Fermi Gas

Derivation of the Fermi Energy

Consider a crystal lattice with an electron gas as a 3 dimensional infinite square well with dimensions l_x, l_y, l_z. The wavefunctions of individual fermions (pretending they are non-interacting) can be seperated as ψ(x,y) = ψ_x(x)ψ_y(y)ψ_z(z). The solutions will be the usual ones to the Schrodinger equation:

$$\frac{-\hbar^2}{2m}\frac{d^2 \psi_x}{dx}=E_x \psi_x$$

with the usual wave numbers $k_x=\frac{\sqrt{2mE_x}}{\hbar}$, and quantum numbers satisfying the boundry conditions k_xl_x = n_xπ. The full wavefunction for each particle will be:

$$\psi_{n_{x}n_{y}n_{z}}(x,y,z)=\sqrt{\frac{4}{l_{x}l_{y}}}\sin\left(\frac{n_{x}\pi}{l_{x}}x\right)\sin\left(\frac{n_{y}\pi}{l_{y}}y\right)\sin\left(\frac{n_{z}\pi}{l_{z}}z\right)$$

and the associated energies (with E = E_x + E_y + E_z):

$$E_{n_{x}n_{y}n_z}=\frac{\hbar^{2}\pi^{2}}{2m}\left(\frac{n_{x}^{2}}{l_{x}^{2}}+\frac{n_{y}^{2}}{l_{y}^{2}}+\frac{n_{z}^{2}}{l_{z}^{2}}\right)=\frac{\hbar^2|\vec{k}|^2}{2m}$$

where |k⃗|² is the magnitude of the particle’s k-vector in k-space. This k-space can be imagined as a grid of blocks, each representing a possible particle state (with a double degeneracy for spin). Positions on this grid have coordinates (k_x,k_y,k_z) corresponding to the positive integer quantum numbers. These blocks will be filled from the lowest energy upwards: for large numbers of occupying particles, the filling pattern can be approximated as an expanding spherical shell with radius $|\vec{k_F}|^2$.

Note that we’re “over counting” the number of occupied states because the “sides” of the quarter sphere in k-space (where one of the associated quantum numbers is zero) do not represent valid states. These surfaces can be ignored for very large N because the surface area to volume ratio is so low, but the correction can be important. There will then be a second correction due to removing the states along the individual axes twice (once for each side-surface), u.s.w.

The surface of this shell is called the Fermi surface and represents the most excited states in the gas. The radius can be derived by calculating the total volume enclosed: each block has volume $\frac{\pi^3}{l_x l_y l_z}=\frac{\pi^3}{V}$ and there are N/2 blocks occupied by N fermions, so:

$$\frac{1}{8}(\frac{4\pi}{3} |k_{F}|^{3}) = \frac{Nq}{2}(\frac{\pi^{3}}{V}) $$ $$|k_{F}| = \sqrt{\frac{3Nq\pi^2}{V}}^3=\sqrt{3\pi^2\rho}^3$$

ρ is the “free fermion density”. The corresponding energy is:

$$E_{F}=\frac{\hbar^{2}}{2m}|k_{F}|^{2}=\frac{\hbar^{2}}{2m}(3\rho \pi)^{2/3}$$