The most common system of many fermions is a metal. A metal can be thought of as a gas of electrons and following Sommerfeld’s approach, we will discuss some of the properties of an electron gas. We will focus on the non-interacting Fermi gas, commonly known as the ideal Fermi gas.
The value of the chemical potential at $T=0$ is known as the Fermi energy, $\epsilon_F$. The Fermi temperature, $T_F$ is the temperature at which the thermal energy is $\epsilon_F$.
\[T_F=\frac{\epsilon_F}{k_B}.\]This and other related quantities are summarized below.
| Quantity | Symbol | Defining Equation |
|---|---|---|
| Fermi temperature | $T_F$ | $k_BT_F=\epsilon_F$ |
| Fermi momentum | $p_F$ | $\frac{p_F^2}{2m}=\epsilon_F$ |
| Fermi wave vector | $k_F$ | $\hbar k_F=p_F$ |
| Fermi pressure | $P_F$ | $P_F=\frac{2}{5}n\epsilon_F$ |
At low temperatures ($T\ll T_F$), the Fermi-Dirac distribution function looks like a step function and the Fermi gas is said to be degenerate. At absolute zero, it is completely degenerate and all states below the Fermi energy are filled and all states above are empty. At very low temperatures, the Fermi gas is said to be strongly degenerate. With increasing temperature, the Fermi gas becomes more and more non degenerate and the distribution resembles the Maxwell-Boltzmann distribution at high temperatures.
Conversion from summation over all wave vectors to an integral over all of $k$-space
Consider a system defined on a finite cubic volume $V=L^3$. To avoid boundary effects, it is convenient to define the system using periodic boundary conditions. In such a setting, free particles have plane wave eigenfunctions.
\[\psi_{\vec{k}}(\vec{r}) = \frac{e^{i\vec{k}\cdot\vec{r}}}{\sqrt{V}}\]Imposing periodic boundary conditions along the $x$-direction, we have,
\[\psi_{\vec{k}}(x+L,y,z) = \psi_{\vec{k}}(x,y,z) ~\Rightarrow~ k_x L=2\pi n_x,\]where $n_x$ is an integer. The same holds for the other components of $\vec{k}$. Thus,
\[\vec{k} = \frac{2\pi}{L}(n_x,n_y,n_z),~~ n_x,n_y,n_z\in\mathbb{Z}\]Now, a sum over all allowed $\vec{k}$ of some function $f(\vec{k})$ can be simplified as follows.
\[\sum_{\vec{k}}f(\vec{k}) = \sum_{k_x}\left[ \sum_{k_y}\left( \sum_{k_z} f(\vec{k}) \right) \right] = \frac{1}{\Delta k_x} \frac{1}{\Delta k_y} \frac{1}{\Delta k_z} \sum_{k_x}\left[\Delta k_x \sum_{k_y}\left\lbrace\Delta k_y \sum_{k_z}\left(\Delta k_z f(\vec{k}) \right) \right\rbrace \right],\]where $\Delta k_\alpha$ represents the amount by which $k_\alpha$ changes for consecutive terms in the sum over $k_\alpha$. Thus, $\Delta k_\alpha=\frac{2\pi}{L}$ for $\alpha\in\lbrace x,y,z\rbrace$. When $L$ is large, we can therefore convert the sum to an integral.
\[\sum_{\vec{k}}f(\vec{k}) = \frac{V}{(2\pi)^3} \int\limits_{-\infty}^\infty dk_x \int\limits_{-\infty}^\infty dk_y \int\limits_{-\infty}^\infty dk_z f(\vec{k}) = \frac{V}{(2\pi)^3} \int\limits_{\begin{array}{c}{\sf all}\\k{\sf -space}\end{array}} d^3k f(\vec{k})\] \[\Rightarrow \sum_{\vec{k}} = \frac{V}{(2\pi)^3} \int\limits_{\begin{array}{c}{\sf all}\\k{\sf -space}\end{array}} d^3k\]