A fermion system has a density of states that increases with energy. With particle number fixed, which statements about its chemical potential $\mu$ and Fermi–Dirac occupation are correct?
The chemical potential decreases as temperature increases.
The chemical potential increases as temperature increases.
The chemical potential is independent of temperature.
The occupation probability of a state at energy equal to the chemical potential is 1/2.
Correct answer: (A) The chemical potential decreases as temperature increases.; (D) The occupation probability of a state at energy equal to the chemical potential is 1/2.
Explanation
At low temperature, the Sommerfeld expansion gives $\mu(T)\simeq E_F-\frac{\pi^2}{6}(k_BT)^2g\prime(E_F)/g(E_F)$. An increasing density of states has $g\prime(E_F)>0$, so the chemical potential decreases. Also $f(E)=[e^{(E-\mu)/(k_BT)}+1]^{-1}$ gives $f(\mu)=1/2$ directly. Hence A and D. The temperature trend is the usual fixed-number, low-temperature result underlying this question.
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