Fermi-Dirac Distribution Law
This statistics is obeyed by indistinguishable particles of half-integral spin that have anti-symmetric wave function and obey Pauli exclusion principle (e.g. electron, proton and neutron).
Consider a system having n distinguishable particles. Let these particles be divided into groups or lavels such that there are (n1, n2, n3,……ni) number of particles in groups whose approximate constant energy (ϵ1, ϵ2, ϵ3,…… ϵi) respectively. Let gidenote the degeneracy or statistical weight of ith level.
The number of distinguishable arrangements of (ni) particles in (gi) cells
`\frac{g_i!}{n_i!(g_i-n_i)!}` ........(1)
The total number of eigen states for the whole system is given by
`P=\π\frac{g_i!}{n_i!(g_i-n_i)!}` ........(2)
The Fermi-Dirac distribution law can now be obtained by determining the most probable distribution.
Taking log of eq. (2), we get
`\log P=\log\π\frac{g_i!}{n_i!(g_i-n_i)!}` ......(3)
The conditions of most probable distribution gives
`\delta(\log P)=-\sum_i\{\log\frac{g_i!}{(g_i-n_i)!}\}\delta n_i` .......(4)
For the equilibrium state, we must have δ(log P) = 0
`\sum_i\{\log\frac{g_i!}{(g_i-n_i)!}\}\delta n_i=0` ......(5)
Subject to the limitations
`\sum_i\delta n_i=0` ......(6)
`\sum_i\in_i\delta n_i=0` .......(7)
Applying the lagrange method of undetermined multiplier i.e., multiplying eq. (6) by α and eq. (7) by β and adding the resulting expression to eq. (5), We get
`\sum_i\{\log\frac{n_i}{g_i-n_i}+\alpha+\in_i\beta\}\delta n_i=0` ......(8)
As the variation (δni) are independent of each other,
`\log\frac{n_i}{(g_i-n_i)}+\alpha+\beta\in_i=0`
`\log\frac{n_i}{(g_i-n_i)}=-(\alpha+\beta\in_i)`
`\frac{n_i}{(g_i-n_i)}=e^{-(\alpha+\beta\in_i)}`
`\frac{(g_i-n_i)}{n_i}=e^{(\alpha+\beta\in_i)}`
`\frac{g_i}{n_i}-1=e^{(\alpha+\beta\in_i)}`
`\frac{g_i}{n_i}=e^{(\alpha+\beta\in_i)}+1`
`n_i=\frac{g_i}{e^{(\alpha+\beta\in_i)}+1}`
This is known as Fermi-Dirac distribution Law. This equation represents the most probable distribution of the particles among various energy levels for a system obeying Fermi-Dirac statistics.