## Quantum free electron theory of metals:

- Quantum free electron theory was proposed by Arnold Sommerfeld.
- As per his investigation, the behavior of free electron in different possible energy states is explained, how the large number of electrons are disturbed in the energy states.

Assumptions :

- The energies of free electrons are quantized.
- The distribution of electrons is as per the Pauli's exclusion principle.
- Electrons travel under constant potential and confine to the boundaries of metal.
- All the attractive and repulsive forces are neglected.

## Fermi energy level and Fermi energy:

- Electrons on this energy level are also called Fermions and obey Pauli's exclusion principle.
- Each energy level can accommodate at most two electrons.
- The highest occupied energy level by the electron at zero kelvin is called Fermi energy level.
- At zero kelvin energy levels below the Fermi level are completely filled and above Fermi level all the energy levels are empty.
- If the temperature increases above zero kelvin, the probability occupation of electrons at that level is half.
- That particular energy of that level is called Fermi energy.
- The value of Fermi energy in metal is 5eV.

## Success of quantum free electron theory of metals:

1.

At zero kelvin energy levels under Fermi energy level are filled by electrons. When the temperature is increased to room temperature few number lo electrons conduct heat or electricity.

If we consider one kilo mole of metal

__Specific heat of solid:__At zero kelvin energy levels under Fermi energy level are filled by electrons. When the temperature is increased to room temperature few number lo electrons conduct heat or electricity.

If we consider one kilo mole of metal

2.

We know that the electrical conductivity is given by

__Temperature dependence of electrical conductivity:__We know that the electrical conductivity is given by

We know the relaxation time can be written as

from the above equation we can write,

Substituting this equation in electrical conductivity expression we get,

In this equation there is no direct relation between the electrical conductivity and the temperature but we know

So, from both the above equations we get,

So this theoretical result is agreed with the experimental result.

3.

As per quantum theory the electrical conductivity is given by,

__Dependence of electrical conductivity on the concentration of electron:__As per quantum theory the electrical conductivity is given by,

from this equation we have,

This theory also fails to explain the direct relation between the electrical conductivity and the electron density.

4. The concepts of electrical conductivity and thermal conductivity are explained by this theory.

5. The phenomenon of Ferro magnetism, Para magnetism and susceptibility are explained by this theory.

6. The concept of photoelectric effect, Compton effect and black body radiation are explained by this theory.

5. The phenomenon of Ferro magnetism, Para magnetism and susceptibility are explained by this theory.

6. The concept of photoelectric effect, Compton effect and black body radiation are explained by this theory.

## Fermi Dirac distribution F(E) :

Distribution of electrons among various energy levels is described by the distribution function is called Fermi Dirac distribution F(E). The distribution at any temperature T as be expressed by Fermi Dirac distribution

If we raise the temperature above zero kelvin, those electrons at the Fermi level absorb the energy and moves to higher energy states.

This equation mean that the probability of an electron at Fermi level is half above zero kelvin. If the temperature increases to infinity the electrons will loose their quantum mechanical character and Fermi Dirac distribution and reduces to classical theory. If we plot a graph between F(E) and E we get,

## Expression for electrical conductivity on the basis of quantum free electron theory:

With the application of electrical field, electron acquires addition velocity which is called drift velocity. According to Newton's second law of motion,

## Scattering of electron and electrical resistivity:

The main factors effecting electrical resistivity of solid are

1. Temperature.

2. Impurities.

In general the resistivity change is represented by

1. Temperature.

2. Impurities.

In general the resistivity change is represented by

__Temperature dependence:__

As temperature of the solid increases, atoms vibrate with larger amplitudes.

We also know,

From these equations, we get the relation between the resistivity and temperature

There may be a chance to dislocate from its mean position, which leads to increase in the number of scattering electrons.