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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.

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  • $\begingroup$ 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. $\endgroup$ – Anyon Feb 11 at 1:48

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