Download discrete bose-einstein systems in a box with low adiabatic invariant

PUBLISHING HOUSE
OF THE ROMANIAN ACADEMY
PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A,
Volume 3, Number 1-2/2002.
DISCRETE BOSE-EINSTEIN SYSTEMS IN A BOX WITH LOW ADIABATIC
INVARIANT
Valentin I. VLAD, Nicholas IONESCU-PALLAS
Institute of Atomic Physics, NILPRP- Laser Dept., P.O.Box, MG-36,
Bucharest and The Romanian Academy –CASP, Bucharest, Romania
Corresponding author : Valentin I.VLAD : E-mail: [email protected]
Abstract. The Bose-Einstein energy spectrum of a quantum gas, confined in a rigid (cubic) box, is
discrete and strongly dependent on the box geometry and temperature, for low product of the atomic
mass number, Aat and the adiabatic invariant, TV2/3, i.e. on  = AatTV2/3. Even within the approximation of noninteracting particles in the gas, the calculation of the thermodynamic properties of BoseEinstein systems turns out to be a difficult mathematical problem. It is solved in the textbooks and
most papers by approximating the sums by integrals. The present study compares the total number of
particles and the total energy obtained by summing up the exact contributions of the eigenvalues and
their weights, for defined values of , to the results of the approximate integrals. Then, the passage
from sums to integrals is done in a more rigorous manner and better analytical approximations are
found. The corrected thermodynamic functions depend on . The critical temperature is corrected also in order to describe more accurately the discrete Bose-Einstein systems and their onset of the phase
transition.
1. INTRODUCTION
The Bose-Einstein systems are described usually by continuous thermodynamic functions,
which are dependent on kinetic energy, temperature and chemical potential, but is independent on
the container size and shape (considering the
quantum gas with a very large number of identical
particles and stored in a large container) [1-9].
In Bose Einstein condensation (BEC) experiments, the quantum gas is confined in a trap using
an external potential. This confinement may be
reasonably replaced by a free particle gas, which
is placed inside of a virtual three-dimensional rigid box. The previous attempts to calculate the effects of the container size on the boson gas thermodynamics have used the (Weyl-Pleijel) asymptotic state density corrections only. These corrections were relatively small, but increasing with the
box size decrease. More recently, a number of
papers calculated the effect of trap dimension and
size on BEC [10-21] taking account of the system
discreteness in this phenomenon.
When working with systems of identical particles, one can use the grand canonical ensemble in
order to dispense of the particle number conservation. The grand canonical ensemble allows the
number of particles and the energy in the system
to fluctuate, but keeps the chemical potential
fixed. This is useful for systems, which undergo
phase transitions. For an ideal gas of identical particles each of mass m, the grand partition function
can be written under the form [see for example,
8]:
Z (T , V , )  Trexp  (H  N) / kT 
(1)
where H and N are the kinetic energy and total
particle number operators and  is the chemical
potential.
Assuming that the ideal quantum gas is confined in a cubic box, with size L and with periodic
boundary conditions, the momentum operator for
a particle has the following eigenvalues:
p q  k  h / L iq1  jq 2  kq3 
(2)
where k is the wave-vector of the particle and qi
are integer numbers and zero (each triplet defining
a system state).
Valentin I.VLAD, Nicholas IONESCU-PALLAS
The kinetic energy operator for a single particle is p 2 / 2m and has eigenvalues:
q 
pq2
2m
 0  q;
0 
h2 1
2m L2
(3)
where q is an integer state number and 0 – the
inter-level energy. The energy levels, q, of the
particle in a box, can be obtained resorting to the
Brillouin’s model of kinetic energy quantising
through the intermediary of momentum quantising, by imposing periodicity conditions on the
walls. This procedure allows the exchange of energy and particles between the box and the thermostat and accordingly, makes possible the use of
quantum statistics via the grand ensemble partition.
The energies of the box states, (q), are distributed in a discrete spectrum defined by the spatial quantisation rule for the momentum (wavevectors):
q  q12  q22  q32 ;
q j  0,1,2,3,...
( j  1,2,3)
g asy  2 q
(5)
The average of the distribution g(q) follows the
asymptotic trend from Eq.(5).
For particle confinement in relatively
small volumes and at relatively small temperatures (we shall define later what "relatively small"
means), we have to account for the random distribution of the degeneracy. One can expect a discrete and irregular kinetic energy spectrum. We
can define the quantum degeneracy factor:
g (q)
g (q)
 (q) 

g asy (q)
2 q
degeneracy fluctuations are large for a box with
small number of states (and must be taken into
account), the average number of states tends to the
asymptotic value very rapidly. For a number of
states larger than 100, the classical equation (5)
can be safely used.
The total degeneracy in an atomic gas, gT,
is the product of the box degeneracy, gq=g(q) and
the intrinsic spin degeneracy of the atoms, gS
=2S+1 (where S is the total spin of the atoms): gT
= gq  gS..
Let us denote by nq the number of particles with momentum pq. Then, the particle number operator can be written as N =q nq and the
kinetic energy operator is H =q nqq . The grand
partition function becomes:
Z (T , V , ) 
(4)
The number of degenerate states in the box is
strongly and randomly fluctuating. The degeneracy occurs due to the discrete spatial orientations
of the state wave-vectors with the same quantum
number (q).
The weight (degeneracy) of state with a quantum number q can be found, for large level numbers (asymptotic case), as [17-21]:
(6)
which includes the spatial quantization effects.
We have checked that the factor (q) is randomly
fluctuating around the value 1 by the calculation
of the average number of states on constant frequency intervals. The result is that, although the
2

q
Zq 
 
exp  ni ( q  ) / kT

q
nl  0

  

 g 1  exp (
q
q
q
 ) / kT





gT
(7)
 gT
.
Through agency of the grand potential, the particle number in the gas is:
N  kT


q


ln Z  kT


nq 

 g exp(
q
q
q
q
ln Z q
 
 ) / kT  1
1
(8)
and the total energy can be calculated as:
 ln Z
 ln Z

)
T

 ln Z q
 

nq  q
q q
q

E  k T(T
 kT



q

 
(9)
 
g q  q exp ( q  ) / k T  1
1
We can write also:
E
g q
g q
 (   T) / kT
  / kT T  / kT
 e
1 e
e
1
gT  q
 1
A  exp(  q / Aat L2T )  1
(10)
with  - the chemical potential (< 0),
  h 2 / 2m0 p k  9.5060  10 14[cm 2  K ] ,
k – the Boltzmann constant, T – the absolute temperature of the boson quantum gas,
A  exp( / kT) - the fugacity, and Aat – the
3
Discrete Bose-Eimstein systems
atomic mass number (m = Aat m0p). Using the
quantum degeneracy factor defined in (6), the
boson spectrum (10) can be put in the form:
E

2g S  q 3 / 2
  (q).
 exp(  q / Aat L2 T )  1
A 1
A = 0.99 ; Aat * L^2 * T = 4 * 10^ - 13
40
30
u (q, A, Aat L2 T ) 

u
50
(11)
20
10
0
The Bose-Einstein energy spectrum (BEES),
from Eq.(11), is discrete for a small number of
states in the cubic box, as shown in Fig.1. From
this graph, one can observe that the quantum effects may occur in cubic cavities with micrometer
sizes, at temperatures around K, which are presently reached by evaporation and laser cooling
[21]. These spectra show that, the higher the adiabatic invariant, L2T, the higher the number of levels, for specified particles (atoms) in the cavity.
At a certain resolution limit, the spectrum is obtained by averaging the energy lines ( 1) and
the continuous BEES is reached (dashed graph in
Fig.1).
We can introduce a reasonable superior limit
of the number of states in the box, qT , which
brings a significant contribution to the BEES. At
high energies, the exponential term dominates and
(q) 1, so that Eq.(11) can be brought to the
form:
10
20
30
40
q
50
(a)
u
A = 0.99; Aat * L^2 * T = 6 * 10^ - 13
80
60
40
20
10
20
30
40
q
50
(b)
u
A = 0.99; Aat * L^2 * T = 8.7 * 10^ - 13
175
150
125
100
75
50
25
10
20
30
40
q
50
(c)
u( x)  Bx 3 / 2 exp( x)
u
A = 0.99 ; Aat * L^2 * T = 20 * 10^ - 13
700
with :
600
x  ( / Aat L2T )q  ( / )  q
500
Considering A < 1 and neglecting the levels
which bring a contribution of less than 10-2, one
can parametrically truncate the Bose-Einstein distribution at the highest significant level number
(HSL) in the box:
qT  12.5  ( Aat L2T / )  12.5  ( / )
400
300
200
100
50
100
150
200
q
(d)
(12)
One can observe that, in the above conditions, the
fugacity plays a minor role in this truncation and
for any of its values, Eq.(12) ensures an overevaluated value for qT.
For Aat = 87 (Rubidium), L = 10 -4 cm and T =
–6
10 K, Eq.(14) leads to: qT 114.
Fig. 1.
Some conventional Bose-Einstein spectra (dashed lines) and
discrete Bose-Einstein spectra (solid lines, joining the tops of
the energy spectrum lines), for A = 0.99 and for the values of
 = Aat L2 T , which are shown in each graph.
For Li7, one can find some more convenient
conditions for double quantised box, namely: L =
10 m and T  120 nK. For a precision of 10-3,
one can take: qT 15( /) 137.
Valentin I.VLAD, Nicholas IONESCU-PALLAS
2. THE STATISTICAL PROPERTIES OF
THE DISCRETE BE GAS IN THE CUBIC
BOX

N  N0  gS 
N  gS
A
q 0

 
  
 
2q 1 / 2
1 q / 
e
1

E
 g S   
kT

(13)
  h 2 / 2m0 p k  9.5060  10 14[cm2  K ];
2x 1 / 2 
dx
A 1 e x /   1
0
With the above notations, the Eqs.(8) and (9)
can be written under the form:
qT
4

A
(18)
3/ 2
 f 2 ( A)
2x 3 / 2 
dx
e
1
(19)
1 x / 
0
Solving the integral Eq.(18) to obtain the particle number in terms of fugacity, we have generalized the result from [4] as:
  Aat L T
2
and:
3 / 2

2q
E
 g S  
1 q / 
kT
1
   q 0 A e
qT
3/ 2

(14)
The total number of particles and the total energy of the free boson gas are described in many
textbooks passing rapidly from sums to the integrals [see for example, 5]:
N  N0 
3
gS L m
3/ 2 
2 
2 3

0
 d
  
exp
 1
 kT 
1/ 2
4
3/ 2
 
N  

3  

 0.011402640  N 2  

9/2
(16)
 5.1824177  10
-12

 N  

7
3
 
 5.9954498  10  N  

-6
15 / 2
4  
6

 7.0387568  10 -10  N 6  

21 / 2
.
A
1
0.8
3
Nk T[1  f 1 ( A)];
2
f 1 ( A)  0.1767767  A  0.0658001  A 2
E 
0.6
0.4
 0.0364655  A 3  0.0239014  A 4
 0.0171965  A 5  0.0131299  A 6
 0.0104470  A  0.0085675  A
7
8
 0.0071910  A 9  0.0061474  A10
 R10 ;
R10  0.0053339 A11  (1  0.8783629  A
 0.6414814  A 2 ) /(1  0.8887962  A 2 ).
9
(21)
where N0 is the number of condensed particles and
CGS-Gauss unit system was used. In the classical
limit, [21]:
A  0,

A  0.17958712  N  


 7.5083995  10 -8  N 5  


(20)
One can invert this function in order to obtain the
dependence of the fugacity and of the chemical
potential on N and , which are shown in Fig.2.
 3.3332681  10

 3 / 2 d
E 
;
  
2 2 3 0
exp
 1
 kT 

  0  q; 0   kT

g S L3m3 / 2
(15)
 
N 
 f 2 ( A)  A  2 3 / 2 A 2 



3 / 2 3
3
A  4 3 / 2 A 4  5 3 / 2 A 5  ....
0.2
(17)
-12
2
10
-12
4
10
-12
6
 10
8
10-12

-11 
1
10
Fig.2. The dependence of the fugacity on the number of particles in the quantum gas, Na (1000 – continuous line, 100dashed line and 10 atoms – dotted line) and on the adiabatic
invariant (multiplied by the atomic mass number), .
5
Discrete Bose-Eimstein systems
However, our procedure of building the grand
canonical ensemble and the present cooling procedure, which is based on evaporation (which
eliminates progressively the particles with the
highest kinetic energy) lead to consider a variable
particle number in the system. Thus, it is more
correctly to calculate this number, with the approximate series for the integral from Eqs.(18)
and (20), in function of a given fugacity and adiabatic invariant ().
One can remark that N-N0 increases monotonically with A and . higher the fugacity in the DQB,
the stronger the increase of N-N0 with . The
higher the fugacity in the DQB, the stronger the
increase of N-N0 with .
The replacement of sums by integrals must be
done with increased mathematical rigor, for example using the modified Euler-Mac Laurin formula [22], which gives for the particle number:
 A 
N / gS  

1  A 
1/ 2

0


0
N
 A  L3

g 3 / 2 ( A) 

gS
 1  A  3T
7  2  A 
9 
1 
,


1 
24  1  A 
70  1  A 
(23)
where  T  h / 2mkT  L / 1 / 2 and the
function g3/2(A) is proportional to f2(A) and can be
approximated using, for example, the development from (20). However that series is very slowly converging and for precise results, needs some
tens of terms.
Robinson gave formulae for the calculations of
the integrals from (18) and (19), which are
claimed to ensure a precision under 1% [3]. Improved formulae (of relative error  10-7) found
by us have the form:
g3 / 2 ()  2.6123754  3.5449077  1 / 2 
1.4603528    0.1039328  2  0.0042495  3 
2 x dx

1 x / 
A e
1
0.000369  4  0.0000621  5  0.0000014  6 
2 x dx
2 x
 1 
 
1 x / 
1 x / 
A e
 1  24  x A e
 1 x  0.5
(22)
0.0000016  7 ;
(24)
g5 / 2 ()  1.3414872  2.3632718  3 / 2 
2.6123754    0.7301772  2  0.0346543  3 
Considering:
 0    kT ,
0.0010568  4  0.0000734  5
( /   1)
(24' )
where    ln A   / kT (0,1].
our calculations lead to:
N N0















gs
1200
1000
Fig.3. The plot of the active number of particles in Bose-Einstein gas at fixed fugacity,
A=0.99, versus  =AatL2T [cm2K] for direct
summation (dots), our sum approximation
from Eq.(23) (continuous line) and improved
Robinson’s approximation of integral from
Eq.(24) (small dashes) .
800
600
400
200
510-13
110-12
1.510-12
We have compared the sum and integral results
in Fig.3. One can remark that the finite sum gives
smaller values for the active particles in the box
than different evaluations of the integrals. Our
analytical formula for the active particle number

210-12
(23) is the closest to the exact result. The usual
calculations lead to errors, which increase with 
(for example at  = 210-12, when N-N0  1150, the
error of classical formula (20), still used in many
works in these parameter range, arrives to  6%).
6
Valentin I.VLAD, Nicholas IONESCU-PALLAS
3. BEC IN THE CUBIC BOX WITH LOW
ADIABATIC INVARIANT
The onset of condensation occurs in the region,
where the discrete distribution cannot be circumvented. Thus, it is interesting and of pedagogical
value to discuss BEC without the continuous
spectrum approximation in the cubic box with low
adiabatic invariant. Recently, Ketterle and van
Druten have considered this topic for particles in a
3D harmonic potential remarking the advantage of
absence of diverging integrals and of a special
role given to the ground level of the system [13].
Previous attempts to calculate BEC in a cubic box
considered a non-zero ground level energy and an
inter-level energy four times lower than that used
by most authors [15]. This is in contradiction with
explicit Landau’s hypothesis [5] and is difficult to
be supported by the thermodynamic laws and the
correspondence principle.
In Fig. 4a, our result with finite summation:
N0
N
1
Ђ
Ђ
Ђ
Ђ
Ђ
Ђ
Ђ
Ђ
A = 0.99
0.8
0.6
0.4
0.2
-12
1ґ 10
qT

q 1
n
0.8
0.6
0.4
0.2
0.6
0.8
g
Ђ
Ђ
Ђ
Ђ
Ђ
Ђ
Ђ
Ђ
gB
1
(b)
N0
N
1
Ђ
Ђ
Ђ
Ђ
Ђ
Ђ
Ђ
Ђ
is compared to that given by the integral:
N0 / N  1  ( /  B )2 / 3 ;
A = 0.99
0.8
 B  0.167829 N 2 / 3  1.59538  10 13 N 2 / 3
(26)
One can remark that the direct summation has
considerable differences in comparison with the
common considered ratio of condensed and total
particle number. However, if we correct the main
parameter,
the
critical
temperature,
as
 B '  1.10646  B , these graphs come much closer
(Fig.4b).
Our analytical approximation of sum by integral leads also to a corrected expression for the
critical temperature:

0.4
(25)
q
q 0
 B '  1  0.9046 N 0
g
-12
4ґ 10
A = 0.99
qT
nq /
-12
3ґ 10
(a)
N0
N
1
Ђ
Ђ
Ђ
Ђ
Ђ
Ђ
Ђ
Ђ
0.2
N0 / N 
-12
2ґ 10
1 / 2

B
(27)
For the number of particles in the considered box,
the correction becomes  B '  1.0905  B , which is
in a better agreement with the exact calculation.
4. THE TOTAL ENERGY OF DISCRETE
BOSE-EINSTEIN SYSTEM IN THE CUBIC
BOX
In the cubic box, the total energy should be
written by summing the state energies up to the
highest significant one (characterized by qT):
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
g
Ђ
Ђ
Ђ
Ђ
Ђ
Ђ
Ђ
Ђ
gB
(c)
Fig.4. The ratio of condensed particle and total particle numbers for fixed fugacity, A=0.99:
(a) by direct summation, versus  =AatL2T [cm2K];
(b) by direct summation (continuous) and by conventional
equation (dashed), versus  /B (T/TB );
(c) by direct summation (continuous) and by conventional
equation (dashed), versus  /B.
E

 2g s
kT

qT (  )
A
q 1
1
q3/ 2
(q)
 e ( /  ) q  1
(28)
Again, the replacement of sums by integrals, as
found in many text books, could be done with increased mathematical rigor, for example using the
modified Euler – Mac Laurin formula [22], which
gives, with same procedures as in Eqs.(22) and
(24):
Discrete Bose-Eimstein systems
E
3 g 5 / 2 () 
7 2
 A 

 1) g S 

N  (
kT 2 g 3 / 2 () 
24
 1  A 
3
7.6 10-15    7.610-13
(29)
 A  
gS 
 .
40 2  1  A  
We can calculate precisely the ratio of total
energies of the particle gas in the box and in a
classical (large) container with Eqs. (28) and (17):

 α
E
F  A,  

   (3 / 2) NkT[1  f1 ( A)]

4g s
 
3N [1  f1 ( A)]   
qT (  )
A
q 1
q3 / 2
  (q) 
1
 e( /  ) q  1
α
  
3 [1  f1 ( A)] f 2 ( A)   
F 1
4
5 / 2 qT (  )

q 1
q3 / 2
A1  e(  /  ) q  1
(30)
In the asymptotic limit, (q) goes to 1 (by averaging over many and very close modes), F tends to
1, and one arrives to the conventional formalism.
The corrective factor, F, is represented in Fig.5, in
function of the parameter , for A = 0.99.
The best fit of the energy data obtained with
sums can be obtained with accuracy of 0.1% using
the function (Fig.5, with continuous line):
F  exp 5.7955  1015 (1 / )  4.4057  1028 (1 / )2  (31)
Thus, with specified atoms and box size, the total
energy in the box with low adiabatic invariant has
a stronger dependence on temperature than is predicted by the conventional law. As the quantum
gas confined in the box is emptied of kinetic levels, its total energy is strongly decreasing according to the new law:
E  (3 / 2) NkT[1  f 1 ( A)] 
exp 0.0609668 ( /  )  0.0487550 ( /  ) 2 
(32)
Eq. (30) and Fig.5 show that the asymptotic
limit can be set for F( /)  1, at a conventional
limit of qmax/   8, which leads to qmax  76
10-14 [cm2.K] and to qT max  100.On the other
hand, the lowest box mode with non-zero energy
is qT min  12.5  (  / )  1 and set an inferior
limit to qmin  0.76 10-14 [cm2.K].
Thus, we can define the double quantisation
regime of the cubic box in the range:
1  q  100
7
(33)
[cm2.K]
(34)
For given atoms, the box size and the temperature are reciprocal parameters in the box, i.e. the
same effects (in the thermodynamics of the boson
gas) can be obtained either by varying L2 = V2/3 or
by varying T, if their product remains constant.
We can remark that, the discrete Bose-Einstein
systems are characterized by finite and small
number of particles. From Eq.(13) and (23), it is
possible to calculate the number of particles at  =
10-13[cm2.K], where the correction factor of the
total energy is F  0.9, for A = 0.99. In the specified quasi-degenerate gas, there are N-N0  9 particles.
4. CONCLUSIONS
We have shown that the energy spectrum
of a Bose-Einstein system, which is confined in a
rigid cubic box with low adiabatic invariant, is
discrete and depends strongly on the box size and
temperature. The complex aspect of the spectrum,
is the consequence of the random degeneracy distribution in the cubic box, which introduces an
additional energy quantisation controlled by the
adiabatic invariant (multiplied by the atomic mass
number),  = AatL2T and by gas fugacity, A (or
alternatively, by the particle number).
We have introduced a reasonable superior limit
of the number of states in the box, which brings a
significant contribution to this discrete spectrum.
Then, the particle number of the Bose-Einstein
system can be calculated by a finite sum, which
give smaller values for the active particles in the
box than different evaluations of the common
used integrals. Our analytical formula for the active particle number is the closest to the exact result.
In the BEC phase, one can remark that the direct summation is considerably different in comparison to the common considered ratio of condensed and total particle number. However, if we
correct the main parameter, the critical temperature, these results are closer one to the other. Our
analytical approximation of sum by integral leads
also to a corrected expression for the critical temperature, which is in a better agreement with the
exact calculation.
Furthermore, the total energy, obtained in this
case by summing up the exact contributions of the
eigenvalues and their weights for well-defined
values of , shows a faster (exponential) decrease
8
Valentin I.VLAD, Nicholas IONESCU-PALLAS
to zero than it is classically expected, for   0.
For A = 0.99, the quantum effects occur in the
range: 0.7610-14    7610-14 [cm2K]. In this
regime, the box size and the temperature are reciprocal parameters in the sense that the same effects (in the boson gas) can be obtained either by
varying L or by varying T, if their product remain
constant. The lighter the gas atoms, the higher the
temperatures or the box size, for the same effects
in discrete Bose-Einstein systems, which are characterized also by small number of particles.
1.0
F=EXP(P1*(1/)-P2*((1/
)^2))
0.8
Data: Data2_D
Model: user1
Chi^2 = 0.00003
P1
-5.7955E-15
P2
4.4057E-28
0.6
F
±3.5744E-16
±1.2039E-29
0.4
0.2
Fig.5.The ratio between the total energy
of the particles in the box and the total
energy in a conventional container, F, in
function of  [cm2.K] for fixed fugacity,
A =0.99. The points are the results of
sums and the continuous line is the best
fit with the exponential function from
Eq.(30) over this set of points.
0.0
0.00E+000
2.00E-013
4.00E-013

6.00E-013
8.00E-013
2
(cm K)
ACKNOWLEDGEMENTS.
One of the authors (V.I.V.) thanks The “Abdus
Salam” International Centre for Theoretical Physics, Trieste (Italy) for the working stages at the
Centre as a Regular Associate Member. Particularly, he wishes to thank Prof. Gallieno Denardo for
his support in these visits. He wishes to
acknowledge Prof. Herbert Walther for the useful
discussions and for the privilege to be an external
collaborator of Max Planck Institut für Quantenoptik.
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Received : April 11, 2002