# Constant density in Sommerfeld model

In chapter 2 of Ashcroft and Mermin's Solid state physics we reach this expression of the electron energy density:

$$n=\int_{0}^{\varepsilon_{F}} g(\varepsilon) d \varepsilon+\left\{\left(\mu-\varepsilon_{F}\right) g\left(\varepsilon_{F}\right)+\frac{\pi^{2}}{6}\left(k_{B} T\right)^{2} g^{\prime}\left(\varepsilon_{F}\right)\right\}$$

We state that n is independent of temperature but we say:

$$0=\left(\mu-\varepsilon_{F}\right) g\left(\varepsilon_{F}\right)+\frac{\pi^{2}}{6}\left(k_{B} T\right)^{2} g^{\prime}\left(\varepsilon_{F}\right)$$

and not

$$0=\frac{\pi^{2}}{6}\left(k_{B} T\right)^{2} g^{\prime}\left(\varepsilon_{F}\right)$$

Where is the temperature dependence in the first term, is it due to the chemical potential energy being T dependent?

It feels a bit backwards because we do this to get the chem. potential T dependence so it feels circular. If anybody has a better explanation I'm missing that would be great.

• I'm not sure why you think it's circular. Maybe it'll be clearer if you try to take the zero temperature limit in the first equation. – Anyon Feb 11 at 1:48