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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 N j cannot be larger than g j. FD statistic is useful in characterizing free electrons in semi-conductors and metals.

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

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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

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There are two constrains Using the Lagrange multiplier

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See white board for details

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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

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Finding the distribution function

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13.6 Diluted gas and Maxwell- Boltzman distribution Dilute: the occupation number N j is significantly smaller than the available quantum states, g j >> N j. 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.

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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.

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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

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