The free electron theory of metals Energetics

The simplest electronic theory of metals regards a metallic object as abox filled with noninteracting electrons. (A slightly more elaborate picture is the jellium model in which the free electrons are moving on the backgroimd of a continuous positive uniform charge distribution that represents the nuclei.) The Drude model, built on this picture, is characterized by two parameters The density of electrons n (number per unit volume) and the relaxation time r. The density n is sometimes expressed in terms of the radius of a sphere whose volume is the volume per electron in the metal 

The density of states of a free particle as a function of its energy E was obtained in Section 2.8.2. It is given by 

Let N be the total number of free electrons and n their density, so that N = nO,. Being Fermions, we can have at most one electron per state. This implies that at 

7 = 0 the highest occupied energy, the Fermi energy Ep, has to satisfy 

Problem 4.2. Show that the ground state energy of this N electron system is given by 

At finite temperature the picture described above changes slightly. The probability that a single electron level of energy E is occupied is given by the Fermi-Dirac distribution 

---