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Statistical Physics
Quantum Distributions
In quantum systems only certain energy values are allowed.
In quantum theory particles are described by wave functions.
Identical particles cannot be distinguished from one another if
there is a significant overlap of their wave functions.
Pauli principle has a significant impact on how energy states can
be occupied and therefore on the corresponding energy distribution.
Statistical Physics
Quantum Distributions
Fermions: particles with half-integer spins
The Fermi-Dirac distribution, which is valid for fermions:
n( E )  g ( E ) FFD
FFD 
1
B1 exp( E )  1
Statistical Physics
Quantum Distributions
Bosons: particles with zero or integer spins
The Bose-Einstein distribution, which is valid for bosons:
n( E )  g ( E ) FBE
FBE 
1
B2 exp( E )  1
Statistical Physics
Fermi-Dirac Statistics
It provides the basis for our understanding of the behavior
of a collection of fermions.
Applying it to the problem of electrical conduction.
Statistical Physics
Fermi-Dirac Statistics
From Fermi-Dirac distribution:
FFD
1

B1 exp( E )  1
The factor B1 is computed by integrating n(E)dE over all allowed
energies.
The parameter β(=1/kT) is contained in FFD, so that B1
should be temperature dependent.
B1 = exp(-βEF)
Statistical Physics
Fermi-Dirac Statistics
FFD
1

exp  ( E  EF )  1
EF : Fermi energy
FFD = ½ for E = EF
FFD = 1 for E < EF
FFD = 0 for E > EF
Statistical Physics
Fermi-Dirac Statistics
At T=0 fermions occupy the lowest energy levels available
to them.
They cannot all be in the lowest level, since that would violate
the Pauli principle. Rather, fermions will fill all the available
energy levels up to a particular energy ( the Fermi energy ).
At T=0 there is no chance that thermal agitation will kick
a fermion to an energy greater than EF.
Statistical Physics
Fermi-Dirac Statistics
The Fermi-Dirac factor FFD at various temperatures
Statistical Physics
Fermi-Dirac Statistics
As the temperature increases from T=0, more fermions may
be in higher energy levels. The Fermi-Dirac factor “smears out”
from the sharp step function [ Figure (a) ] to a smoother curve
[ Figure (b) ].
A Fermi temperature’s defined as TF= EF /k which shown
in Figure (c).
Statistical Physics
Fermi-Dirac Statistics
When T << TF the step function approximation for FFD
is reasonably accurate.
When T>> TF , FFD approaches a simple decaying
exponential [ Figure (d) ].
As the temperature increases, the step is gradually rounded.
Finally, at very high temperatures, the distribution approaches
the simple decaying exponential of Maxwell-Boltzmann
distribution.
Statistical Physics
Fermi-Dirac Statistics Classical Theory of Electrical Conduction
In 1900 Paul Drude developed a
theory of electrical conduction in
an effort to explain the observed
conductivity of metals.
Drude model assumed that the
electrons in a metal existed as
a gas of free particles.
Paul Drude (1863-1906)
Statistical Physics
Fermi-Dirac Statistics Classical Theory of Electrical Conduction
In this model the metal is thought of as a lattice of positive
ions with a gas of electrons free to flow through it .
The electron have a thermal kinetic energy proportional to
temperature. The mean speed of an electron at room
temperature can be calculated to be about 105 m/s.
The velocities of the particles in a gas are directed randomly.
Therefore, there will be no flow of electrons unless an electric
field is applied to the conductor.
Statistical Physics
Fermi-Dirac Statistics Classical Theory of Electrical Conduction
When an electric field is
applied, the negatively
charged electrons flow
in the opposite direction
to the field.
Drude was able to show that the current in conductor should
be linearly proportional to the applied electric field.
Statistical Physics
Fermi-Dirac Statistics Classical Theory of Electrical Conduction
The principal success of Drude’s theory was that it did
predict Ohm’s law.
Unfortunately, the numerical predictions of the theory were
not so successful.
One important prediction was that the electrical conductivity
could be expressed as
ne 2

m
Statistical Physics
Fermi-Dirac Statistics Classical Theory of Electrical Conduction
n is the number density of conduction electrons
e is the electronic charge
 is the average time between electron-ion collisions
m is the electronic mass
When combined with the other parameters in above Equation,
produced a value of σ that is about one order of magnitude
too small for most conductors. The Drude theory is therefore
incorrect in this prediction.
Statistical Physics
Fermi-Dirac Statistics Classical Theory of Electrical Conduction
Drude model, the conductivity should be
proportional to T-1/2. But for most conductors
the conductivity is nearly proportional to T-1
except at very low temperatures, where it no
longer follows a simple relation.
Clearly the classical model of Drude
has failed to predict this important
experimental fact.
l

v
ne 2l

mv
4
v
2
kT
m
Statistical Physics
Fermi-Dirac Statistics Quantum Theory of Electrical Conduction
How electron energies are distributed in a conductor?
The real problem we face is to find g(E), the number
of allowed states per unit energy.
What energy values should we use?
From assumption of the Drude model about “free electron”
and use the results obtained in Quantum theory for
a three-dimensional infinite square well potential.
Statistical Physics
Fermi-Dirac Statistics Quantum Theory of Electrical Conduction
The allowed energies are
h2
2
2
2
E
(
n

n

n
1
2
3)
2
8mL
Where L is the length of side of the cube and
ni are the integer quantum numbers.
Above Equation can be rewritten as E = r2E1
E1 is just a constant, not the ground-state energy .
Statistical Physics
Fermi-Dirac Statistics Quantum Theory of Electrical Conduction
The number of allowed states up to
“radius” r will be directly related to
The spherical “volume”(4/3)πr3 .
The exact number of states up to r is
1 4 3
N r  (2)( )( 3 r )
8
The extra factor of 2 is due to spin degeneracy.
Statistical Physics
Fermi-Dirac Statistics Quantum Theory of Electrical Conduction
The factor 1/8 is necessary because we are restricted to
positive quantum numbers and, therefore, to one octant
of the three-dimensional number space.
Nr as a function of E :
1 E 3/ 2
Nr   ( )
3 E1
Statistical Physics
Fermi-Dirac Statistics Quantum Theory of Electrical Conduction
At T=0 the Fermi energy is the energy of the highest occupied
energy level. If there are a total of N electrons, then
1 EF 3 / 2
N  ( )
3 E1
EF  E1 (
3N

)2 / 3
h 2 3N 2 / 3

( 3)
8m L
Statistical Physics
Fermi-Dirac Statistics Quantum Theory of Electrical Conduction
The density of states can be calculated by differentiating
Equation of N with respect to energy :
dN r  3 / 2 1/ 2
g (E) 
 2 E1 E
dE
3 N 3 / 2 1 / 2
3 / 2 3 N
1/ 2

g ( E )  2 ( EF
)E 
EF E

2
Statistical Physics
Fermi-Dirac Statistics Quantum Theory of Electrical Conduction
At T = 0 we have
n(E) = g(E) for E < EF , n(E) =0 for E > EF .
With the distribution function n(E) the mean electronic energy
can be calculated easily :
E
E
1
N


0
En( E )dE 
1
N

EF
0
Eg ( E )dE
EF
1
N
3N
3 / 2 3 / 2
(
)
E
0 2 F E dE
E  EF
3
2
3 / 2
EF
3/ 2
3
E
dE

5 EF

0
Statistical Physics
Fermi-Dirac Statistics Quantum Theory of Electrical Conduction
Therefore the internal energy (U) of the system is:
U  NE  53 NEF
The fraction of electrons capable of participating in this
thermal process is on the order of kT/EF. The exact number
of electrons depends on temperature, because the shape
of the curve n(E) changes with temperature.
Statistical Physics
Fermi-Dirac Statistics Quantum Theory of Electrical Conduction
T = 300 K
T= 0 K
Statistical Physics
Fermi-Dirac Statistics Quantum Theory of Electrical Conduction
kT
U  NEF  N
kT ,  1
EF
3
5
U
2 T
CV 
 2Nk
T
EF
T
CV  2R
TF
The heat capacity
Statistical Physics
Fermi-Dirac Statistics Quantum Theory of Electrical Conduction
2 EF
uF 
 1.6 106 m / s
m
ne 2l

 6 107  1.m 1
muF
  l  r 2  U 1  T 1
The electrical conductivity varies inversely with temperature.
This is another striking success for the quantum theory.
Statistical Physics
Bose-Einstein Statistics Blackbody Radiation
The intensity of the emitted radiation as a function of
temperature and wavelength as:
U ( , T ) 
2c 2 h
1
5
e hc / kT  1
Statistical Physics
Bose-Einstein Statistics Blackbody Radiation
The electromagnetic radiation is really a collection of photons
of energy hc/λ.
Use the Bose-Einstein distribution to find how the photons are
distributed by energy, and then use the relationship E=hc/λ to
turn the energy distribution into a wavelength distribution.
The desired temperature dependence should already be
included in the Bose-Einstein factor.
Statistical Physics
Bose-Einstein Statistics Blackbody Radiation
The energy of a photon is pc, so
hc
2
2
2
E
n1  n2  n3
2L
N r  2( 18 )( 43 r 3 )
hc
E
r
2L
8L3 3
Nr  3 3 E
3h c
Statistical Physics
Bose-Einstein Statistics Blackbody Radiation
The density of states g (E) is
dN r 8L3 2
g (E) 
 3 3E
dE h c
n( E )  g ( E ) FBE
8L 2
1
n( E )  3 3 E E / kT
hc
e
1
3
Statistical Physics
Bose-Einstein Statistics Blackbody Radiation
The next step is to convert from a number distribution to
an energy density distribution u(E). To do this it is necessary
to multiply by the factor E/L3 ( that is, energy per unit volume):
En( E ) 8 3
1
u( E ) 
 3 3 E E / kT
3
L
hc
e
1
En( E )
8 3
1
u ( E )dE 
dE  3 3 E E / kT
dE
3
L
hc
e
1
Statistical Physics
Bose-Einstein Statistics Blackbody Radiation
Using E=hc/λ and |dE|=(hc/λ2)dλ, we find
u ( , T )d 
8hc
d
5 e hc / kT  1
In the SI system multiplying by a constant factor c/4 is required
to change the energy density to a spectral intensity:
U ( , T ) 
2hc 2
1
5
e hc / kT  1
Statistical Physics
Bose-Einstein Statistics Blackbody Radiation
u(λ,T) is energy per unit volume per unit wavelength inside
the cavity.
U(λ,T) is power per unit area per unit wavelength for
radiation emitted from the cavity.
Quantum Statistics Summary
Fermi-Dirac distribution
Function
f E 
1
exp E   kBT 1
Energy

Dependence
Bose-Einstein distribution
f E 
1
exp E   kBT 1

Quantum Particles
Fermions
e.g., electrons, neutrons, protons
and quarks
Bosons
e.g., photons, Cooper pairs
and cold Rb
Spins
1/2
integer
Properties
At temperature of 0 K, each energy level is
occupied by two Fermi particles with
opposite spins.
 Pauli exclusion principle
At very low temperature, large numbers of
Bosons fall into lowest energy state.
 Bose-Einstein condensation