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Lecture 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.
- Plancks formula for the distribution of energy

over the black-body spectrum. - Stefan-Boltzmann law of black-body radiation.

Ideal Bose gas.

We shall now study the properties of a perfect

gas of bozons of non-zero mass.

The Pauli 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

displays most remarkable quantal features. The

properties of BE gas follow from Bose-Einstein

distribution.

If the zero of energy is taken at the lowest

energy state, we must have

or

At absolute zero all the particles will be in the

ground state, and we have for no

This is satisfied by

In this limit

We now consider the situation at finite

temperatures. Let g(?)d? be the number of states

in d? at ?. We have

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

where

The power ½ in ? is coming from the following

consideration

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 then we will

get an incorrect result for N.

This is because g(0)0 in the approximation we

are using, whereas there is actually one state at

?0. If this one state is going to be important

we should write

where ?(?) is the Dirac delta function. We have

then, instead of (8.8)

It is convenient to write

where ?e?/? and 0? ? ? 1 , from (8.3).

If ?ltlt1, the classical Boltzmann distribution is

a good approximation. If ??? the distribution is

degenerate and most of the particles will be in

the ground state.

In the treatment of BE gas we are going to need

integrals of the form

We have

The last integral is equal to

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

We have

where

Further,

where

Because ??1 these series always converge. We

note that

From (8.13)

or taking the spin to be zero

and from (8.9)

Here

is the number in the ground state, and

is the number of particles in excited states.

At high temperatures ?ltlt1 we obtain the usual

classical result for the energy

Einstein Condensation

Let us consider equation (8.25) in the quantum

region.

For ?1 we have

where ? is the Riemann zeta function

If N0 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(1).

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 is

appreciably greater than N?, N 0 must be large

and the number of particles in excited states

must approach (8.31)

Let us define a temperature T0 such that

where ?0 is the thermal de Broglie wavelength at

T0. Then, from (8.31)

The number of particles in excited states varies

as T3/2 for Tlt T0, in the temperature region for

which F(?)?F(1)2.612. Further, the number of

particles in the ground state is given

approximately by

Thus for T even a little less than T0 a large

number of particles are in the ground state,

whereas for TgtT0 there are practically no

particles in the ground state. We call T0 the

degeneracy temperature or the condensation

temperature. It can be calculated easily from the

relation

where VM is the molar volume in cm3 and M is the

molecular weight. For liquid helium VM27.6 cm3

M4, and T03.1oK.

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

Figure 8.1 Comparison of the Einstein

condensation of bosons in momentum space with the

ordinary condensation of a liquid in coordinate

space.

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.

It is believed that the lambda-point transition

observed in liquid helium at 2.19 0K is

essentially an Einstein condensation. Remarkable

physical properties described as superfluidity

are exhibited by the low-temperature phase, which

is known as liquid He II. It is generally

believed that the superflow properties are

related to the Einstein condensation in the

ground state.

- When a material does become a superfluid, 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

The Lambda Point

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 ?.

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

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.

Black body radiation and the Plank radiation law

The term "black body" was introduced by Gustav

Kirchhoff in 1862. The light emitted by a black

body is called black-body radiation

We now consider photons in thermal equilibrium

with matter. Among the important properties of

photons are

They are Bose particles, with spin 1, having two

modes of propagation. The two modes may be taken

as clockwise and counter-clockwise circular

polarization. We are therefore to replace the

factor (2I1) in the density states by 2. A

particle traveling with the velocity of light

must look the same in any frame of reference in

uniform motion.

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.

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).

Photons have zero rest mass. This suggests,

recalling the definition

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.

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

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.

The number of states having wave vector ? ?k? is

where k2?/? is a wave vector. The de Broglie

relation ??/p may be written as pk. Now for

photons

The zero of energy is taken at the ground state,

so that the usual zero-point energy does not

appear below. Defining

we have from (8.37)

Plank radiation law

Thus the number of photons s(?)d? in d? at ? in

thermal equilibrium is

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

volume ?(?,T)d? in d? at ? is V-1h?s(?)d?, so

that

This is the Plank radiation law for the energy

density of radiation in thermal equilibrium with

material temperature T.

The total energy density is

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

to

where ? is the gamma function and ? the Riemann

zeta function. We have for the radiant energy per

unit volume

where the constant ? (which is not the entropy)

is given by

This is the Stefan-Boltzmann law.

BLACK-BODY RADIATION

Let us consider another approach to the

black-body radiation. We consider a radiation

cavity of volume V at temperature T.

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

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 ), would obey

Maxwell-Boltzmann statistics however, the

expression for single-oscillator partition

function Z1(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).

Now the number of normal modes of vibration per

unit volume of the enclosure in the frequency

range (?,?d?) is given by Rayleigh expression

where the factor 2 has been included to take into

account the duplicity of the transverse modes c

here denotes the velocity of light.

Plancks formula

which is the Plancks 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.

Radiation Curves

Radiation Curves

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.

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.

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.