The Nuclear Shell Model Toward the Drip Lines ALFREDO POVES Departamento de Física Teórica and IFT, UAM-CSIC Universidad Autónoma de Madrid (Spain) Universidad Internacional del Mar Aguilas, July 25-28, 2011
OUTLINE Spherical mean field and Correlations The physics of very neutron rich nuclei at N=20 and N=28 A new region of deformation south of 68 Ni Interlude; Aligned T=0 pairs in 92 Pd
Monopole anomalies and Multipole universality The different facets of the nuclear dynamics depend on the balance of the two main components of the nuclear hamiltonian; the Monopole which produces the effective spherical mean field and the Multipole responsible for the correlations Large scale shell model calculations have unveiled the monopole anomalies of the two-body realistic interactions, i.e that they tend to produce effective single particle energies which are not compatible with the experimental data and which, if used without modifications, produce spectroscopic catastrophes
Monopole anomalies and Multipole universality Already in the late 70 s Pasquini and Zuker showed that the Kuo Brown interaction could not produce neither a magic 48 Ca nor a magic 56 Ni. In this last case it made a nearly perfect rotor instead. A few monopole corrections (mainly T=1) restored high quality spectroscopy Holt, Otsuka and Schwenk have recently shown that the monopole component of the three body force may explain the monopole anomalies relevant for 28 O and 48 Ca. The Multipole Hamiltonian of the realistic two body interactions (dominated by L=0 pairings, quadrupole and octupole) does not seem to require any substantial modification and is universal in the sense that all the interactions produce equivalent multipole hamiltonoans
The fate of magic closures Magic numbers are associated to energy gaps in the spherical mean field. Therefore, to promote particles above the Fermi level costs energy. However, some intruder configurations can overwhelm their loss of monopole energy with their huge gain in correlation energy. Several examples of this phenomenon exist in stable magic nuclei in the form of coexisting spherical, deformed and superdeformed states in a very narrow energy range, Nuclear Allotropy? In the case of 40 Ca they have described in tha spherical shell model framework
The Monopole Hamiltonian H m = ǫ i n i + [ 1 (1+δ ij ) a ij n i (n j δ ij ) + 1 2 b ij ( T i T j 3n )] i 4 δ ij + A ijk n i n j n k The coefficients a and b are defined in terms of the centroïds: V T ij = J V JT ijij [J] J [J] as: a ij = 1 4 (3V ij 1 + Vij 0), b ij = Vij 1 Vij 0, the sums run over Pauli allowed values.
The Monopole Hamiltonian The evolution of effective spherical single particle energies with the number of particles in the valence space is dictated by H m. Schematically: ǫ j ({n i }) = ǫ j ({n i = 0})+ i a ij n i + i,k A ijk n i n k
Valence Spaces; sd-pf The valence space of two major shells 1f 5/2 2p 1/2 2p 3/2 1f 7/2 1d 3/2 2s 1/2 1d 5/2 pf -shell sd-shell can encompass the physics of nuclei between 18 O and 64 Ge including the islands of inversion which appear at N=20 (around 31 Na) and N=28 (around 42 Si) as well as the excited superdeformed bands of N=Z magic nuclei like 40 Ca. And, indeed, using essentially a single effective interaction, SDPF-U.
N=20 far from stability The region around 31 Na provides a beautiful example of intruder dominance in the ground states, known experimentally since long (Detraz, Thibault, Guillemaud, Klotz, Walter). Early shell model calculations (Poves and Retamosa (87), Warburton, Becker and Brown (90)) pointed out the role of deformed intruder configurations 2p-2h neutron excitations from the sd to the pf -shell and started the study of the boundaries of the so called island of inversion and the properties of its inhabitants. Similar mechanisms produce the other known islands of inversion centered in 11 Li (N=8), 42 Si (N=28), and 64 Cr (N=40)
The Drift of the Single Particle Energies: N=20 10 ESPE (MeV) 0-10 -20 d5/2 s1/2 d3/2 f7/2 p3/2 p1/2 f5/2 8 14 16 Proton number 20
Quadrupole Collectivity vs. Magic Closures N=20 Four protons away from doubly magic 40 Ca, 34 Si is a new doubly magic nucleus because the proton Z=14 and the neutron N=20 gaps reinforce each other. To go even more neutron rich, one needs to remove protons from the 0d 5/2 orbit. This causes two effects; a reduction of the N=20 neutron gap and the onset of proton collectivity. Both conspire in the sudden appearance of an Island of Inversion in which Deformed Intruder states become ground states, as in 32 Mg, 31 Na and 30 Ne.
The drift of the single particle energies, N=28 10 ESPE (MeV) 0-10 f7/2 p3/2 p1/2 f5/2 8 14 16 Proton number 20
Quadrupole Collectivity vs. Magic Closures N=28 As we remove protons from doubly magic 48 Ca, the N=28 neutron gap slowly shrinks. In 46 Ar the collectivity induced by the action of the four valence protons in the almost degenerate quasi-spin doublet 1s 1/2-0d 3/2, is not enough to beat the N=28 closure. 46 Ar is non-collective. In 44 S, the quadrupole collectivity sets in. The N=28 closure blows out and prolate and non collective states coexist. The ground state and the first excited 2 + form the germ of a prolate rotational band. The 0 + isomer, predicted by the shell model calculations has been recently found at Ganil. In turn 42 Si is an oblate well deformed, rotor with a first 2 + state at 770 kev and 40 Mg is predicted to be a very collective prolate rotor, with a 2 + at 680 kev. In addition it could well develop a neutron halo because more than two neutrons are, in average, in p wave.
The Magnesium isotopes from the proton to the neutron dripline; SDPF-U interaction 2 + excitation energy in MeV 2 1.8 1.6 1.4 1.2 1 0.8 EXP TH 0.6 8 10 12 14 16 18 20 22 24 26 28 30 32 N
The Silicon isotopes from the proton to the neutron dripline; SDPF-U interaction 4 2 + excitation energy in MeV 3 2 1 EXP TH 0 8 10 12 14 16 18 20 22 24 26 28 30 32 N
The Magnesium isotopes; B(E2) s SDPF-U int. 150 100 e 2 fm 4 50 B(E2) exp B(E2) th 0 30 32 34 36 A
The Neon isotopes; SDPF-U int. 2.5 2 + excitation energy in MeV 2 1.5 1 exp th 0.5 8 10 12 14 16 18 20 22 24 26 28 N
The island of inversion south of 68 Ni Figure credit, Carin Cain
The Valence Space for 68 Ni and its neighbors 2d 5/2 1g 9/2 1f 5/2 1f 5/2 2p 1/2 2p 1/2 2p 3/2 2p 3/2 1f 7/2 Neutrons Protons 48 Ca acts as the inert core
The island of inversion south of 68 Ni The situation at N=40 is similar to the one found at N=20 except that 68 Ni is not a bona fide magic nucleus. Removing protons from the 0f 7/2 orbit, activates the quadrupole collectivity, which, in turn, favors the np-nh neutron configurations across N=40, that take advantage of the quasi-su3 coherence of the doublet 0g 9/2-1d 5/2. Large scale SM calculations in the valence space of the full pf -shell for the protons and the 0f 5/2 1p 3/2 1p 1/2 0g 9/2 and 1d 5/2 orbits for the neutrons, predict a new region of deformation centered at 64 Cr.
68 Ni shell model 8 + 6 + 4399 4244 exp. 8 + 4208 6 + 3999 4 + 3184 5 2779 0 + 2450 2 + 1990 0 + 1400 4 + 3147 5 2848 (0 + ) 2511 2 + 2034 0 + 1770 40 52 0 + 0 68 Ni 0 + 0
The neutron ESPES at N=40 and N=20 10 5 N=40 5 0 N=20 ESPE (MeV) 0-5 -10-15 f7/2 p3/2 f5/2 p1/2 g9/2 d5/2 20 Z 28 32 ESPE (MeV) -5-10 -15-20 d5/2 s1/2 d3/2 f7/2 p3/2 8 Z 14 16
The N=40 isotones E(2 + ) (MeV) 3 2.5 2 1.5 1 0.5 0 SM EXP N=40 (a) 20 22 24 26 28 B(E2;2 + -> 0 + ) (e 2 fm 4 ) 500 400 300 200 100 0 SM EXP (b) 20 22 24 26 28 Z Z
The Iron Isotopes E(2 + ) (MeV) 1 0.8 0.6 0.4 0.2 0 Fe SM EXP (a) 36 38 40 42 B(E2; J+2 -> J) (e 2 fm 4 ) 800 700 600 500 400 300 200 100 0 J=0 J=2 J=4 EXP, J=0 (c) 36 38 40 42 N N
The Chromium Isotopes E(2 + ) (MeV) 1 0.8 0.6 0.4 0.2 0 Cr SM EXP (a) 36 38 40 42 B(E2; J+2 -> J) (e 2 fm 4 ) 800 700 600 500 400 300 200 100 0 J=0 J=2 J=4 (c) 36 38 40 42 N N
The Nickel Isotopes E(2 + ) (MeV) 3 2.5 2 1.5 1 0.5 0 Ni SM EXP (a) 36 38 40 42 B(E2;2 + -> 0 + ) (e 2 fm 4 ) 250 200 150 100 50 0 SM EXP (b) 36 38 40 42 N N
The yrast bands, Iron and Chromium 3 (b) 3 (b) E(J+2)/E(J) 2.5 2 1.5 1 0.5 SM, J=2 SM, J=4 EXP, J=2 36 38 40 42 E(J+2)/E(J) 2.5 2 1.5 1 0.5 SM, J=2 SM, J=4 EXP, J=2 36 38 40 42 N N
Conclusions State of the art Shell Model calculations encompassing two major oscillator shells make it possible to describe complete series of isotopes from the proton to the neutron drip lines They can also cross the islands of inversion at N=8, 20, and 28 And predict new ones as that south of 68 Ni, centered in 64 Cr