Nuclear Electromagnetic MomentsElectric MultipolesPowerPoint PresentationSlide 4QM Analog for the NucleusElectric Multipole MomentsSlide 7Magnetic Multipole MomentsSlide 9Slide 10g-factorsSlide 12Nuclear ElectromagneticMomentsElectric Multipoles•The electric energy associated with the electric charge distribution in the nucleus is determined by the interaction of the nuclear charge distribution with electric fields.€ Ee= eiVi=1Z∑xi, yi,zi( )Vx'≡∂V∂xEe= eiV0+ V1+ V2+ ...[ ]i=1Z∑V0= V 0( )V1= Vx'xi+ Vy'yi+ Vz'zi( )V2=12Vxx'xi2+ Vyy'yi2+ Vzz'zi2+ Vxy'xiyi+ ...( ) € ei xi, yi,zi( )€ Vr r ( )=14πεo1rn+1( )n=0∞∑r'( )∫nPncosθ( )ρdτVr r ( )=14πεoV0+ V1+ V2+ ...[ ]V0=1rρdτ∫V1=1r2r'cosθ ρdτ∫V2=1r3r'( )232cos2θ −12 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ρdτ∫Parity of Vn goes as (-1)nQM Analog for the Nucleus•Vn is the multipole operator of order n is the nuclear wave function •For all fixed-parity states, the contribution from all odd multipole operators is zero! € ρ x, y,z( )= eiPix, y,z( )i=1Z∑E ∝ ψ *Vnψ ; ψr r 1,r r 2,r r 3,r r 4,...( )∫n∑Electric Multipole Moments •All odd electric multipole moments must vanish 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 must vanish if the nuclear system is time-reversible - i.e., if it obeys time reversal symmetry.•Find an electric dipole moment for neutron implies time reversal symmetry violation!Magnetic Multipole Moments •Classically, a circulating current induces a magnetic dipole moment --• where A is the area enclosed by i.•If i is due to a single charge e moving with velocity v, we get --€ μ =iA€ μ =eTA ; μ =e2πr vπr2 ; μ =evr2μ =emvr2m ; μ =epr2m ; μ =e2mLMagnetic Multipole Moments •If i is due to a single charge e moving with velocity v, we get --€ μ =iA€ μ =eTA ; μ =e2πr vπr2 ; μ =evr2μ =emvr2m ; μ =epr2m ; μ =e2mL€ μB≡eh2me ; μN≡eh2mN € r μ * =eh2ml (l +1) ; μz=eh2mml μ =eh2ml•In the QM regime, this becomes --g-factors € r μ l*= gleh2ml (l +1) ; μlz= gleh2mml μl= gleh2ml € gl=1 gl= 0For the protonFor the neutrong-factors € r μ s*= gseh2ms(s +1) ; μsz= gseh2mms μs= gseh2ms ; gs= 2€ gs= gs=For the protonFor the neutronFor the