# 13.4 Fermi-Dirac Distribution

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13.4 Fermi-Dirac Distribution
Fermions are particles that are identical and indistinguishable. Fermions include particles such as electrons, positrons, protons, neutrons, etc. They all have half-integer spin. Fermions obey the Pauli exclusion principle, i.e. each quantum state can only accept one particle. Therefore, for fermions Nj cannot be larger than gj. FD statistic is useful in characterizing free electrons in semi-conductors and metals.

For FD statistics, the quantum states of each energy level can be classified into two groups: occupied Nj and unoccupied (gj-Nj), similar to head and tail situation (Note, quantum states are distinguishable!) The thermodynamic probability for the jth energy level is calculated as where gj is N in the coin-tossing experiments. The total thermodynamic probability is

W and ln(W) have a monotonic relationship, the configuration which gives the maximum W value also generates the largest ln(W) value. The Stirling approximation can thus be employed to find maximum W

There are two constrains
Using the Lagrange multiplier

See white board for details

13.5 Bose-Einstein distribution
Bosons have zero-spin (spin factor is 1). Bosons are indistinguishable particles. Each quantum state can hold any number of bosons. The thermodynamic probability for level j is The thermodynamic probability of the system is

Finding the distribution function

13.6 Diluted gas and Maxwell-Boltzman distribution
Dilute: the occupation number Nj is significantly smaller than the available quantum states, gj >> Nj. The above condition is valid for real gases except at very low temperature. As a result, there is very unlikely that more than one particle occupies a quantum state. Therefore, the FD and BE statistics should merge there.

The above two slides show that FD and BE merged.
The above “classic limit” is called Maxwell-Boltzman distribution. Notice the difference They difference is a constant. Because the distribution is established through differentiation, the distribution is not affected by such a constant.

Summary Boltzman statistics: Fermi-Dirac statistics:
Bose-Einstein statistics: Problem 13-4: Show that for a system of N particles obeying Maxwell-Boltzmann statistics, the occupation number for the jth energy level is given by