1 / 8

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

edward
Download Presentation

Nuclear Shell Model

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related