Effect of dtmu quasi-nucleus structure on energy l

arXiv:physics/0403116v2[physics.atom-ph]31Mar2004Effectofdtμquasi-nucleusstructureonenergylevelsofthe(dtμ)XeeexoticmoleculeO.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.Theeffectoftheinternalstructureandmotionofthedtμquasi-nucleusonenergylevelsisstudiedusingthethree-bodydescriptionofthe(dtμ)Xeemoleculebasedonthehierarchyofscalesandcorrespondingenergiesofitsconstituentsubsystems.Foranumberofrovibrationalstatesof(dtμ)deeand(dtμ)tee,theshiftsandsplittingsofenergylevelsarecalculatedinthesecondorderoftheperturbationtheory.PACSnumbers:36.10.-k,36.10.Dr,33.20.Wr1I.INTRODUCTIONItisknownthatonestoppedmuoninadeuterium-tritiummixtureyieldsmorethan100nuclearfusionreactions.Theprocessofmuon-catalyzedfusionhasbeenintensivelystudiedandadetaileddescriptionexistsintheliterature,e.g.,inreviewarticles[1,2,3,4].Oneofthekeyprocessesinthemuon-catalyzedfusioncycleistheformationofthehydrogen-likeexoticmolecule(dtμ)Xee(forthesakeofgeneralityXstandsforeitherisotoped,t,orp),inwhichadtμmesicmoleculesubstitutesforoneofthenucleiinthehydrogenmolecule.ItiswidelyacceptedthattheresonancemechanismproposedbyVesman[5]isresponsibleforthehighrateofthedtμformation.Duetothismechanism,adtμmesicmoleculeinalooselyboundexcitedstatetobeproducedbylow-energycollisionsoftμmesicatomsandDXmoleculesinaresonanceprocesstμ+DX−→(dtμ)Xeefollowedbydtμtransitiontothetμgroundstate.Therateoftheresonanceprocessissensitivetothepreciseresonancepositionandanaccuracybettera1meVisnecessarytoobtainreasonabletheoreticalestimatesoftheformationrate[1,2,3,4].Resonanceformationcantakeplaceiftheenergyreleasedindtμbindingistransferredtotherovibrationalexcitationoftheexoticmolecule(dtμ)Xee.Thisisactuallythecaseasdtμhasalooselyboundexcitedstatewithanangularmomentumλ=1andbindingenergywhichiscomparabletovibrationalquantumofthe(dtμ)Xeemolecule.Inanon-relativisticapproximation,differentcalculationsdeterminewithagoodaccuracythebindingenergyoftheisolateddtμmesicmolecule[1,2,3,4].Toobtaintheprecisevalueofthebindingenergyonehastocorrectthenon-relativisticenergyforrelativisticeffects,hyperfineeffects,finitenuclearsize,vacuumpolarization,andothers.Theresonancepositionisdetermined,besidesthebindingenergyofisolateddtμ,bytheenergyoftherovibrationalexcitationofthehydrogen-likemolecule(dtμ)XeewithonenucleusbeingtheparticleXandtheothertheexciteddtμmesicmolecule.Asthe”size”oftheexciteddtμmesicmoleculewithλ=1isoftheorderof0.05a.u.[8],whichismuchsmallerthantheinternucleardistanceinthewholemolecule,therovibrationalspectrumof(dtμ)Xeecanbecalculatedtoagoodapproximationbytreatingdtμasapoint-likechargedparticle[6,7].Nevertheless,toreachanaccuracyoftheorderofatenthofameVoneshouldtakeintoaccounttheenergyshiftwhicharisesduetotheinternalstructureandmotionofadtμmesicmolecule.2Theeffectofthedtμfinitesizewaspreviouslyinvestigatedinasimpleapproach[8,9,10]wheretheenergyshiftsforthe(dtμ)deewereobtainedbymultiplyingby1.45theshiftcalculatedfortheatom-likesystem(dtμ)einthesecondorderperturbationtheory(PT).Withintheframeworkofthissimpleapproachitisnotpossibletotakeaccountofthemolecularstructure;inparticular,thecalculatedenergyshiftisindependentoftherovibrationalquantumnumbers.Theeffectofthemolecularstructure,i.e.,thedependenceonangularmomentum,wasexplicitlydemonstratedintheelaboratesix-bodycalculation[8]ofthe(dtμ)deeenergyshiftsinthefirstorderoftheperturbationtheory.Note,however,thatthefirst-andsecond-orderPTcontributionstotheenergyshiftarecomparable.Recently,resonancepositionsinthelow-energytμ+D2scatteringhavebeenobtainedintheelaboratethree-bodycalculation[11,12].Onlyfewresonancestateswiththezerototalangularmomentumhavebeenconsideredinthispaper.Themainaimofthepresentpaperistocalculatetheenergyshiftswhichariseduetotheinternalstructureandmotionofthedtμmesicmoleculeembeddedinthehydrogen-like(dtμ)Xeemolecule.Thecalculationisreducedtosolutionofathree-bodyproblemforheavyparticlestμ,dandX.Thisapproachisbasedonthehierarchyofscalesandcorrespondingenergiesofconstituentsubsystemsofthe(dtμ)Xeethusreliablytakingintoaccountthespecificfeaturesofthismolecule.Asaresult,theenergyshiftsareobtainedforanumberofvibrationalandrotationalstatesof(dtμ)deeand(dtμ)teeinthesecond-orderPT.II.METHODThestructureoftheexoticmolecule(dtμ)Xeeischaracterizedbyahierarchyofscalesandcorrespondingenergiesofitsconstituentsubsystems.Inthisrespect,atμmesicatomissmallincomparisonwithitsmeanseparationfromadeuteroninthelooselybounddtμmesicmolecule,whichallowstμtobetreatedasapoint-likeneutralparticleinteractingwithadeuteronbytheshort-rangeeffectivepotential.ThereisalsointeractionoftμwiththesecondnucleusX;however,thismightbeneglectedduet