aa r X i v : . [ qu a n t - ph ] D ec Temperature dependence of the Casimir force
Iver Brevik and Johan S Høye Department of Energy and Process Engineering, Norwegian University of Scienceand Technology, N-7491 Trondheim, Norway. Department of Physics, Norwegian University of Science and Technology, N-7491Trondheim, Norway.E-mail: [email protected]
E-mail: [email protected]
Abstract.
The Casimir force - at first a rather unexpected consequence of quantumelectrodynamics - was discovered by Hendrik Casimir in Eindhoven in 1948. It predictsthat two uncharged metal plates experience an attractive force because of the zero-pointfluctuations of the electromagnetic field. The idea was tested experimentally in the1950’s and 1960’s, but the results were not so accurate that one could make a definiteconclusion regarding the existence of the effect. Evgeny Lifshitz expanded the theoryin 1955 so as to deal with general dielectric media. Much experimental work has laterbeen done to test the theory’s predictions, especially with regards to the temperaturedependence of the effect. The existence of the effect itself was verified beyond doubt bySabisky and Anderson in 1973. Another quarter century had to pass before Lamoreauxand collaborators were able to confirm - or at least make plausible - the temperaturedependence predicted by Lifshitz formula in combination with reasonable input data forthe material’s dispersive properties. The situation is not yet clear-cut, however; thereare recent experiments indicating results in disagreement with those of Lamoreaux. Inthe present paper a brief review is given of the status of this research field.
1. Introduction
Let us begin by citing Hendrik B. G. Casimir himself:”Inside a metal there are forces of cohesion and if you take two metal plates andpress them together these forces of cohesion begin to act. On the other hand you canstart from one piece and split it. Then you have first to break chemical bonds and nextto overcome van der Waals forces of classical type, and if you separate the two pieceseven further there remains a curious little tail. The Casimir force, sit venia verbo, isthe last but also the most elegant trace of cohesion energy”.This extract is taken from Casimir’s modestly formulated introductory talk at theFourth Workshop on Quantum Field Theory under the Influence of External Conditions(QFEXT98), held in Leipzig in September 1998 [1]. One of us was fortunate enoughto be attending this remarkable event. Casimir was then almost 90 years old, andthe workshop quite appropriately took the opportunity to celebrate the 50th years’anniversary of Casimir’s pioneering paper published in 1948 [2]. The last-mentionedpaper gave a very simple derivation based upon quantum electrodynamics of how theattractive force between two neutral parallel metal plates at small separation can beenvisaged as a result of the change of electromagnetic field energy in the region betweenthe plates. Casimir used to emphasize that the very idea of linking electromagneticfield energy to mechanical forces, in principle observable, was brought up during aconversation with Niels Bohr in Copenhagen in 1946 or 1947: After Casimir had toldBohr about his latest works on van der Waals forces, Bohr thought this over, and thenmumbled something like ”this must have something to do with zero-point energy”. Thatwas all, but in retrospect Casimir said he owed much to this remark.So, in a strict sense and in conformity with the above statement of Casimir in hisintroductory talk, one might say that the Casimir effect concerns the case of relativelylarge separations between plates only (called the ”tail” above), where the so-calledretardation effects due to the finite velocity of light play a role. If that view were tobe upheld, the practical importance of the Casimir effect would be rather limited. Incommon usage the Casimir effect has however been taken to mean also cases where theseparations between media are small. That means, one also incorporates situationsin which retardation effects are unimportant. The latter class of phenomena goestraditionally under the name of van der Waals forces. Thus, Casimir forces and vander Waals forces are concepts used for the most part interchangeably nowadays. As isknown, these kind of forces are the dominant interactions between neutral particles onnanometer to micrometer length scales. This makes the effects ubiquitous in physics,chemistry, and also biology. The effects are encountered, for instance, in the action ofdetergents, in the self-assembly of viruses, and even in the abilities of geckoes to climbflat surfaces.Elementary introductions to the Casimir effect can be found in many books onquantum mechanics and quantum electrodynamics, for instance, that of Power [3].Readers interested in more advanced treatises may consult books of Bordag et al. [4],of Milton [5], or also extensive review articles of Milton [6] and of Plunien et al. [7]. Anice presentation of regularization schemes for the Casimir effect was given by Reuterand Dittrich in this journal [8].Whereas the main properties of the Casimir effect are well known by now, thereare issues related to the temperature dependence of the effect that are still insufficientlyunderstood and subject to lively discussion in the contemporary literature. We havetherefore found it useful to give a brief review of the state of art in this field, at a levelthat we think should be accessible for general physicists as well as for graduate students.Also, undergraduates ought to be able to follow the essentials from our presentationbelow.
2. Basic theory. The Lifshitz formula
The typical Casimir setup is illustrated in Fig. 1; two parallel metal plates areseparated by a gap of width a . We shall assume the plates to be nonmagnetic. The a ε εε =1 Figure 1.
Definition sketch: Vacuum gap of width a between two nonmagnetic plateswith permittivity ε where metals correspond to ε → ∞ . electric field between the plates must satisfy the boundary conditions, saying thatthe electric field component parallel to the metal surfaces is equal to zero. Thisimplies that the electromagnetic field will have discrete eigenfrequencies analogous tothe eigenfrequencies of a violin string. These oscillations are quantized as harmonicoscillators which have ground state energy ~ ω/ ω is the angular frequencyand ~ = h/ (2 π ) the reduced Planck’s constant. The difference in ground state energybetween the cases where a is infinite and where a is finite leads to an attractive forcebetween the plates. This is the Casimir force. Casimir found that the force per unitsurface area (the pressure) for metals at zero temperature is f c = − π ~ c a (1)(negative sign means an attractive force).The Casimir force has turned out to be quite difficult to measure experimentally.This is understandable, all the time that the force is small - according to equation (1)only 1.3 mPa (about 10 − atmospheres) when the separation is a = 1 µ m.In 1955 Lifshitz [9] derived a more general expression for the Casimir force betweentwo identical parallel dielectric plates of (relative) permittivity ε . As before, we assumethe media to be nonmagnetic. The expression also holds for finite absolute temperatures T . The expression is relatively complicated, f c = − k B Tπ ∞ X m =0 ′ ∞ Z ζ m q dq (cid:20) A m e − qa − A m e − qa + B m e − qa − B m e − qa (cid:21) . (2)Here ζ m = 2 πmk B T / ~ with m integer are called the Matsubara frequencies, and playan important role when T is finite. The ζ m = − iω m (or ζ m = iω m , dependent on whichconvention is used) are caused to be discrete because of quantum mechanics. Moreover, c is the velocity of light in vacuum, k B is Boltzmann’s constant, and the prime on thesummation sign means that the term with m = 0 is to be taken with half weight. Theconstants A m and B m are defined by A m = (cid:18) εp − sεp + s (cid:19) , B m = (cid:18) s − ps + p (cid:19) , (3)with s = ε − p , p = qcζ m . (4)Expression (2) makes use of imaginary frequencies. As the physical frequencies are ω m = iζ m , the permittivity ε is represented as ε = ε ( iζ m ). The coefficients A m and B m are the squares of the reflection coefficients for respectively the TM (transversemagnetic) and the TE (transverse electric) electromagnetic waves in the region betweenthe plates. In each of the two cases either the magnetic field, or the electric field, areparallel to the plates. Expression (1) holds for all densities of the material, and for all T ≥ • in the prefactor of the sum; • in the lower limit ζ m of the integral, and • in a possible temperature dependence of the dissipation parameter ν ; see equation(11) below.We shall in the following consider the case of metals. This has conventionally beentaken to mean that ε ( iζ m ) → ∞ . At first one might think that this case is unproblematic:simply plug in the appropriate value of ε and calculate the sum and integral in (2)analytically or numerically. However, here one encounters a mathematically delicateproblem, for the TE mode in the limit of zero frequency. Specifically,if first ε → ∞ and then ζ → , the B → , while (5)if first ζ → ε → ∞ , the B → . (6)These two options have given rise to two different models of a metal, namely the IdealMetal-model (IM), and the Modified Ideal Metal model (MIM). We will now considerthese two cases more closely.
3. Ideal metals
This served as the standard model for the Casimir effect of metals for several years.The IM model was introduced in the classic paper of Schwinger et al. from 1978 [10]. Itfollows option (5) above, and means that the contributions from the TE and TM modesare equal to each other, including the case of zero frequency, A m = B m = 1 , m = 0 , , , , ... (7)Mathematically, the integral in equation (2) can be calculated analytically when T = 0.Here we give the results for T = 0, and include the corrections for low temperatures.The Casimir force takes the form f c = − π ~ c a " (cid:18) k B T a ~ c (cid:19) , ak B T ~ c ≪ . (8)For the free energy F per unit surface, determined by f c = − ∂F/∂a , the result is F = − π ~ c a " ζ (3) π (cid:18) ak B T ~ c (cid:19) − (cid:18) ak B T ~ c (cid:19) , ak B T ~ c ≪ , (9)where ζ (3) means the Riemann zeta-function with 3 as argument. (Actually the middleterm in this equation, independent of a , requires separate attention; cf., for instance,Refs. [5] or [14].)Finally, we shall be interested in the entropy S , which is given by thethermodynamic relation S = − ∂F/∂T . We get S = 3 k B ζ (3)2 π (cid:18) k B T ~ c (cid:19) − k B π a (cid:18) k B T ~ c (cid:19) , ak B T ~ c ≪ . (10)From this it is seen that S = 0 when T = 0. This is Nernst’s theorem, also called thethird law of thermodynamics. (Actually it is stated more correctly by saying that S =constant, independent of other parameters, at T = 0.) This theorem will be a centralpoint in the present discussion. The IM model thus satisfies this basic requirement fromthermodynamics right away.
4. Modified ideal model, and its generalizations
In view of the satisfactory behavior of the IM model noted above, one may ask: Whyshould there be any reason for changing this model at all? A problem is that a realmaterial has to satisfy a realistic dispersion relation. In practice, the following dispersionrelation, called the Drude relation, is followed by metals to a reasonably good accuracy, ε ( iζ ) = 1 + ω p ζ ( ζ + ν ) . (11)Here ω p is the plasma frequency, and ν is the dissipation parameter (describing ohmicresistance in the metal). In all real metals, ν stays finite, this being related to impuritieswhich are always present. It turns out that the Drude relation very accurately fits opticalexperimental data for ζ < × rad s − [11, 12]. A typical example is gold, for which ω p = 9 .
03 eV, ν = 0 . ζ → ζ [ ε ( iζ ) −
1) = 0 , (12)the zero-frequency TE mode does not contribute to the Casimir force. The firstto emphasize this kind of behavior were Bostr¨om and Sernelius [13], and detaileddiscussions were given in [14] and [15]. There are several other papers arguing alongsimilar lines. From a different viewpoint Jancovici and Samaj [16] and Buenzli andMartin [17] considered a classical plasma of free charges in the high-temperature limit,and found the linear dependence in T in the Casimir force to be reduced by a factor of2 from the IM model prediction.According to the information coming from the dispersion relation we thus ought touse A = 1 , B = 0 , (13)as input values in the Lifshitz formula (2). At first sight the above equations (8), (9)and (10) are then replaced by f c = − π ~ c a " (cid:18) ak B T ~ c (cid:19) + k B T πa ζ (3) , (14) F = − π ~ c a " ζ (3) π (cid:18) ak B T ~ c (cid:19) − (cid:18) ak B T ~ c (cid:19) + k B T πa ζ (3) , (15) S = 3 k B ζ (3)2 π (cid:18) k B T ~ c (cid:19) − k B π a (cid:18) k B T ~ c (cid:19) − k B ζ (3)16 πa . (16)The most striking property of these expressions is that S (0) = − k B ζ (3) / (16 πa ), thusviolating Nernst’s theorem.A lively discussion on this point has taken place in the literature. Arguments haveeven been given to give up the Drude dispersion model as such and replace it withthe ”plasma model” which effectively means setting the dissipation parameter ν in (11)equal to zero. Discussions along these lines can be found, for instance, in [18]. Likemany other researchers we think, however, that such changes of the electrodynamictheory of media should be avoided. Rather, more accurate calculations are needed. Theclash with Nernst’s theorem is a consequence of over-idealized assumptions. It appearsnatural to perform more accurate calculations of the expressions (14)-(16), within theframework of the Drude model, taking into account measured values of ε ( iζ ) and ν .Thus equation (11) is used as basis, for small ζ . Such calculations were actual done,and reported in [14, 19, 20], for the case of gold. Figure 2 shows the free energy versustemperature for low T . The linear term actually changes into a parabolic form withhorizontal slope at T = 0. The Nernst theorem is thus not broken after all.The story is however many-facetted. Without going into great detail here, wemention that the property of the entropy becoming negative for small T may appeardisturbing. This is actually related to the circumstance that Casimir quantities representphysical subsystems only. Therefore they are not subject to the usual thermodynamicrestrictions that hold for closed systems. A detailed analysis of this point, making useof a harmonic oscillator model, is given in [14]. Temperature (K) -2 mFree energy pr. unit area (J ) Plate material: GoldSeparation: 1000 nm
Temperature (K) 800600400200-3.8-4.0-3.0-3.2-3.4-3.6-2.8 0.5Free energy -2 ) (Jm-3.9182-3.9179x 10 -10 x 10 -10 Figure 2.
Free energy between gold plates as a function of temperature. The insetshows the variation for small T . From Ref. [20]. One special effect of the negative contribution to the entropy is that the Casimirforce for metals (or more generally for large values of ε ) decreases with increasingtemperature in a certain temperature interval before it again increases to reach theclassical limit for T → ∞ where only the m = 0 term in (2) contributes. Thisbehavior is illustrated in Fig. 3. Here the Casimir force, multiplied with the factor a for convenience, is shown versus a at constant temperature T = 300 K. Since the forceessentially depends on the product aT (strictly speaking this is true for nondispersivemedia only), the figure effectively shows how the force varies versus T for a fixed valueof a . The approximately linear decrease between 1 and 3 µ m is clearly shown, as is thelinearly increasing curve for a > µ m. (The large deviation from the linear behaviorbelow 1 µ m is due to the dispersion.) −27 a( µ m) − a F T ( N m ) T = 300 K
Figure 3.
Casimir pressure for gold plates, multiplied with a , as function of a when T = 300 K. From Ref. [14].
5. On experiments
As mentioned above, the Casimir attractive force is small, in practice much smallerthan the electrostatic force due to the so-called ”patch potentials” on the metallic testbodies, and the influence from the latter kind of forces has to be eliminated by calibrationprocedures. This is quite a demanding task for the experimentalists. Usually one willmeasure the force between a microsphere and a plane, instead of between two planes,because of the strict restriction to geometric parallelism in the latter case.One might think: Would it not be possible in principle to find the temperaturedependence of the Casimir force simply by measuring the force at some temperature T and then repeat the measurement at some other temperature T + ∆ T ? However, theexperimentalists tell us that this is not possible in practice, because of disturbances andlack of stability. So all experiments to date have been carried out at room temperature.We mentioned above the importance of the parameter aT . This means that at roomtemperature measurements at large distances a will be of great interest in connectionwith the temperature dependence of the effect as then dispersion plays a decreasing role.The problem, of course, is that at large gap widths the force becomes much smaller thanit is at a typical width of 1 µ m.The Casimir force was first definitely confirmed for dielectrics by Sabisky andAnderson in 1973 [21]. A quarter of a century later, Lamoreaux demonstrated thatthe Casimir theory for metal plates held true [22]. The measurements have later beenreproduced by several others. In our context a most valuable property of the Lamoreauxexperiment is that it was carried out at large distances. Lamoreaux also was involvedin the newer version of this experiment [23] (see also Milton’s comments in [24]), wheredistances a between 0.7 µ m and 7 µ m were tested. Quite remarkable, the theoreticalpredictions based upon the Drude model were found to agree with the observed resultsto a high accuracy.If this experiment stands the test of time, it will be important as it helps usunderstand better the electromagnetic and thermodynamic properties of real materialsas well as the underlying quantum vacuum. The need for making large subtractionsbecause of the mentioned patch potentials implies however uncertainties in theinterpretation of the data in this experiment. However there are other experiments,like as the very accurate one of Decca [18] carried out at small separations, which yieldresults apparently in accordance with the plasma model ( ν = 0) rather than the Drudemodel. The reason for this conflict between experimental results is not understoodin the community. One might suggest that the explanation has to do with the effectcalled Debye shielding, known from solid state physics and plasma physics, which canchange the effective gap with between plates from the geometrically measured width.But people doing the experiments tell us that such explanations seem unlikely. Also,due to the atomic structure, surfaces are not sharply defined. After all, and perhapssurprisingly, we can hardly do anything else than to conclude that the thermal Casimireffect has managed to escape from an unambiguous explanation for quite a long time.One might only hope, that when the explanation eventually turns up, it will reflect somedeep physical property and not merely a triviality! References [1] Casimir H B G 1998 Preface, in
The Casimir Effect 50 Years Later , Proceedings of the FourthWorkshop on Quantum Field Theory under the Influence of External Conditions, edited byMichael Bordag (Singapore: World Scientific)[2] Casimir H B G 1948
Proc. K. Ned. Akad. Wet. Introductory Quantum Electrodynamics (London: Longmans), Appendix 1[4] Bordag M, Klimchitskaya G L, Mohideen U and Mostepanenko V M 2009
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