy gaps in the band structure and possibly superconductivity is ignored. The electron gas can be in one dimension (1D), 2D, or 3D. The low- and high-temperature T limits of the thermal and magnetic properties of a 3D free-electron Fermi gas are well known [1-8]. Other aspects and consequences of the Fermi statistics have also been considered [9-21]. In the limit of high T the heat capacity at constant volume C_v and the pressure p are the same as for a monatomic ideal gas and the magnetic spin susceptibility χ is the same as for N isolated electrons following the Curie law C/T , where C is the Curie constant for S = 1/2. At low T in the degenerate regime, CV is proportional to T, and χ and p saturate to constant value as T→0 K. Here expressions for the chemical poten tial µ(T) are discussed [22] that allow the crossovers of the dimension-dependent χ, internal energy U and C_v in 1D, 2D, and 3D, and of p, enthalpy H, heat capacity at constant pressure C_p, isothermal bulk modulus K_T , and thermal expansion coefficient α in 2D and 3D to be calculated between the low-T degenerate regime and the high-T nondegenerate regime. Illustratrative plots of the T dependences of these properties are provided.

There are many instances in which free-electron Fermi gas properties are observed in real metals. An important example is the T –dependendent C_v of metals at low T compared to the Fermi temperature T_F = E_F/K_B (E_F is the Fermi energy) which is proportional to T irrespective of the dimensionality of the electron gas in the metal. However, the proportionality constant γ, known as the Sommerfeld electronic heat capacity coefficient, is proportional to the electronic density of states at E_F which depends on the dimensionality of the metal for a given electron concentration as discussed below. The free-electron model as reflected in the theoretical value of γ is often fairly close for a particular metal to the measured value and hence is a good starting point for in terpreting the γ value. Deviations occur due to features not included in the free-electron Fermi-gas model such as the influences of the interactions of the conduction electrons with the lattice (the electron-phonon interaction) and of electron-electron interactions, such as described in Refs. [19,21].

In superconductors, the free-electron Fermi-gas model is not appropriate below the superconducting transition temperature T_c even for simple metals exhibiting so called conventional superconductivity because within the BCS theory [23] the superconductivity arises from indirect attractive interactions between the conduction electrons mediated by phonons. Here an energy gap opens in the quasiparticle (electron/hole) excitation spectrum below T_c, so that the electronic heat capacity decreases faster than linearly below T_c and approaches zero exponentially for T → 0.

Interestingly, the free-electron Fermi-gas theory also sometimes applies to nonmetals containing nuclei with nonzero spin, such as liquid [3] He for which the nuclei have spin S = 1/2. Indeed, as shown in Figure 3 below, the magnetic susceptibility χ versus T of liquid well by 3D S = 1/2 noninteracting-fermion theory all the way from the degenerate (T << T_F) to the nondegenerate (T >> T_F) temperature regime, and a fit to the data yielded the value of T_F for this material. Additional examples of 3D Fermi gases include the electron gas in white-dwarf stars and the neutrons in neutron stars [5], where the Fermi energies were estimated. Important examples of 2D or quasi-2D electron gases occur in metaloxide-semiconductor field-effect transistors (MOSFETS) in which the Quantum Hall Effect was discovered [24] for which the 1985 Nobel Prize in Physics was awarded [25], as well as in other materials [26].

Fundamental expressions and the notation used here are given in the Sec. II. The equations from which µ(T) can be calculated for 1D, 2D, and 3D electron Fermi gases are presented in Sec. III. The χ is obtained and plotted versus T from the degenerate to the nondegenerate regime for electron Fermi gases in 1D, 2D, and 3D in Sec. IV, and the dimension-dependent U and CV versus T are calculated and plotted in Sec. V. The pressure p, enthalpy H, heat capacity at constant pressure C_p, isothermal bulk modulus K_T, and thermal expansion coefficient α in 2D and 3D are derived and plotted from the degenerate to nondegenerate temperature regime in Secs. VI–IX, respectively. Concluding remarks are given in Sec. X.

Some of our results for the free-electron Fermi gas were already known from either published or unpublished sources. The µ(T ) in 1D, 2D, and 3D was thoroughly investigated in Ref [22]. The U(T ) and C_v(T) at low T for 1D, 2D, and 3D were obtained in Ref [18]. Sommerfeld expansions of the low-T properties are routinely obtained for all three dimensions in many sources. Here we include our calculations and plots of these previouslystudied quantities for completeness. The calculations of µ(T ) in 1D, 2D, and 3D are required for the calculations of all other thermodynamic properties versus dimensionality from the degenerate to the nondegenerate temperature regime.

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