The Free-Electron Fermi Gas in Metals

Comprehensive vocabulary flashcards covering the classical Drude picture, its failures, the quantum Sommerfeld model, Fermi gas kinematics, state counting, density of states, degenerate heat capacity, and Pauli paramagnetism.

Drude model

A classical microscopic model that treats conduction electrons as classical particles moving through a stationary background of positive ions, accelerating under an applied electric field and undergoing randomizing collisions after an average relaxation time $\tau$.

Relaxation time ($\tau$)

The average time interval between collisions that randomize the electron velocity; it serves as the primary microscopic quantity controlling electrical resistivity.

Drude electrical conductivity ($\sigma$)

The conductivity predicted by the classical Drude model, given by $\sigma = \frac{ne^2\tau}{m}$, where $n$ is electron density, $e$ is the elementary charge, $\tau$ is relaxation time, and $m$ is electron mass.

Electrical resistivity ($\rho$)

The inverse of electrical conductivity, given in the free-electron framework by $\rho = \frac{m}{ne^2\tau}$.

Classical electronic heat capacity

The constant electronic heat capacity predicted by classical equipartition for $N$ free conduction electrons, given by $C_{V,e} = \frac{3}{2} N k_{\text{B}}$, which fails to match experimental low-temperature measurements.

Degenerate electron gas

A quantum regime of conduction electrons where the thermal energy is much smaller than the Fermi energy ($k_{\text{B}}T \ll E_{\text{F}}$), requiring quantum state filling and Fermi–Dirac statistics rather than classical Maxwell–Boltzmann statistics.

Sommerfeld model

A model of metals that retains the free-electron concept of Drude but replaces classical particles with quantum-mechanical fermions obeying the Pauli exclusion principle and Fermi–Dirac statistics.

Free-electron periodic boundary conditions

The boundary conditions $\psi(x + L, y, z) = \psi(x, y, z)$ applied to plane-wave electron states in a cubic box of volume $V = L^3$, which quantize allowed wave vectors to $k_i = \frac{2\pi n_i}{L}$, where $n_i$ is an integer.

Free-electron dispersion relation

The quadratic kinetic energy–wave vector relation for a free electron, given by $E_{\mathbf{k}} = \frac{\hbar^2 k^2}{2m}$, where constant-energy surfaces in $\mathbf{k}$-space are concentric spheres.

Fermi wave vector ($k_{\text{F}}$)

The radius of the occupied Fermi sphere in $\mathbf{k}$-space at zero temperature, determined by electron density $n = \frac{N}{V}$ according to $k_{\text{F}} = (3\pi^2 n)^{1/3}$.

Fermi sphere

The sphere of radius $k_{\text{F}}$ in $\mathbf{k}$-space that encloses all occupied one-electron quantum states in a free-electron gas at absolute zero temperature.

Fermi momentum ($p_{\text{F}}$)

The momentum of an electron at the Fermi surface, defined by $p_{\text{F}} = \hbar k_{\text{F}}$.

Fermi velocity ($v_{\text{F}}$)

The speed of an electron located at the Fermi surface, given by $v_{\text{F}} = \frac{\hbar k_{\text{F}}}{m} = \frac{p_{\text{F}}}{m}$.

Fermi energy ($E_{\text{F}}$)

The energy of the highest occupied one-electron state at zero temperature, given for a 3D free-electron gas by $E_{\text{F}} = \frac{\hbar^2 k_{\text{F}}^2}{2m} = \frac{\hbar^2}{2m} (3\pi^2 n)^{2/3}$.

Fermi temperature ($T_{\text{F}}$)

The temperature scale corresponding to the Fermi energy, defined as $T_{\text{F}} = \frac{E_{\text{F}}}{k_{\text{B}}}$, which is typically on the order of $10^4\,\text{K}$ in metals.

Density of states ($g(E)$)

The number of one-electron quantum states per unit energy interval; for a 3D free-electron gas of volume $V$, it is given by $g(E) = \frac{V}{2\pi^2} \left(\frac{2m}{\hbar^2}\right)^{3/2} \sqrt{E}$.

Density of states at the Fermi energy ($g(E_{\text{F}})$)

The density of states evaluated at the Fermi level, which simplifies to $g(E_{\text{F}}) = \frac{3N}{2E_{\text{F}}}$ and directly dictates low-temperature electronic thermodynamic properties.

Fermi–Dirac distribution function ($f(E)$)

The probability that a fermionic state of energy $E$ is occupied at temperature $T$, defined as $f(E) = \frac{1}{e^{(E - \mu)/(k_{\text{B}}T)} + 1}$, which forms a sharp step function at $T = 0\,\text{K}$ with $\mu(0) = E_{\text{F}}$.

Zero-temperature total energy ($U_0$)

The ground-state internal energy of a 3D degenerate Fermi gas at absolute zero, given by $U_0 = \frac{3}{5} N E_{\text{F}}$, corresponding to an average kinetic energy per electron of $\frac{3}{5} E_{\text{F}}$.

Degeneracy pressure ($P_0$)

The non-thermal pressure exerted by fermions at zero temperature due to Pauli exclusion, given by $P_0 = \frac{2}{3} \frac{U_0}{V} = \frac{2}{5} n E_{\text{F}}$.

Electronic heat capacity ($C_{V,e}$)

The heat capacity of degenerate electrons at low temperatures, given by $C_{V,e} = \frac{\pi^2}{2} N k_{\text{B}} \left(\frac{k_{\text{B}}T}{E_{\text{F}}}\right) = \frac{\pi^2}{3} g(E_{\text{F}}) k_{\text{B}}^2 T$, which is linear in $T$ because only a fraction of order $T/T_{\text{F}}$ of electrons are thermally excitable.

Chemical potential shift ($\mu(T)$)

The temperature dependence of the chemical potential for a 3D Fermi gas at $T \ll T_{\text{F}}$, given by $\mu(T) \simeq E_{\text{F}} \left[1 - \frac{\pi^2}{12} \left(\frac{k_{\text{B}}T}{E_{\text{F}}}\right)^2\right]$, which decreases quadratically with increasing $T$ to conserve electron number.

Electronic heat capacity coefficient ($\gamma$)

The linear coefficient in the low-temperature heat capacity $C_{V,e} = \gamma T$, which can be extracted experimentally from the intercept of a plot of $C_V/T$ versus $T^2$.

Mean free path ($\ell$)

The average distance an electron travels between scattering events, estimated in a degenerate metal by $\ell \sim v_{\text{F}} \tau$, where $v_{\text{F}}$ is the large microscopic Fermi velocity.

Pauli paramagnetic susceptibility ($\chi_{\text{P}}$)

The weak, approximately temperature-independent magnetic susceptibility of conduction electrons, given in SI units by $\chi_{\text{P}} = \frac{\mu_0 \mu_{\text{B}}^2 g(E_{\text{F}})}{V} = \frac{3 \mu_0 n \mu_{\text{B}}^2}{2 E_{\text{F}}}$, governed solely by states near the Fermi surface.

Fermi surface

The boundary in $\mathbf{k}$-space separating occupied from unoccupied electronic quantum states at $T = 0\,\text{K}$, which uniquely governs low-energy thermal, transport, and magnetic excitations in metals.