Nuclear Shell Model. P461 - Nuclei II 1

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Patrick Atkins
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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 1

2 Infinite Radial Well Radial part of Scrod Eq h d u h 2 + V( r) + 2m dr 2m u( r) rr( r) P( r) = 4πu Easy to solve if l=0 l( l + 1) 2 u = r 2πn p ( hk) ( hn) u = sinkr k = E = = = 2a 2m 2m 8ma 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 2 Eu P461 - Nuclei II 2

3 Angular part If V(r) then can separate variables ψ(r,θ,φ) = R(r)Y( θ,φ) have spherical harmonics for angular wave function Angular momentum then quantized like in Hydrogen (except that L>0 for n=1, etc) 2 2 L ψ = l( l + 1) h ψ L ψ = mhψ l = 0, 1, 2K m = lkl n = r quantum# 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 Z P461 - Nuclei II 3

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 4

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 5

6 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 II 6

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 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 II 7

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 r r r µ r r N eh µ = µ L + µ S = ( gl L + gs S) µ N = h 2m p 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 8

Properties of Nuclei

Properties of Nuclei Z protons and N neutrons held together with a short-ranged force gives binding energy m 938. 3 MeV / c m 939. 6 MeV / c p 2 2 n M Zm Nm E Am nucleus p n bind N with A Z N m u 9315.