# Finding the chemical potential of a system

For a system of non interacting electrons at temperature T the density of states is given by

$$g(\epsilon)=\begin{cases}\sqrt{\epsilon-\epsilon_0} & \text{for }\epsilon > \epsilon_0 \\ 0 &\text{for }\epsilon_0>\epsilon>0 \\ \sqrt{-\epsilon} & \text{for }\epsilon<0\end{cases}$$

where $\epsilon$ is the energy of the electron and $\epsilon_{0}$ is a constant. At $T=0$ electrons occupy states up to $\epsilon=0$. What is the chemical potential?

so if I try to find the partition function for a N electron system $$Z =\int g(\epsilon)d\epsilon e^{-\beta(\epsilon- N\mu)}$$ now if I try to do the integral for the part of $\epsilon$<0 with an integral limit of 0 to -$\infty$ the integral blows up, so how do I manage that?

• Try an energy cut-off in the conduction band to keep $N$ finite and let the cut-off go to infinity in the end result. – Sebastian Riese May 7 '18 at 16:03
• did you mean the valance band? the integral for the conduction band results in a gamma function but for the valance band i.e. $\epsilon$<0 if I change negetive $\epsilon$ to a positive variable it results in a positive exponential with infinite limit – koushik naskar May 7 '18 at 16:26
• Yes I mean the valence band, sorry for the confusion. – Sebastian Riese May 7 '18 at 17:50