In an attempt to describe the excitation spectra and transitions strengths of light even-even Cd-isotopes (A = 98 - 108), large-scale shell-model calculations in the proton (2p(1/2), 1g(9/2)) and the neutron (2d(5/2), 1g(7/2), 2d(3/2), 3s(1/2), 1h(11/2)) model space were performed. Preliminary results are presented and discussed

Blazhev, A

Heyde, Kristiaan

Jolie, J

Schmidt, T

EPJ Web of Conferences

A. Blazhev

K. Heyde

J. Jolie

T. Schmidt

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A shell-model study of the light Cd isotopes

In an attempt to describe the excitation spectra and transitions strengths of light even-even Cd-isotopes (A = 98 − 108), large-scale shell-model calculations in the proton (2p1/2, 1g9/2) and the neutron (2d5/2, 1g7/2, 2d3/2, 3s1/2, 1h11/2) model space were performed. Preliminary results are presented and discussed

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A shell-model study of the light Cd isotopesA. Blazhev1,a, K. Heyde2, J. Jolie1, and T. Schmidt11Institut für Kernphysik, Universität zu Köln, D-50937 Cologne, Germany2Department of Physics and Astronomy, Ghent University, B-9000 Ghent, BelgiumAbstract. In an attempt to describe the excitation spectra and transitions strengths of light even-even Cd-isotopes (A = 98 − 108), large-scale shell-model calculations in the proton (2p1/2, 1g9/2) and the neutron(2d5/2, 1g7/2, 2d3/2, 3s1/2, 1h11/2) model space were performed. Preliminary results are presented and discussed.1 IntroductionNuclei in which one type of nucleons (protons or neutrons)constitute a closed shell, are characterized by pairing-dominated energy spectra, in which seniority can be usedas an approximate quantum number. This turns out to bethe case for, e.g., the Sn (Z = 50) and Pb (Z = 82) isotopesand the N = 82 isotones. The typical energy scale is ﬁxedby the pair-breaking energy (energy gap 2Δ) resulting ina quickly increasing level-density at an energy of ∼ 1.5 -2 MeV. Moreover, the energy of the ﬁrst excited 2+ statestays remarkably constant with changing neutron (in theZ = 50 and Z = 82 isotopes) or proton number (in theN = 82 isotones). These results are consistent with theBCS model, or, if the model space allows, with large-scaleshell-model calculations [1, 2].Moving away from the closed shell, adding (or remov-ing) a few valence protons or neutrons, both the excitationenergy of the low-lying 2+1 , 4+1 , 6+1 , .. excited states as wellas corresponding B(E2) values, when known, indicate anonset of quadrupole collectivity.It is an interesting issue to explore in detail how nucleiwith just two protons outside the closed shell (or missing)behave, see, e.g., the Cd nuclei (Z = 48), Te (Z = 52),the Hg (Z = 80) and the Po (Z = 84) isotopes, as well asisotones with N = 80 and N = 84.For the Cd nuclei, an extensive set of experimental datahas been obtained over the years, covering essentially thewhole N = 50−82 neutron major shell, and even going be-yond the N = 82 closed shell, both on low- and high-spinstates, B(E2) values, g-factors, as well as the systematicsfor those data (see refs.[3, 4] and the references therein).See also ref.[5] for a recent review on the structure of 100Snand neighbouring nuclei including the light Cd isotopes.2 Shell-Model CalculationsIt is our aim, using the complete (N = 50 − 82) neutronmodel space, i.e. valence neutrons in the 2d5/2, 1g7/2,ae-mail: a.blazhev@ikp.uni-koeln.de2d3/2, 3s1/2 and 1h11/2 orbitals, and having ten valenceprotons in the 2p1/2 and 1g9/2 orbitals (Z = 48) in the(Z = 38 − 50) proton model space, to perform a large-scale shell-model study (LSSM) to calculate and explorea multitude of nuclear structure properties, in particular,the changing structure with increasing number of valenceneutrons moving outside the N = 50 closed shell. Thenucleon-nucleon (NN) interaction that we use is an eﬀec-tive realistic force which was modiﬁed to exhibit the cor-rect monopole characteristics for the propagation of thesingle-neutron energies over the N = 50 − 82 shell [6],i.e. starting with experimental single particle energies of89Sr at N = 51 and moving towards the end of the neutronshell, where the shell model reproduces well the neutronhole states in 131Sn.An important point is the use of the proton and neu-tron eﬀective charges throughout the full set of Cd nuclei.The procedure used is to ﬁx the proton eﬀective charge eπﬁtting the theoretical B(E2) value for the 8+1 → 6+1 tran-sition to the known experimental value in 98Cd. Havingﬁxed this value, the neutron eﬀective charge eν was ﬁxedby comparing the experimentally known and theoreticallycalculated B(E2) values for the 2+1 → 0+1 transitions in102,104Cd [7]. The eﬀective charges used in the currentwork are eπ = 1.7e and eν = 1.1e, i.e. the same as inthe former shell-model calculations with this interaction[6, 7].With these ingredients kept constant in the LSSMstudy of the Cd nuclei, it is the NN interaction actingin the large model space that produces the nuclear struc-ture properties as a function of increasing neutron num-ber. In particular, we concentrate on a study of the in-terplay of the pairing part of the NN force that prefersbroken pair states to be ordered in some particular se-quence (dictated by the seniority quantum number) versusthe quadrupole component (mainly proton-neutron part ifmany neutrons are present) of the NN force, that prefers togenerate quadrupole collective excitation modes.DOI: 10.1051/C©Owned by the authors, published by EDP Sciences, 2015/0101  (2015)201epjconfEPJ Web of Conferences ,010159933   	                  !66Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/2015930101698Cd02468100Cd0244668102Cd02446688104Cd0246 68106Cd024 2668108Cd02423468021416181021414261628102141 42612343628182830214161626381828302141226162638102141222332424361626381820500100015002000250030003500EnergykeVFigure 1. Experimental (black) and LSSM (red) energy spectra of the even-even light Cd isotopes. Experimental data are taken fromrefs.[3, 4, 6] and the NNDC database [8].The large scale shell model calculations were per-formed with the code ANTOINE [1]. The calculationspresented here were performed in the full model spacewithout any additional truncations. The m-scheme matrixdimension for m=0 in 108Cd was about 108.Figure 1 shows the results of the LSSM compared tothe experimental spectra for the even-even 98−108Cd iso-topes. The lowest 2+ and 4+ states energies are well repro-duced by the shell-model calculations while the higher-spin states show deviations in energy. The calculated ra-tios of the energy of the ﬁrst 4+ and 2+ states are knownas the R(4/2) ratios. As shown in ﬁgure 2, the R(4/2) ra-tio for calculated even Cd isotopes compares well with theexperimental values. The R(4/2) ratio is often used as asignature for the collectivity of the nucleus. Values be-low 2 correspond to single-particle structures, while highervalues of R(4/2) correspond to "collective" structures. Avalue of exactly 2 corresponds to the ratio for a harmonicvibrator, while a value of 3.3 corresponds to the case ofa rigid rotor. At N = 50, 98Cd shows a seniority-type ofspectrum (see ﬁgure 1) which also corresponds to a sin-gle particle R(4/2) ratio (see ﬁgure 2), as expected for anucleus only two proton-holes away from a doubly-magicnucleus, i.e. 100Sn. By increasing the number of valenceneutrons outside the closed N = 50 shell, the experimen-tal R(4/2) ratio raises above the pure vibrational value of2, reaches a maximum of about 2.4 at 108Cd and stabi-lizes afterwards around a value of 2.3. The increase ofcollectivity in the light Cd nuclei as demonstrated by theR(4/2) behaviour also shows up in a steady increase of theelectric quadrupole transition strength between the low-lying yrast states. Starting from 104Cd a clear collectiveyrast band, up to the 6+ state, is present in the calculationswith values of B(E2) decently reproducing known experi-mental values [9]. Therefore we have calculated the evo-Figure 2. Experimental (black squares) and LSSM (red circles)R(4/2) ratios for the light even Cd isotopes.lution of the B(E2) reduced transition probabilities alongthe yrast line, in particular concentrating on the 2+1 → 0+1and 4+1 → 2+1 transitions. For transitions from higher spin6+,8+ states, it is of importance to locate the strongestB(E2; 6+i → 4+1 ) and B(E2; 8+i → 6+j ) transitions in or-der to identify strongly correlated levels when attemptingto join them in a given band structure. Consequently wehave studied in some more detail emerging band structuresfrom the shell-model calculations for 106,108Cd and resultsare in preparation for publication [9].The study of the calculated neutron and proton single-particle orbital partial occupations numbers for the groundstate, 0+1 , as a function of the valence neutron number,increasing from 0 to 10 (98Cd to 108Cd), reveals a “si-multaneous” ﬁlling of the ν2d5/2 and ν1g7/2 orbitals withthe ν2d5/2 orbital being the lower one and respectivelyEPJ Web of Conferences01016-p.2Table 1. Occupation numbers of proton and neutron valenceorbitals in the wave function of selected calculated states in theeven 98−108Cd nuclei. In the table only the occupation numberfor the π2p1/2 proton orbital is given, while the occupationnumber for the other proton valence orbital π1g9/2 can be easilycalculated, i.e. the sum of both is equal to 10. The numbers forthe diﬀerent orbitals are given as a function of the Cd isotopemass number. The occupation numbers for the diﬀerent statesare separated by double horizontal lines.0+1 \ A = 98 100 102 104 106 108π2p1/2 1.76 1.88 1.95 1.97 1.98 1.98ν2d5/2 - 1.09 2.38 3.22 3.82 4.41ν1g7/2 - 0.67 1.11 2.01 3.07 3.94ν2d3/2 - 0.14 0.30 0.45 0.62 0.82ν3s1/2 - 0.07 0.16 0.24 0.33 0.49ν1h11/2 - 0.03 0.05 0.08 0.16 0.340+2 \ A = 98 100 102 104 106 108π2p1/2 0.24 1.75 1.93 1.97 1.98 1.98ν2d5/2 - 0.70 2.44 3.67 4.51 4.17ν1g7/2 - 1.20 1.21 1.63 2.51 3.65ν2d3/2 - 0.03 0.17 0.40 0.52 0.70ν3s1/2 - 0.03 0.13 0.25 0.36 0.58ν1h11/2 - 0.04 0.05 0.05 0.10 0.908+1 \ A = 98 100 102 104 106 108π2p1/2 2.00 2.00 2.00 1.99 1.99 1.99ν2d5/2 - 1.19 2.35 3.31 3.89 3.40ν1g7/2 - 0.57 1.20 2.07 3.26 3.59ν2d3/2 - 0.14 0.27 0.33 0.46 0.62ν3s1/2 - 0.08 0.14 0.26 0.30 0.35ν1h11/2 - 0.02 0.04 0.03 0.09 2.04exhibiting an occupation number of about one neutronmore compared to the close-lying ν1g7/2 orbital, while theoccupation numbers for the other neutron orbitals remainrelatively small (see table 1). This implies a strong mixingof conﬁgurations with neutrons in the ν2d5/2 and ν1g7/2orbitals in the description of the 0+1 state. Similar orbitaloccupation behaviour is found also for other states. It isworth noticing that at 108Cd the 0+2 state suddenly showsan increase of the ν1h11/2 occupation number raising closeto 1 (see table 1), manifesting the onset of the role playedby the 1h11/2 neutron orbital. This eﬀect is even morepronounced for the 8+1 state in108Cd where the trend of anincreasing occupation number for ν2d5/2 is reduced anda value larger than 2 for the ν1h11/2 orbital occupationnumber is observed (see table 1). This can be describedusing the picture of an even number (2,4,...) of neutron“particle-hole” excitations mostly from the ν2d5/2 to theν1h11/2 orbital. An occupation number of more than 2 forthe ν1h11/2 orbital means that the wave function containsmainly conﬁgurations with 2 neutrons in ν1h11/2 orbitaland a minor mixing of conﬁgurations with more than2 neutrons in ν1h11/2 orbital, once again showing theincreasing inﬂuence of the ν1h11/2 orbital for the nuclearstructure as one approaches the middle of the neutronN = 50 − 82 shell.3 Conclusions and outlookIn conclusion, we have performed LSSM cal-culations for the light even-even 98−108Cd iso-topes in the proton (2p1/2, 1g9/2) and the neutron(2d5/2, 1g7/2, 2d3/2, 3s1/2, 1h11/2) model space. The en-ergy spectra describing the lowest yrast states are wellreproduced over the whole calculated range, while thehigher-lying yrast and non-yrast states show deviationscomparing with the experimental data. By comparing thecalculated transition strengths and branching ratios withexperimental ones, it is possible to associate a speciﬁcshell-model state with a corresponding experimentalone, neglecting the order of the speciﬁc state within thecalculated set of states [9]. The increasing collectivityin the light Cd isotopes, when moving away from theN = 50 closed shell, is well reproduced. Shell-modeloccupation numbers are discussed and the onset of ﬁllingof the neutron ν1h11/2 orbital is noticed for selected statesin 108Cd. Since the present approach contains the fullmodel space, as discussed before, both collective modesas well as multiple broken-pair states (in particular thehigh-spin states 8+,10+,12+,...) can result within a singleframework. The results will be presented in a forthcomingpaper [9].Financial support from the Interuniversity AttractionPoles Programme of the Belgian State-Federal Oﬃce forScientiﬁc and Cultural Aﬀairs (IAP Grant P7/12) is ac-knowledged.References[1] E. Caurier and F. Nowacki, Act. Phys. Pol. B 30, 705(1999)[2] E. Caurier, G. Martínez-Pinedo, F. Nowacki, A. Povesand A. P. Zuker, Rev. Mod. Phys. 77, 427 (2005)[3] P. Garrett and J. L. Wood, J. Phys. G 37, 064028(2010), and corrigendum ibid 069701[4] K. Heyde and J. L. Wood, Revs. Mod. Phys. 83, 1467(2011)[5] T. Faestermann, M. Górska, and H. Grawe, Prog. Part.Nucl. Phys. 69, 85 (2013)[6] N. Boelaert, N. Smirnova, K. Heyde and J. Jolie, Phys.Rev. C 75, 014316 (2007)[7] N. Boelaert, A. Dewald, C. Fransen, J. Jolie, A. Lin-nemann, B. Melon, A. Moller, N. Smirnova, andK. Heyde, Phys. Rev. C 75, 054311 (2007)[8] http://www.nndc.bnl.gov/ensdf/[9] A. Blazhev et al., in preparation.CGS1501016-p.3

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A shell-model study of the light Cd isotopes

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