1. Oct 2006, Lectures 4&5 1 Lectures 4 & 5 The end of the SEMF and the nuclear shell model

2. Oct 2006, Lectures 4&5 2 4.1 Overview 4.2 Shortcomings of the SEMF magic numbers for N and Z spin & parity of nuclei unexplained magnetic moments of nuclei value of nuclear density values of the SEMF coefficients 4.3 The nuclear shell model choosing a potential L*S coupling Nuclear “Spin” and Parity Shortfalls of the shell model

3. Oct 2006, Lectures 4&5 3 4.2 Shortcomings of the SEMF

4. Oct 2006, Lectures 4&5 4 4.2 Shortcomings of the SEMF (magic numbers in E bind /A) SEMF does not apply for A<20 (2,2)(2,2) 2*(2,2) = Be(4,4) E   94keV (6,6)(6,6) (8,8)(8,8) (10,10) (N,Z)(N,Z) There are systematic deviations from SEMF for A>20

5. Oct 2006, Lectures 4&5 5 4.2 Shortcomings of the SEMF (magic numbers in numbers of stable isotopes and isotones) Proton Magic Numbers N Z Magic Proton Numbers (stable isotopes) Magic Neutron Numbers (stable isotones) Neutron Magic Numbers

6. Oct 2006, Lectures 4&5 6 4.2 Shortcomings of the SEMF (magic numbers in separation energies) Neutron separation energies saw tooth from pairing term step down when N goes across magic number at 82 Ba Neutron separation energy in MeV

7. Oct 2006, Lectures 4&5 7 N=50 Z=50 N=82 Z=82 N=126 iron mountain 4.2 Shortcomings of the SEMF (abundances of elements in the solar system) Complex plot due to dynamics of creation, see lecture on nucleosynthesis no A=5 or 8

8. Oct 2006, Lectures 4&5 8 4.2 Shortcomings of the SEMF (other evidence for magic numbers, Isomers) Nuclei with N=magic have abnormally small n-capture cross sections (they don’t like n’s) Close to magic numbers nuclei can have “long lived” excited states (   >O(10 -6 s) called “isomers”. One speaks of “islands of isomerism” [Don’t make holydays there!] They show up as nuclei with very large energies for their first excited state (a nucleon has to jump across a shell closure) 208 Pb First excitation energy

9. Oct 2006, Lectures 4&5 9 4.2 Shortcomings of the SEMF (others) spin & parity of nuclei do not fit into a drop model magnetic moments of nuclei are incompatible with drops actual value of nuclear density is unpredicted values of the SEMF coefficients except Coulomb and Asymmetry are completely empirical

10. Oct 2006, Lectures 4&5 10 4.2 Towards a nuclear shell model How to get to a quantum mechanical model of the nucleus? Can’t just solve the n-body problem because: we don’t know if a two body model makes sense (it does not make much sense for a normal liquid drop) if it did make sense we don’t know the two body potentials (yet!) and if we did, we could not even solve a three body problem But we can solve a two body problem! Need simplifying assumptions

11. Oct 2006, Lectures 4&5 11 4.3 The nuclear shell model This section follows Williams, Chapters 8.1 to 8.4

12. Oct 2006, Lectures 4&5 12 4.3 Making a shell model (Assumptions) Assumptions: Each nucleon moves in an averaged potential neutrons see average of all nucleon-nucleon nuclear interactions protons see same as neutrons plus proton-proton electric repulsion the two potentials for n and p are wells of some form (nucleons are bound) Each nucleon moves in single particle orbit corresponding to its state in the potential  We are making a single particle shell model Q: why does this make sense if nucleus full of nucleons and typical mean free paths of nuclear scattering projectiles = O(2fm) A: Because nucleons are fermions and stack up. They can not loose energy in collisions since there is no state to drop into after collision Use Schroedinger Equation to compute Energies (i.e. non-relativistic), justified by simple infinite square well energy estimates Aim to get the correct magic numbers (shell closures) and be content

13. Oct 2006, Lectures 4&5 13 4.3 Making a shell model (without thinking, just compute) Try some potentials; motto: “Eat what you know” Coulomb infin. square harmonic desired magic numbers 2 8 20 28 50 82 126

14. Oct 2006, Lectures 4&5 14 4.3 Making a shell model (with thinking) We know how potential should look like! It must be of finite depth and … If we have short range nucleon-nucleon potential then … … the average potential must look like the density flat in the middle (you don’t know where the middle is if you are surrounded by nucleons) steep at the edge (due to short range nucleon-nucleon potential) R ≈ Nuclear Radius d ≈ width of the edge

15. 15 4.3 Making a shell model (what to expect when rounding off a potential well) Higher L solutions get larger “angular momentum barrier”  Higher L wave functions are “localised” at larger r and thus closer to “edge” Radial Wavefunction U(r)=R(r)*r for the finite square well Rounding the edge affects high L states most because they are closer to the edge then low L ones. High L states drop in energy because can now spill out across the “edge” this reduces their curvature which reduces their energy So high L states drop rounding the well!!

16. Oct 2006, Lectures 4&5 16 4.3 Making a shell model (with thinking) The “well improvement program” Harmonic is bad Even realistic well does not match magic numbers Need more splitting of high L states Include spin-orbit coupling a’la atomic magnetic coupling much too weak and wrong sign Two-nucleon potential has nuclear spin orbit term deep in nucleus it averages away at the edge it has biggest effect the higher L the bigger the split

17. Oct 2006, Lectures 4&5 17 4.3 Making a shell model (spin orbit terms) Q: how does the spin orbit term look like? Spin S and orbital angular momentum L in our model are that of single nucleon in the assumed average potential In the middle the two-nucleon interactions average to a flat potential and the two-nucleon spin-orbit terms average to zero Reasonable to assume that the average spin-orbit term is strongest in the non symmetric environment near the edge  Dimension: Length 2 compensate 1/r * d/dr

18. Oct 2006, Lectures 4&5 18 4.3 Making a shell model (spin orbit terms) Good quantum numbers without LS term : l, l z & s=½, s z from operators L 2, L z, S 2, S z with Eigenvalues of l(l+1)ħ 2, s(s+1)ħ 2, l z ħ, s z ħ With LS term need operators commuting with new H J=L+S & J z =L z +S z with quantum numbers j, j z, l, s Since s=½ one gets j=l+½ or j=l-½ (l≠0) Giving eigenvalues of LS [ LS=(L+S) 2 -L 2 -S 2 ] ½[j(j+1)-l(l+1)-s(s+1)]ħ 2 So potential becomes: V(r) + ½l ħ 2 W(r) for j=l+½ V(r) - ½(l+1) ħ 2 W(r) for j=l -½ we can see this asymmetric splitting on slide 16

19. Oct 2006, Lectures 4&5 19 4.3 Making a shell model (fine print) There are of course two wells with different potentials for n and p We currently assume one well for all nuclei but … The shape of the well depends on the size of the nucleus and this will shift energy levels as one adds more nucleons Using a different well for each nucleus is too long winded for us though perfectly doable So lets not use this model to precisely predict exact energy levels but to make magic numbers and …

20. Oct 2006, Lectures 4&5 20 4.3 Predictions from the shell model (total nuclear “spin” in ground states) Total nuclear angular momentum is called nuclear spin = J tot Just a few empirical rules on how to add up all nucleon J’s to give J tot of the whole nucleus Two identical nucleons occupying same level (same n,j,l) couple their J’s to give J(pair)=0  J tot (even-even ground states) = 0  J tot (odd-A; i.e. one unpaired nucleon) = J(unpaired nucleon) Carefull: Need to know which level nucleon occupies. I.e. more or less accurate shell model wanted!  |J unpaired-n -J unpaired-p |<J tot (odd-odd)< J unpaired-n +J unpaired-p there is no rule on how to combine the two unpaired J’s

21. Oct 2006, Lectures 4&5 21 4.3 Predictions from the shell model (nuclear parity in groundstates) Parity of a compound system (nucleus): P(even-even groundstates) = +1 because all levels occupied by two nucleons P(odd-A groundstates) = P(unpaired nucleon) No prediction for parity of odd-odd nuclei

22. Oct 2006, Lectures 4&5 22 4.3 Shortcomings of the shell model The fact that we can not predict spin or parity for odd-odd nuclei tells us that we do not have a very good model for the LS interactions A consequence of the above is that the shell model predictions for nuclear magnetic moments are very imprecise We can not predict accurate energy levels because: we only use one “well” to suit all nuclei we ignore the fact that n and p should have separate wells of different shape As a consequence of the above we can not reliably predict much (configuration, excitation energy) about excited states other then an educated guess of the configuration of the lowest excitation