Casimir interaction between inclined metallic cylinders

0.74 MB·English

by  P. Rodriguez-Lopez

Casimirinteractionbetweeninclinedmetalliccylinders
Pablo Rodriguez-Lopez1,2 and Thorsten Emig3
1DepartamentodeF´ısicaAplicadaIandGISC,FacultaddeCienciasF´ısicas,UniversidadComplutense,28040Madrid,Spain
2DepartamentodeMatema´ticasandGISC,UniversidadCarlosIIIdeMadrid,AvenidadelaUniversidad30,28911Legane´s,Spain
3Laboratoire de Physique The´orique et Mode`les Statistiques,
CNRS UMR 8626, Baˆt. 100, Universite´ Paris-Sud, 91405 Orsay cedex, France
TheCasimirinteractionbetweenone-dimensionalmetallicobjects(cylinders,wires)displaysunconventional
features.HerewestudytheorientationdependenceofthisinteractionbycomputingtheCasimirenergybetween
two inclined cylinders over a wide range of separations. We consider Dirichlet, Neumann and perfect metal
boundary conditions, both at zero temperature and in the classical high temperature limit. For all types of
boundary conditions, we find that at large distances the interaction decays slowly with distance, similarly to
thecaseofparallelcylinders,andatsmalldistancesscalesastheinteractionoftwospheres(butwithdifferent
numericalcoefficients). Ournumericalresultsatintermediatedistancesagreewithouranalyticpredictionsat
2 smallandlargeseparations.Experimentalimplicationsarediscussed.
1
0 PACSnumbers:45.70.-n,45.70.Mg
2

n
a I. INTRODUCTION calobjectsmentionedbeforewhereentropiceffectscompete
J with the direct interaction. In the limit where the length of

1 Collectivephenomenainchargedensityandcurrentfluctu- the objects is much smaller than their distance, the orienta-
3 ationsareknowntogenerateCasimir-Lifshitzinteractions[1, tiondependencehasbeenstudiedformetallicspheroids[13].
2]thatareunconventionallylong-rangedforone-dimensional Here we are interested in the opposite limit where the decay
]
h metallicobjectsascylindersorwires[3,4].Theseinteractions oftheinteractionisexpectedtobemuchslower[4,14]. The
p attractcurrentlysubstantialinterestsinceCasimirforcescould case of non-parallel, infinitely long cylinders has been stud-
- playanimportantroleformicroandnanodevices[5]. Ingen- ied by making additivity assumptions [7]. Recently, the van
t
n eral,thebuildingelementsofsuchdevicescannotbemodeled derWaalsinteractionbetweencrossedcylinders,ignoringre-
a asplanarsurfaces(asinRef.1and2);morecommonshapes tardation effects, has been computed in the asymptotic large
u are one-dimensional structures as wires or beams. Interac- distancelimit[15]. Theforcebetweeninclinedcylindershas
q
tionsofCasimiror(non-retarded)vanderWaalstypebetween been derived from a dilution process for anisotropic dielec-
[
quasione-dimensionalshapesarealsoimportantinbiological tricmedia[16]andtheresultsresemblepair-wisesummation

1 systemswhererodlikeparticlesauchasDNA,viruses,ormi- results. Advanced numerical techniques have been used to
v crotubolesinteractduetocorrelatedchargefluctuations[6,7]. studythecaseoftwoperpendicularcylindersoffinitelength
8
Duetocorrelationeffectsthatareparticularlystrongforcon- L (capsules) [17]. When L is increased, the energy is found
4
6 ducting objects, the interaction is not properly described by to approach an L-independent value. Our numerical results
6 pairwisesummationapproaches. Fornon-conductingobjects, forsufficientlysmalldistances(whereafiniteLbecomesless
. thesesummationschemesusuallyyieldthecorrectscalingof important)areconsistentwiththereportedvalues.
1
theinteractionenergywithdistanceanddimensionoftheob- The Casimir force between crossed metal cylinders has
0
2 jects. However,formetalsthescalingisdifferentandcanbe beenmeasured[18].Morerecently,Deccaetal.discussedthe
1 rather sensitive to the description of the material properties possibilitytomeasurethethermalCasimirforceanditsgradi-
: [8,9]. Proximityforceapproximations(PFA)arerestrictedto entbetweenaplateandamicrofabricatedcylinderattachedto
v
i short separations [10] but for cylindrical shapes it is impor- amicromachinedoscillator[19].
X tanttostudyinteractionsalsoatlargerseparationsduetotheir Beforeweconsidertheinteractionatarbitraryseparations,
r slowdecay. it is instructive to review the prediction of the PFA. This
a approximation is often applied to describe the interaction at
A recently developed scattering approach can be used to
compute the interaction between various shapes, including surface-to-surfacedistancesl=d−2Rmuchshorterthanthe
cylinders,overalargerangeofseparations[11,12].Inthefol- radii of curvature. For parallel cylinders of length L → ∞,
thisapproximationyieldsatzerotemperature[14]
lowingweshallemploythelattertechniquetostudytheorien-
tationdependenceoftheCasimirinteractionbetweenmetallic (cid:114)
π3 R
cylinders, both for a scalar field obeying Dirichlet or Neu- EPFA =− (cid:126)cL , (I.1)
mann boundary conditions and for the electromagnetic field 0,(cid:107) 1920 l5
with perfect metal boundary conditions. The temperature T
andinthehightemperaturelimit
isassumedbeeitherzeroorintheclassicallimitk T (cid:29) (cid:126)c.
B
Anunderstandingofthesensitivitytoorientationisimportant (cid:114)
ζ(3) R
since the measurement of cylinder interactions can be ham- EcPl,FA =− 16 kBTL l3 (I.2)
pered by deviations from parallelism due to external pertur- (cid:107)
bations. Orientationdependenceplaysalsoaroleinmixtures so that one has a finite energy per length in the limit of in-
ofcylindricalshapesascarbonnanotubesandofthebiologi- finitelylongcylinders. Forinclinedcylinderswithinclination
2
angleθ ∈ [0,π/2],cf.Fig.1,weobtain(seeApp.B)atzero
temperature
π3 (cid:126)c R
EPFA =− , (I.3)
0,θ 720sin(θ)l2
andinthehightemperaturelimit
ζ(3) k T R
EPFA =− B . (I.4)
cl,θ 4 sin(θ) l
TheenergyisnolongerextensiveinLbuthasasimpleorien-
tationdependencewithadivergence∼ 1/θ fortheapproach
of the parallel configuration. In the limit of close approach,
l → 0, the energy of inclined cylinders is less divergent in l
thanintheparallelsetup. Forinclinedcylindersatshortdis-
tance one would expect a scaling of the PFA energy that re-
semblestheonefortwocompactobjectsas,e.g.,twospheres.
Indeed, the scaling of the PFA energies of two spheres with
d
l and R is the same as in Eqs. (I.3), (I.4). In the following,
weshalldescribehowtheinteractionatlargerseparationsde-
viates from this simple PFA estimates. We find that the in-
teractions are no longer simple power-laws in l but acquire Figure1.(coloronline)Definitionofdistanceandorientationoftwo
logarithmicfactorsandthatforinclinedcylindersthedecayat inclinedcylinderswithaxis-to-axisseparationdandinclinationan-
verylargel(cid:29)Risevenslowerthanforparallelcylinders. gleθ.
The rest of this work is organized as follows. In Sec. II
we describe the system and review briefly the scattering ap-
proachthatweuseinthefollowingsectionsandderivetheso- inthecoordinatesystemofobjectitooutgoingwavesinthe
calledtranslationmatricesthatcouplemultipolemomentsof systemofobjectj.
inclinedcylinders.InthefollowingSec.IIIweobtainanalytic In the high temperature limit, k T (cid:29) (cid:126)c, only the first
B
resultsatasymptoticallylargeseparationswhileinSec.IVwe Matsubarafrequencycontributes,andtheenergycanbewrit-
computenumericallytheinteractionatintermediatedistances tenas
andcompareittoPFApredictions. Finally,wepresentacon-
clusionanddiscussionofexperimentalimplicationsinSec.V. lim E = kBT logdet[I−N(0)]=−kBT (cid:88)∞ 1Tr[Np(0)],
Mathematicaldetailsofthederivationoftranslationmatrices T 2 2 p
→∞ p=1
and the PFA formulae are provided in App. A and App. B,
(II.6)
respectively.
wheretherelationlogdetA = TrlogAandtheexpansionof
log(1−x)forsmallxhavebeenused. Thelatterexpression
correspondstoamultiplescatteringexpansionsinceeachfac-
II. SCATTERINGAPPROACH tor of N describes two scattering events, one at each of the
two objects. The Casimir energy at zero temperature can be
A. GeometryandCasimirenergy writtenalsoasamultiplescatteringexpansion,
Recently,ascatteringapproachforCasimirinteractionsbe- E = (cid:126)c (cid:90) ∞dκlogdet[I−N(κ)]
0
2π
tween non-planar objects has been developed [11, 12]. It 0
relates the Casimir energy to the electromagnetic scattering =−(cid:126)c (cid:88)∞ 1(cid:90) ∞dκTr[Np(κ)] . (II.7)
propertiesofeachobject. TheCasimirfreeenergyattemper-
2π p
atureT canbewrittenas p=1 0
For cylinders, the T matrices are generally known [12]. The
E =k T(cid:88)∞ (cid:48)logdet[I−N(κ )], (II.5) Umatricesareonlyknownforcylindricalcoordinatesystems
B n
withparallelaxes[12]. Fornon-parallelcylinders,Eqs.(II.6)
n=0
and(II.7)remainsvalid. ButnowtheUmatricesdependnot
wheretheprimedsumrunsoverMatsubarafrequenciesκ = only on the distance d between the cylinder axes but also on
n
2πk T/((cid:126)c)withthen=0termweightedby1/2.Forasys- their relative orientation, described by the inclination angle
B
temoftwoobjects,thismatrixisN=T U T U . HereT θ. Inparticular, wewillstudythecaseofatranslationalong
1 12 2 21 i
istheT-matrixoftheith object, whichaccountsforthegeo- androtationaboutthey-axiswithx = R x+dˆywherethe
(cid:48) θ
metricalandmaterialpropertiesoftheobject. U aretransla- matrixR describesarotationaboutthey-axisbyanangleθ,
ij θ
tionmatricesthatdescribetheconversionfromregularwaves seeFig.1.
3
B. Translationmatrices coordinatesystems. Theoutgoingvectorwavesaredefinedin
termsoftheoutgoingscalarcylinderwavesas
In this section we derive the translation matrices U for
scalarandelectromagneticpartialwavesinnon-parallelcylin- Mout = 1∇×(cid:0)φout ˆz(cid:1) , (II.13)
n,kz p n,kz
der coordinate systems. In particular we transform outgoing
cylindrical waves with imaginary frequency ω = iκ to reg- Nout = 1 ∇×(cid:0)φout ˆz(cid:1) (II.14)
ular cylindrical wave in a coordinate system translated by a n,kz κp n,kz
distancedandrotatedbyanangleθ.
andequivalentlyforregularvectorwaves. Toderivethetrans-
lationmatrices,wefirstmultiplybothsidesofEq. (II.10)by
1. Scalarwaves ˆz(cid:48) andapply p1(cid:48)∇(cid:48)×onbothsidesofthisequation. Thenthe
left hand side gives the components of Mout in the coordi-
n,kz
nateframe(x,y ,z ). Toobtainthecomponentsintheframe
(cid:48) (cid:48) (cid:48)
We consider regular and outgoing cylindrical wave func-
(x,y,z)wemultiplybothsideswiththeinverserotationma-
tionswhichontheimaginaryfrequencyaxisaregivenby trixR−θ1. Thisyields,using∇(cid:48) =Rθ∇,
φreg (x)=I (ρp)einθeikzz, (II.8) (cid:90)
n,kz n Monu(cid:48)t,kz(cid:48)(x(cid:48))=R−θ1M(cid:48)onu(cid:48)t,kz(cid:48)(x(cid:48))= p1 (cid:88)∞ 2Lπ ∞ dkz
φonu,tkz(x)=Kn(ρp)einθeikzz, (II.9) ·Un(cid:48)kz(cid:48),nkz(d,θ)∇×(cid:104)φrne,gkz(cid:48)(xn)=(−c∞os(θ)ˆz−−∞sin(θ)ˆx)(cid:105),
(cid:112) (cid:112) (II.15)
withp= κ2+k2andρ= x2+y2.Theoutgoingwaves
z
atpositionx =R x+dˆycanbeexpandedintermsofregular
(cid:48) θ where we have used that ˆz(cid:48) = −sin(θ)ˆx + cos(θ)ˆz. This
wavesatxbythelinearrelation (cid:16) (cid:17)
showsthatweneedtoexpress∇× φreg (x)ˆx intermsof
n,kz
φonu(cid:48)t,kz(cid:48)(x(cid:48))= (cid:88)∞ 2Lπ (cid:90) ∞ dkzUn(cid:48)kz(cid:48),nkz(d,θ)φrne,gkz(x), Mˆxirnne,gkczy(lxin)daenrdcoNornre,gdkizn(axt)e.s)Otnheatcanshow(byexpressing∇and
n=−∞ −∞ (II.10) (cid:16) (cid:17) ik (cid:16) (cid:17)
mwhatihchemdaetfiicnaelsdtehtaeiltsraonfsltahteiodnermivaattriioxnUofn(cid:48)tkhz(cid:48)e,ntkrazn(dsl,aθti)o.n Tmhae- ∇× φrne,gkz(x)ˆx =− 2z Mrne−g1,kz(x)+Mrne+g1,kz(x)
iκ(cid:16) (cid:17)
trix aregiven in App.A. For θ > 0, thematrix elements are + Nreg (x)−Nreg (x) .
givenby 2 n−1,kz n+1,kz
(II.16)
2π(−i)n+n(cid:48) (cid:18) (cid:113) (cid:19)n(cid:48) WhenthisresultissubstitutedintoEq.(II.15),weneedtoshift
Un(cid:48)kz(cid:48),nkz(d,θ)= L sinθ ξ(cid:48)− ξ(cid:48)2+1 theindexninordertoexpresstherighthandsideintermsof
√ vector waves with the same index n. This can be done by
×(cid:16)ξ−(cid:112)ξ2+1(cid:17)−n e−d k(cid:48)2x+p(cid:48)2, using
(cid:113)
2 k(cid:48)2x+p(cid:48)2(II.11) Un(cid:48)kz(cid:48),n±1kz(d,θ)=∓i(cid:16)ξ−(cid:112)1+ξ2(cid:17)∓1Un(cid:48)kz(cid:48),nkz(d,θ).
(II.17)
(cid:113) Thisyieldsaftersomeelementaryalgebrathetranslationfor-
where p = κ2+k 2, ξ = k /p, ξ = k /p, k =
(cid:48) z(cid:48) x (cid:48) x(cid:48) (cid:48) x mulaforvectorwaves,
(cos(θ)k −k )/sin(θ) and k = (k −cos(θ)k )/sin(θ).
z z(cid:48) x(cid:48) z z(cid:48)
sT(FreohoeretaAθttrip<aopnn.s0laAatnht.dieWotnorhvameennrasaltwalrlitxeisoiiegnnln)edmaiocrfeaeUnteotnsbt(cid:48)htfkaoaz(cid:48)irt,nnttekhhdzee(bitdnry,avθnde)srls→maetuitors−antndbms,efθaocthrr→maixnagct−ieoodθrn-., (cid:32)(cid:18)McNoononsuu(cid:48)((cid:48)t,t,θkkz(cid:48))z(cid:48)−(cid:19)s(ixn(cid:48)()θ=)knz=(cid:88)ξ∞−∞sin2Lπ(θ(cid:90))−κ∞(cid:112)∞1dk+zUξ2n(cid:48)kz(cid:33)(cid:48),(cid:18)nkzM(dre,gθ)pp(cid:19)(cid:48)
respondstotheinversecoordinatetransformationbytheargu- −sin(θ)κ(cid:112)1+pξ2 cos(θ)−psin(θ)kzξ Nrneg,kz (x).
ments(−d,−θ),wehave p p n,kz
(II.18)
Un(cid:48)kz(cid:48),nkz(−d,−θ)=(−1)n+n(cid:48)Un(cid:48)kz(cid:48),nkz(d,θ). (II.12) HerewehaveusedthatN = κ1∇×Mand κ1∇×N = −M
to obtain the translation formula for N. The translation for-
mula for the inverse coordinate transformation is given by
2. Electromagneticwaves Eq. (II.18) with Un(cid:48)kz(cid:48),nkz replaced by (−1)n+n(cid:48)Un(cid:48)kz(cid:48),nkz,
see Eq. (II.12). Notice that the inverted sign of sinθ in the
FortheEMfieldoneexpectsthatthetranslationmatrixcou- expression for z is compensated by sign changes that result
(cid:48)
ples the polarizations due to the different orientation of the fromEqs.(II.17)and(II.12).
4
C. Scatteringamplitudes(T-matrices) sothataneffectivelengthL overwhichtheinclinedcylin-
eff
ders interact in the limit θ → 0 can be determined by the
1. Scalarfield relationsinθ =πd/Leff.
Inordertostudythehightemperaturelimitk T (cid:29) (cid:126)cwe
B
need to consider only the zero Matsubara frequency contri-
For Dirichlet (D) and Neumann (N) boundary conditions,
bution to the energy. A simple scaling analysis shows that
the scattering amplitudes of a cylinder of radius R are given
N (κ = 0) ≈ 1/|k | for k → 0 with logarith-
bytheexpressions, 0kz,0kz z z
mic corrections. Hence the trace of N is not well defined,
TDn(cid:48)kz(cid:48),nkz =−2LπKInn((ppRR))δn,n(cid:48)δ(kz−kz(cid:48)), (II.19) idK.eel.ti,c[IhN−s0hkoNzw,00ekkdzz,(0tκhkazt=(κN0=is)a0ist)r]naicosetncaolattrswasceoelplcedlraeasfitosnreofdpo.errKcaoteomnrn,pesatochttaohnbad-t
2π I (pR)
TNn(cid:48)kz(cid:48),nkz =− L Kn(cid:48)n(cid:48)(pR)δn,n(cid:48)δ(kz−kz(cid:48)), (II.20) pjercotbsl[e2m0]o.fWtwhiolecpoamraplalcetlcdyislicnsd,etrwsocotinlstetidtucteyleinffdeecrtsivaerleyano2nD-
compactobjectswhosegeometrycannotbereducedtoalower
whereI ,K aremodifiedBesselfunctions.
n n dimensionaloneofcompactobjects. However,weexpectthe
forcebetweentiltedcylinderstobewelldefined. Thisimplies
that the operator ∂ N (κ = 0) should be a trace class
2. Electromagneticfield operator.Thisisinddee0dktzh,0ekczase.InthehighT limitweobtain
theforce
Foraperfectmetalcylinder, thescatteringamplitudedoes
not couple electric (E) and magnetic (M) polarizations. The F =− πkBT . (III.24)
E(M)polarizationisdescribedbyD(N)boundaryconditions T 4dsin(θ)log2(d/R)
sothatthescatteringamplitudeisgivenby
The energy can be obtained from the force by integration,
TEn(cid:48)Ekz(cid:48),nkz =TDn(cid:48)kz(cid:48),nkz, TMn(cid:48)kMz(cid:48),nkz =TNn(cid:48)kz(cid:48),nkz. (II.21) ET =(cid:82)d∞dd(cid:48)FT(d(cid:48)),leadingto
πk T
E =− B . (III.25)
III. LARGEDISTANCEEXPANSION T 4sin(θ)log(d/R)
This form of Casimir interaction appears to be the one with
In this section we obtain a large distance expansion of the
theweakestdecayknowntodate.
scalarandelectromagneticCasimirenergiesatzeroandhigh
temperature. The approximations performed in this section ForNeumannboundaryconditionsnologarithmicinterac-
are valid if the cylinder radius that is small compared to the tion appears but the energy is proportional to the product of
distancebetweenthecylinders. Toperformtheexpansionwe the cross-sectional areas of the cylinders. In Eqs. (II.6) and
apply the relation logdetA = TrlogA and expand log(I− (II.7)wehavetoconsiderthetermswithp = 1,|n| ≤ 1and
N) ≈ −N in Eqs. (II.6) and (II.7). In addition to that, we |n| ≤ 1 since I (z)/K (z) = −z2 +O(cid:2)z4(cid:3) for |n| ≤ 1.
(cid:48) n(cid:48) n(cid:48) 2
perform an expansion of the scattering amplitudes T in the WiththisexpansionwegettolowestorderinR/dtheenergy
radiusR. atzerotemperature,
(cid:126)cR4
E =− [167+cos(2θ)], (III.26)
A. Scalarfield 0 320d5sin(θ)
To obtain the asymptotic Casimir energy for Dirichlet whichcanbecomparedtothecorrespondingenergyforpar-
boundary conditions at large distances d (cid:29) R, we need to allelcylinders[14],
consider only the terms with n = n = 0 and p = 1
of Eqs. (II.6) and (II.7). Using for the(cid:48)small radius expan- 7(cid:126)cLR4
sion of the scattering amplitude T that for small z one has E0(cid:107) =− 5πd6 (III.27)
I0(z)/K0(z) ≈ −log−1(z), we get for the energy at zero
so that the effective length L in the limit θ → 0 is deter-
temperature eff
minedbysinθ =(3π/8)d/L . Inthehightemperaturelimit
eff
(cid:126)c theenergycanbeobtaineddirectlysinceN(κ = 0)isatrace
E =− . (III.22)
0 8dsin(θ)log2(d/R) classoperator,yielding
The corresponding energy for parallel cylinders of length E =− 3πkBTR4 [98+cos(2θ)]. (III.28)
L→∞is[12] T 1024d4sin(θ)
(cid:126)cL IncontrasttotheDirichletcase, theenergyhasamorecom-
E0(cid:107) =−8πd2log2(d/R), (III.23) plicatedorientationdependence.
5
B. Electromagneticfield peraturelimitwhereallmatrixelementsarecomputedinthe
limit κ → 0, the polarizations are decoupled at all distances
since the matrix elements that couple different polarizations
areproportionaltoκ,seeEq.(II.18). Wefindthattheasymp-
We calculate the energies with the help of the expansions totic Casimir energy for perfect metal cylinders at zero tem-
given in Eqs. (II.6), (II.7). In general, the translation ma- peraturecanbewrittenas
trix of Eq. (II.18) mixes the two polarizations. However, the
leading part of the energy at far distance is given by the E- (cid:126)cΩ(θ)
E =− , (III.29)
polarizedwavesonlyduethelogarithmicdependenceonfre- 0 8dsin(θ) log2(d/R)
quency of the scattering amplitude for E modes with n = 0
andapower-lawdependenceforM modes. Inthehightem- wherethefunctionΩ(θ)isdefinedas
1 (cid:90) π (cid:90) 2π (cosθsinϑ+cosϑsinϕsinθ)2
Ω(θ)= dϑsinϑ dϕ (III.30)
4π cos2ϕsin2ϑ+(cosϑsinθ+sinϑsinϕcosθ)2
0 0
The orientation dependence described by Ω(θ) results from canbeseenbyconsideringthelimitκ → 0ofthegeometric
thegeometricfactorcosθ−(k ξ/p)sinθ thatappearsinthe factor(p/p)[cosθ−(k ξ/p)sinθ]whichisunity. Hence,in
z (cid:48) z
couplingoftheEpolarizationinEq.(II.18).WehaveΩ(0)= thehightemperaturelimit,theinteractionispreciselythesum
1 so that for θ → 0 the Dirichlet result given in Eq. (III.22) of the Dirichlet and Neumann energy. At large separations,
isrecovered. ForperpendicularcylindersonehasΩ(π/2) = theDirichletcontributiondominates. Thisleadsfortheelec-
1−log(2). AtintermediateanglesthefunctionΩ(θ)canbe tromagneticcasetothesameproblemasforaDirichletscalar
computedbynumericalintegration. Thecorrespondingresult fieldduetothefactthatN atκ=0isnotatraceclass
0kz,0kz
is shown in Fig. 2. The function Ω(θ) can be also expanded operator.
(cid:80)
asaFourierseriesΩ(θ) = ∞n=0Ω2ncos(2nθ)withthelow However,asinthescalarcase,weareabletoobtainthehigh
ordercoefficientsgivenbyΩ =0.6137,Ω =0.3333,Ω = temperaturelimitoftheforce,because∂ N (κ=0)is
0 2 4 d 0kz,0kz
0.0333andΩ =0.0096. a trace class operator. The far distance approximation of the
6
forceis
(cid:87)(cid:72)Θ(cid:76)
1.0 F =− πkBT , (III.31)
T 4dsin(θ)log2(R/d)
0.8
andthecorrespondingenergyisobtainedbyintegrationas
πk T
0.6 E =− B , (III.32)
T
4sin(θ)log(d/R)
0.4 which is identical to the Dirichlet result of Eq. (III.25). It
should be mentioned that this slow decay with distance is a
consequence of the metallic response of the cylinders. For
0.2
non-conductingcylindersapower-lawdecayisexpected.
0.0 Θ
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
IV. NUMERICALRESULTSFORSMALLER
SEPARATIONS
Figure2. AmplitudefunctionΩ(θ)oftheasymptoticelectromag-
neticCasimirenergyofEq.(III.30)asfunctionofinclinationangle
θ. InthisSectionwepresentnumericalresultsfortheinterac-
tionbetweeninclinedcylinders. Thisallowsustogobeyond
thelimitofasymptoticallylargeseparations. Wefocusonthe
Next,weconsiderthehightemperaturelimit. Inthislimit, zero temperature with the energy given by Eq. (II.7). The
at all separations the polarizations are decoupled since the broken translational symmetry of inclined cylinders leads to
couplingisproportionaltoκ. Thisimpliesthattheinteraction amatrixNthatisnon-diagonalinthecontinuousindexk . In
z
is the sum of the energy for Dirichlet and Neumann bound- contrasttocompactobjectsorsystemswithtranslationalsym-
ary conditions. However, the presence of a (n-independent) metry(parallelcylinders),onehastocomputethedeterminant
geometricfactorinthetranslationmatrixelementsforbothE of a non-diagonal operator over a continuous index. There
andM polarizations, seeEq.(II.18), couldmodifytheinter- are basically two approaches to obtain an approximation for
action in the electromagnetic case. That this is not the case thedeterminant. Onecanapproximatethedeterminantbythe
6
traceofaseriesofpowersofN[seeEq.(II.7)]whichisknown E0/E0P,FθA θ = π PFA
asmultiplescatteringexpansionorperformadiscretizationin 2
2.0 PFA-corr
k andcomputethedeterminantovertheresultingfinitesetof
z
discrete variables. While the first approach is limited to suf- Numeric
ficientlylargeseparationssincethemultiplescatteringseries 1.5 d R
(cid:29)
needstobetruncatedatsomeorder, thesecondapproachre-
quires enough discretization points in order to minimize the
1.0
discretization error. We have decided to follow the second
route. This discussion shows that the evaluation of the in-
teractionbetweeninclinedcylindersissubstantiallymoreex- 0.5
pensive in terms of numerical operations than that of paral-
lelcylinders. TheseconddiscreteindexnofthematrixNis
R
truncatedatsomevaluenmaxasinthecaseofcompactobjects 0.1 0.2 0.3 0.4 0.5 d
[11].Thechoiceofn dependsontherangeofdistancesfor
max
which one wouldlike to compute theenergy. n increases
max
withdecreasingminimaldistance. Forourresultsshownbe- Figure3.(coloronline)ElectromagneticCasimirenergynormalized
tothePFAenergyasfunctionoftheinversedistanceforperpendic-
low, we found it sufficient to consider |n| ≤ 3 = n . We
max ularcylinders(θ = π/2). Thedashedcurverepresentstheasymp-
studiedtwodifferentvaluesfortheinclinationangle,θ =π/2
toticenergyofEq.(III.29)andthedottedcurveshowstheenergyof
andθ =π/4,forseparationsd≥2.22R.
Eq.(IV.33).
OurnumericalresultsfortheelectromagneticCasimirener-
giesareshowninFigs.3and4(dotsandsolidcurves). They
are scaled by PFA energy for cylinders with θ = π/2 and
with θ = π/4, respectively. Regarding the approach of the E0/E0P,FθA θ = π PFA
numericalresultstotheasymptoticexpressionfortheenergy 3.0 4 PFA-corr
of Eq. (III.29) (dashed curves), we observe similar behavior
2.5 Numeric
as in the case of parallel cylinders [14]: Since the energy
d R
decays logarithmically (with sub-leading logarithmic correc- 2.0 (cid:29)
tions), the actual energy converges to the asymptotic result
only at extremely large separations that are not shown in the 1.5
plots. At intermediate distances, higher order partial waves
1.0
need to be included and are expected to lead to an approach
of the energy to the PFA result at small separations. This
0.5
tendency indeed can be observed for our numerical data. It
should be noted that beyond the distance d ≈ 3.33R where R
0.1 0.2 0.3 0.4 0.5 d
the curve for the numerical result starts to bend downwards,
theresultbecomesinaccurateandmorepartialwavesmustbe
included. Insteadofperformingcomputationswithmorepar- Figure4. (coloronline)SameplotsasinFig.3butforinclination
tialwaves,wecompareournumericalresultstotheprediction θ=π/4.
ofarecentlydevelopedgradientexpansionoftheinteraction
energyforgentlycurvedsurfaces[21,22].Thelatterapproach
yieldsforthefirstcorrectiontoPFAenergytheresult
ofω(r,θ)onθisclosetotheasymptoticresultofEq.(III.29),
(cid:20) (cid:18) (cid:19) (cid:21)
E =EPFA 1− 1 10 − 7 l +... (IV.33) while for reduced distances ω(r,θ) becomes more flat, indi-
0,θ 2 π2 24 R catingthatitconvergestotheconstantvalueofthePFAresult.
where l = d − 2R is the surface-to-surface distance. This
resultisshownasdottedcurvesinFigs.3and4. Itcanbeob-
servedthatournumericalresultsnicelyapproachthepredic- V. SUMMARYANDCONCLUSION
tionofEq.(IV.33)forshortdistances. Hence, ournumerical
resultstogetherwithEq.(IV.33)providetheoverallbehavior We have studied the interaction of inclined cylinders with
oftheCasimirinteractionbetweeninclinedcylinders. Dirichlet, Neumann and perfect conductor boundary condi-
Wearealsointerestedintheorientationdependenceofthe tions. In order to study this system, we have obtained the
energy at a fixed distance. As can be seen from Eq. (III.29) translationmatricesbetweenoutgoingandregularcylindrical
and Eq. (I.3), the energies have a different angular depen- wavesininclinedcoordinatesystems,bothforscalarandelec-
dence. To study the angular dependence of the energy at in- tromagneticwaves. Theinteractionenergieshavebeencom-
termediate distances, we show in Fig. 5 the rescaled energy puted over wide range of separations, using analytic and nu-
ω(r,θ) = E (r,θ)sin(θ)asfunctionoftheinclinationangle meric approaches. The zero temperature and classical high
0
comparedtothesamefunctionatθ = π/2,fordifferentdis- temperature limits were considered. In the latter, we have
tances. One observes that at large distances the dependence found that the zeroth order Matsubara term corresponds to
7
ACKNOWLEDGMENTS
3.0 WeacknowledgehelpfuldiscussionswithR.Brito,M.Kar-
darandR.Zandi. P.R.-L.’sresearchissupportedbyprojects
2.5 MOSAICO, UCM/PR34/07-15859, MODELICO (Comu-
nidaddeMadrid)andaFPUMECgrant.
2.0
1.5
AppendixA:Derivationoftranslationmatrices
1.0
Inthisappendix,weprovidethederivationoftheresultin
0.5 Eq. (II.11). We start from the Fourier transform of outgoing
wavesinthexy-plane,
0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 (cid:90) (cid:90)
K (ρp)einθ =2π(−i)n ∞ kx ∞ ky
n
Figure 5. (color online) Ratio ω(θ)/ω(π/2) with ω(θ) = 2π 2π
Eθ.0T(rh,eθd)assinhe(dθ)cuarnvderrep=resRen/tdstahsefausnycmtiopntotoicf trheesuilntcolfinEaqti.o(nIIaI.n2g9l)e. ×(cid:18)kx+i−k∞y(cid:19)n ei−(k∞xx+kyy) . (A.1)
The solid curves connect numerical data for r = 0.01, r = 0.1, p kx2+ky2+p2
r = 0.2,andr = 0.3(fromtoptobottom). ThePFAyieldsforthe
The integration over k can be easily performed using the
ratiounity. y
residue theorem. Since the integrand has poles at k =
y
(cid:112)
±i p2+k2,weobtain
x
K (ρp)einθ =(−i)n(cid:90) ∞ dk (cid:32)kx∓(cid:112)kx2+p2(cid:33)n
a non-trace class operator N(κ = 0) for a scalar field with n x p
Dirichlet boundary conditions and for the electromagnetic −∞√
field. However, we could obtain the energy by integration eikxx∓y kx2+p2
× , (A.2)
of the force which is well-defined since ∂dN(κ = 0) is a 2(cid:112)kx2+p2
trace-class operator. We should remark that the zeroth order
MatsubaratermcontributestoCasimirenergyatanynonzero where the minus (plus) sign applies to y > 0 (y < 0). With
temperaturesothatthisresultisrelevantnotonlyinthehigh the coordinate transformation x(cid:48) = Rθx+dˆy and the corre-
temperaturelimit. Thenon-traceclasspropertyofN(κ = 0) sponding wave vector components kx = kx(cid:48) cosθ−kz(cid:48) sinθ,
(cid:112)
can be related to the metallic response of the cylinders. As ky(cid:48) = ky = ±i κ2+kx2+kz2, kz = kx(cid:48) sinθ+kz(cid:48) cosθ we
soonasoneofthetwocylindersisnon-conductingoroneof get the outgoing wave in the primed coordinate system for
the cylinders has a scalar field boundary condition different y(cid:48) >0(y(cid:48) <0)
fromtheDirichletcase,N(κ=0)isatraceclassoperator.
 (cid:113) n(cid:48)
(cid:90) k ∓ k 2+p2
beWeaesinlyoteexttheantdoeudrtoandaileylseicstrfiocrcpyelrinfedcetrsmseitnaclecythlientdrearnsslcaotiuolnd φonu(cid:48),tkz(cid:48)(x(cid:48))=(−i)n(cid:48) ∞ dkx(cid:48)  x(cid:48) px(cid:48) (cid:48) 
(cid:48)
matricesremainunchanged. Inthiscaseitisexpectedthatat −∞ √
large distances the energy decays according to a power-law, ×ei(kxx+kyy+kzz)e∓ kx(cid:48)2+p(cid:48)2d, (A.3)
the zero Matsubara frequency contribution leads to a trace (cid:113)
2 k 2+p2
class operator, and the interaction is generally weaker than x(cid:48) (cid:48)
intheperfectmetalcase.
(cid:113)
withp = κ2+k 2.Nowweusetheexpansionof2Dplane
The Casimir interactions between cylinders as a function (cid:48) z(cid:48)
wavesincylindricalwaves,
of their inclination angle θ could be experimentally probed.
The force between cylinders with fixed inclination could be (cid:88) (cid:16) (cid:113) (cid:17)
ei(xkz+kyy) = inJ ρ k2+k2 einθ
measured directly or by measuring the effect of the Casimir n x y
forceonthemechanicalresponse(oscillations)ofthinmetal- n∈Z
(cid:32) (cid:33)
lic wires. The rather slow decay of the interaction between inarccos √kx
inclinedcylindersmightallowtoconsiderseparationsthatare ×e− kx2+ky2 . (A.4)
well beyond the validity range of the PFA. Another experi-
(cid:113)
mentallyinterestingquantityisthetorqueτ = −∂E/∂θ be- Withtherelationsip= k2+k2and
x y
tween the cylinders for which a torsion pendulum might be
employed. For future work, it would be interesting to study (cid:32) (cid:33)
theinfluenceoffiniteconductivityandtherelatedanomalous e−inarccos √kkx2x+ky2 =in(cid:18)kx+iky(cid:19)−n , (A.5)
scalingoftheenergy[9]ontheorientationdependence. p
8
theplanewavecanbewrittenas ByinsertionofthissuminEq.(A.3),wecanexpresstheout-
goingwaveintermsofincomingwaves,
(cid:32) (cid:112) (cid:33) n
ei(xkz+kyy) = (cid:88)(−i)nI (ρp)einθ kx− kx2+p2 − .
n
p
n Z
∈ (A.6)
φonu(cid:48),tkz(cid:48)(x(cid:48))=n(cid:88)∈Z(−i)n+n(cid:48)(cid:90)−∞∞dkx(cid:48) kx(cid:48) −(cid:113)pk(cid:48)(cid:48)2x+p(cid:48)2n(cid:48)(cid:32)kx−(cid:112)pkx2+p2(cid:33)−n 2e−(cid:113)dk√(cid:48)2xk(cid:48)+2x+pp(cid:48)(cid:48)22In(ρp)einθeikzz, (A.7)
whereweassumedy >0,d>0,andθ >0. InordertobringtheresultintotheformofEq.(II.10),wechangethevariableof
(cid:48)
integrationfromk tok usingk =cos(θ)k +sin(θ)k anddk =dk /sinθ,whichyields
x(cid:48) z z z(cid:48) x(cid:48) x(cid:48) z
φonu(cid:48),tkz(cid:48)(x(cid:48))=n(cid:88)∈Z(cid:90)−∞∞dkz(−siin)(nθ+)n(cid:48) kx(cid:48) −(cid:113)pk(cid:48)(cid:48)2x+p(cid:48)2n(cid:48)(cid:32)kx−(cid:112)pkx2+p2(cid:33)−n 2e−(cid:113)dk√(cid:48)2xk(cid:48)+2x+pp(cid:48)(cid:48)22φrne,kgz(x), (A.8)
where we have used φrne,kgz(x) = In(ρp)einθeikzz, and it is wheredisthedistancebetweenthecylinderaxes. Takinginto
understoodthat account that a surface element of the parallelogram is given
k k by sin(θ)dudv, the PFA energy at zero temperature can be
kx =kzcotθ− sinz(cid:48)θ, kx(cid:48) = sinzθ −kz(cid:48) cotθ. (A.9) writtenas
With the definitions ξ = kx/p and ξ(cid:48) = kx(cid:48)/p(cid:48) Eq. (A.8) is (cid:126)cπ2 (cid:90) si2nR(θ) (cid:90) si2nR(θ) sin(θ)dudv
equivalenttoEq.(II.11). EPFA =− . (B.2)
0,θ 720 h3(u,v)
For θ < 0 the overall sign of Un(cid:48)kz(cid:48),nkz(d,θ) must be 0 0
changed. The translation matrix elements for the inverse of When we introduce the surface-to-surface distance l = d−
the transformation x(cid:48) = Rθx+dˆy is obtained by assuming 2R,afterperformingachangeofintegrationvariablestheen-
y(cid:48) <0andd→−d,θ →−θ. Whenweindicatebytheargu- ergycanbewrittenas
ments(−d,−θ)thatthetranslationmatrixcorrespondstothe
inversecoordinatetransformation,wehave π2(cid:126)c 1 l
EPFA =−
Un(cid:48)kz(cid:48),nkz(−d,−θ)=(−1)n+n(cid:48)Un(cid:48)kz(cid:48),nkz(d,θ). (A.10) 0,(cid:90)θ √R/l (cid:90)7√20R/slinθR2 dsdt
Finally,wenotethatthetranslationmatricesforparallelcylin-
× √ √ .
drical coordinate systems follow from a direct integration of (cid:20) (cid:113) (cid:113) (cid:21)3
Eq.(A.7)overkx(cid:48) inthelimitθ →0. − R/l − R/l Rl +2− 1− Rls2− 1− Rlt2
(B.3)
AppendixB:ProximityForceApproximation This expression has the advantage that we can expand the
squarerootsforsmalll/Rwhichleadsto
InthisappendixwederivethePFAfortheenergyoftwoin-
π2(cid:126)c 1 R
clinedcylindersatzerotemperature,Eq.(I.3),andinthehigh EPFA =−
temperature limit, Eq. (I.4). The area across which the two 0,θ 720 sin(θ)l2
√ √
inclined cylinders overlap, viewed along the axis that is per- (cid:90) R/l (cid:90) R/l dsdt
pendiculartothetwocylinderaxesandthatintersectstheaxes × √ √ .(B.4)
in their crossing point, forms a parallelogram of edge length R/l R/l (cid:2)1+ 1(s2+t2)(cid:3)3
− − 2
2R/sinθ. Let us denote the coordinates along the edges of
Inthisexpressionwecanextendtheintegrationlimitstoinfin-
thisparallelogramasuandv. Thenthelocaldistanceh(u,v)
itytoobtainthelimitingbehaviorforsmalll/R. Thisyields
between the two cylinder surfaces, measured normal to the
pfulanncetiotnhat is spanned by the cylinder axes, is given by the lim EPFA =−π2(cid:126)c 1 R(cid:90) 2πdϕ(cid:90) ∞ ρdρ
h(u,v)=d−(cid:113)R2−(usinθ−R)2 Rl→0 0,θ 720 sin(θ)l2 0 0 (cid:16)1+ ρ22(cid:17)3
(cid:113) π3(cid:126)c 1 R
− R2−(vsinθ−R)2. (B.1) =− 720 sin(θ)l2 , (B.5)
9
which is the result of Eq. (I.3). For small l/R the latter ap- peraturesisgivenby
proximation deviates from the exact integral of Eq. (B.3) by
less than 1% for l/R < 0.01. It is instructive to compare ζ(3)(cid:90) si2nR(θ) (cid:90) si2nR(θ) sin(θ)dudv
EPFA =−k T . (B.7)
this PFA energy to the one for two spheres of radius R and cl,θ B 8π h2(u,v)
0 0
surface-to-surfacedistancelwhichis
Performing the same transformations and approximation as
EPFA =−π3(cid:126)cR (B.6) forT =0,wefindforsmalll/RthePFAenergy
spheres 1440 l2
lim Ecl =−k T ζ(3) 1 R(cid:90) 2πdϕ(cid:90) ∞ ρdρ
apnedrpheenndciceurleadruccyeldinbdyeraswfaictthorθo=f1π//22c.omparedtothecaseof Rl→0 PFA B 8π sin(θ) l 0 0(cid:16)1+ ρ22(cid:17)2
In analogy to the above computation, it is possible to ob- ζ(3) 1 R
tainthePFAenergyforthehightemperaturelimit. Usingthe =−kBT , (B.8)
4 sin(θ) l
hightemperatureinteractionbetweentwoperfectlyreflecting
plates,thePFAenergyfortwoinclinedcylindersathightem- whichcorrespondstoEq.(I.4).
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