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13.4 Fermi-Dirac Distribution

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.

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13.4 Fermi-Dirac Distribution

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  1. 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.

  2. 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

  3. 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

  4. There are two constrains • Using the Lagrange multiplier

  5. See white board for details

  6. 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

  7. Finding the distribution function

  8. 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.

  9. 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.

  10. 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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