Edge effects in electrostatic calibrations for the measurement of the Casimir force

0.17 MB·English

by  Qun Wei

Edge effects in electrostatic calibrations for the measurement of the Casimir force
Qun Wei1 and Roberto Onofrio2,1
1Department of Physics and Astronomy,Dartmouth College,6127 Wilder Laboratory,Hanover,NH 03755,USA
2Dipartimento di Fisica “Galileo Galilei”,Universit`a di Padova,Via Marzolo 8,Padova 35131,Italy
(Dated: January 10, 2011)
We have performed numerical simulations to evaluate the effect on the capacitance of finite size
boundaries realistically present in the parallel plane, sphere-plane, and cylinder-plane geometries.
The potential impact of edge effects in assessing the accuracy of the parameters obtained in the
electrostatic calibrations of Casimir force experiments is then discussed.
PACSnumbers: 12.20.Fv,03.70.+k,04.80.Cc,11.10.Wx
1
1
1. INTRODUCTION 2. PARALLEL PLATE GEOMETRY
0
2

n
The Casimir force [1] has been demonstrated in a va- We start our analysis with the parallel plates configu-
a
J riety of experimental setups and geometries, yet there ration since this is the simplest geometry even in terms

7 is an ongoing reanalysis of the level of accuracy with of possible deviations from the ideal, infinite plane case.
which it has been determined, which is crucial for as- Moreover,besideshavinganexactexpressionintheideal

]
sessing reliable limits to the existence of Yukawa forces caseofinfiniteplates,analyticalapproximateexpressions
h
p of gravitationaloriginpredicted by variousmodels [2–4]. are also available for the capacitance including edge ef-
- In performing Casimir force measurements a crucial role fects [19], allowing to obtain reliable numerical bench-
t
n is played by the related electrostatic force calibrations, marks. We consider two identical parallel square plates
a and mastering all possible systematic effects in the lat- of length L, and boundary conditions set in such a way
u
ter is mandatory to assess the precision of the former. thatthetwoplatesareataconstantelectricpotentialdif-
q
[ The presence of anomalous exponents in the power-law ference. The total electrostatic potential energy is then
dependence upon distance and the dependence on dis- computed by numerically solving the Laplace equation
1
tanceoftheminimizingpotential[5]havebeenidentified usingadedicatedfiniteelementanalysissoftware(COM-
v
as a possible source of systematic effects, to be carefully SOL). We evaluate the total electrostatic energy W by
0 el
8 scrutinizedineachexperimentalsetup[6]. Thedistance- summing the electrostatic energy density over a selected
4 dependenceoftheminimizingpotentialhasbeenrecently volume surrounding the two equipotential surfaces. The
1 modelizedtheoretically,afterafirsteffortreportedin[7], capacitance C can then be calculated from the relation-
1. in terms of non-equipotentialconducting surfaces due to ship C = 2WelV2. The parameters of the mesh (mesh
0 random patterns of patch charges [8]. Here we report size, rate of growth etc.) are carefully chosen and ex-
1
results on a further potential source of systematic error tensively tested to ensure the accuracy of the numeri-
1
by studying, by means of numerical simulations using fi- cal results. When the distance between the two plates
:
v nite element analysis, the influence of the unavoidable is much smaller than L, the capacitance obtained from
i
X presence of edges on electrostatic calibrations in various the numerical simulation agrees within 0.01% with the
geometries (for the influence of edge effects on Casimir well known analytical formula C = ǫ A/d, where A
r pp 0
a forces in peculiar geometries see [9, 10]). We limit the is the surface area of the plates, d their separation, and
attention to the three geometries that, apart from the ǫ the vacuum electric permittivity. However, when the
0
crossed-plane investigated in [11], have been extensively distancebecomeslarger,the valueofthe capacitancebe-
discussed for measuring Casimir force so far, i.e. the ginstodeviatefromtheanalyticalformula. Theleftplot
parallelplates [12, 13], the sphere-plane [14–17], and the in Fig. 2 shows the capacitance of two parallel square
recently proposed cylinder-plane [18] (see Fig. 1). We plates vs. distance, together with the expected capaci-
discuss the deviations from ideality in all these geome- tancefortwoplatesfollowingtheinfinitesurfaceformula,
tries through the essential knowledge of the capacitance appearing as a dashed line. The data at large distance
dependence on distance, and its general implications for deviate significantly from the dashed line, and the fact
determining the parameters used in the measurement of that they are all above implies that the power-law expo-
the Casimir force. This is performed under the simplyf- nent is softer, i.e. in between 0 and -1. This deviation
ing assumptions that the surfaces are equipotential, i.e. is ascribed to the finite size of the parallel plates. In
by omitting any superimposed effect due to electrostatic fact,ifonlythefieldlinesinthevolumedelimitedbythe
patches,andbyneglectingthetensorialnatureoftheca- two plates are used for the sum of the electrostatic en-
pacitance among all the conducting surfaces realistically ergy, which corresponds to ignoring the contribution of
involved in concrete experimental setups. the outer region, the capacitance obtained from the nu-
2
FIG.1: Selected geometries for thenumerical studyof finitesize effects. Parallel platesof finitesize (a);a spherein front of a
finitesizeplane(b);acylinderinfrontofaplane,withwidthsmaller(c)andlarger(d)thanthesizeoftheplane. Inthelatter
three configurations, a truncated sphere and short and long truncated cylinders, as shown bythe dashed lines, havebeen also
studied.
merical simulation would still agree with the analytical For the whole sphere, it is worth remarking that neither
formula even at the largest explored distances. In order theexactnortheapproximateexpressioncangiveanac-
to better quantify this deviation we have fitted the ca- curate value for the capacitance between a sphere and
pacitance curve by progressively removing points at the a finite plane, even at small distance. We have checked
largestdistance. AsshownintherightplotinFig.2,the thattheidealizedcaseofasphereandaninfiniteplaneis
optimalexponentbecomessmallerthanunitywhendata approachedby considering squareplates of progressively
at large distance are progressivelyincluded in the fit. largersize. Thecapacitanceintherealisticcasestillpre-
serves a logarithmic dependence at small distance, with
the slopes very close to each other. For the truncated
3. SPHERE-PLANE GEOMETRY
sphere,thecapacitanceissignificantlysmallerthanboth
the exact and the approximate expressions, which is ex-
In the case of the sphere-plane geometry, the exact pected considering that there is less conducting surface
expression for the capacitance between a sphere and an available in this case. Moreover, much smaller distance
infinite plane is written in terms of a series [20]: is required for the capacitance to be approximated by a
logarithmic dependence with a slope comparable to the
+∞
1 one of the whole sphere case. The truncation aspect ra-
C =4πǫ Rsinh(α) (1)
sp 0 sinh(nα) tio has been chosen in close analogy on the case of the
nX=1
lenses used in the sphere-plane experiment reported in
with cosh(α) = 1 + d/R, R is the sphere radius, and [23], and it should also be similar to the one used in
d the separation distance. In the limit of small sepa- the firstmodern Casimir force experimentin the sphere-
rations, d/R << 1, an approximate expression for the plane configuration [14].
capacitance can be obtained [20]:
4. CYLINDER-PLANE GEOMETRY
R 23 θ
C 2πǫ R ln +ln2+ + (2)
sp 0
≈ (cid:18) d 20 63(cid:19)
Forthecylinder-planegeometry,theexpressionforthe
where0 θ 1. Byneglectingthedistance-independent capacitance between a cylinder and an infinite plane is
≤ ≤
terms one obtains an expression often appearing in rela-
2πǫ L
tionshipto the so-calledProximityForceApproximation 0
C = (3)
(PFA)[21,22]inelectrostatics. Theformulasaboveboth cp cosh−1(1+d/R)
assume an infinite plane and a whole sphere. In real ex-
where L and R are the length and radius of the cylin-
perimentsinvolvingmicroresonatorsthesizeoftheplane
der,anddthedistance. Inthelimitofsmallseparations,
is not necessarily large enough to be considered infinite
d/R << 1, an approximate expression for the capaci-
andthusedgeeffectscouldbepresent. Insomemeasure-
tance can be obtained by expanding the inverse hyper-
ments [17] the sphere is located close to one end of the
bolic cosine function with Puiseux series [24]:
squared plane, to increase the torque exerted on the un-
derlying microresonator. Also, in other measurements a
√2Rπǫ L
0
lens,schematizedasatruncatedspherewasusedinlieuof C . (4)
cp ≈ d0.5
awholesphere[14,23]. Fig.3showsthenumericalresults
for configurations taking into account these deviations Besidethe finite sizeofthe plane,the finite lengthofthe
from the idealized case, as well as the curves expected cylinder may also be a source of deviation from ideality.
from the exact (1) and the approximate (2) expressions. Therefore we have considered two cases, differing in the
3
1.000
1×10-10
0.995
nt
e
n0.990
o
) p
F x
C(pp1×10-11 al e0.985
m
pti0.980
O
0.975
1×10-12 0.970
1.0×10-5 1.0×10-4 5.0×10-5 1.0×10-4 5.0×10-4
d(m) d (m)
max
FIG. 2: (Left) Capacitance vs. distance for two parallel square plates. Dots indicate the results of the numerical simulation,
whilethedashedlineisthecapacitanceexpectedusingtheformulaCpp =ǫ0A/d. ThelengthofthesquareplatesLwaschosen
tobe8.86 mm. (Right)Optimalexponentǫc comingfrom thebestfitofasetofnumericaldatawith Cpp=ǫ0A/(d−d0)ǫc vs.
the distance of thefarthest data point used in thefitting. Red dots are obtained by fixingd0=0, corresponding to an a priori
knowledge of the absolute distance, while the black squares are obtained if d0 is considered as a fitting parameter, as usually
donewhen analyzing realexperimentaldatawith noa priori independentknowledgeof theabsolute distancebetweenthetwo
surfaces. Considering the size of the square plates used in the experiment reported in [13] of 1.1 mm, all separation distances
in that case are obtained by scaling down thehorizontal axis by a factor ≃65.
length of the cylinder chosen as smaller or larger than Let us consider the cylinder-plane geometry as an ex-
the side length of the plane, and the analysis has been ample. As already mentioned, in a real experiment it is
repeated for truncated cylinders. The capacitance per hard to determine directly the absolute separation dis-
unit length C /L vs. distance for different configura- tance, and only data fitting allows to obtain the sepa-
cp eff
tions areshownin Fig. 4. Inthis figure L is defined as ration if considered as a fitting parameter (see also the
eff
the minimum betweenthe length ofthe cylinder andthe detailed discussion in [25] on the same definition of dis-
length of the plane along the cylinder axis, i.e. defining tancebetweentwomacroscopicbodies). Intheidealcase,
the overlappingareabetweenthe cylinderandthe plane. by fitting the capacitance data with a relationship like
AscanbeseeninFig.4,truncated,widercylindersdevi-
atelessfromtheexactexpressionincomparisontowhole, C =C +K /(d d )ǫc (5)
cp 0 c 0
−
narrowercylinders. However,longerwidthmeansharder
parallelization and consequently worse achievable mini- with the exponent ǫ = 0.5 and C , K and d as fit-
c 0 c 0
mumdistance. FromFig.4, anarrowtruncatedcylinder ting parameters, the absolute distance can be obtained
seems to be a good compromise, which is the configura- as d = d d . Here C represents a constant fit-
abs 0 0
tion used in realistic experiments. ting paramete−r due to background, parasitic, distance-
independent capacitance added to the cylinder-plane ca-
pacitance,obtainable in the limit of very largegapsepa-
5. IMPLICATIONS FOR THE ELECTROSTATIC ration. In the case of our numerical simulation data, we
CALIBRATIONS IN CASIMIR FORCE
canfixC =0. Ifthecorrectfittingequationisused,the
0
EXPERIMENTS
valueofd obtainedshouldagreewithinthefittingerrors
0
withthe expectednullvalue. By fitting withEq. (5)the
The discussion aboveshows that when the sizes of the simulation data of the narrow truncated cylinder up to
samplesarefiniteandtheedgeeffectscannotbeignored, distance10µm, the maximumdistanceatwhichelectro-
the capacitance will not scale with distance through the static calibrations are usually carried out, d = 18 3
0
− ±
exponent expected from the ideal formula. If the expo- nmisobtainedwiththeexponentǫ isfixedat0.5,while
c
nent is incorrectly forced to the value from the ideal for- weobtaind =12 1nmifǫ isleftasafreeparameter,
0 c
±
mulainthe datafitting procedure,systematicerrorswill with an optimal value of 0.4849 0.0005. In both cases,
±
be propagated to all the fitting parameters. In fact, the the value of d from the fitting is different from zero by
0
optimal exponent is not a constant and depends on dis- severalstandarddeviations. Thesmallestdistanceinthe
tance as evident in the plane-plane configuration shown data is 400 nm, -18 nm and 12 nm represent errors with
inFig.2,thereforeassumingaconstantoptimalexponent respecttothesmallestgapseparationof4.5%and3.0%,
over the entire range of explored distances will generate respectively. Similaranalysiscanberepeatedforthepar-
systematic errors in the electrostatic calibrations. allel square plates, and the whole and truncated sphere
4
6×10-14 1×10-8
Centered
Lateral Truncated (w=4 mm)
Truncated Whole (w=4 mm)
Exact Truncated (w=12 mm)
PFA
IPFA Whole (w=12 mm)
4×10-14 m) Exact
F) F/
C(sp /L (p 1×10-9
c
2×10-14 C
0 1.0×10-7 1.0×10-6 1.0×10-5 1×10-10 1.0×10-6 1.0×10-5 1.0×10-4
d(m) d(m)
FIG. 3: Capacitance dependence on distance in the sphere- FIG. 4: Simulations in finite-size cylinder-plane configura-
plane configurations. The case of a whole sphere located tions. Capacitance per unit of effective length vs. distance
abovethecenterofasquareplate(blackdots)andontheedge for a truncated cylinder (black dots) and a whole cylinder
of the plate (red squares) are shown, as well as the case of a (red squares) of width 4 mm in front of a square plane of
truncated sphere (blue diamonds). The radius of the sphere size 10 mm × 28 mm, and similar plots for truncated (blue
ischosentobe0.15mmandthelengthofthesquareplate0.5 diamonds) and whole (green triangles) cylinders of width 12
mm. Thecontinuouslinecorrespondstotheexactexpression mm. The radius of the cylinder is chosen to be 12 mm. The
asin (1),and thedashed dotlinecorresponds totheapprox- continuouslinecorrespondstotheexactexpression asin (3),
imate expression as in (2), named Improved PFA (IPFA), and the approximate expression (4) coincides with the exact
while the dashed line is the capacitance expected from the expression in this distance range, confirming that the PFA
PFA method. holds toa high degree of accuracy up to thelargest explored
distances.
configuration by fitting the data respectively with
the capacitance is in general given by
C =C +K /(d d )ǫc,
pp 0 c 0
− (6) ∂E (d) 1∂C(d)
Csp =C0+Kcln(d−d0). Fel =− ∂edl =−2 ∂d V2 (7)
The results are shown in Table I, where the values of
d0 are all significantly different from the expected null 1 ∂F(d) 1 ∂2C(d)
value. Toshowthatthisisnotafittingartifact,thesame ∆νe2l =−4π2m ∂d = 8π2m ∂d2 V2. (8)
eff eff
fitting procedures have been repeated with simulation
dataatmuchsmallerdistancewherethe edgeeffects can Then the absolute value of the capacitance in itself is ir-
be safely ignored. In this check the values of d agree relevantforforceorsquarefrequencyshiftmeasurements
0
with the expected value of zero within the associated and,due to the firstandsecondderivatives,the distance
error from the fitting procedure. Therefore it is evident dependenceoftheforceorfrequencyshiftsignalmayend
thatthe presenceofedgeeffects mayresultinsignificant up being less sensitive to the edge effects.
systematicerrorsinthebestfit,affectingtheaccuracyof Tochecktheinfluenceoftheedgeeffectsonforcemea-
crucial parameters such as the absolute distance. surements,numericaldifferentiationhasbeencarriedout,
With regardto Casimir experiments, other crucialpa- using the Lagrange three-point formula [27], on the sim-
rameters of the experiment beside the absolute distance ulation capacitance data. The numerical differentiation
between the two surfaces are determined through the method works better when the data points are equally
electrostatic calibrations, for instance the effective mass spaced, therefore in the case of the sphere-plane con-
m of the resonator or, equivalently, its stiffness. The figuration the numerical differentiation is performed in
eff
calibrations are usually done at relatively large distance the semilogarithmic domain, while for plane-plane and
to ensure that the Casimir force can be neglected. How- cylinder-plane configurations in the logarithmic domain.
ever,in this regime the edge effects become more signifi- Then we fit the force vs. distance data with the corre-
cantleadingtosystematicerrorsinthefittingprocedures sponding equations for each geometry, obtaining the re-
as shown above. However, in most Casimir experiments sults shownin Table II.It seemsthat interms ofthe op-
what is measured in the calibrations, rather than the timal exponent, the edge effects are less significantwhen
capacitance, is the electrostatic force between the two the force data are considered, apart from the case of the
surfacesor,ifaresonatoris usedfordynamicalmeasure- truncated sphere. More specifically, the results for the
ments, its square frequency shift [26]. truncated sphere can be understood as if it behaves like
The relationship between these two observables and asphereonlyatsmalldistance,insteadresemblesaplane
5
Parallel plates SphereI SphereII Cylinder edge effects in the theoretical modelization of the exper-
d0(nm) −90±20 13±3 144±13 −18±3 imentalsetupforatargetedlevelofaccuracyofthe elec-
d0/dmin 1.8% 6.5% 72% 4.5% trostaticcalibrations. Ifgeometricalconfigurationsman-
d′0(nm) 140±20 N/A N/A 12±1 ifestly differing from the idealized cases are also consid-
′
d0/dmin 2.8% N/A N/A 3.0% ered, such as a small portion of a sphere or a truncated
ǫc 0.977±0.002 N/A N/A 0.4849±0.0005 cylinder,thedeviationsinthecapacitanceareevenmore
pronounced and a full numerical evaluation of the elec-
TABLE I: Parameters of the best fit obtained for the capac-
trostatics is necessary even at relatively moderate levels
itance data in the case of the four configurations considered
in the text with Sphere I being whole sphere and Sphere II of desiredaccuracy. We have also shownthat force (gra-
truncated sphere. For parallel plates and cylinder-plane con- dient force) measurements, based on the first (second)
figurations, the top two rows are the parameters obtained derivativewithrespecttodistanceofthecapacitance,are
when the exponent ǫc is fixed at 0.5 and 1, and the bottom in general more robust than absolute capacitance mea-
threerowswhenǫc isleftasafreeparameter. Forthesphere- surement in itself with respect to the geometrical devia-
planeconfiguration,duetothelogarithmicdependenceofthe
tionsfromideality,confirmingfromanotherperspectivea
capacitance scales with distance, no exponent can be intro-
pointrecentlydiscussedin[28]. Finally,wewanttopoint
duced.
outthatdetectingedgeeffectsprovidesalsoasimpleand
reliablemethodtoassesstheprecisionoftheelectrostatic
calibrations, for instance determining the minimum sep-
when considered at large distance from the plane. This
arationgapabovewhich deviationsattributable to finite
result is at variance with respect to the behavior actu-
edges are effectively observed.
ally observed in [5], since a square frequency shift power
lawdependenceondistancewithanexponentlessthan2
(therefore corresponding to a force exponent ǫf <1) has ACKNOWLEDGMENTS
beenobserveduniformlyovertheentirerangeofexplored
distances. Therefore the origin of the anomaly observed
We acknowledge useful discussions with D. A. R.
in [5] cannot be ascribedto edge effects. Analogouscon-
Dalvit.
siderations can be repeated for the first derivative of
theforcewheneversquarefrequencyshifts aremeasured.
The influence of the edge effects on the values of fitting
REFERENCES
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0 f
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the case of the truncated sphere the error is much more
significant. However, although the edge effects are not
as strong in force measurement as in capacitance mea-
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Geometry Fitting range (µm) Kf(NmǫfV2) ǫf d0(nm) d0/dmin
Parallel plates 7−50 (3.59±0.03)×10−16 1.9971±0.0007 10.5±2.9 0.15%
50−450 (4.4±1.0)×10−16 1.973±0.025 800±760 1.6%
Whole sphere 0.2−3 (6.1±6.8)×10−15 0.97±0.08 2±19 1.0%
3−25 (1.4±2.0)×10−15 1.10±0.12 -430±410 14.3%
Truncated sphere 0.2−3 (4.3±2.8)×10−17 1.30±0.04 -35±10 17.5%
3−25 (1.4±0.6)×10−21 1.92±0.04 -480±80 16.0%
Truncated cylinder 0.5−10 (36.4±2.2)×10−16 1.513±0.005 −5.8±2.0 1.2%
10−160 (29.2±6.2)×10−16 1.538±0.020 −370±180 3.7%
TABLE II: Parameters of the best fit obtained considering an electrostatic-like force obtained leaving the power exponent for
thedistance dependence, as a free parameter in thecases of the four configurations considered in thetext. The parameter Kf
is the electrostatic coefficient, ǫf the optimal exponent for the force measurement, and d0 an offset in distance, such that the
electrostatic force can be expressed as Fel =F0+KfV2/(d−d0)ǫf.
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