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Review

For cases when Z 1>> N condition is not met:

• Fermi-Dirac Distribution (fermions)

• Bose-Einstein Distribution (bosons)

nFD =

1

e(ε −μ ) / kT  +1

n BE  =1

e(ε −μ )/kT  −1

Fermi-Dirac Distribution• Consider cases looking at ε and μ 

nFD =1

e(ε −μ )/ kT  +1

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Fermi-Dirac Distribution

• Consider cases looking at ε and μ 

Relation to Boltzmann Distribution• Calculate occupancy n

 Boltzmann = NP(s)

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Comparison of Distributions

Quantum Gas• For ideal gas, single-particle partition function

• For to hold, Z 1 >> N 

 Z 1 =VZ int

υ Q

 Z  = Z 1 N   N !

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Application of Fermi-Dirac Distribution• Modelling behavior of conduction electrons in metal

 – Gas of fermions at very low temperature

• Condition for Boltzmann statistics is not met

V 

 N 

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At Zero Temperature

• At T =0, Fermi-Dirac distribution becomes step function

 –   ε F  ≡ μ (T =0)

• When nearly all states below ε F are occupied and those above

are empty

 – Called degenerate gas

•   ε F depends on total number of electrons in system

Degenerate Fermi Gas• Electrons in the system are free particles

 – Ignoring attractive forces from ions in the crystal lattice

• Wavefunctions for free electron in metal block 

 – Represented by standing wave modes

λ n =

2 L

n ;  pn =

h

λ n =

hn

2 L

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Degenerate Fermi Gas

• Wavefunctions for free electron in metal block 

 – Represented by standing wave in a box wavefunctions

 – Each “lattice point” (n x, n y, n z) is pair of electron states

• One for each spin orientation

 – Surface of sphere => radius nmax

 

ε  =

r

 p

2

2m= h

2

8mL2n x

2 + n y2 + n z

2(