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Nuclear Shell Model

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

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Nuclear Shell Model

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  1. 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 II

  2. 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 II

  3. Angular part • If V(r) then can separate variables y(r,q,f) = R(r)Y( q,f) 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 II

  4. 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 II

  5. 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 II

  6. Energy Levels in Nuclei • Levels in ascending order (both p,n) State n L degeneracy(2j+1) sum 1S1/2 1 0 2 2*** 1P3/2 1 1 4 6 1P1/2 1 1 2 8*** 1D5/2 1 2 6 14 2S1/2 2 0 2 16 1D3/2 1 2 4 20*** 1F7/2 1 3 8 28*** 2P3/2 2 1 4 32 1F5/2 1 3 6 38 2P1/2 2 1 2 40 1G9/2 1 4 10 50*** *** “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 II

  7. 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-16 20 Ca 20 28 Ni 28 Cr-52(24,28) 50 Sn 50 Ni-78 82 Pb 82 126 136 • 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 II

  8. 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 P461 - Nuclei II

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