FERMI-DIRAC STATISTICS

The particles are indistinguishable. The total wave function is anti-symmetric with respect to interchange particles. Fermions obey the Pauli's exclusion principle. Restriction on the number of fermions which can occupy an energy level as given by the Pauli's principle. In any system of indistinguishable particles, each molecular quantum state can be occupied by not more than one molecule at a time. This restriction is a distribution quite different from the Blotzmann-Distribution.

                Consider the ni particles are distributed among the gi states where gi is the degeneracy of the energy states. Since there can be only one particle in each energy level ni, the number of particles can be equal to gi. The first particle can be placed in any one of the available states. The second can be arranged in (gi1) different ways. Thus the number of arrangement is given by the expression,

W'=.      giǃ       ........(1)

Since the particles are indistinguishable and ni, particles can be arranged in niǃ ways..... The above expression has to be divided by the possible number of permutations of ni particles. Hence the number of arrangement in the i-th energy level is given by expression,

W'=    giǃ     

      niǃ( ɡi−ni) ǃ ...........(2)

Thus the thermodynamic probability W for the system of N particles among the various energy levels is given by

W=∏    gi      

       i  niǃ( ɡi−ni)ǃ..........(3)

Take log on both sides of equation (3) we have

lnw=∑i [ lnɡiǃ− ln niǃ− ln (ɡi−ni) ǃ]

Applying Stirling's approximation,

ln W= ∑i ɡi ln ɡi−ɡi−ni ln ni+ni −( ɡi−ni) ln (ɡi−ni) +( ɡi−ni)........(4)

The most probable distribution is one for which dlnW=0

Differentiate equation (4) w.r.t ni and equate to zero for maximum value of W

dln W= ∑i [−ni ln ni + (ɡi−ni)     1     +ln (ɡi −

                                                 (ɡi−ni)                ni)] dni = 0..........(5)

∑i ln   ɡi−ni   dni =0..........(6)

            ni                                                             Since N= ∑i ni= constant and ∑i ni¡= constant,

Hence dN= ∑i dni=0...........(7) and 

∑i∈idni =0......(8)

Applying Lagrange's undetermined multipliers, multiply equation (7) by α and (8) β and subtract from the equation (6) we obtain

∑i [ln    ɡi−ni    − α−β] dni=0.....(9)

               ni                                                         ln   ɡi   ni    −α−β =0.......(10).                                    ni                                                          ln     gi-ni       = α+ β:       gi-ni       =α+β εi : 

         ni                             ni                                    gi   - 1=eᵅ⁺β εi:      gi    = eᵅ⁺β εi +1                ni                            ni                                  ni =         gi       .............. (11)

        eᵅ⁺βεi+1    

Equation (11) is known as Fermi-Dirac distribution law.

      If eᵅ⁺βεi >>1, then equation (11)

can be written as ni=      ɡi           which

                                       eᵅ⁺βεi                         is M-B distribution law. Thus M-B distribution appears to be the classical limit of F-D distribution law.