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The Nuclear Shell Model: Feats and Challenges. Alfredo Poves o Universidad Aut´noma de Madrid Bordeaux, October 2006 Strasbourg-Madrid collaboration OUTLINE Introduction Extreme correlations Exotic nuclei Double beta decay The three pillars of the shell model The Eﬀective Interaction The Valence Space The Algorithms and their Codes ınez-Pinedo, F. Nowacki, A. Poves E. Caurier, G. Mart´ and A. P. Zuker. “The Shell Model as a Uniﬁed View of Nuclear Structure”, Reviews of Modern Physics, 77 (2005) 427-488 The Eﬀective Interaction: Key aspects The evolution of the Spherical mean ﬁeld in the valence spaces. What is missing in the monople hamiltonian derived from the realistic NN interactions, be it through a G-matrix, Vlow−k or other options? The multipole hamiltonian does not seem to demand major changes with respect to the one derived from the realistic nucleon-nucleon potentials Do we really need three body forces? Would they be reducible to simple monopole forms? The Valence Space(s) Miscellanea of computationally accessible valence spaces: (note: in a major HO shell of principal quantum number p the orbit j=p+1/2 is called intruder and the remaining ones are denoted by rp) • Classical 0¯ ω valence spaces are the p, sd and pf h shells • r2-pf : intruders around N and/or Z=20 76 82 • r3-g9/2(d5/2): Ge, Se, and the region around 80 Zr • r3-g9/2(d5/2) for the neutrons and pf for protons: neutron rich Cr, Fe, Ni, Zn • r4-h11/2 for neutrons and p1/2 − g9/2-r4 for protons: 96 Zr, 100Mo,110Pd, 116Cd 124 • r4-h11/2 for neutrons and protons: Sn, 128−130Te, 136 Xe The Algorithms and the Codes Algorithms include Direct Diagonalisation, Lanczos, Monte Carlo Shell Model, Quantum Monte Carlo Diagonalization, Projected Shell-Model, DMRG and diﬀerent extrapolation ansatzs The Strasbourg-Madrid codes can deal with problems involving basis of 1010 Slater determinants, using relatively modest computational resources Some Physics Goals Precision Spectroscopy towards larger masses Changing Magic Numbers far from Stability: The competing roles of spherical mean ﬁeld and correlations Double β decay, the key to the nature of the neutrinos, the absolute scale of their masses and their hierarchy No core shell model for light nuclei. Ab initio description of the low-lying intruder states and of the origin of the Gamow-Teller quenching Nuclear Structure and Nuclear Astrophysics Extreme Correlations. Extreme Coexistence. Spherical, Deformed and Superdeformed states in the same nucleus; The case of 40Ca In the valence space of two major shells 1f5/2 2p1/2 2p3/2 1f7/2 pf -shell 1d3/2 2s1/2 1d5/2 sd-shell The relevant conﬁgurations are: [sd]24 0p-0h in 40 Ca, SPHERICAL [sd]20 [pf]4 4p-4h in 40 Ca, DEFORMED [sd]16 [pf]8 8p-8h in 40 Ca, SUPERDEFORMED 1f5/2 2p1/2 2p3/2 1f7/2 pf -shell 1d3/2 2s1/2 1d5/2 sd-shell The superdeformed band: 8p-8h 20 18 exp 16 sdpf-8p-8h 14 12 J 10 8 6 4 2 0 0 1000 2000 3000 4000 Eγ (in keV) The Superdeformed band: Mixed calculation 20 exp 18 sdpf-full 16 14 12 J 10 8 6 4 2 0 0 1000 2000 3000 4000 Eγ (in keV) Huge correlation energies!!! -260 Lowest Slater Determinant Lowest energy at fix np-nh -265 + The lowest 0 ’s after mixing -270 -275 -280 -285 -290 0 2 4 6 8 10 np-nh The np-nh energies as a function of J + -266 0 + 2 -268 4 + + -270 6 + 8 + -272 10 + 12 -274 14 + + -276 16 -278 -280 -282 -284 0 2 4 6 8 10 np-nh Transition Quadrupole Moments 250 Transition quadrupole moment (e fm ) 2 200 150 100 0 2 4 6 8 10 12 14 16 18 J Quasi-SU3 + Pseudo-SU3 interpretation In the 4p-4h intrinsic state of 40Ca, the two neutrons and two protons in the pf -shell can be placed in the lowest K=1/2 quasi-SU3 level of the p=3 shell. This gives a contribution Q0=25 b2. In the pseudo-sd shell. p=1 we are left with eight particles, that contribute with Q0=7 b2. For the 8p-8h state the values are Q0=35 b2 and Q0=11 b2 Using the proper values of the oscillator length it obtains: 40 Ca 4p-4h band Q0=125 e fm2 (Q0=148 e fm2) 40 Ca 8p-8h band Q0=180 e fm2 (Q0=226 e fm2) In very good accord with the data. The values in blue assume strict SU3 symmetry in both shells. The SM results almost saturate the quasi-SU3 bounds. The SU3 values are a 25% larger. The geometry of the spherical mean ﬁeld orbits giving rise to deformed rotors pertains to variants of Elliott’s SU3 (pseudo-SU3, quasi-SU3). np-nh conﬁgurations across N=Z=20 produce superdeformed shapes that can be explained in the pseudo-SU3+quasi-SU3 scheme. This scheme will also apply to other mass regions either protron rich as in 80 Zr, or neutron rich as in 42S. Shell Model at the limits of the stability At the very neutron rich or very proton rich edges, the T=0 and T=1 channels of the eﬀective nuclear interaction weight very diﬀerently than they do at the stability line. Therefore the eﬀective single particle structure may suﬀer important changes, leading in some cases to the vanishing of established shell closures or to the appearance of new ones. Shell Model at the limits of the stability N=20 The region around 31Na provides a beautiful example of intruder dominance in the ground states. This has been known experimentally (Detraz, Thibault, Guillemaud, Klotz, Walter) since long. Early mean ﬁeld (Campi) and shell model (Warburton, Retamosa) interpretations pointed out the role of deformed intruder conﬁgurations 2p-2h neutron excitations from the sd to the pf -shell and started to study the boundaries of the so called “island of inversion” Recently there has been a renewal of interest in this region, triggered by the advent of new radioactive ion beam facilities that pave the way toward spectroscopic studies far from stability. The limits of the “island of inversion” depend crucially of the eﬀective spherical single particle energies (ESPE’s) The evolution of the spherical mean ﬁeld at N=20 Strasbourg-Madrid 10 2p3/2 5 1f7/2 1d3/2 0 2s1/2 1d5/2 ESPE (MeV) -5 -10 -15 -20 -25 28 30 32 34 36 38 40 A The evolution of the spherical mean ﬁeld at N=20 Tokyo 10 2p3/2 5 1f7/2 2d3/2 0 2s1/2 2d5/2 ESPE (MeV) -5 -10 -15 -20 -25 28 30 32 34 36 38 40 A The saga of the Calcium isotopes; How many are magic? 40 48 The classics: Ca and Ca The proton-rich challengers: 36 34 Ca and Ca 34 (why not?, Si is doubly magic, isn’t it?) The very neutron rich ones: 52Ca, 54 Ca and, perhaps, 60Ca The monopole drift in the Calcium chain 0 ESPE (MeV) -5 -10 20 28 32 34 40 Neutron number KB3G-A (solid line) GXPF1-A (dashed line) They show almost identical behavior at N=28 and N=32. 52Ca is (weakly) magic. 54 Is Ca doubly magic? 0 ESPE (MeV) -5 -10 20 28 32 34 40 Neutron number The monopole drift caused by the ﬁlling of the orbit 1p3/2 is very diﬀerent for both interactions. GXPF1-A increases strongly the 1p3/2-0f5/2 gap,while KB3G-A reduces it slightly Therefore, N=34 should be magic with GXPF1-A and not at all with KB3G-A A word of caution on the monopole drifts All these monopole drifts involve solely the neutron- neutron interaction. Therefore, once the single particle spectrum of 41Ca is ﬁxed, the magicity of N=34 depends only of the neutron-neutron interaction. Mutatis mutandis, the same applies to the N=14 and N=16 closures in the Oxygen isotopes Our present knowledge of the eﬀective interactions is not precise enough as to predict the details of the monopole drifts far from stability. Some phenomenological ingredients have to be extracted from key experiments. The N=34 case is a good example. 54 Is Ca doubly magic? Exhibits from Isolde and elsewhere 52 Ca 4+ 6.48 (4+) 5.95 4+ 5.93 4+ 5.47 2+ 4+ 5.32 2+ 5.18 2+ 5.09 2+ 4+ 5.30 4+ 5.14 4.48 4.90 5.22 2+ 0+ 4.43 2+ 4.39 + + 0 4.28 2+ (3-) + 2 4.11 0+ 3.99 4 4.34 + 3.94 0+ 3.69 3.90 3.42 (1+,2+) 3.15 1+ 3.29 41+ 3.08 1+ 2.38 2.26 2+ 2.56 2+ 2.55 2+ 2+ 2+ 2.35 2.35 1+ 2.25 0+ 0 0+ 0 0+ 0 0+ 0 0+ 0 GXPF1 GXPF1A Exp. KB3GA KB3G F. Perrot, F. Marechal et al. (Isolde) 54 Is Ca doubly magic? Exhibits from Isolde and elsewhere 53 Ca 5/2- 2.85 5/2- 2.94 (3/2-) 2.22 3/2- 2.12 3/2- 3/2- 2.16 2.00 3/2- 1.46 5/2- 1.23 5/2- 1.03 1/2- 0 1/2- 0 1/2- 0 1/2- 0 1/2- 0 GXPF1 GXPF1A Exp. KB3GA KB3G F. Perrot, F. Marechal et al. (Isolde) 54 Is Ca doubly magic? Exhibits from Isolde and elsewhere 55 Ti 5/2- 0.89 1/2- 1/2- 1/2- 0.04 5/2- 0.19 5/2- GXPF1A KB3GA KB3G 54 Is Ca doubly magic? Exhibits from Isolde and elsewhere 56 8+ Ti 5.33 6+ 4.89 8+ 8+ 4.37 4.51 8+ 6+ 4.21 6+ 4.17 4.31 4.09 6+ 3.85 6+ 6+ 3.04 6+ 2.98 6+ 2.89 6+ 2.87 2.86 4+ 2.53 4+ 4+ 2.27 2.29 4+ 2.16 4+ 2.00 2+ 1.51 2+ 1.17 2+ 1.13 2+ 1.05 2+ 0.89 0+ 0 0+ 0 0+ 0 0+ 0 0+ 0 GXPF1 GXPF1A Exp. KB3GA KB3G R. Janssens, B. Fornal et al. MSU Is 54Ca doubly magic? The strange behavior of the B(E2)’s in the Titanium isotopes 800 exp gxpf1 700 kb3g B(E2) (in e fm ) 4 600 2 500 400 300 200 46 48 50 52 54 56 58 A isovector eﬀective charge equal to zero Is 54Ca doubly magic? The strange behavior of the B(E2)’s in the Titanium isotopes 800 exp gxpf1 700 kb3g B(E2) (in e fm ) 4 600 2 500 400 300 200 46 48 50 52 54 56 58 A isovector eﬀective charge equal to 0.6e 54 Predicted Ca low-lying spectrum 54 + Ca 2 3.83 + 2 2.95 2+ 1.77 2+ 1.32 0+ + 0 0 + 0 + 0 0 GXPF1 GXPF1A KB3GA KB3G 42 N=28: Si; doubly magic or well deformed? There have been recent claims of double magicity for 42Si by Friedman et al. (Nature 2005) based in indirect evidence from a two proton knock-out experiment at MSU. Previous shell model calculations in a model space comprising the full sd-shell for protons and the full pf -shell for neutrons that reproduced successfully the spectroscopy of the Sulfur isotopes, predicted a heavily mixed 42Si, oblate, with Q0– intrinsic quadrupole moment– equal to –56 e fm2, a 2+ excitation energy of 1.49 MeV and a doubly closed (N=28, Z=14) component that amounts only to 28% [Caurier, Nowacki and Poves (2004)]. With this interaction, the proton gap at Z=14 has a value of 6.2 MeV, and the N=28 closure vanishes below 46Ar, due to the combined eﬀect of a slight reduction of the N=28 neutron gap when sd-protons are removed, and the availability of valence protons that favor the neutron excitations across N=28 via the quadrupole- quadrupole neutron proton interaction 42 Si, an oblate, well deformed, rotor In an experiment of in-beam γ-spectroscopy performed at GANIL, S. Grevy et al. have measured the excitation energy of the 2+ of 42Si, (770 keV). We have tried to understand whether 42Si can have such a low-lying 2+, a feature associated to deformed rotors, without a massive breaking of the Z=14 proton closure, as implied by the MSU result. In order to lower the 2+ excitation energy we have reduced the pairing in the pf -shell orbits 300 keV, to bring the 2+ excitation energy of 36Si to its experimental position. Then we have modiﬁed the d5/2–pf -shell monopoles equally, to obtain a smaller value of the Z=14 gap, 5.8 MeV. With this interaction, 42Si is more clearly an oblate rotor; the 2+ excitation energy is now 810 keV, and the intrinsic quadrupole moment –87 e fm2, corresponding to β=–0.45. The doubly magic N=28, Z=14 component represents only 20% of the ground state wave-function. The ground state has (in average) 2.2 neutrons above N=28 and 1.1 protons above Z=14. This means that the massive breaking of the N=28 shell closure can indeed be achieved with a relatively modest opening of the Z=14 one. N=28: from doubly magic 48Ca toward well deformed drip line 40Mg; The physical picture As we remove protons from doubly magic 48Ca, the N=28 neutron gap slowly shrinks. In 46Ar the collectivity induced by the action of the four valence protons in the quasi-degenerate quasi-spin doublet 1s1/2-0d3/2 is not enough to beat the N=28 closure. 46 Ar comes out oblate non-collective. Notice that, were the N=28 gap wiped out, it would be oblate and collective. This can be easily understood assuming the development of quasi-SU3 in the neutron orbits and pseudo-SU3 in the proton orbits. In 44S, the quadrupole collectivity has already set in. The N=28 closure blows out and prolate and oblate states coexist. The ground state and the ﬁrst excited 2+ form the germ of a prolate rotational band. However, it dies out already at the 4+ state. No regular band appears on top of the oblate 0+ isomer, predicted by the shell model calculations and recently found also at Ganil. N=28: from doubly magic 48Ca toward well deformed drip line 40Mg; The physical picture The scene changes suddenly in 42Si, because the pseudo-spin doublet is not relevant any more. The proton collectivity can only develop promoting particles through the Z=14 closure. As we have seen, even with relatively large values of the gap it does. The collective coupling scheme is now quasi-SU3, and the favored shape oblate. The calculations produce a very regular oblate band that resists up to J=8+. An excited 0+ state, prolate, predicted at 1.3 MeV, does not generate any band. The situation is even better in 40Mg –that should be the heavier magnesium isotope according to some calculations– because now the proton orbit 0d5/2 is open. The calculations produce a very collective prolate rotor, with a deformation similar in absolute value to that of 42Si, a 2+ excitation energy even lower (∼600 keV) and a very regular rotational band up to J=8+. N=28: from doubly magic 48Ca toward well deformed drip line 40Mg; The physical picture In conclusion, the combined eﬀects of the erosion of the semi-magic closures N=28 and Z=14, and the action of the quadrupole interaction, produce a very rich variety of behaviors and shapes in the even N=28 isotones; spherical 48Ca; oblate non-collective 46Ar; coexistence in 44S, and two rotors, oblate 42Si and prolate 40Mg. SHELL MODEL CALCULATIONS OF THE NEUTRINOLESS DOUBLE BETA DECAY The “crisis” of the calculations of the 0ν, ββ nuclear matrix elements The QRPA “explosion” gpp, the miraculous factor Does a good 2ν m.e. guarantee a good 0ν m.e.? To quench or not to quench . . . The quest for better wave functions Quality indicators • Good spectroscopy for parent, daughter and grand- daughter, even better if its extend to a full mass region • GT-strengths and strength functions, 2ν matrix elements, etc. Large scale shell model calculations (LSSM) vs QRPA, the pros and cons • Interaction • Valence space • Pairing • Deformation Update of the 0ν results In the valence spaces r3-g9/2 (76Ge, 82Se) and r4- h11/2 (124Sn, 128−130Te, 136Xe) we have obtained high quality eﬀective interactions by carrying out multi- parametrical ﬁts whose starting point is given by realistic G-matrices. In the valence space proposed for 96Zr, 100Mo, 110Pd and 116Cd, the results are still subject to further improvement mν for T 1 = 1025 y. MGT 0ν 1-χF 2 48 Ca 0.85 0.67 1.14 76 Ge 0.90 2.35 1.10 82 Se 0.42 2.26 1.10 (110Pd) 0.67 2.21 1.15 (116Cd) 0.27 2.49 1.18 124 Sn 0.45 2.11 1.13 128 Te 1.92 2.36 1.13 130 Te 0.35 2.13 1.13 136 Xe 0.41 1.77 1.13 Dependence on the eﬀective interaction The results depend only weakly on the eﬀective interactions provided they are compatible with the spectroscopy of the region. For the lower pf shell we have three interactions that work properly, KB3, FPD6 and GXPF1. Their predictions for the 2ν and the neutrinoless modes are quite close to each other KB3 FPD6 GXPF1 MGT (2ν) 0.083 0.104 0.107 MGT (0ν) 0.667 0.726 0.621 Similarly, in the r3g and r4h spaces, the variations among the predictions of spectroscopically tested interactions is small (10-20%) Learning from the 48Ca →48Ti and the (ﬁctitious) 48Ti →48Cr decays The inﬂuence of deformation Changing adequately the eﬀective interaction we can increase or decrease the deformation of parent, grand- daughter or both, and so gauge its eﬀect on the decays. We have artiﬁcially changed the deformation of 48Ti and 48Cr adding an extra λQ · Q term to the eﬀective interaction. A mismatch of deformation can reduce the ββ matrix elements by factors 2-3. This exercise shows that the eﬀect of deformation is very important and cannot be overlooked The inﬂuence of the spin-orbit partner Similarly, we can increase artiﬁcially the excitation energy of the spin-orbit partner of the intruder orbit. Surprisingly enough, this aﬀects very little the values of the matrix elements, particularly in the neutrinoless case. Even removing the spin-orbit partner completely produces minor changes 48 Ca →48Ti 48 Ti →48Cr MGT (2ν) 0.083 0.213 MGT (0ν) 0.667 1.298 Without spin-orbit partner 48 Ca →48Ti 48 Ti →48Cr MGT (2ν) 0.049 0.274 MGT (0ν) 0.518 1.386 The contributions to the 0ν matrix element as a function of the J of the of the decaying pair : A=82 6 5 82 Se 4 3 2 1 MGT 0 -1 -2 -3 -4 -5 -6 0+ 1+ 2+ - 3+ - 4+ - 5+ - 6+ - 7+ - 8+ - 9+ 0- 1- 2 3 4 5 6 7 8 10 82 8 Se, (9 lev.) 2ν 0ν 6 gpp=0.87: M =0.11, Μ =3.09 0ν 4 M 2 0 -2 0 1 2 3 4 5 6 7 8 9 J The contributions to the 0ν matrix element as a function of the J of the of the decaying pair : A=130 7 6 130 5 Te 4 3 2 1 MGT 0 -1 -2 -3 -4 -5 -6 -7 0+ 1+ 2+ 3+ 4+ 5+ 6+ 7+ 8+ 9+ 10+ 11+ - - - - - - - - - - - 0 1 2 3 4 5 6 7 8 9 10 12 130 10 Te, (13 lev.) 2ν 0ν 8 gpp=0.84: M =0.017, Μ =2.12 0ν 6 M 4 2 0 -2 0 1 2 3 4 5 6 7 8 9 J The multipole structure of the 0ν matrix element The transformation of a two body interaction from the p-p to the p-h form is highly non-unique [(a† a† )J · (a1a2)J ]0 can go to 1 2 † [(a1a1)λ · (a† a2)λ]0 or to 2 † † γ [(a1a2) · (a2a1)γ ]0 We make the choice of keeping all the orderings of the matrix elements even if they are redundant. Notice however that other choices may lead to diﬀerent decompositions, that have the same physical content Our results diﬀer markedly of those of the QRPA calculations The multipole structure of the 0ν matrix element: A=76 1 0.8 76 Ge 0.6 0.4 0.2 MGT 0 -0.2 -0.4 -0.6 -0.8 -1 0+ - 1+ - 2+ - 3+ - 4+ - 5+ - 6+ - 7+ - 8+ - 9+ 0 1 2 3 4 5 6 7 8 The multipole structure of the 0ν matrix element: A=82 1 0.8 82 Se 0.6 0.4 0.2 MGT 0 -0.2 -0.4 -0.6 -0.8 -1 0+ - 1+ - 2+ - 3+ - 4+ - 5+ - 6+ - 7+ - 8+ - 9+ 0 1 2 3 4 5 6 7 8 The multipole structure of the 0ν matrix element: A=130 1 0.8 130 Te 0.6 0.4 0.2 MGT 0 -0.2 -0.4 -0.6 -0.8 -1 0+ 1+ 2+ 3+ 4+ 5+ 6+ 7+ 8+ 9+ 10+ 11+ - - - - - - - - - - 0 1 2 3 4 5 6 7 8 9 10- • Large scale shell model calculations with high quality eﬀective interactions are available or will be in the immediate future for all but one of the neutrinoless double beta emitters • The theoretical spread of the values of the nuclear matrix elements entering in the lifetime calculations is greatly reduced if the ingredients of each calculation are examined critically and only those fulﬁlling a set of quality criteria are retained • A concerted eﬀort of benchmarking between LSSM and QRPA practitioners would be of utmost importance to increase the reliability and precision of the nuclear structure input for the double beta decay processes

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