Transcript 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.

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.
•Planck’s formula for the distribution of energy over the
black-body spectrum.
•Stefan-Boltzmann law of black-body radiation.
1
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 lowtemperature 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.
nj 
e ( 
1
j
 )
1
(8.1 )
We must always have n j 0 , as the number of particles in a state
cannot be negative. We require accordingly that
e  (   )  1
(8.2 )
2
If the zero of energy is taken at the lowest energy state, we must have
e
or
 
1
 0
(8.3)
(8.4)
At absolute zero all the particles will be in the ground state, and we
have for no
lim
 0
1
e
 /
1
N
(8.5)
This is satisfied by
   / N
(8.6)
In this limit
G  N  -
(8.7)
3
We now consider the situation at finite temperatures. Let g()d be the
number of states in d at . We have

 N   n (i )   g( )n( )d
i
(8.8)
0

E    i n ( i )    g ( )n( )d
i
(8.9)
0
The density of states g() can be presented as
g( )  C
1/2
(8.10)
where
C  (2I  1)(V / 3 )(2M )3/ 2
(8.11)
4
The power ½ in  is coming from the following consideration:

p2

2M
N
2
n

n

4

n
p
dp

p
p

V p
0
dp  
1/ 2
p  (2 M )1/2
p dp ~  d
2
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; 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
g(  )   (  )  C 1/ 2
where
(8.12)
() is the Dirac delta function. We have then, instead of
(8.8)
5

 N  n(0)   C 1/ 2 n( )d
(8.13)
0
It is convenient to write
n(  ) 
1
1

where =e/ and
e /   1
(8.14)
e    1
0   1 , from (8.3).
If <<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.
In the treatment of BE gas we are going to need integrals of the form

Is  
0
We have
 s d
1

e
 /
1
(8.15)
6

Is     e
s
 / 
0


0
m 1

1

 / 
1


e


d 

(8.16)
   s d   m e  m /


m 1
0
=   m ( / m) s 1  (m /  ) s e  m / d (m /  )
The last integral is equal to

s u
u
 e du  (s  1)
(8.17)
0
where (x) is the gamma function. From (8.16)
Is   ( s  1 )
s1

  mm( s1 )
(8.18)
m1
We have
7
I1/ 2  12  1/ 2 3 / 2 F ( )
where

m
1
m3/ 2
F(  )  
(8.19)
(8.20)
Further,
I3 / 2  34  1/ 2 5 / 2 H ( )
where

m
1
m5 / 2
H(  )  
(8.21)
(8.22)
Because 1 these series always converge. We note that
H ' ( ) 
1

F ( )
(8.23)
8

 N  n(0)   C 1/ 2 n( )d
From (8.13)
(8.13)
0
 N 

C 1/ 2 d

 1 0 1  /
e 1
1
e
  /

or taking the spin to be zero
N

(8.24)
1

C  ( V / 3 )(2M )3/ 2
V
3
F(  )  N 0  N'
(8.25)

E    i n ( i )    g ( )n( )d
and from (8.9)
i
E  kT
3
2
V

3
H ( )
0
(8.26)
9
Here
N0 

1 
(8.27)
is the number in the ground state, and
N' 
V
3
F(  )
(8.28)
is the number of particles in excited states.
At high temperatures
energy:
<<1 we obtain the usual classical result for the
H ( ) 3
E  NkT
 2 NkT
F ( )
3
2
(8.29)
10
Einstein Condensation
N
Let us consider equation (8.25) in the quantum region.
F( 1 )   m3 / 2   ( 23 )  2.612
m
where
 is the Riemann zeta function

1

V
3
F(  )  N 0  N'
For =1 we have
(8.30)

1
s
n
n 1
(s) 
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).
N' 
V
3
F(  )
(8.28)
11
N'  2.612
V

3
(8.31)
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
N  2.612
V
o3
(8.32)
where 0 is the thermal de Broglie wavelength at T0. Then, from
(8.31)
12
 
T 
N'
 0  0
N

T 
3
3/ 2
(8.33)
The number of particles in excited states varies as T3/2 for T< 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

N0  N  N'  N 1 T / T0 
3/ 2

(8.34)
Thus for T even a little less than T0 a large number of particles are in
the ground state, whereas for T>T0 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
13
2
3
T0  115/ VM M
where VM is the molar volume in cm3 and M is the molecular weight.
For liquid helium VM=27.6 cm3; M=4, and T0=3.1oK.
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
14
p
p
q
Ordinary liquid condensation
q
Einstein condensation
Figure 8.1 Comparison of the Einstein condensation of bosons in
momentum space with the ordinary condensation of a liquid in coordinate
space.
15
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.
16
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
17
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 .
18
It took 70 years to realize Einstein's concept of BoseEinstein 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.
19
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 blackbody 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 counterclockwise circular polarization. We are therefore to replace the factor
(2I+1) 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.
20
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
 1 
T0  

 2.612 
3/2
h  N
 
2Mk  V 
2
2/3
(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.
21
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
n( ) 
1
e
 / kT
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.
22
The number of states having wave vector  k is
4 3
N V
k
3
 2  3
2
(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
dN  g(  )d  G(  )d
we have from (8.37)
g ( ) 
V
2
 2 (c) 3
8V 2
G ( )  3 
c
(8.39)
(8.40)
(8.41)
23
Plank radiation law
Thus the number of photons s()d in d at
equilibrium is
8V
2
s( )  n( )G( )  3  / kT
c e
1

in thermal
(8.42)
where n() is given by (8.1). The energy per unit volume (,T)d in
d at  is V-1hs()d, so that
8h  3
 ( , T )  3  / kT
c e
1
(8.43)
This is the Plank radiation law for the energy density of radiation in
thermal equilibrium with material temperature T.
24
The total energy density is

E
8
   ( , T )d  3
V
c
0

 3 d
e 

/ kT
0
8  kT 
= 3 

c   
By (8.18) the last integral on the right is equal to
1
4
x 3 dx
0 e x  1 (8.44)
Is   ( s  1 )
s1

  mm( s1 )
m1
(8.45)
(4)1 / n4  6 (4)   4 / 15
where  is the gamma function and  the Riemann zeta function. We
have for the radiant energy per unit volume
E
 T 4
V
(8.46)
where the constant  (which is not the entropy) is given by
8 5 k 4

15c 3h 3
This is the Stefan-Boltzmann law.
Stefan constant
(8.47)
25
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
(ns  21 )s
where ns=0,1,2,…., and
s 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
s of the radiation mode) being  s.
26
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).
 

1
1
1

Z1    exp   M 2 q 2 
p 2 dqdp 
h   
2M
2


1/ 2
=
1  2 


2 
h  M 
Z1 (  ) 

e
n0
 2M

 
  ( n  ) 
1
2
1/ 2




1

 21 


e

 2 sinh
 
1 e
1
2

 
1
27
The expectation value of a energy of a Planck oscillator of frequency
s is then given by eqn. (4.68), excluding the zero-point term 21  s:
E
1
2
N coth( 21  )
  s 
 s
 
1
 N  2   
e
 1

e s / kT  1
(8.48)
Now the number of normal modes of vibration per unit volume of the
enclosure in the frequency range (,+d) is given by Rayleigh
expression
2
 1   1   d
2  4   d    2 3
     c
2
(8.49)
where the factor 2 has been included to take into account the duplicity
of the transverse modes; c here denotes the velocity of light.
28
Planck’s formula
  d
 ( )d  2 3  / kT
 c e
1
3
(8.50)
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.
29
Radiation Curves
30
Radiation Curves
31
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.
32
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.
33
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.
34