4. The Partition Function of an Ideal Fermi Gas If the particles are fermions , n can only be 0 or 1 : The grand partition function for all particles in the i th single-particle state (the sum is taken over all possible values of n i ) : Putting all the levels together, the full partition function is given by: 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

5. Fermi-Dirac Distribution Fermi-Dirac distribution The mean number of fermions in a particular state: The probability of a state to be occupied by a fermion: (  is determined by T and the particle density) 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

6. Fermi-Dirac Distribution At T = 0 , all the states with  <  have the occupancy = 1, all the states with  >  have the occupancy = 0 (i.e., they are unoccupied). With increasing T , the step-like function is “smeared” over the energy range ~ k B T . T =0 ~ k B T  =  ( with respect to  ) 1 0 n=N/V – the average density of particles The macrostate of such system is completely defined if we know the mean occupancy for all energy levels, which is often called the distribution function : While f ( E) is often less than unity, it is not a probability: 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

7. The Partition Function of an Ideal Bose Gas If the particles are Bosons , n can be any #, i.e. 0, 1, 2, … The grand partition function for all particles in the i th single-particle state (the sum is taken over all possible values of n i ) : Putting all the levels together, the full partition function is given by: 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

8. Bose-Einstein Distribution Bose-Einstein distribution The mean number of Bosons in a particular state: The probability of a state to be occupied by a Boson: The mean number of particles in a given state for the BEG can exceed unity, it diverges as   min(  ) . 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

9. Comparison of FD and BE Distributions Maxwell-Boltzmann distribution: 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

10. Maxwell-Boltzmann Distribution (ideal gas model) Maxwell-Boltzmann distribution The mean number of particles in a particular state of N particles in volume V : MB is the low density limit where the difference between FD and BE disappears. Recall the Boltzmann distribution (ch.6) derived from canonical ensemble: 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

11. Comparison of FD, BE and MB Distribution 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

12. Comparison of FD, BE and MB Distribution (at low density limit) MB is the low density limit where the difference between FD and BE disappears. The difference between FD, BE and MB gets smaller when  gets more negative. 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

13. Comparison between Distributions Boltzmann Fermi Dirac Bose Einstein indistinguishable Z=(Z 1 ) N / N! n K <<1 spin doesn’t matter localized particles  don’t overlap gas molecules at low densities “ unlimited” number of particles per state n K <<1 indistinguishable integer spin 0,1,2 … bosons wavefunctions overlap total  symmetric photons 4 He atoms unlimited number of particles per state indistinguishable half-integer spin 1/2,3/2,5/2 … fermions wavefunctions overlap total  anti-symmetric free electrons in metals electrons in white dwarfs never more than 1 particle per state 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

15. TERIMA KASIH 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |