Thermal Physics: Fermi Gas and chemical potential

[WWCY]

Hi all, I have an issue trying to understand the following paragraph from Blundell's book.

How, exactly, does the definition of $\mu_0 = E_F$ "make sense"? In the sentence after 30.21, it seems to say that the mean energy for a system with $N$ particles differs from that of a system with $N-1$ particles by the highest occupy-able energy level at $T = 0$, which is $E_f$. What does this mean?

Many thanks in advance.

 

[DrClaude]

The chemical potential can be seen as the (Gibbs free) energy per particle, or the energy necessary to add one more particle to the system. With fermions at T = 0, all states up to $E_F$ are occupied, therefore the next fermion will add an energy of $E_F$ to the system. It does make sense to equate $\mu(T=0) = E_F$.