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Theory-experiment comparison for the Casimir force between metallic test bodies: A

spatially nonlocal dielectric response

G. L. Klimchitskaya1, 2 and V. M. Mostepanenko1, 2, 3

1Central Astronomical Observatory at Pulkovo of the Russian Academy of Sciences, Saint Petersburg, 196140, Russia2Peter the Great Saint Petersburg Polytechnic University, Saint Petersburg, 195251, Russia

3Kazan Federal University, Kazan, 420008, Russia

It has been known that the Lifshitz theory of the Casimir force comes into conflict with themeasurement data if the response of conduction electrons in metals to electromagnetic fluctuations isdescribed by the well tested dissipative Drude model. The same theory is in a very good agreementwith measurements of the Casimir force from graphene whose spatially nonlocal electromagneticresponse is derived from the first principles of quantum electrodynamics. Here, we propose thespatially nonlocal phenomenological dielectric functions of metals which lead to nearly the sameresponse, as the standard Drude model, to the propagating waves, but to a different response inthe case of evanescent waves. Unlike some previous suggestions of this type, the response functionsconsidered here depend on all components of the wave vector as is most natural in the formalismof specular reflection used. It is shown that these response functions satisfy the Kramers-Kronigrelations. We derive respective expressions for the surface impedances and reflection coefficients.The obtained results are used to compute the effective Casimir pressure between two parallel plates,the Casimir force between a sphere and a plate, and its gradient in configurations of the most preciseexperiments performed with both nonmagnetic (Au) and magnetic (Ni) test bodies. It is shown thatin all cases (Au-Au, Au-Ni, and Ni-Ni test bodies) the predictions of the Lifshitz theory found byusing the dissipative nonlocal response functions are in as good agreement with the measurementdata, as was reached previously with the dissipationless plasma model. Possible developments andapplications of these results are discussed.

INTRODUCTION

As predicted by Casimir [1], two parallel unchargedideal metal planes separated by a distance a at zero tem-perature should attract each other by the force

F (a) = − π2

240

~c

a4, (1)

which depends on the Planck constant ~ and the speed oflight c. According to Casimir, this force is caused by thezero-point oscillations of quantized electromagnetic fieldwhose spectrum is altered by the presence of ideal metalplanes. Within the formalism of quantum electrodynam-ics, in the absence of planes the zero-point energy is givenby an integral over a continuous wave vector k. In thepresence of planes, however, the tangential componentof electric field vanishes on the plane surfaces and thecomponent kz in direction perpendicular to the planesbecomes discrete. The modified zero-point energy of theelectromagnetic field is an integral over kx and ky, but adiscrete sum in kz . Although both zero-point energies inthe absence and in the presence of planes are infinitelylarge, their difference is finite. As a result, the negativederivative of this difference with respect to a is equal tothe Casimir force (1).More recently, it was understood that the Casimir

force, as well as the more familiar van der Waals force, be-longs to a wide class of physical phenomena determinedby the zero-point and thermal fluctuations of the electro-magnetic field. Lifshitz [2] created the general theory offorces of this kind acting between two parallel material

plates (semispaces) kept at any temperature in thermalequilibrium with the environment. In the framework ofthe Lifshitz theory, the ideal metal boundary conditionsare replaced by the electrodynamic continuity conditionswhich take into account real material properties by meansof the frequency-dependent dielectric permittivity and(for magnetic plates) magnetic permeability.

There are two main approaches to a derivation of ba-sic expressions of the Lifshitz theory for the free en-ergy and force of fluctuation origin. One of them isbased on the fluctuation-dissipation theorem of statis-tical physics [2, 3] and another one, which goes backto Casimir, on quantum field theory with appropri-ate boundary conditions [4–7]. By the frequently usedpresent-day terminology, the name van der Waals forcerefers to the fluctuation-induced forces at separations ofa few nanometers which do not depend on c. The termCasimir force is used in all remaining cases, i.e., whenthe interaction of fluctuation origin depends on ~, c, ma-terial properties and temperature. By now the Lifshitztheory is generalized for the case of boundary surfaces ofarbitrary geometric shape [8–12].

A systematic investigation of the thermal Casimir forcebetween parallel plates made of real metals traces back to2000. In Ref. [13] it was shown that if the low-frequencydielectric response of metals is described by the realisticDrude model taking a proper account of the relaxationproperties of conduction electrons, the Lifshitz theorypredicts large thermal correction which arises at shortseparations of a few hundred nanometers at room tem-perature and decreases the force magnitude. The thermal

2

correction of this type does not arise if the low-frequencydielectric response of metals is described by the plasmamodel which disregards the relaxation properties of con-duction electrons [14]. It should be remembered, how-ever, that the plasma model is in fact applicable onlyat high frequencies in the range of infrared optics wherethe relaxation properties do not play any role. The sur-prising thing is that the Casimir entropy calculated withthe Lifshitz theory employing the Drude model violatesthe third law of thermodynamics (the Nernst heat theo-rem) for metals with perfect crystal lattices [15–21] (theNernst heat theorem is followed for metals with the de-fects of structure [22–24], but this does not solve theproblem because perfect crystal lattice is a system withthe nondegenerate ground state, so that it should sat-isfy the third law of thermodynamics). If, however, theCasimir entropy is found using the plasma model dielec-tric response, the Nernst heat theorem is satisfied with-out trouble [15–21].

Of even greater surprise is that the theoretical pre-dictions of the Lifshitz theory using the Drude dielectricresponse at low frequencies have been excluded by themeasurement data of many experiments performed withboth nonmagnetic (Au) and magnetic (Ni) metallic testbodies by the two different experimental groups [25–37].The same measurement data were found to be in a verygood agreement with the predictions of the Lifshitz the-ory employing the plasma dielectric response [25–37]. Inthe most striking experiment of Ref. [33] using the dif-ferential measurement scheme, a difference between theexcluded and confirmed theoretical predictions was by upto a factor of 1000. Thus, the Drude mode, which pro-vides an adequate description of numerous optical andelectrical physical phenomena, does not work in applica-tion to the Casimir force.

Note that the measurement data of one experimentperformed at separations of a few micrometers were foundto be in better agreement with theoretical predictions us-ing the Drude model [38]. To obtain this conclusion, theCasimir force calculated using the Drude or the plasmamodel was subtracted from by the order of magnitudelarger measured force. The obtained differences were fit-ted to the theoretical electric force originating from thesurface patches. The results of Ref. [38] were shown to be,however, uncertain because in this experiment the surfaceimperfections, which are unavoidably present on a sur-face of the used spherical lens of centimeter-size radius,were ignored [39]. Recent Casimir experiment performedin the micrometer separation range, where the patch po-tentials were directly measured by means of Kelvin probemicroscopy, demonstrated an agreement with theoreticalpredictions using the plasma model and excluded thoseusing the Drude model [37].

An apparent disagreement of theoretical predictions ofthe fundamental Lifshitz theory using the Drude model,which was fully validated in the area of electromagnetic

and optical phenomena other than the Casimir effect,with the measurement data, as well as with the basicprinciples of thermodynamics, is puzzling and calls for asatisfactory explanation. This subject was hotly debatedin the literature starting from 2000 and many attemptshave been made directed to reaching an agreement withthermodynamics, looking for some unaccounted system-atic effects in measurements of the Casimir force, or de-veloping the more exact theory for a sphere-plate geom-etry used in experiments. Considerable advances havebeen made in this way (see Refs. [40–45] for a review),but the ultimate resolution of the Casimir puzzle stillremains to be found.

One possible approach to understanding of this prob-lem is that all models of the electromagnetic responseof macroscopic bodies are in some sense phenomenologi-cal, either relying on assumptions about bulk propertiesor upon very simplified models of the response of indi-vidual atoms to an applied field. This means that nomodel derived in any macroscopic framework using, e.g.,the Boltzmann transport equations or Kubo theory canbe expected to work under all circumstances, especially,in the very extreme conditions in Casimir force experi-ments. From this standpoint, it is not at all surprisingthat some physical situations give results in contrast withthe predictions of commonly used models. There are,however, some limitations following from fundamentalphysics that exacerbate the situation. Thus, accordingto Maxwell equations, the dielectric response of metalsto electromagnetic field in the quasistatic limit is inverseproportional to the frequency. The Drude model satisfiesthis demand and describes the relaxation properties ofconduction electrons whereas the dielectric permittivityof the plasma model is inverse proportional to the secondpower of frequency and does not describe the relaxationproperties.

At one time it was hoped that a resolution of this prob-lem may come from the investigation of graphene whichis a two-dimensional sheet of carbon atoms packed in thehexagonal crystal lattice. At energies below a few eV,which are characteristic for the Casimir effect at not tooshort separations, graphene is well described by the Diracmodel as a set of massless or very light electronic quasi-particles satisfying the Dirac equation where c is replacedwith the Fermi velocity vF [46, 47]. The spatially non-local dielectric response of graphene to electromagneticfield was found precisely on the basis of first principles ofquantum electrodynamics using the polarization tensor[48–51]. The Lifshitz theory employing this dielectric re-sponse turned out to be in agreement with experimentson measuring the Casimir force in graphene systems [52–55] and with the Nernst heat theorem [56–60]. The ques-tion arises of whether the spatially nonlocal dielectricresponse could be helpful for a resolution of the Casimirpuzzle.

Unfortunately, an application of the conventional spa-

3

tially nonlocal dielectric permittivities derived in the lit-erature for theoretical description of the anomalous skineffect, screening effects etc. [61–67], although leaves roomfor a resolution of thermodynamic problems, does notremedy a contradiction between the Lifshitz theory andthe measurement data [68–72]. In this situation, somephenomenological models are worth consideration.

With this approach, Ref. [73] proposed the spatiallynonlocal dielectric permittivities which describe nearlythe same response, as the Drude model, to electromag-netic waves on the mass shell (i.e., to the propagatingwaves), but quite a different response, than the Drudemodel, to the off-the-mass-shell waves, which are alsocalled evanescent. These permittivities depend only onthe magnitude of the wave vector projection on the planeof Casimir plates k⊥. It was shown that they satisfythe Kramers-Kronig relations [73], and the respectiveCasimir entropy goes to zero with vanishing temperaturein agreement with the Nernst heat theorem [74]. Withthe aim of solving the above problems, spatially nonlocalpermittivities were also introduced in Ref. [75].The most important thing is that the proposed nonlo-

cal permittivities bring the Lifshitz theory in agreementwith the measurement data of experiments performedwith two nonmagnetic (Au) [73] and two magnetic (Ni)[76] test bodies. To perform the theory-experiment com-parison, the expressions for the reflection coefficients en-tering the Lifshitz formula via the surface impedanceswere used. The latter have been found in Refs. [62, 63]for nonmagnetic metals in the approximation of specularreflection of electrons on the boundary surfaces as somefunctionals of nonlocal dielectric permittivities. For thecase of magnetic metals, similar surface impedances wererecently derived in Ref. [76].

In this paper, we suggest the spatially nonlocal re-sponse functions which, similar to Ref. [73], lead to thesame results, as the standard Drude model, for electro-magnetic waves on the mass shell, but to alternative re-sults for the off-the-mass-shell fields. However, unlikeRef. [73], the response functions considered below dependon all components of the wave vector k what is fully con-sistent with the used formalism of surface impedancesin the approximation of specular reflection developed inRefs. [62, 63] (see Sec. II for more detail).

We calculate the surface impedances and reflection co-efficients for two independent polarizations of the elec-tromagnetic field in the case of Au and Ni surfaces. Theobtained reflection coefficients for Au surfaces are usedto perform computations of the effective Casimir pres-sure between two Au-coated plates [27] and the Casimirforce between an Au-coated sphere and an Au-coatedplate [37] in the experiments performed by means of amicromechanical torsional oscillator. Using the same re-flection coefficients, we also compute the gradient of theCasimir force between an Au-coated sphere and an Au-coated plate in the experiments performed in different

separation regions by means of an atomic force micro-scope [29, 36]. With the help of reflection coefficients onboth Au and Ni surfaces, we perform computations ofthe gradient of the Casimir force between an Au-coatedsphere and a Ni-coated plate measured in the experiment[30]. Finally, the gradient of the Casimir force betweena Ni-coated sphere and a Ni-coated plate is computedwhich was measured in Refs. [31, 32]. According to ourresults, in all cases the predictions of the Lifshitz theoryusing the suggested nonlocal response functions, whichtake the proper account of the relaxation properties ofconduction electrons in the region of propagating waves,are in a very good agreement with the measurement data.Future applications of these results are discussed.

The paper is organized as follows. In Sec. II, we presentthe formalism of the Lifshitz theory in the approximationof specular reflection. In Sec. III, the spatially nonlo-cal response functions depending on all components ofthe wave vector are introduced and their properties areinvestigated. Section IV is devoted to computations ofthe effective Casimir pressure and Casimir force in dif-ferent experiments using a micromechanical torsional os-cillator. In Secs. V and VI computations of the gradientof the Casimir force are performed between nonmagneticand with magnetic test bodies, respectively, measuredby means of an atomic force microscope. In Sec. VII, thereader will find our conclusions and a discussion.

FORMALISM OF THE LIFSHITZ THEORY IN

THE APPROXIMATION OF SPECULAR

REFLECTION

The Casimir free energy per unit area and pressurein the configuration of two thick parallel plates (semis-paces) spaced at separation a at temperature T in ther-mal equilibrium with the environment are given by thefamous Lifshitz formulas [2, 3] (see also [40, 41] for mod-ern notations in terms of the reflection coefficients usedbelow)

F(a, T ) =kBT

2π

∞∑

l=0

′∫

∞

0

k⊥ dk⊥

×∑

α

ln[

1− r(1)α (iξl, k⊥)r(2)α (iξl, k⊥)e

−2aql]

,

P (a, T ) = −kBT

π

∞∑

l=0

′∫ ∞

0

qlk⊥ dk⊥

×∑

α

[

e2aql

r(1)α (iξl, k⊥)r

(2)α (iξl, k⊥)

− 1

]−1

. (2)

Here, kB is the Boltzmann constant, k⊥ = (k2x+k2y)1/2

is the magnitude of the wave vector projection on the

4

plane of plates, ξl = 2πkBT l/~ are the Matsubara fre-quencies, ql = (k2

⊥+ ξ2l /c

2)1/2, and the prime on thesummation sign divides by 2 the term of the first sumswith l = 0. The reflection coefficients of electromagneticwaves with the transverse magnetic (α = TM) and trans-verse electric (α = TE) polarizations on the first and sec-

ond plates are r(1)α (iξl, k⊥) and r

(2)α (iξl, k⊥), respectively.

In the original version of the Lifshitz theory [2, 3], it isassumed that the plate materials possess only the tempo-ral dispersion, i.e., their dielectric permittivities, εn(ω),and magnetic permeability, µn(ω), where n = 1, 2 forthe first and second plates, depend on the frequency ω (adependence on T , e.g., for metals is also allowed). In thiscase the reflection coefficients are given by the familiarFresnel formulas considered at ω = iξl

r(n)TM(iξl, k⊥) =

εn(iξl)ql − kn(iξl, k⊥)

εn(iξl)ql + kn(iξl, k⊥),

r(n)TE(iξl, k⊥) =

µn(iξl)ql − kn(iξl, k⊥)

µn(iξl)ql + kn(iξl, k⊥), (3)

where

kn(iξl, k⊥) =

[

k2⊥ + εn(iξl)µn(iξl)ξ2lc2

]1/2

. (4)

The reflection coefficients in Eq. (2) can also beexpressed in terms of the exact surface impedances

Z(n)TM(iξl, k⊥) and Z

(n)TE (iξl, k⊥)

r(n)TM(iξl, k⊥) =

cql − ξlZ(n)TM(iξl, k⊥)

cql + ξlZ(n)TM(iξl, k⊥)

,

r(n)TE(iξl, k⊥) =

cqlZ(n)TE (iξl, k⊥)− ξl

cqlZ(n)TE (iξl, k⊥) + ξl

, (5)

where for materials possessing only the temporal disper-sion the impedances are connected with the dielectricpermittivities and magnetic permeabilities as [69, 76]

Z(n)TM(iξl, k⊥) =

ckn(iξl, k⊥)

ξlεn(iξl),

Z(n)TE (iξl, k⊥) =

ξlµn(iξl)

ckn(iξl, k⊥). (6)

It is evident that the substitution of Eq. (6) in Eq. (5)returns us back to the reflection coefficients (3).According to generalizations of the Lifshitz theory in

the framework of the scattering approach [10–12], Eq. (2)with appropriately defined reflection coefficients remainsvalid for any planar structures. If materials of the plates,besides temporal, possess the spatial dispersion, deriva-tion of the exact expressions for the reflection coeffi-cients runs into problems. The point is that the re-sponse of spatially dispersive material filling in the entire3-dimensional space to electric fields parallel and perpen-dicular to the wave vector k = (kx, ky, kz) is described

by the longitudinal, εL(ω,k), and transverse, εTr(ω,k),dielectric permittivities [66, 77].In the strict sense, these permittivities can be intro-

duced only under a condition of translational invariancewhich is violated by the presence of Casimir plates sep-arated by a vacuum gap [78–80]. The standard Lifshitztheory deals with plate materials possessing only the tem-poral dispersion. Therefore the dielectric permittivitiesdepend only on ω and the violation of translational in-varianse makes no problem. For the plate materials withspatial dispersion, this violation, however, makes impos-sible an employment of the standard continuity boundaryconditions and derivation of the Fresnel reflection coeffi-cients (3).This difficulty can be circumvented as follows. Since

the Casimir force is determined by the dielectric prop-erties in the bulk and appropriate boundary conditions,it is possible to preserve the translational invariance infictitious homogeneous medium by assuming the specu-lar reflection of charge carriers (electrons) on the bound-ary surfaces of Casimir plates. In doing so an electronreflected on an interface between the plate and the vac-uum gap is indistinguishable from an electron comingfreely on the source side of a fictitious medium. Thenone can introduce the longitudinal and transverse dielec-tric permittivities εL and εTr which depend on ω and allcomponents of the wave vector k because the fictitiousmedium is translationally invariant in all directions, andnot only in the plane of Casimir plates [62, 63]. Physicallythere is no sufficient reason to exclude one of the wavevector components. This would be justified for grapheneand other two-dimensional materials but not for a bulkymatter possessing the spatial dispersion. Note that theresponse functions depending on all components of k arefully consistent with the Lifshitz theory where the reflec-tion coefficients depend only on k⊥. As a result, thesecoefficients preserve the form of Eq. (5) whereas the sur-face impedances are obtained from the nonlocal bulk per-mittivities depending on ω and k by the integration withrespect to kz as:

Z(n)TM(iξl, k⊥) =

cξlµn(iξl)

π

∫

∞

−∞

dkz

k2

[

k2⊥

εLn (iξl,k)µn(iξl)ξ2l

+k2z

εTrn (iξl,k)µn(iξl)ξ2l + c2k2

]

,

(7)

Z(n)TE (iξl, k⊥) =

cξlµn(iξl)

π

∫ ∞

−∞

dkz

εTrn (iξl,k)µn(iξl)ξ2l + c2k2 .

For nonmagnetic metals Eq. (7) was derived in Refs. [62,63] (see also Ref. [81] for a review) and generalized forthe case of magnetic ones in Ref. [76]. There is also ageneralization of Eq. (7) for the case of diffuse reflec-tion [67, 82]. Note, however, that for sufficiently smoothboundary surfaces used in experiments on measuring theCasimir force the approximation of specular reflection is

5

well applicable.As mentioned in Sec. I, the surface impedances (7) with

µn(iξl) = 1 and the dielectric permittivities εL(ω,k) andεTr(ω,k) describing the anomalous skin effect [63] wereused to calculate the Casimir force between Au surfacesin Refs. [70, 71]. It was found, however, that correctionsto the force due to spatial nonlocality in the region ofthe anomalous skin effect are too small and incapable toexplain a disagreement between the measurement dataand theoretical predictions.

SPATIALLY NONLOCAL DIELECTRIC

FUNCTIONS PROVIDING AN ALTERNATIVE

RESPONSE TO THE OFF-THE-MASS-SHELL

FIELDS

The phenomenological nonlocal dielectric functions de-pending on k⊥, which bring the Lifshitz theory in agree-ment with experiments on measuring the Casimir forcebetween two similar plates and with thermodynamics donot disregarding the dissipation of conduction electrons,were introduced in Refs. [73, 74, 76]. Here, we proposethe nonlocal dielectric permittivities depending on allcomponents of the wave vector

εTrn (ω, k) = ε(n)c (ω)−ω2p,n

ω(ω + iγn)

(

1 + ivTrn k

ω

)

,

(8)

εLn (ω, k) = ε(n)c (ω)−

ω2p,n

ω(ω + iγn)

(

1 + iv Ln k

ω

)−1

,

where ε(n)c (ω) is the contribution determined by the core

electrons, ωp,n is the plasma frequency, γn is the relax-ation parameter, k = |k| is the magnitude of the wavevector, and vTrn , v L

n are constants of the order of Fermivelocity vF,n (as before, n = 1, 2 for materials of the firstand second plates).The response functions introduced in Refs. [73, 74, 76]

are obtained from Eq. (8) if the magnitude of the wavevector k is replaced with the magnitude of its projec-tion on the plane of the Casimir plates k⊥. It wasshown [45, 73] that the Lifshitz theory using the result-ing dielectric functions is in good agreement with themeasurement data of experiments [27, 36] measuring theCasimir interaction between two similar Au test bodiesif vTr1,2 = vL1,2 = 7vF,Au. With the same numerical coef-

ficient, vTr1,2 = vL1,2 = 7vF,Ni, the Lifshitz theory usingthese permittivities was found in agreement [76] withmeasurements of the Casimir interaction between twosimilar magnetic (Ni) test bodies [31, 32]. However, asdiscussed in previous section, the nonlocal permittivitiesdepending only on k⊥ are not fully consistent with theused formalism of surface impedances in the approxima-tion of specular reflection and represent only some kindof a simplified particular case.

We return to the permittivities (8) which depend on allcomponents of the wave vector. For the electromagneticwaves on the mass shell (the propagating waves) theydescribe approximately the same response as the stan-dard Drude model supplemented by the oscillator terms

ε(n)c (ω) taking into account the interband transitions [83].This is because the additions to unity depending on k inthe parantheses in Eq. (8) under a condition ck 6 ω be-come negligibly small

vTr,Ln k

ω∼ vF,n

c

ck

ω6

vF,nc

≪ 1. (9)

In doing so, the response functions (8) take into accountdissipation of conduction electrons by means of the relax-ation parameter γn as does the Drude model. If, however,ck > ω, as it holds for the evanescent waves which areoff the mass shell in free space, the permittivities (8) canlead to an electromagnetic response differing from thatof the Drude model.Below we demonstrate that the response functions (8),

which depend on all the three wave vector components,provide a complete agreement with all measurements ofthe Casimir interaction including that one between dis-similar (Au and Ni) plates [30]. The case of dissimi-lar metals is of special interest because in the range ofexperimental separations the Lifshitz theory using theDrude and plasma dielectric permittivities at low fre-quencies leads to almost coinciding results which arein agreement with the measurement data in the lim-its of the experimental errors [30]. Calculations showthat the theoretical predictions using nonlocal dielectricfunctions of Refs. [73, 74, 76] depending only on k⊥ are

also in agreement with experiment if vTr,LAu = 7vF,Au and

vTr,LNi = 7vF,Ni. An advantage of the nonlocal responsefunctions (8) depending on all wave vector components is,however, that they are not only fully consistent with theformalism of surface impedances in the approximation ofspecular reflection, but lead to by a factor of 4.7 smallervTr,LAu and vTr,LNi (see Secs. IV–VI). This significantly de-creases any possible differences between theoretical pre-dictions obtained using the standard Drude model andits nonlocal modification in optical experiments exploit-ing the propagating electromagnetic waves.We begin with calculation of the reflection coefficients

(5) at zero Matsubara frequency ξ0 = 0 which plays themajor role in the problems of Casimir physics discussedin Sec. I. Substituting ω = iξl in Eq. (8), one obtains theexpressions for nonlocal permittivities (8) at the pureimaginary Matsubara frequencies

εTrn (iξl, k) = ε(n)c (iξl) +ω2p,n

ξl(ξl + γn)

(

1 +vTrn k

ξl

)

,

(10)

εLn (iξl, k) = ε(n)c (iξl) +

ω2p,n

ξl(ξl + γn)

(

1 +v Ln k

ξl

)−1

.

6

Substituting Eq. (10) to the first expression in Eq. (7)and considering vanishing ξ0, we find the asymptotic be-

havior of Z(n)TM in this case

Z(n)TM(iξ0, k⊥) =

2ck2⊥

πξ0

∫

∞

0

dkz√

k2⊥+ k2z(

√

k2⊥+ k2z + bn)

,

(11)where bn ≡ ω2

p,n/(γnvLn ). Performing here the change

of integration variable from kz to k = (k2⊥+ k2z)

1/2, werewrite Eq. (11) in the form

Z(n)TM(iξ0, k⊥) =

2ck2⊥

πξ0

∫ ∞

k⊥

dk√

k2 − k2⊥(k + bn)

. (12)

This integral can be calculated using 1.2.45.3 and1.2.45.5 in Ref. [84] with the result

Z(n)TM(iξ0, k⊥) =

2ck2⊥

πξ0

1√b2n−k2

⊥

lnbn+

√b2n−k2

⊥

k⊥

, k⊥ < bn,

1√k2

⊥−b2

n

arccos bnk⊥

, k⊥ > bn.

(13)Substituting this result in the first expression of Eq. (5)

and putting ξ0 = 0, we finally obtain

r(n)TM(0, k⊥) =

π√

b2n − k2⊥− 2k⊥ ln

bn+√

b2n−k2

⊥

k⊥

π√

b2n − k2⊥+ 2k⊥ ln

bn+√

b2n−k2

⊥

k⊥

(14)

for k⊥ < bn and

r(n)TM(0, k⊥) =

π√

k2⊥− b2n − 2k⊥ arccos bn

k⊥

π√

k2⊥− b2n + 2k⊥ arccos bn

k⊥

(15)

for k⊥ > bn. For k⊥ = bn both Eqs. (14) and (15) leadto the result

r(n)TM(0, k⊥) =

π − 2

π + 2. (16)

Now we consider the value of the TE reflection coef-ficient at zero Matsubara frequency. For this purpose,we substitute the first expression in Eq. (10) to the sec-ond formula in Eq. (7) and find the following asymptoticexpression in the case of vanishing ξ0:

Z(n)TE (iξ0, k⊥) =

2ξ0µn(iξ0)

πc

∫

∞

0

dkz

Bn

√

k2⊥+ k2z + k2

⊥+ k2z

,

(17)

where Bn ≡ µn(iξ0)ω2p,nv

Trn /(γnc

2).

The integral in Eq. (17) has the same form as inEq. (11). Calculating it in the same way as above, oneobtains

Z(n)TE (iξ0, k⊥) =

2ξ0µn(iξ0)

πc

1√B2

n−k2

⊥

lnBn+

√B2

n−k2

⊥

k⊥

, k⊥ < Bn,

1√k2

⊥−B2

n

arccos Bn

k⊥

, k⊥ > Bn.

(18)Substituting this equation in the second expression of

Eq. (5) and putting ξ0 = 0, we find the following results:

r(n)TE(0, k⊥) =

2µn(0)k⊥ lnBn+

√B2

n−k2

⊥

k⊥

− π√

B2n − k2

⊥

2µn(0)k⊥ lnBn+

√B2

n−k2

⊥

k⊥

+ π√

B2n − k2

⊥

(19)for k⊥ < Bn and

r(n)TE(0, k⊥) =

2µn(0)k⊥ arccos Bn

k⊥

− π√

k2⊥−B2

n

2µn(0)k⊥ arccos Bn

k⊥

+ π√

k2⊥−B2

n

(20)

for k⊥ > Bn. For k⊥ = Bn, Eqs. (19) and (20) lead to

r(n)TE(0, k⊥) =

2µn(0)− π

2µn(0) + π. (21)

From Eqs. (14), (15) and (19), (20) one can see that atzero Matsubara frequency the magnetic properties makean impact only on the TE reflection coefficient in theLifshitz formula (2).

The values of impedances and reflection coefficients atall Matsubara frequencies with l > 1 are more compli-cated. We again substitute Eq. (10) in the first expressionin Eq. (7) and introduce in the obtained integrals the fol-lowing dimensionless integration variable and projectionmagnitude of the wave vector on the plane of plates:

x =kzc

ξl, pl =

k⊥c

ξl. (22)

Then the impedance ZTM takes the form

Z(n)TM(iξl, k⊥) =

2k2⊥c2

πξ2l

∫

∞

0

(c+ v Ln

√

p2l + x2)dx

(p2l + x2)[cA(n)l + v L

n

√

p2l + x2](23)

+2µn(iξl)

π

∫

∞

0

x2 dx

(p2l + x2)[µn(iξl)(A(n)l +D

(n)l

√

p2l + x2) + p2l + x2],

7

where

A(n)l ≡ ε(n)c (iξl) +

ω2p,n

ξl(ξl + γn), D

(n)l ≡

ω2p,nv

Trn

ξl(ξl + γn)c.

(24)

In a similar way, substituting the first expression inEq. (10) in the second formula of Eq. (7) and usingEq. (22), one obtains

Z(n)TE (iξl, k⊥) =

2µn(iξl)

π

∫

∞

0

dx

µn(iξl)(A(n)l +D

(n)l

√

p2l + x2) + p2l + x2. (25)

Equations (23) and (25) are convenient for numericalcomputations performed in the next sections.In the end of this section, we note that the spatially

nonlocal dielectric permittivities (8) introduced abovesatisfy the Kramers-Kronig relations as it should be inaccordance with the condition of causality. This can beproven similar to Ref. [73] if to take into account that thedielectric permittivity of core electrons in Eq. (8) takesthe form [41, 83]

ε(n)c (ω) = 1 +

Kn∑

j=1

gn,jω2n,j − ω2 − iγn,jω

, (26)

where ωn,j 6= 0 are the oscillator frequencies, gn,j are theoscillator strengths, γn,j are the relaxation parametersandKn are the numbers of oscillators for the first (n = 1)and second (n = 2) Casimir plates, respectively. Thepermittivity (26) satisfies the standard Kramers-Kronigrelations [66, 77].In this case the derivation presented in Ref. [73] is

repeated with the only replacement of k⊥ with k =(k2

⊥+ k2z)

1/2. As a result, for εTrn one obtains the fol-lowing Kramers-Kronig relations

Re εTrn (ω, k) = 1 +1

π−∫

∞

−∞

dxIm εTrn (x, k)

x− ω−

ω2p,n

ω2

vTrn k

γn,

Im εTrn (ω, k) = − 1

π−∫

∞

−∞

dx

x− ω

[

Re εTrn (x, k) +ω2p,n

x2

vTrn k

γn

]

+4πReσTr

n,0(k)

ω, (27)

where the integrals are understood as the principal valuesand the real part of the static transverse conductivity isgiven by [73]

ReσTrn,0(k) =

ω2p,n(γn − vTrn k)

4πγ2n

. (28)

It should be mentioned that the last term on the right-hand side of the first equality in Eq. (27) originates fromthe second-order pole of the dielectric permittivity εTrn (ω)at ω = 0. The last term on the right-hand side of the

second equality in Eq. (27) is caused by the first-orderpole so that in the local limit, k → 0, Eq. (28) representsthe static conductivity of the standard Drude model [77].The dielectric permittivity εL

n (ω) defined in Eq. (8) hasno poles at ω = 0. For this reason, the Kramers-Kronigrelations for this permittivity take the same simplest formas for the permittivity of core electrons [66, 77]

Re εLn (ω, k) = 1 +

1

π−∫

∞

−∞

dxIm εL

n (x, k)

x− ω,

Im εLn (ω, k) = − 1

π−∫

∞

−∞

dxRe εL

n (x, k)

x− ω. (29)

From Eqs. (27) and (29), it is easy to find the ex-pressions for nonlocal dielectric permittivities along theimaginary frequency axis [73, 77]

εTrn (iξ, k) = 1 +2

π

∫

∞

0

dxxIm εTrn (x, k)

x2 + ξ2+

ω2p,n

ξ2vTrn k

γn,

εLn (iξ, k) = 1 +

2

π

∫

∞

0

dxxIm εL

n (x, k)

x2 + ξ2, (30)

which are useful for computations by means of the Lif-shitz formula (2).

COMPARISON WITH THEORY FOR

MEASUREMENTS BETWEEN NONMAGNETIC

TEST BODIES BY MEANS OF A

MICROMECHANICAL TORSIONAL

OSCILLATOR

In the series of dynamic experiments performed in highvacuum in the configuration of an Au-coated sphere of ra-dius R and an Au-coated plate by means of a microme-chanical torsional oscillator at room temperature [25–28],the measurement data for the gradient of the Casimirforce F ′

sp(a, T ) was represented in terms of the effectiveCasimir pressure between two Au plates

P expt(a, T ) = − 1

2πRF ′

sp(a, T ). (31)

This was done by using the proximity force approxima-tion [40, 41] which leads to a relative error of less than

8

a/R (all measurements were performed at the sphere-plate separations a < 750 nm with a sphere radiusR = 151.2 µm [27]).It was found [25–28] that the theoretical predictions of

the Lifshitz theory computed by Eq. (2) with taken intoaccount surface roughness [40, 41, 85–87] are excludedby the measurement data if the dielectric response of Auis described by the Drude model supplemented by thepermittivity of core electrons. This response function isobtained from Eq. (10) with k = 0

εDn (iξl) = εTrn (iξl, 0) = εLn (iξl, 0) = ε(n)c (iξl)+

ω2p,n

ξl(ξl + γn).

(32)

In doing so the values of ε(n)c (iξl) were found from the

optical data for the complex index of refraction of Au[88] using the Kramers-Kronig relations. If, however, theresponse function (32) with γn = 0 is used in compu-tations (i.e., the plasma-like model which disregards therelaxation properties of conduction electrons), the theo-retical predictions turn out to be in good agreement withthe measurement data [25–28].Here, we compare the measurement data of the most

precise experiment of these series [27] and theoretical pre-dictions of the Lifshitz theory using the nonlocal dielec-tric permittivities (10). Numerical computations of theCasimir pressure were performed by Eqs. (2), (5), (7),and (10). In so doing the reflection coefficients at zeroMatsubara frequency were calculated by Eqs. (14), (15)and (20), (21). The values of impedances at all Matsub-ara frequencies with l > 1 were computed by Eqs. (23)and (25). The surface roughness was accounted for in thesame way as in Ref. [27], i.e., with the help of an addi-tive approach. Its contribution to the effective Casimirpressure turns out to be negligibly small.The following values of all parameters have been used

in computations. Taking into account that both test bod-ies were made of Au, one should put ~ωp,1 = ~ωp,2 =~ωp,Au = 8.9 eV and ~γ1 = ~γ2 = ~γAu = 35.7 meV[27, 40, 41]. Since Au is a nonmagnetic metal, we

have µ1(iξl) = µ2(iξl) = 1. The values of ε(1)c (iξl) =

ε(2)c (iξl) = ε

(Au)c (iξl) were calculated in Refs. [25–28] us-

ing the tabulated optical data for the complex index ofrefraction of Au [88] and used here. For the constants ofthe order of Fermi velocity in Eqs. (8) and (10) we haveused the equal values vTrn = v L

n = vn. In so doing, in thisexperiment and in all other experiments considered be-low, the best agreement between experiment and theoryis reached for vn = 3vF,n/2 with the value of vF,n foundunder an assumption of the spherical Fermi surface

mev2F,n

2= ~ωp,n, (33)

whereme is the electron mass. Thus, for Au in this exper-iment from Eq. (33) one obtains vF,1 = vF,2 = vF,Au =

1.77 × 106 m/s. The numerical factor between vn andvF,n, which is equal to 3/2 for the dielectric permittivities(8) and (10), is in fact the single fitting parameter of ourphenomenological model. The chosen value of this pa-rameter leads to the best agreement between experimentand theory. We recall that for the nonlocal permittivitiesof Refs. [73, 74, 76] depending only on k⊥ this parameteris equal to 7 resulting in larger (but as yet sufficientlysmall) deviations from the standard Drude model in op-tical experiments.

The computational results for the magnitude of the ef-fective Casimir pressure are shown by the upper (red)theoretical bands in Figs. 1(a)–1(d) over four separationregions from 162.03 to 745.98 nm. The experimental dataare shown as crosses. The horizontal and vertical arms ofthese crosses indicate the total measurement errors foundat the 95% confidence level as a combination of system-atic and random errors [27]. The lower (black) theoreticalbands in Fig. 1 indicate the theoretical predictions of theLifshitz theory obtained by Eqs. (2) and (3) using thestandard local Drude response function (32). The widthof each theoretical band is determined by the errors in allthe above theoretical parameters used in computations.

As is seen in Fig. 1, the measurement data are in a verygood agreement with theoretical predictions of the Lif-shitz theory made using the spatially nonlocal responsefunctions (8), (10) which take into account the dissipa-tion properties of conduction electrons. Almost the sametheoretical predictions in agreement with the measure-ment data were made by the Lifshitz theory using theplasma response function given by Eq. (32) with γn = 0,i.e., with the relaxation properties of free electrons disre-garded. The respective theoretical bands cannot be dis-tinguished from the upper (red) bands shown in Fig. 1.As to the lower (black) theoretical bands in Fig. 1 com-puted using the Drude response function (32), they areexcluded by the measurement data over the entire rangeof separations.

We are coming now to the recently performed experi-ment on measuring the differential Casimir force betweenan Au-coated sphere of R = 149.7 µm radius and top andbottom of the Au-coated rectangular trenches [37]. Asmost of precise measurements of the Casimir interaction,this one was made at room temperature in high vacuum.Thanks to the differential character of this measurement,it has been made possible to obtain the meaningful dataup to separation distances of a few micrometers usingthe same setup of a micromechanical torsional oscillator.Due to the sufficiently deep trenches used, the effectivelymeasured Casimir force was that acting between a sphereand a plate which served as the trench top.

In the framework of the proximity force approximation,the Casimir force acting between a sphere and a plate isgiven by

Fsp(a, T ) = 2πRF(a, T ), (34)

9

160 170 180 190 200 210 220 230

400

600

800

1000

350 400 450 500

20

30

40

50

60

70

240 260 280 300 320 340

100

150

200

250

300

550 600 650 700

4

6

8

10

12

14

a (nm) a (nm)

|P|(mPa)

|P|(mPa)

(a) (b)

(c) (d)

FIG. 1: The magnitudes of the effective Casimir pressure between two Au-coated parallel plates computed using the spatiallynonlocal response functions and the Drude model are shown as functions of separation by the upper (red) and lower (black)bands, respectively. The experimental data [27] are shown as crosses whose arms indicate the total errors determined at the95% confidence level.

where the Casimir free energy in the configuration oftwo parallel plates is presented in Eq. (2). In Ref. [37],the force Fsp was computed both approximately usingEqs. (2), (3), and (34) and precisely on the basis of firstprinciples of quantum electrodynamics at nonzero tem-perature using the scattering approach [89–92] and thegradient expansion [93–96]. It was shown [37] that alldifferences between the approximate and exact resultsare well below the measurement errors within the sepa-ration region from 0.2 to 8 µm, irrespective of whetherthe Drude or plasma response function is used in compu-tations.

The obtained theoretical results employing the plasmamodel given by Eq. (32) with γn = 0 were found to be ina good agreement with the measurement data over the

entire range of separations. The results computed simi-larly, but using the Drude model, were excluded by thedata over the separation region from 0.2 to 4.8 µm. Inso doing the background electric force due to patch po-tentials was investigated with the help of Kelvin probemicroscopy [97] and included in the total experimentalerror of the Casimir force determined at the 95% confi-dence level.

Here, we compute the Casimir force in the configura-tion of the experiment [37] by the first equality in Eq. (2)and Eq. (34) using the proposed spatially nonlocal dielec-tric functions (10). For the reflection coefficients withl = 0, Eqs. (14), (15) and (20), (21) have been used,and for l > 1 numerical computations were performed byEqs. (23) and (25) as described above with the following

10

values of all parameters of Au [37]: ~ωp,Au = 9.0 eV,~γAu = 35.0 meV, vF,Au = 1.78× 106 m/s. The obtainedresults were multiplied by a correction factor accountingfor an inaccuracy of the proximity force approximationwhich was calculated in Ref. [37].

The obtained computational results in the range of sep-arations from 1 to 4.9 µm are shown by the upper (red)band in Fig. 2. In the same figure, the lower (black) bandis computed by Eqs. (2), (3) and (34) using the Druderesponse function (32). The measurement data are indi-cated as crosses. As is seen in Fig. 2, the upper and lowervertical arms of the crosses differ from one another. Thisis because the attractive electric force due to patch po-tentials is included as part of the error in measuring theCasimir force. According to Fig. 2, the theoretical predic-tions of the Lifshitz theory using the proposed nonlocalresponse functions (8), (10) are in agreement with themeasurement data. Good agreement also holds over theranges of separations from 0.2 to 1 µm and from 4.8 to8 µm which are not shown in Fig. 2. The predictions ofthe same theory using the Drude model are excluded overthe range of separations from 1 to 4.8 µm (in Ref. [37] itis shown that they are also excluded in the measurementrange from 0.2 to 1 µm but here we are more interestedin the region of large separations exceeding 1 µm).

The Casimir forces computed by Eqs. (2), (3) and (34)using the plasma model given by Eq. (32) with γn = 0are almost the same as the ones computed above with thespatially nonlocal response functions. Thus, at separa-tions of 1, 3, 5, and 7 µm the pairs of force magnitudes (infN) computed using the nonlocal response functions andthe plasma model are (372.98, 374.62), (21.44, 21.67),

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50

100

200

300

400

a (µm)

|F|(fN)

FIG. 2: The magnitudes of the Casimir force between an Au-coated sphere and an Au-coated plate computed using thespatially nonlocal response functions and the Drude modelare shown as functions of separation by the upper (red) andlower (black) bands, respectively. The experimental data [37]are shown as crosses whose arms indicate the total errors de-termined at the 95% confidence level.

(7.24, 7.30), and (3.69, 3.71), i.e., only 0.44%, 1.06%,0.8%, and 0.54% relative differences, well below the re-spective experimental errors. Although these results arenot experimentally distinguishable, that ones obtainedusing the nonlocal response functions should be consid-ered as preferable as they are obtained with taken intoaccount relaxation properties of conduction electrons.

MEASUREMENTS BETWEEN NONMAGNETIC

TEST BODIES BY MEANS OF AN ATOMIC

FORCE MICROSCOPE

Another experimental setup for measuring the Casimirinteraction is an atomic force microscope whose sharptip is replaced with a sphere of sufficiently large radius[98]. Here, we compare the measurement data of threemost precise experiments on measuring the gradient ofthe Casimir force between an Au-coated sphere and anAu-coated plate obtained by means of a dynamic atomicforce microscope [29, 36] with theoretical predictions us-ing the nonlocal dielectric functions (10). All measure-ments were performed in high vacuum at room tempera-ture. In interpretation of all these experiments the same

parameters of Au, i.e., the values of ωp,Au, γAu, and ε(Au)c ,

have been used as already listed in Sec. IV when de-scribing measurements of the differential Casimir forcebetween a sphere and a plate with rectangular trenches.We start with the experiment of Ref. [29] which em-

ployed the sphere of R = 41.3 µm radius. The theoreticalforce gradients were computed using the Lifshitz theoryand the proximity force approximation with taken intoaccount correction for its inaccuracy [29]

F ′

sp(a, T ) = −2πRP (a, T )[

1 + θ(a, T )a

R

]

, (35)

where the Casimir pressure P is given by the second ex-pression in Eq. (2) with the Fresnel reflection coefficients(3) and at separations below 1 µm the coefficient θ is neg-ative and does not exceed unity (see Refs. [93–95] and themore complete results for different dielectric functions inRefs. [92, 96]). The effect of surface roughness was takeninto account perturbatively and shown to be negligiblysmall.According to the results of Ref. [29], the theoretical

predictions obtained using the Drude model (32) are ex-cluded by the measurement data within the range of sep-arations from 235 to 420 nm. The same data turned outto be in a good agreement with theoretical predictionsfound by using the plasma model which does not takeinto account the relaxation properties of free electrons,i.e., put γAu = 0. Thus, the results obtained earlier inRefs. [25–28] by means a micromechanical torsional os-cillator were confirmed independently by using quite dif-ferent laboratory setup.We have computed the gradients of the Casimir force

(35) in the experimental configuration of Ref. [29] using

11

the spatially nonlocal response functions (10) and reflec-tion coefficients (5) expressed via the surface impedancesas described in Secs. III and IV. The same parameters ofAu, as in Ref. [29], have been used and vTr,L

n = v Tr,LAu =

3vF,Au/2 as indicated above. The computational resultsare shown by the upper (red) bands in Fig. 3 where theexperimental data are presented as crosses whose armsindicate the total measurement errors determined at the67% confidence level. The lower (black) bands in Fig. 3reproduce the computational results of Ref. [29] obtainedusing the Drude model (32). In fact, our computationalresults using the nonlocal response functions are almostcoinciding with the results of Ref. [29] using the dissi-pationless plasma model. In doing so, our results are ina good agreement with the measurement data over theentire range of separations which exclude the theoreticalpredictions using the Drude model over the separationrange from 235 to 420 nm.

An upgraded setup employing the atomic force micro-scope with increased sensitivity of the cantilever througha decrease of its spring constant was used in Refs. [35, 36]in more precise measurements of the Casimir force gra-dients up to larger separation distances. The importantproperty of an upgraded setup was an employment of thetwo-step cleaning procedure of the vacuum chamber andtest body surfaces by means of UV light followed by Ar-

240 260 280 300 320

20

30

40

50

60

70

350 400 450 500

5

10

15

20

a (nm)

F′ (µN/m

)F

′ (µN/m

)

(a)

(b)

FIG. 3: The gradients of the Casimir force between an Au-coated sphere and an Au-coated plate computed using thespatially nonlocal response functions and the Drude modelare shown as functions of separation by the upper (red) andlower (black) bands, respectively. The experimental data [29]are shown as crosses whose arms indicate the total errors de-termined at the 67% confidence level.

ion bombardment. The radius of the sphere used wasR = 43.47 µm.

The comparison between experiment and theory inRef. [36] was made as described above in this section us-ing Eq. (35) and the dielectric response (32) with γAu 6= 0(the Drude model) and γAu = 0 (the plasma model). Inthe measurements with smaller oscillation amplitude ofthe cantilever equal to 10 nm, the theoretical predictionsusing the Drude model were excluded over the range ofseparations from 250 to 820 nm whereas those using theplasma model were found to be in agreement with themeasurement data over the entire measurement range.

We have computed the gradient of the Casimir force inthe experimental configuration of Refs. [35, 36] using theproposed dielectric functions (10) with the same param-

eters of Au, as in these references, and vTr,LAu = 3vF,Au/2

as above. The computational results are shown by theupper (red) bands in Fig. 4 as a function of separation.The measurement data with their errors determined atthe 67% confidence level are shown as crosses. The the-oretical predictions obtained using the Drude model [36]are presented by the lower (black) bands. As is seen inFig. 4, our results, which take into account the dissipa-tion of free electrons, are in agreement with the data overthe entire separation region from 250 to 950 nm.

Another set of measurements was performed in

0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.4010

20

30

40

50

60

0.4 0.5 0.6 0.7 0.8 0.9

2

4

6

8

10

a (nm)

F′ (µN/m

)F

′ (µN/m

)

(a)

(b)

FIG. 4: The gradients of the Casimir force between an Au-coated sphere and an Au-coated plate computed using thespatially nonlocal response functions and the Drude modelare shown as functions of separation by the upper (red) andlower (black) bands, respectively. The experimental data [36]obtained with smaller oscillation amplitude of the cantileverare shown as crosses whose arms indicate the total errors de-termined at the 67% confidence level.

12

Ref. [36] with a larger oscillation amplitude of cantileverequal to 20 nm. This made it possible to get the meaning-ful measurement data at larger separation distances andexclude theoretical predictions using the Drude model upto 1.1 µm.

Our computational results for the gradient of theCasimir force obtained with the nonlocal dielectric func-tions (10) are presented by the upper (red) bands inFig. 5 over the range of separations from 0.6 to 1.3 µmof the data set in Ref. [36] measured with a larger os-cillation amplitude. They are in a good agreement withthe measurement data indicated as crosses. Our resultsaccounting for the dissipation properties of free electronsare almost coinciding with those obtained in Ref. [36]using the dissipationless plasma model, but deviate sig-nificantly from those obtained by means of the Drudemodel. The latter are shown by the lower (black) bands.

One can conclude that the Lifshitz theory employ-ing the proposed nonlocal dielectric permittivity is inequally good agreement with the measurement data ofprecise experiments performed by two different experi-mental groups by means of a micromechanical torsionaloscillator and an atomic force microscope using test bod-ies made of a nonmagnetic metal.

0.60 0.65 0.70 0.75 0.80 0.85 0.900.5

1.0

1.5

2.0

0.9 1.0 1.1 1.2 1.3

0.1

0.2

0.3

0.4

0.5

0.6

a (nm)

F′ (µN/m

)F

′ (µN/m

)

(a)

(b)

FIG. 5: The gradients of the Casimir force between an Au-coated sphere and an Au-coated plate computed using thespatially nonlocal response functions and the Drude modelare shown as functions of separation by the upper (red) andlower (black) bands, respectively. The experimental data [36]obtained with larger oscillation amplitude of the cantileverare shown as crosses whose arms indicate the total errors de-termined at the 67% confidence level.

THEORY-EXPERIMENT COMPARISON WITH

MAGNETIC TEST BODIES

The action of magnetic properties of the plate mate-rials on the Casimir force has attracted considerable at-tention in the literature. Thus, although in Refs. [2] and[3] the Lifshitz formulas were derived for nonmagnetictest bodies, they were rewritten with account of mag-netic properties in Ref. [99]. The Casimir force actingbetween an ideal metal plate and an infinitely permeableone was found in Ref. [100]. Thereafter the Casimir forcebetween one magnetic and one nonmagnetic plates, aswell as between two magnetic plates, was considered bymany authors (see, e.g., Refs. [12, 78, 101–103]). Thisproblem attracted special attention in connection withthe possibility of repulsive Casimir forces [42].

As was emphasized in Ref. [104], the magnetic proper-ties of boundary plates make an impact on the Casimirfree energy and pressure entirely through the zero-frequency terms of the Lifshitz formulas (2). This iscaused by the fact that the frequency-dependent mag-netic permeability becomes equal to unity at frequencieswhich are much smaller than the first Matsubara fre-quency at not too low temperature [105, 106].

We begin with an experiment of Ref. [30] where thegradient of the Casimir force was measured between anAu-coated sphere of R = 64.1 µm radius and a platecoated with a magnetic metal Ni by means of an atomicforce microscope. Note that in measuring of the Casimirforce using magnetic metals they are not magnetizedand do not give rise to a gradient of any additionalforce of magnetic origin [32]. In Ref. [30] the theoreti-cal force gradients were computed by Eq. (35) with theFresnel reflection coefficients (3) and spatially local di-electric permittivities (32) with γAu 6= 0 (the Drudemodel) and γAu = 0 (the plasma model) at room tem-perature. The following values of parameters for Au(n = 1) and Ni (n = 2) have been used: ~ωp,1 =~ωp,Au = 9.0 eV, ~γ1 = ~γAu = 35.0 meV [40, 41, 88]and µ1(iξl) = µAu(iξl) = 1; ~ωp,2 = ~ωp,Ni = 4.89 eV,~γ2 = ~γNi = 43.6 meV [88, 107] and µ2(0) = µNi(0) =110, µ2(iξl) = µNi(iξl) = 1 for l > 1. The values of

ε(1)c (iξl) = ε

(Au)c (iξl) and ε

(2)c (iξl) = ε

(Ni)c (iξl) were ob-

tained from the optical data of Au and Ni, respectively,as described in Refs. [29, 30, 32, 40, 41].

According to the results of Ref. [30], within the mea-surement range from 220 to 500 nm the theoretical pre-dictions of the Lifshitz theory using the Drude and theplasma models are almost coinciding and are in a goodagreement with the measurement data. (Note that atlarger separations of a few micrometers the gradients ofthe Casimir force between a nonmagnetic-metal sphereand a magnetic-metal plate computed using the Drudeand plasma models are different [104].) The predictionsobtained by means of the plasma model with magnetic

13

properties of the Ni plate disregarded [µNi(iξl) = 1 forall l > 0] are excluded by the measurement data over therange of separations from 220 to 420 nm. If the Drudemodel is used in computations, the obtained results donot depend on whether the magnetic properties of Niplate are included or omitted.

Here, we compute the gradient of the Casimir force inthe experimental configuration of Ref. [30] using Eq. (35)and the spatially nonlocal dielectric permittivities (10).When computing the Casimir pressure, we have usedthe impedance reflection coefficients (5) for Au and Nileading to Eqs. (14), (15) and (19), (20) for l = 0 andEqs. (23), (25) for l > 1. The same parameters for Auand Ni, as indicated above have been used and the Fermivelocity for Ni was found from Eq. (33) under an assump-tion of the spherical Fermi surface, vF,Ni = 1.31×106 m/s(for Au we used vF,Au = 1.78 × 106 m/s employed inSec. IV and V). As in all previously considered exper-iments, the best agreement between the measurementdata and theoretical predictions is reached for vTrAu =vLAu = 3vF,Au/2 and vTrNi = vLNi = 3vF,Ni/2.

Computational results obtained using the spatiallynonlocal dielectric permittivities (10) are shown by thesolid (red) bands in Fig. 6 where the experimental datawith their total errors determined at the 67% confidencelevel are presented as crosses. As is seen in Fig. 6, the the-oretical predictions of the Lifshitz theory employing theproposed nonlocal dielectric functions are in a very good

220 240 260 280 300 320

40

60

80

100

120

350 400 450 5005

10

15

20

25

30

a (nm)

F′ (µN/m

)F

′ (µN/m

)

(a)

(b)

FIG. 6: The gradients of the Casimir force between an Au-coated sphere and a Ni-coated plate computed using the spa-tially nonlocal response functions are shown by the solid (red)bands as a function of separation. The experimental data [30]are shown as crosses whose arms indicate the total errors de-termined at the 67% confidence level.

agreement with the measurement data over the entirerange of experimental separations. For the configurationof Au-Ni test bodies in the range from 220 to 500 nmalmost the same theoretical predictions in equally goodagreement with the measurement data are obtained whenthe conduction electrons are described by the spatiallylocal Drude or plasma model.

Next, we consider the experiment of Refs. [31, 32] onmeasuring the gradient of the Casimir force between asphere of R = 61.71 µm radius and a plate both coatedwith a magnetic metal Ni performed by means of anatomic force microscope. In these references computa-tions of the force gradient using the standard Lifshitztheory were performed as in Sec. V but with the pa-rameters ωp,1 = ωp,2 = ωp,Ni, γ1 = γ2 = γNi, and

ε(1)c (iξl) = ε

(2)c (iξl) = ε

(Ni)c (iξl) presented above. An im-

portant result found in Ref. [31] is that for two magneticmetals the gradients of the Casimir force calculated usingthe Drude model are larger than those found by means ofthe plasma model. This is different from the case of twoAu plates (see Figs. 3–5). It was shown [31, 32] that thetheoretical predictions obtained with the Drude modelare excluded by the data over the separation region from223 to 420 nm, whereas similar predictions made with thehelp of a plasma model are in a very good agreement withthe measurement results. Similar to the case of test bod-ies coated with Au films, this result is puzzling becauseat low frequencies the relaxation properties of conductionelectrons are well studied in many physical phenomenaother than the Casimir effect.

Here, we computed the gradient of the Casimir forcebetween the Ni-coated surfaces of a sphere and a plateby Eq. (35). The Casimir pressure in this equation wasfound from the second line in Eq. (2), reflection coef-ficients (5), and impedance functions (7), as describedabove, using the spatially nonlocal dielectric permittivi-ties (10) with the same parameters of Ni as above. Thecomputational results are shown by the lower (red) bandsin Fig. 7 as a function of separation. They are in a verygood agreement with the measurement data indicated ascrosses. As in all other experiments using an atomic forcemicroscope, the total experimental errors are determinedat the 67% confidence level. The upper (black) bands inFig. 7 reproduce the results of Refs. [31, 32] computed bythe Lifshitz formula and the spatially local Drude model(32). It is seen that these results are excluded by themeasurement data over the separation region from 223to 420 nm in accordance with the conclusion made inRefs. [31, 32]. The point is that at short separations upto a few hundred nanometers the force gradients com-puted using the local plasma model, which disregardsthe relaxation of free electrons, and the spatially non-local dielectric functions (10), which take relaxation intoaccount, are almost coinciding. In this situation, a failureof the Drude model may be explained by an inadequate

14

240 260 280 300 320 34020

40

60

80

100

120

360 380 400 420 440 460 480 500

5

10

15

20

a (nm)

F′ (µN/m

)F

′ (µN/m

)(a)

(b)

FIG. 7: The gradients of the Casimir force between a Ni-coated sphere and a Ni-coated plate computed using the spa-tially nonlocal response functions and the Drude model areshown as functions of separation by the lower (red) and upper(black) bands, respectively. The experimental data [31, 32]are shown as crosses whose arms indicate the total errors de-termined at the 67% confidence level.

description of the dielectric response to electromagneticwaves off the mass shell.

CONCLUSIONS AND DISCUSSION

In this paper, we have proposed the phenomenologi-cal spatially nonlocal dielectric functions which providenearly the same response to electromagnetic waves onthe mass shell, as does the standard Drude model, butrespond differently to the off-the-mass-shell fields. Unlikethe previously suggested response functions of this kind[73, 74, 76], the permittivities presented here depend onall the three components of the wave vector which is amore general case in the approximation of specular re-flection used.

As discussed in Sec. I, the problem of disagreementbetween theoretical predictions of the fundamental Lif-shitz theory using the well-tested Drude model with theexperimental data for metallic test bodies remains unre-solved for almost 20 years. Many attempts of its resolu-tion have been undertaken (including a consideration ofthe frequency-dependent relaxation parameter [108], i.e.,the so-called Gurzhi model), but the problem is as yet un-resolved. Similar problem arises for dielectric materials[45, 109]. All this makes it warranted to consider somephenomenological approaches suggested by analogy with

graphene which, due to its simplicity in comparison withmetallic materials, allows the fundamental calculation ofits spatially nonlocal dielectric response based on the firstprinciples of quantum electrodynamics at nonzero tem-perature.

Using this line of reasoning, the surface impedancesand reflection coefficients determined by the proposednonlocal dielectric functions have been found in the ap-proximation of specular reflection. This made it possi-ble to calculate the effective Casimir pressure betweentwo parallel metallic plates, the Casimir force betweena sphere and a plate, and its gradient predicted by thesuggested approach in configurations of several preciseexperiments performed during the last 15 years by twodifferent experimental groups by means of micromechan-ical torsional oscillator and atomic force microscope. Indoing so the experiments with both nonmagnetic andmagnetic test bodies were considered.

It was shown that the suggested spatially nonlocaldielectric functions taking into account the dissipationproperties of conduction electrons bring the Lifshitz the-ory to equally good agreement with the measurementdata of all the performed experiments as does the plasmamodel which disregards the dissipation of conductionelectrons. Good agreement with seven considered ex-periments, which were performed between both nonmag-netic and magnetic metallic surfaces (Au-Au, Au-Ni, andNi-Ni), was reached with the velocity parameters com-mon to the transverse and longitudinal permittivitiesvTrn = v L

n = vn = 3vF,n/2, where the Fermi velocitiesvF,n are determined in the approximation of a sphericalFermi surface. In doing so, the theoretical predictionsare rather sensitive to the value of vTrn but are almostindependent on v L

n in the region from 0 to 10vF,n. In theregions of experimental separations these predictions dif-fer from the previously made experimentally consistentpredictions using the dissipationless plasma models onlyin the limits of measurement errors.

The single precise experiment which was not comparedwith theoretical predictions of the suggested nonlocal ap-proach to calculation of the Casimir force is the mea-surement of differential Casimir force between the Ni-Nisurfaces of a sphere and a plate performed by means ofa micromechanical torsional oscillator [33]. Taking intoaccount, however, that experiments by means of both anatomic force microscope and micromechanical torsionaloscillator are in a good agreement with this calculationapproach for Au surfaces, the results of Sec. VI for twomagnetic surfaces can be safely extended to the experi-ment of Ref. [33].

To conclude, the suggested spatially nonlocal dielectricfunctions include the dissipation of conduction electronsleading to only negligibly small deviations from the stan-dard Drude dielectric response in the area of propagat-ing waves on the mass shell, satisfy the Kramers-Kronigrelations, and bring the theoretical predictions of the Lif-

15

shitz theory in agreement with the measurement data ofall precise experiments on measuring the Casimir force.In the future it is desirable to provide some grounding intheory to these permittivities based, e.g., on the polariza-tion tensor in (3+1)-dimensions [110], or quantum fieldtheoretical approach to correlation functions and Boltz-mann kinetic theory [111].

ACKNOWLEDGMENTS

This work was partially supported by the Peter theGreat Saint Petersburg Polytechnic University in theframework of the Russian state assignment for basic re-search (project No. FSEG-2020-0024). This paper hasbeen supported by the Kazan Federal University Strate-gic Academic Leadership Program. The authors aregrateful to C. Henkel and V. B. Svetovoy for useful dis-cussions.

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