Deformation of the proton emitter 113Cs from electromagnetic transition and proton-emission rates

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Deformation of the proton emitter 113Cs from electromagnetic transition and proton-emission rates Hodge, D.; Cullen, D. M.; Taylor, M. J.; Singh, B. S. Nara; Ferreira, L. S.; Maglione, E.; Smith, J. F.; Scholey, Catherine; Rahkila, Panu; Grahn, Tuomas; Braunroth, T.; Badran, Hussam; Capponi, L.; Girka, A.; Greenlees, Paul; Julin, Rauno; Konki, Joonas; Mallaburn, M.; Nefodov, O.; O'Neill, G. G.; Pakarinen, Janne; Papadakis, Philippos; Partanen, Jari; Ruotsalainen, Panu; Sandzelius, Mikael; Sarén, Jan; Smolen, M.; Sorri, Juha; Stolze, Sanna; Uusitalo, Juha Hodge, D., Cullen, D. M., Taylor, M. J., Singh, B. S. N., Ferreira, L. S., Maglione, E., Smith, J. F., Scholey, C., Rahkila, P., Grahn, T., Braunroth, T., Badran, H., Capponi, L., Girka, A., Greenlees, P., Julin, R., Konki, J., Mallaburn, M., Nefodov, O., . . . Uusitalo, J. (2016). Deformation of the proton emitter 113Cs from electromagnetic transition and proton-emission rates. Physical Review C, 94(3), Article 034321. https://doi.org/10.1103/PhysRevC.94.034321 2016 PHYSICAL REVIEW C 94, 034321 (2016) Deformation of the proton emitter 113Cs from electromagnetic transition and proton-emission rates D. Hodge,1D. M. Cullen,1M. J. Taylor,1,*B. S. Nara Singh,1L. S. Ferreira,2E. Maglione,3J. F. Smith,4C. Scholey,5 P. Rahkila,5T. Grahn,5T. Braunroth,6H. Badran,5L. Capponi,4A. Girka,5P. T. Greenlees,5R. Julin,5J. Konki,5 M. Mallaburn,1O. Nefodov,5G. G. O’Neill,7J. Pakarinen,5P. Papadakis,5J. Partanen,5P. Ruotsalainen,5M. Sandzelius,5 J. Sar´ en,5M. Smolen,4J. Sorri,5S. Stolze,5and J. Uusitalo5 1School of Physics and Astronomy, Schuster Building, The University of Manchester, Manchester M13 9PL, United Kingdom 2Centro de F´ ısica e Engenharia de Materiais CeFEMA and Departmento de F´ ısica, Instituto Superior T´ ecnico, Universidade de Lisboa, Av Rovisco Pais, 1049 001, Lisboa, Portugal 3Dipartimento di Fisica e Astronomia “G. Galilei” Via Marzolo 8, I-35131, Padova, Italy and Istituto Nazionale di Fisica Nucleare, I-35131, Padova, Italy 4School of Engineering and Computing, University of the West of Scotland, Paisley PA1 2BE, United Kingdom 5University of Jyvaskyla, Department of Physics, P.O. Box 35, Jyvaskyla FI-40014, Finland 6Institut f¨ ur Kernphysik, Universit¨ at zu K¨ oln, D-50937 K¨ oln, Germany 7Department of Physics, Oliver Lodge Laboratory, University of Liverpool, Liverpool L69 7ZE, United Kingdom (Received 23 May 2016; published 21 September 2016) The lifetime of the (11/2+) state in the band above the proton-emitting (3/2+) state in 113Cs has been measured to be τ=24(6) ps from a recoil-decay-tagged differential-plunger experiment. The measured lifetime was used to deduce the deformation of the states using wave functions from a nonadiabatic quasiparticle model to independently calculate both proton-emission and electromagnetic γ-ray transition rates as a function of deformation. The only quadrupole deformation, which was able to reproduce the experimental excitation energies of the states, the electromagnetic decay rate of the (11/2+) state and the proton-emission rate of the (3/2+) state, was found to be β2=0.22(6). This deformation is in agreement with the earlier proton emission studies which concluded that 113Cs was best described as a deformed proton emitter, however, it is now more firmly supported by the present measurement of the electromagnetic transition rate. DOI: 10.1103/PhysRevC.94.034321 I. INTRODUCTION Proton emission can provide valuable nuclear-structure information and allow sensitive tests of theoretical nuclear models for proton-rich nuclei in the region beyond the drip line [1,2]. Advances in experimental techniques [3,4] and the associated new experimental data can allow extensions to these models to be made [5,6], especially those models which explore the coupling of weakly bound and unbound states to the continuum. Proton emission from 113Cs was first discovered in 1984 by Faestermann et al. [7]. In that work and others [7–10], the cross section of 113Cs was established to be ∼30 μbinthe58Ni(58Ni ,p2n) reaction at 230 MeV. The (3/2+) ground state of 113Cs is now established to decay with the emission of a single 969(8)-keV proton with a half-life of 17.1(2) μs[11]. Several theoretical studies have been performed to explain proton emission from nuclei. Many of these studies have considered the proton tunneling through a spherical nuclear potential within various models (for example, the WentzelKramers-Brillouin (WKB), two-potential, and distorted wave *Present address: Institute of Cancer Sciences, The University of Manchester, Manchester M20 4BX, United Kingdom. Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Born approaches [12,13]). Although these early approaches were found to work well for nuclei near to the spherical closed shells, for other nuclei, like 113Cs, the calculated protontransition rates failed to match the experimental half-lives of the proton-emitting ground state. For example in the spherical WKB calculations of ˚ Aberg et al. [12,13], proton emission from the d5/2state in 113Cs was predicted to have a half-life of ∼0.5μs, which became ∼0.9μs when the d5/2state spectroscopic factor was included. This remains a factor of ∼19 times shorter than the established 17.1(2)-μs experimental half-life [7–11]. Already, more than 25 years ago, it was suggested that the absence of deformation in these models may explain the discrepancies [14]. In an attempt to alleviate these problems, Bugrov and Kadmensky included configuration-mixing effects associated with deformation into their calculations for 113Cs [14]. Proton-emission half-lives were calculated as a function of quadrupole deformation for the d5/2,g7/2,g9/2(extruder) or h11/2(intruder) orbits that were predicted to be occupied by the odd proton above the Z=50 closed shell [14]. With these spherical configuration-mixed calculations, agreement with the 17.1(2)-μs experimental proton emission half-life was only found for a Kπ=3/2+orbital at a quadrupole deformation, β2=0.2. In 1995, M¨ oller and Nix performed a series of global deformation calculations using a finite-range droplet macroscopic and folded-Yukawa single-particle microscopic model [15]. The results of these microscopic calculations turned out to be in agreement with the earlier spherical configuration-mixed calculations of Bugrov et al. [14] and predicted that 113Cs most likely had a deformation of β2=0.21. The first attempt to fully account for proton emission in a deformed potential 2469-9985/2016/94(3)/034321(8) 034321-1 Published by the American Physical Society D. HODGE et al. PHYSICAL REVIEW C 94, 034321 (2016) was performed for 113Cs in 1998 by Maglione et al. [16]. In that work, the absolute decay width of a single-particle level in a deformed Woods-Saxon potential was calculated which, for the first time, allowed the underlying nature of the 3/2+ proton-emitting state to be more fully evaluated as a function of deformation in 113Cs [16]. The correct experimental decay width for proton emission in 113Cs was reproduced with a quadrupole deformation β2=0.15–0.20 [16]. Further developments of that work allowed improved wave functions to be obtained using a nonadiabatic quasiparticle model [17–19]. These nonadiabatic calculations have been used for 113Cs in the present work to allow for any non rigid-rotational components in the daughter nucleus, 112Xe. In order to validate the theoretical predictions of deformation in these nuclei, any experimental information on the deformation of the proton-emitting states in 113Cs, or even the states in the bands built upon them, is necessary. To date, three separate recoil-decay tagging experiments have been performed to study 113Cs at Oak Ridge and Argonne National Laboratories [20] and more recently at the University of Jyv¨ askyl¨ a[11]. In the latter recoil-decay tagging experiment, the level scheme above the proton emitting state in 113Cs was more firmly assigned [11]. From the aligned angular momentum behavior, observed band crossings and blocking arguments, a rotational band, band 1, was assigned to be based upon a g7/2[422]3/2 configuration and a second band, band 2, upon a d5/2[420]1/2 configuration. In the present work, the lifetime of the (11/2+) state in band 1 has been measured using a differential-plunger device [3]. The lifetime has been interpreted within a new theoretical framework which considers both electromagnetic transition and proton emission rates using a common deformation and a common set of wave functions from a nonadiabatic quasiparticle model [17–19]. This approach has demonstrated that 113Cs is best described as a deformed nucleus with a quadrupole deformation parameter, β2=0.22(6). II. EXPERIMENTAL DETAILS The 58Ni(58Ni ,p2n) reaction was used for 12 days to populate excited states in 113Cs. A 230-MeV, 4-pnA 58Ni beam was provided by the University of Jyv¨ askyl¨ a K130 cyclotron. This beam was directed onto a 1.1mg/cm258Ni target foil, which was mounted in the differential plunger for unbound states (DPUNS) [3] along with a downstream, 1.5 mg/cm2Mg degrader foil. Nuclei recoiling from the target foil, traveling with a velocity measured to be v/c =0.038(2), were slowed by the degrader foil to a velocity of v/c = 0.018(2). The two resultant Doppler-shifts split the intensity of each γ-ray transition into respective fully shifted, Is, and degraded, Id, components. This allowed the implementation of the differential decay curve method (DDCM) within the recoil distance Doppler-shift (RDDS) technique [4,21]. The JUROGAM-II array of 39 Compton-suppressed germanium detectors surrounded the differential plunger at the target position [22–24]. The germanium detectors were arranged into four distinct “rings” with angles of θ=76◦, 105◦, 134◦, and 158◦with respect to the beam axis. Ten EUROGAM Phase I-type detectors situated at θ=134◦are collectively referred to as Ring 2. This ring of detectors had an absolute efficiency of ∼1.0% at 1.3 MeV and was found to be the only detector ring with sufficient statistics and a sufficiently large detection angle to resolve both the 113Cs fully shiftedand degraded-peak components of the transitions used in the RDDS analysis (see later). After passing through the degrader foil, the recoiling nuclei were transmitted through the gas-filled recoil ion transport unit (RITU) [25], triggering a multiwire proportional counter (MWPC) before implanting into one of the two double-sided silicon strip detectors (DSSD) of the GREAT spectrometer [26]. The average DSSD rate was ∼9.5 kHz throughout the experiment. All detector signals were time stamped by the total data readout (TDR) system [27] and the data were sorted with the grain software package [28]. In this work, recoil-tagged events were created by imposing conditions on the energy loss of the recoils in the MWPC and the time-of-flight between the MWPC and pair of DSSDs. These conditions allowed the separation of recoil events from any scatteredor primary-beam background in the DSSDs. Recoil-decay tagging (RDT) [29] was used to correlate prompt γrays detected in JUROGAM-II with proton decays of 113Cs detected up to ∼12 half-lives (∼0.2 ms) after a recoil implant in the same DSSD pixel. In order to distinguish between recoil-decay events and recoil implants from subsequent beam pulses in the same DSSD pixel, recoil-decay events were identified by their detection in anti-coincidence with the MWPC signal. Figure 1(a) shows a two-dimensional spectrum of time after recoil implant versus DSSD energy for these decay events. 50 100 150 200 Time (µs) 500 1000 1500 Counts (a) 050 100 150 200 Channels 102 103 104 Counts 113Cs 112Cs/109I 113Xe (b) Time (µs) 0 0 FIG. 1. (a) Two-dimensional histogram of DSSD energy versus time for decay events in anticoincidence with an MWPC signal. The area contained within the red line in (a) was used as the twodimensional RDT gate for 113Cs in this work. (b) The one-dimensional total energy projection of the two-dimensional histogram. Proton decays are identified from 113Cs [11], 109I(813(4)-keV, T1/2∼100 μs [31]), 112Cs (807(7)-keV, T1/2∼500 μs[30]) and internal conversion electrons from the T1/2=6.9(3)−μs isomer decay in 113Xe [32]. The inset to (b) shows the number of events within the 113Cs two-dimensional RDT gate as a function of time, demonstrating the 16.9(1) μs113Cs proton-decay half-life. 034321-2 DEFORMATION OF THE PROTON EMITTER 113Cs . . . PHYSICAL REVIEW C 94, 034321 (2016) The decay events corresponding to the proton emission from 113Cs are enclosed by the red two-dimensional RDT gate shown in Fig. 1(a). This subset of events covered a proton energy range of 850–1030 keV and time difference range between 10–200 μs, overlapping the established 113Cs proton decay properties [11]. This two-dimensional gate was used to construct the proton-tagged RDT spectra for the RDDS analysis in this work. Figure 1(b) is an energy projection of this matrix which shows proton decays from 113Cs [11] (from the p2nreaction exit channel), 109I(from the αp2n reaction exit channel), 112Cs (from the p3nreaction exit channel) [30,31], and internal conversion electrons from the 6.9(3)-μs isomer decay half-life in 113Xe [32]. Note that within the energy resolution of the DSSD detector, the 813(4)-keV proton-decay energy of 109I[31] overlaps with the 807(7)-keV proton-decay energy of 112Cs [30]. However, the proton decay half-life is dominated by the more intensely produced 109I (T1/2∼100 μs) compared with the more weakly produced 112Cs (T1/2∼500 μs) [11]. The inset to Fig. 1(b) shows the 16.9(1)-μs proton-decay half-life extracted for the proton decay of 113Cs from the present data. This value is consistent with the previously accepted value 17.1(2) μsfromRef.[11]. III. RESULTS A partial level scheme for 113Cs is shown in Fig. 2from Ref. [11]. This figure highlights the low-lying levels of the two previously identified bands 1 and 2 which are relevant to the present work. RDDS lifetime analysis was performed on the 384-keV, (11/2+)to(7/2+) transition in band 1. The 384-keV transition proved to be the only transition in the proton-tagged γ-ray spectra which had sufficient statistics and separation from other transitions to allow a full DDCM analysis to be (3/2 ) (5/2 ) (5/2 ) (7/2 ) (11/2 ) (15/2 ) (19/2 ) (9/2 ) (13/2 ) (17/2 ) 384 596 737 166 91 74 511 610 719 (92) 1 2 Band 1 Band 2 FIG. 2. A partial level scheme of 113Cs showing the lower energy states of rotational bands 1 and 2 [11]. The width of each γ-ray transition arrow represents its relative intensity with the white part representing the internal-conversion component. The RDDS analysis was performed using the 384-keV, (11/2+)to(7/2+) transition in band 1. performed. Data were collected at five different target-todegrader separations of 135, 210, 300, 590, and 3000 μm during the experiment. These distances were chosen to provide a large variation of intensities in both the fully-shifted and degraded components of the 384-keV transition, as required by the RDDS method [4,21]. Figure 3(a) shows the total proton-tagged Ring 2 γ-ray spectrum corresponding to all target-to-degrader distances, x, used throughout the experiment. All of the spectra shown in Fig. 3have been Doppler corrected using v/c =0.038(2) so that the fully shifted components of the 113Cs transitions lie at the correct energy. However, it should be noted that this backward-facing ring of detectors corresponds to a laboratory angle of θ=134◦and therefore, the degraded component of each transition lies at a higher energy than the fully-shifted component. Figures 3(b) to 3(e) show the variation of the 384keV transition component intensities, Isand Id, as a function of distance and illustrate how the dominant component changes from degraded at 389 keV to fully shifted at 384 keV as the distance is increased from 135 μm [Fig. 3(b)] to 3000 μm [Fig. 3(e)]. The corresponding spectra for the higher-lying, (15/2+)to (11/2+) feeding transition, are shown in Figs. 3(f) and 3(g). These spectra show that the fully shifted 596-keV component contains the majority of the transition intensity at the 135-μm [Fig. 3(f)] and 210-μm distances [Fig. 3(g)] with negligible intensity in the degraded component at 604-keV, even at these shortest target-to-degrader distances. This balance of shifted and degraded component intensities indicates that the lifetime of the (15/2+) state is much shorter than that of the (11/2+) state. As a consequence of this, a full RDDS lifetime analysis for the (15/2+) state could not be performed with the targetdegrader foil distances used in this experiment for the (11/2+) state. For all of the spectra shown in Figs. 3(b) to 3(g), Gaussian fits were used to extract the intensities of the fully shifted (green or purple) and degraded (red or cyan) components of the 384-keV (11/2+)to(7/2+) and 596-keV (15/2+)to(11/2+) transitions. The centroids and widths of the fully shifted 384keV and 596-keV components were fixed using values deduced from a higher statistics summed spectrum of the two longest distances, 590 μm [Fig. 3(d)] and 3000 μm [Fig. 3(e)]. The degraded component of the (11/2+)to(7/2+) transition at 389 keV in this sum spectrum remained small and did not interfere with establishing the centroid and width of the fully shifted 384-keV component. Once the centroid and width of the fully shifted 384-keV peak were fixed, the 389-keV degraded component peak width and centroid could then be extracted from the sum of all target-to-degrader distances, see insert to Fig. 3(a). Table Isummarizes the centroids and widths that were used to fit the intensities of the transitions shown in Fig. 3. In contrast, for the 596-keV (15/2+)to(11/2+) transition, the small intensity of the degraded component at 604 keV meant that its centroid and width could not be fixed from the longest distance spectra in the same manner that was used for the 384-keV transition. Instead, the centroid of the degraded component of the (15/2+)to(11/2+) transition was calculated using the measured degraded v/c =0.018(2) to be 604.2 keV from its Doppler-shift relative to the fully shifted 596-keV 034321-3 D. HODGE et al. PHYSICAL REVIEW C 94, 034321 (2016) 0 4 8 12 16 Counts/keV 380 390 0 4 8 12 16 20 380 390 Energy (keV) 200 400 600 800 1000 1200 0 20 40 60 590 600 610 0 4 8 12 16 0 4 8 12 16 376 384 392 0 20 40 60 384 389 384 389 389 389 384 384 (b) 135 μm(c) 210 μm (d) 590 μm (e) 3000 μm 74 91 384 508, 511 596 610 616, 618 (a) Sum of all distances 658 735, 737 719 811, 814 896 745 819 (f) 135 μm (g) 210 μm 596 604 610 596 604 610 166 389 384 389 616 616 FIG. 3. 113Cs proton-tagged fully Doppler-corrected Ring 2 (θ=134◦) JUROGAM-II spectra. (a) A summed spectrum containing events from measurements made at all the distances. The inset is focused on the 384and 389-keV peaks corresponding to the respective fully shifted and degraded components of the transition from the (11/2+) state. (b)–(g) Spectra corresponding to individual target-to-degrader distances, focused on the 384-keV [(b)–(e)] and 596-keV [(f) and (g)] γ-ray transitions from the (11/2+)and(15/2+) states in band 1. Gaussian fits to the fully shifted (red or purple) components and degraded (green or cyan) components are shown as well as the fully shifted component of the 610-keV transition from band 2 (orange). peak. The peak width of the unobserved 604-keV component was calculated to be 0.3-keV wider than the 5-keV width of the observed 389-keV component from the increase in Doppler broadening expected at the higher 604-keV energy [22,33]. The widths and centroids of the fully shifted and degraded components of the (15/2+)to(11/2+) and (11/2+)to(7/2+) transitions were then fixed at the values shown in Table I across all distances for the RDDS [4] analysis. The intensities extracted from the fits in Figs. 3(b) to 3(g) were normalized to the total counts in the 957-keV 112Te peak in a recoil-tagged Ring 2 γ-ray spectrum for each target-to-degrader distance. This normalization was applied to account for the differing data collection periods used for the different target-to-degrader TABLE I. The widths and centroids used to fit the shifted and degraded components of the (15/2+)→(11/2+)and(11/2+)→ (7/2+) transitions shown in Fig. 3. See text for details. Transition Component Centroid (keV) Width (keV) (11/2+)→(7/2+) Fully-Shifted 384.1(4) 5.0(6) (11/2+)→(7/2+) Degraded 388.9(3) 5.0(6) (15/2+)→(11/2+) Fully-Shifted 596.3(5) 5.1(7) (15/2+)→(11/2+) Degraded 604.2(9) 5.3(6) distance settings. The 957-keV peak was chosen due to its high intensity, short lifetime and separation from other transitions in the recoil-tagged spectrum [34,35]. The normalized shifted and degraded component intensities of the (15/2+)to(11/2+) and (11/2+)to(7/2+) transitions are listed in Table II. The lifetime of the (11/2+) state was determined using the differential decay curve method (DDCM) [21] using intensities listed in Table II according to τi(x)=− 1 v Qs ij (x)+α(x)Qs ih(x) d dt Qs ij (x).(1) The subscripts h,i, and jrepresent the (15/2+), (11/2+), and (7/2+) excited states, respectively. Qs ij (x) are the normalized intensities of the fully shifted component of the 384-keV transition, Qs ih(x) are the normalized intensities of the fully shifted component of the 596-keV transition. xrepresents the target-to-degrader distance, and the velocity of the fully shifted recoils, v=0.038(2)c.Thetermα(x) in Equation (1) reflects the difference in intensity between the feeding and depopulating transitions given by α(x)=Qs ij (x)+Qd ij (x) Qs ih(x)+Qd ih(x).(2) 034321-4 DEFORMATION OF THE PROTON EMITTER 113Cs . . . PHYSICAL REVIEW C 94, 034321 (2016) TABLE II. 113Cs normalized fully-shifted, Qs, and degraded, Qd, component intensities of the 384-keV (ij ) and 596-keV (ih) transitions in band 1 for each target-to-degrader distance, see text for details. Target-to-degrader Qs ij (Counts) Qd ij (Counts) Qs ih (Counts) Qd ih (Counts) distance (μm) 384-keV component 389-keV component 596-keV component 604-keV component 135 13(5) 34(6) 43(7) 4(2) 210 18(6) 25(6) 35(6) 2(2) 300 24(7) 16(6) 36(7) 1(1) 590 46(7) 6(5) 37(6) 0(2) 3000 52(8) 5(5) 41(8) 4(2) A value of α=1.17(23) was determined from an average and standard deviation of the individual values of α(x) at each target-to-degrader distance. This indicates that there is likely some unobserved side-feeding populating the (11/2+) state, although the calculated value is consistent with the αvalue of 1.00(6) extracted from Ref. [11]. Figure 4shows the DDCM fitting procedure used to extract the lifetime, τi,ofthe(11/2+) state based on Eq. (1)[21]. Figure 4(b) shows a piece-wise second-order polynomial fit to the variation of the normalized intensity, Qs ij , of the fully shifted component of the 384-keV transition as a function of target-to-degrader distance. Figure 4(c) shows the difference in fully shifted intensity as a function of distance, x, between the (11/2+) component and the (15/2+) component multiplied by the side-feeding parameter α. In the DDCM procedure, the 0 10 20 30 40 50 τ (ps) 0 20 40 60 Qij (Counts) 100 200 500 1000 2000 Target to Degrader Distance, x (μm) -20 0 20 40 -Qij+αQih (Counts) (c) (b) (a) ss s τ (11/2+) = 24(6) ps FIG. 4. Lifetime analysis for the (11/2+) state in 113Cs based on the DDCM. (a) shows the 24(6)-ps lifetime of the (11/2+) state. This lifetime was calculated from a weighted average of the individual τi(x), corresponding to distances, x, of 135, 210, and 300 μm in the region of sensitivity. (b) shows the variation of the normalized intensity, Qs ij , of the fully shifted component of the 384-keV transition which depopulates the (11/2+) state as a function of distance. (c) shows the difference in fully shifted intensity as a function of distance between the (11/2+) component and the (15/2+) component multiplied by the side feeding parameter α.In(b)and (c), the piece-wise polynomial fit used to extract the lifetime in the DDCM are shown by the solid lines, as discussed in the text. data shown in Figs. 4(b) and 4(c) are fitted simultaneously and the results of these fits are shown as solid lines in panels (b) and (c). The value of τi(x) was calculated at each distance within the “region of sensitivity” from a χ2minimization of the simultaneous fits. The region of sensitivity is defined as the range of target-to-degrader distances over which there is the largest change in intensity of the fully shifted component of the transition depopulating the level of interest and this is where the errors on the individual lifetimes are minimized [21]. A weighted average of the individual lifetimes τi(x)in the region of sensitivity was taken to provide a final value of τi=24(6) ps for the lifetime of the 11/2+state, see Fig. 4(a). The full DDCM analysis could not be performed for the (15/2+)to(11/2+) transition, as the intensity of the 596-keV shifted component did not vary at the target-to-degrader distances optimized for the lifetime of the (11/2+) state in this experiment. However, the lifetime of the (15/2+) state has instead been estimated from the shifted and degraded component intensities of the (15/2+)to(11/2+) 596-keV transition from the shortest target-to-degrader distance, 135 μm. This was the shortest distance where some sensitivity to the lifetime of the (15/2+) state and a nonzero value for the number of counts in the degraded peak was observed (see Table II). Using this value, the lifetime of the (15/2+) state was estimated under the assumption that the (19/2+) state has a significantly shorter lifetime than the (15/2+) state and that no side-feeding was present. In particular an upper limit of τh<5psforthe (15/2+) state lifetime was calculated according to Qd ih =Qs ih +Qd ihexp−x vτh,(3) using the data in Table II. Any lifetime value greater than this for the (15/2+) state would have been revealed by the observation of statistically significant nonzero counts in the 604-keV degraded component, Qd ih, at the larger target-todegrader distances. Table II shows that this is not the case. IV. DISCUSSION In order to determine the magnitude of the deformation of the proton-emitting state in 113Cs from the experimental results, theoretical quasiparticle model calculations have been performed using a deformed mean-field Woods-Saxon potential with spin-orbit interaction as discussed in Refs. [17,18]. In the present work, the theoretical calculations were based on the same approach with some additional improvements. In particular, these calculations were extended 034321-5 D. HODGE et al. PHYSICAL REVIEW C 94, 034321 (2016) to allow for nonadiabatic cases where the core in the model is allowed to be less rigid. This is of particular use where nonrotational level schemes or components are experimentally established in the daughter nucleus [6]. In the current 113Cs calculations, the experimental rotational spectrum of 112Xe [36] was used as the core. Wave functions extracted from this model were then fixed and used consistently in both proton emission codes based on the approaches discussed in Ref. [18] and standard electromagnetic transition-rate calculations based on Refs. [37,38]. In this way, the half-lives for both proton emission and electromagnetic decays were predicted using a common set of wave functions. 1. Predicted excitation energies of the states Figure 5shows the predictions from the nonadiabatic quasiparticle model for the excitation energy of the states in Band 1 of 113Cs (dashed lines) as a function of quadrupole deformation. Also shown are the experimental values (solid lines) from Ref. [11]. In the figure, the energies of the states have been plotted relative to the excitation energy of the 3/2+state in Band 1 which is predicted to be the lowest state in energy for moderate deformations, β2>0.2. Such deformation is consistent with both the previous predictions of M¨ oller and Nix [15] and the single-particle decay width in the deformed Woods-Saxon calculations of Maglione et al. [16]. The excitation energies calculated from the nonadiabatic quasiparticle model in this work are in reasonable agreement with the experimental values, see Fig. 5. The predicted excitation energy of the 7/2+state in Band 1 intersects the experimental value at a deformation of β2=0.24. This deformation is consistent with the deformation of the (3/2+) state from Maglione et al. and also that of M¨ oller and Nix discussed above. At higher angular momentum in Band 1, the quadrupole deformation predicted from the intersection of the experimental and theoretical lines appears to decrease. Figure 5shows that the theoretical values only overlap with 0.1 0.15 0.2 Quadrupole Deformation, β2 -0.5 0 0.5 1 1.5 2 2.5 Excitation Energy [E-E(3/2+)] (MeV) (7/2+) (11/2+) (15/2+) (19/2+) FIG. 5. Relative excitation energy of the states in Band 1 of 113Cs as a function of quadrupole deformation. Solid lines represent the experimental values and dashed lines the theoretical predictions. the experimental values with a lower deformation β2=0.18 for the (11/2+) state and at β2=0.15 for the (15/2+) state. This apparent reduction in deformation may, however, be reconciled from the aligned angular momentum plots for 113Cs shown in Ref. [11]. At rotational frequencies of 0.37 MeV, the (19/2+) state was shown to already be influenced by the back-bending process and therefore, may no longer correspond to the more pure Band 1 configuration of the lower-spin (3/2+), (7/2+), and (11/2+) states calculated in the model. Such behavior may have lowered the experimental energy of the states (solid lines) relative to the theoretical (dashed) lines shown in Fig. 5, leading to an artificially reduced intersection at smaller deformation for the higher-spin states in Band 1. With this reasonable agreement of the experimental excitation energy of the states with those predicted from the nonadiabatic quasiparticle model, the wave functions extracted from the model for the (11/2+) and (15/2+) states were then used to independently extract the γ-ray and proton-emission transition rates as a function of quadrupole deformation parameter. 2. Predicted B(E2) transition rates Figures 6(a) and 6(b) show the predicted results of a standard electromagnetic reduced transition probability B(E2) calculation [37,38] for the lifetimes of the (11/2+) and (15/2+) states in 113Cs as a function of β2deformation, respectively. The calculations were based on the wave functions extracted from the nonadiabatic quasiparticle model calculation for the (11/2+) and (15/2+) states of 113Cs. Also shown in Fig. 6(a) is the 24(6) ps lifetime for the (11/2+) state from the 0.1 0.2 0.3 β2 1 10 100 1000 τ (ps) 0.1 0.2 0.3 1 10 100 1000 (b)(a) FIG. 6. Predicted excited-state lifetimes (a) (11/2+)and(b) (15/2+) as a function of quadrupole deformation in 113Cs employing the quasiparticle wave functions that have been used to calculate the proton-emission half-life in Fig. 7(see later). The experimental lifetime of the (11/2+) state and its uncertainty are denoted by the red and dashed lines in (a). The τ<5 ps limit for the (15/2+) state is denoted by the dashed line in (b), as discussed in the text. At deformations β2>0.25 in the calculation, Band 1 is crossed by another configuration based on the [404]9/2 level, beyond which the lifetime is no longer reflective of the underlying configuration of Band 1. 034321-6 DEFORMATION OF THE PROTON EMITTER 113Cs . . . PHYSICAL REVIEW C 94, 034321 (2016) 0.1 0.2 0.3 0.4 β 2 0.1 1 10 100 1000 Half-life of Proton Emission (μs) 3/2+ 5/2+ 1/2+ T1/2=16.9(1) μs FIG. 7. Theoretical proton emission half-life as a function of quadrupole deformation for the low-spin states in 113Cs using the non-adiabatic quasiparticle wave functions [17]. Also shown is the 16.9(1)-μs experimental value from this work which only overlaps with the calculated half-life of the (3/2+) state. experimental DDCM analysis by the red and black dashed lines. From Fig. 6(a), it can be seen that the theoretical lifetime for the (11/2+) state is only consistent with the experimental τ=24(6)-ps RDDS experimental lifetime, deduced in this work for a quadrupole deformation parameter, β2∼0.22–0.25. In agreement with the analysis of the (11/2+) state, Fig. 6(b) shows the experimental τ<5 ps limit on the (15/2+) state estimated using the 4(2) counts in the 596-keV transition degraded component at the 135-μm distance shown in Table I.InFig.6, the theoretical values are only seen to overlap with this <5 ps experimental limit for a quadrupole deformation β2>0.19. At deformations β2>0.25 in the calculation, Band 1 is crossed by another configuration based on the [404]9/2 level, beyond which the lifetime is no longer reflective of the underlying configuration of Band 1. 3. Predicted proton-emission transition rates Using the same nonadiabatic quasiparticle model wave functions that were employed for the γ-ray transition rates discussed above, the lifetimes for proton emission have been calculated as a function of quadrupole deformation for the 1/2+,3/2+, and 5/2+states below the (11/2+) state [16]. Figure 7shows the results of these calculations, along with the experimental half-life extracted in this work, 16.9(1) μs (solid black line). Agreement between the theoretical and experimental proton emission half-life is found for the 3/2+ state at deformation β2=0.08 and also at β2=0.22. The smaller β2value of 0.08 is discounted as a solution as it does not agree with the predicted deformations from neither the excitation energies of the states of Band 1, shown in Fig. 5, nor with the deformation predicted from the lifetime of the (11/2+) TABLE III. A summary of the deformation estimates for the (11/2+), (15/2+), and (3/2+) states from the various methods used in this work. State Result from β2 (11/2+) Excitation energy ∼0.18 (11/2+) Lifetime measurement 0.22–0.26 (15/2+) Excitation energy ∼0.15 (15/2+) Lifetime measurement >0.19 (3/2+) Proton emission lifetime ∼0.22 or (15/2+) states discussed above. The higher β2=0.22 value is however, in good agreement with the deformations extracted from the electromagnetic transition rates and also from the excitation energies of the states. In order to deduce the most consistent deformation from the present work, the results from the various methods were used to extract deformations for the (11/2+), (15/2+), and (3/2+) states in 113Cs, as summarized in Table III. From Table III it can be seen that for the (11/2+) state, the range of deformations, from the excitation-energy calculation and the B(E2) calculation, gives an average and standard deviation, β2=0.22(6). This deformation and its uncertainty is dominated by the B(E2) calculation which is the most sensitive part our method. The deformation limit extracted for the (15/2+) state lifetime limit is in agreement, giving β2> 0.19. Finally, the proton lifetime measurement for the (3/2+) state in this work is also consistent with this deformation β2=0.22(6). In summary in this work, the lifetime of the (11/2+) state in 113Cs has been measured for the first time to be τ=24(6) ps. A limit of τ<5 ps was also placed on the higher lying (15/2+) state. The half-life of the proton emitting (3/2+) state has been remeasured to be 16.9(1) μs and is found to be in agreement with the previously accepted value. In order to interpret these results, wave functions from a nonadiabatic quasiparticle model have been used to calculate both proton-emission and γ-ray transition rates for 113Cs. The only deformation which matches the experimental lifetime and excitation energy of the (11/2+) and (15/2+) states, and the proton-emission lifetime is found to be β2=0.22(6). This deformation is consistent with the earlier predictions for 113Cs [14–16] but is now, for the first time, more firmly based on an approach supported by an experimental RDDS lifetime measurement. ACKNOWLEDGMENTS This work was supported by the EU 7th Framework Programme, Integrating Activities Transnational Access, Project No. 262010 ENSAR, and by the Academy of Finland under the Finnish Centre of Excellence Programme (Nuclear and Accelerator Based Physics Programme at JYFL). The authors acknowledge GAMMAPOOL support for the JUROGAM detectors and D.H., D.M.C., B.S.N.S., M.J.T., M.M., J.F.S., and M.S. acknowledge support of the Science and Technology Facilities Council, Grant No. ST/L005794/1. 034321-7 D. HODGE et al. PHYSICAL REVIEW C 94, 034321 (2016) [1] P. J. Woods and C. N. Davids, Annu. Rev. Nucl. Part. 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