First measurement of the temperature dependence of muon transfer rate from muonic hydrogen atoms to oxygen

Contents lists available atScienceDirect

Physics

Letters

A

www.elsevier.com/locate/pla

First

measurement

of

the

temperature

dependence

of

muon

transfer

rate

from

muonic

hydrogen

atoms

to

oxygen

E. Mocchiutti

a

,

∗

,

A. Adamczak

b

,

D. Bakalov

c

,

G. Baldazzi

d

,

R. Benocci

e

,

f

,

R. Bertoni

e

,

M. Bonesini

e

,

g

,

V. Bonvicini

a

,

H. Cabrera Morales

a

,

F. Chignoli

e

,

M. Clemenza

e

,

g

,

L. Colace

h

,

i

,

M. Danailov

a

,

j

,

P. Danev

c

,

A. de Bari

k

,

l

,

C. De Vecchi

l

,

M. De Vincenzi

h

,

m

,

E. Furlanetto

a

,

n

,

F. Fuschino

d

,

o

,

K.S. Gadedjisso-Tossou

a

,

p

,

q

,

D. Guffanti

a

,

r

,

K. Ishida

s

,

C. Labanti

d

,

o

,

V. Maggi

e

,

f

,

R. Mazza

e

,

A. Menegolli

k

,

l

,

G. Morgante

d

,

o

,

M. Nastasi

e

,

g

,

J. Niemela

p

,

C. Pizzolotto

a

,

A. Pullia

t

,

u

,

R. Ramponi

t

,

v

,

L.P. Rignanese

d

,

M. Rossella

l

,

N. Rossi

n

,

M. Stoilov

c

,

L. Stoychev

a

,

p

,

L. Tortora

h

,

E. Vallazza

a

,

G. Zampa

a

,

A. Vacchi

a

,

n

,

s a

SezioneINFNdiTrieste,viaA.Valerio2,Trieste,Italy

b_{InstituteofNuclearPhysics,PolishAcademyofSciences,Radzikowskiego152,PL31342Kraków,Poland}

c_{InstituteforNuclearResearchandNuclearEnergy,BulgarianAcademyofSciences,blvd.Tsarigradskoch.72,Sofia1142,Bulgaria} d_{SezioneINFNdiBologna,vialeBertiPichat6/2,Bologna,Italy}

e_{SezioneINFNdiMilanoBicocca,PiazzadellaScienza3,Milano,Italy}

f_{DipartimentodiScienzedell’AmbienteedellaTerra,UniversitàdiMilanoBicocca,PiazzadellaScienza1,Milano,Italy} g_{DipartimentodiFisicaG.Occhialini,UniversitàdiMilanoBicocca,PiazzadellaScienza3,Milano,Italy}

h_{SezioneINFNdiRomaTre,ViadellaVascaNavale84,Roma,Italy}

i_{DipartimentodiIngegneria,UniversitàdegliStudiRomaTre,ViaV.Volterra62,Roma,Italy} j_{SincrotroneElettraTrieste,SS14,km163.5,Basovizza,Italy}

k_{DipartimentodiFisica,UniversitàdiPavia,viaA.Bassi6,Pavia,Italy} l_{SezioneINFNdiPavia,ViaA.Bassi6,Pavia,Italy}

m_{DipartimentodiMatematicaeFisica,UniversitàdiRomaTre,ViadellaVascaNavale84,Roma,Italy}

n_{DipartimentodiScienzeMatematiche,InformaticheeFisiche,UniversitàdiUdine,viadelleScienze206,Udine,Italy} o_{INAF-OASBologna,viaP.Gobetti93/3,Bologna,Italy}

p_{TheAbdusSalamInternationalCentreforTheoreticalPhysics,StradaCostiera11,Trieste,Italy}

q_{LaboratoiredePhysiquedesComposantsàSemi-conducteurs(LPCS),Départmentdephysique,UniversitédeLomé,Lomé,Togo} r_{GranSassoScienceInstitute,viaF.Crispi7,L’Aquila,Italy}

s_{RikenNishinaCenter,RIKEN,2-1Hirosawa,Wako,Saitama351-0198,Japan} t_{SezioneINFNdiMilano,viaCeloria16,Milano,Italy}

u_{DipartimentodiFisica,UniversitàdegliStudidiMilano,viaCeloria16,Milano,Italy}

v_{IFN-CNR,DipartimentodiFisica,PolitecnicodiMilano,piazzaLeonardodaVinci32,Milano,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received23March2020

Receivedinrevisedform31May2020 Accepted9June2020

Availableonline17June2020 CommunicatedbyS.Khonina Keywords:

Muonichydrogen Transferrate Oxygen

We reportthe firstmeasurementofthe temperaturedependenceofmuon transferrate frommuonic hydrogenatomstooxygenbetween100and300 K.DatawereobtainedfromtheX-rayspectraofdelayed eventsinagaseoustarget,madeofaH2/O2mixture,exposedtoamuonbeam.Thisworksetsconstraints

ontheoreticalmodelsofmuontransferandisoffundamentalimportanceforthemeasurementofthe hyperfinesplittingofmuonichydrogengroundstateasproposedbytheFAMUcollaboration.

©2020TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*

Correspondingauthor.

E-mailaddress:Emiliano.Mocchiutti@ts.infn.it(E. Mocchiutti).

1. Introduction

We present the results of a systematic experimental investi-gationof the temperaturedependence ofthe muon transfer rate from the ground-state muonic hydrogen atom

μ

p to oxygen. A

https://doi.org/10.1016/j.physleta.2020.126667

preciseknowledgeofthisprocessis ofkey importanceforthe fi-nalobjectiveoftheFAMUCollaboration–themeasurementofthe hyperfinesplitting(HFS)ofthe 1S stateof

μ

pby meansoflaser spectroscopy [1,2].

Inthe experimentplannedby theFAMU collaboration,a laser tunedtothehyperfine-splittingresonanceenergyinduces singlet-to-triplettransitionsbetweenthetotal-spinstatesF

=

0 andF

=

1 of1S muonic hydrogen,whichisformedandthenthermalizedin agaseousH2 target.Afterthelaserexcitation,the

μ

p(F

=

1)atom

quicklyde-excitesbacktotheF

=

0 stateincollisionswithprotons boundin thesurroundingH2 molecules.As aresult, theHFS

en-ergyisconvertedintoadditional

μ

p kineticenergyupto 0.12 eV with a wide distribution duethe simultaneous rotational transi-tionsintheaffectedH2 molecules.

A clearand strong signal from the “laser-accelerated” neutral

μp

atomsisnecessaryinordertoperformapreciseHFS measure-ment.Ourideaistouseanadmixture ofahigh-Zelement inthe H2 target and detect characteristic X-rays from the de-excitation

ofthehigh-Zmuonicatom formedwhenthemuonistransferred from

μ

p. Since muon transfer takes place both from the ther-malizedandlaser-accelerated

μ

p atoms, itis crucialto choosea higher-Zelementwhichischaracterizedbyastrongenergy depen-denceofthemuon-transferratebelowabout0.1 eV.

AccordingtothePSIexperiments [3–5], oxygenisagood can-didate.Althoughtheseexperimentswere performedonlyatroom temperature,anunexpectedlyhighvalueofthemuontransferrate was found for the (μp)1S atoms not yet thermalized

(“epither-mal”).Energetic

μ

p atomsaredueto theaccelerationvia atomic cascade [6,7] afterthenegativemuoncaptureinhydrogenand

μ

p formationinexcitedCoulombstates(n

≈

14).Astrongdependence of the muon transfer rateto oxygen to the collision energy was alsoanticipated inthequantum-mechanicalcalculationsreported inRef. [8,9].

The PSI data on the energy-dependent muon transfer rateto oxygen were fitted with a two-step function of the collision en-ergy [5].Thisfunction,however,isinappropriateformodeling the HFSmeasurement because(a) thefitis toorough an approxima-tion to the data and (b), most important, it provides no infor-mation about the energy dependence of the transfer ratein the energy range below the threshold of 0.12 eV (corresponding to temperatures

∼

1000K),whichinfactistherangeofexperimental interest.Thisinformation canonlybe obtainedby measurements ofthemuontransferratetooxygenfrom

μ

pthermalizedatomsin asufficientlywidetemperaturerangebelow1000 K,asperformed inthepresentwork.

InthermalequilibriumattemperatureT ,theobservable trans-ferrate



pO

(

T

)

isexpressedintermsofthefunctionthatdescribes

thedependence ofthetransfer rateon thecollision energy E by

meansoftherelation



pO

(

T

)

=

∞



0 dE fMB

(

E

;

T

) λ

pO

(

E

),

(1) where fMB

(

E

;

T

)

=

2



E

/

π

(

kBT

)

−3/2e−E/(kBT) (2) is the Maxwell–Boltzmann distribution, with kB the Boltzmann constant.The main contributionto theintegral inthe right-hand side ofEq. (1) comes fromtheregion around E

=

kBT ; therefore themeasurementof



pO

(

T

)

intherangebetween100and300 K

willprovide importantinformation aboutthe behavior of

λ

pO

(

E

)

forenergiesinandaroundtherange0.01

<

E

<

0.05 eV.

The obtainedresults allow us to test the available theoretical calculationsofthemuon transferrateandwillbeusedtochoose optimalconditionsfortheFAMUHFSmeasurement.

2. Experiment

TheFAMUapparatus,usedforthetransferratemeasurement,is builtaroundapressurizedandthermalizedgastargetcontainedin an aluminiumvesselkept atthermalequilibriumandsurrounded byX-raydetectors.TheFAMUexperimentalmethodrequiresa de-tection system suited for time resolved X-ray spectroscopy [2]. Eightscintillatingcountersare used,basedon1”Lanthanum Bro-mide crystalsandread by Hamamatsuhighspeed

photomultipli-ers. Data are acquired and processed by 500 MHz CAEN V1730

digitizers,withintheframeworkoftheFAMUDataAcquisition Sys-tem [10].Outputsignalsfromthedetectorsprovidemeasurements of boththe energyandtimespectrum ofthe recordedevents.In this configuration,thedetectors providea good resolution,about 8.5% (FWHM) at 133 keV, and an excellent time resolution, sig-nalrise timeofabout12 ns [11]. Theexperimentwasperformed attheRIKENmuon facilityoftheRutherfordAppletonLaboratory (UK) [12],usingmuonsproducedinbuncheswitharepetitionrate

of 50 Hz. More details about the FAMU apparatus can be found

in [11].

The datausedinthisanalysiswere acquiredusingagas mix-tureof H2 andO2 withan oxygen concentration cOof 190 ppm.

The target was filled atroom temperatureto 41 bar and subse-quentlybrought tosixdifferenttemperatures,from300to104 K degrees.The lowesttemperaturewassignificantlyhigherthanthe oxygen condensation point forthe mixture, 54 K, obtainedfrom Ref. [13]. A thermally insulated vertical tube provided external connection to the target.After filling,the target was sealedby a valveplacedjustoutsidethecryogenicvessel.Theportionoftube between thetarget andthe valve hada volume aboutten times smallerthanthetargetitself.Duetogravity,thetemperature gra-dientinsidethetuberangedfromroomtemperature,atthevalve, tothegastarget temperature,atthevessel.Thevariationof den-sity inside the target was sufficiently small that any associated systematiceffectswerenegligible.

Anaccuratedeterminationofthemuontransferrateasa func-tion of kinetic energy requires measurements under steady-state conditions, when the kinetic-energy distribution of

μ

p atoms is well known.The distributionof

μ

pinitial energy,afterthemuon capture in H2 andsubsequent atomic cascade, is broad and can

coverhundredsofeVdepending onthetargetdensity [6,7,14,15]. The shape of this distribution is not well known. Therefore, we havemeasuredthemuontransferrateonlyaftertheslowingdown and thermalizationof

μ

patoms, whentheir kinetic energies are described bytheMaxwell–Boltzmanndistribution foragiven tar-gettemperature.

Data were takenwitha targetnumber densitywhichenabled fastthermalizationof

μ

patoms(manytimesfasterthanthemuon lifetime) and rapid quenching ofthe initial statistical population of the spin state F

=

1. For H2 gas at 41 bar and 300 K, the

MonteCarlosimulationgivesathermalizationtimeof150 nsand a quenching time of 10 ns [16]. In order to observe the char-acteristic X-rays from muonic oxygen over long times, the oxy-gen concentration fora given target density cannot be toohigh. For cO

=

190 ppm atthis temperatureandpressure, the average

muon transfer rate fromthe thermalized

μ

p atomsto oxygen is 0.78

×

106 s−1 (usingthe experimentalrateof 8.5

×

1010s−1 at liquid-hydrogendensity [5]),whichiscomparable withthe muon decay rate. This choice enabled us to observe the muon-transfer processfromthermalized

μ

patomsforseveralmicroseconds.

Muonic oxygen atoms are formed in excited Coulomb states,

thiswaythe large backgroundof X-rayemissions dueto the in-teraction between muons and all the elements of the target — mostly aluminium, nickel,gold, andcarbon — was strongly sup-pressedandcouldbeneglected.

Undertheseconditions,atagiventemperature,T ,thevariation ofthenumberNμp of

μ

patomsinthetarget inthetimeinterval

dt isgivenby:

dNμp

(

t

)

= −

Nμp

(

t

)λ

dis

(

T

)

dt

,

(3)

where

λ

dis

(

T

)

isthetotaldisappearancerateofmuonichydrogen

atomsattemperatureT ,givenby:

λ

dis

(

T

)

= λ

0

+ φ [

cp



ppμ

+

cd



pd

(

T

)

+

cO



pO

(

T

)

].

(4)

Here

λ

0

= (

4665.01

±

0.14)

×

102 s−1 [17,18] is the

disappear-ancerateofthemuonsboundtoprotons(thatincludesbothmuon decayandnuclearcapture),



ppμ

=

2.01

×

106s−1 [17] isthe

for-mation rate of the pp

μ

molecular ion in collisions of

μ

p with ahydrogen nucleus(normalized to liquidhydrogendensity, LHD,

N0

=

4.25

×

1022 atom/cm3) and



pO

(

T

)

is the muon transfer

rate from

μ

p to oxygen atoms. The muon transfer rate



pd

(

T

)

from

μ

ptodeuterium,boundmostlyinHDmolecules,variesfrom 8.65

×

109_s−1 _{at100 Kto8.20}

_×

₁₀9 _s−1 _{at300 K(normalizedto}

2.125

×

1022_{HD molecules/cm}3_{). Therate}

_

pd formuontransfer

tobaredeuteriumnucleiispracticallyconstantatthelowest ener-gies [19].The energy-dependentelectronscreening inhydrogenic molecules [20] leadsto the above appreciable changeof



pd

(

T

)

intheconsideredtemperatureinterval.Thenumberdensityofthe atomsinthetargetgasis

φ

= (

4.869

±

0.003)

×

10−2inLHDunits, andcp,cO,andcdarethenumberconcentrationsofhydrogen,

oxy-gen,anddeuteriumrespectively.Weusedhydrogenwithmeasured naturaldeuterium abundance cd

= (

1.358

±

0.001)

×

10−4 [21].1

TheoxygenconcentrationwascO

= (

1.90

±

0.04)

×

10−4 andcp

=

1

−

cO

−

cd.

Letusnotethattherateofnonresonant pp

μ

formationis prac-tically constantbelow0.1 eV [22]. The

μ

d atoms,which are cre-atedviathemuontransferfrom

μ

patomstothesmallnatural ad-mixtureofdeuterium,haveaninitialenergyofabout42 eV.They areneverthermalizedsinceadeepRamsauer-Townsendminimum inthecrosssectionofscattering

μ

d+H2 ispresentat7 eV [23].As

aresult, fortarget densities corresponding to 300 K hydrogenat 41 bar,the mean kinetic energyof

μ

d atomsis 26 eV andonly a very small fraction of these atoms is thermalized. The muon transfer rate



dO from

μ

d atoms to oxygen is negligible at

en-ergies



1 eV,asconfirmedbytheabsenceofcorrespondingX-ray spectrafrommuonicoxygenatshorttimesintheexperiment per-formed ina pure D2 target witha small admixture of SO2 [24].

Therefore, a contribution to the X-ray spectra due to the muon transferfrom

μ

datomstooxygenisneglectedinthepresent anal-ysis. Our Monte Carlo simulation, which used the experimental results of Ref. [24] for the rate



dO at thermal, epithermal and

higherenergies,has shownthat thiscontributionis smallerthan 1%fortimesuptoseveralmicroseconds.

3. Dataanalysis

Inouranalysis,onlythesteady-statedelayedX-rayeventsdue tothethermalizedatomsareconsidered.Theonlyunknown vari-ableisthetransferrate



pO

(

T

),

whichisdeterminedusingEqs. (3)

1 _{Asamplebottleofthesamebatchofpurehydrogenusedtoproducethe} mix-turewasprovidedbythegassupplier.Thissamplegaswastestedusingaprecise massspectrometertodeterminethedeuteriumabundance.Unfortunately,thesame studycouldnotbedonewiththegasmixtureusedduringtheacquisitionto deter-minebettertheoxygenabundance.

and(4) bynumericallyfittingthetimeevolutionoftheoxygen X-rays at temperature T andleaving



pO a free parameter. Details

aboutthemethodcanbefoundinRef. [25].

The experimental sample used in thiswork consistsof about 2.6

×

106 muon triggers, andcorresponds to

≈

7.8

×

107 recon-structedX-rays.Thetime,innanoseconds,associatedtoeachX-ray is relative to the trigger generated by the accelerator and beam control system. In thistime reference frame, thetwo muon spill

bunches peak at about 530 and850 ns. Each muon spill bunch

consistsofabout103 _muons.

Dataweretakenatsixtargettemperatures:104,153,201,240, 272,and300 K.Twosensorswereusedtomeasurethetarget tem-perature.Thecryogenic systemwas abletokeep thetemperature stableby limitingfluctuationsto about10 mK/h,within the sys-tematicerrorsofthetemperaturesensors. Moredetailsaboutthe cryogenicsystemcanbefoundin [11].Eachtemperaturewaskept stableforanacquisitiontimeofthreehours.

X-ray signalswere identifiedandreconstructed usinga fitting procedureonthedetectorwaveforms.Acleandatasamplewas ob-tainedbyapplyinglightselectioncriteriabasedonthereduced

χ

2

ofthewaveformfitandonthedistancebetweentwoconsecutive reconstructed signals. The reduced

χ

2 _{selection (}

_χ

_˜

2

_<

_{100) was}

used to rejectnot-converged fit andpoorly reconstructed events. Therequirementonthedistancebetweentwoconsecutivesignals to be greater than 30 ns was chosen by testing the reconstruc-tionalgorithm bymeans ofaGEANT4 simulation.The simulation

showedthat when two peaks are separatedby morethan 30 ns

the software reconstruction efficiency and accuracy (correct en-ergyreconstruction)isbetterthan99.9%.Selectionefficienciesand fractional livetime were estimatedandtakenintoaccountinthe analysisofthe time evolutionof theoxygen lines.The combined efficienciesandfractionallivetimewereabout95%,constantabove 2000ns,andsmoothly decreasedtoabout92% at1200ns. A de-taileddiscussionofthedataselectionandselectionefficienciescan befoundinRef. [25].Foreachtemperature,theenergyspectrumof delayedeventswasstudiedasafunctionoftime.Thedelayedtime window –from

≈

1200to 10000 nsfrom thetrigger –was split into 20 bins of increasing width, the narrowestbeing

≈

140 ns. The time resolution of the reconstructed events was better than 1 ns,henceanymigration effectonneighboring binswas negligi-bleandnotimedeconvolutionwasneeded.

Anestimation ofthebackgroundbelowtheoxygen-linesignal was the most important aspect of the data analysis. The

back-ground was estimated for each time and temperature bin using

thedatatakenwithapureH2-gastargetandwiththesame

selec-tioncriteriaandsamegascondition,butwithasmallerstatistical sampling.

Fig. 1. Energyspectraat104 Kintwodifferenttimebins:[1140,1283] nsintheleftpaneland[3717,4183] nsintherightpanel.Solidgreenline:energyspectra;dottedred line:estimatedbackground;shadedarea:signalafterbackgroundsubtraction.(Forinterpretationofthecolorsinthefigure(s),thereaderisreferredtothewebversionof thisarticle.)

both backgroundand signal eventsin the normalizationinterval. A further consideration of systematicerrors in the shape of the backgroundwas estimatedbycomparingthepurehydrogen back-grounddistributionwithanalyticalfunctionsfittedtothedataand by using gas target mixture withdifferent composition (i.e., CO2

inhydrogenandArinhydrogen).Theoverallsystematicerrorwas quadraticallysummedto thestatisticalerrorforeach signal spec-trum(see,e.g.,numbersreportedinFig.1),beforeperformingthe fittothedatathatprovidesthetransfer-ratemeasurement.

Notice that, due to the background subtraction, a time mea-surementcannotbeassociatedto eachoxygenx-ray, i.e.,itisnot possibletotagthe singleevent. Hence,afterbackground subtrac-tioninagiventimebin,thereisnotanaveragetimemeasurement coupledtothesignalspectrum.

The time evolution of the oxygen line X-ray spectra, at the two temperaturelimitsof300 K(red circles)andat 104 K(blue circles),isshowninFig.2andreportedinTable1forallthe tem-peratures. Each point represents the integrated signal, after the background subtraction and efficiency correction, divided by the time binwidth.Data point arecentered inthetime binsand er-rorbarsonthexaxisrepresentthetimebinwidth.Toperforma fitof thedataversus time, a step-likefit functionwas used: the fitfunctionisan exponentialfunction,whichisintegratedwithin each time bin that was defined for the collected data. Function anddatacanthenbeconsideredastwohistograms,whichcanbe compareddirectly. Forthepurposes ofthisquantitative compari-sonthequestionofchoosingthecorrecttimevaluewithinthebin is irrelevant. The fit was performed starting at

≈

1200 ns, about 350 nsafterthe second muonspill inorderto take intoaccount onlythethermalizedphase,givenathermalizationtimeof150 ns. Theupperlimitofthetimewindowwas chosen accordingtothe available statistics: data points were used if the measured inte-grated signal was greater than three times the associated error.

Fig. 2. Timedependenceofoxygenlineintensitiesat300 Kand104 K.Two expo-nentialfunctions(solidlines)correspondtothetransferratefit.Dashedlinesarean extrapolationofthefitatshortertimes.Verticalerrorbarsassociatedtothepoints includethestatisticalandbackground-systematicerrors.Horizontalerrorbars rep-resentthebinwidth.

Table 1

TimeevolutionoftheoxygenlineX-rayspectrameasuredatsixdifferenttemperatures.Uncertainties asso-ciatedtothemeasurementsincludestatisticalandbackground-systematicerrors.Noticethatthefirstthree timebinscoveranon-thermalizedphasewhichcoincidewiththesecondmuonbeamspillarrival,low de-tectionefficiency,andhighpile-upeffects.Thefitwasperformedstartingfromthebin[1140,1283]ns.

Time bin Oxygen line time evolution [X-rays/ns]

[ns] 300 K 272 K 240 K 201 K 153 K 104 K 800 – 900 435±29 477±28 451±29 407±28 365±30 279±28 900 – 1013 326±15 333±14 296±15 266±14 220±15 202±14 1013 – 1140 257±11 252±10 243±11 224±10 182±11 146±10 1140 – 1283 228±8 216±8 215±8 192±8 156±8 111±8 1283 – 1444 166±7 165±7 161±7 133±7 123±7 83±7 1444 – 1626 136±5 124±5 133±5 119±5 109±6 88±5 1626 – 1829 104±5 104±4 103±5 96±4 87±5 68±4 1829 – 2059 84±4 85±4 90±4 83±4 76±4 63±4 2059 – 2317 53±3 60±3 58±3 57±3 55±3 47±3 2317 – 2608 34±3 36±3 36±3 40±3 37±3 34±3 2608 – 2935 28±2 26±2 26±2 30±2 31±2 30±2 2935 – 3303 14.8±1.9 15.1±1.9 14±2 17.1±1.9 20±2 21.9±1.9 3303 – 3717 6.1±1.6 7.1±1.6 9.9±1.6 10.2±1.6 11.9±1.7 13.3±1.6 3717 – 4183 — — 4.2±1.3 4.6±1.3 6.2±1.4 6.9±1.4 4183 – 4708 — — — 4.2±1.0 4.5±1.1 6.1±1.1 4708 – 5298 — — — — — 4.4±0.8 5298 – 5963 — — — — — 2.2±0.7 Table 2

Summaryoftransferratesfrommuonichydrogentooxygen.Thefirsterror repre-sentsthestatisticalandbackgroundrelatedsystematicerrorsquadraticallysummed. Theseconderrorreportsgeneralsystematicuncertainty.Thethirdcolumnreports thereducedχ2 _{ofthefit.Thefourthandfifthcolumnsshowtheresultsofthe} Kolmogorov-Smirnov(K-S)testanditsderivedmaximumdistance,respectively, per-formedonthepullsincomparisonwithastandardGaussiandistribution.

Mean pO(T) Reduced K-S test K-S max temperature T [K] [1010_s−1_{] χ}2 _{on pull distance} 104 3.07±0.29±0.07 1.21 1.00 0.14 153 5.20±0.33±0.10 0.81 0.85 0.25 201 6.48±0.32±0.13 1.95 1.00 0.17 240 8.03±0.35±0.16 1.64 0.81 0.27 272 8.18±0.37±0.17 1.69 0.76 0.30 300 8.79±0.39±0.18 1.80 0.76 0.30

arestill energeticandin an epithermalstate. The same breakin thespectrawasobservedatPSIandwas helpfulinunderstanding theexistence ofa kinetic energydependenceofthe transferrate frommuonichydrogentooxygen [5].The errorbarsassociatedto thepointsincludethestatisticalandbackground-systematicerrors, asdescribedpreviously.

4. Results

Theexperimental results are presentedin Table2. Duringthe measurements the temperaturewas kept stable, witha variation aroundthemeanvalue T lessthan0.1%(column 1),towhichthe measuredrate



pO(column 2)isreferred.Foreachfit,thereduced

χ

2 _{isreported(thirdcolumn).Thequalityofthefitwas assessed}

also by studying the pull distribution, where the pull is defined astheratiobetweentheresidual andtheerrorassociated tothe point((Xmeasured

−

Xfitted

)/σ

X).Ifthefitisgoodandthefluctuation purelystatistical,thenthepullsaredistributedasastandard nor-maldistribution(aGaussianwithzeromeanandunitwidth).For each fit,the unbinned andordered pulls data set was compared toastandardnormaldistributionusingtheKolomogorov-Smirnov test.Resultsofprobability andmaximumdistanceare reportedin columnsfourandfiveof Table2.The resultsofthe Kolmogorov-Smirnov test appliedto the sixmeasurements could not exclude normallydistributedpulls.

The six data sets andcorresponding fits are reported also in Fig.3,leftpanel, whereeach oneis artificiallyadjusted vertically forclearervisualization.Itcanbeseenthattheepithermal compo-nentbecomesmoreevidentatlowtemperatureswhenthethermal

transferrateissmaller.TherightpanelofFig.3showstheoverall distributionofthepullsforthesixtemperaturefits.Thesolidline representsa Gaussian fittothe distributionandit isinexcellent agreementwithastandardnormaldistribution.

Fig.4showstheresultsobtainedinthiswork. Thisisthefirst measurement ofthe transfer ratefrommuonic hydrogento oxy-gen asfunction ofthe temperature. The errorbars represent the quadratic sumof statisticaland systematicerrors, asreported in Table 2. The analysis of systematic errors shows that the main sourceofuncertaintyderivesfromthegascomposition.TheH2/O2

gas mixturewas preparedby the supplier witha relative uncer-taintyof3%.Othersourcesofsystematiceffects,including temper-atureandpressuremeasurements,timing,parameterserror propa-gationwereestimatedtobesmallerthan1% eachandconsidered negligible. Results are in excellent agreement with the PSI mea-surementat294 K [5].The linesrepresentthetheoretical results presented in [8,9] as function of kinetic energy, which we con-verted totemperatureusing Eq. (1).Experimental dataarenot in agreementwiththeoretical calculations,however.Thisis not sur-prising,sincetheserough calculationsdonottakeintoaccountin details the electron screening effects and also other aspects are simplified,e.g.,oxygenistreatedasafreeatomandnotaspartof oxygenmolecule.

5. Conclusions

Inthefirstinvestigationofthetemperaturedependenceofthe muon-transferprocessfromthethermalized

μ

patomstooxygen, wehaveobservedastrongmonotonicrisebyafactorof

∼

3 ofthe rate



pO

(

T

)

inthetemperatureinterval104–300 K.

The measurements were performed in conditions of thermal

equilibrium that allows usto use Eq. (1) and anticipate a much greater rise of themuon transferrate

λ

pO

(

E

)

fromenergies E

∼

0.01 eV up to energies of the order of E

∼

0.1 eV. The work

onextractingtheexplicitenergydependenceandtheuncertainty of

λ

pO

(

E

)

fromthe experimental data in Table 2 is currently in

progress;theresultswillbepresentedelsewhere.

Such a strong change enables us to employ the muon

trans-ferratetooxygenasasignature ofthekinetic-energygainofthe

μp

atominthe plannedFAMUspectroscopymeasurementofthe

hyperfinesplittingof1S stateofthisatom.

Fig. 3. Leftpanel:timedependenceoftheoxygenlineintensityforthesixtemperaturebins.Eachsetofdatahasbeenscaledbythefactorreportedinthefigure.Errorbars, solidanddashedlineshavethesamemeaningasinFig.2.Rightpanel:overallpulldistribution.SolidlineisaGaussianfitofthedata.Meanandwidthareconsistentwith astandardnormaldistribution.

Fig. 4. Transferratefrommuonichydrogentooxygen:comparisonofthepresent workwiththeexperimental [5],andthetheoreticalresults [8,9] convertedintheir temperaturedependenceusingMaxwell–Boltzmanndistributions.

scattering and will hopefully stimulate the development of new andmoreefficientcomputationalapproaches.

Declarationofcompetinginterest

Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

Acknowledgements

Theresearch activity presentedinthispaperhasbeen carried outinthe frameworkoftheFAMUexperimentfundedby Istituto Nazionale di Fisica Nucleare (INFN). The use of the low energy

muons beam has been allowed by the RIKEN RAL Muon

Facil-ity. We thanktheRAL staff (cooling, gas,andradioactivesources sections) andespecially Mr.Chris Goodway,PressureandFurnace SectionLeader,fortheirhelp,suggestions,professionalismand pre-ciouscollaborationintheset-upoftheexperimentatRIKEN-RAL.

We thank the Criotec company and especially Ing. Adriano

Mussinattoforthetechnicalhelpandsupportintheconstruction oftheFAMUtarget.

WethankourcolleaguesChiaraBoschiandIlariaBaneschi(IGG, CNR Pisa) for their help in the measurement of the gas isotopic composition.

D. Bakalov,P. DanevandM. Stoilovacknowledgethesupportof GrantDN08-17oftheBulgarianScienceFund.

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