# Applications and problems of FD – Statistics

Fermi–Dirac statistics has many applications in studying electrical and thermal conductivity, thermo-electricity, thermionic and photoelectric effects, specific heat of metals, and many more. Using the assumption that metals contain free electrons constituting like a perfect gas known as ‘electron gas’. Though many scientists worked on this, but Sommerfeld in 1928 rescued quantum electron theory of metals. The conductivity of metals is due to the presence of free electrons inside a metallic conductor moving freely inside them and colliding with fixed atoms behaving like an electron gas. These are called ‘fermions’ because electrons abide by Pauli’s exclusion principle.

1. Average energy of electron, at absolute zero temperature.
2. Average energy of electron at T > 0 and electronic specific heat.
3. Richardson – Dushman equation (thermionic emission)

## 1.Fermi change of electron at absolute zero temperature (T = 0) :

Since electrons are indistinguishable and spin half particles, Fermi- Dirac statistics can be applicable to electrons.

At absolute zero temperature, Fermi energy

Number of electrons in the energy range and E and E+dE,

N(E)dE = f(E)g(E)dE  …………………….. 1

where f(E) = FD distribution function

g(E)dE = number of quantum states of electron having energy between E and E+dE

Thus total number of electrons having energy E=0 to E=∞

Substituting the value of c, we get

where n = N/V ; number of electron per unit volume.

## 2. Average energy of each electron at absolute zero temperature (T=0)-

The highest energy level which an electron can hold in the valence band at 0 K, is known as Fermi energy level.

Since electrons are indistinguishable and spin-half particles, so FD- statistics may be applicable to electrons,

Thus average energy of an electron at absolute zero,

For electron

Total energy of electrons at absolute zero temperature,

Energy of  conduction electrons at any temperature (T>0) –

## 3. Specific heat of conduction- electrons in metals –

Electrons are indistinguishable, spin half particles, hence , free electrons in metal are considered as ” fermions”.

Number of electrons having energy between E and E+dE,

N(E)dE = f(E)g(E)dE  …………………….. 1

where f(E) = F.D. distribution function

g(E)dE = number of quantum states having energy between E and E+dE,

again

Total number of electrons having energy from 0 to ∞

……………….. 2

Total energy of electrons having energy from 0 to ∞,

…………………….. 3

To solve above two integrals we introduce a function λ(E), such that

λ(E) → 0 when E → 0

and λ(E) → 0 when E → ∞

Integral :

……………………. 4

Expanding λ(E) by Taylor’s series expansion about E = FF ,

……………………………. 5

Solutions of above integrals

as f(E) → 0 when E → ∞

and f(E) → 1 when E → 0

Substituting above solutions in equation 5, we get ,

Let us consider another function ϕ(E), such that

…………………. 6

I. Using this standard integral ; taking ϕ(E) = cE^1/2, we get N from equation 2, as

………………. 7

Since EF >> kT , hence we may take the crude approximation as

on the fraction term

…………………. 8

II. Using standard integral 6 , taking ϕ(E) = cE^3/2, we get E from equation 3

with crude approximation

…………………. 9

Average energy of electron at any temperature T,

…………………………… 10

III. Electronic specific heat / Electronic heat capacity :

………………………… 11

In terms of Fermi temperature TF ; which is defined as ,

……………………….. 12

FD statistics thus  successfully resolves the discrepancy regarding specific heat of the classical limit

Cv = 3/2 Nk

Equation 11 shows that electronic specific heat is linearly proportional to temperature as

Cv = αT   where

Again at low temperatures, the specific heat due to lattice vibration is proportional to T^3 ; called Debye T^3 law.

Thus total specific heat of metals at low temperatures,

## Richardson – Dushman thermionic emission equation

In 1911 Richardson proved that electrons are emitted from hot metal and not from the surrounding air, as some had thought. That same year he proposed a mathematical equation that relates the rate of electron emission to the absolute temperature of the metal. This equation, called Richardson’s law or the Richardson- Dushman equation, became an important tool in electron-tube research and technology.

If electrons within a conductor come up to the surface with sufficient energy such that they may escape through the surface leading to the phenomena of thermionic emission of electrons.

The electrons will give rise to a current which was derived lay Richardson and Dushman ; known as Richardson – Dushman equation.

On heating a metal to a temperature T, electrons will escape into vaccum from the metal surface i.e. emission of elections take place when Kinetic energy of election ≥ Es

Where

Es is the surface energy

EF is the Fermi energy Er

ϕ is the work function of metal

Derivation :

Since electrons are indistinguishable spin half particles hence they obey Fermi- Dirac statistics.

Number of electrons having energy between E and E+dE,

N(E)dE = f(E)g(E)dE  …………………….. 1

where f(E) = FD distribution function

when

Since the elections are free, hence energy = KE + 0

…………….. 2

m is the mass of electron.

Number of electrons with velocity components

…………………… 3

as for electron :

Number of electrons per unit volume wilt velocity components

………………… 4

In order to escape from the metal surface along + x direction (normal to metal surface) at temperature T,

Thus number of electrons escaping per unit area with energy

Hence current density ,

This is known as Richardson – Dushman equation for thermionic emission.