1. The Nuclear Shell Model – Past and Present Igal Talmi The Weizmann Institute of Science Rehovot Israel

2. The main success of the nuclear shell model Nuclei with “magic numbers” of protons and neutrons exhibit extra stability. In the shell model these are interpreted as the numbers of protons and neutrons in closed shells, where nucleons are moving independently in closely lying orbits in a common central potential well. In 1949, Maria G.Mayer and independently Haxel, Jensen and Suess introduced a strong spin-orbit interaction. This gave rise to the observed magic numbers 28, 50, 82 and 126 as well as to 8 and 20 which were easier to accept.

3. 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.

4. 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 (e.g. to obtain the neutron effective charge!) Only recently this effort seems to yield some reliable results. This does not solve the major problem - how independent nucleon motion can be reconciled with the strong and short ranged bare interaction.

5. 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.

6. The first successful calculation – low lying levels of 40 K and 38 Cl In the simplest shell model configurations of these nuclei, the 12 neutrons outside the 16 O core, completely fill the 1d 5/2, 2s 1/2, 1d 3/2 orbits while the proton 1d 5/2 and 2s 1/2 orbits are also closed. The valence nucleons are in 38 Cl: one 1d 3/2 proton and a 1f 7/2 neutron in 40 K: (1d 3/2 ) 3 J p =3/2 proton configuration and a 1f 7/2 neutron. In each nucleus there are states with J=2, 3, 4, 5 Using levels of 38 Cl, the 40 K levels may be calculated and vice versa.

7. Comparison with 1954 data only the spin 2 - agreed with our prediction

8. We were disappointed but not surprized. Why? The assumption that the states belong to rather pure jj-coupling configurations may have been far fetched. Also the restriction to two-body interactions could not be justified a priori. There was no evidence that values of matrix elements do not appreciably change when going from one nucleus to the next. Naturally, we did not publish our results but then…

9. Comparison of our predictions with experiment in 1955

10. Conclusions from the relation between the 38 Cl and 40 K levels The restriction to two-body effective interaction is in good agreement with some experimental data. Matrix elements (or differences) do not change appreciably when going from one nucleus to its neighbors (Nature has been kind to us). Some shell model configurations in nuclei are very simple. It may be stated that Z=16 is a proton magic number (as long as the neutron number is N=21).

11. This was the first successful calculation and it was followed by more complicated ones carried out in the same way. The complete p-shell, p 3/2 and p 1/2 orbits (Cohen and Kurath), Zr isotopes (mostly the Argonne group), the complete d 5/2,d 3/2,s 1/2 shell (Wildenthal, Alex Brown et al) and others. This series culminated in more detailed calculations including millions of shell model states, with only two-body forces (Strasbourg and Tokyo). Not all matrix elements determined. Only simple examples will be shown. Effective interactions, no longer restricted by the bare interactions, have been adopted

12. General features of the effective interaction extracted even 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. The average T=0 interaction, between protons and neutrons, is strong and attractive. It breaks seniority in a major way.

13. 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 11 Be, and in more detail in a review article in 1962. It will be considered later but first let us look at identical valence nucleons..

14. Seniority in j n Configurations The seniority scheme is the set of states which diagonalizes the pairing interaction V J = =0 unless J=0, M=0 States of identical nucleons are characterized by v, the seniority which in a certain sense, is the number of unpaired particles. In states in with seniority v in the j v configuration, no two particles are coupled to states with J=0, i.e. zero pairing. From such a state, with given value of J, a seniority v state may be obtained by adding (n-v)/2 pairs with J=0 and antisymmetrizing. The state, in the j n configuration thus obtained, has the same value of J.

15. Seniority in Nuclei Seniority was introduced by Racah in 1943 for classifying states of electrons in atoms. There is strong attraction in T=1, J=0 states of nucleons. Hence, seniority is much more useful in nuclei. The lower the seniority the higher the amount of pairing. Ground states of semi-magic nuclei are expected to have lowest seniorities, v=0 in even nuclei and v=1 in odd ones.

18. A useful lemma

22. Properties of the seniority scheme ground states If a two-body interaction is diagonal in the seniority scheme, its eigenvalues in states with v=0 and v=1 are given by αn(n-1)/2 + β(n-v)/2 where α and β are linear combinations of two- body energies V J =. This leads to a binding energy expression (mass formula) for semi-magic nuclei B.E.(j n )=B.E.(n=0)+nC j +αn(n-1)/2+ β(n-v)/2 It includes a linear, quadratic and pairing terms.

23. Neutron separation energies from calcium isotopes

24. 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.

27. Properties of the seniority scheme excited states If the two-body interaction is diagonal in the seniority scheme in j n configurations, then level spacings are constant, independent of n.

28. Levels of even (1h 11/2 ) n configurations

29. Levels of odd (1h 11/2 ) n configurations

30. Properties of the seniority scheme moments and transitions Odd tensor operators are diagonal and their matrix elements in j n 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.

31. E2 matrix elements in (1h 11/2 ) n configurations

32. B(E2) values in even (1h 9/2 ) n configurations

33. The rates of E2 transitions usually increase as more valence nucleons are added. Here, the opposite behavior is evident.

34. Generalized seniority There are semi-magic nuclei where there is evidence that valence nucleons occupy several orbits. Still, they exhibit seniority features, in binding energies and constant 0 - 2 spacings. A generalization of seniority to mixed configurations is needed.

35. Generalized seniority Pair creation operators in different orbits S j’ †, S j” †,… may be added to S † =Σ S j † which has, with its hermitean conjugate the same commutation relations as in a single j-orbit, with 2j+1 replaced by 2Ω=Σ(2j+1). A more realistic pair creation operator is S † =Σα j S j † If not all coefficients are equal the algebra is more difficult. Still,

36. Generalized seniority:ground states There are states with seniority-like features. Assume that (S † ) N |0> are exact eigenstates H(S † ) N |0>=E N (S † ) N |0> HS † |0>=VS † |0> In fact, using the lemma, it follows, H(S † ) N |0>=(S † ) N H|0>+N(S † ) N−1 [H,S † ]|0>+ ½N(N−1)(S † ) N−2 [[H,S † ],S † ]|0>= {NV+½N(N−1)W}(S † ) N |0>

37. Generalized seniority mass formula Here, V is the (lowest) eigenvalue of the two nucleon J=0 states. It is obtained by diagonalization of the Hamiltonian matrix for N=2, J=0, including single nucleon energies and two-body interactions. The formula for binding energies of even-even nuclei is BE(N)=BE(N=0)+NV+½N(N−1)W Two-nucleon separation energies should lie on a straight line.

38. Generalized seniority in Sn isotopes Binding energies of tin isotopes were measured in all isotopes from N=50 to N=82. Two neutron separation energies lie on a smooth curve and no breaks due to possible sub- shells are present. Still, the line has a slight curvature indicating a possible small cubic term in binding energies. Including a term proportional to N(N−1)(N−2)/6 led to very good agreement. Such a term could be due to weak 3-body effective interactions or due to polarization of the core by valence nucleons.

39. Generalized seniority: excited states When not all α j coefficients are equal, it is no longer true that all level spacings are constant. Still, there is one state for any given J>0 which may lie at a constant spacing above the J=0 ground state. Consider the operator D J † =(1+δ jj’ ) −½ Σ(jmj’m’|jj’JM)a jm † a j’m’ † which creates an eigenstate of H in a two particle system, HD J † |0>=V J D J † |0>. If HS † D J † |0>=[H,S † ]D J † |0>+S † HD J † |0>= [[H,S † ],D J † ]|0>+(V+V J )S † D J † |0> is an eigenstate, the double commutator condition must be satisfied, [[H,S † ],D J † ]=WS † D J † from which follows H(S † ) N−1 D J † |0>={(N−1)V+V J +½N(N−1)W(S † ) N−1 D J † |0>+ {E N +V 2 −V}(S † ) N−1 D J † |0> This means that V 2 −V spacings are independent of N

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

41. Yrast states of odd Sn isotopes

42. Proton-neutron interactions- level order and spacings From a 1959 experiment it was “concluded that the assignment J= ½ - for 11 Be, as expected from the shell model is possible but cannot be established firmly on the basis of present evidence.” We did not think so. In 13 C the first excited state, 3 MeV above the ½ - g.s. has spin ½ + attributed to a 2s 1/2 valence neutron. The g.s.of 11 Be is obtained from 13 C by removing two 1p 3/2 protons. Their interaction with a 1p 1/2 neutron is expected to be stronger than their interaction with a 2s 1/2 neutron, hence, the latter’s orbit may become lower in 11 Be.

43. Experimental information on p 3/2 p 1/2 and p 3/2 s 1/2 interactions in 12 B 7

44. How should this information be used? Removing two j-protons coupled to J=0 reduces the interaction with a j’-neutron by TWICE the average jj’ interaction (the monopole part). The average interaction is determined by the position of the center-of-mass of the jj’ levels, V(jj’)=SUM J (2J+1) /SUM J (2J+1)

45. Shell model prediction of the 11 Be ground state and excited level

46. Comments on the 11 Be 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 s neutron wave function should be appreciably extended and 11 Be 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 11 Be, implied by the shell model, is a real effect.

48. 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.

49. Levels of Ba (Z=56) isotopes Drastic reduction of 0-2 spacings

50. Ab initio calculations of nuclear many body problem Ab initio calculations of nuclear states and energies are of great importance. Their input is the bare interaction. If it is sufficiently correct and the approximations made are satisfactory, accurate energies 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. There are still many difficulties to overcome.

51. Shell model wave functions are real to an appreciable extent We saw some evidence in the halo nucleus 11 Be and in E2 transitions. This is also evident from pick-up and stripping reactions. Other evidence is offered by Coulomb energy differences.

52. Coulomb energy differences of 1d 5/2 and 2s 1/2 orbits

53. Will simplicity emerge from complexity? 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? 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.

55. 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. Still, three-body interactions with core nucleons could contribute to effective two-body interactions between valence nucleons and to single nucleon energies.

56. Which nuclei are magic? In magic nuclei, energies of first excited states are rather high as in 36 S (Z=16, N=20) where the first excited state is at 3.3 MeV, considerably higher than in its neighbors. Shell closure may be concluded if valence nucleons occupy higher orbits like 1d 3/2 protons in the 38 Cl, 40 K case. Shell closure may be demonstrated by a large drop in separation energies (no stronger binding of the closed shells!)

57. Two inaccurate statements are often made The first is that the realization “that magic numbers are actually not immutable occurred only during the past 10 years”. This was realized almost 50 years ago. The other is that “the shell structure may change” ONLY “for nuclei away from stability”, “nuclear orbitals change their ordering in regions far away from stability”. Such changes occur all over the nuclear landscape.

58. Various approaches to nuclear structure physics The simple Ab initio shell model Theoretical derivation calculations of the The original effective interaction Recent approach developments

59. Ab initio calculations new developements A typical theory of this kind is the No Core Shell Model (NCSM). The input of all ab initio calculations is the bare interaction between free nucleons. After 60 years of intensive work, there are several prescriptions which reproduce rather well, results of scattering experiments but none has a solid theoretical foundation. No core is assumed and harmonic oscillator wave functions are used as a complete scheme. Some results of NCSM will be presented and discussed in the following, showing the present status and some problems.

60. Merits of ab initio Calculations Ab initio calculations of nuclear states and energies are of great importance. If the input, the bare interaction, is sufficiently correct and the approximations made are satisfactory, accurate energies of nuclear states, also of nuclei inaccessible experimentally, will be obtained. More important would be the knowledge of the real nuclear wave functions. This will enable the calculation of various moments and transitions, electromagnetic and weak ones, including double beta decay. There are still many difficulties to overcome.

61. Proton separation energies from N=28 isotones