A measurement of the neutron to ^{199}He magnetic moment ratio

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Physics Letters B

www.elsevier.com/locate/physletb

A measurement of the neutron to ¹⁹⁹ Hg magnetic moment ratio

S. Afach

^a,b,c

, C.A. Baker

^d

, G. Ban

^e

, G. Bison

^b

, K. Bodek

^f

, M. Burghoff

^g

, Z. Chowdhuri

^b

, M. Daum

^b

, M. Fertl

^a,b,1

, B. Franke

^a,b,2

, P. Geltenbort

^h

, K. Green

^d,i

,

M.G.D. van der Grinten

^d,i

, Z. Grujic

^j

, P.G. Harris

ⁱ

, W. Heil

^k

, V. Hélaine

^b,e

, R. Henneck

^b

, M. Horras

^a,b

, P. Iaydjiev

^d,3

, S.N. Ivanov

^d,4

, M. Kasprzak

^j

, Y. Kermaïdic

^l

, K. Kirch

^a,b

, A. Knecht

^b

, H.-C. Koch

^j,k

, J. Krempel

^a

, M. Ku ´zniak

^b,f,5

, B. Lauss

^b

, T. Lefort

^e

, Y. Lemière

^e

, A. Mtchedlishvili

^b

, O. Naviliat-Cuncic

^e,6

, J.M. Pendlebury

ⁱ

, M. Perkowski

^f

, E. Pierre

^b,e

, F.M. Piegsa

^a

, G. Pignol

^l,∗

, P.N. Prashanth

^m

, G. Quéméner

^e

, D. Rebreyend

^l

, D. Ries

^b

, S. Roccia

ⁿ

, P. Schmidt-Wellenburg

^b

, A. Schnabel

^g

, N. Severijns

^m

, D. Shiers

ⁱ

, K.F. Smith

^i,7

, J. Voigt

^g

, A. Weis

^j

, G. Wyszynski

^a,f

, J. Zejma

^f

, J. Zenner

^a,b,o

, G. Zsigmond

^b

aETHZürich,InstituteforParticlePhysics,CH-8093Zürich,Switzerland bPaulScherrerInstitute(PSI),CH-5232Villigen-PSI,Switzerland

cHansBergerDepartmentofNeurology,JenaUniversityHospital,D-07747Jena,Germany dRutherfordAppletonLaboratory,Chilton,Didcot,OxonOX110QX,UnitedKingdom eLPCCaen,ENSICAEN,UniversitédeCaen,CNRS/IN2P3,Caen,France

fMarianSmoluchowskiInstituteofPhysics,JagiellonianUniversity,30-059Cracow,Poland gPhysikalischTechnischeBundesanstalt,Berlin,Germany

hInstitutLaue–Langevin,Grenoble,France

iDepartmentofPhysicsandAstronomy,UniversityofSussex,Falmer,BrightonBN19QH,UnitedKingdom jPhysicsDepartment,UniversityofFribourg,CH-1700 Fribourg,Switzerland

kInstitutfürPhysik,Johannes-Gutenberg-Universität,D-55128Mainz,Germany lLPSC,UniversitéGrenobleAlpes,CNRS/IN2P3,Grenoble,France

mInstituutvoorKern- enStralingsfysica,KatholiekeUniversiteitLeuven,B-3001Leuven,Belgium nCSNSM,UniversitéParisSud,CNRS/IN2P3,OrsayCampus,France

oInstitutfür Kernchemie,Johannes-Gutenberg-Universität,D-55128 Mainz,Germany

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received8July2014

Receivedinrevisedform19September 2014

Accepted19October2014 Availableonline23October2014 Editor:V.Metag

Keywords:

Ultracoldneutrons Mercuryatoms Magneticmoment Gyromagneticratio

Theneutrongyromagneticratiohasbeenmeasuredrelativetothatofthe¹⁹⁹Hgatomwithanuncertainty of0.8 ppm.Weemployedanapparatuswhereultracoldneutronsandmercuryatomsarestoredinthe samevolumeandreporttheresultγn/γHg=³.8424574(30).

©^{2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/3.0/).FundedbySCOAP³.

*

Correspondingauthor.

E-mailaddress:pignol@lpsc.in2p3.fr(G. Pignol).

1 NowatUniversityofWashington,SeattleWA,USA.

2 NowatMax-Planck-InstituteofQuantumOptics,Garching,Germany.

3 OnleavefromINRNE,Sofia,Bulgaria.

4 OnleavefromPNPI,St. Petersburg,Russia.

5 NowatQueen’sUniversity,KingstonON,Canada.

1. Introduction

After Chadwick’sdiscovery ofthe neutronin 1932, it became clear that nuclei are made out of protons and neutrons. In this

6 NowatMichiganStateUniversity,East-Lansing,USA.

7 Deceased.

http://dx.doi.org/10.1016/j.physletb.2014.10.046

0370-2693/©^{2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/3.0/).Fundedby SCOAP³.

picture the neutron hadto bear a nonzero magneticmoment in order to account for the magnetic moments of nuclei. The ob- servation that the neutron, an electrically neutral particle, has a nonzeromagneticmomentisinconflictwiththepredictionofthe relativistic Diracequation valid forelementaryspin 1/2 particles.

Thisfactwasanearlyindicationoftheexistenceofasub-structure forneutronsandprotons.

Thefirstdirectmeasurementoftheneutronmagneticmoment wasreportedbyAlvarezandBlochin1940[1]withanuncertainty ofone percent. The frequency f_n ofthe neutron spin precession in a magnetic field B0 was measured using the Rabi resonance method, from which the gyromagnetic ratio

γ

n=²

π

fn/B0 and themagnetic moment

μ

n= ¯^h/2

γ

n were extracted. The precision ofthismethodwas thenimproved byusing theprotonmagnetic resonance technique to measure the magnetic field B0 [2,3]. In 1956, Cohen, Corngold and Ramsey achieved an uncertainty of 25 ppm, using Ramsey’s resonance technique ofseparated oscil- latingfields [4]. Then, challenged by the possible discovery of a nonzeroneutron electric dipole moment (nEDM), Ramsey’s tech- niquewas further developed.Profiting fromthesedevelopments, Greeneandcoworkers[5]measuredin1977theneutron-to-proton magnetic moment ratio with an improvement of two orders of magnitudeinaccuracy.Inthelatterexperiment,theseparatedfield resonancetechniquewasappliedsimultaneouslytoabeamofslow neutrons anda flow of protons in waterprecessing in the same magneticfield.

Since1986theneutrongyromagneticratiohasbeenconsidered bytheCommitteeonDataforScienceandTechnology(CODATA)as afundamentalconstant.Inthe2010evaluationofthefundamental constants[6],theacceptedvalue

γ

n

2

π ⁼

²⁹

^.

^1646943

⁽

⁶⁹

⁾

^MHz

^/

^T

^[

⁰

^.

^{24 ppm}

^]

⁽¹⁾

was obtained by combining Greene’s measurement [5] of

γ

n/

γ

_p^

withthedeterminationoftheshieldedprotongyromagneticratio inpurewater

γ

_p^.

Inthisletterwereportonameasurementoftheneutronmag- netic moment with an accuracy ofbetter than 1 ppm, using an apparatusthat was builttosearch fortheneutron electricdipole moment[7].Theexperimentalmethoddiffersintwoessentialas- pectsfromtheonereportedinRef.[5].First,weusestoredultra- coldneutrons(UCNs)ratherthanabeamofcoldneutrons.Second, themagneticfieldismeasuredusingaco-magnetometerbasedon thenuclear spin precession of¹⁹⁹Hg atoms. The resultpresented is,in fact, a measurement of theratio

γ

n/

γ

Hg ofthe neutron to mercury¹⁹⁹Hgmagneticmoments.

Althoughnotassuitableastheproton,the¹⁹⁹Hgatomcanalso beusedasamagneticmomentstandard.In1960Cagnacmeasured

γ

Hg/

γ

_p^ inKastler’slabusingthenewly inventedBrossel’s double resonancemethod[8].Sincethemagneticfield wascalibratedus- ing nuclear magnetic resonance in water, the mercury magnetic momentwaseffectivelymeasuredinunitsoftheprotonmagnetic moment.Cagnac’sresult

γ

Hg/

γ

_p^₌0.1782706(3)canbecombined withthe current accepted value ofthe shieldedproton moment (

γ

_p^/2

π

=⁴².5763866(10)MHz/T[6]) toextract a mercurygyro- magneticratioof

γ

_Hg

2

π ⁼

⁷

^.

⁵⁹⁰¹¹⁸

⁽

¹³

⁾

^MHz

^/

^T

^[

¹

^.

^{71 ppm}

^].

⁽²⁾

Ourresultfor

γ

n/

γ

Hgwithanaccuracybetterthan1ppmpro- vides an interesting consistency check on results (1) and (2). It confirms the currently accepted value of the neutron magnetic moment,whichwas inferred froma single experiment[5].Alter- natively,ourresultcanbeusedtoproduceamoreaccurate value forthemercurymagneticmoment.

Fig. 1. Verticalcutthroughtheinner partofthe nEDMapparatus.Schematically indicatedare theprecessionchamber, the mercurycomagnetometerand the Cs magnetometerarray.

2. ThemeasurementwiththenEDMspectrometer

The measurementwas performedin fall2012withthe nEDM spectrometer, currently installed at the Paul Scherrer Institute (PSI). A precursor of this room-temperature apparatus, operated atInstitutLaue–Langevin (ILL),hasproduced thecurrentlylowest experimental limit for the neutron EDM [9]. A detailed descrip- tionoftheapparatus,whenoperatedatILL,canbe foundin[10].

A schematicviewofthecoreoftheapparatus,asinstallednowat PSI,isdepictedinFig. 1.

ThespectrometerusesUCNs,neutronswithkinetic energiesof

≈100 neV thatcanbestoredinmaterialbottlesforafewminutes.

Inatypicalmeasurementcycle,UCNsfromthenewPSIsource[11]

are guidedtotheprecession chamber, fillingthevolumefor34 s beforeclosingtheUCNvalve.The neutronsare fullypolarizedon their way fromthe source to the apparatus by a 5 T supercon- ducting magnet. The precession chamber is a 22 liter cylindrical storage volume of height H=^{12 cm made out of a deuterated} polystyrenering and two diamond-like-carboncoatedflat metal- licplates.Theentireapparatusisevacuatedtoaresidualpressure below 10⁻⁵mbar during the measurements. After the comple- tion ofthe Ramsey procedure described below,the UCNvalve is opened again and the neutrons fall down into a detector where they are counted. Onthe wayto the detectorthey passthrough a magnetizediron foilthat servesasa spin analyzer,providinga spin-dependentcountingoftheneutrons.

The precession chamber is exposed to a static vertical mag- netic field of B0≈^{1030 nT, either pointing upwards or down-} wards, corresponding to a neutron Larmor precession frequency of f_n=₂^γ_{π B}ⁿ 0≈^{30 Hz. This highly} homogeneous magnetic field (δB/B≈^{10−3) is generated by a cos}θ coil. In addition,a set of trim coils permits the optimization ofthe magnetic field unifor- mityortheapplicationofmagneticfieldgradients.Thesecoilsare woundontheoutsideofthecylindricalvacuumtank,whichissur- roundedbyafour-layermu-metalmagneticshield.

During the storage of polarized UCN, the Ramsey method of separatedoscillatoryfieldsisemployedinordertomeasure f_nac- curately.Atthe beginningofthestorageperiod,a firstoscillating horizontalfieldpulseoffrequency fRF≈ ^fn isappliedfort=^{2 s,} thereby flippingthe neutron spinsby

π

/2. Next, the UCN spins are allowed to precess freely in thehorizontal plane around the B0 field,fora precessiontime ofT =^{180 s.Asecond}

π

/2-pulse, at the same frequency fRF and in phase withthe first pulse, is thenapplied.TheRamseyprocedureisresonantinthesensethat itflipstheneutronspinsbyexactly

π

onlywhen fRF=^fn.

In order to monitorthe magnetic field B0 within the preces- sion chamber, a cohabiting mercury magnetometer is used [12].

A gasof¹⁹⁹Hgmercuryatomsiscontinuouslypolarizedbyoptical pumpinginapolarizationchamber situatedbelowtheprecession chamber.Atthebeginningofameasurementcycle,theprecession

Fig. 2. Ramseyfitfortherun6043.Inthisrunthegoodness-of-fitisquantifiedby χ²/d.o.f.=1.3.

chamberisfilledwithpolarizedatoms,anda

π

/2 mercurypulse isappliedimmediatelybeforethefirstneutronpulse.Thus,during theprecessiontime,bothneutronspinsandmercuryspinsprecess in the horizontal plane, samplingthe same volume. The Larmor frequency fHg=^γ₂^Hg_{π B}0≈^{8 Hz of themercuryatomsismeasured} by optical means: the modulation of the transmission of a cir- cularly polarizedresonant UV lightbeam froma²⁰⁴Hg discharge lampismeasuredwithaphotomultipliertube.Foreachcycle,the mercurycomagnetometerprovidesameasurementofthemagnetic fieldaveragedoverthesameperiodoftimeasfortheneutrons,at anaccuracyof

σ

(fHg)≈^{1 μHz.}

Atypicalrunconsistsofasuccession of∼^{20cycles,inwhich} theneutronpulsefrequency fRF israndomlyvaried.Weanalyzed each run to extract the neutron to mercuryfrequency ratio R= fn/fHg, which, in absence of systematic effects, would be equal to

γ

n/

γ

Hg. In a Ramsey experiment, the phase of the spin after the effective precession time T +^4t/

π

is φ=²

π

(f_RF− ^fn)(T+ 4t/

π

)(seee.g. [13]foraderivationofthefactor4/

π

).Whenthe magneticfieldactingontheneutronismonitoredviathemercury precessionfrequency,thentheanalysisofarunconsistsinfitting aRamseyfringepatterntothedata,accordingto

N^up_i

=

^N0



1

− α

cos



K

(

f_RF,i

/

f_Hg,i

−

^R

) 

,

(3)

whereN^up_i isthenumberofneutroncountsincyclei, f_RF,i is the frequencyoftheneutron

π

/2-pulsesforcyclei and fHg,ithemea- suredmercuryprecessionfrequency.Theparameter K definedby

K

=

²

π (

T

+

^4t

/ π ) 

^fHg



⁽⁴⁾

representstheperiodofthefringes,where^fHg^{isthemeanmer-} cury frequency during the run. For each run the three parame- ters

α

(visibility of the Ramsey fringe), N₀ and R are extracted from the fit. An example of a run is shown in Fig. 2. The ver- tical error bars follow from counting statistics and the horizon- tal error bars are negligible. The sensitivity of the technique is thus limited by neutron counting statistics and was on average

σ

(R)=¹.5×^{10−6perrun.}

Different runs correspond to different trim coil current set- tings and thus different magnetic field gradients. The measure- mentswithdifferentmagneticfieldconfigurationsservetocorrect thesystematiceffects described inthe next section.Forthe sake of simplicity we have retained for the final analysis only runs withgradients smallerthan200 pT/cm,i.e.7runswiththemag- neticfieldpointingdownwardsand9runswiththemagneticfield pointingupwards.

3. Systematiceffects

There are fourknown effects that could significantly shiftthe ratio R= ^fn/fHg from its unperturbed value

γ

n/

γ

Hg, either by affecting the neutron frequency or the mercury frequency. We writethecombinationofalltheseeffectsas

R

=

^fn f_Hg

= γ

n

γ

_Hg

⁽

¹

^{+ δ}

^Grav

^{+ δ}

^T

^{+ δ}

^Light

^{+ δ}

^Earth

^),

⁽⁵⁾

andaddressthemindetailinwhatfollows.

The most important effectis known as the gravitationalshift δGrav.Theroomtemperaturegasofmercuryatomsfillsthestorage volumewithauniformdensityinallspatialdirections.Conversely, the much colder UCNgas isstrongly affected by gravityand the UCN density is significantly higher at the bottom of the storage chamber thanatthetop.Thisresultsinadifferenceh ofthever- ticallocationsofthecentersofmassofthetwospecieswhich,in turn,resultsinashiftoftheratio R inthepresence ofavertical magneticfieldgradient:

δ

_Grav^↑/↓

= ±

^h B₀

∂

B

∂

z

,

(6)

where the arrows and the ± ^{sign refer to the direction of the} magnetic field. Adirect measurement ofthiseffect was reported in[14].

To correctforthe gravitationalshift,themagnetic field gradi- ent wasmeasured usingan arrayof cesiummagnetometers (four magnetometers onthetopofthechamber andseven onthebot- tom; seeFig. 1).Theworkingprincipleofthesemagnetometersis describedin[16].Thetransversecomponentsofthemagneticfield aresmall,thereforethescalarmagnetometersareeffectivelymea- suring the longitudinalcomponent: B≈^Bz.For each run, a field valuewasextractedforeachmagnetometerbyaveragingthemag- netometer readings over theduration of therun. A second-order parameterizationofthefieldwasthenfittedtothefieldmeasured at thepositions of the elevenmagnetometers. The parameteriza- tion

B

(

x

,

y

,

z

) =

^b0

+

^gxx

+

^gyy

+

^gzz

+

^gxx



x²

−

^z2

 +

gy y



y²

−

z²



+

gxyxy

+

gxzxz

+

gyzyz

,

(7) isthemostgeneralsecondorderpolynomialsatisfyingconstraints imposed by Maxwell’s equation. The nine parameters were ex- tracted by performing an unweighted least-squares minimization.

The mainparameter ofinterestis g_z,since thedifferencesofthe UCNandmercuryvolumeaverages forthe othertermsare negli- gible. Foran errorevaluationon gz,the jackknifeprocedurewas applied,byremovingonemagnetometeratatimeandperforming thefitalways withtenmagneticfieldvalues.Thestandarddevia- tionofthoseelevenvaluesforg_z wasthenusedastheuncertainty ongz.Thejackknifeerrorobtainedbythisprocedurewas8 pT/cm.

Thisanalysismethodwas testedusingtoydata[15],itwasfound that the jackknife procedure accounts for the incompleteness of thesecond orderparameterization(7)andforthepossibleoffsets ofthemagnetometers.

Fig. 3 shows the R value for each run as a function of the gradient gz. We extrapolate the R value to the limit of vanish- ing gradient by performing a combinedfit of thedata according to

R^↑

=

^R↑₀



1

+

^h

B₀g^↑_z



and R^↓

=

^R↓₀



1

−

^h

B₀g^↓_z



,

(8)

andobtainthefollowingvaluesforthethreefitparameters:

Fig. 3. NeutrontomercuryfrequencyR=fn/fHgplottedasafunctionofthefield gradient.Upwards-pointing(red)anddownwards-pointing(blue) trianglesrepre- sentsrunsinwhich B0 waspointingupwardsand downwards,respectively.The verticalerrorbarsaresmallerthanthesizeofthetriangles.

h

= −

⁰

.

235

(

5

)

cm

,

(9)

R^↑₀

=

3

.

8424580

(

23

),

(10)

R^↓₀

=

³

.

8424653

(

27

).

(11)

Thegoodness-of-fitisquantifiedby

χ

²/d.o.f.=²³/13,whichsup- portsthevalidityofthegradienterrors.Ananalysisofanextended set of data, including runswith higher magnetic field gradients, hasbeenperformedindependently[17],confirmingtheresultpre- sentedhere.ThevaluesR^↑₀ andR^↓₀ areintermediatequantitiesfor whichEq.(5)holdswiththeδGravtermsettozero.Theotherthree systematiceffects,whichdependsolelyatfirstorderonthemag- neticfielddirection,stillneedtobecorrected.

ThesecondsystematiceffectδT arisesfromresidualtransverse magneticfieldcomponents BT.Theneutronmagnetometerissen- sitive to the volume average of the magnetic field modulus, viz.

f_n∝ |^B| ^{sincethe neutron Larmorfrequency islarger than the} wallcollisionfrequency(adiabaticregime).Ontheotherhand,the mercuryLarmorfrequency is muchlower than the wall collision frequency. As a consequence the mercury magnetometer is sen- sitive to the vectorial volume average of the field: f_Hg∝ |^B|^. Fromthisessentialdifferencethe resultingfrequencyshiftdueto atransversefieldisgivenby

δ

T

= 

BT2



2B²₀

,

(12)

where^BT2^{isthevolumeaverageovertheprecessionchamberof} thesquaredtransversefield.

Tocorrectfor the transverseshift we performedan extensive magnetic-field mapping of the precession chamber region using a three-axis fluxgate magnetometer attached to a robot mapper.

Fromthe magnetic-fieldmapswe were abletoextract thetrans- versefieldcomponentsinbothconfigurationsB₀upandB₀down:



B_T²



_↑

= (

²

.

1

±

⁰

.

5

)

nT²

; 

B_T²



_↓

= (

¹

.

7

±

⁰

.

7

)

nT²

.

(13) Fromthisweinfercorrectionsgivenby

δ

^↑_T

= (

¹

.

0

±

⁰

.

2

) ×

¹⁰⁻⁶

; δ

_T^↓

= (

⁰

.

8

±

⁰

.

3

) ×

¹⁰⁻⁶

.

(14) The third systematic effect δ_Light arises from a possible light shift of the mercury precession frequency induced by the light beamthatdetectstheHgfree-inductiondecay.Thisphenomenon,

discoveredin1961by Cohen-Tannoudji[18],isashiftoftheres- onance frequencyproportional to the UV light intensity.We per- formed adedicated test toquantify thiseffect,by measuringthe mercuryfrequencywhilevaryingthelightintensity.Thelightshift has avectorial component, i.e.it could depend on the anglebe- tweenthelightbeamaxisandthemagneticfielddirection.There- fore the associated frequency shift could in principle depend on theB0 direction.Weperformedthetestforboth B0 polaritiesand wereporttheresult:

δ

_Light^↑

= (

⁰

.

34

±

⁰

.

18

) ×

¹⁰⁻⁶

;

δ

_Light^↓

= (

⁰

.

21

±

⁰

.

14

) ×

¹⁰⁻⁶

.

(15)

ThelastsystematiceffectδEarth isashiftduetotheEarth’sro- tation(see[19]foradiscussionofthiseffectinthecontextofthe nEDM).Theprecessionfrequenciesoftheneutronandthemercury resultsfromtheLarmorfrequenciesinanon-rotatingframecom- binedwiththerotationoftheEarth.Onecanderivethefollowing expressionfortheassociatedfrequencyshift

δ

_Earth^↑/↓

= ∓



f_Earth

fn

+

^fEarth fHg



sin

(λ) = ∓

¹

.

4

×

¹⁰⁻⁶

,

(16)

where f_Earth =¹¹.6 μHz is the Earth’s rotation frequency and sin(λ)=⁰.738 thesineofthelatitudeofthePSIlocation.Thecon- ventionsare suchthat fn and fHg arepositive frequencies.Inthe derivation of Eq.(16)it was importantto consider that the true neutron andmercury gyromagneticratios are negative and posi- tivequantities,respectively.

The relative difference between the R ratios measured with B0 up and B0 down,after correcting forthe gravitational, trans- verse,andlightshifts,shouldamounttotheEarth’srotationeffect, whoseanticipatedvalueisδ^↑_Earth− δ_Earth^↓ = −².7×^{10−6.Indeedwe} find

2R^↑₀

−

R^↓₀ R^↑₀

+

^R↓₀

− 

δ

^↑_T

− δ

_T^↓



− 

δ

^↑_Light

− δ

_Light^↓



= (−

²

.

2

±

¹

.

0

) ×

¹⁰⁻⁶

,

(17)

inagreementwiththeexpectedvalue.

Two additional minorsystematic effects were considered and neglected. First,the Bloch–Siegert shiftofthe neutronresonance, associatedwiththeuseofalinearlyoscillating ratherthanrotat- ing field for applying the

π

/2-pulses,is calculated to be 2 μHz.

Second, possible biases induced by the DAQ electronicswere in- vestigated.Adedicated multifunctionelectronic module[20] syn- chronizedonan atomicclockserves bothtogeneratetheneutron pulsesandto sample themercuryprecession signal. We checked thateffectsthatcouldmodifythefrequencyratio,suchasalossof phasecoherencebetweenthetwoneutronpulses,arenegligible.

The error budget and the correction procedure are summa- rized in Table 1 and the key data for each run is presented in Table 2.Asafinalstepwecombinetheresultsobtainedfor B0 up and B0 down.Theagreementbetweenthesetworesultsprovides a nontrivial consistency check. To avoid double use of data, we conservativelyquoteforthefinaluncertaintythelargestofthetwo individualerrors:

γ

n

/ γ

Hg

=

³

.

8424574

(

30

) [

⁰

.

78 ppm

].

⁽¹⁸⁾

4. Discussion

Ourresult(18)canbeconsideredasaconsistencycheckofthe acceptedvaluesforthe neutronand¹⁹⁹Hg magneticmoments as showninFig. 4.Equivalently, thisresultprovidesa newaccurate

Table 1

Errorbudgetforthemeasurementofγn/γHg.

Effect B0↑ ^B0↓

Counting statistics ±⁰.5×¹⁰⁻⁶ ±⁰.5×¹⁰⁻⁶ Gravitationalshift

(3.84× δGrav)

(−⁸.9±².3)×¹⁰⁻⁶ (−¹.8±².7)×¹⁰⁻⁶

Intermediate R0 3.8424580(23) 3.8424653(27) Transverseshift

(3.84× δ^T)

(3.7±⁰.8)×¹⁰⁻⁶ (3.0±¹.2)×¹⁰⁻⁶ Lightshift

(3.84× δLight)

(1.3±⁰.7)×¹⁰⁻⁶ (0.8±⁰.6)×¹⁰⁻⁶ Earthrotation

(3.84× δEarth)

−⁵.3×¹⁰⁻⁶ +⁵.3×¹⁰⁻⁶

Corrected value 3.8424583(26) 3.8424562(30) Combined finalγn/γHg 3.8424574(30)

Table 2

Keydataforeachrunselectedfortheanalysis:thefittedvisibilityα,theratioR, theaveragefieldBHgextractedfromthemercurycomagnetometer,thegradientgz extractedfromtheCsmagnetometerarrayandthesquaredtransversefieldBT2 extractedfromthefieldmapsarereported.

Run # α R BHg

[nT]

gz [pT/cm]

BT2 [nT²] B0↓

6015 0.41 3.842321 1031.86 −^{175 2}.5

6016 0.48 3.842587 1031.33 130 1.8

6023 0.66 3.842435 1031.60 −^{23 1}.1

6027-8 0.57 3.842508 1030.18 32 1.0

6030 0.55 3.842415 1030.32 −43 1.0

6031 0.38 3.842622 1029.92 185 2.6

6033 0.48 3.842357 1030.45 −120 1.7

B0↑

6040-1 0.54 3.842445 1027.97 9 1.7

6042 0.36 3.842325 1028.23 161 2.6

6043 0.38 3.842582 1027.70 −^{144 3}.0

6047 0.46 3.842520 1027.82 −67 2.1

6049 0.43 3.842378 1028.14 86 1.9

6058 0.53 3.842434 1027.82 17 1.7

6059 0.42 3.842511 1026.82 −^{44 2}.0

6060 0.55 3.842455 1028.25 5 1.7

6064 0.44 3.842392 1029.47 68 1.9

andindependentmeasurementoftheneutrongyromagneticratio.

Using(2),weobtain:

γ

n

2

π ⁼

²⁹

^.

¹⁶⁴⁷⁰⁵

⁽

⁵⁵

⁾

^MHz

^/

^T

^[

¹

^.

^{89 ppm}

^],

⁽¹⁹⁾

thus confirming the accepted value which, until now, was based onasinglemeasurement[6].

Alternatively,wecan usethe results(1)and(18)toproposea new,moreaccuratevalueofthe¹⁹⁹Hgatomicgyromagneticratio:

γ

Hg

2

π ⁼

⁷

^.

^5901152

⁽

⁶²

⁾

^MHz

^/

^T

^[

⁰

^.

^{82 ppm}

^].

⁽²⁰⁾

Acknowledgements

WearegratefultothePSIstaff(the acceleratoroperatingteam andtheBSQgroup)forprovidingexcellentrunningconditions and

Fig. 4. 1-sigmaallowedregionsintheγn,γHgplane.Ourfinalvaluefortheneutron tomercurymagneticmomentratio(18)herelabeledas“PSI2012”formsthedi- agonalband.Thehorizontalbandistheneutronmagneticmoment(1)valuefrom Greeneetal.andtheverticalbandisfromthemeasurementofthemercurymag- neticmoment(2)byCagnac.

acknowledge the outstanding support of M. Meier and F. Burri.

SupportbytheSwissNationalScienceFoundationProjects200020- 144473,200021-126562,200020-149211(PSI)and200020-140421 (Fribourg) isgratefullyacknowledged.TheLPCCaenandtheLPSC acknowledge the support of the French Agence Nationale de la Recherche(ANR)underreferenceANR-09–BLAN-0046.Polishpart- nerswanttoacknowledgeTheNationalScienceCentre,Poland,for the grant No. UMO-2012/04/M/ST2/00556. This work was partly supportedbytheFundforScientificResearchFlanders(FWO),and ProjectGOA/2010/10oftheKULeuven.Theoriginalapparatuswas fundedbygrantsfromtheUK’sPPARC.

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