unit-iv(physics)

7/30/2019 UNIT-IV(physics)

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Statistical mechanics can be applied to systems

such as

The subject which deals with therelationship between the overall behaviorof the system and the properties of theparticles is calledStatistical Mechanics.

molecules in a gas

photons in a cavity

free electrons in a metal


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Macro stateAny 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 ) .

NOTE

For N particle system , there may be always

possible N+1 Macro states.


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Micro state

The state of system as specified by the actual

properties of each individual, elemental

components, in the ultimate detail permitted

by the uncertainty principle is known as Microstate .

NOTE

For N particle system , there may be always

possible 2n Micro states.


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Phase space

The three dimensional space in which thelocation of a particle is completely specified by

the three position co-ordinates, is known asPosition space.

Small volume in a position space dV = dx dy dz


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The three dimensional space in which themomentum of a particle is completely specified bythe three momentum co-ordinates Px Pyand Pzis known asMomentum space.

Small volume in a momentum spaced= dpxdpydpz


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The combination of theposition space andmomentum space is known as

Phase space.

Small volume in a phase space d= dV d


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Draw a sphere with an origin O as centreand the maximum momentum pmas radius.

pz

0

px

py

pm


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All the points within this sphere will have theirmomentum lying between 0 and pm.

The momentum space volume =

volume of the sphere of radius pm .

Momentum volume is given by

= 4/3pm3


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Similarly phase space volume is given by

= V

= 4/3pm3

Vwhere,

Vposition space volume


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The set of possible states for a systemof Nparticles is referred as ensemble

in statistical mechanics.

ENSEMBLE

(OR)


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A collection of large number of microscopically

identical but essentially independent systems is

calledensemble.

NOTE:

An ensemble satisfies the same macroscopic

condition.

Example:In an ensemble the systems play the role of as

the non-interactive molecules do in a gas.


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There are 3-types of ensembles, those are

1) MICRO CANONICAL ENSEMBLE

2) CANONICAL ENSEMBLE

3) GRAND CANONICAL ENSEMBLE


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1) MICRO CANONICAL ENSEMBLE

It is the collection of a large number of essentially

independent systems having the same energy (E),

volume(V) and the number of particles(N).

We assume that all the particles are identical and the individual

systems of micro canonical ensemble are separated by rigid,

well insulated walls such that the values of E, V & N for

a particular system are not affected by the presence of

other systems.

NOTE:


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E,V,N E,V,N E,V,N

E,V,N E,V,N E,V,N

E,V,N E,V,N E,V,N


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2) CANONICAL ENSEMBLE


independent systems having the same temperature

(T),volume(V) and the same number of identical

particles(N).

NOTE:The equality of temperature of all the systems can be

achieved by bringing each in thermal contact at temperature(T).

The individual systems of a canonical ensemble are separated by

rigid, impermeable but conducting walls as a result all the

systems will arrive at the common temperature(T).


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T,V,N T,V,N T,V,N

T,V,N

T,V,NT,V,N

T,V,N T,V,N

T,V,N


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3) GRAND CANONICAL ENSEMBLE


independent systems having the same temperature

(T),volume(V) and the chemical potential ().

NOTE:

The individual systems of a grand canonical ensemble are

separated by rigid & conducting walls.

Since the separating walls are conducting, the exchange of heatenergy takes place between the systems of particles such a way

that all the systems arrive at a common temperature (T) &

chemical potential ().


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T,V, T,V, T,V,

T,V,

T,V,T,V,

T,V,N T,V,

T,V,


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Statistical distributionStatistical mechanics determines the most probableway of distribution of total energy E among the N particles of a system in thermal equilibrium at absolutetemperature 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)


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Where

g(E) is the number of states of energy

And f(E) is the probability of occupancy of

each state of energy E .

E


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(Classical statistics)

Let us consider a system consisting ofmolecules of an ideal gas under ordinaryconditions of temperature and pressure.

Assumptions: The particles are identical and distinguishable.

Maxwell Boltzmann


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Since cells are extremely small, each cell can haveeither one particle or no. of particles though there

is no limit on the number of particles which canoccupy a phase space cell.

The volume of each phase space cell chosen isextremely small and hence chosen volume has

very large number of cells.


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The state of each particle is specifiedinstantaneous position and momentum

co-ordinates.

Energy levels are continuous.

The system is isolated which meansthat both the total number of particlesof the system and their total energyremain constant.


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MAXWELL - BOLTZMANN

DISTRIBUTION

This distribution is applied to a macroscopic systemconsisting of a large number n of identical but

distinguishable particles, such as gas molecules in a

container.

This distribution tells us the way of distribution of total

energy E of the system among the various identical

particles.

Let us consider that the entire system is divided into

groups of particles, such that in every group the

particles have nearly the same energy.


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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 [or

the number of electron states] in the 1st, 2nd

,3rd,,ith, groups be g1,g2,g3,.gi,. and so

on respectively.

In a given system the total number of particles is

constant.


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Hence its

derivative]1.....[0

in

The total energy of the particles present indifferent groups is equal to the energy of the

system(E).

constnEnEnEnEnEEei iiii ..........,.332211

Hence its derivative

]2.....[0 iinE


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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 1ncn

)!(!

!..

11nnn

nei

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

we arrange n2 particles in the second group. This isdone in

2

)(1 nCnn


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)!(!

)!(..

212

1

nnnn

nnei

The number of ways in which the particles in all groups

are chosen is

......)!(!

)!()!(!

!21

1

11

1

nnnn

nnnnn

nw

1 2 3

!

! ! !...... !....

!

!

i

i i

n

n n n n

n

n

i

is the multiplication factor


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The second factor is the distribution of particles over the

different states and is independent of each other.

Of the ni particles in the ith group the first particle can

occupy any one of the gi states. So there are gi ways ,

and each of the subsequent particles can also occupy

the remaining states in gi ways. So, the total number ofthe ways the ni particles are distributed among the gi

states is gini ways.

The probability distribution or the total number of waysin which n particles can be distributed among the

various energy states is W2


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ii n

ii

n

i

nnngggggW )(....).....()()()( 321 3212

The number of different ways by which n particles of thesystem are to be distributed among the available electron

states is

]!

[!

)(

!

!

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

!321

321

321

21

i

n

ii

n

ii

ii

n

i

nnn

n

gn

g

n

n

ggggnnn

n

WWW

i

i

i


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Where i represents the multiplication factor.

Taking natural logarithms on both sides of equationin above.

!ln!lnln

!ln!lnlnln

,ln!ln

ln!ln!lnln

i

i

ii

i

i

i

i

i

i

i

iiii

gnnnnn

gnnnnnnW

xxxx

ionapproximatStrilingApplying

gnnnW


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For the most probable distribution, W is maximum provided

n and E are constants.

Differentiate equation and equate to zero for the maximum

value of W.

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

)(ln1

)(ln

1

0ln max

i

i

iii

i i

i

iiii

i ii

n

ngnn

n

ngnnnW


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

and adding to the (3) equation and there by

sidesbothononentialTaking

Eg

n

or

Egn

or

nEgn

i

i

i

iii

iii

i

i

exp

)ln(

0lnln

.0]lnln[


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iE

i

i eeg

n

The above equation is called Maxwell-Boltzmann law. The value of

has been extracted separately and is equal to 1/kBT.

Where KB is called Boltzmann constant and T is called absolute

temperature.

TKEi

iiMB

B

i

eeg

nEf

1)(


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2) Quantum Statistics

According to quantum statistics the particles ofthe system are indistinguishable, their wave

functions do overlap and such system of

particles fall into two categories

Bose - Einstein distribution

Fermi - Dirac distribution.


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According to Bose-Einstein statistics the particles ofany physical system are identical, indistinguishable

and have integral spin, and further those are called

as Bosons.

Assumptions: The Bosons of the system are identical and

indistinguishable.

Sir J.Bose A . Einstein


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The Bosons have integral spin angular

momentum in units of h/2.

Bosons obey uncertainty principle.

Any number of bosons can occupy a single

cell in phase space.

Bosons do not obey the Pauli's

exclusion principle

Energy states are discrete.


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Wave functions representing the bosons

are symmetric

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

The probability of Boson occupies a state

of energy E is given by

1)}(exp{

1)(

kT

EEfBE

This is called Bose Einstein distributionfunction.

The quantity is a constant and depends on

the property of the system and temperature T.


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Let us divided a box into gisections & the particles

are distributed among these sections.

Once this has been done, the remaining (gi-1) compartments

& niparticles, the number of ways doing this will equal to(ni+ gi-1)!


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Thus the total number of ways realizing the distribution

will be gi(n

i+ g

i 1)! ------- (1)

There are nipermutations which corresponds to thesame conservative function we thus obtain the

required number of ways as

gi(ni + gi -1)!/gi! ni!

(OR)

(ni + gi -1)!/ni! (gi -1)!


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G = (n1

+ g1-1)!/n

1! (g

1-1)! . (n

2+ g

2-1)!/n

2! (g

2-1)!..... (n

i+ g

i-1)!/n

i! (g

i-1)!

= i (ni + gi-1)!/ni! (gi-1)! ----- (2)

We have the probability W of the system for occurring

with the specified distribution to the total number of

Eigen states.

W = i (ni + gi -1)!

ni !(gi -1)!X constant ---- (3)


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Taking the log of eq(3), we have

log W = log (ni+gi-1)! log ni! log (gi -1)! + constant ------- (4)

Using the stirling approximation eq(4) becomes as

log W = (ni+gi) log( ni + gi )ni log ni gi log gi + constant (5)

log w = i (ni+gi) log(ni + gi) ni logni gi log gi)

Where,

ni , gi >> 1. Hence 1 is neglected


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log w = i ni log(ni + gi) + (ni + gi ) nini log ni) - ni/ni ni

( ni + gi )

log w = i ni log(ni + gi) ni log ni)

log w = -i log ni / (ni + gi ) ni ---- (6)

The condition for maximum probability gives as

log ni / (ni + gi ) ni = 0 ---- (7)


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The auxillary condition to be satisfied

(OR)

n =ni = 0 --- (8)

E =Eini = 0 --- (9)

Multiplying eq(8) by & eq(9) by & addingthe resultant expression to eq(7) , we get


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ni = gi / (e+Ei- 1) ---- (11)

This represents the most probable distribution

of the elements for a system obeyingBose Einstein statistics.

Therefore,


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According to Fermi - Dirac statistics the

particles of any physical system areindistinguishable and have half integral

spin. These particles are known as Fermions.

Assumptions: Fermions are identical and indistinguishable.

They obey Pauli s exclusion principle

Fermi Dirac


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Fermions have half integral spin.

Wave function representing fermions

are anti symmetric

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

Uncertainty principle is applicable.

Energy states are discrete.


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The distribution function & Fermi level is valid

only in equilibrium.

The distribution function changes only with

temperature.

It is valid for all fermions.

Electrons & holes follow the FermiDirac

statistics and hence they are called

Fermions.


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The value of F(E) never exceeds unity.

The probability finding energy for electron is

Fe(E) = Pe(E) =1

1 + e (EEF) / KBT

The probability finding energy for hole is

Ph(E) = 1 - Pe(E) =1 + e (EF

E) / KB

T

1

Wh


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Explanation(FD statistics)

P(E)

E

O

At

T = 0 KEF

Figure-11.0

P(E)

E

O

AtT = 0 KEF

Figure-21.0

E1

E3E2

T1T2

T3

Where,

E3 > E2 > E1 > EF

T3 > T2 > T1 > T


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From figure the following points are noted.

When the material is at a temperature higher than

0 K, it receives the thermal energy from surroundings

and they excited.

As a result, they move into higher energy levels

which were unoccupied at 0 K. The occupation

obeys a statistical distribution called FermiDiracdistribution law.


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According to this law, the probability F(E) or P(E)

that a given energy state E is occupied at a

temperature T is given by

Where,

F(E) = P(E) = FermiDirac probability function

KB = Boltzmann constant

F(E) = P(E) =1

1 + e (EEF) / KBT

--------- (1)


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And Ef is called Fermi energy and is a constant for a

given system.

The maximum energypossessed by an electron at

absolute temperature isknown as Fermi energy( EF).


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From equation (1) we may discuss the following 2 cases.

CASE:1If E > EF,

then the exponential term becomes infinite and F(E) = 0

i.e. there is no probability of finding an occupied state ofenergy greater than EF at absolute zero.

Hence, Fermi energy is the maximum energy that any

electron may occupy at 0 K

As temperature increases the electrons are occupied

the higher energy states which are unoccupied at 0 K

as shown in figure(2).


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CASE:2

If E = EF ,then the F(E) = P(E) = at any temperature.

Hence, Fermi level represents the energy state with

50% probability of being filled.

Electrons in a crystal(metal) obey the FermiDirac

(F-D) distribution (statistics)


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Let E1, E2, E3 Eibe the energy of the particles,

g1, g2, g3---- gi be the energy states & n1, n2, ---- ni .

The number ways in which niparticles can

be put in ginumber of states are given by

gi(gi-1)(gi-2)(gi-3)-----(gini+1)

(OR) gi!/(gini)! ----- (1)


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Dividing the eq(1) with ni! , we get number of Eigen

energy states

gi! / ni! (gini)!------- (2)

Thus, for the whole system, the total number of Eigen

energy states can be written as

G = i gi!/ni !( gi ni )! ---- (3)

(OR)

The probability W of a specific state is

proportional to the total number of energy (G)


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log W = i log gi! log (ni! (gi -ni)! + log C

log W = i log gi! log ni! + log (gi-ni)! + log C

log W = i log(g1, g2,-- gi ) log(n1 , n2 -- ni ) + log (g1-n1)----

---log (gini ) + log C

log W = i gi log gi ni log ni (gi ni ) log (gi ni) + log c

log W = i (ni gi ) log (gi ni ) + gi log gi

ni log ni + log c --- (6)


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Therefore,

(log w) = -i

log ni

- log( gi

ni

) ni

log ni

( gini)

-i (log w) = ni--- (7)

For maximum probability,

(log w) = 0

Hence, eq(7) becomes as


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( gini)i ni = 0 --------(8)

log ni

To evaluate the distribution function, two auxillary conditions

has to be introduce in eq(8)

n =ni = 0 --- (9)1-----

E =Eini = 0 --- (10)2-----

Multiplying eq(9) by & eq(10) by & addingthe resultant expression to eq(8) , we get


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( gini)i ni = 0

log ni ni+ ni + Ei

(OR)

( gini)i ni = 0log ni + + Ei

Since, Since, ni is arbitrary it is 0

Therefore,

( gini)= 0log ni + + Ei


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(OR)

( gini)

log ni = - ( + Ei )

( gini)

ni = e-( + Ei )

- ni( 1 - gi/ ni )

ni

e( + Ei )

1=

gi/ ni- 1

1

e( + Ei )

1=

(OR)


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gi/ ni- 1 = e( + Ei )

(OR)

gi/ ni = 1 + e( + Ei )

Therefore, the FermiDirac function can be written as

F (E) = ni/gi = 11 + e + Ei

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