Prompt-delayed γ -ray spectroscopy of neutron-rich ^{119,121} In isotopes
S. Biswas, A. Lemasson, M. Rejmund, A. Navin, Y.H. Kim, C. Michelagnoli, I. Stefan, R. Banik, P. Bednarczyk, Soumik Bhattacharya, S. Bhattacharyya, E. Clément, H. L. Crawford, G. de France, P. Fallon, G. Frémont, J. Goupil, B. Jacquot, H.J. Li, J. Ljungvall, A. Maj, L. Ménager, V. Morel, R. Palit, R.M. Pérez-Vidal, J. Ropert
aa r X i v : . [ nu c l - e x ] J u l Prompt-delayed γ -ray spectroscopy of neutron-rich , In isotopes
S. Biswas, A. Lemasson, ∗ M. Rejmund, A. Navin, Y.H. Kim, † C. Michelagnoli, † I. Stefan, R. Banik,
3, 4
P. Bednarczyk, Soumik Bhattacharya,
S. Bhattacharyya,
E. Cl´ement, H.L. Crawford, G. de France, P. Fallon, G. Fr´emont, J. Goupil, B. Jacquot, H.J. Li, J. Ljungvall, A. Maj, L. M´enager, V. Morel, R. Palit, R.M. P´erez-Vidal, and J. Ropert GANIL, CEA/DRF-CNRS/IN2P3, Bd Henri Becquerel, BP 55027, F-14076 Caen Cedex 05, France Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700064, India Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai-400094, India Institute of Nuclear Physics PAN, 31-342 Krak´ow, Poland Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA Department of Nuclear and Atomic Physics, Tata Institute of Fundamental Research,Mumbai, 400005, India Instituto de F´ısica Corpuscular, CSIC-Universitat de Val`encia, E-46980 Val`encia, Spain (Dated: July 21, 2020)
Background:
The Z = 50 shell closure, near N = 82, is unique in the sense that it is the onlyshell closure with the spin-orbit partner orbitals, πg / and πg / , enclosing the magic gap. Theinteraction of the proton hole/particle in the above mentioned orbitals with neutrons in the νh / orbital is an important prerequisite to the understanding of the nuclear structure near N = 82 andthe νπ interaction. Purpose:
To explore the structural similarity between the high-spin isomeric states in In ( Z =49), Sn ( Z = 50) and Sb ( Z = 51) isotopes from a microscopic point of view. In addition, tounderstand the role of a proton hole or particle in the spin-orbit partner orbitals, πg / and πg / ,respectively, with neutron holes in the νh / orbital on these aforementioned isomers. Methods:
The fusion and transfer induced fission reaction Be(
U, f) with 6.2 MeV/u beamenergy, using a unique setup consisting of AGATA, VAMOS++ and EXOGAM detectors, was usedto populate through the fission process and study the neutron-rich , In isotopes. This setupenabled the prompt-delayed γ -ray spectroscopy of isotopes in the time range of 100 ns − µ s. Results:
In the odd- A , In isotopes, indications of a short half-life 19 / − isomeric state, inaddition to the previously known 25 / + isomeric state, were observed from the present data. Further,new prompt transitions above the 25 / + isomer in In were identified along with reevaluation ofits half-life.
Conclusions:
The experimental data were compared with the theoretical results obtainedin the framework of large-scale shell-model calculations in a restricted model space. The h πg / νh / ; I | ˆ H| πg / νh / ; I i two-body matrix elements of residual interaction were modifiedto explain the excitation energies and the B ( E
2) transition probabilities in the neutron-rich In iso-topes. The (i) decreasing trend of E (29 / + ) − E (25 / + ) in odd-In (with dominant configuration πg − / νh − / and maximum aligned spin of 29 / + ) and (ii) increasing trend of E (27 / + ) − E (23 / + )in odd-Sb (with dominant configuration πg +17 / νh − / and maximum aligned spin of 27 / + ) withincreasing neutron number could be understood as a consequence of hole-hole and particle-holeinteractions, respectively. I. INTRODUCTION
The protons and neutrons in an atomic nucleus areheld together by the dominant strong nuclear force. How-ever, due to the limited theoretical knowledge on thestrong force (governed by QCD), the structure of the nu-cleus is being described using a variety of models. Outof these, the most successful models are the nuclear shellmodel by Maria Goeppert Mayer and Jensen, and thecollective model by Bohr and Mottelson. The empiricalone-body nuclear spin-orbit force, introduced by Mayer ∗ [email protected] † Present address: Institut Laue-Langevin, F-38042 GrenobleCedex, France and Jensen, was able to successfully explain all the exper-imentally observed magic numbers [1, 2]. The spin-orbitterm led to the splitting of the spin-orbit partner orbitals( j > = l + s and j < = l − s ), with one of the partners( j > ) generally moved into the lower oscillator shell, andcalled the intruder orbital. Such examples of intruder or-bitals are the f / (leading to the magic number 28), g / (leading to magic number 50), h / (leading to the magicnumber 82), and i / (leading to the magic number 126).The magic number 50 is unique as the nuclei around thisshell closure have holes and particles occupying adjacentspin-orbit partners, j < = g / and j > = g / .In neutron-rich In ( Z = 49) and Sb ( Z = 51) isotopes,the valence proton hole and particle occupy the adja-cent spin-orbit partners g / and g / , respectively, whilethe valence neutron holes dominantly occupy the h / orbital. Even- A − Sn isotopes possess 7 − and 10 + isomers, with dominant neutron νh − / d − / and νh − / configurations, respectively [3–6]. Odd- A − Sb iso-topes have 19 / − and 23 / + isomers with dominant πg / νh − / d − / and πg / νh − / configurations [7–11],with an additional proton particle in g / coupled to the7 − and 10 + isomers in even- A Sn isotopes, respectively.It should be noted that the 23 / + isomers in Sb isotopesdo not correspond to maximally aligned configurations( I πMax = 27 / + ) but rather to I πMax − / + , il-lustrating the influence of spin and seniority mixing inthese isotopes. From this analogy, the odd- A − Inisotopes would be expected to have 21 / − and 25 / + isomers with dominant πg − / νh − / d − / and πg − / νh − / configurations, having an additional proton hole in g / coupled to the 7 − and 10 + isomers in even- A Sn, respec-tively.In the odd- A − In isotopes, several isomeric statesassociated to the above mentioned configurations havebeen reported in Refs. [12–17] and are summarized inTable I. In
In, a (25 / + ) isomeric state was observedwith a half-life of 240(25) ns but there was no indica-tion of negative-parity isomer [12]. In In, an isomericstate, decaying by the 99 keV transition and decayingthrough a γ cascade involving 99-214-953-1181 keV wasreported in Ref. [12] and Ref. [13] with two different half-lives (350 (50) ns and 17 (2) µ s respectively) and tentativeassignments (25 / + and 21 / − respectively). In heavier − In isotopes, several positive parity (23 / + , 25 / + ,29 / + ) and negative parity (17 / − and 21 / − ) isomericstates were reported in Refs. [13–17]. These experimentaldata indicate the need to perform prompt-delayed γ -rayspectroscopy of the neutron-rich In isotopes for the reli-able assignment of states.In the present manuscript, new high-spin prompt γ -raytransitions above the (25 / + ) isomeric state are reported TABLE I. Isomeric states half-lives and tentative assignmentsreported in the literature for odd- A In isotopes , In.Nucleus J π T / Reference
In (25 / + ) 240 (25) ns [12] In (25 / + ) 350 (50) ns [12](21 / − ) 17 (2) µ s [13] In (17 / − ) 1.4 (2) µ s [13, 14](21 / − ) > µ s [13] In (25 / + ) 5.0 (15) ms [14](21 / − ) 9.4 (6) µ s [13, 14] In (29 / + ) 9.0 (2) µ s [14](21 / − ) 1.04 (10) s [15] In (29 / + ) 110.0 (15) ms [14, 16](17 / − ) 8.5 (5) µ s [14](17 / − ) 8.7 (7) µ s [16](17 / − ) 11.3 +22 − µ s [17](23 / − ) 700 ms [14](23 / − ) 670 (10) ms [16] in the odd- A In isotope along with new determinationof the lifetime of the isomeric (25 / + ) state. Further,the existence of new (19 / − ) isomers in , In isotopesis also discussed. These new experimental results arediscussed using large-scale shell-model calculations.
II. EXPERIMENTAL DETAILS A U beam at an energy of 6.2 MeV/u impinging ona Be target was used to populate the neutron-rich fis-sion fragments , In via fusion- and transfer-inducedfission reactions at GANIL. Two targets were used dur-ing the experiment : the 1 . µ m and 5 µ m targets werebombarded for 32 hours and 160 hours respectively withtypical beam intensities of 1 pnA and 0.45 pnA respec-tively. The two datasets were combined for the presentdata analysis. The experimental setup consisted of acombination of AGATA γ -ray tracking array [18, 19], VA-MOS++ spectrometer [20] and the EXOGAM array [21].The AGATA array was placed at 13 . ◦ relative to the beam axis, con-sisted of a Multi-Wire Proportional Counter (MWPC),two drift chambers and a segmented ionization cham-ber. It was used for the unambiguous isotopic identifica-tion of the fission fragments ( Z , A , q ) [20, 22–24]. Thevelocity vector of the recoiling ions (measured by DualPosition-Sensitive MWPC (DPS-MWPC) detector [25]),placed at the entrance of the VAMOS++ spectrometer,and the γ -ray emission angle (determined using AGATA)were used to obtain the energy of the Doppler correctedprompt γ rays ( γ P ), on an event-by-event basis. The de-layed γ rays ( γ D ) were detected using seven EXOGAMHPGe Clover detectors [21], arranged in a wall like con-figuration at the focal plane of VAMOS++, behind theionization chamber. The decays curves were obtainedfrom the time difference ( t decay ) between the signals fromthe DPS-MWPC detector placed at the entrance of VA-MOS++ and the EXOGAM HPGe detectors placed atthe focal plane, and thus the time measured is indepen-dent of the time of flight. Due to the logic delays usedduring the experiment the presented time ( t decay ) has800 ns offset, the true reaction time is located thereforeat t decay = 800 ns. Additional information on the back-ground subtraction, efficiency evaluation, half-life evalu-ation are given in the Refs. [24, 26]. The uncertaintiesin the energy of prompt and delayed γ ray transitions is ∼ γ rays were determined separately. The spin-parities wereassigned based on systematics and shell model calcula-tions. The tabulated values of the energies and relativeintensities of the prompt and delayed γ -ray transitionsare provided as Supplemental Material [27]. (13/2 ) (11/2 ) (17/2 ) (21/2 ) (25/2 ) (27/2 ) (29/2 ) (15/2 ) (17/2 ) (19/2 ) (21/2 ) (23/2 ) (25/2 ) (27/2 ) (29/2 ) (7/2 ) (11/2 ) (15/2 ) (19/2 ) In FIG. 1. (Color online) The level scheme of
In. The widthsof the arrows represent the intensities of the transitions. Thenewly identified and the previously known isomeric states areindicated by a thick red and black line, respectively. Thepreviously known half-life but remeasured in this work andconsistent with the previous work is marked by a red dashedbox.
III. EXPERIMENTAL RESULTSA. In The level scheme for the
In isotope, obtained fromthe present work is shown in Fig. 1. Previous measure-ment on the high-spin states in
In was reported inRef. [12]. The A - and Z -gated tracked Doppler correctedprompt singles γ -ray ( γ P ) spectrum, for the 1 . × In ions identified in VAMOS++ in coincidence withprompt γ -ray, is shown in Fig. 2(a). The prompt γ -ray transition emitted by the complementary fission frag-ments [22, 23] (mainly from the fusion-fission process) areobserved and identified (marked by ”c”) in the low energypart of the γ -ray spectra. The random coincidences withX-ray emitted by U is also observed and is markedby an @ symbol. All the previously known γ rays wereobserved in the present spectrum, except for the 60 keV(19 / − → / − ) and 64 keV (19 / − → / + ) γ -raytransitions suggesting a lifetime of the (19 / − ) state. Inaddition, two new prompt γ rays, 313 and 511 keV areobserved and are marked with an asterisk. The trackedDoppler corrected prompt γ P - γ P coincidence spectrumwith a sum gate on the 145, 207, and 289 keV γ -raytransitions is shown in Fig. 2(b). The newly observed313 and 511 keV transitions along with the previouslyobserved 145, 207, 289, 499, 599 and 1098 keV transi-tions are marked in this figure. However, due to lack ofenough statistics in the coincidence spectrum, the 313and 511 keV transitions were not placed in the level Energy (keV) C oun t s / k e V C oun t s / k e V C oun t s / k e V C oun t s / k e V (d) γ D (t decay < 3 µ s) on 336 keV C oun t s / k e V (c) γ P - γ P with gate on 928 keV transition C oun t s / k e V γ P - γ P with gate (b) γ P - γ P with sum gate on Energy (keV) (a) γ P
422 499 (e) γ P in coincidence with γ D = 152 + 218 keV (t decay < 3 µ s)
499 599 1098 @ c
178 1020313 * 511 *511 *313 * C oun t s / k e V Energy (keV) γ D - γ D with gate on 218 keV FIG. 2. A − and Z − gated γ -ray spectrum for In: (a) Thetracked Doppler corrected prompt singles γ -ray ( γ P ) spec-trum with the new γ -ray transitions marked with asterisk. (b)Tracked Doppler corrected prompt γ P - γ P coincidence spec-trum with sum gate on the 145, 207, and 289 keV γ -raytransitions. (c) Tracked Doppler corrected prompt γ P - γ P coincidence spectrum with gate on the 928 keV γ -ray. (d)The delayed singles γ -ray ( γ D ) spectra for t decay < µ s. (e)Tracked Doppler corrected γ P in coincidence with γ D = 152and 218 keV γ rays (for t decay < µ s). The inset shows thetracked Doppler corrected prompt γ P - γ P coincidence spec-trum with gate on the 336 keV γ -ray. scheme in Fig. 1. The 94, 928 and 1204 keV transitionsare not observed, pointing to the fact that the (19 / − )state could be an isomeric state. This was confirmedby the observation of only 94 and 1204 keV transitionswith gate on 928 keV transition in the prompt γ P spec-trum, as shown in Fig. 2(c). The tracked Doppler cor-rected prompt γ P - γ P coincidence spectrum with a gateon the 721 keV γ -ray transition, yielding 422, 636 and740 keV transitions (not shown in this figure). The de-layed γ -ray spectrum ( γ D ) for t decay < µ s is shown inFig. 2(d), yielding 94, 152, 178, 218, 928, 1020, 1025 and1204 keV transitions. The 60 and 64 keV transitions arebelow the detection threshold of the present delayed- γ -ray setup and cannot be observed. The inset of Fig. 2(d) Time ( µ s) C o un t s / . µ s γ D = 152 keV γ D = 1204 + 928 keVT = 270(11) nsT = 304(10) ns FIG. 3. Decay curves along with the fits for the differenttransitions in
In: (a) 152 keV and (b) sum of 1204 and928 keV transitions. shows the delayed γ D - γ D coincidence spectrum with gateon the 218 keV γ -ray transition, yielding 94, 152 and1204 keV transitions. The tracked Doppler corrected γ P in coincidence with any γ D (for t decay < µ s) is shownin Fig. 2(e). This spectrum yields the previously iden-tified prompt γ -ray transitions, 336 and 460 keV tran-sitions. The tracked Doppler corrected prompt γ P - γ P coincidence spectrum with a gate on the 336 keV γ -rayalso yields 460 keV in coincidence, as shown in the insetof Fig. 2(e). This shows that the 336 and 460 keV transi-tions are indeed in coincidence and lie above the (25 / + )isomer.Figure 3 shows the half-life fit (one-component) for thedecay spectrum upon gating on 152 keV transitions yield-ing a value of T / = 270(11) ns for the (25 / + ) state(in agreement with the value of 240(25) ns reported inRef. [12]). The presently reported half-life corresponds to B ( E
2; 25 / + → / + ) = 19(1) e fm . Even with a veryshort time gate, it was not possible to obtain a γ D spectracontaining only the 94, 928, and 1204 keV transitions, in-dicating that the (19 / − ) state has a short half-life, whichis below the sensitivity of the present setup. In order tohave an estimation of the half-life of the (19 / − ) state,the half-life fit (one-component) for the decay spectrumupon gating on the 1204 keV transition was determinedto be T / = 304(10) ns. This shows that the half-life ofthe (19 / − ) state should be around 3ns < T / <
10 ns,which is compatible with the half-life estimate for an M
160 keV or E γ -ray spectrum. B. In The level scheme for the
In isotope, obtained fromthe present work is shown in Fig. 4. The A - and Z - gated tracked Doppler corrected prompt singles γ -ray( γ P ) spectrum, for the 4 . × In ions identified inVAMOS++ in coincidence with prompt γ -ray, is shownin Fig. 5(a). The prompt γ -ray transition emitted bythe complementary fission fragment [22, 23] (mainly fromthe fusion-fission) are observed and identified (marked by”c”) in the low energy part of the γ -ray spectra. Therandom coincidences with X-ray emitted by U is alsoobserved and is marked by an @ symbol. All the previ-ously known γ rays were observed in the present spec-trum. In addition, four new prompt transitions 326, 352,763, and 1052 keV transitions were observed and theseare marked with an asterisk in Fig. 5(a) and denoted byred in the level scheme (Fig. 4). The tracked Dopplercorrected prompt γ P - γ P coincidence spectrum with sumgate on the 110, 169, and 361 keV γ -ray transitions isshown in Fig. 5(b). All the previously observed 110, 169,361, 543, 573 and 1116 keV transitions are labeled. The953 and 1181 keV transitions are not observed, point-ing to the fact that the (19 / − ) state is likely to bean isomeric state with a short half-life, which is belowthe sensitivity of the present setup. This was confirmedby the observation of 1181 keV transition only with thegate on 953 keV transition in the prompt γ P spectrum,as shown in Fig. 5(c). The tracked Doppler correctedprompt γ P - γ P coincidence spectrum with a gate on the988 keV γ -ray transition, yielding 420 keV transition (notshown in this figure). The delayed γ -ray spectrum ( γ D )for t decay < µ s is shown in Fig. 5(d), yielding 99, 160,214, 953, and 1181 keV transitions. The inset of Fig. 5(d)shows the delayed γ D - γ D coincidence spectrum with agate on the 214 keV γ -ray transition, yielding 99, 953and 1181 keV transitions. The tracked Doppler corrected γ P in coincidence with any γ D (for t decay < µ s) isshown in Fig. 5(e). This spectrum yields the newly iden-tified prompt γ -ray transitions, 326 and 352 keV tran-sitions. The tracked Doppler corrected prompt γ P - γ P coincidence spectrum with the sum gate on the newlyobserved 352 keV γ -ray also yields 326 and 763 keV in co-incidence, as shown in the inset of Fig. 5(e). This provesthat the 326, 352 and 763 keV transitions are indeed incoincidence and lie above the (25 / + ) isomer.Figure 6(a) shows the half-life fit (one-component) forthe decay spectrum upon gating on 99 keV transitionyielding a value of T / = 7 . µ s for the (25 / + ) state(in disagreement with the value of 350(50) ns reported inRef. [12] and 17(2) µ s reported in Ref. [13]), as mentionedin the introduction. The newly reported half-life yielded B ( E
2; 25 / + → / + ) = 3 . fm . Similar fits werecarried out for the other delayed transitions in the samecascade, namely the 214, 953, and 1181 keV transitionsyielding 7 . µ s, 7 . µ s, and 7 . µ s as shown inFigs. 6(b), (c), and (d), respectively. That the half-life ofthe (25 / + ) state cannot be 350(50) ns (Ref. [12]), is alsomentioned in Ref. [13] and evident from our measure-ment. However, the disagreement between the presentwork and Ref. [13] cannot be firmly understood as thepresent data set confirms the measurement of long-lived (13/2 ) (11/2 ) (17/2 ) (21/2 )(25/2 )(27/2 )(29/2 )(31/2 )(15/2 ) (17/2 )(19/2 )(21/2 )(23/2 )(25/2 )(27/2 )(29/2 ) (7/2 ) (11/2 ) (15/2 ) In XY FIG. 4. (Color online) The level scheme of
In. The widthsof the arrows represent the intensities of the transitions. Thenewly identified transitions are shown in red. The newly iden-tified and the previously known isomeric states are indicatedby a thick red and black line, respectively. The newly re-determined half-life is marked by a red box. The X and Yenergies refers respectively to the energies of the unobserved(17 / − ) → (15 / − ) and (19 / − ) → (17 / − ) transitions asreported in Ref. [12]. Because of these unobserved transi-tions, excitation energies of levels above the 15 / − level arenot determined. isomers in isotopes of Sb (Ref. [26]) and thus a long half-life of 17(2) µ s could not be seen in the case of In.Further, it was not possible to obtain a γ D spectrawith a very short time gate containing only the 953 and1181 keV transitions. This suggests that the (19 / − )state has a short half-life, which is below the sensitivityof the present setup, similar to what is observed in In.An estimate for the half-life could not be carried out inthis case as the half-life of the (25 / + ) state is relativelylarge. IV. DISCUSSION
Large-scale shell-model calculations, with a modelspace consisting of valence neutrons ( ν ) in d / , s / and h / and protons ( π ) in p / , g / and g / or-bitals, near the Fermi surface, were carried out to under-stand the structure of the high-spin states in neutron-rich , In isotopes. The isotones of In and Sb rangingfrom N = 70 to N = 80 were also studied. An interac-tion, derived from the jj45pn and jj55pn interactions [28]was adjusted to account for the missing correlations inthe restricted model space used in this work to improvethe agreement with both the level energies and B ( E Energy (keV) C oun t s / k e V C oun t s / k e V C oun t s / k e V C oun t s / k e V (d) γ D (t decay < 30 µ s) on 352 keV C oun t s / k e V (c) γ P - γ P with gate on 953 keV transition Energy (keV) C oun t s / k e V γ P - γ P with gate (b) γ P - γ P with sum gate on C oun t s / k e V *352 * 1181 Energy (keV) (a) γ P
361 543 (e) γ P in coincidence with γ D = 214 keV (t decay < 30 µ s)
361 573
988 1116 *763 *640 c @ γ D - γ D with gate on 214 keV FIG. 5. (Color online) A - and Z -gated γ -ray spectra for In:(a) The tracked Doppler corrected prompt singles γ -ray ( γ P )spectrum with the new γ -ray transitions marked with aster-isk. (b) Tracked Doppler corrected prompt γ P - γ P coincidencespectrum with sum gate on the 110, 169, and 361 keV γ -raytransitions. (c) Tracked Doppler corrected prompt γ P - γ P co-incidence spectrum with gate on the 953 keV γ -ray. (d) Thedelayed singles γ -ray ( γ D ) spectra for t decay < µ s. Theinset in (d) shows the decay curve along with the fit for the99 keV transition. (e) Tracked Doppler corrected γ P in coin-cidence with γ D = 214 keV γ -ray (for t decay < µ s). Theinset shows the tracked Doppler corrected prompt γ P - γ P co-incidence spectrum with gate on the 352 keV γ -ray. then used, in the present work, to explain the energiesand the B ( E
2) values in the In isotopes. The calcu-lations were performed using the NATHAN code [29].The theoretical B ( E
2; 25 / + → / + ) = 42 . . fm values for In and
In, respectively, ob-tained from this interaction are not in agreement withthe corresponding experimental values (the green dashedline in Fig. 7(b)). The reason behind this could be at-tributed to the fact that in the Ref. [26], no changeswere made in the h πg / νh / ; I | ˆ H| πg / νh / ; I i two-body matrix elements of residual interaction. To improvethe calculated values of the B(E2)’s in the In isotopes,the h πg / νh / ; I | ˆ H| πg / νh / ; I i ( I = 6 , ,
10) ma-
Time ( µ s) T = 7.2(2) µ s γ D = 1181 keV C oun t s / . µ s (d)(c) T = 7.0(2) µ s γ D = 953 keV (b) T = 7.5(2) µ s γ D = 214 keV (a) T = 7.3(2) µ s γ D = 99 keV FIG. 6. Decay curves along with the fits for the differenttransitions in
In: (a) 99 keV, (b) 214 keV, (c) 953 keV,and (d) 1181 keV transitions. trix were modified by -200, -400 and +500 keV, respec-tively.Figure 7 shows the experimental excitation energiesfor the positive-parity and negative-parity levels (panel(a)) and B ( E
2; 25 / + → / + ) values (panel (b)) in , In, along with the calculations (SM) using theabove mentioned modified interaction. It can be seen inFig. 7(a) that the experimental energies are reproducedwithin ∼
500 keV. The newly calculated B ( E
2; 25 / + → / + ) = 19 . . fm values (red solid line) wereobtained for In and
In isotopes respectively, are inreasonably good agreement with the present experimen-tal values of 19(1) and 3 . fm (red triangle), respec-tively, as shown in Fig. 7(b). The shell model calculationsshow that the 25 / + levels in odd- A In have a dominant πg − / νh − / configuration. The B ( E
2; 25 / + → / + )in the odd- A In isotopes are expected to follow a similartrend as for the B ( E
2; 10 + → + ) in even- A Sn ( νh − / )and B ( E
2; 23 / + → / + ) in odd- A Sb ( πg / νh − / )isotopes. Both these levels involve the ( νh − / ) config-uration. It was shown in Fig. 17 of Ref. [26] that boththe experimental and the calculated B ( E
2; 10 + → + ) ineven- A Sn and B ( E
2; 23 / + → / + ) in odd- A Sb showa parabolic behavior with a minimum around N = 72 , B ( E
2; 25 / + → / + ) values for the , In isotopes reported in Ref. [12] show an increase (black squares in Fig. 7(b)). However, the present ex-perimental values (red triangles in Fig. 7(b)) follow theexpected behavior inherited from the Sn isotopes.Based on a good agreement of the shell model re-sults and the present experimental data, a systematicshell model analysis, using the interaction discussedabove, was made for the isotopic chains of − Sb and − In. The results of the calculations are shown inFig. 8. The 10 + seniority isomer in even- A − Sn iso-topes is known to be dominantly arising from νh − / con-figuration [3–6]. Stretched angular momentum couplingof πg / particle in − Sb ( πg / hole in − In)to the νh − / configuration leads to a maximum alignedspin of I πMax (Sb) = 27 / + in Sb and I πMax (In) = 29 / + in In. In the case of Sb, I πMax (Sb) − / + isomerswere reported for − Sb isotopes [26]. However, inthe case of In, I πMax (In) − / + isomers are ob-served only in the lighter odd- A , , In isotopes,while I πMax (In) = 29 / + isomers are reported in heavierodd- A , In isotopes [12, 14]. The evolution of theenergy of levels involved can be used to explain these ob-servations. The results of shell model calculations for the29 / + , 25 / + and 21 / + levels in − In are shown inFig. 8(a). In Fig. 8(b) the 27 / + , 23 / + and 19 / + lev-els in − Sb are shown. Panels (a) and (b) includealso the calculated 10 + state in − Sn. The energyof 10 + levels in Sn, associated with the νh − / configu-ration, is calculated to be relatively constant. In In, theinversion of the 25 / + and 21 / + levels beyond In, isconsistent with the reported long-lived 25 / + isomer thatdoes not decay to 21 / + level in − In [14]. Further-more, the energy of the high lying 29 / + level decreasessteeply as the neutron number increases, it lies below the21 / + and 25 / + in , In. A similar conclusion wasobtained from shell model calculations in Ref. [14] for , In isotopes showing that the energy of the 29 / + is lower than the 25 / + level due to strong νπ interac-tion in the ( πg − / νh − / ; 10 − ) state. In the case of theSb (Fig. 8(b)), the calculations for the − Sb isotopesshow that the ordering of the 27 / + , 23 / + and 19 / + level is preserved, in agreement with the experimental ob-servations. An opposite behavior of the I πMax (In) = 29 / + in In and I πMax (Sb) = 27 / + in Sb can be clearly seen fromthe Fig. 8(a) and (b): while the energy of the 29 / + levels in In is decreasing with increasing neutron num-ber, the energy of the 27 / + levels in Sb increases. Thiscould be understood as a consequence of the hole-holeand particle-hole nature of the νπ interaction. At highparticle occupancy of the νh / orbital the πgνh / interaction is of the hole-hole nature in In and of theparticle-hole nature in Sb while at low particle occupancyof the νh / orbital it is of the hole-particle nature in Inand particle-particle in Sb.Beyond the energy levels, an in-depth analysis of theprobability of the neutron angular momentum decom-position, P ( I ν = 8 + and 10 + ) and neutron seniority
120 122 124 126 (cid:2)(cid:3)(cid:4)
Expt. (Ref [12])Expt. (This work)SM (Ref. [26])SM (This work)
Mass Number, A B ( E ) ( e f m ) (b) (a) (25/2 - )(11/2 + )9/2 + (13/2 + )(17/2 + )(21/2 + )(25/2 + )(27/2 + )(29/2 + ) (23/2 - ) SM Expt. (21/2 - ) In SM SM Expt. Expt. (15/2 - )(19/2 - )(17/2 - )(29/2 - )(27/2 - ) In Expt. E x c it a ti on E n e r gy ( M e V ) + + ( π g ν h )Odd-A In π = + SM + (11/2 + )(13/2 + )(17/2 + )(29/2 + )(27/2 + )(25/2 + ) (19/2 - )(31/2 + ) (29/2 - )(27/2 - )(25/2 - )(21/2 - )(23/2 - )(15/2 - )(17/2 - ) (19/2 - )(21/2 + ) π = + π = - π = - π = + π = + π = - π = - FIG. 7. (Color online) (a) Comparison of the experimental (Expt.) level schemes with those obtained from shell modelcalculations using the interaction (SM) in the present work for both positive- and negative-parity states in , In. Thedifferent states with the same spin are joined by dotted lines. (b) The experimental B ( E
2) values for 25 / + → / + in odd- A , In from the previous work (Ref. [12]) and the present work are shown by the black square and red triangle respectively.The shell model (SM) calculations from Ref. [26] and the present work are shown by green dashed and red solid lines, respectivelyfor − In isotopes. P ( υ ν = 2 and 4), for the 29 / + and 25 / + states in − In; 27 / + and 23 / + states in − Sb, using theshell model calculations, was made. These probabilitydistributions are shown in Figs. 8(c,d,e,f), respectively.Figure 8(c) shows that for − In, the 29 / + level hasa dominant contribution from I ν = 10 + ( I ν = 8 + is notpossible for this spin). The contribution from I ν = 10 + to the 29 / + level reaches the unity only in In, inthe − In isotopes there is an admixture arising from I ν > + with υ ν = 4 as can be seen in Fig 8(e). For the25 / + state in − In, both the contributions from I ν = 8 + and 10 + are shown by red hatched and redfilled bars. It can be seen from Fig 8(c) that the con-tribution from I ν = 8 + dominates at high mass numberand that the contribution from I ν = 10 + is increasing with decreasing mass number. At lower mass numberthe contribution to the 25 / + state form the υ ν = 2dominates, while with increasing mass number υ ν = 4increases. This trend suddenly breaks beyond A = 125,since at A = 129 υ ν = 4 is no longer possible. This couldbe also a reason of the presence of the discontinuity ofthe increasing energy trend of the 21 / + and 25 / + withincreasing mass number (see Fig. 8(c)).A similar decomposition is shown in Figs. 8(d) and (f)for Sb isotopes. Figure 8 (d) shows that for − Sb,the 27 / + state also has a dominant contribution from I ν = 10 + . For the 23 / + levels in − Sb, both thecontributions from I ν = 8 + and 10 + are shown by redhatched and red filled bars respectively. It can be seenfrom Fig. 8 (d) that on the contrary to In, the contri- In
70 121 In
72 123 In
74 125 In
76 127 In
78 129 In
80 121 Sb
70 123 Sb
72 125 Sb
74 127 Sb
76 129 Sb
78 131 Sb + + + + + + + + (a) Z = 49 (b) Z = 51 P (I ν ) P (I ν )
119 121 123 125 127 129 131
Mass Number, A P ( υ ν )
119 121 123 125 127 129 131
Mass Number, A P ( υ ν ) (e) (f) + + + + (c) (d) + (I ν = 10 + )(I ν = 8 + )(I ν = 10 + ) (I ν = 10 + )(I ν = 10 + )(I ν = 8 + )23/2 + + ( υ ν = 2)25/2 + ( υ ν = 2)25/2 + ( υ ν = 4)29/2 + ( υ ν = 4) 27/2 + ( υ ν = 2)27/2 + ( υ ν = 4)23/2 + ( υ ν = 2)( υ ν = 4)( υ ν = 2)( υ ν = 4) 23/2 + FIG. 8. (Color online) Evolution of the calculated (SM) energies and the probabilities for the neutron angular momentum andneutron seniority for the, spin-orbit partners of In and Sb isotopes (a) Energies of 21 / + (black), 25 / + (red), and 29 / + (blue)states in Z = 49 − In isotopes; (b) Energies of the 19 / + (black), 23 / + (red), and 27 / + (blue) level in Z = 51 − Sbisotopes. The evolution of the 10 + states in Z = 50 − Sn isotopes are also shown in green in these panels. The probabilityof the neutron angular momentum, P ( I ν = 8 + and 10 + ) for the (c) 29 / + (blue filled), 25 / + levels (red hatched filled) in In;and (d) for the 27 / + (blue filled), 23 / + (red filled and hatched) states in Sb isotopes. The occupancy of the νh / orbitalfor each level is also shown on top of each bar in these panels. The probability of the neutron seniority, P ( υ ν = 2 and 4) forthe (e) 29 / + (blue), 25 / + states (red) in In; and (f) 27 / + (blue filled and hatched), 23 / + (red filled hatched) states in Sbisotopes. bution from I ν = 10 + dominates at high mass numberand that the contribution of I ν = 8 + increases with de-creasing mass number. Similarly in Fig 8(f) the invertedtrends, as compared to 25 / + states in In, can be seen inthe contribution of υ ν = 2 and 4 to the 23 / + states.Summarizing, the shell model calculations, the pres-ence of the 25 / + isomers in lighter In isotopes was repro-duced and the B ( E
2) values of the corresponding γ -decaywere well described. An in-depth analysis and the com-parison of the I πMax (In) = 29 / + and I πMax (In) − / + in In and I πMax (Sb) = 27 / + and I πMax (Sb) − / + in Sb revealed several “mirror”-like symmetries betweenthe corresponding states in In and Sb, as a function ofthe mass number: • The energy of the I πMax (In) = 29 / + is stronglydown sloping while the energy of the I πMax (Sb) =27 / + is up sloping, with increasing mass number.Both states result dominantly from the stretchedcoupling of proton-hole or proton-particle with theneutron 10 + with seniority υ ν = 2 state of the un-derlying Sn core. • The energies of the I πMax (In) − / + and the I πMax (Sb) − / + are similar and increaseslightly with increasing mass number. Both re-sult dominantly from the coupling of proton-hole orproton-particle coupled to a mixture of the neutron8 + and 10 + states with strong mixing of seniorities υ ν = 2 and 4. In In, at lower mass number thedominant contribution is from the 10 + state andthe seniority υ ν = 2, both decrease with increas-ing mass number. On the contrary, in Sb, at lowermass number the dominant contribution is from the8 + state and the seniority υ ν = 4, and the trendsare opposite to those observed in In.Those “mirror”-like symmetries could be understood asresulting from the particle-hole and the hole-hole sym-metry of the νπ interaction. V. SUMMARY AND CONCLUSIONS
The neutron-rich odd- A , In isotopes were pro-duced as fission fragments in the reaction Be (
U, f) atenergies around the Coulomb barrier. New prompt tran-sitions were identified above the 25 / + isomer in Inalong with the remeasurement of the half-life of this iso-mer. Additionally, the possibility of a short half-life ofthe (19 / − ) isomeric states in odd- A , In isotopeswas demonstrated. These results were possible using theunique combination of AGATA, VAMOS++ and EX-OGAM detectors, measuring the prompt-delayed spec-troscopy of isotopically identified fission fragments. Shell model calculations, using a modification of the interac-tion used in Ref. [26], were used to interpret the dataand also study the evolution of the energy of the 21 / − ,25 / − and 29 / − levels in isotopic chains of odd- A In andthe corresponding 19 / − ; 23 / − and 25 / − levels in odd- A Sb, using a decomposition in terms of the probabilitydistributions of the neutron angular momentum and neu-tron seniority. A good agreement was obtained betweenthe measured energies and the B ( E
2) transition proba-bilities and shell model calculations. In addition, in theodd- A In isotopes (having a πg − / νh − / configuration)the newly measured B ( E
2; 25 / + → / + ) were shownto follow a similar behavior to those for the B ( E
2; 10 + → + ) in even- A Sn ( νh − / ) isotopes and B ( E
2; 23 / + → / + ) in odd- A Sb isotopes ( πg / νh − / ). Shell modelcalculations presented in this work show that the πg / hole coupled to the νh − / configuration leads to the low-ering of the energy of the maximum aligned spin 29 / + level as a function of increasing neutron number, thusleading to the observation of 29 / + isomeric states inthe , In isotopes. The πg / particle coupling tothe νh − / configuration leads to the increase of the en-ergy of maximum aligned spin 27 / + level as a functionof increasing neutron number leading to 23 / + isomericstates in − Sb isotopes. These results illustrate therole of the particle-hole and the hole-hole symmetry ofthe νπ interaction. Future experiments using prompt-delayed spectroscopy that are sensitive to short lifetimescould allow to determine the lifetime of the 19 / − statein , In that would further clarify the high spin struc-ture of these isotopes.
VI. ACKNOWLEDGMENTS
The authors would like to thank the AGATA Collab-oration for the availability of the AGATA γ -ray track-ing array at GANIL. We acknowledge the importanttechnical contributions of GANIL accelerator staff. Wethank C. Schmitt for help during the experiment andcareful reading of the manuscript. We acknowledge A.O. Macchiavelli for help during the experiment. Wealso thank P. Van Isacker for his valuable discussionson the theoretical interpretation and careful reading ofthe manuscript. PB and AM acknowledge support fromthe Polish National Science Centre (NCN) under Con-tract No. 2016/22/M/ST2/00269 and the French LEACOPIGAL project. SBi, RB, SBh, SBh and RP acknowl-edge support from CEFIPRA project No. 5604-4 andthe LIA France-India agreement. HLC and PF acknowl-edge support from the U.S. Department of Energy, Of-fice of Science, Office of Nuclear Physics under ContractNo. DE-AC02-05CH11231 (LBNL). 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