Some Mathematical Aspects of the Lifshitz Formula for the Thermal Casimir Force

Transcription

1 Some Mathematica Aspects of the Lifshitz Formua for the Therma Casimir Force, A. O. Caride, G. L. Kimchitskaya and S. I. Zanette Centro Brasieiro de Pesquisas Físicas, Rio de Janeiro, Brazi E-mai: The appicabiity of the Lifshitz formua is discussed to the case of two thick parae pates made of rea meta. The usua description of the zero-point vacuum osciations on the background of the frequency-dependent dieectric permittivity is shown to be in contradiction with thermodynamics. Instead, the Lifshitz formua for the Casimir free energy shoud be reformuated in terms of the refection coefficients containing the surface impedance instead of the dieectric permittivity. This approach is presenty confirmed experimentay by precision measurements of the van der Waas and Casimir forces in micromechanica systems and it is in agreement with thermodynamics. Fourth Internationa Winter Conference on Mathematica Methods in Physics 9-13 August 24 Centro Brasieiro de Pesquisas Fisicas (CBPF/MCT), Rio de Janeiro, Brazi Speaker. On eave from Noncommercia Partnership Scientific Instruments, Moscow, Russia. On eave from North-West Technica University, St.Petersburg, Russia. Pubished by SISSA

2 Mathematica Aspects of the Lifshitz Formua 1. Introduction It is common knowedge that Lifshitz formua [1] describes the van der Waas and Casimir force acting between two thick pane parae materia pates separated by a gap of width a. According to this formua, the free energy of the van der Waas and Casimir interaction can be represented in terms of refection coefficients F R = k BT = { [ ] [ ]} n 1 r 2 (ξ,k )e 2aq + n 1 r 2 (ξ,k )e 2aq. (1.1) Here prime means the addition of a mutipe 1/2 near the term with =, and the Lifshitz refection coefficients take the form r 2 (ξ,k ) r,l 2 (ξ ε q k 2,k ) =, ε q + k r 2 (ξ,k ) r,l 2 (ξ q k 2,k ) =, (1.2) q + k where ε ε(iξ ), ε(ω) is the dieectric permittivity of the pate materia, ξ = k B T/ h ( =,1,2,...) are the Matsubara frequencies, and k 2 k 2 + ε ξ 2 /c2, q 2 k 2 + ξ2 /c2. Beginning in 2, the behavior of the therma correction to the Casimir force between rea metas has been hoty debated. It was shown that Lifshitz formua eads to different resuts depending on the mode of meta conductivity used. For rea metas at ow frequencies ω, the dieectric permittivity ε varies as ω 1. After substituting ε ω 1 into the Lifshitz formua, the resut is a therma correction which is severa hundred times greater than for idea metas at separations of a few tenths of a micrometer [2, 3] The attempt [4] to modify the zero-frequency term of the Lifshitz formua for rea metas, assuming that it behaves as in the case of idea metas, aso eads to a arge therma correction to the Casimir force at short separations. It is important to note that in the approaches of both [2, 3] and aso of [4] a thermodymanic puzze arises, i.e., the Nernst heat theorem is vioated for a perfect attice [5, 6]. (See aso [7] where it is shown that for the preservation of the Nernst heat theorem in the approach of [2, 3] it is necessary to have metas with defects or impurities; it is common knowedge, however, that thermodynamics must be vaid for both perfect and imperfect attices.) This puzze casts doubt on the many appications of the Lifshitz theory of dispersion forces, and thus represents a potentiay serious chaenge to both experimenta and theoretica physics. By contrast, the use of ε ω 2, as hods in a free eectron pasma mode negecting reaxation, eads [8, 9] to a sma therma correction to the Casimir force at short separations. This is in quaitative agreement with the case of an idea meta and is consistent with the Nernst heat theorem. It shoud be borne in mind, however, that the pasma mode is not universa, and is appicabe ony in the case when the characteristic frequency is in the domain of infrared optics. The present paper demonstrates that the main reason why the Drude mode in combination with the Lifshitz theory had faied to describe the therma Casimir force is the inadequacy of the standard concept of a fuctuating eectromagnetic fied on the background of ε depending ony on frequency inside a ossy rea meta. To avoid a contradiction with thermodynamics, one shoud use the refection coefficients expressed in terms of the surface impedance. / 2

3 Mathematica Aspects of the Lifshitz Formua 2. The fuctuating fied and the surface impedance The concept of a fuctuating eectromagnetic fied works we for the description of zeropoint osciations in media with a frequency-dependent dieectric permittivity where no rea eectric current does arise. We wi now consider a conductor in an externa eectric fied, which varies with some frequency ω satisfying the conditions δ n (ω), v F ω, (2.1) where is the mean free path of a conduction eectron, δ n (ω) = c/ σω is the penetration depth of the fied inside a meta, σ is the conductivity, and v F is the Fermi veocity. Eqs. (2.1) determine the domain of the norma skin effect [1]. In this frequency region the externa fied eads to the initiation of a rea current of the conduction eectrons and the dieectric permittivity is modeed by the Drude function ε ω 1 eading to the difficuties with the Lifshitz formua mentioned in Introduction. The physica reason for these difficuties becomes cear when one observes that the aternating eectric fied with frequencies characteristic for the norma skin effect inevitaby eads to heating of a meta when it penetrates through the skin ayer. By contrast, the therma photons in therma equiibrium with a meta pate or the virtua photons (giving rise to the Casimir effect) can not ead to the initiation of a rea current and heating of the meta (this is prohibited by thermodynamics). Hence the concept of a fuctuating eectromagnetic fied penetrating inside a meta cannot describe virtua and therma photons in the frequency region (2.1). As a consequence, the Lifshitz formua can not be appied in combination with the Drude dieectric function in the domain of the norma skin effect. As is evident from the foregoing, another theoretica basis is needed to find the therma Casimir force between rea metas different from the approach used in the case of dieectrics. Here we show that this basis is given by the surface impedance boundary conditions introduced by M. A. Leontovich [1, 11]. The fundamenta difference of the surface impedance boundary conditions from the other approaches is that they permit not to consider the eectromagnetic fuctuations inside a meta. Instead, the foowing boundary conditions are imposed taking into account the properties of rea meta E t = Z(ω)[B t n], (2.2) where Z(ω) = 1/ ε(ω) is the Leontovich surface impedance of the conductor, E t and B t are the tangentia components of eectric and magnetic fieds, and n is the unit norma vector to the surface (pointed inside a meta). The boundary condition (2.2) can be used to determine the eectromagnetic fied outside a meta. Note, that the impedance Z(ω) and the condition (2.2) suggest a more universa description than the one by means of ε. They sti hod in the domain of the anomaous skin effect where a description in terms of the dieectric permittivity ε(ω) is impossibe. For idea metas it hods Z. The use of the Leontovich impedance in Eq. (2.2) which does not depend on the poarization state and transverse momentum, is of prime importance. Note that in [12] the exact impedances depending on a transverse momentum were used. This has ed to the same concusions as were obtained previousy from the Lifshitz formua combined with the dieectric permittivity ε ω 1. / 3

4 Mathematica Aspects of the Lifshitz Formua As was aready mentioned above, these concusions are in vioation of the Nernst heat theorem for a perfect attice [5, 6, 7]. Athough a recent review [12] caims agreement with the Nernst heat theorem in [2, 3], no specific objections against the rigorous anaytica proof of the opposite statement in [6] are presented. The faacy in the cacuations of [12] concerning the type of the impedance is that it disregards the requirement that the refection properties for virtua photons on a cassica boundary shoud be the same as for rea photons. Paper [6] demonstrates in detai that by enforcing this requirement the exact and Leontovich impedances coincide at zero frequency and ead to the concusions of [13] which are in perfect agreement with the Nernst heat theorem. 3. Lifshitz formua in terms of surface impedance Let us consider the case of rea eigenfrequencies ω k,n, ω k,n (i.e., the pure imaginary impedance). The tota free energy of the eectromagnetic osciations is given by the sum of the free energies of osciators over a possibe vaues of their quantum numbers, [ hωα ) F = α 2 + k BT n (1 ] ( e hωα k B T = k B T n 2sinh hω ) α. (3.1) α 2k B T At T, the vaue of F from Eq. (3.1) coincides with the sum of the zero-point energies which is usuay considered at zero temperature. Appying this to the eectromagnetic osciations between meta pates, where α = {p,k,n}, and p = or abes the poarization states, we obtain F = k B T n n 2sinh hω k,n 2k B T + n ( 2sinh hω k,n 2k B T ). (3.2) Using the impedance boundary conditions, it can be easiy shown that the eigenfrequencies of the eectromagnetic fied between pates with parae and perpendicuar poarizations are determined by the equations (ω,k ) 1 ( 2 e aq 1 η 2)( sinhaq 2iη ) 1 η 2 coshaq =, (3.3) (ω,k ) 1 2 e aq ( 1 κ 2)( sinhaq + I 1 n i C 1 2iκ ) cosh aq 1 κ2 =, (3.4) where η = η(ω) = Zω/(cq), κ = κ(ω) = Zcq/ω, and q 2 = k 2 ω2 /c 2. The expression in the right-hand side of Eq. (3.2) is evidenty divergent. Before performing a renormaization, et us equivaenty represent the sum over the eigenfrequencies ω, k,n by the use of the argument theorem [14]. Then Eq. (3.2) can be rewritten as F = k B T 2sinh d [ n (ω,k ) + n (ω,k ) ]. (3.5) hω 2k B T Here, the cosed contour C 1 is bypassed countercockwise. It consists of two arcs, one having an infinitey sma radius ε and the other one an infinitey arge radius R, and two straight ines L 1, L 2 incined at the anges ±45 degrees to the rea axis. The quantities, (ω,k ) have their roots at / 4

5 Mathematica Aspects of the Lifshitz Formua the photon eigenfrequencies and are defined in Eqs. (3.3) and (3.4). Unike the usua derivation of the Lifshitz formua at T [15] the function under the integra in (3.5) has branch points rather than poes at the imaginary frequencies ω = iξ. The contour C 1 is chosen so as to avoid a these branch points and encose a the photon eigenfrequencies. After the integration by parts and some rearrangement [13], we find the equivaent but more simpe expression for the Casimir free energy F = k BT = [ n (ξ,k ) + n (ξ,k ) ]. (3.6) Expression (3.6) is sti infinite. To remove the divergences, we subtract from the right-hand side of Eq. (3.6) the free energy in the case of infinitey separated interacting bodies (a ). Then the physica, renormaized, free energy vanishes for infinitey remote pates. From Eqs. (3.3) and (3.4) after the substitution ω iξ in the imit a it foows (ξ,k ) = 1 ( 1 + η η ) 1 + η 2, (ξ,k ) = 1 ( 1 + κ κ ) 1 + κ 2. (3.7) The renormaization prescription is equivaent to the change of, (ξ,k ) in Eq. (3.6) for R, (ξ,k ), (ξ,k ), (ξ,k ) = 1 r2, (ξ,k )e 2aq, (3.8) where the quantities r, (ξ,k ) have the meaning of refection coefficients and are given by ( r 2 (ξ cq Z ξ,k ) = cq + Z ξ ) 2, r 2 (ξ,k ) = ξ Z cq 2. (3.9) ξ + Z cq Here Z Z(iξ ). The refection coefficients (3.9) are in accordance with [11] where the refection of a pane eectromagnetic wave incident from vacuum onto the pane surface of a meta was described in terms of the Leontovich surface impedance. Thus, the fina renormaized expression for the Casimir free energy in the surface impedance approach is given once more by the Lifshitz formua (1.1), where the refection coefficients are expressed in terms of impedance according Eq. (3.9). The above derivation was performed under the assumption that the photon eigenfrequencies are rea. This is, however, not the case for arbitrary compex impedance. If the photon eigenfrequencies are compex, the free energy is not given by Eq. (3.1) (which is aready cear from the compexity of the right-hand side of this equation). For arbitrary compex impedance the correct expression for the free energy shoud be determined from the soution of an auxiiary eectrodynamic probem [16]. In fact the Casimir free energy is the functiona of the impedance even when the impedance has a nonzero rea part taking absorption into account. The soution of the auxiiary eectrodynamic probem eads to concusion [16] that the correct free energy is obtained from Eqs. (1.1), (3.9) by anaytic continuation to arbitrary compex impedances, i.e., to arbitrary osciation spectra. The quaitative reason for the vaidity of this statement is that the free energy depends ony on the behavior of Z(ω) at the imaginary frequency axis where Z(ω) is aways rea. / 5

6 Mathematica Aspects of the Lifshitz Formua It must be emphasized that the surface impedance approach is in perfect agreement with thermodynamics. In the impedance approach the entropy, defined as S(a,T ) = F R(a,T ), (3.1) T is positive and equa to zero at zero temperature in accordance with the Nernst heat theorem (remind that this is not the case in the approaches based on the use of a frequency dependent dieectric permittivity). 4. Concusions and discussion In the above we demonstrated that the Lifshitz formua for the Casimir free energy can be derived starting from the boundary condition on the surface of rea meta containing the Leontovich impedance. In doing so there is no need to use the concept of a fuctuating eectromagnetic fied inside a meta. We argued that the standard concept of a fuctuating fied inside a meta, described by the dieectric permittivity depending ony on frequency cannot serve as an adequate mode for the zero-point osciations and therma photons. It foows from the fact that the vacuum osciations and therma photons in equiibrium can not ead to a heating of a meta as do the eectromagnetic fuctuations on the background of ε(ω). If this fact is overooked, contradictions with the thermodynamics arise when one substitutes into the Lifshitz formua the Drude dieectric function taking into account the voume reaxation and, consequenty, the Joue heating. In the impedance approach the entropy is in a cases nonnegative and takes zero vaue at zero temperature. Thus, Eqs. (1.1) and (3.9) ay down the theoretica foundation for the cacuation of the therma Casimir effect. In fact the approaches of [2, 3, 4] and the impedance approach of [6, 13] predict quite different magnitudes of the therma corrections to the Casimir force. Up to separation distances of a few hundred nanometers, the therma correction predicted by the impedance approach is negigiby sma. As to the therma corrections of [2, 3, 4], they may achieve severa percent of force magnitude. Recent experiment [17] on measuring the Casimir force by means of a micromechanica torsiona osciator is consistent with the theoretica predictions of the impedance approach. At the same time, the experimenta data of this experiment is in drastic contradiction with the approaches of [2, 3, 4]. The experiments on measuring the Casimir and van der Waas forces by means of an atomic force microscope aso suggest good opportunity to distinguish between the two approaches. To concude, different derivations of the Lifshitz formua for the Casimir free energy in the case of rea metas ead to one and the same mathematica expression containing, however, two different pairs of refection coefficients. Recent resuts demonstrate quite cear that for rea metas the use of refection coefficients, expressed in terms of the frequency-dependent dieectric permittivity, is not ony in vioation of thermodynamics but is aso in contradiction with experiment. By this reason the refection coefficients in terms of the Leontovich impedance are evidenty preferabe. Acknowedgments V.M.M. is gratefu to the participants of the Seminar at Centro Brasieiro de Pesquisas Físicas (CCP) for discussion. The authors acknowedge FAPERJ for financia support. / 6

7 Mathematica Aspects of the Lifshitz Formua References [1] E. M. Lifshitz and L. P. Pitaevskii, Statistica Physics, Part II, Pergamon Press, Oxford, 198. [2] M. Boström and B. E. Serneius, Therma effects on the Casimir force in the.1 5µm range, Phys. Rev. Lett. 84 (2) [3] J. S. Høye, I. Brevik, J. B. Aarseth and K. A. Miton, Does the transverse eectric mode contribute to the Casimir force for a meta? Phys. Rev. E67 (23) [4] V. B. Svetovoy and M. V. Lokhanin, Linear temperature correction to the Casimir force, Phys. Lett. A28 (21) 177. [5] V. B. Bezerra, G. L. Kimchitskaya and, Correation of energy and free energy for the therma Casimir force between rea metas, Phys. Rev. A66 (22) [6] V. B. Bezerra, G. L. Kimchitskaya, and C. Romero, Vioation of the Nernst heat theorem in the theory of therma Casimir force between Drude metas, Phys. Rev. A69 (24) [7] M. Boström and B. E. Serneius, Entropy of the Casimir effect between rea meta pates, Physica A339 (24) 53. [8] C. Genet, A. Lambrecht and S. Reynaud, Temperature dependence of the Casimir effect between metaic mirrors, Phys. Rev. A62 (2) [9] M. Bordag, B. Geyer, G. L. Kimchitskaya and, Casimir force at both nonzero temperature and finite conductivity, Phys. Rev. Lett. 85 (2) 53. [1] E. M. Lifshitz and L. P. Pitaevskii, Physica Kinetics, Pergamon Press, Oxford, [11] L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Eectrodynamics of Continuous Media, Pergamon Press, Oxford, [12] K. A. Miton, The Casimir effect: Recent controversies and progress, J. Phys. A37 (24) R29. [13] B. Geyer, G. L. Kimchitskaya and, Surface impedance approach soves probems with the therma Casimir force between rea metas, Phys. Rev. A67 (23) [14] M. Bordag, U. Mohideen and, New deveopments in the Casimir effect, Phys. Rep. 353 (21) 1. [15] P. W. Mionni, The Quantum Vacuum, Academic Press, San Diego, [16] Yu. S. Barash and V. L. Ginzburg, Eectromagnetic fuctuations ia a substance and moecuar (van der Waas) interbody forces, Sov. Phys. Usp. 18 (1975) 35. [17] R. C. Decca, E. Fischbach, G. L. Kimchitskaya, D. E. Krause, D. López and, Improved tests of extra-dimensiona physics and therma quantum fied theory from new Casimir force measurements, Phys. Rev. D68 (23) / 7