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Ideal Bose gas.

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Title: Ideal Bose gas.


1
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.

2
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.
3
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
4
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
5
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)
6
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
7
The last integral is equal to
where ?(x) is the gamma function. From (8.16)
We have
8
where
Further,
where
Because ??1 these series always converge. We
note that
9
From (8.13)
or taking the spin to be zero
and from (8.9)
10
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
11
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).
12
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)
13
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
14
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
15
Figure 8.1 Comparison of the Einstein
condensation of bosons in momentum space with the
ordinary condensation of a liquid in coordinate
space.
16
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.
17
  • 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

18
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 ?.
19
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.

20
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.
21
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.
22
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.
23
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)
24
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.
25
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.
26
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

27
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).
28
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.
29
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.
30
Radiation Curves
31
Radiation Curves
32
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.

33
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.
34
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.
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