Does the Transverse Electric Zero Mode Contribute to the Casimir Effect for a Metal?
Abstract
The finite temperature Casimir free energy, entropy, and internal energy are considered anew for a conventional parallelplate configuration, in the light of current discussions in the literature. In the case of an “ideal” metal, characterized by a refractive index equal to infinity for all frequencies, we recover, via a somewhat unconventional method, conventional results for the temperature dependence, meaning that the zerofrequency transverse electric mode contributes the same as the transverse magnetic mode. For a real metal, however, approximately obeying the Drude dispersive model at low frequencies, we find that the zerofrequency transverse electric mode does not contribute at all. This would appear to lead to an observable temperature dependence and a violation of the third law of thermodynamics. It had been suggested that the source of the difficulty was the behaviour of the reflection coefficient for perpendicular polarization but we show that this is not the case. By introducing a simplified model for the Casimir interaction, consisting of two harmonic oscillators interacting via a third one, we illustrate the behavior of the transverse electric field. Numerical results are presented based on the refractive index for gold. A linear temperature correction to the Casimir force between parallel plates is indeed found which should be observable in roomtemperature experiments, but this does not entail any thermodynamic inconsistency.
pacs:
11.10.Wx, 05.30.d, 73.61.At, 77.22.ChI Introduction
In spite of the numerous treatises on the Casimir effect during the past decade—for some books and review papers see, for instance, Milton milton01 , Mostepanenko and Trunov mostepanenko97 , Milonni milonni94 , Plunien et al. plunien86 , Bordag et al. bordag01 —it is somewhat surprising that such a basic issue as the temperature dependence of this effect is still unclear and has recently given rise to a lively discussion. This issue is not restricted to the case of curvilinear geometry, but is present even in the simplest conventional geometry of two parallel metal plates separated by a gap of width . Thus Klimchitskaya and Mostepanenko in their detailed investigation klimchitskaya01 , and also Bordag et al. bordag00 , and Fischbach et al. fischbach01 , have argued that the Drude dispersion relation for a frequencydispersive medium leads to inconsistencies in the sense that the reflection coefficient for perpendicular polarization (the TE mode) becomes discontinuous as the imaginary frequency goes to zero. As is well known, the Drude dispersion relation reads for imaginary frequencies
(1) 
where is the plasma frequency and the relaxation frequency. (Usually, is taken to be a constant, equal to its roomtemperature value.) The mentioned authors, instead of the Drude relation, give preference to the plasma dispersion relation, since no such discontinuity is then encountered. (In Ref. bezerra02 , the plasma relation together with the socalled surface impedance approach is argued to be the method best suited to describe the thermal Casimir force between real metals.) The plasma relation is
(2) 
The arguments in Refs. klimchitskaya01 ; bordag00 ; fischbach01 ; bezerra02 are interesting, since they raise doubts not only about the applicability of the Drude model as such, but even more, doubt about the applicability of the fundamental Lifshitz formula at low temperatures (see, for instance, Ref. lifshitz80 ).
The essence of the problem appears to be the following: For a metal, does the transverse electric (TE) mode contribute to the Casimir effect in the limit of zero frequency, corresponding to Matsubara integer ? It is precisely for this mode that the purported discontinuity of the reflection coefficient , mentioned above, can occur. The problem is most acute in the high regime (the contribution becomes increasingly important as increases), but is present at moderate and low temperatures as well. The conventional recipe for handling the twolimit problem for a metal, , , has been to take the limits in the following order:

Set first ;

then take the limit .
This way of proceeding was advocated in the early paper of Schwinger, , and Milton schwinger78 (we will call it the SDM prescription), and was followed also in one of the recent papers by some of the current authors hoye01 , and in Milton’s recent book milton01 . It seems to escaped recent notice that the physical basis for this prescription, namely the necessity of enforcing the correct electrostatic boundary conditions, was explicitly stated in Ref. schwinger78 .
Boström and Sernelius bostrom00 seem to have been the first to inquire whether this prescription is right: They argued that in view of a realistic dispersion relation at low frequencies the TE mode should not contribute. And three of the present authors arrived recently at the same conclusion, in two papers dealing with the case of two concentric spherical surfaces brevik02 ; brevik02a .
The BoströmSernelius paper gave rise to a heated debate in the literature bordag00 ; lamoreaux00 ; sernelius01 ; bordag01a on the role of the TE mode for a metal. The advent of accurate experiments in recent years, by Lamoreaux lamoreaux97 , Mohideen et al. mohideen98 ; roy99 ; harris00 ; chen02 , Ederth ederth00 , Chan et al. chan01 , and Bressi et al. bressi02 (cf. also the recent review paper of Lambrecht and Reynaud lambrecht02 ), represents important progress in this field. Especially the experiment of Bressi et al. is of interest in the present context, since it deals directly with the Casimir force between metal surfaces that are parallel, and so avoids use of complicating factors such as the proximity force theorem blocki77 , which nevertheless seems well understood. This experiment is fraught with experimental difficulties (related to keeping the plates sufficiently parallel), so the accuracy is claimed by the authors to be moderate (15%), but it is to be hoped that this accuracy will soon be improved. Several other related papers have appeared recently, discussing the interpretation of the mentioned experiments as well as more general aspects of finite temperature Casimir theory lamoreaux98 ; lambrecht00 ; genet02 ; svetovoy00 ; barton01 ; feinberg01 ; bezerra02a .
Our purpose in the present paper is to analyze the Casimir temperature problem anew, assuming conventional parallelplate geometry from the outset, therewith avoiding the spherical Bessel functions that become necessary if spherical geometry is contemplated. In particular, we will focus attention on the TE mode. Let us summarize our results:
It is useful to distinguish between two different classes of metals. The first class, which we will call “ideal” metals, is characterized by a refractive index for all frequencies. It implies that the reflection coefficient mentioned above is unity for all . This corresponds to the traditional recipe 1 and 2 above when handling the twolimit problem for metals. It means that the TE mode contributes to the Casimir force just the same amount as does the transverse magnetic (TM) mode.
The obvious drawback of this “ideal” metal is that it does not occur in nature. And this brings us to the second class, which is the one of real metals, in which case we must observe an appropriate dispersion relation, especially at low frequencies. It is most commonly assumed that the most appropriate dispersion when is the Drude relation, Eq. (1). As we will show, the Drude model implies that the TE mode does not contribute. The total free energy for a real metal becomes accordingly one half of the conventional expression. In contradistinction to recent statements in the literature klimchitskaya01 ; bordag00 ; fischbach01 we find that there exists no physical difficulty or ambiguity associated with the vanishing coefficient at . This is so because goes to zero smoothly when , as long as the transverse wave vector is nonvanishing. (If is precisely zero, there occurs a singularity in the reflection coefficient, but this has no physical importance since this point is of measure zero in the integral over .) Our present results are in agreement with Refs. brevik02 ; brevik02a , as well as with Boström and Sernelius bostrom00 .
A different view has recently been put forward by Torgerson and Lamoreaux torgerson . They argue that the Drudemodel behavior does not accurately represent the TE zero mode, which necessarily has a vanishing tangential component at the surface of a perfect conductor. They point to the necessity of taking the finite thickness of the metallic coatings into account. Their arguments seem to imply that the conventional temperature dependence is correct. However, in our opinion electrostatic considerations of this kind do not solve the zero temperature problem; what is required to incorporate temperature dependence is an analytic continuation into imaginary frequencies of Green’s functions referring to nonzero wavenumber.
Before embarking on the calculations let us emphasize the following point: The occurrence of the mode only once instead of twice is understandable physically. This mode is precisely the TM static mode, corresponding to the electric field being perpendicular to the two metal plates. It is the natural groundstate mode present when . Actually, in Sec. III of Ref. hoye01 we showed how the uniqueness of the static mode emerges naturally, using statistical mechanical considerations.
The outline of our paper is the following. In the next section we show why the exclusion of the TE zero mode seems to lead to an observable temperature correction to the force between real metal plates, and worse, seems to imply a violation of the third law of thermodynamics. In Sec. III we expand on the situation of an “ideal” metal in the sense described above, and calculate the Casimir free energy, entropy, and internal energy via a somewhat unconventional route. Equivalence with earlier results is demonstrated. In Sec. IV we introduce a new and simplified model to illustrate the Casimir problem, based essentially on statistical mechanics. In this model the system is replaced by two harmonic oscillators (the two media) that interact via a third oscillator (the electromagnetic field). Depending upon the form of the interaction we then have two situations. The first is the one where the induced interaction (or free energy), which is negative, increases linearly in magnitude with temperature in the classical limit. The other situation, which is more unexpected, is where the induced interaction vanishes in the classical limit. These two situations can be regarded as analogous to the behavior of the TM and TE modes. We also consider a strongly simplified case of real metals, and show how in such a case the contribution to entropy goes to zero smoothly as . Arguing on basis of the EulerMaclaurin formula we find this to be a general property (except in the idealized metal limit). We then go on to present numerical results based on the dispersion relation for gold, and obtain results qualitatively in accord with our analytical model. In the Appendices the smoothness of the reflection coefficient , and of the TE Green’s function, in the limit is explicitly demonstrated. We also discuss the temperature dependence of the relaxation frequency, . We conclude that a linear temperature dependence should be observable in room temperature experiments.
In this paper we use natural units, .
Ii Temperature Effect for Metal Plates
We begin by reviewing how temperature effects are incorporated into the expression for the force between parallel dielectric (or conducting) plates separated by a distance . To obtain the finite temperature Casimir force from the zerotemperature expression, one conventionally makes the following substitution in the imaginary frequency,
(3a)  
and replaces the integral over frequencies by a sum,  
(3b) 
This reflects the requirement that thermal Green’s functions be periodic in imaginary time with period ms . Suppose we write the finitetemperature force/area as [for the explicit form, see Eq. (16) below]
(4) 
where the prime on the summation sign means that the term is counted with half weight. To get the low temperature limit, one can use the EulerMaclaurin (EM) sum formula,
(5) 
where is the th Bernoulli number. This means here, with halfweight for the term,
(6) 
It is noteworthy that the terms involving cancel in Eq. (6). The reason for this is that the EM formula equates an integral to its trapezoidalrule approximation plus a series of corrections; thus the for in Eq. (4) is built in automatically. For a perfect conductor
(7) 
Of course, the integral in Eq. (6) is just the inverse of the finitetemperature prescription (3b), and gives the zerotemperature result. The only nonzero odd derivative occurring is
(8) 
which gives a Stefan’s law type of term, seen in Eq. (12) below.
The problem is that the EM formula only applies if is continuous. If we follow the argument of Ref. bostrom00 ; brevik02 ; brevik02a , and take the limit at the end ( are the permittivities of the two parallel dielectric slabs), this is not the case, and for the TE mode
(9a)  
(9b) 
Then we have to modify the argument as follows:
(10)  
where is defined by continuity,
(11) 
Then by using the EM formula,
(12)  
The same result for the lowtemperature limit is extracted through use of the Poisson sum formula, as, for example, discussed in Ref. milton01 . Let us refer to these results, with the TE zero mode excluded, as the modified ideal metal model.
Exclusion of the TE zero mode will reduce the linear dependence at high temperature by a factor of two, but this is not observable by present experiments. The main problem, however, is that it adds a linear term at low temperature, which is given in Eq. (12), up to exponentially small corrections milton01 .
There are apparently two serious problems with the result (12):

It would seem to be ruled out by experiment. The ratio of the linear term to the term is
(13a) or putting in the numbers (300 K eV, MeV fm) (13b) or as Klimchitskaya observed klim , there is a 15% effect at room temperature at a separation of one micron. One would have expected this to have been been seen by Lamoreaux lamoreaux97 ; his experiment was reported to be in agreement with the conventional theoretical prediction at the level of 5%.

Another serious problem is the apparent thermodynamic inconsistency. A linear term in the force implies a linear term in the free energy (per unit area),
(14) which implies a nonzero contribution to the entropy/area at zero temperature:
(15)
Taken at face value, this statement appears to be incorrect. We will discuss this problem more closely in Sec. IV, and will find that although a linear temperature dependence will occur at room temperature, the entropy will go to zero as the temperature goes to zero. The point is that the free energy for a finite always will have a zero slope at , thus ensuring that at . The apparent conflict with Eq. (15) or Eq. (12) is due to the fact that the curvature of near becomes infinite when . So Eqs. (14) and (15), corresponding to the modified ideal metal model, describe real metals approximately only for low, but not zero temperature—See, for example, Eq. (71).
Iii Casimir free energy, entropy, and internal energy
The Casimir surface force density between two dielectric plates separated by a distance can be written as
(16) 
(We follow the conventions of Ref. hoye98 and further references therein; here we further set .) The relation between and the transverse wave vector is , where . Furthermore
(17a)  
(17b) 
with being the permittivity. Note that whenever is constant, the and depend on and only in the combination ,
(18) 
(This result may also be found in standard references such as Ref. milton01 .)
The free energy per unit area can be obtained from Eq. (16) by integration with respect to since . We get hoye01
(19a)  
where  
(19b) 
(In the notation of Ref. hoye01 , .)
From thermodynamics the entropy and internal energy (both per unit area) are related to by , implying
(20) 
As mentioned above the behaviour of as has been disputed, especially for metals where . We now see the mathematical root of the problem: The quantities in the limit except that for any finite . So the question has been whether or or something in between should be used in this limit as results will differ for finite , producing, as we saw above, a difference in the force linear in . The corresponding difference in entropy will thus be nonzero. Such a difference would lead to a violation of the third law of thermodynamics, which states that the entropy of a system with a nondegenerate ground state should be zero at . Inclusion of the interaction between the plates at different separations cannot change this general property. We will show that this discrepancy vanishes when the limit is considered carefully, by using the EulerMaclaurin summation formula. Also, we will perform explicit analytic evaluation for any for metallic plates in the case where for all .
We will consider this latter case first. It is the case of “ideal” metals mentioned in Sec. I and already considered briefly in Sec. II.
iii.1 “Ideal” metals
With we have where we now also put , i.e., . To remove the dependence in the lower limit of integration in Eq. (19a), it is convenient to use the quantity of Eq. (17b) as a new variable. Expanding the logarithmic terms in Eq. (19a) and keeping only the leading term, we get the task of calculating
(21) 
where
(22) 
Carrying out the integration in Eq. (21) we obtain
(23a)  
with  
(23b)  
(It is easy to check that this result is correct at , where is not defined.) 
We encounter the following sums
(24a)  
(24b) 
so that
(25a)  
(25b)  
(25c) 
The quantity is given by the first two of these sums,
(26) 
Alternatively, one could just first perform the summation in Eq. (21) (for ) and then integrate. This summation yields . By subsequently integrating by parts the quantity in Eq. (26) is recovered (adding the term separately).
By further expansion of the logarithm in Eq. (19a) one obtains terms to be integrated and summed like Eq. (21). Performing the same steps as before, we find that the result (26) generalizes to
(27) 
valid for arbitrary temperature.
The surface force per area (16) can now be obtained via utilizing [Eq. (22)]. This yields
(28) 
The same result is also obtained by evaluating expression (16) (with ) in the same way as expression (19a) for was evaluated above. Using the second method, mentioned below Eq. (26), one finds that the integration of yields the combination of present in Eq. (28).
Considering the limit, which implies the limit, one obtains
(29) 
using the limiting values of expressions (24a), (25a), and (25b). This is the well known Casimir result for idealized metallic plates at , seen in Eq. (12).
The internal energy is now found from Eqs. (20), (22), and (27) to be
(30) 
and similarly an expression for the entropy follows from
(31) 
Now we can analyze the thermodynamic quantities in the low temperature limit using the properties of as defined by Eqs. (24a)–(25c). We have for low temperature,^{1}^{1}1Actually, for a roomtemperature experiment, need not be small. For K and m, . where
(32a)  
(32b)  
(32c)  
(32d) 
Inserting this into expressions (27), (28) or (30) one finds that the terms linear in vanish.^{2}^{2}2This is actually stronger than necessary to insure vanishing entropy, since such terms would give terms in the energy or free energy. Thus the entropy (31) vanishes, as it should in accordance with the third law of thermodynamics.
To obtain the leading correction to the result for finite one must consider the term in the power series expansion of the summand in Eq. (28). However, the summation of this term with respect to diverges,^{3}^{3}3For this reason, the alternate expression (3.35) in Ref. milton01 might be preferred. See Eq. (54) below. because the expansion of is not valid for large . For small one can instead integrate, without expanding, using the EulerMaclaurin summation formula (5) to obtain a finite correction to the zerotemperature result. Using Eq. (5) to evaluate expression (28), the expression (29) has to be subtracted to make finite. Putting we have, apart from a prefactor,
(33) 
with in view of the expansions (32a)–(32c). Integrating and using expressions (24a), (25a), (25b), we obtain
(34) 
Including the result (29) we thus find
(35)  
where we have inserted expression (22) for and noted that there is no term in Eq. (28), i.e., is to be subtracted from expression (5). All the odd derivatives in the EulerMaclaurin formula vanish because is even. It should be noted that the expression for is in agreement with what has been found earlier [cf. Eq. (12)], via alternative methods, by Milton milton01 , Klimchitskaya and Mostepanenko klimchitskaya01 , Sauer sauer62 , Mehra mehra67 , and others, where the exponentially small correction to the above formula is also given.
The free energy (27) can be obtained from , but this leaves a temperature dependent constant of integration. So instead we make use of the method above, where from Eq. (27)
(36) 
and where now . With Eq. (36) we get a nonzero integral
(37)  
using partial integration. The integral (37) may be easily evaluated by contour methods. Due to symmetry the integral can be extended to minus infinity and then the contour of integration can be distorted into one which encircles the poles along the positive imaginary axis. Since has poles at with integer we get^{4}^{4}4This low temperature dependence in , which does not contribute to the force, is determined by the linear high temperature behavior of —see Ref. milton01 , Sec. 3.2.1.
(38) 
In view of this result as well as Eq. (29) we obtain for the free energy ()
(39)  
This result, including its exponentially small correction, is given in Ref. milton01 and references therein. The internal energy , which can be most easily be evaluated using Eq. (20), can also be computed by the method above, starting from the sum (30). Then
(40) 
with . Partial integration replaces the of Eq. (37) with , and we obtain
(41) 
With Eq. (31) the entropy thus becomes (recall that is assumed)
(42) 
iii.2 Equivalence with earlier results
Equivalence with previous derivations can be shown for any . It is then convenient to utilize the Poisson summation formula. If is the Fourier transform of , defined by
(43) 
then
(44) 
With one finds
(45) 
Thus
(46) 
the familiar cotangent expansion, which can be verified in many different ways (cf. Ref. hoye81 ).
In Eqs. (27) and (28) one of the sums is []
(47a)  
where with Eq. (46)  
(47b) 
Summation first with respect to where also the result (46) is utilized then gives
(48) 
In the limit only the term remains, and we get the result if we use the expansion (32a) ()
(49) 
which is consistent with the sum occurring in Eq. (29).
To obtain the free energy and the force there are sums and that follow from the and of Eqs. (25a) and (25b). And like Eqs. (47a) the relations between the various lead to
(50) 
where . Also:
(51) 
So to obtain we need, because
(52) 
the combination
(53)  
Altogether, restricting to positive values due to symmetry, the expression (28) can be reexpressed as ()
(54) 
which is the desired known expression. (For example, compare Eq. (3.35) of Ref. milton01 .)
To calculate the free energy (27) one likewise needs
(55)  
Thus the free energy becomes
(56) 
Compared with the small or expansion (39) it is clear that the last term of Eq. (56) gives the term of (39). The coefficient can also be identified from Eq. (56). As when we must have, when comparing with Eq. (39),
(57a)  
or  
(57b) 
which is in agreement with Eq. (38).
Iv Finite permittivity. Real metals
iv.1 Two harmonic oscillator models
With finite permittivity the and of Eq. (17a) will vary with . Especially as or (). In the high temperature or classical limit only the Matsubara frequency (or ) can contribute as . Thus, in the classical limit one has the result that the TE mode does not contribute at all. Physically, this means that the temperature becomes so high that only the static dipoledipole interaction contributes (the limit of the TM mode). In our opinion this somewhat unexpected behaviour is related to the peculiar type of interaction that exists between the canonical momentum of a particle and the electromagnetic vector potential , which for a particle of mass and charge is . In addition to the standard cross term interaction this also implies an interaction .
As an illustration of the above we can consider two models, in each of which two harmonic oscillators interact via a third one. These oscillators represent a simplified picture of our polarizable parallel plates interacting via the electromagnetic field. The classical partition function of a harmonic oscillator with frequency is const/, which gives a free energy . Thus for three noninteracting harmonic oscillators the inverse partition function is proportional to , where
(58a)  
with  
(58b) 
(The quantity corresponds to above.) By quantization using the path integral method hoye81 ; brevik88 , the classical system is split into a set of harmonic oscillator systems described by Matsubara frequencies. Expression (58a) is replaced by
(59a)  
where  
(59b) 
(For real frequencies, , determines the response to an external oscillating force acting on the oscillator.)
Now add interactions, of strength proportional to , between the third oscillator and the other two. The usual form of this interaction is , where and are coordinates. Let this constitute the first model, which is analogous to the TM mode. Then the quantity becomes the determinant of the matrix,
(60a)  
where  
(60b) 
The quantum free energy for this system of three coupled oscillators is given by summing over the Matsubara frequencies, as in Eq. (19a):
(61) 
where and is replaced by in the term of Eq. (60a). The limiting procedure is required to make the full free energy well defined. This means that the path integral representation of a harmonic oscillator is discretized by dividing the imaginary time of periodicity into pieces each of length as done in Ref. hoye81 . There, in an appendix an explicit evaluation was performed for one single oscillator.
The various factors in Eq. (60a) can be interpreted as follows: The product corresponds to the noninteracting system, the next two factors represent the result of interaction of single oscillators with the third one, while the last one is the contribution from the induced interaction between the two single oscillators via the third one. The logarithm of the last term is the analogue of the Casimir free energy. In this respect the term represents the induced interaction. Furthermore the () represents the “bare” polarizability of noninteracting particles which for nonzero becomes . Due to interaction with the “radiation” field this polarizability is modified into (), where represents a “radiation” reaction from the “field” upon each single oscillator.
The above represents the ordinary situation, analogous to the TM mode. To model the TE mode, we can consider an analogy with the electromagnetic interaction in which the third oscillator can interact with the momenta of the first two. The analogous interaction will be is mass), including the unperturbed term. By evaluation of the classical partition function one now finds that the interaction from const. has no influence. (This is the analogue of classical diamagnetism which is equal to zero, as const. is seen to have no influence on the result when is integrated first.)
Quantum mechanically, the problem is a bit more complex. However, we can now exchange the roles of momenta and coordinates of the first two oscillators, i.e., we introduce a momentum representation. Then the interaction with the third oscillator can be written as ()
(62) 
Now the last quadratic term adds to the energy of the third oscillator alone. Thus, compared to the first model considered above, is changed while the other remain unchanged:
(63a)  
Likewise in the quantum case  
(63b) 
The quantity can still be written in the form (60a), but due to the change of , the () is replaced by when evaluating , i.e.,
(64) 
The induced (analogous to the Casimir) free energy is again given by the logarithm of the third term in Eq. (60a). At zero and finite temperatures the latter logarithm is negative, and the free energy
(65) 
is negative. Note that here the limiting procedure of Eq. (61) is not needed as sums for free energy differences converge, without difficulties. In the classical limit, however, the induced free energy becomes equal to zero ( implies that we get the logarithm of unity). We note the analogy: At high temperatures the same is true for the TE mode in the Casimir effect. There exists thus at least somewhere a finite temperature interval for which the Casimir free energy increases with increasing temperature. In turn, this means that the Casimir entropy becomes negative in this interval.
This is a counterintuitive effect, but is physically due to the fact that we are dealing with the induced interaction part of the free energy of a composite system. We cannot apply usual thermodynamic restrictions such as positiveness of entropy to a “subsystem” of this sort. There exists actually a striking analogy with the peculiar formal properties one encounters in connection with the theory of the electromagnetic field in a continuous medium. The electromagnetic energymomentum tensor that experimentally turns out to be definitely the best alternative when dealing with highfrequency effects, is the Minkowski tensor (cf., for instance, Ref. brevik79 ). This tensor is however nonsymmetric, apparently breaking general conservation principles for angular momentum. The reason why this peculiar behaviour is yet quite legitimate physically, is that phenomenological electrodynamic theory is dealing only with a subsystem (the field itself plus its interaction with matter), and we cannot apply the same formal restrictions on it as we could if the system were closed.
iv.2 Real metal
In the limit of an ideal metal () the traditional (SDM) prescription, as mentioned in the Introduction, implies that for all . In addition, as also mentioned previously, thermodynamic arguments have been given, claiming that the entropy does not become zero at in violation of the third law of thermodynamics if is used bezerra02a . However, we do not find this to be the case; as we will show below, the entropy will be zero as required at , even for a metal that is not idealized and where one bases the analysis on the value .
Let us go back to Eq. (21). That equation was obtained by expanding Eq. (19a) to first order in under the assumption that . Doing the same expansion for finite permittivity, we obtain an integrand which contains a term with a factor (or ) that varies with such that when . Expanding Eq. (19a) to higher order one obtains likewise powers of which, because , become less important as compared to the case of an ideal metal (where ). One can first consider the case where is independent of . When is large one can use as a rough approximation
(66) 
This simple expression for is intended to show essential features that will be obtained more accurately in a detailed numerical calculation. With this, Eq. (24a) (neglecting the influence of ) will turn into
(67a)  
with similar modifications for . Here  
(67b) 
is an effective sharp cutoff limit for the integral, a crude model for what should be a gradual cutoff for the integral of interest. [A gradual cutoff will only modify the last term of (67a) into a sum or integral over terms with varying . Namely, with varying , Eq. (21), if we recall the comment below Eq. (26), changes into ( for large)
(68)  
using partial integration. The approximation (66) means that ]
As we did to obtain Eq. (42), we carry out the sum over in Eq. (31) while assuming sufficiently large such that approximation (66) can be used. Then as in Eq. (67a) one obtains the previous result minus a term with . Keeping only the leading term, Eq. (42) is modified into
(69) 
[However, to be more accurate for and thus for . When this is taken into account, we find that in a more narrow region, , but that Eq. (69) holds for