Competing single-particle and collective states in the low-energy structure of 113 I

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I. INTRODUCTION
The systematic comparison of reduced transition probabilities, determined from lifetime measurements, can reveal trends in the evolution of nuclear shapes in isotopes spanning regions of single-particle and collective behavior.Reduced transition rates between the ground and first excited 2 + states in the even-A Sn isotopes showed excellent agreement with those calculated within the framework of the shell model using a Hamiltonian based on the the state-of-the-art chargedependent Bonn (CD-Bonn) nucleon-nucleon interaction [1].Although the best agreement was observed when a 100 Sn core was adopted, a deviation was observed for isotopes with N < 66 which has been interpreted as an increase in collectivity as the N = Z = 50 doubly-magic shell closure is approached [2][3][4][5].However, any such increase in collectivity or weakening of the N = 50 shell closure currently lacks a stringent theoretical interpretation [6].
The recently determined B(E2; 0 + → 2 + ) = 0.10(4) e 2 b 2 value for 104 Sn, using the PRESPEC setup at GSI [7], is significantly lower than that measured for 106 Sn [5] and heavier isotopes [2][3][4][5] and shows that the previously proposed anomalous B(E2) values for the lightest tin isotopes should be treated with caution.The measured B(E2) for 104 Sn is well reproduced by shell-model calculations indicating a fairly robust N = Z = 50 shell closure.In addition, the measured B(E2; 2 + → 0 + ) value for 108 Te [8] indicates that no enhancement of collectivity relative to the standard large-scale shell model (LSSM) predictions was found for this N = 56 nucleus.Further spectroscopic data in neighboring nuclei, such as the neutron deficient iodine isotopes, may help to shed some light on this important issue and provides the motivation for the present work.
In a previous study of 113 I, Starosta et al. [9] observed excited states to very high spin (∼60 h) thus establishing a comprehensive decay scheme.In subsequent work, Petkov et al. [10] measured lifetimes for some of the negative-parity states in the yrast band; however, lifetimes were not reported for the low-lying positive-parity states.Lifetime data for the yrast 9/2 + state in the odd-A isotopes is only available for A > 113 and the only other measurement for a low-lying 11/2 + state in these isotopes is 109 I which was performed by this collaboration [11].For 109 I, the 11/2 + → 7/2 + experimental reduced transition probability was found to be considerably smaller than that predicted by shell-model calculations using the CD-Bonn nucleon-nucleon potential.As 109 I is unbound to proton decay, the discrepancy was explained by the inability of the shell-model calculations to accurately account for the behavior of the unbound nuclear states [11].It was clear from the previous studies of the tin and iodine isotopes that further spectroscopic information on the low-energy structure of 113 I was needed.

II. EXPERIMENTAL DETAILS AND DATA ANALYSIS
The recoil distance Doppler shift (RDDS) technique [12] has proven to be an extremely successful method for determining the lifetimes of excited nuclear states down to ∼1 ps.
When sufficient statistics are available, the use of coincident γ -ray transitions have allowed the effects of unobserved side feeding to be eliminated from the lifetime derivation, yielding a highly accurate measurement of the states of interest.
Excited states in 113 I were populated using the 58 Ni( 58 Ni, 3p) reaction at a beam energy of 210 MeV.The K130 cyclotron, at the Accelerator Laboratory of the University of Jyväskylä, Finland, accelerated the 58 Ni beam onto a 1.05 mg/cm 2 58 Ni target.The target, along with a 1.2 mg/cm 2 Mg degrader foil, were mounted within the Differential Plunger for Unbound Nuclear States (DPUNS) [13] at the center of the Jurogam-II γ -ray spectrometer [14].The DPUNS degrader foil acted to reduce the full velocity of the recoiling reaction products from v/c = 0.040(1) to v/c = 0.025 (1).DPUNS was operated in "gas mode" with the recoil ion transport unit (RITU) [15,16] being used as a beam dump.The helium gas of RITU was extracted from the beamline by a Roots differential pumping system (see Ref. [13] for further information).Prompt γ rays de-exciting the 113 I nuclei were recorded by the Jurogam-II array which comprised 15 Eurogam Phase-1-type single-crystal germanium detectors, located at backward angles of 157.6 • (5 ring 1 detectors) and 133.6 • (10 ring 2 detectors) to the beam axis and 24 clover detectors located in two detector rings (rings 3 and 4) either side of 90 • (±14.5 • ).All γ rays detected by the Jurogam-II detectors were time stamped by a 100 MHz clock from the total data readout acquisition system [17].Data were sorted into one-dimensional histograms and two-dimensional matrices for offline analysis.The lifetimes of the low-lying states in 113 I were determined using the differential decay curve method (DDCM) [18].In the present work, the lifetime of the level of interest, τ (x), was derived at each target-to-degrader distance x using the relationship where I s A and I d A correspond to the normalized intensities of the shifted (s) and degraded (d) components of the depopulating transition that correlate with the shifted component (I s B ) of the transition feeding the state of interest.The intensities were normalized across each distance to the number of γ rays in the corresponding, ungated, total projection.Finally, v is the mean velocity of the recoiling nuclei.The lifetime of the state under investigation is then determined from a piecewise fit to the normalized shifted intensity data.The derivative of this function, multiplied by the value of the lifetime τ (x) is simultaneously fitted to the denominator in Eq. (1) and a χ 2 minimum found.The weighted average of each individual lifetime value, within the so-called "region of sensitivity" [19], was used to deduce the final lifetime value.450 keV transition from 114 Xe (2p channel) can be seen along with numerous decays from A ≈ 80 nuclei produced in 58 Ni reactions on the Mg degrader foil.Figure 1(b) was created from the same γ -γ matrix as that used for Fig. 1(a) and shows decays that correlate with the 531 keV 15/2 − → 11/2 − transition in 113 I.All of the transitions in Fig. 1(b) have been identified as belonging to 113 I [9,10].

III. RESULTS
Previous studies of 113 I managed to establish a comprehensive level scheme to high spins [9], thus the γ -γ coincidence matrix produced in this work was only used to confirm the transition intensities and low-spin feeding patterns.
Figure 2 shows the resulting partial level scheme highlighting the states and transitions that are of particular importance to the subsequent excited-state lifetime analysis.The γ -ray intensities shown in Fig. 2 have been efficiency corrected and are summarized along with the transition energies in Table I.

A. Negative-parity band
Previous work by Petkov et al. [10], measured lifetimes for some of the states in the negative-parity yrast band which is understood to be built upon a low-K prolate πh 11/2 orbital.In particular, an accurate measurement of the 15/2 − state lifetime, τ = 7.22(43) ps, was made [10].A remeasurement of this lifetime was performed with the current data set to reveal any potential systematic errors associated with the experimental setup.For excited-state lifetime determination a γ -γ correlation matrix was produced using data recorded in the Jurogam-II ring 1 (157.5 • ) against ring 2 (137.5 • ) detectors.The large angles, relative to the recoil direction, that these detectors subtend, maximizes the separation between the shifted and degraded components of the photopeak.By gating on the shifted component of the 638 keV transition (ring 2) and projecting onto the ring 1 axis, the intensities of the 531 keV shifted and degraded components were measured as a function of target-to-degrader distance.Figure 3 shows the resulting ring 1 spectra for four target-to-degrader distances ranging from 62 to 760 μm.The shifted (left peak) and degraded (right peak) intensity change is clear with increasing distance, with the 531 keV decay being fully shifted by 760 μm.The Gaussian fits used to extract the relative intensities are  also shown.The measured intensities were normalized to the number of counts in the ring 1 total projections.A simultaneous fit to the shifted and degraded data was performed and the result used to evaluate the state lifetime at each target-to-degrader distance in the region of sensitivity.A weighted mean resulted in a lifetime of 7.21(43) ps, in excellent agreement with that measured previously, 7.22(43) ps [10].The agreement between the literature value for the 15/2 − state lifetime and that measured in this work shows that no significant systematic errors are present on any lifetime values extracted from these data.
The current literature value for the lifetime of the 11/2 − state, τ = 229(52) ps [10], has a 23% uncertainty, prompting a remeasurement using this data set.The analysis for the 11/2 − state lifetime is analogous with that performed for the 15/2 − state.A ring 2 gate on the shifted component of the 531 keV transition was used to produce a ring 1 spectrum containing the shifted and degraded components of the 388 keV decay.Figure 4(a) shows the 11/2 − state lifetime evaluated for targetto-degrader distances in the region-of-sensitivity along with the weighted mean of τ = 216(7) ps. Figure 4(b) shows the normalized shifted intensities along with the result of a leastsquares fit (solid line) which had a χ 2 value of 1.25.The newly measured value is consistent with the current literature value but has a much improved uncertainty (∼3%).

B. 11/2 + state lifetime
The previous RDDS study of 113 I did not report on any measurement of the lifetimes of the low-lying positive-parity states that are populated by decays out of the 11/2 − band head.In the present work, a measurement of the positiveparity 11/2 + state was performed using a direct gate on the shifted component of 864 keV decay (ring 2) and measuring the component intensities of the 690 keV transition in the ring 1 projection.A gate on the 263 keV feeding transition could not be used in this analysis, because the degraded component of the 684 keV 23/2 − → 19/2 − transition, present in the decay path, interferes with the shifted component of the 690 keV decay that depopulates the 11/2 + state.Figure 5 shows the resulting ring 1 spectra for three target-to-degrader distances ranging from 7 μm to 10 mm.The relative change in the shifted (left peak) and degraded (right peak) photopeak intensities with increasing distance is clear with the 690 keV decay being fully shifted at 10 mm.The Gaussian fits used to extract the relative intensities are also shown.
Figure 6(a) shows the 11/2 + state lifetime evaluated for target-to-degrader distances in the region-of-sensitivity along with the weighted mean of τ = 3.7(7) ps. Figure 6(b) shows the normalized shifted intensities along with the result of a least-squares fit (solid line) which had a χ 2 value of 0.35.

C. 9/2 + state lifetime
The 9/2 + state lifetime spectra could not be obtained by gating on a direct feeding transition (as was used for the 15/2 − , 11/2 − , and 11/2 + states) due to contamination from higher-lying transitions.Instead, a gate was placed on the degraded component [20] of the 629 keV 9/2 + → 5/2 + decay and the shifted and degraded intensities of the populating 388 keV decay were measured.Figure 7 shows the resulting ring 1 spectra for three target-to-degrader distances ranging from 217 μm to 10 mm.The Gaussian fits used to extract the relative intensities are also shown.Figure 8(a) shows the 9/2 + state lifetime evaluated for target-to-degrader distances in the region-of-sensitivity along with the weighted mean of τ = 28(4) ps. Figure 8(b) shows the normalized shifted intensities along with the result of a least-squares fit (solid line) which had a χ 2 = 1.6.

IV. DISCUSSION
Reduced quadrupole transition probabilities can be deduced from measured state lifetimes using where E γ is in MeV, τ is in picoseconds, and the (1 + α) term corrects for internal conversion.Using Eq. ( 2

A. Shell-model calculations
In a simple shell-model picture one may expect the lowenergy structure of 113 I to arise from configurations dominated by the three valence protons outside of the Z = 50 close shell.The unpaired proton predominantly residing in the d 5/2 orbital and the remaining two protons (and the ten valence neutrons outside of the N = 50 closed shell) coupled to spin zero would constitute the 5/2 + ground state.Starting from this simple picture, the first excited 9/2 + state could then be created from a recoupling of a pair of spin-zero coupled protons to spin 2. The observed 7/2 + state, however, cannot be explained in this way.A significant occupation of another orbital, such as the g 7/2 , needs to be invoked in order to create an excited 7/2 + state.The observation of the 566 keV decay from the 9/2 + state may suggest that the g 7/2 orbital plays a significant role in the generation of the 9/2 + state, because a proton or neutron pair recoupling back to spin zero could not produce the final 7/2 + state.A more likely scenario is that the 9/2 + state arises from a pair of spin-2 coupled protons predominately occupying the g 7/2 orbital as opposed to the d 5/2 orbital.The 11/2 + state could then be understood in terms of the odd proton occupying the g 7/2 orbital along with a spin-2 coupled proton pair.It is therefore evident that the low-lying positive-parity states of interest in this work could be explained by a simple shell-model picture involving only the d 5/2 and g 7/2 orbitals and the recoupling of proton spins to zero or two.
for the 1g 7/2 , 2d 5/2 , 2d 3/2 , and 1h 11/2 orbitals, respectively.The residual two-body interaction was obtained from a G matrix derived from the CD-Bonn nucleon-nucleon potential [1,23].The aforementioned arguments were used to restrict the neutron and proton occupations; all nucleons were restricted to the d 5/2 and g 7/2 orbitals with neutrons filling the d 5/2 orbital.The calculated energy eigenvalues are compared to those determined experimentally in Fig. 9.Although the shell-model overestimates the excitation energy of the 7/2 + state, good agreement is observed between the calculated and experimental partial level schemes.
Using the calculated wave functions, the quadrupole reduced transition probability for the 9/2 + → 5/2 + transition was determined.A Woods-Saxon potential along with standard effective proton and neutron charges of 1.5e and 0.5e, respectively, were adopted.A B(E2) value of 45 W.u. was calculated for the 629 keV transition which agrees, within 3σ , with the B(E2) = 32(5) W.u. measured in this work.An intrinsic quadrupole moment of −0.12 e b was also calculated for the 9/2 + state.The dominant wave-function configurations for the excited 7/2 + , 9/2 + , and 11/2 + states are represented graphically in Fig. 10.The calculated wave function for the 9/2 + state exhibits a significant g 7/2 contribution, as postulated.The similarity between the configurations for the  In the limit of rigid rotation the transition quadrupole moment Q t equals the intrinsic quadrupole moment Q 0 .The intrinsic quadrupole moment of the 11/2 + state can, therefore, be determined within the framework of the rigid rotor model [24], using B(E2 : The measured transition probabilities and shell-modelcalculated wave functions show a clear difference between the structures of the 11/2 + and 9/2 + states.This difference should also manifest itself in the E1 transitions that feed into these structures from the 11/2 − state.Taking into account a weak 179 keV decay to a second 9/2 + 2 state [10], partial lifetimes of 304 (25) and 1075(94) ps were deduced for the 388 and 263 keV decays, respectively, using the 216 (7) ps lifetime for the 11/2 − state remeasured in this work.Using the deduced partial lifetimes and conversion coefficients taken from BrIcc [25], B(E1) values of 5.5(5) × 10 −4 and 4.9(5) × 10 −4 W.u. were calculated for the 388 and 263 keV decays, respectively.The similarity between the decay probabilities for the 388 and 263 keV transitions, assumed to be populating states with different structures, is startling.This similarity was alluded to, but not discussed in detail, in the previous study of 113 I by Petkov and co-workers [10].In that study the lifetimes of the 15/2 − and 19/2 − states were measured along with that of the 11/2 − state.The deduced transition probabilities were explained using the rigid rotor model assuming incomplete alignment of the odd proton.The best description of the data was obtained using quadrupole deformation parameters in the range β 2 = 0.24 − 0.26.The quadrupole deformation for the 11/2 − state is much larger than that for the 11/2 + state.This difference in deformation could be a manifestation of a change in shape between the 11/2 − and 11/2 + states which would hinder the decay somewhat.To try and understand the origin of this similarity between strengths for transitions populating two apparently different structures, configuration-constrained total Routhian surface calculations have been performed in this work.

D. Configuration-constrained total Routhian surface calculations
Total Routhian surface (TRS) calculations provide a theoretical means of describing collective states in rotating nuclei.As the extent of nuclear deformation can change with rotational frequency, total Routhian surfaces can yield important information on the evolution of collectivity within and between excited bands.Calculations were performed for 113 I using the model detailed in Ref. [26].The results showed that the valence neutrons do not contribute significantly to the wave functions of states with excitation energies below 1.6 MeV.
Previous TRS calculations by Paul et al. [27] stated that the low-spin (−, −1/2) configuration with deformation parameters β = 0.19 and γ ∼ +7 • could be associated with a band built on the πh 11/2 , 1/2 − [550] orbital and was identified with the negative-parity band built upon on the 11/2 − state.The present TRS calculations could not produce a one-quasiparticle, 11/2 − bandhead state with an excitation energy of around 1 MeV.The 1/2 − [550] configuration resided at an excitation energy of 468 keV without cranking which was pushed up to 560 keV once cranked.Three-quasiparticle configurations all had excitation energies in excess of 2 MeV, at least double the experimental value of 1017 keV for the 11/2 − state.
It is well known that E1 transitions in medium mass nuclei are often hindered [28]; for example, the 1 + → 2 − E1 transition in 120 I has a reduced transition probability of 5.3(4) × 10 −4 W.u. It may be expected that both the 388 k and 263 keV decays would be hindered due to the configuration changes taking place; however, a larger hindrance may be expected for a stretched decay from a deformed collective state to a shell-model-like configuration than for an unstretched decay to a prolately deformed state.Figure 11 shows a slight γ softness for the 11/2 + state which could result in a fairly mixed configuration for this state.This mixing could possibly increase the overlap between the initial and final state wave functions thus reducing any potential decay hindrance.Therefore, it is still not fully clear why similar E1 strengths have been observed for decays from the 11/2 − state to apparently dissimilar, competing structures.Further lifetime measurements in neighboring nuclei, for example 111 I, are required to help shed light on this issue.

Figure 1 (FIG. 1 .
Figure 1(a) shows an ungated γ -ray spectrum obtained from a Jurogam-II clover versus clover γ -γ matrix (ring 3 versus ring 4 detectors).The most prominent yrast decays in 113 I are denoted by their energies.The most intense decays originating from other reaction channels are denoted by symbols.A

FIG. 2 .
FIG.2.Partial level scheme for 113 I, produced from a γ -γ correlation matrix, showing the states and transitions that are of particular importance in this work.The observed correlations confirm the transition ordering reported in Refs.[9,10].The state lifetimes shown are those measured in this work.

FIG. 4 .
FIG. 4. (Color online) (a) The lifetime of the 11/2 − state evaluated for each target-degrader separation in the region-of-sensitivity along with the weighted mean of 216 ps (solid line) and its uncertainty ±7 ps (dashed lines).(b) Normalized shifted intensities for the 11/2 − → 9/2 + 388 keV decay along with the results of a least-squares fit (solid line).

FIG. 6 .
FIG. 6. (Color online) (a)The lifetime of the 11/2 + state evaluated for each target-degrader separation in the region-of-sensitivity along with the weighted mean of 3.7 ps (solid line) and its uncertainty ±0.7 ps (dashed lines).(b) Normalized shifted intensities for the 11/2 + → 7/2 + 690 keV decay along with the results of a leastsquares fit (solid line).

FIG. 8 .
FIG. 8. (Color online) (a)The lifetime of the 9/2 + state evaluated for each target-degrader separation in the region-of-sensitivity along with the weighted mean of 28 ps (solid line) and its uncertainty ±4 ps (dashed lines).(b) Normalized shifted intensities for the 11/2 − → 9/2 + 388 keV decay along with the results of a leastsquares fit (solid line).

TABLE I .
113I efficiency corrected γ -ray intensities measured in this work for the transitions shown in Fig.2.Intensities have been normalized to the 531 keV 15/2 − → 11/2 − transition.

TABLE II .
Summary of the lifetimes and reduced quadrupole transition probabilities measured in this work.Shell-model calculated transition probabilities are also shown.+ and 11/2 + states is clear from Fig. 10 as is the difference between those configurations and that of the 9/2 + state.B(E2) = 59 W.u. was calculated for the 11/2 + → 7/2 + 690 keV transition, which is considerably less than the experimental value of 209(39).The measured and calculated reduced transition probabilities are summarized in Table II.It is interesting to note that the shell-model overestimated the experimental B(E2 : 11/2 + → 7/2 + ) = 710(90) e 2 fm 4 [B(E2) SM = 1146 e 2 fm 4 ] for 109 I, whereas the shell model underestimates the corresponding B(E2) for 113 I. B.

11/2 + state
• .The calculated excitation energy and deformation parameter for this configuration are in good agreement with those of the 11/2 + FIG.11. Results from configuration-constrained total Routhian surface calculations for the 11/2 + state in 113 I.The calculated surface shows a minimum at β 2 = 0.18, γ = 7.4 • for the 11/2 + state at an excitation energy of 690 keV with a configuration of 1/2 + [420].