Broken seniority symmetry in the semimagic proton mid-shell nucleus

Lifetime measurements of low-lying excited states in the semimagic ( N = 50) nucleus 95 Rh have been performed by means of the fast-timing technique. The experiment was carried out using γ -ray detector arrays consisting of LaBr 3 (Ce) scintillators and germanium detectors integrated into the DESPEC experimental setup commissioned for the Facility for Antiproton and Ion Research ( fair ) Phase-0, Darmstadt, Germany. The excited states in 95 Rh were populated primarily via the β decays of 95 Pd nuclei, produced in the projectile fragmentation of a 850 MeV/nucleon 124 Xe beam impinging on a 4 g/cm 2 9 Be target. The deduced electromagnetic E2 transition strengths for the γ -ray cascade within the multiplet structure depopulating from the isomeric I π = 21 / 2 + state are found to exhibit strong deviations from predictions of standard shell model calculations which feature approximately conserved seniority symmetry. In particular, the observation of a strongly suppressed E2 strength for the 13 / 2 + → 9 / 2 + ground state transition cannot be explained by calculations employing standard interactions. This remarkable result may require revision of the nucleon-nucleon interactions employed in state-of-the-art theoretical model calculations, and might also point to the need for including three-body forces in the Hamiltonian.

Lifetime measurements of low-lying excited states in the semimagic (N = 50) nucleus 95 Rh have been performed by means of the fast-timing technique.The experiment was carried out using γ-ray detector arrays consisting of LaBr3(Ce) scintillators and germanium detectors integrated into the DESPEC experimental setup commissioned for the Facility for Antiproton and Ion Research (fair) Phase-0, Darmstadt, Germany.
The excited states in 95 Rh were populated primarily via the β decays of 95 Pd nuclei, produced in the projectile fragmentation of a 850 MeV/nucleon 124 Xe beam impinging on a 4 g/cm 2 9 Be target.The deduced electromagnetic E2 transition strengths for the γ-ray cascade within the multiplet structure depopulating from the isomeric I π = 21/2 + state are found to exhibit strong deviations from predictions of standard shell model calculations which feature approximately conserved seniority symmetry.In particular, the observation of a strongly suppressed E2 strength for the 13/2 + → 9/2 + ground state transition cannot be explained by calculations employing standard interactions.This remarkable result may require revision of the nucleon-nucleon interactions employed in state-of-theart theoretical model calculations, and might also point to the need for including three-body forces in the Hamiltonian.
Introduction.One of the most intriguing aspects of nuclear phenomenology is the emergence of regular and simple structural patterns from the complex nuclear many-body correlations.In particular, the vast majority of semimagic nuclei (those with either the number of neutrons, N , or the number of protons, Z, corresponding to a filled quantum shell) are characterized by the pairwise coupling of the valence nucleons in the unfilled shell [1].The seniority quantum number, ν, is defined as the number of neutrons or protons that are not coupled in pairs to angular momentum J = 0 [2].Characteristic, regular energy spectra and special patterns of the electric quadrupole transition strengths between the member states of the corresponding seniority multiplets arise as a combined effect of the strong spin-orbit coupling and residual pairing correlations in the nuclear mean field.
Even though seniority is an approximate symmetry it hence has a profound impact on the description of the spectroscopic and electromagnetic transition properties of nuclei near closed quantum shells.It is a strictly conserved quantum number for systems with identical particles in a single-j angular momentum subshell with j ≤ 7/2 in the presence of an attractive two-body pairing force.Even for systems with higher j values, like for particles in the g 9/2 , j = 9/2 subshell, the seniority-violating interaction matrix elements are expected to be negligible in most empirical shell model interactions.The j = 9/2 case has recently received particular interest with respect to the special partial conservation of seniority in systems with four proton particles/holes [3][4][5][6][7][8][9][10].
A striking consequence of seniority symmetry is that the squares of the ∆ν = 0 matrix elements of even-tensor oneand two-particle operators, such as the electromagnetic quadrupole (E2) operator, are symmetric with respect to the middle of the angular momentum subshell, i.e. when the number of particles occupying levels in the angular momentum subshell is n j = (2j +1)/2.This can be considered to be a result of the combination of particle-hole conjugation and seniority conservation, intertwined through a Berry phase [11].The corresponding transition probabilities between such states therefore vanish when the Fermi level is situated in the middle of the subshell, as is the case for 95 Rh.Converseley, the seniority changing ∆ν = 2 matrix elements of the same operators, which mirror the behavior of the seniority conserving matrix elements as a function of shell filling, are maximal at mid-shell.
The few observed exceptions from the general rule of energy level spectra and E2 transition rates following the predictions of shell model calculations with a seniority conserving nuclear interaction are particularly interesting since they may reveal otherwise hidden details of the nuclear force.Such cases include the semimagic (N = 50) nuclei 94 Ru and 96 Pd, for which there are large discrepancies between the observed 4 + → 2 + electromagnetic E2 transition strengths and theoretical predictions in different directions [7,[12][13][14][15][16].For these nuclei the valence protons occupy the upper half of the N/Z = 28 − 50 major shell which is characterized by the relative isolation of the g 9/2 subshell.Mach et al. reported a lower limit, B(E2:4 + → 2 + ) ≥ 46 e 2 f m 4 for 94 Ru and a value B(E2:4 + → 2 + ) = 3.8(4) e 2 f m 4 for 96 Pd, corresponding to strongly enhanced and retarded 4 + → 2 + E2 transition strengths, respectively, compared with seniority-conserving shell model predictions [7].They suggested that the observed anomalous E2 strengths in these N = 50 isotones and the inferred seniority symmetry breaking is due to residual neutron-proton interactions in combination with neutron particle-hole (ph) excitations across the N = 50 shell gap.Das et al. recently reported a value B(E2:4 + → 2 + ) = 103(24) e 2 f m 4 for the same transition in 94 Ru [12] and proposed, alternatively, the observed seniority symmetry breaking effect in 94 Ru to be a result of a subtle interference between the wave functions of the initial and final states induced by cross-orbital interactions within the major valence shell.Subsequently, Pérez-Vidal et al. reported a value B(E2:4 + → 2 + ) = 38(3) e 2 f m 4 for 94 Ru, differing significantly from the previously reported results [7,12], and arrived at a different conclusion, namely that seniority is largely conserved in the first πg 9/2 orbital [13].This situation requires further investigation, both from an experimental and theoretical stand point.
The nucleus 95 45 Rh 50 is located exactly at the πg 9/2 midshell and should therefore exhibit approximate particle-hole symmetry, i.e. it can be described both as five valence protons and five proton holes in the g 9/2 orbital.A puzzling enhancement of the B(E2) strength in this nucleus has previously been reported [7,[14][15][16] for the 21/2 + → 17/2 + 1 transition instead of the near-vanishing E2 transition rate expected from approximate seniority conservation for a ∆ν = 0 transition.
In this Letter we present lifetime measurements on the low-lying yrast states of 95 Rh using the direct, fast-timing method.The extracted E2 transition strengths are compared with the results from large-scale shell-model and single-j shell calculations.A strong violation of the seniority coupling scheme is observed, which cannot be reproduced by state-of-the-art empirical shell model interactions applied in different model spaces.
Experiment Details and Data Analysis.Lifetime measurements of low-lying yrast states of 95 Rh were performed using the DEcay SPECtroscopy (despec) [17] setup commissioned for the Facility for Antiproton and Ion Research (fair) [18] Phase-0.The results presented here were obtained from the same measurement as those previously reported for 94 Ru [12].
124 Xe ions were accelerated to a kinetic energy of 850 MeV/u by the SIS-18 synchroton at the GSI Helmholtzzentrum für Schwerionenforschung accelerator facility, Darmstadt, Germany and impinged on a 9 Be target of 4 g/cm 2 areal density.Nuclear fragments produced in the reactions were identified and transported to the final focal plane of the FRagment Separator (frs) using the Bρ-∆E-Bρ and ToF-Bρ-∆E methods [20,21].The Advanced Implantation Detector Array (aida) [22,23], composed of three doublesided silicon strip detectors (dsssd), was used to stop the product nuclei and to measure their subsequent chargedparticle decays.The White Rabbit (wr) [24] clock was used to save the implant timing.WR is a common clock to all DESPEC subsystems and used to synchronize these subsystems with 1 ns precision.As a result, the timing information of each implanted 95 Pd ion was saved as a function of its position, (x, y), where x and y are the horizontal and vertical strip numbers of the DSSSD respectively.After implantation of the 95 Pd ions, population of the 21/2 + isomer leads to β-decay into the analogue spin-parity state of the 95 Rh nucleus [25], which then de-excites towards the 9/2 + ground state via a cascade of stretched γ-ray transitions.To identify such β-delayed γ rays, β decays correlated with implants of the 95 Pd ions were searched for within the DSSSD pixels.The time correlation was obtained using the WR clock with a ∼ 3×T 1/2 (T 1/2 = 14(1)s [25]) time window, within which the β-decay pixel position was validated if an ion had been implanted in the same pixel or in any of its immediate neighbors, taking into account that the highly penetrating β rays may scatter to the neighboring pixels from the implantation point depositing partial or full energy.
The FATIMA LaBr 3 (Ce) detectors were used to measure direct γ-γ time differences, with a resolution of 25 ps least significant bit [17].Mean level lifetimes (τ ) for the yrast states of 95 Rh were deduced using the Generalised Centroid Difference (gcd) method [30,31].The centroid difference for a coincident γ-ray pair , ∆C, is directly related to τ , following the relation with the symmetry conditions [31] ∆C(∆E γ ) = −∆C(−∆E γ ), where, ∆E γ is the energy difference between the feeding and the decaying γ rays of the level and PRD is the Prompt Response Difference [30].The calibration measurement of the PRD function for the present setup has been described in Ref. [12].The delayed and anti-delayed time distributions obtained for the 13/2 + and 17/2 + 1 states, are shown in FIG.2(a) and 2(d), respectively, from which the generalized time centroids [33] were obtained.The background contribution to the time distribution was substracted according to the method described in Ref. [34].The time spectra due to background coincidences are shown in FIG.2(b,  c) and FIG.2(e, f) for the 13/2 + and 17/2 + 1 states respectively.The P RD values of −321(±6 ± 14 ± 16)ps and −290(±6 ± 14 ± 15)ps were obtained from the fit shown in FIG.2(b) of Ref. [12], for the 13/2 + and 17/2 + 1 states respectively.The P RD errors in bracket include, the time uncertainty introduced by the large energy width of LaBr 3 detectors, along with the fit residual errors for start and stop energies respectively.One may note that, this is the only source of systematic errors built into the GCD method [30].The cancellation of systematic errors from different sources therefore makes it advantageous over other fast-timing methods.The obtained lifetime values are τ (13/2 + 1 ) = 36 (15)ps and τ (17/2 + 1 ) = 8(18)ps, the latter corresponding to a limit τ (17/2 + 1 ) ≤ 26ps with 1σ uncertainty.The uncertainties include uncorrelated errors added in quadrature [35].The use of the GCD method with the present experimental setup was also validated by remeasuring the mean lives of the 4 + 1 and 2 + 1 states in 94 Ru and 96 Pd from the same experiment.For 94 Ru, the lifetimes of the 4 + 1 and 2 + 1 states were observed to be τ = 32 (11) ps and τ ≤ 14 ps, respectively [12] and consistent with the respective τ ≤ 72 ps and τ ≤ 14 ps limits previously established by Mach et al. [7].For 96 Pd, A. Yaneva et al., [37] have measured the value τ (4 + 1 ) = 1.44 (7) ns and the limit τ (2 + 1 ) ≤ 20ps, which also confirms the previous measurements [7].It is to be noted that, despite the similar ∆E γ values for the first excited states in these nuclei, the delayed nature of the 13/2    Discussion.To understand the observed E2 transition properties of the low-lying yrast structure in 95 Rh, we have carried out configuration interaction model calculations including extensive Large-Scale Shell Model (lssm) calculations in a variety of model spaces and compared the results with those from a pure single-j shell calculation.In FIG. 3 the experimental low-lying yrast spectrum and the associated E2 transition strengths between the members of the πg 9/2 seniority multiplet are compared with theoretical calculations using the single πg 9/2 (labeled "g") model space with the same seniority-conserving empirical interaction as used in Ref. [38], the πf 5/2 p 1/2,3/2 g 9/2 (labeled "fpg") model space with the jun45 interaction [39], and the πνg 9/2,7/2 d 5/2,3/2 s 1/2 model space (labeled "gds"), employing the SDG CD-Bonn based G-matrix renormalized SDGN interaction [7,45], limiting the model space to allow a maximum of t = 6 particles that can be excited across the N = Z = 50 major shell.Effective charges of e p (e n ) = 1.5e(0.5e)were used for the calculations in the "g" and "fpg" model spaces and e p (e n ) = 1.11e(0.84e)[40] in the "gds" model space.Numerical results are listed in TABLE.II.The ν = 3-dominated, 17/2 + state is predicted to be lower in excitation energy than the ν = 5-dominated state in the "g" and "gds" model spaces whereas an inversion between the two states is seen in the "fpg" space calculation.However, predicted energy spectra and electromagnetic transitions strengths for the different model calculations are in general quite similar, in excellent agreement with approximate seniority symmetry.On the other hand, it may be immediately recognized that the calculated electromagnetic transitions strengths are in stark contrast with the experimental observations.In particular, none of the calculations are even close to reproducing the experimentally observed E2 transition rate for the 13/2 + 1 → 9/2 + ground-state transition.
The five protons in the 0g 9/2 subshell can couple to three I π = 9/2 + ; ν = 1, 3, 5, states, two 13/2 + ; ν = 3, 5 and two 17/2 + ; ν = 3, 5 states.The I π = 9/2 + ground state and the first I π = 13/2 + state (which is dominated by ν = 3) are predicted to be well separated in energy in calculations employing a variety of standard effective interactions in different model spaces while the two I π = 17/2 + states can be close to each other [9].A special property of mid-shell nuclei like 95 Rh is that the two-body interaction, operating within a single-j shell, can only mix configurations with seniority differing by ∆ν = 4.This means that the two 13/2 + and 17/2 + states with ν = 3 and 5 never mix in such calculations and the seniority symmetry is strictly conserved.In general, the 13/2 + 2 state (dominated by ν = 5) is predicted at much higher excitation energy than the first, 13/2 + 1 (ν = 3) state; at 3.03 MeV in the g calculation and 2.94 MeV in the fpg model space.The state is calculated to decay primarily to the 17/2 + ν = 3-dominated state or a 11/2 + state.
In general, the wave function of a state |α with angular momentum quantum number I can be expressed as a superposition of single-j configurations with different seniority and configurations as The three I π = 9/2 + configurations can mix with each other due to the allowed ∆ν = 4 mixing between the ν = 1 and ν = 5 configurations and interactions with the interme- diate ν = 3 state via mainly the close-lying p 1/2 subshell.However, the mixing between the ν = 1 I π = 9/2 + ground state and the ν = 3 and ν = 5 I π = 9/2 + states is predicted to be quite small (1.16% and 2.14%, respectively) due to the dominance of the pairing matrix element which increases the energy of the latter states to around 1.8 MeV and 2.3 MeV, respectively, for the calculation employing the realistic jun45 interaction [39].The ν = 3 and ν = 5 admixtures in the ground state wave function are consistently zero or of similar magnitude for all the interactions and model spaces employed in this work.The same applies to the predictions for the mixing between the 13/2 + 1 and 13/2 + 2 states.A particularly striking feature of the electromagnetic properties of 95 Rh is the observed strong hindrance of the 13/2 + 1 → 9/2 + transition, in contradiction to the predictions within the seniority coupling scheme as well as all of our LSSM calculations.The experimentally deduced B(E2:13/2 + 1 → 9/2 + ) value of 5.0 +3.6 −1.6 e 2 f m 4 is reduced by a factor more than 30 compared with the lowest theoretical prediction obtained in this work (TABLE.II).This indicates a strong violation of seniority symmetry and that the two ν = 3 and 5, 13/2 + states, unexpectedly, might be strongly mixed.We have therefore evaluated the influence of various non-diagonal matrix element contributions to Eq. 3 by varying their strengths.No significant mixing has been found in our calculations without invoking an unrealistic adjustment of those matrix elements that would lead to strong perturbations in, e.g., the predicted level energies.
We have also extended our calculations to include neutron cross-shell excitations including d 5/2 and g 7/2 configurations as in Refs.[41,42].However, no significant contributions to the wave functions from neutron core excitations across the N = 50 shell closure were found for the lowest I π = 9/2 + , 13/2 + and 17/2 + states, primarily due to the large energy gap (∼ 4 MeV) between the g 9/2 subshell and higher-lying shells.It is also noteworthy that no evidence for significant cross-shell excitations was observed for the neutron analogue system, 213 Pb, with five neutrons in the 1g 9/2 shell [11].There is very limited experimental information on the E2 transition probabilities in similar systems.We note, however, that the deduced limits on the E2 strengths for the 2 + 1 → 0 + g s transitions in the neighboring N = 50 isotones 94 Ru [7,12] and 96 Pd [7], which are expected to be similar to the 13/2 + 1 → 9/2 + transition in 95 Rh, are in agreement with the predicted behavior for conserved seniority symmetry.
The observed 13/2 + 1 → 9/2 + gs E2 transition strength in 95 Rh appears to be extremely difficult to reproduce using standard effective two-body shell model interactions.Although highly challenging and beyond the current state-ofthe-art in computational capabilities, it is possible that the inclusion of three-body forces into the shell-model Hamiltonian [43,44] could elucidate the mechanism behind this unexpected observation.For a more complete picture, experimental data on similar systems are also required.

FIG. 1 .
FIG. 1. (Color online) (a)The γ-energies of β − γ − γ events in coincidence with 1351 keV transition.The decay of I π = 21/2 + is depicted at the inset, while the 169 keV tansition is coming from the decay of I π = 17/2 − isomer.(b) The WR time difference between AIDA and FATIMA for the β-decay events from 95 Pd.

TABLE II .
Theoretical B(E2) strengths in 95 Rh calculated in different model spaces.The states are labeled by the dominant seniority component in the wave function.See text for details.
Experimental and theoretical level energies and electromagnetic transition strengths for the low-lying excited states in95Rh.The widths of the arrows are proportional to the experimentally deduced (a) and calculated (b, c and d) B(E2) values given in TABLE.I and TABLE.II.Dashed arrows correspond to a vanishing transition strength.The theoretically calculated energy levels are labeled by the dominant seniority quantum number in the wave function.For the calculation employing the jun45 interaction (c) the squared amplitude of the dominant signature component is also given.