# What is the temperature dependence of the Casimir effect?

###### Abstract

There has been recent criticism of our approach to the Casimir force between real metallic surfaces at finite temperature, saying it is in conflict with the third law of thermodynamics and in contradiction with experiment. We show that these claims are unwarranted, and that our approach has strong theoretical support, while the experimental situation is still unclear.

###### pacs:

11.10.Wx, 05.30.-d, 73.61.At, 77.22.Ch## I Introduction

Recently there has been a comment Bezerra:2005hc that has strongly criticized our work Brevik:2004ue on the temperature dependence of the Casimir force between a spherical lens and a plate, both coated with real metals (Au for example). This follows a long paper by some of the same authors decca , which also criticizes our work as being in conflict with fundamental thermodynamical requirements, particularly the third law of thermodynamics or the Nernst heat theorem. Furthermore, they claim that their recent experimental results are completely consistent with their theoretical approach, and in contrast completely rule out our theory. The purpose of this note is to respond to these criticisms, and emphasize that their theoretical claims are invalid, while the experimental situation is still too indecisive to draw definitive conclusions.

First, let us make a statement about the physical nature of the controversy. The conventional approach, dating back to Lifshitz lifshitz80 , and reinforced by Schwinger et al. schwinger78 , describes an ideal metal by a formal limit, where is the permittivity of the material. Mathematically, the limit is taken in such a way that both the TE and TM zero modes (that is, the term in the Matsubara sum) contribute. Recently, however, it has been recognized bostrom that this is not correct, and that the TE zero mode cannot contribute either for finite or for a real metal. Consequently, in the ideal metal limit, this recognition gives rise to a linear temperature term in the pressure at low temperature, and a nonzero value of the entropy at zero temperature. However, real metals are not described by this ideal limit, and even though only one zero mode, the TM one, can be present, as one sees by either thermodynamic or electrodynamic considerations, the contradiction with the Nernst theorem is not present for actual conductors.

In contrast, the authors of Bezerra:2005hc ; decca propose two alternatives to describe the reflection coefficients that enter into the Lifshitz theory, the impedance approach and the plasma model. In the former the transverse momentum dependence in the surface impedance is simply disregarded (the so-called Leontovich approach), while in the second alternative the plasma dispersion relation with no relaxation is used for the dielectric constant. Neither of these omissions is in accordance with the properties of real materials, as we will detail in the following.

## Ii Lifshitz formula

We start by remarking that the Lifshitz formula for the force between two parallel dielectric slabs is essentially geometrical, being determined entirely by multiple reflections at the interfaces between the parallel media (for example, see Sec. 3 of Ref. Milton:2004ya ). Now there is no doubt that one can use surface impedances to describe the reflection coefficients appearing in the theory, for these are merely the linear relations between transverse electric and magnetic fields at the surface of the metal. It is well known that use of either bulk permittivity or surface impedance are equivalent in describing the reflectivity properties, provided that one incorporates transverse momentum, . The boundary conditions implied by Maxwell’s equations provide the precise connection between these alternative descriptions embook . In principle both bulk permittivity and surface impedance should depend on , although theory and optical data suggest that the dependence of permittivity on is rather small. This would then imply a substantial dependence on in the surface impedance. Indeed, in Ref. Brevik:2003rg we show that the exact impedance also leads to zero contribution of the TE zero mode.

### ii.1 Transverse momentum dependence of the surface impedance

The authors of Ref. Bezerra:2005hc ; decca ignore the transverse momentum dependence at zero frequency, because they believe that the “mass shell” condition must be satisfied. However, this relation can only be valid for real photons. In computing the temperature dependence, one is evaluating Green’s functions periodic in imaginary time, with period Martin:1959jp . The Matsubara frequencies, , being imaginary, evidently break the mass-shell condition. Physically, evanescent waves or virtual photons are responsible for the thermal Green’s functions. The inclusion of spatial dispersion, as required by Maxwell’s equations, has been considered in detail in recent publications svetovoy ; sernelius and the effects of the exclusion of the TE zero mode at both high and low temperatures are manifested.

### ii.2 Plasma model

The authors of Ref. Bezerra:2005hc ; decca also state that the plasma model is equally good at describing the optical properties of real metals. The plasma model is a special case of the Drude model, with the relaxation parameter neglected. In fact, optical data clearly show that this limit is inadequate: The Drude model describes the permittivity of real metals up to rad/s, while the plasma model fails for frequencies below rad/s. It is precisely the low-frequency part of the spectrum that is relevant for the discussion of the disputed zero-mode contribution. Those authors argue that the neglect of relaxation is appropriate because the zero temperature limit of the relaxation parameter for an ideal metal (the Bloch-Grüneisen law) is zero. To this we reply that the relaxation parameter for real materials cannot vanish at zero temperature because of the scattering by impurities, and further that all present and nearly all proposed experiments are carried out at room temperature, so the temperature dependence of this parameter should be irrelevant. Moreover, it has recently been demonstrated svetovoy that at sufficiently low temperature the residual value of the relaxation parameter does not play a role, as the frequency characteristic of the anomalous skin effect becomes dominant. In contrast to the ad hoc procedure advocated in Ref. Bezerra:2005hc ; decca , which in addition to using physically inadequate models, employs an unjustified extrapolation from the infrared region (that is, the plasma model, which disregards relaxation, is extrapolated from that region down to zero frequency), we use real optical data at room temperature for the bulk permittivity. Elsewhere it has been emphasized that a dielectric function valid over a wide frequency range and not merely the infrared must be employed, and indeed that frequencies very small compared to the characteristic frequency play a dominant role in the temperature dependence svetovoy . Of course, there is no data at zero frequency, but physical requirements (thermodynamics and Maxwell’s equations) do not allow the TE zero mode to be included in the Lifshitz formula.

## Iii Thermodynamics

As far as we can understand there is no disagreement about the validity of the Lifshitz formula for the Casimir force as long as the dielectric constant is finite. For this latter situation there is no or zero frequency TE (transverse electric) mode, in agreement with Maxwell’s equations of electrodynamics, and inclusion of such a term would violate the third law of thermodynamics svetovoy .

The controversy arises in the limit of a perfect metal. It is well known that in this limit the mathematics of the limiting process can be ambiguous due to its singular nature. So the limit of a perfect metal can in one way be taken by letting when regarding the contribution from the term. Since this term for all finite is zero, its limiting value is also zero. However, the other way is to take the limit while schwinger78 . In the latter case a non-zero contribution arises. The opinion of Bezerra et al. Bezerra:2005hc is that the latter procedure is the proper one while we have the opposite opinion.

So the controversy left is for perfect metals of infinite extension. One of our arguments is that this special case is a limiting case of more realistic models with relaxation. As mentioned earlier, this latter limit can come in conflict with the third law of thermodynamics. But even in this limit the possible violation is ambiguous or not obvious. On one hand the entropy remains zero for temperature in the limit . On the other hand taking the limit with will result in entropy different from zero. The latter is connected to the increasing and diverging slope of the entropy function as close to .

However, for realistic models for metals with relaxation (or finite conductivity) this controversy is not present. As we have found earlier the entropy then goes smoothly towards zero as . Bezerra et al. Bezerra:2005hc correctly point to the fact that the so-called MIM (modified ideal metal) violates the third law of thermodynamics. (This we stated explicitly in Ref. Hoye:2002at .) This violation is then in the sense discussed above, i.e., it is somewhat ambiguous depending upon how the limiting process is performed. It is also correct that according to our approach real metals approach MIM for increasing separation as then lower frequencies become more important. (MIM is the limit where the dielectric constant with the TE zero-mode equal to zero.) But the crucial difference between real metals and MIM is that the former includes relaxation by which there will be no violation of the third law of thermodynamics (at least for finite separation of metal plates and for infinite separation where there is no force or interaction energy left). Moreover, numerical studies have shown that the MIM model disagrees strongly at short distances with values for the Casimir force calculated for real metals even for zero temperature.

Bezerra et al. Bezerra:2005hc further claim that our unstated assumption is that perfect crystals with no defects or impurities do not or cannot exist, and, therefore, that thermodynamics can be violated for them. At best this is a misinterpretation of our arguments. There is no violation of the third law in this connection as quantization of lattice vibrations on a perfect lattice yields no problem; in fact it can be performed exactly. A reason for this is that the standard lattice has a finite dielectric constant and there is no ambiguity. However, if the perfect lattice turns into a perfect metal (with no relaxation) we can again see the problem discussed above. Then there will be no well defined thermal equilibrium as any steady current can stay on forever. And we do not think that Nernst had this latter very special and unrealizable situation in mind when he formulated his theorem.

Thus, we believe that the results obtained by excluding the TE zero mode are consistent with the third law of thermodynamics. The entropy at zero temperature vanishes, except in the limit of a perfect conductor of infinite extent, for which thermal equilibrium can never occur. There is a region in which the Casimir entropy is negative, but this is a rather familiar phenomenon, reflecting the fact that we are describing only part of the complete physical system. (Although Svetovoy and Esquivel svetovoy and Sernelius sernelius agree with our conclusion that the entropy vanishes at zero temperature for real metals with no TE zero mode, and thus there is no contradiction with the Nernst theorem, the former authors believe that there is a thermodynamic difficulty with the negative entropy region.)

## Iv Theoretical support

Two other recent papers also lend support to our point of view. Jancovici and Šamaj jancovici and Buenzli and Martin buenzli have examined the Casimir force between ideal-conductor walls with emphasis on the high-temperature limit. Not surprisingly, ideal inert boundary conditions are shown to be inadequate, and fluctuations within the walls, modeled by the classical Debye-Hückel theory, determine the high temperature behavior. The linear in temperature behavior of the Casimir force is found to be reduced by a factor of two from the behavior predicted by an ideal metal. This is precisely the signal of the omission of the TE mode. Thus, it is very hard to see how the corresponding modification of the low-temperature behavior can be avoided.

Further support for our conclusions can be found in the very recent paper of Sernelius sernelius05 , who calculates the van der Waals-Casimir force between gold plates using the Lindhard or random phase approximation dielectric function. Spatial dispersion plays a crucial role in his calculations. For large separation, the force is one-half that of the ideal metal, just as in the calculations in jancovici ; buenzli . For arbitrary separations between the plates, Sernelius’ results numerically nearly exactly coincide with his earlier ones sernelius01 ; bostrom where dissipation (i.e., nonzero relaxation parameter) is included but no spatial dispersion. In his new calculation, the inclusion of dissipation has negligible additional effect. His new results thus essentially coincide with ours.

## V Experimental constraints

Thus, we claim that there is now overwhelming theoretical evidence that a TE zero-mode should not be included in the Lifshitz formula when describing the interaction between real metal surfaces. This makes the purported exclusion of this theory by recent experiments, as forcefully stated in Refs. Bezerra:2005hc ; decca , most difficult to understand. We believe that the resolution of this conundrum lies in an insufficient appreciation of the backgrounds making Casimir measurements so difficult. We and others iannuzzi have mentioned the difficulty in accurately determining the absolute sphere-plate separation. This may be especially so since the roughness of the surfaces is much larger than the precision stated in the determination of the separation. Nonlocality in the electric properties of the surfaces, as for example seen in the calculation of Ref. jancovici , means that the location of the surface cannot be specified to better than some effective shielding length. Interferometric methods of determining distance may also be somewhat uncertain because of penetration of the surface by light. Also accurate determination of a small difference between experimental values at room temperature and purely theoretical values at gives rise to further difficulties. In short, fitting data precisely to a preferred theory wherein various unknown experimental parameters must be determined does not constitute decisive evidence in favor of that theory; nor does the poorness of fit to an alternative theory when only one of many parameters is allowed to vary constitute decisive evidence against that theory.

In our view, the issue of temperature dependence cannot be settled until experiments are able to detect a variation of the Casimir force with temperature. Clearly, such experiments are difficult. The precise experiments at very short distances ( nm), where the Casimir forces are largest, are not the best place to look for temperature variation, for the temperature dependence is relatively small there. Rather, experiments should be conducted at the micrometer scale, where the effects, if our theory is correct, are above the 10% level, and approach a factor of two reduction from the ideal metal limit for larger separations. (The Lamoreaux experiment lamoreaux , conducted at the micrometer separation scale, was probably not as accurate as claimed.) Proposals to perform measurements of the force between a cylinder and a plane brown-hayes and between eccentric cylinders dalvit have advantages because the forces are stronger than between a sphere and a plane, yet the difficulties in maintaining parallelism is not so severe as with two plane surfaces. In this way the possiblitity of measuring the temperature dependence at relatively large distances where thermal effects are largest may be accomplished.

## Vi New calculations

To this end, we present some new calculations of the force between parallel plates based on our theory, compared with that of Ref. decca . The numerical computations are done on the basis of Eq. (4.18) in our paper Hoye:2002at , and follow the recent paper of Bentsen et al. Bentsen:2005yi . The maximum value of the quantity , being the plate separation, is chosen to be . In all calculations we impose the tolerance for the integrals to be . The overall tolerance in the calculated sum over Matsubara numbers is taken to be . This last tolerance determines the highest Matsubara frequencies occurring in the sum. These values are shown in the tables.

As for permittivities, we are using the (new) data received from Astrid Lambrecht. These data extend from about rad/s to about rad/s. An important virtue of our tables is that they show which frequency domains we actually use. The largest frequency region naturally occurs when the temperature is low. Thus for K, the temperature that we associate with on physical grounds, we use the region from about rad/s to about rad/s. That means, we are, even at this low temperature, working with frequencies that lie entirely within the region of Lambrecht’s data. This implies that we do not have to involve the Drude relation at all, for the finite frequencies. Of course, the Kramers-Kronig relation, required by causality, is employed. There is no analytical approximation involved in our formalism, for any finite frequency. The only exception is the zero frequency case. We then need the Drude relation to assure that there is no contribution to the force from the zero frequency TE mode. This zero frequency contribution to the Casimir force is found analytically, not numerically.

For comparison, we show in Table 1 the results given in Ref. decca for the Leontovich impedance and plasma models. These results are for zero temperature. In their approach the temperature dependence is negligible. It will be noted that their two models do not agree, even though the authors seem to imply that either model is equally good. Our view is that the difference between these two models may be taken as a rough gauge of the accuracy of their predictions. Our results, with details about the frequency range used and the number of terms in the Matsubara sum, are given in Tables 2–4 for K (sufficiently close to K, but a temperature at which our numerical technique is stable), K, and K. The latter is given in the hope that the temperature variation over a 50 K range near room temperature may soon become accessible to experiment.

Separation | Impedance Method | Plasma Method |
---|---|---|

nm | Ref. decca | Ref. decca |

160 | 1144 | 1114.9 |

200 | 509.3 | 501.8 |

250 | 224.7 | 223.1 |

400 | 38.90 | 38.98 |

500 | 16.70 | 16.76 |

700 | 4.605 | 4.628 |

Separation | Pressure | Highest frequency | Number of terms |
---|---|---|---|

nm | mPa | rad/s | in sum |

160 | 1144 | 25674 | |

200 | 508.2 | 20824 | |

250 | 223.7 | 16869 | |

400 | 38.61 | 10789 | |

500 | 16.56 | 8714 | |

700 | 4.556 | 6305 | |

1000 | 1.143 | 4466 |

Separation | Pressure | Highest frequency | Number of terms |
---|---|---|---|

nm | mPa | rad/s | in sum |

160 | 1127 | 86 | |

200 | 497.8 | 70 | |

250 | 217.6 | 57 | |

400 | 36.70 | 36 | |

500 | 15.49 | 30 | |

700 | 4.127 | 22 | |

1000 | 0.9852 | 15 |

Separation | Pressure | Highest frequency | Number of terms |
---|---|---|---|

nm | mPa | rad/s | in sum |

160 | 1124 | 74 | |

200 | 495.7 | 60 | |

250 | 216.4 | 49 | |

400 | 36.35 | 31 | |

500 | 15.30 | 25 | |

700 | 4.052 | 19 | |

1000 | 0.9590 | 13 |

## Vii Conclusions

Were the controversy over the temperature dependence a purely theoretical one, we believe that a consensus along the lines we have been advocating, justified by realistic model calculations by Jancovici and Šamaj jancovici , Buenzli and Martin buenzli , and Sernelius sernelius05 , would now be nearly universally accepted. Evidently what has prevented this consensus from developing is the apparent conflict that the resulting predictions have with experiment. However, we believe it is premature to regard that conflict as decisive. It is imperative to perform room-temperature experiments at larger separations, since the relative temperature corrections are then much larger. The optimum separation seems to lie around 2 m ba . Indeed, the early experiments of Lamoreaux lamoreaux support the Drude theory at large separation, although the error bars are large. The experimental situation when is about 1 m or less (which is the main region of the Decca experiment decca ) may be influenced by extra factors coming into play at short distances, such as surface plasma effects il . At short distances, the results are moreover very sensitive to differences in separation; an error of 1 nm may give rise to errors in the pressure of about 10 mPa or 2% at nm.

###### Acknowledgements.

We thank V. M. Mostepanenko and G. L. Klimchitskaya for alerting us to some numerical discrepancies in the data we had used in an earlier version of this paper. We also thank the other participants of the Seventh Workshop on Quantum Field Theory Under the Influence of External Conditions in Barcelona for numerous helpful comments. We thank Astrid Lambrecht for supplying us with new data. We are grateful to R. Cowsik for conversations on the difficulty of Casimir force measurements. And we thank Bo Sernelius for making his preprint sernelius05 available to us. The work of K. A. M. is supported in part by grants from the US Department of Energy. He also thanks the Department of Physics of Washington University for its hospitality and support.## References

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