Download unit-3 - SNIST Utilities

STATISTICAL
MECHANICS
Introduction
The subject which deals with the relationship
between the overall behavior of the system and the
properties of the particles is called ‘Statistical
Mechanics’.
Statistical Mechanics can be applied to classical
systems such as ‘Molecules in gas’ as well as
‘Photons in a Cavity’ and ‘Free Electrons in a
Metal’.
Physical System
Consider a system composed of ‘N’ identical, non–
interacting particles in a volume ‘ V ’.
let ‘ n1’ particles posses energy ‘ E1’ ,
‘ n2’ particles posses energy ‘ E2’ …. and so on.
Total energy of the system
E  n1 E1  n2 E2  n3 E3  .....
E   ni Ei
and
N   ni
Macro States and Microstates of Systems
Any state of a system as described by actual or
hypothetical observations of its Macroscopic statistical
properties is known as ‘ Macro State ’ and it is specified
by ‘ ( N, V and E ) ’.
The state of system as specified by the actual properties of
each individual, elemental components and it is permitted
by the uncertainty principle is known as ‘ Micro State ’.
For ‘ N ’ particle system , there may be always possible
‘N+1’ Macro States and ‘ 2n ’ Micro States.
Phase Space
1.The three dimensional space in which the location of a particle is
completely specified by the three position coordinates, is known as
‘Position Space’.
small volume in a Position space dV  dxdydz
2.The three dimensional space in which the momentum of a
particle is completely specified by the three momentum
coordinates Px, Py and Pz is known as Momentum Space.
small volume in a momentum space d  dp x dp y dp z
3.The combination of the position space and momentum space is
known as ‘Phase Space’.
small volume in a phase space d  dVd
Phase Space Volume
Consider Let ‘ pm’ be the maximum value of the
momentum of the particles in the system.
Let px ,py, pz represents the three mutually
perpendicular axes in the momentum space as shown
in figure.
Draw a sphere with an origin ‘ O ’ as centre and the
maximum momentum ‘ pm’ as radius.
All the points within this sphere will have their
Momenta lying between ‘ 0 ’ and ‘ pm’.
The momentum space volume 
pz
volume of the sphere of radius p m .
4 3
momentum volume   pm
3
phase space volume   V
4 3
  pmV .....
3
( where V is a position space volume )
0
px
pm
py
1.The small volume d in phase space is called a cell .
2.The total number of cells in phase space 
Total available volume in phase space
volume of one cell
V
3.Total number of cells n 
d
Ensemble
An Ensemble is defined as a collection of very large
number of Assemblies which are essentially independent
of one another. There are three types of ensembles.
1.Microcanonical ensemble
2.Canonical ensemble
3.Grand canonical ensemble
Micro Canonical Canonical
Ensemble
Ensemble
Grand Canonical
Ensemble
All assemblies
1. Same Volume ,
2. Same number of
particles N,
3. Same energy E.
All assemblies
1. Same Volume ,
2. Same temperature
T and Same
Chemical potential
µ.
All assemblies
1.Same Volume ,
2.Same Number of
particles N,
3. Same
temperature T.
Statistical Distribution
Statistical Mechanics determines the most probable way
of distribution of total energy ‘ E ’ among the ‘ N ’
particles of a system in thermal equilibrium at absolute
temperature ‘ T ’.
In statistical mechanics one finds the number of ways ‘
W ’ in which the ‘ N ’ number of particles of energy ‘ E ‘
can be arranged among the available states is given by.
N(E) = g(E) f(E)
Where ‘ g(E) ’ is the number of states of energy ‘ E ’ and
‘ f(E) ’ is the probability of occupancy of each state of
energy ‘ E ’.
Statistical Distribution
Classical Statistics
( Maxwell – Blotzmann distribution)
Quantum Statistics
Bose - Einstein distribution
Fermi - Dirac distribution
Classical Statistics
(Maxwell - Blotzmann Distribution)
Let us consider a system consisting of molecules of an
ideal gas under ordinary conditions of temperature and
pressure.
Such a system is governed by the laws of Classical
Maxwell- Blotzmann Statistics.
Assumptions:
1. The particles are identical and distinguishable.
2. The volume of each phase space cell chosen is
extremely small and hence chosen volume has very
large number of cells.
3. Since cells are extremely small, each cell can have either one
particle or no particle though there is no limit on the number of
particles which can occupy a phase space cell.
4.
The system is isolated which means that both the total number
of particles of the system and their total energy remain constant.
5.
The state of each particle is specified instantaneous position and
momentum co-ordinates.
6.
Energy levels are Continuous.
Failures of Classical Statistics
1. The observed energy distribution of electrons in
Metals.
2. The observed energy distribution of Photons inside the
cavity.
3. The behavior of Helium at low temperatures.
Quantum Statistics
According to Quantum Statistics the particles of
the system are indistinguishable, their wave
functions do overlap and such system of particles
fall into two categories
1.Bose - Einstein Distribution
2.Fermi - Dirac Distribution.
Bose – Einstein Statistics
According to Bose-Einstein statistics the particles of any physical
system are identical, indistinguishable and have integral spin, and
further those are called as Bosons.
Assumptions
1. The Bosons of the system are identical and indistinguishable.
2. The Bosons have integral spin angular momentum in units of
h/2π.
3. Bosons obey uncertainty principle.
4. Any number of bosons can occupy a single cell in phase space.
5. Bosons do not obey the Pouli’s Exclusion principle.
6. Wave functions representing the bosons are Symmetric
i.e, Ψ(1,2) = Ψ(2,1)
7. Energy states are discrete.
8.The probability of boson occupies a state of energy E is given by
f BE ( E ) 
exp(
1
E  Ef
kT
) 1
Fermi - Dirac Statistics
 According to Fermi - Dirac statistics the particles of any physical
system are indistinguishable and have half integral spin. These
particles are known as Fermions.
Assumptions
1.
Fermions are identical and indistinguishable.
2.
They obey Pauli’ s exclusion principle.
3.
Fermions have half integral spin.
4.
Wave function representing fermions are anti symmetric
 (1,2)   (2,1)
6. Uncertainty principle is applicable.
7. Energy states are discrete.
8. The probability of a fermions occupies a state of energy E is
given by
1
F (E) 
E  Ef
1  exp(
)
kT
Density of Energy States
Now we have to calculate the carrier concentration i.e,
the number of electron per unit volume in a given
energy range.
This is given by summing the product of the density of
states ‘ Z(E) ’ and the occupancy probability ‘ F(E) ’
that is
nc 
 Z ( E ) F ( E )dE
energyband
Therefore the number of energy states within a sphere of radius n
4 3
n  n
3
nz
E
Since n1,n2 and n3 can have only
positive integer values, we have
to consider only one octant of
the sphere.
1 4 3
 { n }
8 3
n
nx
dn
E+ dE
ny
 In order to calculate the number of states within a small
energy interval ‘E ’ and ‘E + dE’, we have to construct two
spheres with radii ‘n’ and ‘n + dn’ and calculate the space
occupied within these two sphere.
 Thus the number of energy states having energy values
between ‘E’ and ‘E + dE’ is given by…
1 4
1 4
3
3
Z ( E )dE  { }(n  dn)  { }(n)
8 3
8 3
 2
Z ( E )dE  n dn
2
The expression for the energy
of electron in potential well
is given by
n2h2
E
8mL2
8mL2 E
2
n 
........(1)
2
h
1
8mL2 E 2
n(
)
2
h
differntiating ., eq n (1)
8mL2
2ndn 
dE
2
h
1 8mL2
dn  {
}
dE
2
2n
h
1
8mL2  1  h 2
dE
2

{
}
1
2
h2 
2
 8mL
E2
1
2
 1  8mL 2 dE
dn   {
}
1
2
2 h
E2
Substituting the values of ‘ n2 ’ and ‘ dn ’ from in eq’n.. (1)
Z ( E )dE  2 
 8mL2
4
[
h
2
3
2
1
2
] E dE
3
1
4
 3 (2m) 2 L3 E 2 dE
h
Density of energy states is given by number of energy states
per unit volume.
Density of states
3
1
4
Z ( E )dE  3 (2m) 2 E 2 dE
h