The occupation of orbitals
Now consider the electronic structure of a solid formed from atoms each able to con tribute one electron (for example, the alkali metals). There are N atomic orbitals and therefore N molecular orbitals packed into an apparently continuous band. There are Nelectrons to accommodate. At T = 0, only the lowest 
N molecular orbitals are occupied (Fig. 20.53), and the HOMO is called the Fermi level. However, unlike in molecules, there are empty orbitals very close in energy to the Fermi level, so it requires hardly any energy to excite the uppermost electrons. Some of the electrons are therefore very mobile and give rise to electrical conductivity.
At temperatures above absolute zero, electrons can be excited by the thermal motion of the atoms. The population, P, of the orbitals is given by the Fermi–Dirac distribution, a version of the Boltzmann distribution that takes into account the effect of the Pauli principle:
P=
The quantity µ is the chemical potential, which in this context is the energy of the level for which P = (note that the chemical potential decreases as the temperature increases). The chemical potential in eqn 20.23 has the dimensions of energy, not energy per mole.
The shape of the Fermi–Dirac distribution is shown in Fig. 20.54. For energies well above µ, the 1 in the denominator can be neglected, and then P≈e−(E−µ)/kT . The population now resembles a Boltzmann distribution, decaying exponentially with increasing energy. The higher the temperature, the longer the exponential tail. The electrical conductivity of a metallic solid decreases with increasing temperature even though more electrons are excited into empty orbitals. This apparent paradox is resolved by noting that the increase in temperature causes more vigorous thermal motion of the atoms, so collisions between the moving electrons and an atom are more likely. That is, the electrons are scattered out of their paths through the solid, and are less efficient at transporting charge.
Fig. 20.53 When N electrons occupy a band of Norbitals, it is only half full and the electrons near the Fermi level (the top of the filled levels) are mobile.
Fig. 20.54 The Fermi–Dirac distribution, which gives the population of the levels at a temperature T. The high-energy tail decays exponentially towards zero. The curves are labelled with the value of µ/kT. The pale green region shows the occupation of levels at T=0.
