Write down basic postulates of sommerfield’s free electron gas model. Show that the number of energy states per unit energy is given by g(E) = 4Rv/h3 (2m)3/2 E1/2 . What is the density of energy states in metals? Obtain an expression for the density of states for free electron gas in metals.
Sommerfield's Free Electron Theory
Sommerfeld investigated the behavior of electron gas taking into account the quantum theory and the Pauli's exclusion principle. In Sommerfeld's model the free electrons are assumed to be the valence electrons of the composing atoms. The alkali metals are assumed to contribute one free electron per atom. These electrons are free to move in a constant potential in the metal. They cannot come out of the metal surface, which suggests the presence of high potential barrier supposed to be zero. Thus the conductor acts as a potential well of potential box for the free electrons. To understand this, first we will consider a one-dimensional model and then three-dimensional model of electron gas. According to the free electron model of a solid, all the valence electrons of the constituent atoms of the solid are able to move freely through the volume of the specimen of the solid. These electrons are distinct from those filling the electronic shells of the ion cores. Besides, a s a first approximation it is assumed that those electrons, termed as conduction electrons are unaffected by the periodic fluctuating potential of the ion cores. These electrons therefore, move freely in a region of constant potential and hence their motion can be described by the motion of a free particle in a large box. Where the energy levels lie very close to each other.