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

The lifetime of the (11 / 2 + ) state in the band above the proton-emitting (3 / 2 + ) state in 113 Cs 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 β 2 = 0 . 22(6). This deformation is in agreement with the earlier proton emission studies which concluded that 113 Cs was best described as a deformed proton emitter, however, it is now more ﬁrmly supported by the present measurement of the electromagnetic transition rate.


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 113 Cs was first discovered in 1984 by Faestermann et al. [7]. In that work and others [7][8][9][10], the cross section of 113 Cs was established to be ∼ 30 μb in the 58 Ni( 58 Ni ,p2n) reaction at 230 MeV. The (3/2 + ) ground state of 113 Cs 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 Wentzel-Kramers-Brillouin (WKB), two-potential, and distorted wave * Present address: Institute of Cancer Sciences, The University of Manchester, Manchester M20 4BX, United Kingdom. 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 113 Cs, the calculated protontransition rates failed to match the experimental half-lives of the proton-emitting ground state. For example in the spherical WKB calculations ofÅberg et al. [12,13], proton emission from the d 5/2 state in 113 Cs was predicted to have a half-life of ∼0.5 μs, which became ∼0.9 μs when the d 5/2 state spectroscopic factor was included. This remains a factor of ∼19 times shorter than the established 17.1(2)-μs experimental half-life [7][8][9][10][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 113 Cs [14]. Proton-emission half-lives were calculated as a function of quadrupole deformation for the d 5/2 , g 7/2 , g 9/2 (extruder) or h 11/2 (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 K π = 3/2 + orbital at a quadrupole deformation, β 2 = 0.2. In 1995, Möller 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 113 Cs most likely had a deformation of β 2 = 0.21. The first attempt to fully account for proton emission in a deformed potential was performed for 113 Cs 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 113 Cs [16]. The correct experimental decay width for proton emission in 113 Cs was reproduced with a quadrupole deformation β 2 = 0.15-0.20 [16]. Further developments of that work allowed improved wave functions to be obtained using a nonadiabatic quasiparticle model [17][18][19]. These nonadiabatic calculations have been used for 113 Cs in the present work to allow for any non rigid-rotational components in the daughter nucleus, 112 Xe.
In order to validate the theoretical predictions of deformation in these nuclei, any experimental information on the deformation of the proton-emitting states in 113 Cs, 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 113 Cs at Oak Ridge and Argonne National Laboratories [20] and more recently at the University of Jyväskylä [11]. In the latter recoil-decay tagging experiment, the level scheme above the proton emitting state in 113 Cs 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 g 7/2 [422]3/2 configuration and a second band, band 2, upon a d 5/2 [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][18][19]. This approach has demonstrated that 113 Cs is best described as a deformed nucleus with a quadrupole deformation parameter, β 2 = 0.22 (6).

II. EXPERIMENTAL DETAILS
The 58 Ni( 58 Ni ,p2n) reaction was used for 12 days to populate excited states in 113 Cs. A 230-MeV, 4-pnA 58 Ni beam was provided by the University of Jyväskylä K130 cyclotron. This beam was directed onto a 1.1 mg/cm 2 58 Ni target foil, which was mounted in the differential plunger for unbound states (DPUNS) [3] along with a downstream, 1.5 mg/cm 2 Mg 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, I s , and degraded, I d , 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][23][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 113 Cs 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 γ rays detected in JUROGAM-II with proton decays of 113 Cs 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. The area contained within the red line in (a) was used as the twodimensional RDT gate for 113 Cs in this work. (b) The one-dimensional total energy projection of the two-dimensional histogram. Proton decays are identified from 113 Cs [11], 109 I (813(4)-keV, T 1/2 ∼ 100 μs [31]), 112 Cs (807(7)-keV, T 1/2 ∼ 500 μs [30]) and internal conversion electrons from the T 1/2 = 6.9(3)−μs isomer decay in 113 Xe [32]. The inset to (b) shows the number of events within the 113 Cs two-dimensional RDT gate as a function of time, demonstrating the 16.9(1) μs 113 Cs proton-decay half-life.

034321-2
The decay events corresponding to the proton emission from 113 Cs 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 113 Cs 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 113 Cs [11] (from the p2n reaction exit channel), 109 I (from the αp2n reaction exit channel), 112 Cs (from the p3n reaction exit channel) [30,31], and internal conversion electrons from the 6.9(3)-μs isomer decay half-life in 113 Xe [32]. Note that within the energy resolution of the DSSD detector, the 813(4)-keV proton-decay energy of 109 I [31] overlaps with the 807(7)-keV proton-decay energy of 112 Cs [30]. However, the proton decay half-life is dominated by the more intensely produced 109 I (T 1/2 ∼ 100 μs) compared with the more weakly produced 112 Cs (T 1/2 ∼ 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 113 Cs from the present data. This value is consistent with the previously accepted value 17.1(2) μs from Ref. [11].

III. RESULTS
A partial level scheme for 113 Cs is shown in Fig. 2 from 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  [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. 3 have been Doppler corrected using v/c = 0.038(2) so that the fully shifted components of the 113 Cs 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. 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 384-keV 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 I summarizes 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 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 112 Te 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 The subscripts h,i, and j represent the (15/2 + ), (11/2 + ), and (7/2 + ) excited states, respectively. Q s ij (x) are the normalized intensities of the fully shifted component of the 384-keV transition, Q s ih (x) are the normalized intensities of the fully shifted component of the 596-keV transition. x represents the target-to-degrader distance, and the velocity of the fully shifted recoils, v = 0.038(2)c. The term α(x) in Equation (1) reflects the difference in intensity between the feeding and depopulating transitions given by 034321-4 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 α value of 1.00(6) extracted from Ref. [11]. Figure 4 shows the DDCM fitting procedure used to extract the lifetime, τ i , of the (11/2 + ) state based on Eq. (1) [21]. and (c). The value of τ i (x) was calculated at each distance within the "region of sensitivity" from a χ 2 minimization 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 τ i (x) in the region of sensitivity was taken to provide a final value of τ i = 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 τ h < 5 ps for the (15/2 + ) state lifetime was calculated according to 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, Q d 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 113 Cs 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  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 113 Cs calculations, the experimental rotational spectrum of 112 Xe [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. Figure 5 shows the predictions from the nonadiabatic quasiparticle model for the excitation energy of the states in Band 1 of 113 Cs (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, β 2 > 0.2. Such deformation is consistent with both the previous predictions of Möller 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 β 2 = 0.24. This deformation is consistent with the deformation of the (3/2 + ) state from Maglione et al. and also that of Möller 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. the experimental values with a lower deformation β 2 = 0.18 for the (11/2 + ) state and at β 2 = 0.15 for the (15/2 + ) state. This apparent reduction in deformation may, however, be reconciled from the aligned angular momentum plots for 113 Cs 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.

Predicted excitation energies of the states
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.  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 β 2 > 0. 25 FIG. 7. Theoretical proton emission half-life as a function of quadrupole deformation for the low-spin states in 113 Cs 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, β 2 ∼ 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. In Fig. 6, the theoretical values are only seen to overlap with this < 5 ps experimental limit for a quadrupole deformation β 2 > 0.19. At deformations β 2 > 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.

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 7 shows 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 β 2 = 0.08 and also at β 2 = 0.22. The smaller β 2 value 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 + ) or (15/2 + ) states discussed above. The higher β 2 = 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 113 Cs, 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, β 2 = 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 β 2 > 0. 19. Finally, the proton lifetime measurement for the (3/2 + ) state in this work is also consistent with this deformation β 2 = 0.22 (6).
In summary in this work, the lifetime of the (11/2 + ) state in 113 Cs 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 113 Cs. 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 β 2 = 0.22 (6). This deformation is consistent with the earlier predictions for 113 Cs [14][15][16] but is now, for the first time, more firmly based on an approach supported by an experimental RDDS lifetime measurement.