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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 canbe 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:
1)exp(
1
)()(
1
EBF
FEgEn
FD
FD
Statistical Physics
Quantum Distributions
Bosons: particles with zero or integer spins
The Bose-Einstein distribution, which is valid for bosons:
1)exp(
1
)()(
2
EBF
FEgEn
BE
BE
Statistical Physics
Fermi-Dirac Statistics
It provides the basis for our understanding of the behaviorof a collection of fermions.
Applying it to the problem of electrical conduction.
Statistical Physics
Fermi-Dirac Statistics
From Fermi-Dirac distribution:1)exp(
1
1
EBFFD
The factor B1 is computed by integrating n(E)dE over all allowedenergies.
The parameter β(=1/kT) is contained in FFD, so that B1
should be temperature dependent.
B1 = exp(-βEF)
Statistical Physics
Fermi-Dirac Statistics
1)(exp
1
FFD EEF
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 maybe 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 shownin 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 StatisticsClassical Theory of Electrical Conduction
Paul Drude (1863-1906)
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.
Statistical Physics
Fermi-Dirac StatisticsClassical Theory of Electrical Conduction
In this model the metal is thought of as a lattice of positiveions with a gas of electrons free to flow through it .The electron have a thermal kinetic energy proportional totemperature. 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 electricfield is applied to the conductor.
Statistical Physics
Fermi-Dirac StatisticsClassical Theory of Electrical Conduction
When an electric field is applied, the negatively charged electrons flow in the opposite directionto 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 StatisticsClassical Theory of Electrical Conduction
m
ne 2
One important prediction was that the electrical conductivitycould be expressed as
The principal success of Drude’s theory was that it did predict Ohm’s law.
Unfortunately, the numerical predictions of the theory werenot so successful.
Statistical Physics
Fermi-Dirac StatisticsClassical Theory of Electrical Conduction
n is the number density of conduction electrons
e is the electronic chargeis 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 magnitudetoo small for most conductors. The Drude theory is thereforeincorrect in this prediction.
Statistical Physics
vm
lne
v
l
2
Fermi-Dirac StatisticsClassical Theory of Electrical Conduction
m
kTv
24
Drude model, the conductivity should be proportional to T-1/2. But for most conductorsthe conductivity is nearly proportional to T-1
except at very low temperatures, where it nolonger follows a simple relation.
Clearly the classical model of Drude has failed to predict this important experimental fact.
Statistical Physics
Fermi-Dirac StatisticsQuantum 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 StatisticsQuantum Theory of Electrical Conduction
)(8
23
22
212
2
nnnmL
hE The allowed energies are
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 StatisticsQuantum Theory of Electrical Conduction
The number of allowed states up to“radius” r will be directly related toThe spherical “volume”(4/3)πr3 . The exact number of states up to r is
))(8
1)(2( 3
34 rN r
The extra factor of 2 is due to spin degeneracy.
Statistical Physics
Fermi-Dirac StatisticsQuantum 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 :
2/3
1
)(3
1
E
ENr
Statistical Physics
Fermi-Dirac StatisticsQuantum Theory of Electrical Conduction
At T=0 the Fermi energy is the energy of the highest occupiedenergy level. If there are a total of N electrons, then
3/23
23/2
1
2/3
1
)3(
8)
3(
)(3
1
L
N
m
hNEE
E
EN
F
F
Statistical Physics
Fermi-Dirac StatisticsQuantum Theory of Electrical Conduction
The density of states can be calculated by differentiating Equation of N with respect to energy :
2/12/32/12/3
2
2/12/312
2
3)
3()(
)(
EEN
EN
EEg
EEdE
dNEg
FF
r
Statistical Physics
Fermi-Dirac StatisticsQuantum 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 energycan be calculated easily :
F
F
F
E
FF
E
FN
E
NN
EdEEEE
dEEEN
E
dEEEgdEEEnE
0532/32/3
23
0
2/32/31
0 0
11
)2
3(
)()(
Statistical Physics
Fermi-Dirac StatisticsQuantum Theory of Electrical Conduction
Therefore the internal energy (U) of the system is:
FNEENU 53
The fraction of electrons capable of participating in this thermal process is on the order of kT/EF. The exact numberof electrons depends on temperature, because the shapeof the curve n(E) changes with temperature.
Statistical Physics
Fermi-Dirac StatisticsQuantum Theory of Electrical Conduction
T= 0 K
T = 300 K
Statistical Physics
Fermi-Dirac StatisticsQuantum Theory of Electrical Conduction
FV
FV
FF
T
TRC
E
TNk
T
UC
kTE
kTNNEU
2
2
1,
2
53
The heat capacity
Statistical Physics
Fermi-Dirac StatisticsQuantum Theory of Electrical Conduction
112
1172
6
.106
/106.12
TUrl
mmu
lne
smm
Eu
F
FF
The electrical conductivity varies inversely with temperature. This is another striking success for the quantum theory.
Statistical Physics
Bose-Einstein StatisticsBlackbody Radiation
The intensity of the emitted radiation as a function of temperature and wavelength as:
1
12),(
/5
2
kThce
hcTU
Statistical Physics
Bose-Einstein StatisticsBlackbody Radiation
The electromagnetic radiation is really a collection of photons of energy hc/λ.
Use the Bose-Einstein distribution to find how the photons aredistributed 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 StatisticsBlackbody Radiation
The energy of a photon is pc, so
333
3
334
81
23
22
21
3
8
2
))((2
2
Ech
LN
rL
hcE
rN
nnnL
hcE
r
r
Statistical Physics
Bose-Einstein StatisticsBlackbody Radiation
The density of states g (E) is
1
18)(
)()(
8)(
/2
33
3
233
3
kTE
BE
r
eE
ch
LEn
FEgEn
Ech
L
dE
dNEg
Statistical Physics
Bose-Einstein StatisticsBlackbody Radiation
The next step is to convert from a number distribution to an energy density distribution u(E) . To do this it is necessaryto multiply by the factor E/L3 ( that is, energy per unit volume):
dEe
Ech
dEL
EEndEEu
eE
chL
EEnEu
kTE
kTE
1
18)()(
1
18)()(
/3
333
/3
333
Statistical Physics
1
8),(
/5
kThce
dhcdTu
Bose-Einstein StatisticsBlackbody Radiation
Using E=hc/λ and |dE|=(hc/λ2)dλ, we find
In the SI system multiplying by a constant factor c/4 is requiredto change the energy density to a spectral intensity:
1
12),(
/5
2
kThce
hcTU
Statistical Physics
Bose-Einstein StatisticsBlackbody 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 Bose-Einstein distribution
Function
Energy Dependence
Quantum ParticlesFermions
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
f E 1
exp E kBT 1
f E 1
exp E kBT 1
