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

Tjj

ZNkTN

ln