Download 13.7 The Connection between Classical and Statistical

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 halfinteger 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
wj 
g j!
N j !g j  N j !
where gj is N in the coin-tossing experiments.
• The total thermodynamic probability is
n
wFD  
j 1
g j!
N j !g j  N j  !
• 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
n
ln( wFD )  ln( 
g j!
)
j 1 N !g  N  !
j
j
j
ln( wFD )   ln(
j
g j!
)
N j !g j  N j  !
ln( wFD )   ln( g j !)   ln g j  N j  !   ln( N j !)
j
j
j
• There are two constrains
n
N
j 1
j
N
n
N E
j 1
j
j
U
• Using the Lagrange multiplier
(ln( WFD )
N
U
a

0
N j
N j
N j
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
Wj 
( N j  g j  1)!
N j !( g j  1)!
• The thermodynamic probability of the system is
 ( N j  g j  1)! 

 


j 1  N j !( g j  1)! 
n
WBE
Finding the distribution function
13.6 Diluted gas and MaxwellBoltzman 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 MaxwellBoltzman distribution.
• Notice the difference
N! g j
n
wB  
Nj
Nj
J 1
n
wMB  
gj
Nj
Nj
• They difference is a constant. Because the
distribution is established through differentiation,
the distribution is not affected by such a constant.
J 1
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
  ln Z 

N j   NkT
  
j T
