3.39 Sspd_chapter 1_part 16_fermions_part 17_density of states in  (Page 9/2)

SSPD_Chapter 1_ Part 16 + Part 17.

1.16. FERMION AND FERMI DIRAC STATISTICS.

In nature, particles with (n+1/2)ћ spin angular momentum are distinguishable particles and hence a pair of particles with opposite spin occupy an elemental phase space and no more. Hence at one time, two particles will occupy different energy states or at one spatial coordinate two particles will be at different momenta.

These particles obey Heisenberg’s Uncertainty Principle:

∆E.∆t ≥ ћ 1.161

∆p x .∆x ≥ ћ

Heisenberg’s Uncertainty Principle leads us to Pauli Exclusion Principle. Pauli Exclusion Principle directly comes from Quantum Mechanics. Until the advent of Quantum Mechanics we could not arrive at the correct definition of Specefic Heat.

According to Quantum Mechanics, fermions obey the following distribution function also known as Fermi-Dirac Statistcs:

P(E) = 1/[ Exp{(E-E F )/(kT)} +1 ] ................................1.162

This distribution function or probability function tells us that at 0 Kelvin absolute temperature the probability of occupancy till E _F energy level is Unity and above E _F probability of occupancy is zero. This implies that energy levels till E _F are occupied and above are empty. Each level till E _F are occupied by two oppositely spinning electrons. At 0 Kelvin the probability distribution function is rectangular as shown in Figure(1.86).

Figure(1.86) At 0 Kelvin, fermions having rectangular probability distribution function.

At T Kelvin, probability distribution is skewed and the probability of occupancy at E _F is 50% . This means that half of the time E _F is occupied and half of the time it is empty and electrons are occupying a higher energy state. The probability of occupancy distribution at T Kelvin is shown in Figure(1.87).

Figure(1.87) Skewed Probability Distribution of Fermions at T Kelvin.

As shown in Figure(1.87), around Fermi-level E _F and below number of electrons, n, is only slightly less than the available allowed energy states N(E)dE. This is known as DEGENERATE STATES and fermions follow Fermi-Dirac Statistics.

At energies E- E _F >>kT and at 300 Kelvin kT= 0.026eV that is at E equal or greater than (E _F + 0.26eV), the distribution function is the following:

P(E)=Exp[-(E- E F )/(kT)] ...................................1.163

The above distribution function is the classical distribution function known as Maxwell- Boltzmann Distribution Function followed by ideal gas. Therefore Electrons at E ≥ (E _F + 0.26eV) behave like ideal gas molecules in a closed container. If we look closely we find that at energies equal or greater than (E _F + 0.26eV) ,

n<<N(E)dE

That is number of particles is much less than the available allowed states and this is known as NON-DEGENERATE STATES.

Therefore we can generalize that fermions in degenerate state follow Fermi-Dirac Statistics whereas fermions in non-degenerate states follow Maxwell- Boltzmann Statistics.