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1

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.