EXACT EIGENVALUE DENSITIES AND THERMODYNAMIC PROPERTIES OF PERFECT QUANTUM GASES IN FINITE SYSTEMS

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EXACT EIGENVALUE DENSITIES AND

THERMODYNAMIC PROPERTIES OF PERFECT

QUANTUM GASES IN FINITE SYSTEMS

W. Eckhardt, R. Lautenschläger

To cite this version:

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JOURNAL DE PHYSIQUE _{Colloque C 2 , supplkment au no 7, Tome 38, Juillet 1977, page C2-139}

EXACT EIGENVALUE DENSITIES

AND THERMODYNAMIC PROPERTIES

OF

PERFECT

QUANTUM

GASES IN FINITE SYSTEMS

W. ECKHARDT and R. LAUTENSCHLAGER

Abt. fiir Theoretische Physik 11, Universitat Ulm D-7900 Ulm, Oberer Eselsberg, R.F.A.

R4sum4. - I1 est donne un sommaire bref sur des densitts exactes des valeurs propres de l'tquation d'onde dans des domaines rectangulaires. A l'aide de ces densit6s de valeurs propres, 1'6nergie libre d'un gaz de photons est calcul6e et une formule d'interpolation de Debye gkn6raliste est construite. Le domaine de taille et de temptrature, dans lequel cette thtorie am8ior6e peut &re employ6e, est indiqu6. La validit6 d'approximation T3 de Debye est discutte.

Abstract.

-

A short review about exact eigenvalue densities of the wave equation in rectangular domains is given. With the aid of these eigenvalue densities the free energy of a photon gas bounded in one dimension is calculated and a generalized Debye interpolation formula for finite solids is constructed. The range of size and temperature is reported in which this improved theory may be used. The validity of the ~ ~ - 1 a w of the specific heat is discussed.

1. Introduction. - The calculation of the thermodynamic properties of finite systems is based on the knowledge of their eigenvalue distributions. For the special problem of rectangular geometries the exact eigenvalue densities are well known for different physical systems from the work of Balian and Bloch [1-31 and Baltes [4].

Part 2 of our paper is a review about the eigenvalue densities of three problems : Blackbody radiation, scalar waves and Bose-Einstein particles. In part 3 and 4 we report some new calculations based on two of these eigenvalue densities : Free energy of a photon gas in a thin film and the extension of Debye's theory to finite, cube shaped systems.

2. Eigenvalue densities.

-

Formally the exact eigenvalue densities may be written as distributions :

D

( E ) = S ( E - E

(k)).

{k} denotes the sum-

ik>

mation range according to the allowed eigenvalues which are determined by the wave equation

(A

+

k2) +k = 0 with its corresponding boundary

conditions (b-c.). The separation of D(E) into a smooth and an oscillating part has been fully discussed in the comprehensive work of Balian and Bloch [I-31. As Baltes and Steinle [S] have shown, the mode densities of rectangular systems listed below are implied already in the 50-years-old mathematical results of Walfisz [6, 71 on the lattice-point-problem.

2.1 BLACKBODY RADIATION. - The general case with L I , L 2 , L3 < G O has been treated first by

Baltes [4]. The result for the flat parallelepiped is

known from the earlier work of Barone [8] and Agarwal [9] and reads

The second term in (1) represents the oscillations. Due to only one finite dimension these oscillations are of finite amplitude and show a saw-tooth formed spectrum.

2.2 SCALAR WAVES.

-

Balian and Bloch derived the eigenvalue distribution of the scalar wave equation with two different b.c.'s on a closed surface. In the case of a cube-shaped volume with edge length L they obtain [3] :

VE sin ( yr )

+

Ef

-

2 7r2(ch)3 mI.m2.m3=-" Y 2 with yi = - mi Li and y = c h

gers. The upper sign refers to Dirichlet b.c., the lower sign to Neumann b.c. Jo denotes the Bessel function of order zero ; c is the phase velocity of the scalar waves. The primes at the sums mean that the terms with m = m2 = m3 = 0 and mi = 0 are

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C2-140 W. ECKHARDT AND R. LAUTENSCHLAGER excluded. In contrast to ( I ) , eq. ( 2 ) contains surface

terms.

2 . 3 BOSONS AND FERMIONS. - We consider thin

films of thickness L3. Without taking into account a possible spin degeneracy we find :

D ( E ) 1 2 m 3 / 2 1 2 m

l+

lim

-

with y = (fi2/2

m

)-'I2 2 L p .

Eq. ( 3 ) is based on a result of Balian and

Bloch [3] which is modified by the dispersion relation E = (fi2/2 m ) k 2 . The fluctuations are finite again but in contrast to (1) their amplitudes do not depend on E . They may be neglected if the

thermal wavelength is small compared to L :

(fi2/2 mk, T)'12

*

L .

An extended treatment of finite Bose-Einstein assemblies has been done by Greenspoon and Pathria [ l o ] . The problem of electrons in metallic films has been considered by Kenner and Allen [ l 11 and Paasch and Wonn [12].

3. Free energy of a photon gas (L1, Lz

-

a).

-

The free energy is given by

F = - k . ~ ] ~ " D ( ~ ) l n [ l -exp(- s / k B T ) ] d e

Inserting (1) for D ( e ) we obtain :

-- ( k ~

T I 2

2

4 cash'(%

mkE T)

4 c h L :

,=,

m c f i

In the limit T-+ 0 the last three terms in (4) cancel. From the Gibbs function (4) all thermodynamic properties can easily be derived : For instance the well known internal energy calculated by Fierz [13]

and, by differentiation of the last three terms of (4)

with respect to L s , the T-dependent van der Waals forces, derived by Mehra [14].

4 . Debye theory of finite solids.

-

Many approa-

ches to the hitherto unsolved problem to calculate the exact eigenvalue distribution of the elastic continuum with stress-free boundaries have been made in the last thirty years. Breger and Zhukhovitskii [ I S ] treated an incompressible semi- infinite continuum, Stratton [16] and Dupuis et

al. [I71 considered a slab. The low-temperature

specific heat for a slab-shaped crystal has been calculated by Maradudin and Wallis [18]. All the

authors have rather dealt with surface effects of thin films than with the size effect of a small particle, this means e.g., that the influence of the particle shape cannot be studied in such models.

Numerical methods have been applied to small clusters [19, 201 but these methods are restricted to clusters of about 150 atoms (computer capacity). Therefore other authors use a scalar continuum model, which was first used by Montroll [21],

subsequently by Jura and Pitzer [22], Baltes and Hilf [23], Nonnenmacher [24] and Lau- tenschlager [25]. In this scalar approach the unknown eigenvalue density of the elastic continuum is replaced by the eigenvalue density of the scalar wave problem. This model takes not into account e.g. the scrambling of longitudinal and transversal modes. Therefore the phase velocity c

of eq. (2) should be interpreted as an effective sound velocity in the sense of a helpful empirical parameter that cannot be justified a priori but may be chosen to fit the experiments and it turns

out [23, 251 that it is of the order of magnitude of

the sound velocities that are measured in solids. Our calculations are based on this scalar model, but in contrast to Montroll's calculations we discuss quantitively the influence of the contributions due to the oscillating part in the mode density on the internal energy.

Starting from the eigenvalue density (2), we take into account the discrete character of an N-atomic solid by defining a cut-off energy E M via the

r e l a t i o n : N =

1.'

-

D ( E ) ~ E .

Without the fluctuating parts of (2) we obtain a cubic equation for E M which has the solution :

with the abbreviations :

A = b 2 / 4

+

a 3 / 2 7 ;

a

= 9 / 2 v - 2 7 / 1 6 ;

b = 2 2 7 / 3 2 ~ 2 7 / 8 7~ + ( 6 / 7 ~ ) ( 2 1 / 8 - N ) .

Numerical calculation of e M shows, that for N > SO the deviation of (5) from the exact E M is smaller

than 5 % ; in the limit V + a and N -+ oo formula ( 5 ) leads to Debye's cut-off energy :

sM = ch (6 .;rr2

N /

V)'13.

The internal energy of our system is given by

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THERMODYNAMICS O F FINITE QUANTUM GASES C2-141

E (L, T, EM) = k, T [2 p2(C h)l

v

E D3(xM)

*

with X M = kg T and the Debye functions Dn (xM) = rn

:

x

(ex - I)-' dx. In the limit of very high temperatures (7) yields the Dulong-Petit-Law ( E = 3 Nk, T). In (7) we have neglected the contributions due t o the oscillating part of (5). These contributions become very important for small L T / c ; therefore, we need a lower limit for the validity of (7).

N u m e r i c a l c a l c u l a t i o n s y i e l d :

L T I c

<

6 x 1 0 - l o K s f o r D i r i c h l e t - a n d LT/c> 15 x 10-lOKs for Neumann b.c.'s respecti- vely. In this range the total amount of all neglected contributions is less than 5 % of the smallest term in (7). A similiar estimation has been performed in the related problem of blackbody radiation for the first time by Baltes [26]. To interpret measurements on smaller systems at lower temperatures [27] (quantum size regime) one has to sum up the energies of each mode k. This must be done numerically [22, 23, 251.

In the literature, one often finds a polynom representation in T of the internal energy of finite solids 117, 18, 21, 241 ; for thermodynamic expan- sions see especially the work of Hilf [28]. From (7) a T-expansion can be derived. We write (7) in the form : E =

2

(an T4

+

6

(T)). The quantities

n = l

pn(T) are proportional to the difference of the Debye integrals with finite and infinite upper limit. Then we determine the T-range in which the inequality pn(T)/an T n

<

5 % is valid for each n. We obtain : K , T

<

E, /8. Since we find in this

temperature range, that &(T)

>

2

an Tn, (7) can be

n = l

written in the simple form :

.rr2

v

E(L, T ) = - _{10.(c h)}---7 (k, T)4

.

Eq. (8), of validity as N> 5

x

lo6 heat in the

course, has the same lower limit of (7). Therefore, finite solids with obey the familiar T3-law for the specific

given LT/c range.

Acknowledgment. The authors wish to express their graditude to Pr. H. P. Baltes, Landis & Gyr AG Zug, Swizerland, for helpful comments and literature hints.

References

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[2] BALIAN, R., BLOCH, C., Ann. Phys. 64 (1971) 271. U.R.S.S. 21 (1946) 1001 ; J . Phys. Chem. 20 (1946)

131 BALIAN, R., BLOCH, C., Ann. Phys. 69 (1972) 76. 1459.

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(1976) 1866. [ l q DUPUIS, M., MAZO, R., ONSAGER, L., J. Chem. Phys. 33

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(Comments & Addenda). _{[IS] MARADUDIN,}_{A. A., WALLIS, R. F., Phys. Rev.}₁₄₈₍₁₉₆₆₎

[6] WALFISZ, A., Thesis (Gottingen) 1922.

[7] WALFISZ, A., Math. 2. 19 (1924) 300. 945.

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GREENSPOON, s., PATHRIA, R. K., P ~ Y S . Rev. A 8 (1973) [231 BALTES, H. P., HILF, E. R., Solid State Commun. 12 (1973)

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[ l l ] KENNER, V. E., ALLEN, R. E., Phys. Rev. B 11 (1975) 2858. [24] NONNENMACHER, Th. F., Phys. Lett. 51 A (1975) 213. [I21 PAASCH, G., WONN, H., Phys. Status Solidi (b) 70 (1975) [25] LAUTENSCHLAGER, R., Solid State Commun. 16 (1975) 1331.

555. [26] BALTES, H. P., Appl. Phys. 1 (1973) 39.