arXiv:quant-ph/0008088v3 9 Feb 2001
THE CASIMIR PROBLEM OF SPHERICAL DIELECTRICS: QUANTUM STATISTICAL AND FIELD THEORETICAL APPROACHES J. S. Høye1 , I. Brevik2 , and J. B. Aarseth2 1
Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway 2
Division of Applied Mechanics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway PACS numbers: 05.30.-d; 05.40.+j; 34.20.Gj; 03.70.+k Revised version, February 2001
Abstract The Casimir free energy for a system of two dielectric concentric nonmagnetic spherical bodies is calculated with use of a quantum statistical mechanical method, at arbitrary temperature. By means of this rather novel method, which turns out to be quite powerful (we have shown this to be true in other situations also), we consider first an explicit evaluation of the free energy for the static case, corresponding to zero Matsubara frequency (n = 0). Thereafter, the time-dependent case is examined. For comparison we consider the calculation of the free energy with use of the more commonly known field theoretical method, assuming for simplicity metallic boundary surfaces.
1
1
Introduction
The Casimir problem for dielectrics - for general introductions see, for instance Refs. [1-3] - has turned out to be difficult to solve, in the presence of curved surfaces. The most typical example of a system of this sort is probably that of a single nonmagnetic compact spherical ball, surrounded by a vacuum. (Equivalently, one may imagine a spherical cavity in an otherwise uniform medium, thus dealing just with the situation typical for sonoluminescence.) Formally, in the presence of curved boundaries one is confronted with divergences when summing over all angular momenta up to infinity. This kind of divergence is usually absent when one deals with plane boundaries. Physically, the divergences are coming from the fact that phenomenological electrodynamics, implying use of the permittivity concept, becomes inappropriate at small distances. There exists a natural cutoff in the material, of the order of the intermolecular spacing, and in practice some kind of regularization has to be invoked in order to deal with the divergences in the formalism. More accurately this cutoff is the molecular diameter, as is most easily seen from the statistical mechanical method. By means of this method the electromagnetic Green function can be associated with the pair correlation function between dipole moments. The latter quantity is zero inside the hard cores, and its deviation from from the ”ideal” Green function extends typically a few molecular diameters outwards from the molecule. In the case of nondispersive media, the use of zeta-function methods has proved to be very useful for force, or energy, calculations. The field theory approach to the Casimir problem has been considered at various places; in addition to the references above we may mention Refs. [4-16]. (This list is not intended to be complete; it covers mostly treatments of nonmagnetic media, and does not include the bulk of papers devoted to studies of the special case of media that satisfy the condition εµ = 1. A very extensive list of references is given in the recent report of Nesterenko et al. [14].) The Casimir energy E calculated by field theoretical methods at zero temperature for a dilute nondispersive compact sphere of radius a and refractive index n is positive, E=
23 h ¯c (n − 1)2 , 384 πa
(1)
corresponding to an outward force. Instead of making use of field theoretical methods for continuous matter, 2
one may alternatively use quantum statistical mechanical methods. We shall in the first sections below consider methods that were developed by Høye and Stell, and others. Basic references to this kind of theory are [17] and [18]. In the Casimir context, Høye and Brevik [19] used the quantum statistical mechanical path integral method to calculate the van der Waals force between dielectric plane plates. Recently, we have applied the same method to a single compact spherical ball [20]. This statistical method, although probably not so well known as the field theoretical methods, turns out to be quite powerful. Thus, we can use it to calculate explicitly the short range terms in the single sphere’s free energy, and verify how the repulsive Casimir surface force as calculated by field theoretical methods is simply a residual, cutoff independent, term in a complicated expression containing many terms. Cf. in this context also Refs. [21], [16], and [9]. It is now natural to ask: what is the experimental status in this field? Recently, there has been an impressive improvement of the experimental accuracy as regards force measurements; one has been able to verify the theoretically predicted Casimir forces, lying in the piconewton range, up to an accuracy of about 1 per cent. In Ref. [22] the Casimir force was demonstrated between metallic surfaces of a sphere above a disk using a torsion pendulum, whereas in Refs. [23], [24] an atomic force microscope was used. One important lesson is to be learned from these experimental works is the following: they alway involve two (in principle there may even be more) bodies. The Casimir surface force on a single sphere is not measured. There seems not even to be an idea of how to measure such a force; probably this reflects simply that the force concept as such is not observationally well defined. Thus, in order to keep contact with experiments, at least in principle, one ought to consider at least two bodies. And this brings us to the theme of the present paper, namely to calculate the mutual free energy for a system of two spherically-shaped concentric nonmagnetic dielectrics. We will envisage that there is one compact sphere for r < a, and one semi-infinite similar medium for r > b, so that there is a vacuum gap of width d = b − a in between. There will be an attractive Casimir force between the two media. One may object that there is still no straightforward way to imagine measuring such a force; however this does not create difficulties for our main purpose, which is to calculate the Casimir force in a setting which maintains spherical symmetry and yet avoids the complications with internal, cutoff dependent, forces. 3
In the following four sections we shall deal with the quantum statistical mechanical theory, with an emphasis on the static limit (zero Matsubara frequency, where derivations are more simple). Thereafter, for comparison we consider the alternative field theoretical approach, limiting us for simplicity to the case of perfect metallic walls at r = a, b. The resulting expressions for the free energy are Eq. (18) for the static case (Matsubara frequency equal to zero) and Eq. (40) for the time-dependent case (general Matsubara frequency). These expressions are obtained within the statistical mechanical approach. Within the field theoretical approach, the finite-temperature free energy is given by Eq. (68) assuming, as mentioned, perfectly conducting walls. There is one notable difference between the statistical mechanical approach and the field theoretical approach, as far as the free energy is concerned. In the first of these cases the method is basically more simple, at least in principle, as one needs only knowledge about the mode eigenvalues of the oscillating dipole moments in the dielectric medium; cf. Eq. (3). These eigenvalues are again related to the pair correlation function. In the second case the calculation is more indirect, as one first calculates the surface force (arising from the mutual interaction) on the outer surface implying use of Maxwell’s stress tensor, and thereafter relates the force to the free energy via integration of Eq. (67). That is, the field theoretical method involves use of the two-point functions for the electric and magnetic fields. As we also want to show that the results obtained by these widely different methods are in agreement, we consider some cases that are easy to analyse analytically, by both methods. We ought to stress again the conceptual difference between the two methods studied in this paper. By the field theoretical method the Casimir effect is regarded as the energy shift due to the frequency eigenvalues of the quantized electromagnetic field in the presence of dielectric media. By the more recent statistical mechanical method this energy shift is regarded as a consequence of the dipole-dipole interaction between oscillating dipole moments embedded in the molecules of the media. The latter viewpoint can be realized by exploiting the analogue that exists between phonons in a solid, and electromagnetic waves or photons in vacuum. Then impurities in the solid will be the analogue to dielectric particles. These impurities will couple to the phonons of the solid and modify their frequencies. If, however, one wants to do the statistical mechanics of the latter system, e. g. to calculate its free 4
energy, then one can first integrate out the coordinates of the pure solid that appear in the path integral representation of the quantized problem. With harmonic oscillators one encounters in this way Gaussian integrals which can easily be calculated. The result is independent of the impurities, except that interactions between them are introduced. Thus, the resulting change in free energy can be related to these induced induced interactions (being in general time-dependent) in the system of impurities. Likewise, in this picture the dipole-dipole interactions are related to the Casimir free energy. We employ Gaussian electromagnetic units in this paper.
2
General remarks
Consider the free energy F (T ) due to the mutual interaction between two spherical dielectric bodies with concentric surfaces at r = a and r = b. The attractive Casimir force between the surfaces, per unit area at the outer surface, is equal to f = −1/(4πb2 )∂F/∂b. As shown earlier for the case of plates [19], this Casimir force can be interpreted as the dispersion force arising from thermal fluctuations of molecular dipole moments. By our quantum statistical mechanical considerations this also incorporates the quantum fluctuations at T = 0. That means, all fluctuations can be regarded as purely thermal for any T . The difference between classical and quantum situations is that in the former case these fluctuations vanish at T = 0 while in the latter case they remain finite. In [20] we considered the low density (or small ε − 1) version of the single-body problem showing, as mentioned above, that the divergences are due to the continuum model of the medium. A cutoff in length scale is needed. For realistic systems this cutoff is determined by the molecular hard cores through their influence upon the pair correlation function. For two polarizable particles the free energy due to their mutual attraction can be written as [19, 25] βF = −
∞ 1X 1 1 (α1 ψα2 ψ)n = ln(1 − α1 ψα2 ψ), 2 n=1 n 2
(2)
with β = 1/kB T . The ψ represents the potential energy of the dipole-dipole interaction which for general K (see Eq. (3) below) is given by Eqs. (6) and (7) below. Now the two particles can be generalized to and regarded to be our two spherical bodies, in the same way as two semi-infinite parallel plates 5
were treated in [19]. That means, the two spherical bodies are regarded as two particles with many internal degrees of freedom. In this way Eq. (2) becomes a short hand notation wherein ψ represents the interaction between two points in the two bodies (over which we integrate), and the polarizabilities α1 and α2 become the respective internal correlation functions of the two bodies with their mutual interaction ψ switched off. As noted in [19] the expression (2) is formally exact for coupled harmonic oscillators, i.e., the model that we are employing for the polarizable particles. In terms of graphs, the expression (2) represents the ring graphs in the γ−ordering for the long-range forces, γ being the inverse range of interaction [26]. For coupled oscillators Eq. (2) above and Eq. (3) below are exact results [27]. Extending to the quantum mechanical case, Eq. (2) generalizes to βF =
1X ln(1 − α1K ψK α2K ψK ), 2 K
(3)
where K = 2πn/β with n integer (i.e., n ∈ h−∞, ∞i). Note that K = h ¯ ζn , where ζn is the Matsubara frequency ζn = −iω and ω is the frequency. For low density (or small α) only the first term in the sum (2) is needed. This is the situation considered in [20] and found there, after some transformations, to be in agreement with earlier works. The first term means simply that one takes the (radiating) dipole interaction squared, average (integrate) over the fluctuating dipole moments, and finally integrate over the two media. Thus Z (4) F = ρ2 dr1 dr2 Φ, where ρ is the particle density, r1 < a, r2 > b, and [19] βΦ = −
i 3X 2 h 2 2 αK 2ψDK (r) + ψ∆K (r) , 2 K
(5)
where r = r2 −r1 . (Here r1 and r2 are positions in different media, so double counting does not occur.) The radiating dipole interactions used in (5) can be written as ψ(12) = ψDK (r)DK (12) + ψ∆K (r)∆K (12), with DK (12) = 3(ˆ ra ˆ1K )(ˆ ra ˆ2K ) − a ˆ1K a ˆ2K , ∆K (12) = a ˆ1K a ˆ2K . 6
(6)
Here the hats denote unit vectors, and aiK is the Fourier transform of the fluctuating dipole moment of particle number i in imaginary time; cf. Eq. (5.2) in [25]. Explicitly, from Eq. (5.10) in [25], e−τ ψDK (r) = − 3 r e−τ ψ∆K (r) = − 3 r
1 1 + τ + τ2 , 3 2 2 4π τ + δ(r), 3 3
(7)
with
iωr Kr =− for ℑ(ω) < 0 (or K < 0), c h ¯c and −K → |K| when extending to K > 0 in (7) (see Eq. (5.11) in [25]). For general K we are not able to calculate the integral (4) in a direct way (but we can calculate it indirectly for arbitrary density, as will be argued later, in Sect. 4). We can integrate, however, for K = 0. The integral of interest then becomes Z 1 I= dr1dr2 6 , r 2 = r12 + r22 − 2r1 r2 cos θ. (8) r r1 b τ=
This integral can be evaluated in closed form [7] (second reference). However, in Sect. 3 we want to extend this low density evaluation to the case of arbitrary density or arbitrary ε. To do so, we need the contributions related to the various spherical harmonics. Thus we will here perform an expansion of the integral. This will also be used as an independent verification at the end of Sect. 4, where the more general theory developed there for arbitrary values of K turns out to yield the correct result when K → 0. Using spherical coordinates to integrate over the angle θ between r1 and r2 we first obtain (x = − cos θ) J=
Z
1
−1
"
#
dx 1 1 1 = − 2 2 3 4 (r1 + r2 + 2r1 r2 x) 4r1 r2 (r2 − r1 ) (r2 + r1 )4 ∞ (2l + 2)(2l + 1)2l r1 2l−1 1 X ,(9) = 2r1 r25 l=1 6 r2
performing a series expansion to make it easy to relate to the result for high density. Then, I = 8π 2
Z
0
a
r12 dr1
Z
b
∞
r22 dr2 J = 7
∞ 8π 2 X (l + 1)l σl , 3 l=1 2l + 1
(10)
with σl = (a/b)2l+1 . Inserted in (4) we obtain (α0 = α) ∞ X 3 (l + 1)l 1 βF = βρ2 α2 · 2 · I = − (ε − 1)2 σl . 2 2 l=1 2l + 1
(11)
Note that for small ε − 1, Eq. (11) will be the high temperature result for which only K = 0 contributes. This high temperature result at low density may in itself be of limited interest as it does not go beyond earlier results. But here we use it as a basis to make further developments. So in the next section the formalism is generalized to arbitrary density or ε, although it is still restricted to K = 0. In Sect. 4, a derivation that encompasses both arbitrary ε and K is given.
3
The static case
For simplicity we first consider the static case, by which we mean that the frequency is zero (K = 0). Then the electromagnetic dipole-dipole interaction is the well-known static, time-independent (also called instantaneous) one. By the time-dependent case we mean the general situation with K 6= 0. Then the dipole-dipole interaction will be the radiating or dynamical electromagnetic field where the time delay due to the finite speed of light is involved. Note that in general both the static and the dynamic cases contribute to the Casimir effect. The former, being proportional to T , contains the whole effect when T → ∞, but vanishes when T → 0. For general ε one should sum up the series in Eq. (2). This will not be a simple task. However, one can include a strength factor λ along with the perturbing interaction ψ and differentiate (2) (or (3)) to obtain β
∂F |λ=1 = −ψ(α1 ψc α2 ), ∂λ
where ψc =
ψ . 1 − α1 ψα2 ψ
(12)
Here the α1 ψc α2 will be the pair correlation function for the fluctuating dipole moments. As shown in Appendix A in [19] the ψc (apart from a 8
simple factor) is the Green function for the electromagnetic problem with the dielectric medium present while ψ is the one for vacuum. Thus we can utilize Maxwell’s equations for electrostatics to obtain this zero frequency Green function or ψc in the presence of two dielectric spheres. The electrostatic potential Φ fulfils the Laplace equation ∇2 Φ = 0 with ε =const. Splitting off the spherical harmonic factor Ylm = Ylm (θ, ϕ), Φ = Φl (r)Ylm(θ, ϕ),
(13)
we can write the radially dependent term in the form
Φ=
l a r + B , a l r l+1 + C1 ar C ar l+1 , D ar 1 ε
l+1
r