Nuclear ElectromagneticMomentsElectric Multipoles• The electric energy associated with theelectric charge distribution in the nucleus isdetermined by the interaction of the nuclearcharge distribution with electric fields.† Ee= eiVi=1ZÂxi, yi,zi( )Vx'≡∂V∂xEe= eiV0+ V1+ V2+ ...[ ]i=1ZÂV0= V 0( )V1= Vx'xi+ Vy'yi+ Vz'zi( )V2=12Vxx'xi2+ Vyy'yi2+ Vzz'zi2+ Vxy'xiyi+ ...( ) † ei xi, yi,zi( )† Vr r ( )=14peo1rn+1( )n=0•Âr'( )ÚnPncosq( )rdtVr r ( )=14peoV0+ V1+ V2+ ...[ ]V0=1rrdtÚV1=1r2r'cosqrdtÚV2=1r3r'( )232cos2q-12Ê Ë Á ˆ ¯ ˜ rdtÚParity of Vn goes as (-1)nQM Analog for the Nucleus• Vn is the multipole operator of order n•y is the nuclear wave function• For all fixed-parity states, the contribution fromall odd multipole operators is zero! † rx, y,z( )= eiPix, y,z( )i=1ZÂE µy*Vny ; yr r 1,r r 2,r r 3,r r 4,...( )ÚnÂElectric Multipole Moments• All odd electric multipole moments mustvanish for stationary states (e.g., nuclei,nucleons, etc.)• Therefore, nuclei must not have– Electric dipole moments (n = 1)– Electric octupole moments (n = 3)– Etc…• Search for electric dipole moment for neutronElectric Multipole Moments• In more general terms --• All odd electric multipole moments mustvanish if the nuclear system is time-reversible- i.e., if it obeys time reversal symmetry.• Find an electric dipole moment for neutronimplies time reversal symmetry violation!Magnetic Multipole Moments• Classically, a circulating current induces amagnetic dipole moment --• where A is the area enclosed by i.• If i is due to a single charge e moving withvelocity v, we get --† m= iA† m=eTA ; m=e2pr vpr2 ; m=evr2m=emvr2m ; m=epr2m ; m=e2mLMagnetic Multipole Moments• If i is due to a single charge e moving withvelocity v, we get --† m= iA† m=eTA ; m=e2pr vpr2 ; m=evr2m=emvr2m ; m=epr2m ; m=e2mL† mB≡eh2me ; mN≡eh2mN † r m * =eh2ml(l +1) ; mz=eh2mml m=eh2ml• In the QM regime, this becomes --g-factors † r m l*= gleh2ml(l +1) ; mlz= gleh2mml ml= gleh2ml † gl= 1 gl= 0For the protonFor the neutrong-factors † r m s*= gseh2ms(s +1) ; msz= gseh2mms ms= gseh2ms ; gs= 2† gs= gs=For the protonFor the neutronFor the