statistical modified)

8/3/2019 Statistical Modified)

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STATISTICALSTATISTICAL

MECHANICSMECHANICS


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IntroductionIntroductionThe subject which deals with the relationshipThe subject which deals with the relationshipbetween the overall behavior of the system and thebetween the overall behavior of the system and theproperties of the particles is calledproperties of the particles is called StatisticalStatisticalMechanics.Mechanics.

Statistical Mechanics can be applied to classicalStatistical Mechanics can be applied to classical

systems such assystems such as Molecules in gasMolecules in gas as well asas well asPhotons in a CavityPhotons in a Cavity andand Free Electrons in aFree Electrons in aMetalMetal..


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Physical SystemPhysical System

Consider a system composed ofConsider a system composed ofNN identical, nonidentical, noninteracting particles in a volumeinteracting particles in a volume V . V .

letlet n n11particles posses energyparticles posses energy E E11 ,, n n22

particles posses energyparticles posses energy E E22 . and so on.. and so on.

Total energy of the systemTotal energy of the system

!

!

!

i

ii

nN

and

EnE

EnEnEnE .....332211


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Macro States and Microstates of SystemsMacro States and Microstates of Systems

Any state of a system as described by acAny state of a system as described by actual ortual orhypothetical observations of its Macroscopic statisticalhypothetical observations of its Macroscopic statistical

properties is known asproperties is known as Macro State Macro State and it is specifiedand it is specified

byby ( N, V and E ) .( N, V and E ) .

The state of system as specified by the actual properties ofThe state of system as specified by the actual properties of

each individual, elemental components and it is permittedeach individual, elemental components and it is permitted

by the uncertainty principle is known asby the uncertainty principle is known as Micro State Micro State ..

ForFor N N particle system , there may be always possibleparticle system , there may be always possible

N+1N+1 Macro States andMacro States and 22nn Micro States.Micro States.


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Phase SpacePhase Space1.T1.The three dimensional space in which the location of a particle ishe three dimensional space in which the location of a particle is

completely specified by the three position coordinates, is known ascompletely specified by the three position coordinates, is known asPosition Space.Position Space.

2.The three dimensional space in which the momentum of a2.The three dimensional space in which the momentum of aparticle is completely specified by the three momentumparticle is completely specified by the three momentumcoordinatescoordinates PPxx,, PPyy andand PPzz is known asis known as Momentum Space.Momentum Space.

3.3.The combination of the position space and momentum space isThe combination of the position space and momentum space isknown asknown as Phase Space.Phase Space.

zyxdpdpdpd !+SpaceMomentumainvolumeSmall

dxdydzdV !spacePositionainvolumeSmall

+! dVddXSpacePhaseainvolumeSmall


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Phase Space VolumePhase Space Volume

Consider LetConsider Let p pmmbe the maximum value of thebe the maximum value of themomentum of the particles in the system.momentum of the particles in the system.

LetLet ppxx ,,ppyy,, ppzz represents the three mutuallyrepresents the three mutually

perpendicular axes in the momentum space as shownperpendicular axes in the momentum space as shownin figure.in figure.

Draw a sphere with an originDraw a sphere with an origin O O as centre and theas centre and the

maximum momentummaximum momentum p pmm as radius.as radius.

All the points within this sphere will have theirAll the points within this sphere will have theirMomentaMomenta lying betweenlying between 0 0 andand p pmm..


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pz

0

px

py

pm

)volumespacepositionaisVwhere(

.....3

4

volumespacephase

3

4volumemomentum

.pradiusofspheretheofvolumevolumespacemomentumThe

3

3

m

Vp

V

p

m

m

TX

X

T

!

+!

!+

!


8/39

X

X

d

V

n

+

!

!

cellsofnumber3.Total

celloneofvolumespacephaseinvolumeavailableTotal

spacephaseincellsofnumbertotal2.The

.cellacalledisspacephaseindvolumesmall1.The


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EnsembleEnsemble

AnAn EnsembleEnsemble is defined as a collection of veryis defined as a collection of very largelarge

number ofnumber ofAssembliesAssemblies which are essentiallywhich are essentially independentindependent

of one another.of one another. ThereThere are three types of ensembles.are three types of ensembles.

1.Microcanonical1.Microcanonical ensembleensemble

2.Canonical2.Canonical ensembleensemble

3.Grand3.Grand canonical ensemblecanonical ensemble


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Micro CanonicalMicro Canonical

EnsembleEnsemble

CanonicalCanonical

EnsembleEnsemble

Grand CanonicalGrand Canonical

EnsembleEnsemble

All assembliesAll assemblies1.1. Same Volume ,Same Volume ,

2. Same number of2. Same number of

particles N,particles N,

3. Same energy E.3. Same energy E.

All assembliesAll assemblies1.1.Same Volume ,Same Volume ,

2.Same Number of2.Same Number of

particles N,particles N,

3. Same3. Sametemperature T.temperature T.

All assembliesAll assemblies1.1. Same Volume ,Same Volume ,

2. Same temperature2. Same temperature

T and SameT and Same

Chemical potentialChemical potential..


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Statistical DistributionStatistical Distribution

Statistical Mechanics determines the most probable wayStatistical Mechanics determines the most probable wayof distribution of total energyof distribution of total energy E E among theamong the N N particles of a system in thermal equilibrium at absoluteparticles of a system in thermal equilibrium at absolutetemperaturetemperature T . T .

In statistical mechanics one finds the number of waysIn statistical mechanics one finds the number of ways

W W in which thein which the N N number of particles of energy E number of particles of energy E can be arranged among the available states is given by.can be arranged among the available states is given by.

N(E) = g(E) f(E)N(E) = g(E) f(E)

WhereWhere g(E) g(E) is the number of states of energyis the number of states of energy E E andand

f(E) f(E) is the probability of occupancy of each state ofis the probability of occupancy of each state of

energyenergy E . E .


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Classical StatisticsClassical Statistics

( Maxwell( Maxwell Blotzmann distribution)Blotzmann distribution)

Statistical Distribution

Quantum StatisticsQuantum StatisticsBoseBose -- Einstein distributionEinstein distribution

FermiFermi -- Dirac distributionDirac distribution


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Classical StatisticsClassical Statistics

(Maxwell(Maxwell -- Blotzmann Distribution)Blotzmann Distribution)Let us consider a system consisting of molecules of anLet us consider a system consisting of molecules of an

ideal gas under ordinary conditions of temperature andideal gas under ordinary conditions of temperature andpressure.pressure.

Such a system is governed by the laws of ClassicalSuch a system is governed by the laws of ClassicalMaxwellMaxwell-- Blotzmann Statistics.Blotzmann Statistics.

Assumptions:Assumptions:

1.1. The particles areThe particles are identicalidentical andand distinguishable.distinguishable.

2. The volume of each phase space cell chosen is2. The volume of each phase space cell chosen isextremelyextremely smallsmall and hence chosen volume has veryand hence chosen volume has verylarge number of cells.large number of cells.


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3.3. Since cells are extremely small, each cell can have eitherSince cells are extremely small, each cell can have eitheroneoneparticle or no particleparticle or no particle though there isthough there is no limitno limit on the number ofon the number ofparticles which can occupy a phase space cell.particles which can occupy a phase space cell.

4.4. The system isThe system is isolatedisolated which means that both the totalwhich means that both the total numbernumberof particlesof particles of the system and theirof the system and theirtotal energytotal energy remain constant.remain constant.

5.5. The state of each particle is specified instantaneous position andThe state of each particle is specified instantaneous position andmomentum comomentum co--ordinates.ordinates.

6.6. Energy levels areEnergy levels are CContinuousontinuous..


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Equation for MB Distribution

Let the number of particles in the 1st, 2nd , 3rd

,.,ith,.groups be n1,n2,n3,..ni,.. Respectively.

Also assume that the energies of each particle in the 1st

group is E1, in the 2nd group is E2 and so on.

Let the degeneracy parameter denoted by g [ the number

of molecular states] in the 1st

, 2nd

,3rd

,,ith

, groups beg1,g2,g3,.gi,. and so on respectively.

In a given system the total number of particles is constant.


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...........,.321

constnnnnnneiii

!7!!

Hence its derivative ]1.....[0!7 inH

The total energy of the system is given by

constnEE

nEnEnEnEE

ii

ii

!!

!

.........332211

Hence its derivative ]2.....[0!7 iinEH


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The probability of given distribution W is given by the product of

two factors.

The first factor is, the number of ways in which the groups of n1,n2,n3,ni, particles can be chosen.

To obtain this, first we choose n1 particles which are to be placed in

the first group. This is done in

)!(!

!..

11 nnn

nei

The remaining total number of particles is (n-n1). Now we arrange

n2particles in the second group. This is done in

)!(!

)!(

)(

212

1

12

nnnn

nn

nnnC


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The number of ways in which the particles in all groups are chosen is

factortionmultiplicaiswhere

n

n

w

nnnnn

nw

nnnn

nn

nnn

nw

i

ii

T

T )!

!

(

)!........!!!!

!(

)......)!(!

)!()(

)!(!

!(

1

54321

1

212

1

111

!

!

!

i

T

i

i

n

ii

ni

nnn

gW

ggggW

)(

...).....()()()(

2

3212321

T!

!


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The number of different ways by which n particles of the system

are to be distributed among the available molecular states is

]

!

[!

)(!

!

21

i

n

ii

n

ii

ii

n

gnW

gn

nW

WWW

i

i

T

TT

!

!

!

Taking natural logarithms on both sides of equation in above.

!lngn!lnnnnlnn

!lngnlnn!nnnlnnlnW

x,xlnxlnx!

ionapproximatStrilingApplying

lngn!lnnlnn!lnW

i

i

ii

i

i

i

i

i

i

i

iiii

!

!

!

!


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For the Most probable distribution.

]3].......[0[

)(ln1

ln

)(ln1

ln

0ln

max

max

max

!7

!

!

!

i

i

iii

i i

i

i

iii

i i

i

n

ngn

n

nW

ngnn

nW

W

H

HHH

HHH

H

3


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Multiplying equation (1) with - and equation (2) with

and adding to the (3) equation and there by

TK

EiMB

BTK

E

i

i

E

i

i

B

i

B

i

i

ee

Ef

Tkwhere

eeg

n

eeg

n

E

E

FE

F

1)(

)1

(1

sidesbothonlexponentiaTaking

E)g

nln(

0Elnglnn

0.]Elnglnn[

i

i

i

iii

iiii

i

!

!!

!

!

!

!

3

This distribution tell us the way of distribution of total energy E

of the system among the various identical particles.


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Failures of Classical StatisticsFailures of Classical Statistics

1.1. The observed energy distribution ofThe observed energy distribution of electrons inelectrons inMetalsMetals..

2.2. The observed energy distribution ofThe observed energy distribution of PhotonsPhotons inside theinside thecavity.cavity.

3. The behavior of3. The behavior ofHeliumHelium at low temperatures.at low temperatures.


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Quantum StatisticsQuantum Statistics

According to Quantum Statistics the particles ofAccording to Quantum Statistics the particles ofthe system arethe system are indistinguishableindistinguishable, their wave, their wave

functions dofunctions do overlapoverlap and such system of particlesand such system of particlesfall intofall into twotwo categoriescategories

1.Bose1.Bose -- Einstein DistributionEinstein Distribution2.Fermi2.Fermi -- Dirac Distribution.Dirac Distribution.


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BoseBose Einstein StatisticsEinstein Statistics

According to BoseAccording to Bose--Einstein statistics the particles of any physicalEinstein statistics the particles of any physical

system aresystem are identical,identical, indistinguishableindistinguishable and haveand have integral spinintegral spin, and, and

further those are called asfurther those are called as Bosons.Bosons.

AssumptionsAssumptions1.1. The Bosons of the system areThe Bosons of the system are identicalidentical andand indistinguishableindistinguishable..

2. The Bosons have integral spin angular momentum in units of2. The Bosons have integral spin angular momentum in units of

h/2h/2..

3. Bosons obey3. Bosons obey uncertaintyuncertaintyprinciple.principle.

4. Any number of bosons can occupy a4. Any number of bosons can occupy a single cellsingle cell in phasein phase space.space.


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5. Bosons do not obey the5. Bosons do not obey the PoulisPoulis ExclusionExclusionprinciple.principle.

6. Wave functions representing the bosons are6. Wave functions representing the bosons are SymmetricSymmetric

i.ei.e,, (1,2) =(1,2) = (2,1)(2,1)

7. Energy states are7. Energy states are discretediscrete..

8.The probability of boson occupies a state of energy E is given by

1)exp(

1)(

!

kT

EEEf

fBE


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FermiFermi -- Dirac StatisticsDirac Statistics

According to FermiAccording to Fermi -- Dirac statistics the particles of any physicalDirac statistics the particles of any physical

system aresystem are indistinguishableindistinguishable and haveand have half integral spinhalf integral spin. These. Theseparticles are known asparticles are known as Fermions.Fermions.

AssumptionsAssumptions1.1. Fermions areFermions are identicalidentical andand indistinguishable.indistinguishable.

2.2. They obeyThey obey Pauli s exclusionPauli s exclusionprinciple.principle.

3.3. Fermions haveFermions have half integral spinhalf integral spin..

4.4. Wave function representing fermions areWave function representing fermions are anti symmetricanti symmetric

)1,2()2,1( ]] !


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6.6. UncertaintyUncertaintyprinciple is applicable.principle is applicable.

7. Energy states are7. Energy states are discretediscrete..

8. The probability of a fermions occupies a state of energy E is8. The probability of a fermions occupies a state of energy E is

given bygiven by

)exp(1

1)(

kT

EEEF

f

!


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FermiFermi DiracDirac DistributionDistribution

Consider the system containsConsider the system contains nn numbernumber ofofindistinguishableindistinguishable particlesparticles carryingcarrying different energiesdifferent energies

EE11, E, E22..EEii........

LetLet us considered ,the Systemus considered ,the System be divided into groupsbe divided into groups..If theIf the iithth groupgroup containscontains nnii number of particles, distributednumber of particles, distributedggii no of quantumno of quantum states, all these particles have nearly thestates, all these particles have nearly thesame energysame energy EEii ..

TheThe total number of ways of arrangingtotal number of ways of arranging nniiparticles inparticles in ggiistates..states..

!)!(

!

iii

ii

nng

gw

!


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There fore the total distribution for the completeThere fore the total distribution for the complete systemsystem

isis given asgiven as

)}ln()(lnln{ln)]}()ln()[()ln()ln{(ln

ln!lnbut

)2}.......()!ln(!ln!{lnln

)!)!(

!ln(ln

)1..(....................!)!(

!

iiiiiiiii

iiiiiiiiiiiii

iiiii

iii

ii

iii

ii

ngngnnggWngngngnnngggW

xxxx

ngngW

nnggW

nng

gW

!

!

!

!

!

!


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)5.........(0

)4(..........0equationofdervativetheon,distributiprobablemostfor

henceconstant,areEsystem,theofenergytotaltheand

,systemthe,nparticlesofnumbertotalthem,equilibriuAt

)3.....(..........0}ln){ln(

0)}ln()(

)(

)(ln{

0)}ln()(lnln{

0ln

equationofderivativetheon,distributiprobablemostFor

!x!

!x!

!x

!xx

xx

!x

x

!

x

x

iii

ii

iiiii

iiii

ii

iiiii

i

ii

iiiiiiiii

i

i

nEdE

ndn

nnng

ngnn

ng

ngnnn

n

n

ngngnnggn

Wn


31/39

T

and

T

tk

EEeg

ne

n

g

en

ge

n

ng

E

n

ng

En

ng

Engn

nEngn

B

fE

i

iE

i

i

E

i

iE

i

ii

i

i

ii

i

i

ii

iiii

iiiiii

i

i

ii

BB

F

k

1

k

E-where

..........energy....thatofstatesquantumtheofeach

inparticlesofnumberaveragetherepresents

)exp(1

1

1

11

1)(

)(ln

0)(

ln

0)ln(ln

0))ln(ln[

3.equationtoaddand-by5and-by4equationmultiply

s,multiplieredundeterminmethodLagrangethe,Applying

!!

p

!p!

!p!

!

!

!

!

FE

FE

FE

FE

HFE

FE

FE

FE

FEFE


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Fermi Distribution functionFermi Distribution function Let us consider how the electrons in a real material distribute amongLet us consider how the electrons in a real material distribute among

the different possible energy states.the different possible energy states.

The assembly of electrons considered as electron gas behaving likeThe assembly of electrons considered as electron gas behaving likea system of Fermi particles obeying Fermia system of Fermi particles obeying Fermi -- Dirac statistics.Dirac statistics.

Accordingly, the probabilityAccordingly, the probability F(E) F(E) of an electron occupying anof an electron occupying anenergy levelenergy level E E is given byis given by

WhereWhere EEff is called Fermi energy and is ais called Fermi energy and is a constantconstant for a givenfor a givensystem.system.

)exp(1

1)(

kTEE

EFf

!


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This means that atThis means that at 0k0k, all quantum states with energy, all quantum states with energy belowbelow EEffareare completely occupiedcompletely occupied and thoseand those aboveabove EEffareare unoccupied.unoccupied.

0)(,

1)(,

0

!"

!

!

EFEE

EFEE

forKT

At

f

f

0 Ef E

F(E)

0.5

T=0K

T3>T

2>T

1>0K

T1T2T3


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At any temperature other thanAt any temperature other than 0k,0k,ififE =E = EEff, F(E)=1/2., F(E)=1/2.

Hence, the Fermi level is that state at which theHence, the Fermi level is that state at which theprobability of electron occupation is probability of electron occupation is at by temperatureat by temperature

above 0K and also it is theabove 0K and also it is the highest levelhighest level of the filledof the filledenergy states at 0K.energy states at 0K.

Fermi energy is the energy of the state at which theFermi energy is the energy of the state at which the

probability of electron occupation is probability of electron occupation is at by temperatureat by temperatureabove 0K.above 0K.


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Density of Energy StatesDensity of Energy States

The Number of Electrons present per unit volume in an energyThe Number of Electrons present per unit volume in an energy

level at a given temperature is equal to the product of states ( nolevel at a given temperature is equal to the product of states ( no

of energy levels per unit volume ) and Fermi Dirac distributionof energy levels per unit volume ) and Fermi Dirac distribution

function ( the probability to find an electron ).function ( the probability to find an electron ).

)()( EFdEEgn

bandenergy

c !

Where g(E) density of states and F(E) is the probability

function.

Therefore, to calculate the number of electrons in an energy

level at a given temperature, it is must to know the number of

energy states per unit volume ( Density of states ).


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3

3

4nn T!

Therefore the number of energy states within a sphere of radius n

Since n1,n2 and n3 can have only

positive integer values, we have

to consider only one octant of

the sphere.

}3

4{8

1 3nT!

n

nx

ny

nz

dn

E

E+ dE

The number of energy states with particular energy value E is

depending on how many combinations of the quantum numbers

resulting in the same value n.


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In order to calculate the number of states within a smallIn order to calculate the number of states within a small

energy intervalenergy interval E E andand E +E + dEdE,, we have to constructwe have to construct twotwo

spheresspheres with radiiwith radii nn andand n +n + dndn and calculate the spaceand calculate the space

occupied within these two sphere.occupied within these two sphere.

Thus the number of energy states having energy valuesThus the number of energy states having energy values

betweenbetween EE andand E +E + dEdE is given byis given by

dnndEEg

ndnndEEg

2

33

2)(

)}(3

4{

8

1)}(

3

4{

8

1)(

T

TT

!

!


38/39

The expression for the energy

of electron in potential well

is given by

2

12

1

2

2

2

12

1

2

2

2

2

2

2

2

2

2

1

2

2

2

22

2

22

}8

{2

1

}8

{218

8}

2

1{

82

)1(.,

)8

(

)1........(

8

8

E

dE

h

mLdn

E

dEmLh

hmL

dEh

mL

ndn

dEh

mLndn

eqtingdifferntia

h

EmLn

h

EmL

n

mL

hnE

n

-

!

-

!

!

!

!

!

!


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Substituting the values of n2 and dn from in eqn.. (1)

dEELm

h

dEEh

mLdEEg

2

132

3

3

2

1

2

3

2

2

)2(4

]8[

42)(

T

T

!

v!

Density of energy states is given by number of energy states

per unit volume.

Density of states dEEmh

dEEg 21

2

3

3)2(

4)(

T!

statistical modified)

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