Download Statistical Mechanics Introduction:- The subject which deals with the

Statistical Mechanics
Introduction:- The subject which deals with the relationship between the overall behavior of the system
and properties of the particles is called statistical mechanics.
In statistical mechanics collection of large number of particle is called system.
Phase space:- The six dimensional hypothetical space is called phase space.
This gives complete dynamics of the system.
Here we have 3- position coordinates & 3- momentum coordinates.
Phase space is represented as f (x, y, z, p x , p y , p z )
It represents the positions and momenta of all molecules at given time.
Ensemble:- A collection of large number of essentially independent systems having identical properties
is called an Ensemble.
Depending on interaction of the systems Gibbs classified the following Ensemble.
The three important types of ensembles often considered are
1. Canonical Ensemble,
2. Micro Canonical Ensemble
3. Grand canonical ensemble
Canonical Ensemble: - In Canonical ensemble the systems making up the ensemble are separated from
one another by thermally conducting, rigid and impenetrable ( Impermeable) walls. Hence the volume
of a systems and the number of particle present them remain unchanged and as the walls are thermally
conducting all the systems attain a common temperature. Hence canonical ensembles are also called
TVN ensemble.
Canonical Ensemble corresponds to closed Isothermal system which exchanges energy but not mas with
one another which keeps temperature (T) and number of particles (N) remains constant.
.
TVN
TVN
TVN
TVN
TVN
The Walls separating the systems present in
the ensemble are
TVN
TVN
TVN
TVN
TVN
TVN
TVN
TVN
TVN
TVN
TVN
TVN
TVN
TVN
TVN
1. Thermally conducting
2. Rigid
3. Impenetrable ( Impermeable)
Ex: System in contact with heat reservoir
1
2. Micro Canonical Ensembles: - In micro Canonical Ensemble the systems making up the ensemble are
separated from one another by thermally insulating, rigid and impermeable (impenetrable walls). Hence
the volume of the systems V, the number of the particles present N and total energy remain E remain
unchanged. Micro canonical ensembles are also called NVE ensembles.
In this Ensemble corresponds to isolated systems which exchanges neither heat nor mass with one
another and therefore which keep the total number of particles (N) and total energy as constant.
NVE
NVE
NVE
NVE
NVE
The Walls separating the systems are
NVE
NVE
NVE
NVE
NVE
NVE
NVE
NVE
NVE
NVE
NVE
NVE
NVE
NVE
NVE
1. Thermally insulating
2. Rigid
3. Impenetrable ( Impermeable)
Grand Canonical Ensembles: - In Grand Canonical Ensemble the systems making up the ensembles are
separated from one another by thermally conducting permeable and rigid walls so that they attain a
common temperature T. Since the walls are rigid and permeable, the volume V changes will not be
present; however the number of particles can change due to the permeable property of the separating
walls in such a way that the chemical potentials µ remain same. Grand canonical ensembles are also
called T V µ ensembles.
Consider a system consisting of (N) particles occupying volume (V) temperature constant with which it
can exchange not only energy but also particles.
TV µ
TVµ
TVµ
TVµ
TVµ
TVµ
TVµ
TVµ
TVµ
TVµ
TVµ
TVµ
TVµ
TVµ
TVµ
TVµ
The Walls separating the systems are
1. Thermally conducting
2. Rigid
3. Permeable ( Penetrable)
2
Statistical mechanics is divided into two categories,
1. Classical statistical mechanics
2. Quantum statistical mechanics.
Classical statistical Mechanics or Maxwell Boltzmann statistics: - Classical statistical mechanics is also
called Maxwell-Boltzmann statistics, and is applicable for identical distinguishable particles which can
have any amount of energy and can have any spin. Pauli’s exclusion principle is not valid for the particles
obeying Maxwell-Boltzmann statistics. The phase space in which the particles are present in MaxwellBoltzmann statistics can be divided into infinite number of cells and in a cell any number of particles can
be present. Example of particles obeying Maxwell Boltzmann statistics are molecules of gasses.
Quantum statistical mechanics:- In quantum statistical mechanics particles making up the collection can
have only certain discrete energies. Quantum statistical mechanics is further divided into 1. Bose Einstein statistics and 2. Fermi - Dirac statistics.
1. Bose - Einstein statistics: -Bose – Einstein statistics is applicable when the particles making up the
collection can have only certain discrete energies and are identical, indistinguishable and can have a spin
ℎ
zero or even integral multiple of2𝜋 . Pauli’s exclusion principle is not valid for these particles and hence
in a quantum state or cell in phase space either zero or any number of particles can be present.
Examples for the particles obeying Bose Einstein particles are photons, Helium nuclei etc.
2. Fermi Dirac statistics:- Fermi Dirac statistics is applicable when the particles making up the collection
can have only certain discrete energies and are identical, indistinguishable and can have spin odd
integral multiple of
ℎ
2𝜋
. Pauli’s exclusion principle is valid for these particles and hence ina quantum
state or cell in phase space either zero or only one particle can be present.
3
Comparison of Maxwell Boltzmann, Bose Einstein and Fermi Dirac Statistics
Properties
Type of statistics
Maxwell Boltzmann
Bose Einstein
Fermi Dirac
Particles
Classical
1. Identical
2. distinguishable
Quantum
1. Identical
2. Indistinguishable
Quantum
1. Identical
2. Indistinguishable
Energies of the
particles
Any amount of energy
( Continues)
Only certain discrete
energies
Only certain discrete
energies
Any spin
Even integral multiple
Of spin
Odd half integral multiple
Spin
Pauli’s exclusion
Principle
Heisenberg
Uncertanity
Principle
No. of particle in a
quantum state or
cell
No. of elementary
cells in phase
space
Energy
distribution
function
,3
2
spin
Not Valid
Valid
Not Valid
Valid
Valid
Zero or any number
of particles
Zero or any number
of particles
Zero or only one particle
Infinite
Very large but finite
Very large but finite
f (E) 
f (E) 
1
Ae
E
kT
4 distributions
1. AB 2. B
A
3. AB4. - AB
Molecules of gasses
Application
Gas laws,
Thermodynamics
Behavior of wave
function
2
Not Valid
Different
distribution of a
system with
2particles and 2
cells
Examples
About wave
function
Of
1
e(   E)  1
f (E) 
1
e
(   E)
1
3 Distributions
1. X
X
2. XX
3. XX
Only one
1. X X
Photons , spin zero
particles
Bose Einstein
Condensation, Planks Law
Electrons, protons etc
_______
Overleaping wave function
Overleaping wave function
_______
At high energies leads to
M-B statistics
At high energies leads to
M-B statistics
_______
Wave functions are
symmetric.
Wave functions are
Anti symmetric.
 (1, 2)   (2,1)
4
Electrical & thermal
conductivities etc
 (1, 2)   (2,1)
Concept of electron gas:- In order to explain the high electrical and thermal conductivity of metals.
Somerfield put forward a model known as free electron model.
In 1990 Drude &Lorentz proposed that it is assumed that the valance electrons in metal are not
localized and they move inside the specimen without colliding among themselves.
Conducting metal can be thought of as a collection of non interacting electrons free to roam about
the specimen, like molecules in an Ideal gas obeying F-D statistics also obeying kinetic theory of
gasses.
The free electron gas and ordinary gas are having some differences in particle concentration and
charge. The free electron gas is charged whereas ordinary gas is neutral. The particle concentration
is very high in free electron gas (1029 e
(1025 molecules
m3

m3
) compared to a gaseous system where it is
)
Density of states:The ability of metals to conduct electricity depends on number of quantum states and available
energy levels of the electrons.
[From Planks hypothesis implies that a system can exist only in certain discrete energy states that
energy would not be continuous such a states are quantum states]
The number of quantum states present in a metal between the energy interval E&E dE per unit
volume is known as density of states.
𝑁𝑢𝑚𝑢𝑏𝑒𝑟 𝑜𝑓 𝑞𝑢𝑎𝑛𝑡𝑢𝑚 𝑠𝑡𝑎𝑡𝑒𝑠 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑏𝑒𝑡𝑤𝑤𝑒𝑒𝑛 𝐸 & 𝐸+𝑑𝐸
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒
Density of states g (E) =
Consider the sphere is further divided into number of shells. Therefore each shell is represented by
a set of quantum numbers
 n , n , n  and will have associated energy,[let E be the energy of
x
y
z
the quantum states].
Therefore the radius of the sphere with energy ‘E’ is n  nx  ny  nz
2
2
2
2
Consider that the unit volume represents one energy state therefore number of energy states in a
sphere of radius ‘n’ is
4 3
n
3
The above equation represents the total volume of sphere we know that the quantum numbers
 n , n , n  takes only the +ve integral values and hence one has to take only one octant in the
x
y
z
sphere i.e. 1/8 th part of total volume of the sphere.
5
Every state available within one octant of the sphere of radius ‘n’ and its energy E is g (E) 
1 4 3
n 
8  3

Similarly the energy states available with one octant of the sphere of the sphere of radius ‘n+dn’and its
energy ‘E+dE’
g (E) 
1 4

 (n  dn)3 

4 3

Energy states available within the sphere of radius ‘n’ & ‘n+dn’ is obtained by finding the energy
difference between the two energy levels namely ‘E & E+dE’
1  4

g1 (E) dE    (n  dn)3  n3  
8  3

1 4
    n3  dn3  3n 2 dn  3ndn 2  n3 
8 3


 dn3  3n 2 dn  3ndn 2  66 | P a g e
6
Neglecting the higher powers of dn i.e. dn3 & dn 2
we get
g (E) dE 


3n dn   n dn

6
2
2
2


2
n(ndn)
(1)
Consider a cubic metal piece with cube edge ‘l’ therefore energy of electrons within the cube is
E
Rearranging we get n 2 
n2 h2
8ml 2
8ml 2 E
h2
(2)
To obtain ‘ndn ’value by differentiating eq(2) w.r.t ‘ n’
8ml 2
2ndn  2 dE
h
8ml 2
ndn 
dE
2h 2
(3)
Substituting the values of ‘ndn’ & ‘n’ values in eq(1)
g (E) dE 
1
  8ml E   8ml 
2

2
h2
2
2
 dE 
 
2
  2h 
  8ml 2 E 

2
h
2
Here l 3 is the volume of the metal piece.
6
3
2
1
2
 E dE

According to Poulie’s exclusion principle two opposite spin electrons can only occupies one energy state
Then we get
g ' (E) dE 
2
4
 8 ml 2  1 2

 E dE
 h2 
2 3  8 m 

l  2 
4
h 
3
2
1
E
2
dE
The number of available energy states per unit volume ( l 3 )
i.e. Density of states ( l 3 =1)
g ' (E)dE
l3
2
 g (E) dE 
8 m 
4 h3
2
3
g (E) dE 
 42 m
4 h3
2
3
2
1
E
2
g (E) dE 
1
E
2
dE
dE 
4
h
3
2
 2   2 m
4 h3
 2 m
3
3
2
2
1
2
3
2
1
E
2
dE
2
E dE
From this we can determine the carrier concentration in the metal and semiconductor.
7
Fermi Energy:- Applying Paulie’s exclusion principle to a system of non-interacting electrons, it is seen
that all levels below a certain level will be empty, at absolute zero.
The highest level filled at absolute zero is called the Fermi level. Electrons are fermions.
The probability that an energy state ‘E’ at any temperature (T) the electrons filed in the energy states is
given by the F-D distribution.
1
(E-E )
F
f E 
e
kT
+1
Where E F is Fermi energy. Fermi energy ( E F ) is the maximum kinetic energy possessed by free
electron in a metal at absolute temperature ( T  0 K 0 ).
At absolute zero T  0 K 0
Case 1:
1
E < E  f E 
1
F
1  e 
Case 2: E > E  f  E  
F
Case 3: E = E  f  E  
F
1
0
1  e
1
1
  0.5
0
1 e
2
At ( T  0 K 0 )
At ( T  0 K 0 )
At ( T  0 K 0 )
( e  0)
( e  1)
( e0  1)
From case3 the Fermi energy is defined as the energy level at any temperature in which the probability
of electrons is 50% or 1/2.
0
At T  0K  KT  EF (or) T  
Electrons losses quantum mechanical properties now it obeys traditional M-B Distribution function.
0
At temperature other than ‘ 0K 0 ’ i.e. at T  0K & E  E F we have f  E  
8
1
2
Band Theory of solids:- According to classical free electron theory electrons in a crystal lattice electrons
move in a constant potential.
But according to quantum theory the potential field in a crystal lattice is not constant but it is periodic.
The potential at the site of +ve ions tends to (-  ) and becomes zero in between two ions as shown.
Bloch theorem:- The electrons move in a potential provided by the stationary +ve ions.
Let us consider a move in 1-D crystal consisting of number of +ve ions.
The most important feature of this potential is its periodicity. It has the form
V(x + na) =V(x)
Where n=0, 1,2,3,4 ……………
Where ‘a’ is the periodicity of lattice
The Schrodinger equation to be solved now is
d 2
dx
2

2m
2
E  V ( x ) ( x )  0
If the electrons are completely free, the wave function would be of the form
 (x)  Ae(ikx)
Which has constant amplitude.
The effect of periodic potential is to change the free particle wave function in such a way that its
amplitude changes with the period of the lattice according to ‘Bloch’ the wave function has the form
 (x)  U (x)e(ikx) , K 2 
2m(E  V(x))
2
With the condition U(x)  U (x  a)  U (x  na)
9
Where ‘n’ is integer. Here ‘a’ is periodicity of lattice. The form of’ U(x) ‘depends on the potential ‘U(x)’
&’ K’
Kronig – penny model:- This model describes the behavior of an electron in a periodic potential field
produced by positive ion cores.
The potential of the electron varies periodically with periodicity of ion core and the potential energy
(P.E) of the electron is zero near nucleus of the +Ve ion core and maximum when it is lying between the
adjacent nuclei which are separated by the inter atomic spacing (a). This is shown in fig.
V (x)  0 for the region 0<x<a
V (x)  V0 for the region -b< x <a
Applying the Schrodinger time independent equation for above two regions.
d 2
dx
2

2m
2
2 
Substituting
Then
E ( x )  0
d 2
, dx
2mE
2
2 
2

2m
2
E  V0  ( x )  0
2m(E  V0 )
2
d 2
  2  0 for 0  x  a  (1)
dx 2
d 2
  2  0 for - b  x  a  (2)
dx2
The solutions for the above equations can be written in the Bloch’s form as
 (x)  U(x) eikx  (3)
Here eikx is modulated by periodic function U(x). From Bloch theorem U(x) = U(x+a)
Here k is propagating vector along X- direction K =2π/λ K is also called wave number.
10
Differentiating eq(3) twice with respect to ‘x’ and substituting in eq(1) & (2).
Applying boundary conditions to the solution of the above equation.
We get
  2  2 

 sinh  bsin  a  cosh  bcos  a  cosk(a  b)  (4)
 2 
Let V0   and b  0 Such that V0b remains finit.
Sin h b   b and cos h b  1
 2  2 

2m(V0  E)

2mE
V0  2 E  
2mV0
2
2m
2
V0  E
2
2
V0  E
Substituting all the values in eq (4) we get
2mV0
 b sin  a +cos  a= cos ka
2 2
mV0 ab sin  a
+cos  a= cos ka
2
a
psin  a
+ cos  a= cos ka
a
p=
mV0ab
is called scattering power (or) Potential barrier strength
h2
Suggested that only Here R.H.S can have values between ±1 only. Where L.H.S can exceed these limits.
Kronig _ Penny those values of (αa
electron energy) are acceptable
for which L.H.S also lies between +1 & -1
Thus according to this model there are some forbidden energy gaps between allowed energy bands.
As (αa) is increases width of the allowed energy increases and that of forbidden energy gap decreases.
11
Salient features of K-P Model:1. When ‘p’ is large the allowed region of electrons becomes very narrow.
∴ p → ∝ (or) Sin αa = 0
αa= ±nπ α2 = (n2π2)/a2 but α2 = 2mE/ℏ2
2mE
2

n2 2
 2 2 
 E = n2 
2 
 2ma 
2
This expression shows that energy spectrum of the electron contains discrete energy levels
separated by forbidden region.
2.
When p→ 0 cos αa = cos ka  αa= ka  α = k then α2 = k2 where α2 =
 k2 =
2mE
2
2 2
then E=
k
(∵ k = p/
2m
2
)  E = p2/2m=
2mE
2
1 2
mv
2
The eq shows that all the electrons are completely free to move in crystal without any
constraints. i.e. Here all energies are allowed to the electrons. This case supports the classical
free electrons theory.
3. When p=±1 for an allowed region the values of ‘Cos ka’ ranges from +1 to -1.
Cos ka =±1 ⇒ ka = nπ, k = nπ/a
Thus the discontinuity in electron energy occurs at an interval of nπ/a while n = 1,2,3…….
K=±

a
,±
2 3
4
,±
,±
…………
a
a
a
E-K graphs:- For a free electron total energy is kinetic in nature E = p2/2m
2
From de-Broglie concept we have p =
k
k2
h 2k 2
E

2m 8 2 m2
continuous parabola as shown in fig.
12
The E-K graph for free electrons is a
For an electron in a crystal lattice. Here from the condition of K-P model.
psin  a
+ cos  a= cos ka
a
Discontinuities in energies occour for ka= ±nπ (or) k = ±nπ/a, n= 1,2,3……..
E-K graph for an electron in a crystal lattice is discontinuous parabola as shown with discontinuities.
The region of allowed K-Values from


to
is called 1st Brillouin zone &
a
a
2

2

K
to
 K  is called 2nd Brillouin zone.
a
a
a
a
Classification of materials according to the electrical conductivity of the materials by using band
theory of solids: From Kronig Penney model it follows that the energy level diagram of conduction
electrons consists of groups of allowed energy levels called allowed bands separated from one another
by a forbidden energy or band as shown in the figure.
Conduction electrons occupy these energy levels starting with the levels in lowest allowed band. When
all the levels in one band are completely filled, the electrons begin to occupy levels in next higher
allowed band. This process continues till all the electrons get accommodated.
Valence Band: - Highest occupied allowed band at zero degree Kelvin is
called Valence band. Valence band can be either partly filled or completely
filled and all the allowed bands below Valence band are completely filled and
above Valence band they are completely vacant.
13
Conduction Band: - The next higher allowed band above Valence band is
called Conduction band and conduction is completely vacant at 0 ok.
Energy Gap : - The forbidden energy between Valence and conduction
bands is called Energy Gap.
b). Insulators: - According to band theory of solids, in insulating
materials valence band is completely filled, and conduction band is
completely vacant. A large energy gap of more than 5eV separates
valence band from conduction band. This large energy gap > 5eV
prevents valence band electrons from occupying vacant higher energy
levels of conduction band.
Since electrons cannot acquire higher energies they cannot get accelerated in the direction of the
applied field and the material acts like an insulator.
Eg. Diamond, Sulphur etc.
Condu
Conductors: - In conducting materials like metals the valence band
and conduction band overlap, with zero energy gap. This makes vacant
energy levels of conduction band readily available for electron
occupation. Hence the electrons can get accelerated in the direction of
the applied field gaining higher energies; resulting in the transport of
charge across the material and the material acts like a conductor with
high electrical conductivity.
Eg:- Mg, Zn, etc.
Materials also act like conductors when the valence band is partially
filled, in this case the vacant higher energy levels are available inside the
valence band itself.
eg : - Cu, Ag, Au, Alkali metals
Semiconductors: - Semiconductors are the materials which
behave like insulators at 0 0K and exhibit certain amount of
electrical conductivity at all temperatures greater than 0 0K. In
these materials at 0 0K valence band is completely filled and
conduction band completely vacant. A small energy gap of about
1 eV separates them. At 00k even this small energy gap prevents
the valance band electrons from occupying vacant higher energy
levels of conduction band and hence semiconductors act like
insulators at 00k. At temperatures greater than 00k, some of the
valance band electrons acquire thermal energies greater than
energy gap and jump into conduction band where vacant higher
energies are available. Further the transition of electrons from
VB to CB creates vacant levels called holes in the valance band
14
ctors
Over lapping of Valence
& conduction bands
itself. The electrons in the valance band can acquire higher
energies jumping into the holes. Therefore semiconductors
exhibit small electrical conductivity above 00k.
Concept of effective mass on electron:Quantum mechanics treats electron as a wave packet. The velocity with which this wave packet moves is called
group velocity.
Group Velocity Vg 
d
 (1)
dk
From Plank quantum theory wehave
E  h     
E
Vg 
1 dE
 (2)
dk
dv 1  d  dE  
  

dt
 dt  dk  
1 dk d  dE 
1 d 2 E dk
d
dk
dk 1
a
a

 (3) Fext   k  =

 (Fext )


2
dt dk  dk 
dk dt
dt
dt
dt
a
2
1 d 2E
a 2
*
*
a 2
(Fext )  Fext  2 Fext  m e  a m e  2
 (4)
d E
d E 2
dk 2
dk
dk 2
Since Eq (1) become Vg 
1 dE
 (2)
dk
i.e. velocity of an electron in a crystal lattice is proportional to
dE
.
dk
dv 1  d  dE  
  

dt
 dt  dk  
∴ Acceleration (a) =
1 dk d  dE 
1 d 2 E dk
a
a

 (3)


dt dk  dk 
dk 2 dt
a
We have Fexternal i.e. External force F 
dp
d
dk
dk 1
with p = k , We get Fext   k  =

 (Fext ) .
dt
dt
dt
dt
Substituting above values in eq(3) we get a 
1 d 2E
a 2
(F
)

F

ext
ext
2
d 2E
dk 2
dk 2
15
This is of the form Fext  m*e  a
i.e. electron in a lattice behaves as if its mass is m*e  
when Effective mass of an electron m e 
2
inflection(K0)
m*e  
d 2E
dk 2
0
 (4) is called effective mass of electron
2
infinite effective d E
*
2
d E
2
dk 2
dk
2
 0 m*h , at the point of
i.e electron behaves as if is mass is infinity. This means that electron in an energy
band does not experience any acceleration even when an external electric field is applied.
2
Negative effective mass:- When d E
dk 2
is negative, m *e is negative. After the point of inflection, electron
behaves as its mass is negative. This means that electron experiences acceleration in the direction of electric
field or if it experiences retardation, we can say that electrons behaving as if its mass is negative.
Concept of hole: Hole concept comes in a completely filled energy band only. Absence of an electron or an
electrons vacancy in a completely filled band can be considered as a +ve charge named as a hole.
Its effective mass is denoted as m*h is greater than m *e slightly, movement of electrons in valance band can
be considered as hole movement.
Hole concept is very useful in semiconductors. Movement of free electrons in conduction band gives rise to
electron current and movement of hole in V.B gives rises to hole current in semiconductors.
Physical Significance of effective mass: The variation of Energy, velocity and
effective mass m* with k for the first allowed band is shown in figure.
The velocity can be calculated from the relation Vg =
1 𝑑𝐸
ħ 𝑑𝑘
. This gives a curve of
the type shown fig b. At the bottom (k= 0) of the energy band the velocity vg is
zero and as the value of k increases, the velocity increases reaching a maximum
at k = k0, where k0 corresponds to the point of inflection on E-k curve. Beyond
this point the velocity begins to decrease, finally reaching zero value at k =π /a,
which corresponds to top of the band.
The negative values of wave vector k also exhibit a similar behavior. This
decrease in velocity with increase in energy is a new feature which does not
appear in the behavior of free electrons.
Variation of effective mass with k:- The figure (c)shows the variation of
effective mass m* with k. Near k = 0 the effective mass approaches m the mass
of free electron and as k increases m* also increases, reaching a maximum value
16
at the point of inflexion on the E-k curve. Above the point of inflexion m* is
negative and as k tends to π/a, it decreases to a small negative value.
Near the bottom of all the bands the effective mass m* is positive and m* is negative near the tops of all bands.
From the Fig.b it can be seen that beyond the inflexion point k0 the velocity decreases, i.e. acceleration is negative.
This indicates that in this region lattice exerts a large retarding force on electron. This can be understood by
assuming that the electron behaves like a positively charged particle referred to as hole. Thus the concept of the
hole can be introduced / explained. The degree of freedom fk of electron is defined as the ratio of free electron
mass to effective mass.
fk 
m m  d 2E 
 2 2 
m*
 dt 
fk indicates the extent to which electron is effected by the lattice. If fk =1 the electron in the lattice behaves like
free electron. When fk is small electrons are heavier than free electrons. fk values are positive in lower half of the
band and negative in upper half of the band as shown in fig. d.
17