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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13.4 Fermi-Dirac Distribution Fermions are particles that are identical and indistinguishable. Fermions include particles such as electrons, positrons,
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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
!!
!
jjj
jj NgN
gw
!!
!
1jjj
jn
jFD NgN
gw
• 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
)!!
!ln()ln(
1jjj
jn
jFD NgN
gw
j jjj
jFD NgN
gw )
!!
!ln()ln(
• There are two constrains
• Using the Lagrange multiplier
UEN
NN
j
n
jj
n
jj
1
1
j
jj
jjj
jFD NNggw )!ln(!ln)!ln()ln(
0)(ln(
jjj
FD
N
U
N
Na
N
W
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
)!1(!
)!1(
jj
jjj gN
gNW
n
j jj
jjBE gN
gNW
1 )!1(!
)!1(
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.
j
Nj
n
JB N
gNw
j!
1
j
Nj
n
JMB N
gw
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