High-spin states with seniority v=4,5,6 in {}^{119-126}Sn

High-spin states with seniority in Sn

A. Astier    M.-G. Porquet CSNSM, CNRS/IN2P3 and Université Paris-Sud, Bât 104-108, F-91405 Orsay, France    Ch. Theisen CEA, Centre de Saclay, IRFU/Service de Physique Nucléaire, F-91191 Gif-sur-Yvette Cedex, France    D. Verney IPNO, CNRS/IN2P3 and Université Paris-Sud, F-91406 Orsay, France    I. Deloncle CSNSM, CNRS/IN2P3 and Université Paris-Sud, Bât 104-108, F-91405 Orsay, France    M. Houry Present address: CEA/DSM/Département de recherches sur la Fusion Contrôlée, F-130108 Saint-Paul lez Durance, France CEA, Centre de Saclay, IRFU/Service de Physique Nucléaire, F-91191 Gif-sur-Yvette Cedex, France    R. Lucas CEA, Centre de Saclay, IRFU/Service de Physique Nucléaire, F-91191 Gif-sur-Yvette Cedex, France    F. Azaiez Present address: IPNO, IN2P3-CNRS and Université Paris-Sud, F-91406 Orsay, France IPHC, IN2P3-CNRS and Université Louis Pasteur, F-67037 Strasbourg Cedex 2, France    G. Barreau CENBG, IN2P3-CNRS and Université Bordeaux I, F-33175 Gradignan, France    D. Curien IPHC, IN2P3-CNRS and Université Louis Pasteur, F-67037 Strasbourg Cedex 2, France    O. Dorvaux    G. Duchêne IPHC, IN2P3-CNRS and Université Louis Pasteur, F-67037 Strasbourg Cedex 2, France    B.J.P. Gall IPHC, IN2P3-CNRS and Université Louis Pasteur, F-67037 Strasbourg Cedex 2, France    N. Redon IPNL, IN2P3-CNRS and Université Claude Bernard, F-69622 Villeurbanne Cedex, France    M. Rousseau IPHC, IN2P3-CNRS and Université Louis Pasteur, F-67037 Strasbourg Cedex 2, France    O. Stézowski IPNL, IN2P3-CNRS and Université Claude Bernard, F-69622 Villeurbanne Cedex, France
September 25, 2019
Abstract

The Sn nuclei have been produced as fission fragments in two reactions induced by heavy ions: C+U at 90 MeV bombarding energy, O+Pb at 85 MeV. Their level schemes have been built from gamma rays detected using the Euroball array. High-spin states located above the long-lived isomeric states of the even- and odd-A Sn nuclei have been identified. Moreover isomeric states lying around 4.5 MeV have been established in Sn from the delayed coincidences between the fission fragment detector SAPhIR and the Euroball array. The states located above 3-MeV excitation energy are ascribed to several broken pairs of neutrons occupying the orbit. The maximum value of angular momentum available in such a high-j shell, i.e. for mid-occupation and the breaking of the three neutron pairs, has been identified. This process is observed for the first time in spherical nuclei.

pacs:
25.70.Jj, 27.60.+j, 23.20.-g, 21.60.Cs

I Introduction

Experimental and theoretical investigations of the structure of the Sn nuclei have been the subject of much interest during the last decades. Their low-lying states are textbook examples of shell model approaches, as their description only involves excitations of a few neutrons, all along the chain of known isotopes, Sn.

The high-spin states of Sn nuclei with A can be populated by fusion-evaporation reactions induced by heavy ions. Such experiments, performed many years ago, mainly led to the identification of collective rotational bands up to spin  br79 . These bands are built on ’intruder’ configurations, i.e. two-particle two-hole excitations across the closed shell. At mid neutron shell, this configuration is low in energy, thus the collective rotational band being yrast in the mass range 110-118, dominates the high-spin level schemes. Since this is no longer the case for A 120, the yrast states of the heavy Sn isotopes are expected to be only due to excitations of neutrons moving in a spherical well, particularly the states due to the breaking of several pairs in the orbit. Unfortunately because of the lack of suitable stable projectile-target combinations, high-spin states of heavy Sn isotopes cannot be populated by fusion-evaporation reactions. Thus up to now, medium spin states of the Sn isotopes were only measured up to spin for the even mass and for the odd mass, by using reactions induced by light ions, deep inelastic reactions, isomeric decays of long-lived states of Sn produced by fission of actinides, or -decays of the high-spin long-lived states of heavy In da86 ; ma94 ; br92 ; pi00 ; zh00 ; lo08 ; fo81 . During the completion of the present work, the decay of a new isomeric state in Sn populated in the fragmentation of Xe has been reported pi11 , which has been proposed to be the 15 state expected from the configuration.

For the studies presented in this paper, the Sn isotopes have been produced as fragments of binary fission induced by heavy ions. We have selected two fusion-fission reactions in order to identify unambiguously the -rays emitted by the high-spin states of these nuclei. Moreover angular correlations have been analyzed in order to assign spin and parity values to most of these states. In addition, new isomeric states lying around 4.5 MeV have been established in Sn from the delayed coincidences between fission fragment detectors and the gamma array. All the observed states can be described in terms of broken neutron pairs occupying the orbit. The maximum value of angular momentum available in this high-j shell, i.e. for mid-occupation and the breaking of the three pairs, has been identified.

Ii Experimental methods and data analysis

ii.1 Reactions and -ray detection

The C + U reaction was studied at 90 MeV incident energy. The beam was provided by the Legnaro XTU tandem accelerator. The 47 mg/cm target of U was thick enough to stop the recoiling nuclei. The second reaction, O + Pb at 85 MeV beam energy, was studied at the Vivitron accelerator of IReS (Strasbourg). The thickness of the target was 100 mg/cm. In these two experiments, the gamma-rays were detected with the Euroball array consisting of 71 Compton-suppressed Ge detectors si97 (15 cluster germanium detectors placed in the backward hemisphere with respect to the beam, 26 clover germanium detectors located around 90, and 30 tapered single-crystal germanium detectors located at forward angles). Each cluster detector is composed of seven closely packed large-volume Ge crystals eb96 and each clover detector consists of four smaller Ge crystals du99 . The data were recorded in an event-by-event mode with the requirement that a minimum of five (three) unsuppressed Ge detectors fired in prompt coincidence (within a time window of 50 ns) during the first (second) experiment. About 1.910 (410) coincidence events (within a time window of 300 ns) with a multiplicity greater than or equal to three were registered. The offline analysis consisted of both multi-gated spectra and several three-dimensional ”cubes” built and analyzed with the Radware package ra95 .

ii.2 Isomer selection

To identify new isomeric states in fission fragments, we have performed another experiment using a fission fragment detector to trigger the Euroball array and isolate the delayed -ray cascades. The heavy-ion detector, SAPhIR111SAPhIR, Saclay Aquitaine Photovoltaic cells for Isomer Research., is made of many photovoltaic cells which can be arranged in several geometries th98 . In the present work, it consisted of 32 photovoltaic modules laying in four rings around the target. We have used the C + U reaction at 90 MeV with a thin target, 0.14 mg/cm. Fragments escaping from the target are stopped in the photovoltaic cells of SAPhIR. The detection of the two fragments in coincidence provides a clean signature of fission events. The Euroball time window was [50 ns–s], allowing detection of delayed -rays emitted during the de-excitation of isomeric states.

Time spectra between fragments and -rays were analyzed in order to measure the half-life of isomeric levels. The FWHM of the time distribution for prompt -rays was around 15 ns. In this experiment, new isomeric states were found in Sn nuclei, which will be detailed below.

ii.3 Identification of new -ray cascades

The fusion-fission channel of the above-mentioned reactions leads to the production of the high-spin states of 150 fragments, mainly located on the neutron-rich side of the valley of stability. This gives several thousands of transitions which have to be sorted out. Single-gated spectra are useless in the majority of cases. The selection of one particular nucleus needs at least two energy conditions, implying that at least two transitions have to be known.

The identification of transitions depopulating high-spin levels which are completely unknown is based on the fact that prompt -rays emitted by complementary fragments are detected in coincidence ho91 ; po96 . For each reaction used in this work, we have studied the intensities of -rays emitted by many pairs of complementary fragments with known cascades to establish the relationship between their number of protons and neutrons. The sum of the proton numbers of complementary fragments has been found to be the atomic number of the compound nucleus222In the C + U reaction, we have also identified a weak exit channel: Few pairs of fragments having , instead of 98, indicates a fission process occuring after the transfer of a few nucleons., so that the Sn isotopes are associated to the Cd isotopes in the C + U reaction and to the Zr isotopes in the O + Pb reaction. The number of evaporated neutrons (sum of the pre- and post-fission emitted neutrons) extends from 7 to 14 in the first reaction ho99 ; ho00 and from 2 to 7 in the second one lu02 ; po04a . The distribution and the mean number of emitted neutrons depend slightly on the ratio of the investigated fragments and on the angular momentum of their excited states emitting the -rays. Primary fragments populated at high excitation energy cool down predominantly by neutron evaporation, then the secondary fragments emit -rays. Therefore the number of emitted neutron must be low in order to observe -ray cascades in the most neutron-rich isotopes.

Many new -ray cascades of the Sn nuclei have been identified using the distribution of masses of their partners, as explained below. It is worth noting that the use of two different reactions to produce the various Sn isotopes has turned out to be essential to disentangle the coincidence relationships which are often complicated by the existence of many doublets or triplets of transitions very close in energy.

The relative intensity of the lowest transitions in the new cascades identified in Sn have been measured in the spectra in double coincidences with one new transition and one transition of a partner. As for the other transitions, we have used spectra in double coincidences with two transitions of the new cascades. A loss in intensity occurs when going through an isomeric state. Knowing that the time window was 300 ns for the two experiments, such an effect has been taken into account for the half-lives in the 100-300 ns range, observed in the present work.

ii.4 - angular correlations

In order to determine the spin values of excited states, the coincidence rates of two successive transitions are analyzed as a function of , the average relative angle between the two fired detectors. The Euroball spectrometer had  = 28441 combinations of 2 crystals, out of which only 2000 involved different values of relative angle within 2. Therefore, in order to keep reasonable numbers of counts, all the angles have been gathered around three average relative angles : 22, 46, and 75.

The coincidence rate is increasing between 0 and 90 for the dipole-quadrupole cascades, whereas it decreases for the quadrupole-quadrupole or dipole-dipole ones. More precisely, the angular correlation functions at the three angles of interest were calculated for several combinations of spin sequences, corresponding to typical multipole orders (see table 1). In order to check the method, angular correlations of transitions belonging to the yrast cascades of the fission fragments having well-known multipole orders were analyzed and the expected values were found in all cases.

Spin sequence Multipole R(22) R(46) R(75)
orders
14 – 12 – 10 Q - Q 1.13 1.06 1.00
12 – 11 – 10 D - D 1.06 1.03 1.00
13 – 12 – 10 D - Q 0.92 0.96 1.00
Table 1: Values of the angular correlation functions, R(), normalized to the ones calculated at 75, computed for several combinations of spin sequences and multipole orders (Q = quadrupole, D = Dipole).

When the statistics of our data are too low to perform such a measurement, the spin assignments are based upon (i) the already known spins of some states, (ii) the assumption that in yrast decays, spin values increase with the excitation energy, (iii) the possible existence of cross-over transitions, and (iv) the analogy with the level structures of the other isotopes.

Iii Experimental results

The -rays emitted by the low-lying states of Sn isotopes have been observed in both fusion-fission reactions used in the present work. As for Sn, its yield is so low as only the decay of its 7 state is observed and the transitions of its partners could not be identified. On the other hand, we have measured many new -rays emitted by the high-spin states of Sn, the results are presented in the three following sections.

iii.1 New -ray cascades of Sn isotopes

Up to now, the high-spin level schemes of Sn isotopes could only be established up to the long-lived isomeric states, with I = 10 or 27/2 at about 2-3 MeV excitation energy, since it is difficult to register coincidence relationships between the transitions populating and depopulating states having such half-life values (T) using standard experimental apparatus. On the other hand, in the fusion-fission experiments, all the -ray cascades located above the long-lived isomeric states of the Sn isotopes are easily detected in prompt coincidences with those emitted by their complementary fragments. Three steps are necessary to carry out the search:

  • to find all the rays emitted in prompt coincidences with the transitions of the Cd fragments in the first data set and the Zr fragments in the second one, which do not belong to their respective level schemes.

  • to build all the -ray cascades using their own coincidences.

  • to assign each -ray cascade to one particular Sn isotope, from the distribution of masses of their partners.

The first item is illustrated in Fig. 1(a), gated by the first two transitions of Cd (487 and 676 keV), built from the data set of the C + U reaction.

Figure 1: (Color online) (a) Spectrum of rays detected in coincidence with the first two transitions of Cd (487 and 676 keV), built from the data set of the C + U reaction, in the [950-1250 keV] energy range. All the transitions but two ones, emitted by Cd, belong to its complementary fragments, Sn. They are labelled by their mass A, an asterisk being added when the transition belongs to a new cascade located above a long-lived isomeric state. (b) Spectrum of rays detected in coincidence with the first transition of Cd (487 keV) and the new transition at 1190 keV. Transitions emitted by Cd are labelled by Cd. (c) Spectrum of rays detected in coincidence with the two new transitions at 1190 keV and 241 keV. Transitions emitted by the Cd complementary fragments are labelled by their mass.

Besides two transitions already known in Cd in the [950-1250-keV] energy range, a lot of new transitions are observed and can be assigned to Sn, half of them being known to belong to the cascades built on their ground state (they are labelled by their mass A). Then in order to find the -ray cascades comprising the other transitions, all the spectra in double coincidence with one transition of Cd and one new transition are precisely analyzed. For instance in Fig. 1(b), two new lines at 241 keV and 557 keV are found to be correlated to the new 1190-keV transition and the 487 keV -ray of Cd. Finally, Fig. 1(c) allows to complete the search of a new -ray cascade: It comprises five transitions at 1190 keV, 557 keV, 241 keV, 782 keV and 639 keV. In addition, the fact that this third spectrum shows the transitions emitted by several Cd complementary fragments confirms that this cascade belongs to the level scheme of one Sn isotope. Using the same procedure for each Cd isotope in the first experiment, as well as for Zr in the second one, several new -ray cascades have been observed in the present work.

To assign every -ray cascade to one particular Sn isotope, we have first analyzed the relative intensities of -rays emitted by the partners in each spectrum in double coincidence with two transitions of the new cascades (such as the spectrum shown in Fig. 1(c)). Then we have analyzed all these distributions of masses of the partners. Examples of results obtained in the first experiment are shown in Fig. 2.

Figure 2: (Color online) Relative yields of even- Cd isotopes associated to the new -ray cascades emitted by the Sn isotopes produced in the C + U reaction. The yield of each even- Cd isotope is computed from the number of counts of its line in the spectra gated by the two strongest transitions of the new cascades (their energies are written in each drawing). (a) case of the new cascade assigned to Sn, (b) Sn, (c) Sn, (d) Sn, (e) Sn, (f) Sn. The new cascades assigned to Sn and Sn are discussed in Sec. III.3.1.

The evolution of the Cd yields from the top drawing to the bottom one proves that the six cascades belong to different Sn isotopes. Similar results are obtained from the evolution of the Zr yields associated to each cascade, in the second experiment. Finally these relative distributions of Cd/Zr masses were compared to well known pairs of complementary fragments and the six new cascades discussed in Fig. 2 were assigned to Sn. Their first transitions are marked with an asterisk in the spectrum of Fig. 1(a). Their precise location in the Sn level schemes are discussed in the next sections.

iii.2 Study of the even- Sn isotopes

Very few medium-spin levels were known in the even- Sn isotopes prior to this work. Populated in deep inelastic reactions br92 ; zh00 , two long-lived isomeric states were identified from their -decays to the low-lying states. Lying between 2 and 3 MeV excitation energy, these isomeric states are due to the breaking of one neutron pair, the state with being attributed to a configuration and the one with to a configuration. Using the data of the two fusion-fission reactions of the present work, we have identified the yrast structures of Sn located above their long-lived isomeric states. In the following, we first present the building of each high-spin level scheme and the measurement of isomeric states in the [30-300 ns] range. Then, we discuss the angular momentum and parity assignments of most of the new states of Sn.

iii.2.1 Sn

A cascade starting with the 1190-keV and 557-keV transitions is assigned to Sn, because of the mass distribution of its complementary fragments in the two fusion reactions (Sec. III.1). Besides the three other transitions shown in Fig. 1(b) and (c), namely the 241-keV, 782-keV and 639-keV -rays, a few other -lines have been found to belong to the cascade, thanks to their coincidence relationships. The whole set forms two linked branches which are put on the top of the 10 and 7 states respectively, since two parallel decay paths located just below the 241-keV transition, 557+1190 on the one hand and 253+661+1253 on the other hand, exactly fits the difference in energy between the 10 and 7

Figure 3: (Color online) Level scheme of Sn deduced in the present work. The colored levels are new. The two long-lived isomeric states (s and 11.8(5) s) and their -decays to the low-lying states were already known NNDC . The 355-keV transition, located between two long-lived isomeric states, as well as the very converted 66-keV transition, could not be observed in our work. The width of the arrows is representative of the relative intensity of the rays above the isomeric states.
E(keV)111The number in parenthesis is the error in the last digit. I111The number in parenthesis is the error in the last digit.222The relative intensities are normalized to the sum . JJ E(keV) E(keV)
241.1(2) 95(14) 15 13 4890.1 4649.0
252.9(2) 36(7) 13 11 4649.0 4396.0
453.7(3) 9(3) 16 14 5672.9 5219.3
556.5(2) 83(12) 13 12 4649.0 4092.5
639.0(3) 14(4) (18) 16 6311.9 5672.9
661.1(3) 14(4) 11 9 4396.0 3734.6
782.8(3) 29(6) 16 15 5672.9 4890.1
1127.0(4) 17(4) 14 12 5219.3 4092.5
1190.3(3) 100 12 10 4092.5 2902.2
1253.0(4) 14(4) 9 7 3734.6 2481.6
1350.8(5) 5(2) (17) 15 6240.9 4890.1
1493.8(4) 22(5) 11 10 4396.0 2902.2
Table 2: Properties of the new transitions assigned to Sn in this experiment. The energies of the two long-lived isomeric states at 2481.6 keV (I) and 2902.2 keV (I) (written in bold) are from Ref. NNDC .

isomeric states of Sn (see Fig. 3). All the transitions newly observed in Sn are given in Table 2. The spin and parity of the new states will be discussed and assigned in Sec. III.2.5.

Using the data from the SAPhIR experiment, the transitions involved in the de-excitation of the 4890-keV level have been found to be delayed. The spectrum of -rays which have been detected in the time interval 50 ns-s after the detection of two fragments by SAPhIR and in prompt coincidence with the 241/242 keV transition is drawn in Fig. 4(a). As this line is a triplet, the spectrum exhibits transitions emitted by the isomeric states of three fission fragments, Rb po09 , Sn and Sn. The spectrum shown in Fig. 4(b) is gated by the 253 keV -ray. This line is a doublet, as this energy occurs both in the decay of the 7 isomeric state of Sn (associated with the 1050- and 1230-keV transitions) and in the decay of the isomeric state at 4890 keV, newly established in Sn. The statistics of this spectrum is too low to show the 1253-keV transition, identified in the thick-target Euroball experiment, from double-gated spectra having higher statistics.

Figure 4: Spectra of -rays which have been detected in the time interval 50 ns-s after the detection of two fragments by SAPhIR. (a) -rays in prompt coincidence with the 241/242 keV transition (Sn and Sn). The 458- and 648-keV transitions are pollutions (they belong to the decay of the isomeric 7 state of Rb po09 ). (b) -rays in prompt coincidence with the 253 keV transition, which is a doublet. The 1050- and 1230-keV -rays are the first two transitions of Sn and the 241-, 661-, and 1494-keV ones are assigned to Sn. (c) -rays in prompt coincidence with the 264 keV transition (Sn).

The time distribution between the detection of two fragments by SAPhIR and the emission of the 1190 keV or the 557 keV -ray is shown in Fig. 5(a). In order to reduce the background, we have selected the events containing an additional -ray belonging to the 241-557-1190 cascade. Several least-squares fits of this spectrum were performed, varying the time interval from 70 to 100 ns and shifting the smallest intervals along the time axis. The average of all obtained values is = 36(4) ns, the adopted uncertainty being the observed dispersion (greater than the uncertainties quoted for each fit).

Figure 5: (Color online) (a) Half-life of the 4890 keV state of Sn obtained from the sum of the time distributions of the 1190- and 557-keV transitions. (b) Half-life of the 4720 keV state of Sn obtained from the time distribution of the 1103-keV transition. See text for further details about the gating conditions and procedures.

iii.2.2 Sn

As for Sn, the starting point is the two transitions at 1103 keV and 609 keV which have to be placed above its long-lived isomeric states, because of the mass distribution of its partners in the two fusion reactions (Sec. III.1). Several new transitions are observed in coincidence with the 1103 keV and 609 keV -rays. All the observed relationships result in two linked branches which are placed above the the 10 and 7 isomeric states of Sn, since two parallel decay paths located just below the 242 keV transition, 609+1103 on the one hand and 264+680+1125 on the other hand, exactly fits their difference in energy (see Fig. 6). Examples of coincidence spectra of lines belonging to Sn are given in Fig. 4(a) and (c), which demonstrate that all the transitions involved in the de-excitation of the 4720-keV level are delayed. The time distribution between the detection of two fragments by SAPhIR and the emission of the 1103-keV -ray gives = 146(15) ns. In order to reduce the background we have selected the events containing either the 609-keV transition or the 242-keV one (see Fig. 5(b)).

Figure 6: (Color online) Level scheme of Sn deduced in the present work. The colored levels are new. The two long-lived isomeric states (s and 62(3) s) and their -decays to the low-lying states were already known NNDC . The 281 keV transition, located between two long-lived isomeric states, as well as the very converted 75 keV transition could not be observed in our work. The width of the arrows is representative of the relative intensity of the rays above the isomeric states.

All the transitions newly observed in Sn are given in Table 3. The spin and parity of the new states will be discussed and assigned in Sec. III.2.5.

E(keV)111The number in parenthesis is the error in the last digit. I111The number in parenthesis is the error in the last digit.222The relative intensities are normalized to the sum . JJ E(keV) E(keV)
242.1(2) 52(9) 15 13 4720.5 4478.4
263.8(4) 11(3) 13 11 4478.4 4214.4
487.3(3) 14(4) 16 14 5386.7 4899.4
609.3(2) 62(12) 13 12 4478.4 3869.1
665.9(3) 17(4) 16 15 5386.4 4720.5
680.3(4) 7(3) 11 9 4214.4 3534.0
696.3(3) 14(4)      (17) 6650.8 5954.5
835.9(4) 6(3) (18) 16 6222.6 5386.7
1030.3(3) 38(7) 14 12 4899.4 3869.1
1103.5(3) 100 12 10 3869.1 2765.6
1125.0(4) 7(3) 9 7 3534.0 2409.0
1234.0(4) 21(5) (17) 15 5954.5 4720.5
1448.9(4) 4(2) 11 10 4214.4 2765.6
Table 3: Properties of the new transitions assigned to Sn in this experiment. The energies of the two long-lived isomeric states at 2409.0 keV (I) and 2765.6 keV (I) (written in bold) are from Ref. NNDC .

iii.2.3 Sn

Similar procedures were used to identify the high-spin structures of Sn. By gating on the two coincident transitions at 1047 keV and 620 keV assigned to Sn because of the mass distribution of its partners in the two fusion reactions (Sec. III.1), we found the 229- and the 1178-keV -rays, establishing a cascade of 4 transitions. Using the data from the SAPhIR experiment, the first three transitions have been found to be delayed. The time distribution between the detection of two fragments by SAPhIR and the emission of the 1047- or the 620-keV -ray gives = 260(25) ns. In order to reduce the background, we have selected the events containing an additional -ray of the 229-620-1047 cascade. Since the 229-620-1047 sequence resembles well the 242-609-1103 sequence of Sn and the 241-557-1190 one of Sn, we have placed the cascade of Sn directly above its 10 state at 2657 keV excitation energy (see Fig. 7).

Figure 7: (Color online) Level scheme of Sn deduced in the present work. The colored levels are new. The two long-lived isomeric states (s and 45(5) s) and their -decays to the low-lying states were already known NNDC . The very converted 78 keV and 120 keV transitions could not be observed in our work. The width of the arrows is representative of the relative intensity of the rays above the isomeric states.

Moreover, we have identified other new transitions in Sn by analyzing the spectra in double coincidence with the 1047-keV ray and one transition of its main complementary fragments (either Cd or Zr). An example of coincidence spectrum doubly-gated on some of the new transitions is shown in Fig. 8.

Figure 8: (Color online) Coincidence spectrum double-gated on the 1047-, 996-, and 490-keV transitions of a new cascade identified in Sn, built from the O + Pb data set. The -rays emitted by the Zr complementary fragments are labelled by their masses A, written in italics.

The resulting cascade resembles well those assigned to the lighter isotopes, but the link such as the 666-keV transition (in Sn) and the 782-kev one (in Sn), which is not observed any more.

We have looked for a second branch directly linked to the 7 isomeric state, as measured in Sn. In the spectrum gated by the 229-keV line (SAPhIR experiment), a 252-keV transition is observed, as well as a weak 377-keV transition, these two transitions could be part of the foreseen cascade ending to the 7 state. Unfortunately, no high energy transition could be unambiguously seen in that spectrum, allowing us to complete the cascade. Nevertheless, using the Euroball data sets, we identified a cascade of three transitions (253 keV, 377 keV, and 1116 keV) which does belong to Sn since they are detected in coincidence with the complementary fragments expected in the two fusion reactions, namely Cd and Zr. Then we have assumed that the 253 kev transition measured in the present work is the one decaying the 8 state towards the 7 isomeric state NNDC . All these arguments would lead to a second decay path of the 4323 keV state, 252+377+1116+253. It is worth mentioning that the number of counts of the 253-1116-377 events measured in the Euroball data sets is very high as compared to the very weak number of events containing the 252-keV transition located just above the 4071-keV state. This could be due to a large side feeding of that state. Since some intensities of the coincidence events establishing that second cascade are very weak, we have chosen to draw it with dotted lines in Fig. 7.

All the transitions newly observed in Sn are given in Table 4. The spin and parity of the new states will be discussed and assigned in Sec. III.2.5.

E(keV)111The number in parenthesis is the error in the last digit. I111The number in parenthesis is the error in the last digit.222The relative intensities are normalized to the sum . JJ E(keV) E(keV)
228.5(4) 13(4) 15 13 4551.8 4323.2
251.7(4) 4(2) 13 11 4323.2 4071.4
253.2(3) 22(7) 8 7 2578.4 2325.0
377.3(3) 7(3) 11 9 4071.4 3694.1
490.2(3) 39(6) 16 14 5189.4 4699.2
619.8(2) 43(7) 13 12 4323.2 3703.4
762.7(3) 25(5) (18) 16 5952.1 5189.4
995.8(3) 57(9) 14 12 4699.2 3703.4
1024.3(4) 11(3)      (18) 6976.4 5952.1
1046.8(3) 100 12 10 3703.4 2656.6
1115.7(3) 13(4) 9 8 3694.1 2578.4
1177.7(5) 5(2) (17) 15 5729.4 4551.8
Table 4: Properties of the new transitions assigned to Sn in this experiment. The energies of the two long-lived isomeric states at 2325.0 keV (I) and 2656.6 keV (I), as well as this of the 8 state, (written in bold) are from Ref. NNDC .

iii.2.4 Sn

The two most intense transitions of the new cascade assigned to Sn have energies of 1030 keV and 570 keV (see Sec. III.1). The doubly-gated spectrum exhibits new transitions at 180 keV, 1149 keV and 762 keV.

Figure 9: (Color online) Level scheme of Sn deduced in the present work. The colored levels are new. The two long-lived isomeric states and their -decays to the low-lying states were already known NNDC . The 269-keV transition, located between two long-lived isomeric states, as well as the very converted 76- and 57-keV transitions, could not be observed in our work. The width of the arrows is representative of the relative intensity of the rays above the isomeric states.

Moreover the data of the SAPhIR experiment indicate that the cascade comprising the first three transitions, at 1030 keV, 570 keV, and 180 keV, is delayed. Using the same procedures as previously described, we have measured the half-life, T160(20) ns. Moreover the spectrum doubly-gated by the 1030-keV line and one transition of a complementary fragment reveals other rays belonging to Sn, such as 986 keV, 477 keV and 777 keV, which form another cascade. Thus Sn exhibits two parallel structures having one transition in common, 1030 keV. In comparison to the results obtained in the lighter isotopes, we have placed it directly above the 10 isomeric state (see Fig. 9). Then we have looked for a second decay path of the new isomeric state ending to the 7 state at 2219 keV, but we have found no candidate in our data sets. This is likely due to the lack of statistics, Sn being in the heavy-A tail of the Sn fragment distribution.

Up to now, no 6 state was measured in Sn, whilst such a state is expected to lie below the 8 state, as in the other even- isotopes. The E2 decays of the 8 and 6 states are hindered because of their low energy, thus these levels decay by means of E1 transitions towards negative parity states, 7 and 5 respectively (see Figs. 6 and 7). In the present work, we have observed the 6 5 transitions of Sn in coincidence with the first three transitions of their level schemes. Thus we have looked for the 6 5 ray in Sn, by analyzing the spectra doubly-gated by its first transitions (at 1141 keV, 909 keV, and 112 keV). Figure 10 shows two spectra, built from the O + Pb data set and using similar conditions for Sn and Sn. All the observed lines are assigned to the decay of known states of either Sn (Sn) or their Zr complementary fragments, except the new line at 212 keV, which is assigned to the decay of the 6 state of Sn (see Fig. 9).

Figure 10: (Color online) Coincidence spectra double-gated on the first two transitions of Sn (a) and Sn (b), built from the O + Pb data set. The -rays emitted by Sn are labelled by the spin and parity values of the decaying states, and those emitted by the Zr complementary fragments are labelled by their masses A, written in italics.

Noteworthy is the fact that 212 keV is also the energy of the 2 0 transition of Zr. The observation of prompt coincidence between a ray at this energy and the first transitions of Sn could have been interpreted as the fact that Zr is a complementary fragment of Sn, meaning that the compound nucleus of the O + Pb reaction, Th, may fission before emitting any neutron. Such process has never been observed, as expected since neutron emission from an excited compound nucleus is always faster than the fission. The coincidence relationships measured in the C + U data set corroborate the location of the 212-keV line in the Sn level scheme, which rules out a misinterpretation of the 212-keV line.

All the transitions newly observed in Sn are given in Table 5. The spin and parity of the new states will be discussed and assigned in Sec. III.2.5.

E(keV)111The number in parenthesis is the error in the last digit. I111The number in parenthesis is the error in the last digit.222The relative intensities are normalized to the sum . JJ E(keV) E(keV)
180.5(3) 36(7) 15 13 4345.7 4165.2
211.7(4) 12(4) (6) 5 2373.2 2161.5
476.7(3) 28(7) 16 14 5057.9 4581.2
570.5(3) 56(11) 13 12 4165.2 3594.7
761.7(4) 8(3)      (17) 6256.9 5495.2
777.1(4) 9(3) (18) 16 5835.0 5057.9
986.5(3) 44(9) 14 12 4581.2 3594.7
1030.2(3) 100 12 10 3594.7 2564.5
1149.5(5) 13(4) (17) 15 5495.2 4345.7
Table 5: Properties of the new transitions assigned to Sn in this experiment. The energies of the long-lived isomeric state at 2564.5 keV (I)and of the 5 state at 2161.5 keV (written in bold) are from Ref. NNDC .

iii.2.5 Angular momentum and parity values of the high-spin states of Sn

Given that the new high-spin structures of Sn reported in the previous sections are very close to each other, we assume that the similar states in the four level schemes have the same spin and parity values.

First, we have extracted the internal conversion coefficients of the isomeric transitions of Sn (at 241 keV, 242 keV, 229 keV, and 180 keV respectively) by analyzing the relative intensities of transitions in cascade. The intensity imbalances of the 241-, 242-, and 229-keV rays measured in spectra in double coincidence with at least one transition located above them in their respective level scheme lead to . This is consistent with , , or assignment. On the other hand, the intensity imbalance of the 180-keV ray of Sn gives , in good agreement with the theoretical value for multipolarity,  keV) = 0.20 BRICC . Assuming that the nature of the former transitions are also , we have calculated all the values, which are reported in Table 6. One has to note that these values have the same order of magnitude as those already measured for the isomeric decay of some lower energy states, either in the even- isotopes or in the odd-A ones. This will be discussed in Sec. IV.

Nucleus T111The number in parenthesis is the error in the last digit. 111The number in parenthesis is the error in the last digit. 111The number in parenthesis is the error in the last digit.
keV keV ns W.u.
Sn 4890.1 241.1 36(4) 18(2) 0.51(5)
Sn 4720.5 242.1 146(15) 4.4(4) 0.12(1)
Sn 4551.8 228.5 260(25) 3.2(3) 0.09(1)
Sn 4345.7 180.5 160(20) 16(2) 0.42(5)
Table 6: Properties of the new isomeric states of Sn

Secondly, we have analyzed the angular correlations of strongest transitions. The experimental results are given in Table 7.

E-E R(22)111The number in parenthesis is the error in the last digit. R(46)111The number in parenthesis is the error in the last digit. R(75)111The number in parenthesis is the error in the last digit.
Sn 1190 - 557 0.87(9) 0.95(7) 1.00(5)
1190 - 241 1.1(1) 1.12(8) 1.00(5)
Sn 1103 - 1030 1.17(9) 1.07(7) 1.00(5)
1103 - 609 0.8(1) 0.96(7) 1.00(5)
1103 - 242 1.2(1) 1.14(8) 1.00(5)
Sn 1047 - 996 1.15(9) 1.09(7) 1.00(5)
1047 - 490 1.06(9) 1.11(7) 1.00(5)
996 - 490 1.2(1) 1.05(7) 1.00(5)
Table 7: Coincidence rates between -rays of Sn as a function of their relative angle of detection, normalized to the ones obtained around 75.

The coincidence rates measured for 6 pairs indicate that all their transitions have the same multipole order: The 1190- and 241-keV transitions (in Sn), the 1103-, 1030-, and 242-keV transitions (in Sn), the 1047-, 996-, and 490-keV transitions (in Sn). Since the 241- and 242-keV transitions are , a quadrupole order is also assigned to the 1190-, 1103-, and 1030-keV transitions. Then the 1047-keV ray, located just above the 10 state of Sn is also , as the 1190- and 1103-keV rays in Sn and Sn, respectively . On the other hand, both the 557- and the 609-keV transitions have a different multipole order from that of the 1190- and 1103-keV transitions, respectively (see Table 7). Thus they are dipole transitions.

In conclusion, the cascades of three transitions, placed above the 10 isomeric states of Sn, define the 12, 14, and 16 levels (see Figs. 3, 6, 7, and 9). Moreover the second cascade identifed in each level scheme comprises states with odd spin values, the state decays to the 12 state and the state is isomeric. A negative parity is assigned to the states of this second cascade since in Sn, the state is linked to the long-lived 7 state by means of a cascade of three transitions. This leads to the 9 and 11 states. All these assignments are reported in the level schemes drawn in Figs. 3, 6, 7, and 9.

iii.2.6 Comparison with other results recently published

During the completion of this work, the decay of a new isomeric state in Sn populated in the fragmentation of Xe was reported pi11 . Its deexcitation is very similar to that of the isomeric states we have measured in Sn. The spin and parity values of all the states involved in the isomeric decay were proposed by comparison with results of shell-model calculations.

Moreover, at the very end of the writing of this paper, we became acquainted with a publication on the high-spin states of the even- Sn fo11 . The authors have used the fusion-fission process to populate a few levels lying above the long-lived 10 states, the identification of the first transition of each -ray cascade being confirmed thanks to the behavior of its excitation function in the Sn() reactions. The resulting four(three) new states of Sn(Sn) are part of the structures identified in the present work.

iii.3 Study of the odd-A Sn isotopes

Populated in deep inelastic reactions, a few medium-spin levels were identified in the four odd-A Sn isotopes prior to this work ma94 ; zh00 . The maximum spin values measured in these isotopes were 27/2 in case of negative parity and 19/2 or 23/2 in case of positive parity. Such states correspond to the breaking of one neutron pair in the subshell which gives rise to several sets of states, depending on the subshell occupied by the odd neutron. The state, attributed to the configuration, is located between 3.1 MeV and 2.6 MeV. It is a long-lived isomeric state in Sn (34 ), while the half-lives of the 27/2 states of Sn isotopes are in a 30-230 ns range, i.e. short enough such that coincidence events may be detected across the isomeric states. The maximum spin of the configuration is 23/2. Such a state, which is isomeric, is only known in Sn. On the other hand, the four isotopes exhibit a long-lived isomeric 19/2 state, from the configuration, located around 2 MeV excitation energy.

Using the data of the two fusion-fission reactions of the present work, we have identified, for the first time, several structures of Sn located above 3 MeV excitation energy and spin values greater than 27/2. In the following, we first present the building of each high-spin level scheme. Then, we discuss the angular momentum and parity assignments of most of the new states of Sn.

iii.3.1 Sn and Sn

The high-spin states of Sn and Sn are weakly populated in the two fusion-fission reactions used in the present work, as these two isotopes are in the tails of the Sn fragment distribution. The half-live of their 27/2 states (at 3101 keV and 2624 keV respectively) are short enough to let the detection of coincidence events across the isomers. Therefore the search for -rays populating their 27/2 states is more appropriate in the spectra gated by their low-lying transitions than in the spectra gated by their complementary fragments (namely Cd for Sn and Cd for Sn).

In both data sets registered in our work, all the spectra doubly-gated by the - transitions decaying the 27/2 state of Sn reveal new -lines which have to be placed above it (see Fig. 11).

Figure 11: (Color online) Level scheme of Sn deduced in the present work. The colored levels are new. The width of the arrows is representative of the relative intensity of the rays. The two isomeric states were already known NNDC . In our work, the 818-keV transition could not be observed, nevertheless the 19/2 isomeric state is drawn for the sake of completeness.

As for Sn, a cascade of three new - transitions is observed in coincidence with the transitions decaying the 27/2 state at 2624 keV (see Fig. 12). The new transitions of Sn are also observed in spectra gated by rays emitted by their Cd partners.

Figure 12: (Color online) Level scheme of Sn deduced in the present work. The colored levels are new. The width of the arrows is representative of the relative intensity of the rays. The three isomeric states were already known NNDC . In our work, the 805-keV transition could not be observed and the very converted 167-keV line was extremely weak because of the lifetime of the decaying state.

All the transitions observed in Sn are given in Tables 8 and 9, respectively. The spin and parity of the new states will be discussed in Sec. III.3.4.

E(keV)111The number in parenthesis is the error in the last digit. I111The number in parenthesis is the error in the last digit.222The relative intensities are normalized to . JJ E(keV) E(keV)
174.6(3) 61(9) 27/2 23/2 3101.1 2926.6
511.9(3) 80(12) 23/2 19/2 2926.6 2414.7
827.8(4) 15(5) 5107.9 4280.1
876.8(4) 26(6)        27/2 3977.9 3101.1
1105.2(3) 100 19/2 15/2 2414.7 1309.5
1179.0(5) 21(5)        27/2 4280.1 3101.1
1220.0(3) 100 15/2 11/2 1309.5 89.5
Table 8: Properties of the Sn transitions. The excitation energy of the 11/2 state (written in bold) is from Ref. NNDC
E(keV)111The number in parenthesis is the error in the last digit. I111The number in parenthesis is the error in the last digit.222The relative intensities are normalized to . JJ E(keV) E(keV)
161.3(3) 51(8) 27/2 23/2 2623.3 2462.0
385.6(3) 84(12) 23/2 19/2 2462.0 2076.4
402.8(4) 52(12) 23/2 23/2 2462.0 2059.2
778.4(5) 16(5) 5271.9 4493.5
923.7(4) 34(8)        27/2 3547.0 2623.3
946.5(4) 24(7) 4493.5 3547.5
988.7(3) 100 19/2 15/2 2076.4 1087.7
1087.7(3) 100 15/2 11/2 1087.7 0
Table 9: Properties of the Sn transitions.

We have not identified any -ray cascade which would be placed above the 19/2 or 23/2 states of Sn or Sn. Because of the long lifetime of these isomeric states, the transitions located above them can be only identified from their coincidences with transitions emitted by the partners. When taking into account that the weak population of these two Sn isotopes is shared between several complementary fragments, every coincidence rate was too low in the present work.

iii.3.2 Sn

Prior to this work, the knowledge of the medium spin states of Sn was very similar to the one of Sn, namely the 27/2 and 19/2 states and their decay towards the yrast levels having lower spin values. As in Sn, the half-life of the 27/2 isomer is short enough to measure the coincidence events across it. An example of coincidence spectrum double-gated on the 1151-, 1030-, 470-, and 175-keV transitions of the yrast cascade is drawn in Fig. 13.

Figure 13: (Color online) Coincidence spectrum double-gated on the 1151-, 1030-, 470- and 175-keV transitions of the yrast cascade built on the low-lying 11/2 state of Sn produced in the O + Pb reaction. The -rays emitted by the Zr complementary fragments are labelled by their masses A, written in italics.

Besides the transitions emitted by the complementary fragments, Zr, it exhibits new peaks at 1070, 1083, 1171, 1212, and 1457 keV. In a second step, we have analyzed the coincidence relationships of these new -lines and placed all these transitions in three cascades above the 27/2 state at 2833 keV (see Fig. 14). In addition two other lines were observed in coincidence with a part of the yrast transitions, thus the 889 keV transition defines a new state at 3076 keV and the 1152 keV one a new state at 3809 keV.

Figure 14: (Color online) Level scheme of Sn deduced in the present work. The colored levels are new. The isomeric 27/2 and 19/2 states and their decays were already known NNDC . The 841 keV transition could not be observed in our work. The width of the arrows is representative of the relative intensity of the rays.

The two most intense transitions of the new cascade assigned to Sn thanks to their coincidences with their complementary fragments (see Sec. III.1) have energies of 1238 and 666 keV. In low-energy part of the double gate set on these two transitions (see Fig. 15(a)), two new transitions, at 223 keV and 377 kev, are clearly observed, the intensity of the 223 keV peak being the lowest.

Figure 15: Coincidence spectra double-gated on transitions belonging to two new cascades, built from the C + U data set, in two energy ranges, [180-420] keV and [1060-1880] keV. (a) and (b) Cascade emitted by Sn. The lines labelled by Rh are pollutions (they belong to Rh, the 667- and 378-keV transitions being part of its yrast cascade ve02 ). (c) and (d) Cascade emitted by Sn.

This would indicate that the 223 keV transition has to be located at the top of the cascade. Nevertheless, because of the comparison with the new cascade observed in Sn (see Sec. III.3.3), we have chosen to put this transition at the bottom of the cascade, assuming that it deexcites an isomeric state with a half-life long enough to lower its relative intensity. The measured imbalance leads to T when taking into account the conversion coefficient for an transition, 223 keV)= 0.096 BRICC . Unfortunately, the data from the SAPhIR experiment cannot be used to determine the half-life of this isomeric state, as it only emits one transition (in the time window of Euroball). The corresponding events contain a unique -ray, this is not enough to select unambiguously the emitting nucleus.

In the high-energy part of the spectrum shown in Fig. 15(a), we observe two weak transitions, at 1110 and 1813 keV, which are also present in the spectrum doubly-gated by the 666- and 377-keV transitions (see Fig. 15(b)). They are located at the top of the new cascade. Finally, the whole set is placed above the long-lived isomeric state at 1998 keV, which is the only solution to explain why these transitions do belong to the level scheme of Sn while they are not detected in coincidence with its well-known yrast transitions.

All the transitions observed in Sn are given in Table 10. The spin and parity of the new states will be discussed in Sec. III.3.4.

E(keV)111The number in parenthesis is the error in the last digit. I111The number in parenthesis is the error in the last digit.222The relative intensities are normalized to . JJ E(keV) E(keV)
175.4(2) 38(9) 27/2 23/2 2832.7 2657.3
222.8(3) 48(10)333See text. (23/2) 19/2 2220.8 1998.0
376.9(3) 33(7) (35/2) (31/2) 4501.2 4124.3
470.2(2) 68(13) 23/2 19/2 2657.3 2187.1
665.7(3) 53(11) (31/2) (27/2) 4124.3 3458.6
837.8(4) 3(1) (35/2) (31/2) 5128.0 4290.1
889.4(4) 9(3)        19/2 3076.5 2187.1
1029.8(3) 100 19/2 15/2 2187.1 1157.3
1047.1(5) weak        (39/2) 7361.3 6314.2
1069.7(4) 12(4)        27/2 3902.4 2832.7
1082.9(4) 21(4) (31/2) 27/2 3915.6 2832.7
1109.9(5) 4(3)        (35/2) 5611.1 4501.2
1151.0(3) 100 15/2 11/2 1157.3 6.3
1151.6(4) 8(4)        23/2 3808.9 2657.3
1170.6(5) 6(3) 5073.0 3902.4
1212.5(4) 9(3) (35/2) (31/2) 5128.0 3915.6
1237.8(3) 53(11) (27/2) (23/2) 3458.6 2220.8
1457.4(4) 6(3) (31/2) 27/2 4290.1 2832.7
1813.0(4) 5(2) (39/2) (35/2) 6314.2 4501.2
Table 10: Properties of the Sn transitions. The excitation energies of the 11/2 and 19/2 states (written in bold) are from Ref. NNDC

iii.3.3 Sn

The 23/2 and 27/2 isomeric states were identified in Sn, their very long half-lives being mainly due to the half-filling of the subshell. Thus the identification of all its excited states with I or 23/2 relies on the coincidences with -rays emitted by its complementary fragments.

As for the previous cases, we start with the two coincident transitions at 1140 and 824 keV, assigned to Sn because of the mass distribution of the complementary fragments in the two fusion-fission reactions (see Sec. III.1). The low-energy part of the spectrum doubly-gated on these two transitions reveals one new -line at 261 keV (see Fig. 15(c)) and the high-energy part of the spectrum doubly-gated on the 261- and 824-keV transitions shows new high-energy low-intensity transitions (see Fig. 15(d)).

The resulting structure, which is very similar to the one assigned to Sn, is placed above the positive-parity isomeric state, the 23/2 at 2153 keV (see Fig. 16). It is worth noting that, due to the time range defining the coincidence events (about 300 ns), the already-known E2 transition (208 keV) deexciting this long-lived state, Ts, is never included in an event registered in our experiment. This explains why the new structure of Sn contains one low-energy transition (261 keV) instead of the two transitions (223 and 377 keV) observed in Sn, as discussed in the previous section.

Figure 16: (Color online) Level scheme of Sn deduced in the present work. The colored levels are new. The isomeric 27/2, 23/2 and 19/2 states and their decays were already known NNDC . The 208 and 838 keV transitions could not be observed in our work. as well as several other decays of the 27/2 and 23/2 states ma94 . Two new cascades are tentatively placed above the 27/2 state (see text). The width of the arrows is representative of the relative intensity of the rays, except for that of the decay of the 27/2 long-lived isomeric state (see text).

Noteworthy is the fact that the three transitions decaying the 4378-keV state of Sn have not been measured in the SAPhIR experiment. Thus the half-life of this state is smaller than 30 ns.

In addition, several weak transitions were observed in coincidence with those emitted by the same partners as Sn (namely Te and Zr). They form two cascades of three -rays in mutual coincidences, which look like the new cascade of Sn (located on the top of the 27/2 state because of the coincidence relationships with its yrast transitions, see Fig. 12). Thus we have tentatively placed the two new cascades above the long-lived 27/2 state of Sn (see Fig. 16).

All the transitions observed in Sn are given in Table 11. In our experiments, a small number of events which contain the low-lying transitions of Sn are registered since the prompt fold of the corresponding cascades is strongly lowered by the long half-life of the 27/2 state (34 s). Thus the intensities of the transitions located below that state could not be given relative to that of the ones located above the isomeric states. Nevertheless we have chosen to put the -lines in Table 11, as their energies are now known more precisely than previously ma94 . The spin and parity of the new states will be discussed in Sec. III.3.4.

E(keV)111The number in parenthesis is the error in the last digit. I111The number in parenthesis is the error in the last digit.222The relative intensities are normalized to . JJ E(keV) E(keV)
169.8(2) 333See text. 27/2 23/2 2711.3 2541.5
261.4(3) 43(9) (35/2) (31/2) 4378.0 4116.6
279.1(2) 333See text. 23/2 19/2 2541.5 2262.4
823.6(3) 100 (31/2) (27/2) 4116.6 3293.0
928.3(4) 3(1)        (39/2) 7159.2 6231.0
1106.8(2) 333See text. 15/2 11/2 1106.8 0
1140.0(3) 100 (27/2) 23/2 3293.0 2153.0
1155.6(2) 333See text. 19/2 15/2 2262.4 1106.8
1265.8(4) 12(4)        (35/2) 5643.8 4378.0
1515.3(5) 6(2) 7159.2 5643.8
1853.0(5) 5(2) (39/2) (35/2) 6231.0 4378.0
710.5(3) 444Tentative attribution. 5520.4 4809.9
735.0(3) 444Tentative attribution. 5477.6 4742.6
987.8(3) 444Tentative attribution. (35/2) (31/2) 4742.6 3754.8
991.6(3) 444Tentative attribution. (35/2) (31/2) 4809.9 3818.3
1043.5(3) 444Tentative attribution. (31/2) 27/2 3754.8 2711.3
1107.0(3) 444Tentative attribution. (31/2) 27/2 3818.3 2711.3
Table 11: Properties of the Sn transitions. The excitation energy of the 23/2 state (written in bold) is from Ref. NNDC .

iii.3.4 Angular momentum and parity values of the high-spin states of Sn

The statistics of our data related to the high-spin states of the odd-A Sn nuclei is too low to perform angular correlation analyses. Therefore, the spin assignments shown in Figs. 11, 12, 14, and 16 are based upon a few features:

  • All the transitions with an energy 1 MeV are assumed to have an multipolarity.

  • The 223-keV transition of Sn is assumed to be (see Sec. III.3.2).

  • The 666- and 377-keV transitions of Sn, as well as the 824- and 261-keV transitions of Sn, are assumed to be , as the 35/2 state with the configuration is expected to decay to the 23/2 state with the configuration by means of a cascade of three transitions.

Lastly, we can compute a few transition probabilities. As mentioned above, the half-life of the 2221-keV state of Sn is Ts, then the value of is 1.9(4) . The half-life of the 4378-keV state of Sn is T 30 ns, that leads to the value of the reduced transition probability, .

Iv Discussion

iv.1 General features of configurations

The nuclear shell model (SM) describes the many-body nuclear system in terms of a single-particle Hamiltonian representing the average effect of the strong nucleon-nucleon interactions on a given nucleon, plus residual interactions among a smaller number of particles, the valence nucleons near the Fermi surface, . The identification of states involving many identical nucleons in the same orbit , i.e. states with the configuration, is a straightforward application of SM. These states are expected to exhibit typical features indicating properties of the residual interaction. For instance it is now well known that, when it is assumed that only two-body forces contribute, , the interaction between nucleons in one shell can be expressed in terms of the two-particle matrix elements. Moreover, when using the seniority number, (which can be defined as the number of unpaired nucleons333For a recent review on the use of the seniority quantum number in many-body systems, see Ref. va10 .), we can relate the two-body interaction matrix elements of seniority- states in the configuration to the matrix elements in the configuration.

To illustrate these features, it is instructive to look at results of calculations performed many years ago on the proton configurations la81 . The spectra associated with 3, 4, 5, and 6, the latter corresponding to mid-shell444Due to the particle-hole symmetry, the spectrum associated to the configuration is the same as the one. Thus it is sufficient to discuss the cases with ., were calculated using the residual interactions taken from the experimental spectrum of Dy (with 2). For 3, there are often several states with the same angular momentum. If the two-body residual interactions conserve seniority, the latter can be used as a quantum number to characterize each state. It is worth recalling that to be diagonal in the seniority scheme, a condition involving the five diagonal matrix elements of the interaction has to be fulfilled ta93 . While this condition is not satisfied by most general two-body interactions, it is fulfilled in the present case la81 . Thus the predicted spectra of configurations are very similar, whatever the number of nucleons.

This is illustrated by the theoretical results obtained for the yrast states of configurations with even , drawn in Fig. 17(a). Above the first multiplet (I=2,.., 10) of 2 seniority, we observe a second group of states (I=12, 14, and 16) with =4.

Figure 17: (Color online) Evolution of the yrast states with the configuration as a function of , the number of protons occupying the orbit (see text). (a) Even number of protons. The states with one broken pair (=2) are drawn in black, with two broken pairs (=4) in red and three broken pairs (=6) in green. (b) Odd number of protons. The states with one broken pair (=3) are drawn in black, with two broken pairs (=5) in blue.

When breaking a third pair, we obtain the highest-spin value available in this orbit, I=18. It is worth noting that the =6 14 state is located below the =4 16 state, that has a paramount importance for the decay of the 16 state. While the transition is allowed, the one between states of the same seniority is forbidden as the orbit is half-filled, i.e. =0. As a result, the value of the half-life of the 16 state of the configuration can be even lower than the one of the 16 state of the .

The results obtained for the yrast states of configurations with odd are drawn in Fig. 17(b). While the energy of the 27/2 state is close to the one of the 10, the highest-spin states obtained for an odd number of nucleons are located at lower energy than the ones obtained for an even number. Thus we expect a more compressed spectrum when the number of nucleons occupying the orbit is odd.

Unfortunately the experimental behaviors of the configurations have never been tested, the nuclei of interest lying very close to the proton drip-line. Since they are only produced in reactions with very low cross sections, their high-spin states could not be identified.

Pure configurations occur in a very few nuclei, since an orbit is rarely bounded by two gaps in energy. The closeness of several orbits leads to configuration mixings, nevertheless some of the features due to seniority are found to survive. For instance, there are in semi-magic nuclei fairly constant spacings between the 0 ground state and some states, even though large changes in configuration mixings occur. The Sn nuclei provide a typical example, the energies of their first two states, (I=2 and 4) do not vary much across the major shell (), while the neutron orbits evolve from [] for to [] for (see Fig. 18).

Figure 18: Evolution of the 2 and 4 states of the Sn isotopes, as a function of the mass number.

The highest-spin states of the Sn isotopes with do contain a large component, since the two low- orbits cannot afford large spin values. Therefore their study as a function of the neutron number gives us the opportunity to explore the main features of the configurations. However such a work is restricted to since the yrast states of the lightest-A Sn isotopes are dominated by a collective band coming from 2p-2h excitations across the gap, which hampers identifying states coming from the configurations. The maximum values of angular momentum obtained for various configurations with several broken pairs are given in Table 12.

configuration nucleus
10 even-
27/2 odd-
16 even-
35/2 odd-
18 even-
7 even-
23/2 odd-
15 even-
35/2 odd-
19 even-
39/2 odd-
Table 12: Various configurations with several broken pairs, expected in heavy- Sn isotopes.

iv.2 High-seniority states of the Sn isotopes

The systematics of the excitation energies of the highest-spin states in the Sn even- isotopes are shown in Fig. 19. Noteworthy is the fact that the highest-energy state of Sn measured in the present work is assigned to be the 19 state which is expected above the 17 state, knowing that the 19-17 distance in energy is most likely lower than that the 17- 15 one.

Figure 19: (Color online) Evolution of the highest-spin states of the even- Sn isotopes, as a function of the mass number (this work and Ref. pi11 for Sn). The state having the maximum angular momentum of each configuration is drawn with a filled symbol. (a) Excitation energies of the positive-parity states above the 10 level. For the peculiar behavior of the 18 states (drawn with asterisks), see text. (b) Excitation energies of the negative-parity states above the 7 level.

The evolution of the positive-parity states is very smooth. Likewise, the energies of the negative-parity states display a very regular behavior. This indicates that their main configurations are the same, whatever the number of neutrons. Being very close in energy for , the three neutron orbits () are gradually filled together. Then the occupation number of the subshell does not change by two units from one even- isotope to the next one and the maximum value of seniority does not display the stepwise behavior as a function of , shown in Fig. 17.

Taking into account the maximum values of angular momentum given in Table 12, it is tempting to assume that the positive-parity states have the and configurations, and the negative-parity states, the and configurations. Nevertheless we can notice immediately the peculiar behavior of the 18 states: (i) the irregular variation of its excitation energy as a function of , and (ii) the low value of the 18-16 gap in energy as compared to the results given in Fig. 17. Thus the main configuration of the 18 states is likely not the configuration (this will be discussed in the next section). All the assignments written in Fig. 19 are corroborated by the results of shell model calculations presented in Sec. IV.3.

As above mentioned, when using the proton residual interactions extracted from the Dy spectrum, the 14 state with seniority 6 is located below the 16 state with seniority 4. Then the 16 state decays predominantly towards that 14 state when the subshell is half-filled, since the decay towards the 14 state is hindered. As for the Sn isotopes, we did not observe any delayed component in the -ray cascades located below the 16 states, meaning that their half-lives are smaller than 30 ns. This would lead to for Sn, for instance. This limit is close to the low value of in Sn, which is due to the half-filling of the orbit ma94 (see below). Thus in order to determine whether the 14 states measured in the Sn isotopes have the seniority 4 or 6, we would need to know a more precise limit of the 16 half-lives.

The decays of the isomeric 15 states identified in the even- Sn isotopes allow us to confirm their main configuration, . It is well known that the sign of the transition amplitude between two states with the same seniority depends on the occupation rate of the orbit, being positive for low values and negative for high values. Thus the behavior of as a function of the Sn mass number was used to determine the half-filling of the orbit, i.e. when particle and hole contributions exactly cancel one another. That happens for 123 or 73 ma94 . Results obtained in odd- Sn isotopes corroborated this value br92 ; lo08 . Indeed the transition probability between two states with seniority , such as , displays the same behavior with regard to the orbit filling as the one between two states with seniority , such as . Nevertheless in order to plot the values of both the even- and odd-A Sn isotopes in the same graph, one has to compensate the fractional parentage coefficients entering in the expression of the states with seniority  ta93 : the values corresponding to have to be multiplied by 0.264 ma89 .

The E2 transition amplitudes for these two sets of isomeric transitions are drawn in Fig. 20, as well as the results of the 15 13 transitions obtained in the present work.

Figure 20: (Color online) E2 transition amplitudes for the ( isomeric transitions in the Sn isotopes, for the decay of the 10 states, for the decays of the 27/2 and 15 states. The experimental values for the 27/2 states (in green) and for the 15 states (in blue) have been normalized (see text).

The latters follow exactly the same trend as those of the transitions and the ones This confirms that the main configuration of the 15 and 13 states of the even- Sn isotopes is .

The excitation energies of the negative-parity states above the 27/2 level in Sn are drawn in Fig. 21(a). The highest-spin states of the seniority 5 are more compressed in energy than the ones of seniority 4 shown in Fig. 19(a), as predicted for the configuration (see Sec. IV.1 and Fig. 17).

Figure 21: (Color online) Evolution of the highest-spin states of the odd- Sn isotopes, as a function of the mass number (this work). The state having the maximum angular momentum of each configuration is drawn with a filled symbol. (a) Excitation energies of the negative-parity states above the 27/2 level. (b) Excitation energies of the positive-parity states above the 23/2 level.

The positive-parity states above the 23/2 level in Sn and Sn are due to the breaking of two and three neutron pairs in the orbit [see Fig. 21(b)]. Their almost constant energies indicate once more that the three neutron orbits close to the Fermi level are gradually filled, so the level energies do not depend very much on the total number of neutrons.

iv.3 Results of shell model calculations

We have performed shell-model calculations using the ANTOINE code ca99 , the calculational details being the same as those described in Ref. si09 . Since the present work mainly involves high-spin states, we have restricted the calculations to the excited states with spin values higher than 7/10 for the even- isotopes and 23/2/27/2 for the odd-. In this section, we only present results of Sn and Sn, as those of the lower- isotopes are nearly the same.

The yrast spectra above the long-lived isomeric states are shown in Fig. 22.

Figure 22: (Color online) High-spin levels of Sn and Sn predicted by the SM calculations (see text).

The angular momenta of the states drawn with dotted lines comprise components involving a broken pair in the low- orbits, or . The comparison of results given in Fig. 22 with those obtained for a pure configuration (see Fig. 17) shows the influence of the low- neutron orbits in the energies of these states. For instance, the 31/2 state is now located at mid-distance between the 35/2 and the 27/2 states (see Fig. 22). In the same manner, the 12 state is located at mid-distance between the 14 and the 10 states, while it is located nearer the 14 state in the first calculation.

On the other hand, the states drawn with solid lines are due to the complete alignment of the momenta of the neutron belonging to the broken pairs (cf. the configurations given in Table  12). For instance, the main configuration of the 35/2 state of Sn is