# Fermi energy for a fermion gas

I came across an old exam question for my statistical mechanics course:

Let the energy levels of a gas of $$N$$ fermions be given as $$\varepsilon_n = (n+1)^{\alpha} \ \varepsilon_0 \ , \ \ n = 0,1,2... \ \text{and } \alpha > 1$$, with each level having degeneracy :

$$g_n = (n+1)^{\beta} ,\ \ \beta > 1$$

Calculate the Fermi energy , and the total average energy in the limit $$N >> 1$$.

What I've tried

Since $$N \rightarrow \infty$$ we can use the grand canonical ensemble. The average number of particles that occupies the $$n$$th orbital is (?):

$$\langle N_n \rangle = \dfrac{g(n)}{e^{\beta{(\varepsilon_n + \mu) }} +1 }$$

Hence the total number of particles is:

$$N = \sum^{\infty}_n \langle N_n \rangle$$

If $$g(n) = g$$ (i.e. same degeneracy for every level) , the sum can be written as an integral over the momenta $$\vec{p}$$:

$$\sum_n = g\sum_{\vec{p}} = g\dfrac{V}{h^3} \int d^3p$$

In the limit $$T \rightarrow 0$$ , we find:

$$\dfrac{1}{e^{\beta{(\varepsilon_n + \mu) }} +1 } \rightarrow \Theta(\varepsilon_F - \varepsilon)$$ ($$\Theta$$ being the Heaviside stepfunction, and $$\varepsilon_F$$ the Fermi energy), hence in the same limit, for $$g_n = g$$ we have:

$$N = \sum^{\infty}_n \langle N_n \rangle = g\dfrac{V}{h^3} \int \Theta(\varepsilon_F - \varepsilon) d^3p$$

The integral is just volume of the ball with radius $$p_F = \sqrt{2m \varepsilon_F}$$...

What to do if $$g_n$$ is not constant ?

We get the integral (?):

$$\dfrac{V}{h^3} \int d^3 p \ \Theta(\varepsilon_F - \varepsilon) g(n)$$

I'm not quite sure what to do here...

• I suppose this problem corresponds to a three-dimensional ideal gas in the case of $\alpha = \beta = 2$. But you don't really need this correspondence. I think that due to the lack of volume in this problem, inverse temperature and chemical potential have a scaling like $\beta \sim 1/N^\gamma$, $\mu \sim N^\delta$ and it is possible to consider $\sum_n$ as $\int dn$.
– Gec
May 21 at 11:34

In the case of zero temperature, fermions occupy states with lower energy. Thus, the equation for the number $$n_{\text{max}}$$ of the occupied state with the highest energy has the form $$N \approx \sum_{n=0}^{n_{\text{max}}} g_n$$ Condition $$N \gg 1$$ leads to $$n_{\text{max}} \gg 1$$. The relative change of the power function is slow in the domain of large arguments: $$(n+1)^\beta / n^\beta \approx 1$$ for $$n \gg 1$$. Therefore, the equation for $$n_{\text{max}}$$ can be written as $$N \approx \int_0^{n_{\text{max}}} (n+1)^\beta\ dn \approx \frac1{\beta+1} (n_{\text{max}}+1)^\beta.$$ So we have $$n_{\text{max}}+1 \approx (N(\beta+1))^{\frac1{\beta+1}}$$ and $$\varepsilon_{\text{F}} = \varepsilon_0 (n_{\text{max}} + 1)^\alpha \approx \varepsilon_0 (N(\beta+1))^{\frac{\alpha}{\beta+1}}$$