The Fermi-Dirac Probability Function

Figure 1.1 shows the ith energy level with g_i quantum states. A maximum of one particle is allowed in each quantum state by the Pauli exclusion principle. There are g_i ways of choosing where to place the first panicle, (g_i - 1 ) ways of choosing where to place the second particle, (g_i - 2) ways of choosing where to place the third particle, and so on. Then the total number of ways of arranging N_i particles in the ith energy level (where N_i ≤ g_i) is

(1)

This expression includes all permutations of the N_i particles among themselves.

However, since the particles are indistinguishable, the N_i ! number of permutations that the particles have among themselves in any given arrangement do not count as separate arrangements. The interchange of any two electrons. for example, does not produce a new arrangement. Therefore, the actual number of independent ways of realizing a distribution of N_i particles in the ith level is

(2)

Figure 1.1 The ith energy level with g_i quantum state.

Equation (2) gives the number of independent ways of realizing a distribution of N_i particles in the ith level. The total number of ways of arranging (N₁, N₂, N₃, . . . , N_n) indistinguishable particles among n energy levels is the product of all distributions, or

(3)

The parameter W is the total number of ways in which N electrons can be arranged in this system, where  is the total number of electrons in the system. We want to find the most probable distribution, which means that we want to find the maximum W. The maximum W is found by varying N_i among the E_i levels, which vanes the distribution, but at the same time, we will keep the total number of particles and total energy constant.

We may write the most probable distribution function as

(4)

where E_F is called the Fermi energy. The number density N (E) is the number of particles per unit volume per unit energy and the function g (E) is the number of quantum states per unit volume per unit energy. The function f_F (E) is called the Fermi-Dirac distribution or probability function and gives the probability that a quantum state at the energy E will be occupied by an electron. Another interpretation of the distribution function is that f_F (E) is the ratio of filled to total quantum states at any energy E.