For question 2.8 in the Many atoms per unit cell, what is the definition used for the fermi energy. When looking it up, it is the energy difference between the highest and lowest occupied single-particle states. When applying this definition I don’t see how the fermi energy is E_0.
The fermi energy is the energy of the last occupied state. In this case, if you look at how many states there are, you should find 2N_{\text{atoms}}. Such that if each atom contributes one electron you find that half of the states are filled starting from the bottom of the density of states.
If I am not mistaken, the fermi energy and the concept that is hinted in this exercise will be revisited in much more depth when we work with tight binding for weakly bound electrons. Stay tuned 