Perturbation Theory for Proton-Neutron Scattering Perturbing the Energy

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1、IntJTheorPhysDOI10.1007/s10773-011-0839-2PerturbationTheoryforProton-NeutronScattering:PerturbingtheEnergyM.Al-Sayyed·M.I.JaghoubReceived:20January2011/Accepted:11May2011©SpringerScience+BusinessMedia,LLC2011AbstractUsingthetime-independentSchrödingerequationwithavelocity-de

2、pendentpo-tentialtheenergydependenceofthecorrespondingscatteringphaseshifts,whentheenergyischangedbyasmallamountEfromanarbitraryunperturbedvalueE0,isstudied.WeexpandkcotδasapowerseriesinEandobtainanalyticformulasfortheeffectiverangeexpansionparameterstoallordersinthepertur

3、bingenergy.Formulasforthecorrespond-ingwavefunctionchangesarealsodeveloped.AtlowenergiestheBetheformulafortheeffectiverangeisreproducedandanexpressionfortheshape-dependenttermisobtained.Thederivedformalismisrelevanttofieldslikenuclearandatomicphysics.Examplesthatdemonstrateth

4、eeffectivenessofthederivedformalismarepresented.KeywordsScatteringtheory·Phaseshifts·Effectiverange1IntroductionItiswellknownthatinthelowenergylimitscatteringprocessesoccurinthel=0partialwaveandthescatteringcrosssectionisisotropic.Theenergydependenceofthescatteringphaseshift

5、saswemoveawayfromthezeroenergylimithasbeentheinterestofmanyoldandrecentworks.Inthefieldofnuclearphysicsmuchofwhatweknowaboutthepropertiesofthenuclearforcecomesfromanalyzingthenucleon-nucleonscatteringdataatlowener-gies.Forexample;BlattandJacksonappliedvariationalmethodsdevelo

6、pedbySchwingertoneutron-protonscatteringatenergiesbelow10MeV.Byconsideringthes-wavescatter-ingaloneandignoringthetensorforcetheauthorsobtainedanexpansionforthephaseshiftinpowersofenergy,whichiscommonlyknownastheeffectiverangeexpansion[1].Sim-ilarly,Betheshowedthatthes-wavesc

7、atteringofneutronsupto10MeVbyprotonscanbedescribedbytwoparameters,thescatteringlengthatzeroenergyandtheeffectiverangemultipliedbythesquareofthewavenumber[2].Inamorerecentwork,MilwardandWilkinstartedfromtheprobabilitydensityequationcorrespondingtoalocalpotentialandstudiedM.Al

8、-Sayyed·M.I.Jaghoub()DepartmentofPhysics,UniversityofJordan,11942Amman,Jor