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Bose-Einstein condensation
Masatsugu Sei Suzuki
Department of Physics, SUNY at Binghamton
(Date: November 06, 2018)
_____________________________________________________________________________
A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of weakly interacting
bosons confined in an external potential and cooled to temperatures very near absolute zero (0 K).
Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the
external potential, at which point quantum effects become apparent on a macroscopic scale. This
state of matter was first predicted by Satyendra Nath Bose and Albert Einstein in 1924–25. Bose
first sent a paper to Einstein on the quantum statistics of light quanta (now called photons).
Einstein was impressed, translated the paper himself from English to German and submitted it
for Bose to the Zeitschrift für Physik, which published it. Einstein then extended Bose's ideas to
material particles (or matter) in two other papers.
Seventy years later, the first gaseous condensate was produced by Eric Cornell and Carl
Wieman in 1995 at the University of Colorado at Boulder NIST-JILA lab, using a gas of
rubidium atoms cooled to 170 nK. For their achievements Cornell, Wieman, and Wolfgang
Ketterle at MIT received the 2001 Nobel Prize in Physics. In November 2010 the first photon
BEC was observed.
The slowing of atoms by the use of cooling apparatus produced a singular quantum state
known as a Bose condensate or Bose–Einstein condensate. This phenomenon was predicted in
1925 by generalizing Satyendra Nath Bose's work on the statistical mechanics of (massless)
photons to (massive) atoms. (The Einstein manuscript, once believed to be lost, was found in a
library at Leiden University in 2005.) The result of the efforts of Bose and Einstein is the concept
of a Bose gas, governed by Bose–Einstein statistics, which describes the statistical distribution of
identical particles with integer spin, now known as bosons. Bosonic particles, which include the
photon as well as atoms such as helium-4, are allowed to share quantum states with each other.
Einstein demonstrated that cooling bosonic atoms to a very low temperature would cause them to
fall (or "condense") into the lowest accessible quantum state, resulting in a new form of matter.
http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate
_______________________________________________________________________
1. Bose-Einstein distribution function
The Bose-Einstein distribution function is defined as
1
1),(
)(
eTf
where TkB
1 and kB is the Boltzmann constant. The occupancy of the ground state at = 0 is
2
1
1),0()(0
eTfTN
The total number of particles N should be given by
Tk
eN B
TT
11
1lim
1
1lim
00
For very low temperatures, the chemical potential is very close to zero and should be negative.
0N
TkB
The chemical potential in a boson system must always be lower in energy than the ground state.
Suppose that all the bosons are in the single-particle state of the lowest energy 1 . The total
energy is then
1NUG
Since the ground state is not degenerate
11
0)(
0 1)(1
1lim
1
1lim
1
Tk
eN B
TT
if 1 (in this case )( 1 e >1 and is very close to 1). This equation yields to
N
TkB 1
((Example))
For N = 1022 and T = 1 K, = -1.4 x 10-38 erg <0.
2. Occupancy of the ground state
Density of states for a particle of spin zero is given by
2/3
22)
2(
4)(
ℏ
mgVD (g = 1)
3
where the energy vs k is given by
m
k
2
22ℏ
The number N (fixed) is expressed by
)()()()(),( 0
0
0 TNdfDTNzTNN e
where z is the fugacity (which will be defined later), )(TN e is the number of atoms in excited
states (the number of atoms in the normal phase) and ),0()(0 TfTN is the number of
atoms in the ground state (number of atoms in the condensed phase)
Here we note that we must be cautious in substituting
2/3
22)
2(
4)(
ℏ
mgVD
into
0
)()( dfDN
At high temperature 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 D() = 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
)()()(~
DD
with g = 1, where )( is the Dirac delta function.
),()()()]()([)()(~
0
00
zTNTNdfDdfDN e
Here we have
4
11
1
1
1)(0
z
eTN
and
0
2/3
22
0 11
)2
(4
)()(),(
d
ez
mVdfDzTN e
ℏ
where
ez .
With x , ),( zTNe can be rewritten as
)(
11
2
11
)2
()4(4
11
)()
2(
4),(
2/3
0
0
2/3
2
2/3
2
0
2/12/3
22
zVn
ez
xdxVn
ez
xdx
TmkV
ez
xTkTdxk
mVzTN
Q
xQ
x
B
x
BBe
ℏ
ℏ
Note that the integral (for z<1) is obtained as
00
2/31
2
11
2)(
x
x
x ze
xzedx
ez
xdxz
,
5
61238.2
61238.22
2
)2
3()
2
3(
2
1
2)1(
0
2/3
xe
xdxz
where )(2/3 z is described by
],2
3[PolyLog)(2/3 zz
in Mathematica. Then we get
)()(),( 2/3 zTVnzTN Qe
where )(2/3 z is the zeta function and )1(2/3 z = 2.61238, and nQ(T) is defined as
2/3
2)
2()(ℏTmk
Tn BQ
which is called the quantum concentration. It is the concentration associated with one atom in a
cube equal to the thermal average de Broglie wavelength th. The de Broglie wavelength is given
by
3
1)(
th
Q Tn
where
vmth
ℏ
and v is the average thermal velocity.
6
Fig. Plot of )(2/3 z as a function of z. )1(2/3 z = 2.61238
3. Bose-Einstein-Condensation temperature TE
We start our discussion with the equation given by
)(),( 0 TNzTNN e
We note that )1,( zTNe has a maximum value when z = 1. Since N is constant, N should be
larger than this maximum of )1,( zTNe at z = 1.
)1()()1,( 2/3 zTVnzTNN Qe .
Here we define the temperature TE (the Einstein temperature for Bose-Einstein condensation) at
which
)1,( zTNN Ee .
or
)(61238.2)1()( 2/3 EQEQ TVnzTVnN
The temperature TE can be derived as
3/22
)61238.2
/(
2 VN
MTk EB
ℏ
0.2 0.4 0.6 0.8 1.0z
0.5
1.0
1.5
2.0
3 2 z
7
Below TE,
0)1,()(0 zTNNTN e
We define the molar mass weight as
MA = mNA
and the molar volume as
AM NN
VV
Then the Einstein temperature is rewritten as
3/22
)61238.2
/(
/
2 MA
AA
EB
VN
NMTk
ℏ
((Note))
mnV
N
N
M
V
M
V
M
M
A
Since A
A
N
Mm
M
A
A
A
M
A
V
N
M
N
V
M
mn
8
Fig. Temperature dependence of the thermally excited Bose particle density (blue solid line),
)1,( ztNe for t<1. The remaining density ( 0N ) occupies the ground state (red solid line).
For t>1, NztNe )10,( .
t T TE
Ne t,z 1
Ne t,0 z 1 NN0
N
0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.2
0.4
0.6
0.8
1.0
9
Fig. z vs the normalized temperature ETTt / ( )1t . ez (We use the ContourPlot to get
this plot). NztNe ),( .
((Example))
(a) Liquid 4He;
VM = 27.6 cm3/mol, MA = 4 g/mol.
TE = 3.13672 K.
Note that the density is given by
145.06.27
4
M
A
V
M g/cm3.
(b) Alkali metal Rb atom (which shows the Bose-Einstein condensation at extremely low
temperatures);
z
t T TE
1.0 1.5 2.0 2.5 3.00.4
0.5
0.6
0.7
0.8
0.9
1.0
10
532.1M
A
V
M g/cm3. MA = 85.4678 g/mol
78838.55
AM
MV cm3/mol
TE = 91.8 mK.
(c) Na atom
968.0M
A
V
M g/cm3. MA = 22.98977 g/mol
74976.23
AM
MV cm3/mol
TE = 603.258 mK.
((Mathematica))
4. Physical meaning
In the limited space of the system, the energy of boson (as a single-particle) is quantized. At
high temperature, each boson has an energy as a single-particle state. At T = 0 K, the boson is in
the ground state ( 0 , in Fig.). Suppose that N bosons are in the ground state of the single
particle. This situation is very different from fermions. One fermion is at the ground state. Other
fermions cannot stay in the ground state, because of the Pauli-exclusion principle. The second
fermion will stay at the first excited state.
Clear@"Global`∗"D;
rule1 = 9NA → 6.02214179× 1023, R → 8.314472, kB → 1.3806504× 10−23,
h → 6.62606896 × 10−34, — → 1.05457162853 × 10−34, cm → 10−2,
g → 10−3, VM → 27.6 cm3, MA → 4 g=;
TE =2 π —2
kB MA ê NA
NA ê VM
2.61238
2ê3êê. rule1
3.13672
11
k
E k
O
12
Fig. Bose-Einstein condensation. Each boson has the lowest energy of the single particle state.
The de Broglie wave length beco0mes very large, forming a coherent state.
5. Order parameter N0(T)
We consider the temperature dependence of the occupancy number N0(T) of the ground state
below TE.
)1()()()1,()( 2/300 zTVnTNzTNTNN Qe
The Einstein temperature TE is defined as
)(61238.2)1()( 2/3 EQEQ TVnzTVnN
13
Then we get
2/3
0 )(
ET
TNTNN .
From this Eq., we have
2/3
0 1)(
ET
T
N
TN
We make a plot of N
TN )(0 as a function ot ETTt / . N
TN )(0 becomes zero above TE.
Fig. Plot of N0(T)/N as a function of reduced temperature t = T/TE, showing the critical
behavior in the vicinity of TE.
6. The temperature dependence of z above TE
)()(),( 2/3 zTVnzTNN Qe
TêTE
N0HTLêN
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
14
At T = TE,
61238.2)()1()()1,( 2/3 EQEQEe TVnzTVnzTNN ,
Then we have
)()(),( 2/3 zTVnzTNN Qe
and
61238.2)()1()( 2/3 EQEQ TVnzTVnN
Thus we get
61238.2)()()( 2/3 EQQ TVnzTVn
or
2/32/3
2/3 61238.261238.2)()( tT
Tz E
since
2/3
2
2)(
h
TmkTn B
Q
,
ET
Tt
From the numerical calculation (ContourPlot), the parameter z can be evaluated as a function of t
15
Fig. Plot of z vs a reduced temperature t above TE. ez . The value z tends to z = 1 at t = 1.
Mathematica (ContourPlot): 2/32/3
2/3 61238.261238.2)()( tT
Tz E
7. Possibility of Bose-Einstein condensation in two dimensions
The density of states for the 2D system is given by
kdk
LdD
2
2)(
2
2
2 .
Using the dispersion relation 22
2k
m
ℏ or
2
2
ℏ
mk
z
t T TE
1.0 1.5 2.0 2.5 3.00.4
0.5
0.6
0.7
0.8
0.9
1.0
16
dmL
dmL
dmmL
dD
2
2
2
2
222
2
2
2
2
4
2
1222
2)(
ℏℏ
ℏℏ
Suppose that all bosons are in the excited states,
),( zTNN e
where
0
2
2
0
2
2
0
)(2
2
0
2
2
0
2
12
112
12
)(2
)()(),(
x
x
B
x
B
e
ze
dxzeTLmk
ez
dxTLmk
e
dmL
dfmL
dfDzTN
ℏ
ℏ
ℏ
ℏ
with
ez , x .
Noting that
)1
1ln()1ln(
10
zz
ze
dxzex
x
we get the number density as
)1
1ln()()
1
1ln(
2
),(),( 222 z
Tnz
Tmk
L
zTNzTn D
Bee
ℏ
17
where
222
)(ℏTmk
Tn BD .
Since nzTne ),( =constant, we have
)1
1ln()(2
zTnn D
or
1 2|
1 2ln( )
1
B Ez
mk Tn
z
ℏ
When 1z , we have 0ET . So we do not have any finite critical temperature. In other word,
the Bose-Einstein does not occur in the 2D system.
Fig. Plot of )1ln( z as a function of z. It diverges at z = 1.
In other words, there is no finite critical temperature for the Bose-Einstein condensation in 2D
systems.
In conclusion
3D system: Bose-Einstein condensation
0.2 0.4 0.6 0.8 1.0z
1
2
3
4
ln 1 z
18
2D system: No condensation occurs.
The phenomena of the superconductivity and superfluidity are observed only for the 3D system.