P461 - Nuclei II1 Nuclear Shell Model Potential between nucleons can be studied by studying bound states (pn, ppn, pnn, ppnn) or by scattering cross sections: np -> np pp -> pp nD -> nD pD -> pD If had potential could solve Schrod. Eq. Don’t know precise form but can make general approximation 3d Finite Well with little r-dependence (except at edge of well) Almost spherically symmetric (fusion can be modeled as deformations but we’ll skip) N-N interactions are limited (at high A) due to Pauli exclusion. p + n -> p’ + n’ only if state is available

P461 - Nuclei II2 Infinite Radial Well Radial part of Scrod Eq Easy to solve if l=0 For L>0, angular momentum term goes to infinity at r=0. Reduces effective wavelength, giving higher energy Go to finite well. Wave function extends a bit outside well giving longer effective wavelength and lower energy (ala 1D square wells) In nuceli, potential goes to infinity at r=0 (even with L=0) as that would be equivalent to nucleon “inside” other nucleon

P461 - Nuclei II3 Angular part If V(r) then can separate variables  (r,  = R(r)Y(  have spherical harmonics for angular wave function Angular momentum then quantized like in Hydrogen (except that L>0 for n=1, etc) Energy doesn’t depend on m Energy increases with increasing n (same l) Energy increases with increasing l (same n) If both n,l vary then use experimental observation to determine lower energy Energy will also depend on strong magnetic coupling between nucleons Fill up states separately for p,n

P461 - Nuclei II4 L,S,J Coupling: Atoms vs Nuclei ATOMS: If 2 or more electrons, Hund’s rules: Maximise total S for lowest E (S=1 if two) Maximise total L for lowest E (L=2 if 2 P) Energy split by total J (J=3,2,1 for S=1,L=2) NUCLEI: large self-coupling. Plus if 2 p (or 2 n) then will anti-align giving a state with J=0, S=0, L=0 leftover “odd” p (or n) will have two possible J = L + ½ or J = L – ½ higher J has lower energy if there are both an odd P and an odd n (which is very rare in stable) then add up Jn + Jp Atom called LS coupling nuclei called jj Note that magnetic moments add differently as different g-factor for p,n

P461 - Nuclei II5 Spin Coupling in Nuclei All nucleons in valence shell have same J Strong pairing causes Jz antiparallel (3 and -3) spin wavefunction = antisymmetric space wavefunction = symmetric This causes the N-N to be closer together and increases the attractive force between them e-e in atoms opposite as repulsive force Can also see in scattering of polarized particles Even N, even Z nuclei. Total J=S=L=0 as all n,p paired off Even N, odd Z or odd N, even Z. nuclear spin and parity determined by unpaired nucleon Odd N, odd Z. add together unpaired n,p Explains ad hoc pairing term in mass formula

P461 - Nuclei II6 Energy Levels in Nuclei Levels in ascending order (both p,n) State n L degeneracy(2j+1) sum 1S 1/ *** 1P 3/ P 1/ *** 1D 5/ S 1/ D 3/ *** 1F 7/ *** 2P 3/ F 5/ P 1/ G 9/ *** *** “magic” number is where there is a large energy gap between a filled shell and the next level. More tightly bound nuclei. (all filled subshells are slightly “magic”)

P461 - Nuclei II7 Magic Numbers Large energy gaps between some filled shells and next (unfilled) shell give larger dE/A and more made during nucleosnthesis in stars # protons #neutrons 2 He 2 He-4 6 C 6 C-12 8 O 8 O Ca Ni 28 Cr-52(24,28) 50 Sn 50 Ni Pb Ni-78 (2005) doubly magic. While it is unstable, it is the much neutron rich. Usually more isotopes if p or n are magic. Sn has 20 isotopes, 10 of which are stable

P461 - Nuclei II8 Nuclear Magnetic Moments Protons and neutrons are made from quarks and gluons. Their magnetic moment is due to their spin and orbital angular momentum The g-factors are different than electrons. orbital, p=1 and n=0 as the neutron doesn’t have charge spin, g for proton is 5.6 and for neutron is -3.8 (compared to -2 for the electron; sometimes just 2). A proton is made from 2 up and 1 down quark which have charge 2/3 and -1/3 A neutron is made from 1 up and 2 down and has “more” negative charge/moments No theory which explains hadronic magnetic moments orbital and spin magnetic moments aren’t aligned, need to repeat the exercise in atoms (Zeeman effect) to get values for the z-component of the moment