Nuclear modelsModels we will consider…• Independent particle shell model• Look at data that motivates the model• Construct a model• Make and test predictions from the modelOur approach…• Collective models• Fermi gas modelShell Model - data2p separation energy (between isotones)Becomes much smaller after8, 20, 28, 50, 82, 1262n separation energy (between isotopes)Shell Model - dataARn Æ a + A-4PoTaSudden rise at N = 126Neutron capture cross section ssVery small s at N = 28, 50, 82, 126Abrupt change in nuclear radiusat N = 20, 28, 50, 82, 126DRDRavgShell Model - dataAnd, the observation of discrete photon energies Egemitted from nuclear de-excitationTb show sharp discontinuities near N,Z of 28, 50, 82, 126BE for last n added: sharp discontinuities near, 50, 82, 126 e.g., (d,p), (n,g), (g ,n), (d,t) reactionsShell ModelAssume that the nucleons move (independently) in apotential, V, created by the other nucleons in the nucleus.Assume that the problem can be addressed by the non-relativistic Schrodinger quantum mechanics.Assume that the potential, V, is spherically symmetric andtherefore only a function of r, V(r)† V r( )= -Vo r £ RV r( )= 0 r > R† V r( )µ r2 † V r( )=-Vo1+ exp r - R( )/ a[ ] † Vsor( )r L ⋅r s Spin-orbit potentialShell Model † Vsor( )r L ⋅r s † ˆ H ,ˆ J 2[ ]= 0 ˆ H ,ˆ J z[ ]= 0 † J2=r J ⋅r J r J =r L +r s J2=r L ⋅r L + 2r L ⋅r s +r s ⋅r s J2= L2+ s2+ 2r L ⋅r s r L ⋅r s =J2- L2- s22 † J2= j j + 1( )h2L2= l l + 1( )h2s2= s s + 1( )h2 † r L ⋅r s =12j j + 1( )- l l + 1( )- s s + 1( )[ ]h2 Q.M. † yn,l,s, j,mj good quantum numbersShell Model † 2 ⋅ 2l + 1( ) † r L ⋅r s j=l+1 / 2-r L ⋅r s j=l-1 / 2=2l + 1( )2h2 † yn,l,s, j,mj † 2 j + 1( )Multiplicities --2 spin states † ml different states† mj different states=Energy difference (splitting) increases with † lShell Modelenergy levelsEnergy splittingincreases with † lSpectroscopic statemultiplicitySystematics…A Z NNumber of known stable nucleonsStableRadio- activemI Odd Odd Even 50 50 11 Usually large & pos.II Odd Even Odd 55 36 4 Usually small, neg.III Even Odd Odd 4 4 9 Usually positiveIV Even Even Even 165 12 1 IndeterminateNucleon ClassificationNuclear momentsNuclear magnetic moments† mN= 3.152451 ¥18-18MeV / gaussmb= 5.788378 ¥18-15MeV / gauss† mNmb=meMNª11836† mp= 2.7928mNmn= -1.9131mNIntrinsic (measured) dipole magnetic moments † mL*=e2Mr L ml*=eh2Ml l + 1( )ml*=mNl l + 1( )L is orbital angular momentum for single nucleonM is nucleon mass † ml≡ lmNmax z-axis projectionNuclear magnetic moments† mp*= 2mNs s + 1( )mp= ±mNFrom electron case, you expect to have for this fermion --Does not agree with measurement† mp= 2.7928mNmn= -1.9131mNMeasured dipole magnetic moments† mp*= gpmNs s + 1( )= gpmN32mp= ± gpmN12† gp=mp*mNs s + 1( )gp=2mpmN ; gp= 2 2.7928( )= 5.5856Nuclear magnetic momentsAnd, by the same analysis, one gets --† mp= 2.7928mNmn= -1.9131mNMeasured dipole magnetic moments† mn*= gnmNs s + 1( )= gnmN32mn= ± gnmN12† gn=mn*mNs s + 1( )gn=2mnmN ; gn= 2 -1.9131( )= -3.8262 † gp= 5.5856gn= -3.8262Nuclear magnetic momentsConsider nuclei with odd A.Assume that the pairing interaction causes the “core” of pairednucleons to have net I = 0.Assume that the induced magnetic dipole moment is due to thelast unpaired nucleon.Use this to estimate the nuclear magnetic dipole moment -within this model.Nuclear magnetic moments † l*† s*† j*† s* † l*† s*† j*† s* † j = l - s † j = l + s † gl † gl† gs† gs† g† g† gs† gs† mn*= gnmNs s + 1( )= gnmN32mn= ± gnmN12† mp*= gpmNs s + 1( )= gpmN32mp= ± gpmN12 † ml*= glmNl*= glmNl l +1( )ml= ± glmNl† mj*= gmNj*= gmNj j + 1( )mj= ± gmNj † proton : gl= 1 neutron : gl= 0Nuclear magnetic moments † cos l*j*( )=l*( )2+ j*( )2- s*( )22l*j*cos s*j*( )=s*( )2+ j*( )2- l*( )22 s*j* † mj*=ml*cos l*j*( )+ms*cos s*j*( ) † g j*( )= gll*( )cos l*j*( )+ gss*( )cos s*j*( )[ ]mN † g( )=gl2Ê Ë Á ˆ ¯ ˜ 1 +l*( )2- s*( )2j*( )2È Î Í Í Í ˘ ˚ ˙ ˙ ˙ +gs2Ê Ë Á ˆ ¯ ˜ 1 -l*( )2- s*( )2j*( )2È Î Í Í Í ˘ ˚ ˙ ˙ ˙ † l*† s*† j*† s* † l*† s*† j*† s* † j = l - s † j = l + s † gl † gl† gs† gs† g† g† gs† gsNuclear magnetic moments † g( )=gl2Ê Ë Á ˆ ¯ ˜ 1 +l*( )2- s*( )2j*( )2È Î Í Í Í ˘ ˚ ˙ ˙ ˙ +gs2Ê Ë Á ˆ ¯ ˜ 1 -l*( )2- s*( )2j*( )2È Î Í Í Í ˘ ˚ ˙ ˙ ˙ † j*( )2= j j + 1( )= l +12Ê Ë Á ˆ ¯ ˜ ⋅ l +12+ 1Ê Ë Á ˆ ¯ ˜ j*( )2= l l + 2( )+34s*( )2=34Consider the case: † j = l + s( ) † g( )= gllj+ gs1/2j…some algebra happens here… † l*† s*† j*† s* † l*† s*† j*† s* † j = l - s † j = l + s † gl † gl† gs† gs† g† g† gs† gsNuclear magnetic momentsConsider the case: † j = l + s( ) † g( )= gllj+ gs1/2j † l*† s*† j*† s* † l*† s*† j*† s* † j = l - s † j = l + s † gl † gl† gs† gs† g† g† gs† gs † m= g( )j = g( )Im= gll + gssm= gll +msBut, if † I = l + s( ) † m= gl(I -1/2) +msNuclear magnetic momentsConsider the case: † j = l - s( ) † g( )= gll + 1j + 1- gs1/2j + 1Four cases to consider: both cases shown here for odd proton & odd neutron † l*† s*† j*† s* † l*† s*† j*† s* † j = l - s † j = l + s † gl † gl† gs† gs† g† g† gs† gs † m= gll + gl+ms[ ]II +1Ê Ë Á ˆ ¯ ˜ But, if † I = l - s( ) † m= glI -ms-gl2Ê Ë Á ˆ ¯ ˜ II +1Ê Ë Á ˆ ¯ ˜Nuclear magnetic moments † l*† s*† j*† s* † l*† s*† j*† s* † j = l - s † j = l + s † gl † gl† gs† gs† g† g† gs† gsProton:Neutron: † gl= 1, ms=mp † gl= 0, ms=mn † m= glI -ms-gl2Ê Ë Á ˆ ¯ ˜ II +1Ê Ë Á ˆ ¯ ˜ † j = l - s( ) † m= gl(I -1/2) +ms † j = l + s(