VOLUME 78, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 6 JANUARY 1997

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Demonstration of the Casimir Force in the 0.6 to6 mm Range

S. K. Lamoreaux*Physics Department, University of Washington, Box 35160, Seattle, Washington 98195-156

(Received 28 August 1996)

The vacuum stress between closely spaced conducting surfaces, due to the modification of thpoint fluctuations of the electromagnetic field, has been conclusively demonstrated. The measuemployed an electromechanical system based on a torsion pendulum. Agreement with theorylevel of 5% is obtained. [S0031-9007(96)02025-X]

PACS numbers: 12.20.Fv, 07.07.Mp

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One of the most remarkable predictions of quantelectrodynamics (QED), obtained by Casimir in 1948,that two parallel, closely spaced, conducting plates willmutually attracted [1]. This attractive force is due to texclusion of electromagnetic modes between the pla(as compared to free space) and has magnitude (persurface areaA)

FsadyA ­p2

240hca4

­ 0.0161a4

dynsmmd4ycm2, (1)

wherea is the plate separation; in principle, a QED effecan directly influence a macroscopic, classical, apparaIn spite of the extensive theoretical attention this effhas received over the years (see [2,3] for recent reviethere has been only one attempt at its measurement.measurement, as reported by Sparnaay in 1958, shoan attractive force “not inconsistent with” the predictiogiven by Eq. (1), but with effectively 100% uncertain[4]. A closely related effect, the attraction of a neutratom to a conducting plate, has been recently meas[5]; good agreement with theory was found.

The Casimir force is closely related to the van der Waattraction between dielectric bodies. Formally, Eq. (1)obtained by letting the dielectric constante in the Lif-shitz theory [6] approach infinity, which is an appropate description for a conducting material. However,practical terms, the Casimir and van der Waals forcesquite different; the van der Waals force is always attrtive, whereas the sign of the Casimir force is geomedependent. For example, if a thin spherical conductshell is cut in half, the two hemispheres will experiena mutualrepulsiveforce [7]. These points are discussein Refs. [2,3]. A number of experimental measuremeof short-range forces between dielectric bodies of variforms have been performed; see Ref. [2] for a review.

For our measurement of the Casimir force, the condtors were in the form of a flat plate and a sphere. Ofirst attempts at measurements using parallel plates wunsuccessful; this is because it is very difficult to maintparallelism at the requisite accuracy (1025 rad for 1 cmdiameter plates). There is no issue of parallelism whone plate has a spherical surface; geometrically, thetem is described by the separation at the point of closapproach. However, when one plate is spherical, Eq.

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must be modified; the force for this geometry is simply obtained by the use of the so-called proximity fortheorem (PFT) [8], which in the present case reduceF ­ 2pRE where R is the radius of curvature of thspherical surface, andE is the potential energy per unsurface area which gives rise to the force of attractbetween flat plates. Thus, the magnitude of the Casforce between a sphere and a flat surface is given by

Fcsad ­ 2pR

√13

p2

240hca3

!. (2)

and the result is independent of the plate area.There are at least two corrections to the Casimir for

The first is the effect due to the finite temperatureT ø300 K; this correction has an illustrious history as dicussed by Schwingeret al. [9]; the thermal correctionsfor the Casimir force and van der Waals force are dferent, and are properly derived for the case of conducplates in Refs. [9–11]. Taking the results of Brown aMaclay [11], the surface energy is given byE ­ aT00,whereT00 is the (volume) energy density. Using the PFand Eq. (20a) of [11], the total magnitude of the Casimforce is

FTc sad ­ Fcsad

µ1 1

720p2

fsjd∂

, (3)

wherej ­ kTayhc ­ 0.126a mm21 at T ­ 300 K (k isBoltzmann’s constant) and

fsjd øΩ

sj3y2pdz s3d 2 sj4p2y45d, for j # 1y2 ,sjy8pdz s3d 2 sp2y720d, for j . 1y2 ,

(4)

where z s3d ­ 1.202 . . .. It is interesting to note that inthe largea limit, the correction is independent ofhc andhas the appearance of a classical effect; this is analogto the Rayleigh-Jeans limit of the black body spectrum

The second correction, obtained by Schwinget al. [9], is due to the finite conductivity of the plate(modified by the use of the PFT to the case whereplate is spherical);

F0csad ­ Fcsad

√1 1

4cavp

!, (5)

© 1996 The American Physical Society 5

VOLUME 78, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 6 JANUARY 1997

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wherevp is the plasma frequency for the conductor, ait was assumed that the effective electric susceptibilityesvd ­ 1 2 svpyvd2.

A schematic of the apparatus used in our measuremis shown in Fig. 1, and details of the torsion penduluare shown in Fig. 2. The Casimir force plates compra 2.54 cm diam, 0.5 cm thick quartz optical flat, andspherical lens with radius of curvature11.3 6 0.1 cm anddiameter 4 cm; each was coated (by evaporation) wa continuous layer of Cu of thickness0.5 mm, on allsurfaces. A layer of Au was then evaporated (0.5 mmthick) onto the faces which were subsequently broutogether. As shown in Fig. 1, the flat electrode wmounted on one arm of the torsion pendulum, whthe spherical electrode was placed on a micropositionassembly. The adjustment screws and piezoelectric stranslators (PZTs) form a tripod. A vacuum of ord1024 torr was maintained (to eliminate viscous effectby a small oil (Fomblin) diffusion pump; adjustment rodpassed through simple vacuum rotary feedthroughsallowed coarse adjustment of the plate position, wabout0.5 mm accuracy.

A feedback system was used to keep the torsion pdulum angle fixed; as shown in Fig. 1, two “compensaplates” form a capacitor with respect to the pendulum boAn ac bridge circuit was used to determine whethertwo capacitances are equal; any unbalance was deteby a phase-sensitive detector which provides an errornal to an integral-plus-proportional feedback circuit.dc correction voltage was applied to the compensatas required to keep the torsion pendulum angle fixed.

FIG. 1. Schematic of the apparatus. The vacuum vesdimensions are 55 cm diam by 110 cm tall. The solenactivated plunger was used to press the plates gently toge(during alignment); after such pressing, the plates couldbrought much closer.

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addition, a constant dc voltage ofV0 ­ 7.5 V was ap-plied to the compensators in order to linearize the effof the small correction voltagedV [the force on the pen-dulum due to one compensator isF ­ sV 2y2ddCydx øsV 2

0 y2 6 V0dVddCydx, and the net force from both iF ­ 2V0dVdCydx, where dCydx is the magnitude ofthe change in compensator-pendulum capacitance asgap sizex is varied]. Since the feedback only affecthe torsional mode, a strong magnet (2.5 kG surface fiwas used to overdamp all vibrational modes of the pdulum system. The angle voltage signal from the Phad sensitivity of48 mVymm displacement of the Casimelectrode (1 mm displacement corresponds to an angle2 3 1025 rad). The angular fluctuations were consistewith the expected thermal noise [12];durms ­

pkTya ø

1 mrad, wherea ­ 4.8 dynyrad is the torsion constant fothe tungsten fiber. A microrotation stage allowed turnthe fiber to setdV ­ 0. Before applying the vacuum, thfiber was annealed, with the pendulum hanging, by pasabout 500 mA through it; while the current was flowinthe fiber length increased by about 1%. The drift in appent force, over one month, was less than2 3 1023 dyn.

The Casimir force was measured by simply steppthe voltage applied to the PZTs up and down throughdiscrete and constant steps, and at each step, measthe restoring force, implied by a change indV, requiredto keep the pendulum angle fixed. The maximum dplacement at 92 V was12.3 mm; the relative displacemen

FIG. 2. Details of the pendulum. The body has total m397 g. The ends of the W fiber were plated with a Cu cyansolution; the fiber ends were bent into hairpins of 1 cm lenand then soldered into a 0.5 mm diam, 7 mm deep holethe brass rods. Flat-head screws were glued to the backthe plates; a spring and nut held the plates firmly against tsupports, and ensured good electrical contact.

VOLUME 78, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 6 JANUARY 1997

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as a function of the sixteen steps (up and down serately because of considerable hysteresis) was measur0.01 mm accuracy by use of a laser interferometer, givi32 valuesai for the relative plate position. The averagdisplacement per 5.75 volt step was about0.75 mm. Anupydown sweep sequence typically took 25 min; after eastep, no data were taken for 40 s (about three systemsponse time constants) to ensure that the feedback sywas in equilibrium, after whichdV was averaged for 7 s.

The system calibration was obtained through eltrical measurements based on the variation of thepacitance between the Casimir plates as a functionseparation. A simple relaxation oscillator that only rquired 30 mV rms to be applied between the plates wused to determine the capacitance variation; the oslator frequency was carefully calibrated by a precisivariable capacitor in parallel with the plates, and wtypically 11.24 HzypF. Using the PZTs to change thplate separationa, dCyda ø 0.73 pFymm, with 1% ac-curacy, at the plate separation where the calibrationperformed (about10 mm). Next, a potential differenceV ­ 300 mV was applied between the plates; this genates a forceF ­ sV 2y2ddCyda; the calibration was determined by the change indV from the feedback circuit, andis 0.0420 dyneyvolt, with 1% accuracy. This calibrationis in good agreement with (less accurate) direct measment of the physical dimensions of the compensator plaand their spacing.

With the Casimir plates separated but externally shortogether, there was an apparent shockingly large poteof 430 mV; there are roughly 40 separate electriconnections in this loop and a potential this largeconsistent with what is expected for the various metacontacts. This potential was easily canceled by setan applied voltage between the plates to give a minimdV; this applied voltage was taken as “zero” in regardthe calibration. The contact potential drifted by less th10 mV over the one month of operation of the experimeand was independent of the region of the plates wheremeasurement was performed (this was varied by tiltthe spherical plate with the adjustment screws).

The PZTs give very accurate and reproduciblerelativechanges in the plate separation; the absolute separwas determined by measuring the residual electrattraction between the plates as a function of separa(the contact potential was intentionally not perfeccanceled). The electric force as a function of separacan be obtained by use of the PFT. For two flparallel plates separated by a distancea with a potentialV between them, the potential energy per unit areaE ­

12 sCyAdV 2 ­

12 V 2e0ya. Applying the PFT theorem

gives the electric force as

Fesad ­ pe0RV 2ya ! Csad ­ 2pe0R ln a 1 C0 , (6)

whereCsad is the capacitance between the plates. Tresult agrees with the exact solutions for the force athe capacitance between a sphere and a plane, in the

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Each upydown sweep cycle was fitted to a function oform

Fmsid ­ FTc sai 1 a0d 1

b

ai 1 a01 b , (7)

whereFmsid is the measured force at theith step,a0 isa fit parameter which gives the absolute plate separatandb is a constant; the points of closest approach (clothan 2 mm) are not used in this fit; this allows the truCasimir force, which is insignificant at large distances,be discriminated from the electrical force, as shownFig. 3. Next, the absolute position is assigned toFmsid,and ai is set toai 1 a0. Finally, the electrical force issubtracted, giving

Fmc said ­ Fsaid 2

b

ai2 b , (8)

where Fmc said is the measured residual force (hopeful

the Casimir force). The results of the process, anddirect comparison with the expected Casimir force wno adjustable parameters, is shown in Fig. 3. Typicathe Casimir force had magnitude of at least 20% of telectrical force at the point of closest approach. There wno evidence for a1ya2 force in any of the data; a force othis form could result from patch effects on the surface

The uncertainty ina0 was normally less than0.1 mm.Typically,a0 drifted by about0.1 mm per upydown sweep;this allowed a wide variety of closest approaches to bevestigated, even though the discrete step size was ra

FIG. 3. Top: Measured force as a function of relative positiand least-squares fit toFT

c sai 1 a0d 1 bysai 1 a0d 1 b; thetwo points at closest approach were not used in the fit. BottoMeasured force with electric contribution subtracted; the poiconnected by lines are as expected from the Casimir force.

7

VOLUME 78, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 6 JANUARY 1997

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large. This drift was the result of a variety of environmetal factors, most notably temperature variations. Roug10% of the upydown sweeps were rejected because anomlously large drift resulted in an unsatisfactory convergenfor the fit, as evidenced by an anomalously largex2, anonphysical value fora0, and an inconsistent result forb

which was quite constant. Also, those sweeps wherenet change indV between start and finish correspondto a force greater than4 3 1025 dyn were rejected; in all,the final data set comprises 216 upydown sweeps. Quiteoften, as the absolute separation drifted, the plates wocontact before the end of a complete up sweep. Theat which this occurred could be unambiguously determinby a sudden jump in the feedback signal. Roughly eisteps on the down sweep had to be rejected becauster such a large perturbation, the feedback system requseveral minutes to reestablish equilibrium.

Assuming that the functional form for the Casimir forcis correct, its magnitude was determined by using linleast squares to determine a parameterd for each sweepsuch that

Fmc said ­ s1 1 ddFT

c said 1 b0 . (9)

In this context,b0 should be zero, and for the compledata set,b0 , 5 3 1027 dyn (95% confidence level)The average over the 216 sweeps givesd ­ 0.01 6 0.05,and this is taken as the degree of precision ofmeasurement. There was no evidence for any variaof d depending on the region of the plates used formeasurement.

The most striking demonstration of the Casimir foris given in Fig. 4. The agreement with theory, with n

FIG. 4. Top: All data with electric force subtracted, averaginto bins (of varying width), compared to the expected Casimforce for a 11.3 cm spherical plate. Bottom: TheoreticCasimir force, without the thermal correction, subtracted frotop plot; the solid line shows the expected residuals.

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adjustable parameters, is excellent. It should be noted tthe closest approach is about0.6 mm; this limit could becaused by either dirt on the surfaces, or by an instabiliof the feedback system. The Casimir force is nonlineand increases rapidly at distances less than0.5 mm. Withthe plates separated by10 mm, the feedback loop becameunstable when a 700 mV potential difference between tplates was applied; the change in force with distance (teffective spring constant) in this case isdFeyda ­ 1.5 3

1023 dynymm, which is equal todFcyda at a ­ 0.5 mm.In conclusion, we have given an unambiguous demo

stration of the Casimir force with accuracy of order 5%Our data is not of sufficient accuracy to demonstrate thfinite temperature correction, as shown in Fig. 4(b). Alsogiven a plasma frequency for Au of ordervpy2p ø6 3 1014 Hz, Eq. (5) gives a correction of order 20%at the closest spacings; our data does not support sua deviation. However, the simple frequency dependenof the electrical susceptibility used in the derivation oEq. (5) is not correct for Au, the index of refraction ofwhich has a large imaginary component above the plasmfrequency; a rough estimate using the tabulated complindex [14] limits the conductivity correction as no largethan 3%, which is consistent with our results [15].

I thank Dev Sen (who was supported by the UW NASASpace Grant Program) for contributions to the early stagof this experiment, and Michael Eppard for assistancwith calculations.

*Present address: Los Alamos National LaboratoryNeutron Science and Technology Division P-23, M.SH803, Los Alamos, NM 87545.

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31, 965 (1988).[4] M. J. Sparnaay, Physica (Utrecht)24, 751 (1958).[5] C. I. Sukenik, M. G. Boshier, D. Cho, V. Sangdohar, an

E. A. Hinds, Phys. Rev. Lett.70, 560 (1993).[6] E. M. Lifshitz, Sov. Phys. JETP2, 73 (1956).[7] T. H. Boyer, Phys. Rev.174, 1764 (1968).[8] J. Blocki, J. Randrup, W. J. Swiatecki, and C. F. Tsang

Ann. Phys. (N.Y.)105, 427 (1977).[9] J. Schwinger, L. L. DeRaad, Jr., and K. A. Milton, Ann.

Phys. (N.Y.)115, 1 (1978).[10] J. Mehra, Physica (Utrecht)37, 145 (1967).[11] L. S. Brown and G. J. Maclay, Phys. Rev.184, 1272

(1969).[12] G. Ising, Philos. Mag.1, 827 (1926).[13] W. R. Smythe,Static and Dynamic Electricity(McGraw-

Hill, New York, 1950), pp. 121–122.[14] CRC Handbook of Chemistry and Physics, 76th Ed.(CRC

Press, Boca Raton, 1995), pp. 12–-130.[15] S. Hacyan, R. Jauregui, F. Soto, and C. Villarreal, J. Phy

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