Physica B 165&166 (1990) 595-596 North-Holland

BOSE-EINSTEIN CONDENSATION IN QUASI-TWO-DIMENSIONAL SYSTEMS

Humam B. GHASSIB and Yahya F. WAQQAD

Department of Physics, University of Jordan, Amman, Jordan

Bose-Einstein condensation in an ideal quasi-two-dimensional Bose gas is recon- sidered with a view of establishing a simple criterion for crossover effects from two- to three-dimensional systems. The main attention is focused on the density of states involved. It is deduced that the critical value at which these effects occur is universal, apart from a geometrical factor of the order of unity. The implications for strongly interacting Bose systems are discussed very briefly.

1 . INTRODUCTION It is well known that Bose-Einstein

condensation (BEE) does not occur in strictly two-dimensional ideal Bose systems (l-3), unlike the three-dimension- al case. However, it remains an interest- ing problem to see how, in an (mxmxd) system, EEC reappears and: furthermore, how the critical value d can te deter- mined at which crossoverzfferts from two- to three-dimensional behaviour manifest themselves. This is attempted in Section 2, using the simplest possible techniques. The results obtained are then discussed in Section 3.

2. DEVELOPMENT 2.1 BEC in Strictly Zd-Systems To establish the notation we first

revisit the strictly two-dimensional ideal Bose system.

The number of excited states in this system is gi;en by

dp , (I) r-lexp{6E(p)1-1

where S is the area of the system, h is Planck's constant, 8 is the temperature parameter, l/kBT, kB being Boltzmann's constant and T the temperature, E(p) is the energy of a Bose particle of momentum p, and z is the fugacity 5 exp(-BP), p being the chemical potential.

Performing the angular integration in eq.(l) yields

N_ 2nm In\ “=-_ g1(2), iLi S hZ6

where m is the mass of the boson and gn(z) is the familiar Bose function of order n and argument z (4,5). Clearly, N, is a maximum only when gl(z) is a maximum - i.e., when 2~1. accordingly,

However, gl(l) diverges; all particles In the two-

dimensional ideal Bose gas are excited. This means that no BEC occurs for systems whose density of states is constant, which is the case for infinite two-dimen- sional systems.

2.2 BEC in Quasi-Zd-Systems Suppose the ideal Bose gas is now con-

fined to an infinite plane sheet whose thickness is d. BEC can now be probed in a simple manner by specifying those terms which depend on dimensionality and those which domnot. While N, is still given by

Ne = g(E;d);n(E)>dE, [

(3)

j0 its spatial dependence now enters through the density of states g (E;d). The problem is then reduced to evaluating this last function.

To calculate g(E;d) the total number of states u up to an energy E should be determined first; the derivative ao/aE will then ztve g(E;d). Specifically,

where f(n;d) is 1 if n=O,?l, i n0 is the maximum value of n with the total energy E:

i E(n) ?? hZ kZ - (nn/d)' 7iii[ 1 so that

i nOzdmax=[k.]int;

k being the wavenumber. It follows that

n0

(4)

2 * . f and cA;patible

(5)

a(E(n);d)=

I 4nS mE - h*

0921-4526/90/$03.50 @ 1990 - Elsevier Science Publishers B.V. (North-Holland)

596 H.B. Ghassib, Y.F. Waqqad

”

- s k Co k2

Cf I ,iR(k-n) dRdk

G2n,o 0 -co

f

where V:Sd is the volume of the system. On performing the simple contour in-

tegral involved, this gives u(E(n);d): 8rrV(2mE)'/2+4rrS mE

3h' _-

h'

k: sin(Znnk.) + k0cos(2nnk,)

nn n*71'

- sin(2nnk.j

1

, (7) 2n'n'

where the prime on the summation sign in- dicates that the (n=o) term has been excluded; its value is incorporated into the first term on the right. This is, of course, the exact three-dimensional expression for the total number of energy states. The second term is the analogous expression for the exact two-dimensional case. The third terms is of especial in- terest: as d + ~0, it reduces to zero, as indeed it must. Further, in the zero-d limit, the sum of all terms on the right vanishes except the surface term - again, as expected.

Finally, the sum of the last two terms on the right will be comparable in magni- tude to the first term at d=d,. To deter- mine d,, the summation is first evaluated readily with the aid of Bernoulli poly- nomials (6). The final result is the following quadratic equation for d,:

2 (2mE)f d; _ mE d, - 1 (2mE)3=O;(8)

37 -? 4 3h

from which

- * (9)

3. DISCUSSION The value of d, given by eq. (9) repre-

sents that value of d above which bulk effects begin to dominate. It is simply a natural wavelength reflecting the "con- finement effect". Apart from a geometri- cal factor, which is of the order of unity,

dc is clearly universal, in the sense that

it is independent of the geometry in- volved.

In conclusion, it is observed that the confinement of the ideal Bose gas in space enhances the density fluctuations, thereby destroying the long-range order and, hence, BEC. The disruptive effects in the strongly-interacting system have a rather similar consequence. Accordingly, the above conclusions are a fortiori applicable even to this system.

ACKNOWLEDGEMENTS One of the authors (H.B.G.) wishes to

thank Professor Abdus Salam, the Inter- national Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste, where this work was begun.

REFERENCES

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2.J.M. Ziman, Philos. Mag. 44 (1953) 548.

3. D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Fun- ctions (Benjamin, Reading-Massachusetts, 1975) pp. 214-251.

4. J.E. Robinson, Phys. Rev. 83 (1951)678.

5. F. London, Superfluids, Vol. II (Dover, New York, 1964) Appendix.

6. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1980) p. 1077.