Expression for Electrical Conductivity based on Quantum - Civil Engineering (CE) PDF Download

Expression for electrical conductivity based on quantum theory
By applying Fermi-Dirac statistics and using Boltzmann transport equation, Sommerfield obtained the expression for electrical conductivity of metals. It is given by

 where m∗ is the effective mass of the electron in metals. Expression for resistivity can be written as

Merits of Quantum free electron theory 
The drawbacks of classical free electron theory were removed by Quantum theory. Some examples are given below.
1. Specific Heat:
According to classical theory electrons give significant contribution to specific heat of solids. But experimentally it was found that contribution to specific heat from electrons is negligibly small. According to quantum theory, When heat energy is supplied to solid, only those electron occupying energy levels closer to Fermi level absorb energy and get exited to higher energy levels. Thus only small percentage of electrons is capable of contributing to specific heat of the solid.
It can be shown that only the fraction (kT/ EF ) of electrons contribute to specific heat of solid.
Thus according to the quantum free electron theory of metals energy of all the electrons in one-kilo mole of solid is given by

Where N_A=Avogadro number,k=Boltzmann constant and T=Absolute temperature. Therefore the electronic specific heat is given by

Which is in agreement with the experimental result.

Temperature dependence of electrical conductivity:
According to classical theory σ ∝ 1/ √ T
But experimentally it is found that σ ∝ 1/T
Expression for electrical conductivity can be written as

Since E_F is independent of temperature, v_F is also independent of temperature. Then only λ_F is dependent on temperature. We know that even at very high temperature only electrons closer to Fermi energy get exited. Thus only small percentage of electrons contributes to the electrical conductivity. It can shown that at very temperatures λ_F is inversely proportional to the temperature. Therefore it follows that

σ ∝ 1/T
Which is agreement with the experimental results.

3. Dependence of electrical conductivity on electron concentration:
According to quantum free electron theory expression for electrical conductivity is given by

Note that σ depends on n , but n depends on E_F , also v_F depends on E_F .
Thus there is no direct relationship between σ and n.

Expression for electron concentration from molecular data:
Let M_A=Atomic weight, N=number of free electrons contributed by each atom to solid, D=is the density of the solid and N_A=Avogadro number.
M_A kg of solid contains N_A atoms
1 kg of solid contains N_A/M_A atoms
Unit volume of a solid weighs D kgs
Therefore unit volume of solid contains D(N_A/M_A) atoms Therefore number of electrons per unit volume is given by

n = NN_AD/MA