Effect of dtmu quasi-nucleus structure on energy levels of the (dtmu)Xee exotic molecule

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

E ectofdtµquasi-nucleusstructureonenergylevelsofthe

(dtµ)Xeeexoticmolecule

arXiv:physics/0403116v2 [physics.atom-ph] 31 Mar 2004O.I.Kartavtsev,1A.V.Malykh,1,2andV.P.Permyakov31DzhelepovLaboratoryofNuclearProblems,JointInstituteforNuclearResearch,Dubna,141980,Russia2PhysicsDepartment,NovgorodStateUniversity,NovgorodtheGreat,173003,Russia3BogoliubovLaboratoryofTheoreticalPhysics,JointInstituteforNuclearResearch,Dubna,141980,Russia(Dated:February2,2008)AbstractPreciseenergiesofrovibrationalstatesoftheexotichydrogen-likemolecule(dtµ)Xeeareofimportancefordtµresonantformation,whichisakeyprocessinthemuon-catalyzedfusioncycle.Thee ectoftheinternalstructureandmotionofthedtµquasi-nucleusonenergylevelsisstudiedusingthethree-bodydescriptionofthe(dtµ)Xeemoleculebasedonthehierarchyofscalesandcorrespondingenergiesofitsconstituentsubsystems.Foranumberofrovibrationalstatesof(dtµ)deeand(dtµ)tee,theshiftsandsplittingsofenergylevelsarecalculatedinthesecondorderoftheperturbationtheory.PACSnumbers:36.10.-k,36.10.Dr,33.20.Wr

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

I.INTRODUCTION

Itisknownthatonestoppedmuoninadeuterium-tritiummixtureyieldsmorethan100nuclearfusionreactions.Theprocessofmuon-catalyzedfusionhasbeenintensivelystudiedandadetaileddescriptionexistsintheliterature,e.g.,inreviewarticles[1,2,3,4].Oneofthekeyprocessesinthemuon-catalyzedfusioncycleistheformationofthehydrogen-likeexoticmolecule(dtµ)Xee(forthesakeofgeneralityXstandsforeitherisotoped,t,orp),inwhichadtµmesicmoleculesubstitutesforoneofthenucleiinthehydrogenmolecule.ItiswidelyacceptedthattheresonancemechanismproposedbyVesman[5]isresponsibleforthehighrateofthedtµformation.Duetothismechanism,adtµmesicmoleculeinalooselyboundexcitedstatetobeproducedbylow-energycollisionsoftµmesicatomsandDXmoleculesinaresonanceprocess

tµ+DX →(dtµ)Xee

followedbydtµtransitiontothetµgroundstate.Therateoftheresonanceprocessissensitivetothepreciseresonancepositionandanaccuracybettera1meVisnecessarytoobtainreasonabletheoreticalestimatesoftheformationrate[1,2,3,4].

Resonanceformationcantakeplaceiftheenergyreleasedindtµbindingistransferredtotherovibrationalexcitationoftheexoticmolecule(dtµ)Xee.Thisisactuallythecaseasdtµhasalooselyboundexcitedstatewithanangularmomentumλ=1andbindingenergywhichiscomparabletovibrationalquantumofthe(dtµ)Xeemolecule.Inanon-relativisticapproximation,di erentcalculationsdeterminewithagoodaccuracythebindingenergyoftheisolateddtµmesicmolecule[1,2,3,4].Toobtaintheprecisevalueofthebindingenergyonehastocorrectthenon-relativisticenergyforrelativistice ects,hyper nee ects, nitenuclearsize,vacuumpolarization,andothers.Theresonancepositionisdetermined,besidesthebindingenergyofisolateddtµ,bytheenergyoftherovibrationalexcitationofthehydrogen-likemolecule(dtµ)XeewithonenucleusbeingtheparticleXandtheothertheexciteddtµmesicmolecule.Asthe”size”oftheexciteddtµmesicmoleculewithλ=1isoftheorderof0.05a.u.[8],whichismuchsmallerthantheinternucleardistanceinthewholemolecule,therovibrationalspectrumof(dtµ)Xeecanbecalculatedtoagoodapproximationbytreatingdtµasapoint-likechargedparticle[6,7].Nevertheless,toreachanaccuracyoftheorderofatenthofameVoneshouldtakeintoaccounttheenergyshiftwhicharisesduetotheinternalstructureandmotionofadtµmesicmolecule.

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

Thee ectofthedtµ nitesizewaspreviouslyinvestigatedinasimpleapproach[8,9,10]wheretheenergyshiftsforthe(dtµ)deewereobtainedbymultiplyingby1.45theshiftcalculatedfortheatom-likesystem(dtµ)einthesecondorderperturbationtheory(PT).Withintheframeworkofthissimpleapproachitisnotpossibletotakeaccountofthemolecularstructure;inparticular,thecalculatedenergyshiftisindependentoftherovibrationalquantumnumbers.Thee ectofthemolecularstructure,i.e.,thedependenceonangularmomentum,wasexplicitlydemonstratedintheelaboratesix-bodycalculation[8]ofthe(dtµ)deeenergyshiftsinthe rstorderoftheperturbationtheory.Note,however,thatthe rst-andsecond-orderPTcontributionstotheenergyshiftarecomparable.Recently,resonancepositionsinthelow-energytµ+D2scatteringhavebeenobtainedintheelaboratethree-bodycalculation[11,12].Onlyfewresonancestateswiththezerototalangularmomentumhavebeenconsideredinthispaper.

Themainaimofthepresentpaperistocalculatetheenergyshiftswhichariseduetotheinternalstructureandmotionofthedtµmesicmoleculeembeddedinthehydrogen-like(dtµ)Xeemolecule.Thecalculationisreducedtosolutionofathree-bodyproblemforheavyparticlestµ,dandX.Thisapproachisbasedonthehierarchyofscalesandcorrespondingenergiesofconstituentsubsystemsofthe(dtµ)Xeethusreliablytakingintoaccountthespeci cfeaturesofthismolecule.Asaresult,theenergyshiftsareobtainedforanumberofvibrationalandrotationalstatesof(dtµ)deeand(dtµ)teeinthesecond-orderPT.II.METHOD

Thestructureoftheexoticmolecule(dtµ)Xeeischaracterizedbyahierarchyofscalesandcorrespondingenergiesofitsconstituentsubsystems.Inthisrespect,atµmesicatomissmallincomparisonwithitsmeanseparationfromadeuteroninthelooselybounddtµmesicmolecule,whichallowstµtobetreatedasapoint-likeneutralparticleinteractingwithadeuteronbytheshort-rangee ectivepotential.ThereisalsointeractionoftµwiththesecondnucleusX;however,thismightbeneglectedduetolargeseparationbetweentheseparticles.Inturn,thesizeofadtµmesicmoleculeissmallincomparisonwiththeamplitudeofvibrationsin(dtµ)Xee;therefore,itmovesasapoint-likequasi-nucleusneartheequilibriumposition.Forthisreason,thee ectofthedtµstructureisconsideredwithintheframeworkoftheperturbationtheory.

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

Furthermore,twoelectronsinthehydrogen-likemolecule(dtµ)Xeemovemuchfasterthantheheavyparticlesd,X,andtµ,whichmakesitpossibletousethefamiliarBorn-Oppenheimer(BO)approximation,i.e.,tosolveelectronicproblemwiththe xedchargedparticlesdandXthusobtainingtheBOenergywhichplaysaroleofthee ectivepoten-tialbetweendandX.Theelectronicexcitations,whichrequireaconsiderableamountofenergy[13],arenottakenintoaccountforthelow-energyprocessesunderconsideration.Asaresult,thedescriptionof(dtµ)Xeeisreducedtosolutionofathree-bodyproblemforthreeparticlestµ,dandX.TheinteractionbetweenchargeddandXisdescribedbythewell-knownBOpotentialforthehydrogenmolecule.Inaccordwiththetreatmentofthetµmesicatomasapoint-likeneutralparticle,thepresentcalculationdoesnotexplicitlyusethetµ+de ectivepotential,rathertheresultisexpressedviathelow-energytµ+dscatteringphaseshiftsandcharacteristicsofthetµmesicmoleculeinthelooselyboundexcitedstate.

The(dtµ)Xeestatesareeithertrueboundstatesornarrowresonancesiftheirenergyisbeloworabovethetµ+DXthreshold.Astheenergyshiftsaremainlydeterminedbythecouplingwithclosedchannels,inthepresentcalculationbothresonancesandboundstatesaretreatedonanequalfootingthusneglectingasmallcontributiontotheenergyshiftswhichcomesfromthecouplingwiththeopentµ+DXchannel.

A.Three-bodydescription

Undertheaboveapproximations,theSchr¨odingerequationforthehydrogen-likemolecule(dtµ)Xeereads

1

2µ2 ρ+V1(r)+V2(|ρ βr|)+V(|ρ+αr|) EΨ=0(1)

wheretheJacobicoordinatesrandρarethevectorsfromdtothepoint-likemesicatomtµandfromthesecondnucleusXtothedtµcenterofmass,respectively.Thereducedm1m2m1massesandparametersαandβareµ1=,α=m1+m2+m3

,wherem1,m2andm3arethemassesoftµ,d,andX,respectively.Them1+m2atomicunitsareusedthroughoutthepaperunlessotherisspeci ed.InEq.(1),V(|ρ+αr|)

denotesthewell-knownBOpotentialdescribingtheinteractionbetweenchargeddandXwhiletheshort-rangepotentialsV1(r)andV2(|ρ βr|)describetheinteractionofatµ

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

mesicatomwithadeuteronandX,respectively.Inthefollowing,duetolargeinternuclearseparation(ρ r)in(dtµ)Xee,theshort-rangeinteractionV2(|ρ βr|)ofthetµmesicatomwiththesecondnucleusXisnegligibleandwillbeomitted.

Anaturalzeroth-orderapproximationforthecalculationofthe(dtµ)Xeeenergylevelsistotreatthedtµmesicmoleculeasapointquasi-nucleuswiththedtµmassandtheunitcharge.Thecalculationsoftheenergylevelsinthisapproximationarepresentedin[6,7]fordi erentisotopesXofthehydrogen-likemolecule(dtµ)Xee.Clearly,thetreatmentofdtµasapoint-likeparticleisequivalenttothereplacementoftheexactpotentialV(|ρ+αr|)intheSchr¨odingerequation(1)bythepotentialV(ρ)whichdescribestheBOinteractionbetweenXandthepointparticlelocatedatthedtµcenterofmass.Thus,thee ectofthedtµstructure,whichleadstotheshiftofthezeroth-orderenergylevels,originatesfromtheperturbationpotential

Vp=V(|ρ+αr|) V(ρ).(2)

Inthezeroth-orderapproximationVp=0,thesolutionsofEq.(1)withthetotalangularmomentumLanditsprojectionMarewrittenasaproductofthebisphericalharmonicsofnucleiin(dtµ)Xeewiththeangularmomentuml,andtheradialfunctionofrdescribingLM ,r )describingtheangulardependence,theradialfunctionofρdescribingthemotion(ρYlλtheinternalmotioninamesicmoleculewiththeangularmomentumλ.TheunperturbedenergiesEnlandthecorrespondingsquareintegrableradialfunctionsΦnl(ρ)ofthe(dtµ)Xeevibrationalandrotationalstatessatisfytheequation

1

ρ2 ρ +l(l+1)

2µ1 1 r r2

r2 +V1(r) Eφ(r)=0. (4)

HereE= εvλandφ(r)=φvλ(r)fortheboundstatesandE=k2/2µ1andφ(r)=φkλ(r)forthecontinuumstateswiththewavenumberk.Thefunctionsφvλ(r)aresquareintegrableandthefunctionsφkλ(r)arenormalizedbythecondition

∞ 0r2drφ kλ(r)φqλ(r)=δ(k q).

(5)

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

IncorrespondencewiththeVesmanmechanism,(dtµ)Xeecontainsadtµmesicmoleculeintheweaklyboundstatewiththebindingenergyε11(v=1,λ=1).Otherdtµstates,whosebindingenergiessigni cantlyexceedallthecharacteristicenergiesoftheproblemunderconsideration,willnotbetakenintoaccountinthecalculationoftheenergyshifts.

B.Perturbationtheory

Thee ectofthedtµstructureissmallduetosmallnessofdtµmesicmoleculeincom-parisonwithacharacteristiclengthofdtµmotioninthemolecularpotentialV(ρ).Inotherwords,theperturbationVpissmallincomparisonwithV(ρ)andcanbeexpandedinpowersofthesmallparameterαr.Correspondingly,thedimensionlessparameteroftheperturba-tiontheoryistheratiooftheaveragedistancebetweenthedeuteronandthedtµcenterofmassα r totheaverageamplitudeofvibrations ρ a inthemolecularpotentialneartheequilibriuminternucleardistancea.

Oneshouldnotethatthelowest-ordertermoftheexpansionVp,whichisproportionaltoαr,doesnotcontributetotheenergyshiftsinthe rst-orderPT;therefore,theenergyshiftoforder(αr)2mustbeobtaineduptothesecond-orderPT.Besides,Vpcouplestherotationalstateswithl=L±1whilethestatewithl=Lremainsuncoupled.Astheseparationoftherotationallevelsiscomparativelysmall,thelevelcouplingcannotbeapriorineglectedandrequiresexplicittreatment.Thus,theenergyshiftswillbedeterminedinthesecond-orderdegeneratePTbysolvingasecularequation

det[Vn+Wn+En E]=0(6)

whereVnandWnarethematriceswiththematrixelementsofthe rst-andsecond-order

nnPTVllandW,respectively,thematrixelementsofEnare(Enl+ε11)δll1,andEisthell11

levelenergy.

The rst-orderPTmatrixelementsare

nVll1= ,r )YlLM ,r )d3rd3ρVp|φ11(r)|2Φnl(ρ)Φnl1(ρ)YlLM(ρ(ρ111 (7)

andthesecond-orderPTmatrixelementsincludeasumandanintegraloverintermediatestatesdescribingsimultaneousexcitationsoftheexoticmoleculewiththequantumnumbersνand andadtµmesicmoleculewiththecontinuum-statewavenumberkandtheangular

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

momentumλ

nλλdkZnl,ν (k)Zν ,nl(k)1Wll1= ν λ

ρP1(cosθ)+12

ρ2+ ρ+2 2V V

ρ

α2Q[Unl,nMQL

61l1δll1+Unl,n1l1A2(l1l11)],

Wlln

1= α2UnνDUνnD(l11 λ)Iλ(Eν

ν E+ε11)

where AL1(l1 λ)AL1λ

Iλ( )= ∞[uλ(k)]2dk

ρ2+2

ρ,

UQ

nl,n1l1=2 ρ2dρΦV Vnl(ρ)Φn1l1(ρ) 2ρ

ρ.

andtheangularintegralsALK(lλl1λ1)aregivenintheAppendix.

(11)(12)(16)(18)

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

III.

A.RESULTSOFCALCULATIONMatrixelements

Thesimple,thoughprovidingtherequiredaccuracyexpressionsforthemultipolema-trixelements(16),(17),and(18)areobtainedusing

thefollowingreliableapproximations.Firstly,thematrixelementsarecompletelydeterminedbytheBOpotentialforthehydrogenmoleculeV(ρ)whichisfairlywellknownfromthecalculations[6,7,14,15].As(dtµ)Xeeisproducedinlow-energytµ+DXcollisions,onlythelowestvibrationalstatesshouldbetakenintoaccount.ForthesestatesarelocalizedneartheminimumofV(ρ)attheequilibriuminternucleardistancea≈1.4a.u.,itisnaturaltousetheharmonicapproximation

Vh(ρ)=1

2µ2ω2(ρ a)2[1 αM(ρ a)]+V0.(20)

whichtakesintoaccountthenexttermoftheexpansioninρ a.Theapproximation(20)accuratelyreproducestheexactenergiesofthelowestvibrationalstatescalculatedin[6,7].

1)/2µ2a2≈10 4istwoordersofmagnitudesmallerthanthevibrationalenergyω≈10 2.Secondly,therotationalenergyin(3)forthehydrogen-likemoleculel(l+1)/2µ2ρ2≈l(l+Therefore,underausualapproximation,thecentrifugaltermistreatedperturbatively,i.e.,theeigenenergiesaregivenby

Enl=En0+vrl(l+1)(21)

andthewavefunctionsΦnl(ρ)willbetakenindependentoflinthesameapproximation.Indeed,therotationalspectrumcalculatedin[6,7]isingoodagreementwiththeaboveexpression(21)withvr≈1/2µ2a2 10 4.Thus,undertheaboveapproximations,theradialwavefunctionΦnl(ρ)inthepotential(19)coincideswiththeharmonic-oscillatorwavefunctionandthemultipolematrixelements(16),(17),(18)arereducedtol-independentexpressions

DUnν= 2 √n+1δn+1,ν,

MUnn=µ2ω2,QUnn=2µ2ω2.(22)

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

TheunharmonictermofthepotentialVu(ρ)leadsonlytomodi cationofthedipolematrixelement

DUnν=

η

3 2 √n+1δn+1,ν (n+1)(n+2)δn,ν 2+(2n+1)δn,ν (23)wheretheunharmoniccorrectionisproportionaltothedimensionlessparameterη=2µ2ω≈0.14.

Calculationofthequasi-nucleusmatrixelements(14,15)isbasedonthesmallnessofthetµsizeincomparisonwiththesizeofthelooselybounddtµstate(v=1,λ=1).Thus,almostinallthecon gurationspacetµanddmoveasfreeparticlesandthebound-statewavefunctionisapproximatedby

11(r)=Ca

whereκ=√1+κr κreκr2(24)

2

r

φk2(r)= ,(25)πk[cosδ2(k)j2(kr)+sinδ2(k)y2(kr)](26)

whereδλ(k)arethetµ+dscatteringphaseshiftsandj2(kr)andy2(kr)arethesphericalBesselfunctions.Thed-wavephaseshiftδ2(k)isactuallyverysmall,whichallowseithercoskrreplacingy2(kr)bytheleadingterm

kr

Q=5

µ1ε11(27)oneobtainsthequadrupolemomentum

andtheexpression

Iλ( )=24Ca

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

viathedimensionlessintegrals

J0(z)=∞ 0[sinδ0(k) (k2+3)(k/2)cosδ0(k)]dk2

(k2+1)4(k2+z)

B.Shiftandsplittingofenergylevels.(30)

Energyshiftsareobtainedbysolvingthesecularequation(6)whichisreduced,dueto

theselectionrulesforangularmomentalandl1,toa2×2matrixequationforl,l1=L±1(L=0)andascalarequationforl=l1=L=0.TheenergyshiftswithrespecttotheunperturbedrovibrationalenergiesEnl+ε11aredenotedas 0(n)and ±(nl)forl=Landl=L±1,respectively.NotethatthestatewithL=0andl=l1=1isuncoupled;however,itsenergyshift +(n1)willbedeterminedinthesamemannerasfortheotherL=0states.

nThe rst-orderPTmatrixelementsVllinEq.(6)arecalculatedbysubstitutingthe1

M,QradialintegralsUnn(22),thequadrupolemomentumQ(27),andtheangularintegrals

nAL2(l1l11)(A4)inEq.(11).NotethatVll1appearstobeindependentofthevibrational

quantumnumbernandthisindexwillbeomittedinwhatfollows.Thematrixelementsarescaledbyasingledimensionalparameter

v0=2m1m3ω2Ca

2L+1 1

L(L+1)2 1 inwhichthe rstrowandcolumncorrespondtol,l1=L 1andthesecondonesto

l,l1=L+1.

(33)

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

Thesecond-orderPTmatrixelementsWlln

1inEq.(6)arecalculatedbysubstituting

UnνD(22)andIλ( )(28)inEq.(12),whichgivestheexpression

W32ω

lln

1= v0ε11 (34)

viatheenergyscalev0andthedimensionlessfactors.Solvingthesecularequation(6),onecansafelyreplace,uptoanaccuracyofthesecond-orderPT,theeigenvalueEintheargumentofJλbytheunperturbedvalueEnl.Thus,thecalculationoftheenergyshiftsisbasicallyaccomplishedbyderivationofEqs.(32-34).

However,itisreasonabletomakefurthersimpli cationof(34)byneglectingthedi erenceoftherotationalenergiesintheargumentofJλ,whichallowsobtaininganexplicitandsu cientlyaccuratedependenceoftheenergyshiftsonthequantumnumbersnandl.Astherotationalenergyismuchsmallerthanthevibrationalquantumω,onereplacestheenergydi erencesEn±1l EintheargumentofJλbythel-independentvaluesEn±ingtheangularintegralsAL n0=±ω1(l1l1λ)(A5,A6)andintroducingthenotationJ±

λ=Jλ(1±ω/ε11)forintegralsindependentofnandloneobtains

Wlln

1=v0 1+αn(2 β βn,l,l1=L(L=0)n)δll1+(αl,l(35)n 1)Bll1,1=L 1

where

αn=1 16ωJ2+)+n(J

0+1

3πε 5(n+1)(J0++7J

1152) (37)

determinetheexplicitdependenceonthevibrationalquantumnumbern.Asaresult,thesumofVll(32)andWlln

11(35)takesasimpleform

V,l=l1=L(L=0)

ll1+Wlln

1=v0 βn β αnl,l,(38)

nδll1 αnBll1,1=L 1

i.e.,theparameterβndeterminestheconstantshiftv0βnofalllevelenergiesEnlwhereasαing(38)and(21)inthesecularequation(6)oneobtains

0(n)=v0(βn αn),(39) ±(nl)=v0βn vr[2(l 1)+1]±

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

Thee ectofcouplingoftherotationalstateswithl=L±1isexplicitlytakenintoaccountinexpression(40).

Generally,

the

e ect

decreases

with

decreasing

ratioofthelevelsplittingv0αntotheenergydi erencebetweentherotationalstates

2(l 1)+1 =v0βn± αn

ε11

Thevibrationalquantumωandtherotational-energyconstantvraredeterminedbytheBOinternuclearpotentialofthehydrogenmoleculenearitsminimumor,equivalently,bythelow-lyingpartofthe(dtµ)Xeevibrational-rotationalspectracalculatedin[6,7].FittingtheBOpotentialneartheequilibriumdistancea=1.401totheharmonic,unharmonic,andMorsepotentialsprovidesconsistentdeterminationofbothωandtheparameterαM.Asthe

.

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

BOpotentialisindependentoftheisotopiccomposition,bothµ2ω2andαMareindependentofthemassesofheavyparticlesduetoEq.(20).Theresultofthe tgivesω=321.8meVfor(dtµ)dee(correspondingly,ω=273.1meVfor(dtµ)tee)withafewpercentaccuracyandtheparameterαM=0.7.Fortheseparameters,theenergiesofthelowestvibrationalstatesintheapproximatepotentialareinreasonableagreementwiththeresultsof[6,7].

Therotationalspectracalculatedin[6,7]are ttedtoEq.(21)for1≤l≤10andeach1≤n≤4.Forthelowestvibrationalstaten=1,oneobtainsvr=2.43meVfor(dtµ)deeandvr=1.85meVfor(dtµ)tee.Thesevaluesagreewiththesimpleestimate2µ2vr

determinestheisotopicdependenceofvv≈1/a2thatr.Althoughrslightlydecreasesforthehighervibrationalstates,theabovevalueswillbeusedforn>1,whichleadstoafewpercenterror.

Determinationofthebindingenergyε11ofadtµlooselyboundstatewasasubjectofnumerouselaboratedcalculations.Asaresult,thevalueε11=596meV[1,2,4]isobtainedforthelowesthyper nestatebytakingintoaccountrelativistice ects,hyper nee ects, nitenuclearsize,andvacuumpolarization.TheasymptoticconstantCawasdeterminedinanumberofpapers[9,16,17]byacomparisonoftheasymptoticexpression(24)three-bodywavefunction.Inthefollowing,itisacceptedthevalueCa=0.874/√withthe

(m)m,theparameterv0(31)isindependentofm

isthesameforanyisotopeX1+m.Given2the33,i.e.,it

abovenumericalvaluesoneobtainsv0=1.81meV.

Forthesakeofcompleteness,itisinterestingtoestimatetheenergyscaleforthemolecule(ddµ)Xeetoobyusingthevaluesε11=1975meV,Ca=1.006/√

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

tions[9,10,18]wherethel-independentenergyshiftwasobtainedbyscalingtheresultfortheatom-likefour-bodysystem(dtµ)e.Aspointedoutinthispaper,themonopolecon-tributioncalculatedinRef.[8]dependsonthechoiceofthecoordinatesystemthatdoesnotallowacomparison.Forthisreason,onlythequadrupolecontributiontothe rst-orderPTenergyshiftsofRef.[8]willbecomparedwiththepresentresults.ThequadrupolecontributionofRef.[8]isgiveninTableIIofthatpaper,whileinthepresentapproach E(1)

Q=(10/3)v0AL2(l1l1),asfollowsfromEqs.(11),(22),and(27).Thebothresultsareinexcellentagreementwitheachother,asshowninTableI.Notethatinthepresentapproach

lLAL2(l1l1) E(1)(1)

Q EQ[8] v0

TABLEI:Quadrupolecontributions E(1)

Q(inmeV)tothe rst-orderPTenergyshiftsofthe

presentcalculationandthosefromRef.[8]fordi erentlandL.AlsopresentedaretheangularintegralsAL2(l1l1)andtheparameter v0correspondingtotheenergyshiftsofRef.[8].thedependenceonangularmomentaiscompletelydeterminedbythefactorAL2(l1l1)whichisalsopresentedinTableI.Toagoodaccuracy,theresultsofRef.[8]revealthesamedependenceonangularmomentawhichapprovesthedescriptionofenergyshiftsbyasingleparameterv0.Toemphasizethisfact,thequadrupolecorrectioncalculatedinRef.[8]isexpressedintheform E(1)

Q=(10/3) v0AL2(l1l1)withthevariable v0presentedinTableI.

Indeed,v 0ispracticallyindependentoflandLandagreeswithv0=1.81meV.Agreementbetweenthepresentone-parameterresultforthequadrupolecorrectionandtheelaboratesix-bodycalculation[8]isagoodargumentforthevalidityofthepresentapproach.Atlast,oneshouldobtainWlln

1(34),whichrequiresevaluationofJλ 1+Eν

E

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

oneobtainsJλ1+thesecalculationsandintegrating(29)and

(30)

in

the

energyinterval0≤k2/2µ1≤10eV, Eν E

lLn=2n=3n=2n=3

TABLEII:Energyshifts(meV)forafewstatesof(dtµ)deeand(dtµ)teewiththevibrationalquantumnumbern,thetotalangularmomentumL,andangularmomentumlofthehydrogen-likemoleculewiththepoint-likedtµquasi-nucleus.

thatapplicabilityoftheharmonicapproximationfortheBOpotentialwascheckedbyusingthemodi eddipolematrixelement(23)inthecalculation,whichgivesanestimateoftheunharmoniccorrectionoftheorderof5%intheenergyshifts.Calculationsrevealthattheenergyshiftsareessentiallydependentontheisotopiccompositionandthemolecularquantumnumbersnandl,whichisbasicallyconnectedwiththecancellationofthe rst-andsecond-orderPTcontributions.Inparticular,theenergyshiftsdecreasewithincreasingnsothat ±becomeverysmallorevennegativeforn=4.Thereasonforthisdependenceisanincreasinginthedipolematrixelement(22)withincreasingn,which,inturn,leads

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

toanincreasinginthesecond-orderPTcontribution.Thecancellatione ectwaswidelydiscussed,e.g.,in[9,10,18];nevertheless,thedependenceonthemolecularquantumstatewasbeyondthescopeofthosepaperswhereonlytheatom-likesystem(dtµ)ewascalculated.Ontheotherhand,thecalculation[8]determinedthel-dependenceonlyinthe rst-orderPT.

ThedependenceoftheenergyshiftsonquantumnumbersisillustratedinFigure1forthe(dtµ)deestateswithn=2,3andl=0 4.Inadditiontoadecreasingintheenergy

2

1.6

1.2

0.8

0.401234

l

FIG.1:Energyshifts(meV)of(dtµ)dee.Crosses,squares,andcirclesdenote,respectively, +, ,and 0.Theresultsobtainedbysimpli edformulas(39),(41)aredenotedbythesolid,dashed,anddottedlines,respectively.Threeuppercurvescorrespondtothevibrationalquantumnumbern=2andtheloweroneston=3.

shiftsforhighern,noticetheinverseorderingoflevels,i.e.,thehighestlevelwithL=l 1forn=2becomesthelowestforn=3.Exceptfor +,whosevaluesatsmalll=1,2arequitedi erent,theresultsrevealweakdependenceonlwithsplittingoflevelsoftheorderof0.2meV.

AsdiscussedattheendofSectionIIIB,thedependenceofenergyshiftsandlevelsplittingonquantumnumbersareexpressedtoagoodaccuracybysimpleformulas(39),(41)viafewparameters.Numericalvaluesof16ω

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

X16ω

d0.9170.1130.2360.0510.073

t0.7780.1170.2160.0520.069

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

PTquadrupolecontributiontotheenergyshiftsisinagreementwiththeelaboratesix-bodycalculation[8],whichisagoodargumentforthevalidityofthepresentapproach.Furthermore,itisshownthatforalltheconsideredstatesthee ectofcouplingoftherotationalstateswithl=L 1andl=L+1isbeyondtheaccuracyofthepresentcalculation.Inaddition,theenergyshiftandsplittingisconvenientlyexpressedbysimpleanalyticaldependence(39),(41)ontherotational-vibrationalquantumnumbersnandl.Itisofinteresttocomparethepresentresultwiththedirectnon-perturbativethree-bodycalculationoftµ+D2scattering[11,12]inwhichthee ectofthedtµstructureisexplicitlytakenintoaccount.Thepositionsofnarrowresonancescalculatedinthispapercorrespondtoenergylevelsof(dtµ)deeforL=0(l=1)andn=3,4.Fortwoapproximationsofthee ectivepotentialbetweentµanddusedin[11,12],energyshiftsare,respectively,1meVand4meVforn=3and2meVand5meVforn=4.Thecauseofthenoticeabledi erence(about3meV)isnotclearsincebothpotentialsallowagooddescriptionofthelow-energypropertiesofthetµ+dsystem.Thedependenceonthechoiceofthee ectivepotentialandalimitationonlybyL=0hindersaquantitativecomparisonofthepresentresultsandthoseof[11,12].Qualitatively,theenergyshiftsobtainedinRef.[11,12]exceedthepresentonesand,contrarytoPTconsiderations,thevalueforn=4ishigherthanforn=3.Thisn-dependenceclearlydeservesfurtherinvestigation.

Finally,itshouldbementionedthatthepresentapproach,whichreliablytakesintoac-countthestructureoftheexoticmolecule,ispromisingforwiderapplications,inparticular,fordeterminationofresonancepositionsandformationratesbeyondPTbysolvingthescat-teringproblem.Tillnow,exceptRefs.[11,12],theformationrateshavebeencalculatedonlyinthe rst-orderPT.Inthisrespect,theresultofRef.[26]showsthatthe rst-orderPTdipoleapproximationisquestionableandoneshoulddomorere nedcalculations.Inaddition,itisofinteresttoapplythepresentapproachtotheproblemoftheresonanceformationofmetastabledtµmesicmolecules[27,28]incollisionsofexcitedtµmesicatomswithD2molecules.

APPENDIXA:ANGULARINTEGRALS

Thefollowingangularintegralsarenecessarytocalculatethematrixelements

ALK(lλl1λ1)= LM , , d (ρr)YlLM(ρr)dρrPK(cosθ)Ylλ1λ1 (A1)

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

=ρ/ρ,PK(x)istheLegendrewhereθistheanglebetweentwounitvectors r=r/randρ

polynomial,andthebisphericalharmonicsarede nedas

LM , Ylλ(ρr)= mµ )Yλµ( (lmλµ|LM)Ylm(ρr).(A2)

Evaluatingtheintegral(A1)onecomestotheexpressionintermsoftheClebsh-Gordoncoe cientsand6j-symbols

ALK(lλl1λ1)=( )l1+L

5δlLδl1L+

3(L+2)5(2L+1)δlL 1δl1L 1 L(L+1)

2

1 Lδl,L 1 √15(2l+1)(2l+3),l1=l+1(L l+2)(L l+3)(L+l 2)(L+l 1)

Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $

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