The Nuclear Shell Model- Simplicity from Complexity

1. The Nuclear Shell Model-Simplicity from Complexity IgalTalmi The Weizmann Institute of Science Rehovot Israel

2. To our dear friend and colleague Aldo Covello with our very best wishes

4. The need of an effective interaction in the shell model • In the Mayer-Jensen shell model, wave functions of magic nuclei are well determined. So are states with one valence nucleon or hole. • States with several valence nucleons are degenerate in the single nucleon Hamiltonian. Mutual interactions remove degeneracies and determine wave functions and energies of states. • In the early days, the rather mild potentials used for the interaction between free nucleons, were used in shell model calculations. The results were only qualitative at best.

5. Theoretical derivation of the effective interaction A litle later, the bare interaction turned out to be too singular for use with shell model wave functions. It should be renormalized to obtain the effective interaction. More than 50 years ago, Brueckner introduced the G-matrix and was followed by many authors who refined the nuclear many-body theory for application to finite nuclei. • Starting from the shell model, the aim is to calculate from the bare interaction the effective interaction between valence nucleons. Also other operators like electromagnetic moments should be renormalized. • Only recently this effort seems to yield some reliable results like those obtained by Aldo Covello et al. • This does not address the major problem - how independent nucleon motion can be reconciled with the strong and short ranged bare interaction.

6. Ab initio calculations of nuclear many body problem • Ab initio calculations are of great importance. Their input is the bare interaction. IF it is sufficiently correct and IF the approximations made are satisfactory, accurateenergies of nuclear states, also of nuclei inaccessible experimentally, will be obtained. • The knowledge of the real nuclear wave functions will enable the calculation of various moments and transitions, electromagnetic and weak ones, including double beta decay.

7. The simple shell model • In the absence of reliable theoretical calculations, matrix elements of the effective interaction were determined from experimental data in a consistent way. • Restriction to two-body interactions leads to matrix elements between n-nucleon states which are linear combinations of two-nucleon matrix elements. • Nuclear energies may be calculated by using a smaller set of two-nucleon matrix elements determined consistently from experimental data.

8. General features of the effective interaction extracted from simple cases • The T=1 interaction is strong and attractive in J=0 states. • The T=1 interactions in other states are weak and their average is repulsive. • It leads to a seniority type spectrum. • Theaverage T=0 interaction, between protons and neutrons, is strong and attractive. • It breaks seniority in a major way.

9. Consequences of these features • The potential well of the shell model is created by the attractive proton-neutron interaction which determines its depth and its shape. • Hence, energies of proton orbits are determined by the occupation numbers of neutrons and vice versa. • These conclusions were published in 1960 addressing 11Be, and in more detail in a review article in 1962.

11. Neutron separation energies from calcium isotopes

12. Three-body interactions- where are they? There is no evidence of three-body interactions between valence nucleons. They could be weak, state independent or both. The two-body interactions extracted from experiment, may well include contributions from polarization of the core by valence nucleons. They could also include contributions of three-body interactions between a core nucleon with two valence ones. Single nucleon energies may include contributions from three-body interaction between a valence nucleon and two ones in the core. Still, three-body interactions with core nucleons could contribute to effective two-bodyinteractions between valence nucleons and to single nucleon energies.

13. Neutron separation energies from calcium isotopes

14. A direct result of this behavior is that magic nuclei are not more tightly bound than their preceding even-even neighbors. Their magic properties (stability etc.) are due to the fact that nuclei beyond them are less tightly bound.

16. Which nuclei are magic? In magic nuclei, energies of first excited states are rather high. . Shell closure may be demonstrated by a large drop in separation energies (no stronger binding of the closed shells!)

17. First excited J=2+ levels in some calcium isotopes

19. Neutron separation energies from calcium isotopes

21. Positions of J=2+ levels in nickel and calcium isotopes

22. The difference between low lying levels of 57Ni and 49Ca is due to the attractive interaction between neutrons and 1f7/2 protons. It is stronger between 1f7/2 protons and 1f5/2 neutrons.

25. Two neutron separation energies from even Ni isotopes

26. In oddNi isotopes level spacings change appreciably with the neutron number

28. How real are shell model wave functions? • In view of the apparent complicated calculations of the effective interaction, shell model wave functions, could be just model wave functions. Since they do not include the short range correlations they could be very different from the real ones. • Still, the “wounds” inflicted on the single nucleon wave functions may not change them too much. The correlations are important for short range observables, but may be less for other (“long range”) ones.

29. Shell model wave functions are real to an appreciable extent • Some evidence is in the halo of the 11Be nucleus, due to an extended 2s1/2 valence neutron wave function. • This is also evident from pick-up and stripping reactions. • Other evidence is offered by differences of charge distributions.

30. Charge distribution due to a 1h9/2 proton wave function

32. Other evidence is offered by Coulomb energy differences like between 1d5/2 and 2s1/2 orbits

33. In states of s nucleons the Coulomb energy differences are smaller than in states of nucleons ind- or p-orbits.Levels of 14C and 14N

34. Will simplicity emergefromcomplexity? • Shell model wave functions, of individual nucleons, cannot be the real ones. They do not include short range and other correlations due to the strong bare interaction. Yet the simple shell model seems to have some reality and considerable predictive power. • The bare interaction is the basis of ab initio calculations leading to admixtures of states with excitations to several major shells, the higher the better… • Will the shell model emerge from these calculations as a good approximation? It has been so simple, useful and elegant and it would be illogical to give it up. Reliable many-body theory should explain why it works so well, which interactions lead to it and where it becomes useless.

38. The monopole part of the T=0 interaction affects positions of single nucleon energies. • The quadrupole-quadrupole part in the T=0 interaction breaks seniority in a major way. • In nuclei with valence protons and valence neutrons it leads to strong reduction of the 0-2 spacings – a clear signature of the transition to rotational spectra and nuclear deformation.

39. Neutron separation energies from cadmium isotopes

40. Comments on the 11Be shell model calculation • Last figure is not an “extrapolation”. It is a graphic solution of an exact shell model calculation in a rather limited space. • The ground state is an “intruder” from a higher major shell. It can be said that here the neutron number 8 is no longer a magic number. • The calculated separation energy agreed fairly with a subsequent measurement. It is rather small and yet, we used matrix elements which were determined from stable nuclei. • We failed to see that the sneutron wave function should be appreciably extended and 11Be should be a “halo nucleus”. Shell model wave functions do not include the short range correlations which are due to the strong bare interaction. Thus, they cannot be the real wave functions of the nucleus. Still, some states of actual nuclei demonstrate features of shell model states. The large radius of 11Be, implied by the shell model, is a real effect.

42. Experimental information on p3/2p1/2 and p3/2s1/2 interactions in 12B7

43. Proton separation energies from N=28 isotones

44. Properties of the seniority schemeexcited states • If the two-body interaction is diagonal in the seniority scheme in jn configurations, then level spacings are constant, independent ofn.

45. Levels of even (1h11/2)n configurations

46. Levels of odd (1h11/2)n configurations

47. Properties of the seniority schememoments and transitions • Odd tensor operators are diagonal and their matrix elements in jn configurations are independent of n. • Matrix elements of even tensor operators are functions of n and seniorities. Between states with the same v, matrix elements are equal to those for n=v multiplied by (2j+1-2n)/(2j+1-2v) Simple results follow for E2 transtions.

48. E2 matrix elements in (1h11/2)n configurations

49. B(E2) values in even(1h9/2)n configurations

50. In semi-magic nuclei, with only valence protons or neutrons experiment shows features of generalized seniority Constant 0-2 spacings in Sn isotopes