 Fixed-chemical-potential Sommerfeld density increase

For a three-dimensional parabolic band and fixed positive $\mu -E_c$, the Sommerfeld expansion gives $n(T)=n(0)[1+{\pi }^2(k_{B}T/(\mu -E_c){)}^2/8+O(T^4)]$. This increase differs from the usual fixed-density calculation, in which the chemical potential decreases with temperature. The exact density is proportional to $T^{3/2}{F}_{1/2}[(\mu -E_c)/(k_{B}T)]$.