Q. Give the basic assumptions made in free electron model of solid.
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) mean by Fermi energy level.
Ans. Sommerfield's Free Electron Gas Model
The theory of electrical conductivity in metals based on quantum theory of free electrons was given by sommerfield. According to this theory, the electron is treated as a wave. Therefore, the electrons are bound to obey the quantum concepts. Sommerfield kept the other concepts of the classical free electron theory, which are as follows :
(a) Since the electron is behaving as a wave, the velocity and the energy distribution of the electron is explained by Fermi Dirac statistics.
(b) The electrons are treated as free to move anywhere within the crystal.
(c) There is negligible electrostatic interaction between the electrons and the lattice ions.
(d) The free electrons are totally responsible for electrical conductive nature of the material.
(e) Negligible interaction between the electrons.
(f) The effect of the electric field is to change the distribution of electrons, the electrons move away from the field with a constant velocity and this velocity known as drift velocity.
(1) Success of Free Electron Theory
1. The quantum free electron theory is successful in explaining most of the properties of metal such as the electronic specific heat, electrical conductivity. Electronic par magnetism, Fermi energy and thermionic emission.
2. The most important feature of this theory is that even at very low temperatures the electrons have appreciable energy because of the Pauli exclusion principle onlt those electrons which are close to the Fermi energy take part in excitations either due to thermal energy or due to extreme electric and magnetic fields.
3. Even all success, it fails to explain the positive value of hall coefficient and some transport properties of metals.
(2) Density of Energy States
The total number of available electron state per unit energy range is called density of available electron stat and the number of states the t are filled with electrons per unit energy I s called density of filled electron state.
In order to know the actual number of electron within a given range, we have to know the number of states in the system which has the energy consideration.
(3) Calculation of N (E) Sphere
(4) Fermi Energy, Fermi Level and Fermi Surface : At absolute zero temperature, all states up to a surface are filled with electrons and those above are empty from the surface is called Fermi surface. It simply means that a level which separate the filled unfilled energy levels at 0 K, above the level all the energy states are unfilled and below this all state are completely filled, this level is called Fermi level.