1

Lecture 8Lecture 8

•Ideal Bose gas.

•Thermodynamic behavior of an ideal Bose gas.

•The temperature of condensation.

•Elementary excitation in liquid helium II.

•Thermodynamics of black-body radiation.

•Planck’s formula for the distribution of energy over

the black-body spectrum.

•Stefan-Boltzmann law of black-body radiation.

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Ideal Bose gas.Ideal Bose gas.We shall now study the properties of a perfect gas of bozons perfect gas of bozons of non-zero mass.of non-zero mass.

nej j

1

1 ( ) (8.1 )

We must always have 0 , as the number of particles in a state cannot be negative. We require accordingly that

n j

e ( ) 1 (8.2 )

The Pauli principlePauli principle does not apply in this case, and the low-temperature properties of such a gas are very different from those of a fermion gas discussed in the last lecture. A B-E gas B-E gas displays most remarkable quantal features.displays most remarkable quantal features. The properties of BEBE gas follow from Bose-Einstein distributionBose-Einstein distribution.

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At absolute zero all the particles will be in the ground state,

and we have for no

or

This is satisfied by

In this limit

-NG (8.7)

N/ (8.6)

lim / 0

11e

N (8.5)

0 (8.4)

1 e(8.3)

If the zero of energy is taken at the lowest energy statezero of energy is taken at the lowest energy state, we must have

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N n g n dii

( ) ( ) ( ) 0

(8.8)

We now consider the situation at finite temperatures. Let

g()d be the number of states in dd at . We have

i

ii dngnE0

)()( )( (8.9)

The density of states g(g()) can be presented as

1/2C)g( (8.10)

where

2/33 )2)(/ )(12( MVIC (8.11)

5

p

M

2

2N

Vn n n p dpp

pp

4 2

0

p M( ) /2 1 2

dp 1 2/ p dp d2 1 2~ / We must be cautious in substituting (8.10) into (8.8). At high

temperatures there is no problem. But at low temperatures

there may be a pile-up of particles in the ground state =0;=0;

then we will get an incorrect result for N.N.

This is because g(0)=0g(0)=0 in the approximation we are using,

whereas there is actually one state at =0.=0. If this one state is

going to be important we should write

g C( ) ( ) / 1 2 (8.12)

where (()) is the Dirac delta functionDirac delta function. We have then, instead of (8.8)

The power ½ in is coming from the following consideration:

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It is convenient to write

where =e=e// and 00 1 1 , from

(8.3).

We have

N n C n d( ) ( )/0 1 2

0

(8.13)

ne

( )/

11

1(8.14)

0 / 11

e

dI

s

s (8.15)

In the treatment of BE gas we are going to need integrals of the form

1 e 1 e

If <<1<<1, the classical Boltzmann distribution is a good approximation. If the distribution is degenerate and most of the particles will be in the ground state.

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The last integral is equal to

where (x)(x) is the gamma function. From (8.16)

We have

I s mss m s

m

( ) ( )1 1 1

1 (8.18)

0

)1(sdueu us (8.17)

0

/

1

1

0 1

/

0/

/

)/()/()/(=

1

1

mdemm

ed

de

eI

ms

m

sm

m

mms

ss

(8.16)

8

Fm

m

( ) / 3 21

(8.20)

where

)(2/32/121

2/1 FI (8.19)

)(2/52/143

2/3 HI (8.21)

Further,

where

Hm

m

( ) / 5 21

(8.22)

Because 11 these series always converge. We note that

)(1

)('

FH (8.23)

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NV

F N N 1 3 0( ) ' (8.25)

Ne

C d

e

1

1 11

1 2

0

/

/

/

(8.24)

From (8.13)

or taking the spin to be zero

and from (8.9)

i

ii dngnE0

)()( )(

i

ii dngnE0

)()( )(

)(323

H

VkTE (8.26)

2/33 )2)(/ ( MVC 2/33 )2)(/ ( MVC

N n C n d( ) ( )/0 1 2

0

(8.13)

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(8.27) N0 1

Here

is the number in the ground state, and

(8.28) NV

F' ( ) 3

is the number of particles in excited states.

At high temperatures <<1<<1 we obtain the usual classical result for the energy:

(8.29) NkTF

HNkTE 2

323

)(

)(

11

Einstein Condensation Einstein Condensation Let us consider equation (8.25) in the quantum region.

where is the Riemann zeta function is the Riemann zeta function

1

1

nsn

)s(

1

1

nsn

)s(

If NN00 is to be a large number (as at low temperatures), then

must be very close to 1 and the number of particles in excited

states will be given approximately by (8.28) with F(F()=F(1).)=F(1).

F mm

( ) ( ) ./1 2 6123 2 32 (8.30)

NV

F N N 1 3 0( ) 'N

VF N N

1 3 0( ) '

(8.28) NV

F' ( ) 3

For =1=1 we have

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It should be pointed out that (8.31) represents an upper limit

to the number of particles in states other than the ground

state, at the temperature for which is calculated. If N N is

appreciably greater than NN, , N N 00 must be large and the

number of particles in excited states must approach (8.31)

NV

' .2 612 3(8.31)

NV

o2 612 3. (8.32)

Let us define a temperature T0 such that

where 0 is the thermal de Broglie wavelength at T0. Then,

from (8.31)

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NN

TT

' /

0

30

3 2(8.33)

The number of particles in excited states varies as TT3/23/2 for T< T<

TT00, in the temperature region for which F(F())F(1)=2.612F(1)=2.612.

Further, the number of particles in the ground state is given

approximately by 2301 /

0 T/TN'NNN (8.34)

Thus for TT even a little less than T0 a large number of

particles are in the ground state, whereas for T>TT>T00 there are

practically no particles in the ground state. We call TT00 the

degeneracy temperature or the condensation temperature. It

can be calculated easily from the relation

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where VVMM is the molar volume in cm3 and MM is the molecular

weight. For liquid helium VVMM=27.6 cm=27.6 cm33; ; M=4M=4, and , and TT00=3.1=3.1ooK.K.

It is not correct to treat the atoms in liquid helium as non-

interacting, but the approximation is not as bad in some

respects as one might think.

The rapid increase in population of the ground state below T0

for a Bose gas is known as the Einstein condensation. It is illustrated in Figure 8.1 a condensation in momentum space rather than a condensation in coordinate space such as occurs for liquid-gas phase transformation

MV/T M32

1150

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Ordinary liquid condensation Einstein condensation

p

q

p

q

Figure 8.1 Comparison of the Einstein condensation of bosons in momentum

space with the ordinary condensation of a liquid in coordinate space.

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It is believed that the lambda-point transition observed in

liquid helium at 2.19 2.19 00KK is essentially an Einstein

condensation. Remarkable physical properties described as

superfluiditysuperfluidity are exhibited by the low-temperature phase,

which is known as liquid He II.He II. It is generally believed that

the superflowsuperflow properties are related to the Einstein

condensation in the ground state.

Real gases have no such transition because they all turn into

liquids or solids under the conditions required for Bose

condensation to occur. However, liquid helium (4He) has two

phases called He I and He II, and He II has anomalous thermal

and mechanical properties.

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When a material does become a superfluidsuperfluid, it displays some very strange behaviour;

•if it is placed in an open container it will rise up the sides and flow over the top

•if the fluid's container is rotated from stationary, the fluid inside will never move, the viscosity of the liquid is zero, so any part of the liquid or it's container can be moving at any speed without affecting any of the surrounding fluid

•if a light is shone into a beaker of superfluid and there is an exit at the top the fluid will form a fountain and shoot out of the top exit

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There are other interesting facts about superfluids, the point at which a liquid becomes a superfluid is named the lambda point. This is because at around this area the graph of specific heat capacity against temperature is shaped like the Greek letter .

The Lambda PointThe Lambda Point

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It took 70 years to realize Einstein's concept of Bose-Einstein condensation in a gas. It was first accomplished by Eric Cornell and Carl Wieman in Eric Cornell and Carl Wieman in Boulder, Colorado in 1995.Boulder, Colorado in 1995. They did it by cooling atoms to a much lower temperature than had been previously achieved. Their technique used laser light to first cool and hold the atoms, and then these atoms were further cooled by something called evaporative cooling.

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Black body radiation and the Plank radiation lawBlack body radiation and the Plank radiation law

We now consider photons in thermal equilibrium with matterphotons in thermal equilibrium with matter.

Among the important properties of photons are:

They are Bose particles,Bose particles, with spin 1,spin 1, having two modes of two modes of

propagation.propagation. The two modes may be taken as clockwiseclockwise and

counter-clockwise circular polarizationcounter-clockwise circular polarization. We are therefore to

replace the factor (2I+1)(2I+1) in the density states by 22. A particle

traveling with the velocity of light must look the same in any

frame of reference in uniform motion.

The term "black body""black body" was introduced by Gustav Gustav KirchhoffKirchhoff in 1862. The light emitted by a black body is called black-body radiation

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Because photons are bosons we may excite as many photons into a given state as we like: the electric and magnetic field intensities may be made as large as we like.

Photons have zero rest mass. This suggests, recalling the definition

Th

Mk

N

V0

3/2 2 2 31

2 612 2

.

/

(8.35)

that the degeneracy temperature is infinite for a photon gas.

We can consider photons as the uncondensed portion of a B-E

gas below T0.

It is worth remarking that all fields which are macroscopically

observable arise from bosons; the field amplitude of a

fermion state is restricted severely by the population rule 0 or

1 and so cannot be measured. Boson fields include photons,

phonons (elastic waves), and magnons (spin waves in

ferromagnets).

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We take =0 (=1) in the distribution law as there is no requirement that the total number of photons in the system be conserved. Thus the distribution function (8.1) becomes

ne kT( ) /

1

1(8.36)

We can say it in another way. We recall from the Grand

Canonical ensemble lecture that appears in the distribution

law for the grand canonical ensemble as giving the rate of

change of the entropy of the heat reservoir with a change in

the number of particles in the subsystem. For photons a

change in the number of photons in the subsystem (without

change of energy of the subsystem) will cause no change in

the entropy of the reservoir. Thus we have to put equal to

zero if N refers to the number of photons: this is true for the

grand canonical ensemble and so for all results derived from

it.

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The number of states having wave vector k is

N V k2

2

4

333

(8.37)

where k=2/ is a wave vector. The de Broglie relation =/p may be written as p=k. Now for photons

ck (8.38)

The zero of energy is taken at the ground state, so that the usual zero-point energy does not appear below. Defining

)dG()dg(dN (8.39)

we have from (8.37)

gV

c( )

( )

2

2

3(8.40)

23

8)(

c

VG (8.41)

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Plank radiation law Plank radiation law Thus the number of photons s()d in d at in thermal equilibrium is

where n(n()) is given by (8.1). The energy per unit volume

((,T)d,T)d in dd at is VV-1-1hhs(s()d)d, so that

1

8)()()(

/

2

3

kTec

VGns

(8.42)

1

8),(

/

3

3

kTec

hT

(8.43)

This is the Plank radiation lawPlank radiation law for the energy density of

radiation in thermal equilibrium with material temperature TT..

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The total energy density is

0

34

3

0 0/

3

3

1

8=

1

8),(

x

kT

e

dxxkT

c

e

d

cdT

V

E

(8.44)

By (8.18) the last integral on the right is equal to

I s mss m s

m

( ) ( )1 1 1

1 I s mss m s

m

( ) ( )1 1 1

1

( ) / ( ) /4 1 6 4 154 4n (8.45)

where is the gamma function and the Riemann zeta function. We have for the radiant energy per unit volume

4TV

E (8.46)

where the constant (which is not the entropy) is given by

8

15

5 4

3 3

k

c h(8.47)Stefan

constant

This is the Stefan-Boltzmann law.

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BLACK-BODY RADIATION BLACK-BODY RADIATION Let us consider another approach to the black-body

radiation. We consider a radiation cavity of volume VV at

temperature TT. Historically, this system has been looked upon

from two, practically identical but conceptually different,

points of view:• as an assembly of harmonic oscillators with quantized energies ( )ns s 1

2 where nnss=0,1,2,….,=0,1,2,…., and ss is the angular

frequency an oscillator, or

•as a gas of identical and indistinguishable quanta - the so-

called photons - the energy of a photon (corresponding to

the angular frequency ss of the radiation mode) being .

s

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The first point of view is essentially the same as adopted by

Plank (1900), except that we have also included here the zero-

point energy of the oscillator; for the thermodynamics of the

radiation, this energy is of no great consequence and may be

dropped altogether. The oscillators, being distinguishable from one another (by

the very values of s s ),, would obey Maxwell-Boltzmann

statistics; however, the expression for single-oscillator

partition function ZZ11(V,T)(V,T) would be different from the classical

expression because now the energies accessible to the oscillator are discrete, rather than continuous; see (4.50) and (4.62).

=

1221

2

1

2

1exp

1

2/12/1

2

2221

M

Mh

dqdppM

qMh

Z

Z ee

en

n1

0

12

112

12

12( ) sinh( )

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The expectation value of a energy of a Planck oscillatorPlanck oscillator of

frequency s is then given by eqn. (4.68), excluding the zero-

point term :

12 s

ss

kTe s

/ 1

(8.48)

Now the number of normal modes of vibration per unit volume

of the enclosure in the frequency range ((,,+d+d)) is given by

Rayleigh expressionRayleigh expression

2 41 12 2

2 3

dd

c(8.49)

where the factor 2 has been included to take into account the

duplicity of the transverse modes; cc here denotes the velocity velocity of lightof light.

E N Ne

12

12

12 1

coth( )

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Planck’s formula Planck’s formula

which is the Planck’s formula for the distribution of energy over the black-body spectrum. Integrating (8.50) over all values of , we obtain an expression for the total energy density in the radiation cavity.

1)(

/

3

32

kTe

d

cd

1

)(/

3

32

kTe

d

cd

(8.50)

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Radiation Curves

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Radiation Curves

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Somewhere in the range 900K to 1000K, the blackbody spectrum encroaches enough in the visible to be seen as a dull red glow. Most of the radiated energy is in the infrared.

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Essentially all of the radiation from the human body and its ordinary surroundings is in the infrared portion of the electromagnetic spectrum, which ranges from about 1000 to 1,000,000 on this scale.

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3K Background Radiation A uniform background radiation in the microwave region of the spectrum is observed in all directions in the sky. It shows the wavelength dependence of a "blackbody" radiator at about 3 Kelvins temperature.