Lifetime measurement of the 2 + 1 state in 74 Rb and isospin properties of quadrupole transition strengths at N = Z

nuclei in the A ≈ 70–80 region have attracted a great deal of attention due to phenomena such as shape coexistence and increasing collectivity along the N = Z line. We investigate the structure of nuclei in this region through lifetime measurements using the GRETINA array. The ﬁrst implementation of the Differential Recoil Distance Doppler Shift technique with fast radioactive beams is demonstrated and veriﬁed through a measurement of the well-known B ( E 2 ; 2 + 1 → 0 + 1 ) transition strength in 74 Kr. The method is then applied to determine the B ( E 2 ; 2 + 1 → 0 + 1 ) transition strength in 74 Rb, the heaviest odd– odd N = Z nucleus for which this quantity has been determined. This result and extended systematics along N = Z suggest the dominance of the isoscalar part of the quadrupole transition strengths in self-conjugate nuclei, as well as the possible presence of shape coexistence in 74 Rb.


Introduction
Nuclei along the N = Z line offer a unique opportunity to study nuclear structure. Being symmetric with respect to the isospin degree of freedom, these systems offer an ideal setting in which to study isospin symmetry breaking effects, for example through the differences in the excitation energies of isobaric analogue states in an isospin multiplet [1]. The odd-odd N = Z nuclei are particularly interesting in this regard, as a competition can arise between the isoscalar (T = 0) and isovector (T = 1) pairing modes. These nuclei are also host to many of the most precisely measured superallowed Fermi β decay strengths available, which provide important tests of predictions for the electroweak interaction such as the CVC hypothesis and the unitarity of the CKM matrix [2]. Such tests rely on accurate knowledge of the wave function of the nuclear states involved in the transitions, and so information regarding the structure of these nuclei provides an important input for fundamental physics.
The nuclei near the N = Z line in the vicinity of mass A ≈ 70 exhibit a particularly wide variety of nuclear structure phenomena. These nuclear systems evolve very rapidly as a function of both nucleon number and angular momentum; nuclei in this region have been observed to transition through many different shapes along the N = Z line as well as along isotopic chains [3][4][5][6][7]. This has been understood as a consequence of the many gaps between the Nilsson levels which stabilize different shapes at similar particle numbers [8].   [20], while data for 74 Kr are from [21]. are filling the same orbitals, this can give rise to coexisting structures with different intrinsic deformations, i.e. shape coexistence. This has been observed in 68, 72 Se [9,10] as well as the proton-rich krypton isotopes near N = Z [7]. Such phenomena make this area of the nuclear chart an attractive location to investigate nuclear structure and to test the predictions of nuclear models.
Despite the rich diversity of nuclear structure in this region, the N ≈ Z nuclei near A ≈ 70 − 80 have proven challenging to study. This is due in part to the fact that it is difficult to produce rareisotope beams of these nuclei with sufficient purity and intensity, as well as the large computational effort that is required for theoretical calculations. Of particular relevance for understanding the structure of the nuclei in this region are the B(E2; 2 + 1 → 0 + 1 ) transition rates, which provide an important measure of the collectivity of the nucleus. The even-even nuclei have fared relatively well in this regard, as the B(E2) strengths along the N = Z line have been measured up to 76 Sr [11]. The odd-odd nuclei, on the other hand, have proven more challenging, and until recently the heaviest oddodd N = Z nucleus with a measured B(E2; 2 + → 0 + ) value was 58 Cu [12]. New efforts, however, are beginning to address this issue, with a measurement of the B(E2; 2 + having been recently published [13]. Progress pushing beyond the N = Z line is also being made, with recent measurements of the excited state energies in 70 Kr [14,15] and 74 Sr [16] allowing investigation of isospin symmetry breaking effects. Theoretical efforts are also making inroads in this mass region, with a study of isospin symmetry breaking and shape coexistence near mass A ≈ 70 recently published [17]. In this letter, we report the measurement of the 2 + 1 -state lifetime in the odd-odd N = Z nucleus 74 Rb. This measurement extends the B(E2) systematics of the odd-odd N = Z nuclei and represents the heaviest such system with a measured 2 + 1 -state lifetime. Previous studies of 74 Rb have established the level scheme up to spin-parity J π = 25 + , with further tentative assignments up to J π = (31 + ) [18][19][20]. Comparison with the level scheme of 74 Kr has allowed the identification of the T = 1 isobaric analogue states in 74 Rb up to J π = 8 + [20], a subset of which are shown in Fig. 1 [22,23]. While the isoscalar component is expected to dominate over the isovector, the degree to which this occurs is not necessarily obvious, as the increasing proximity of the N = Z line to the proton drip line can introduce distortions to the wave functions or possibly lead to shape changes as discussed in, for example, Refs. [15,16]. In addition, 74 Kr exhibits strong shape coexistence, with its ground-state wave function composed of nearly equal contributions from intrinsic prolate and oblate configurations [7,[24][25][26].
Comparison of the transition strength of 74 Kr with that measured here for 74 Rb can give an indication of whether such shape coexistence persists in 74 Rb, as the coexisting analogue 0 + 2 state has not been seen [20].

Differential Recoil Doppler Shift technique
For lifetimes in the range of a few picoseconds to about a nanosecond, the Recoil Distance Doppler Shift (RDDS) technique is often the experimental method of choice [27]. In this method, the distance between a nuclear reaction-inducing target and an energy-degrading (or stopping) foil must be varied to observe how the number of decays between the target and degrader changes. Only after data have been taken at several distances can the lifetime be determined reliably, which can necessitate long experiments. High production rates can offset the demanding data requirements, but for nuclei with low production rates the requisite beam time may become prohibitive. To address this problem, a variant of the RDDS technique called the Differential Recoil Distance Doppler Shift (DRDDS) method has been proposed [28,29]. This method modifies the traditional RDDS setup by inserting an additional energy-degrading foil between the target and the degrader. As a result, it is possible to measure the lifetime without systematic variation of the distance between the foils, significantly reducing the data required to measure the lifetime. A summary of the method is given below.
In both the RDDS and the DRDDS, the lifetime of an excited state is measured by determining the unnormalized decay function for that state R(t) and its derivative: In the DRDDS, the three foils in the beam path cause the γ rays emitted by the excited reaction products to have three distinct Doppler-shifted components: a fast component from decays behind the target with an integral I f , a reduced-velocity component behind the first degrader with integral I r , and a slow component behind the second degrader with integral I s . The decay function is the fraction of the beam that remains excited when it passes a given foil, e.g. R(t s ) = I s . The derivative at the first degrader position can be approximated by where x is the distance and v is the beam velocity between the two degraders. Substituting these expressions in Eq. (2) and rearranging, the lifetime can be expressed as In the DRDDS, all of these quantities can be directly measured from one distance setting for the two degraders. Further discussion of the method is warranted in the case that the level of interest suffers from significant feeding from higherlying states. A full discussion is given in Ref. [28], which can be summarized by stating that the effect of feeding is to delay the decay of the excited state which is being studied and causing an excess of counts in the slow peak. It is possible to account for this effect by replacing I s in Eq. (4) with I s − h b h I h s , where I h s is the slow component of the feeding state and b h is the branching fraction from the feeding state to the state of interest. However, if the lifetime of the feeding state is short compared to the flight time between the target and first degrader, I h s approaches zero and the feeding does not affect the lifetime of interest. Because the DRDDS method does not make reference to the fast-peak population I f , it is possible to choose the distance between the target and first degrader such that the feeding states have all decayed before reaching the first degrader. In this way, the fast peak serves as a buffer which isolates the reduced-velocity and slow peaks from the effects of feeding and reduces the sensitivity of the method to systematic uncertainties.

Experiment
The experiment was carried out at the National Superconducting Cyclotron Laboratory at Michigan State University. The Coupled Cyclotron Facility produced a primary beam of 78 Kr ions at an energy of 150 MeV/nucleon which was then impinged on a 9 Be target in order to induce nuclear reactions. The resulting beam was then purified in the A1900 fragment separator [30] with the aid of an aluminum degrader in order to select 74 Kr fragments after the target. The resulting secondary beam had a purity of about 40% 74 Kr with an energy of 93 MeV/nucleon and an average intensity of 1 × 10 5 particles per second.
The 74 Kr secondary beam was transported to the experimental area where the lifetime measurement was performed. This experiment used the TRIple PLunger for EXotic beams (TRIPLEX) [31], which was developed with the DRDDS method in mind and can hold up to three foils as targets or degraders/stoppers. The foils can be positioned between 0-25 mm apart using piezoelectric stepper motors. Three metallic foils were mounted in the TRIPLEX: one 9 Be target which was 139-mg/cm 2 thick, as well as two tantalum degraders of thickness 208 mg/cm 2 and 166 mg/cm 2 . The 9 Be target was used to induce reactions in the secondary beam, which produced 74 Rb via the 9 Be( 74 Kr, 74 Rb) charge-exchange reaction and excited 74 Kr through inelastic scattering, while the degraders were used to incrementally slow the reaction products to lower velocities. This approach avoids the difficulties associated with low beam intensity and purity which can pose a challenge to Coulomb excitation studies. The separations between the target/first degrader and first degrader/second degrader were 1 mm for the lifetime measurement, which corresponds to a flight-time of about 10 ps at a typical velocity of β = 0.35. Data were also taken with the target 10 mm from the first degrader to allow excited states to decay before reaching it. Any decays observed after the first degrader were then attributed to reactions induced in the degraders, which was found to be about 10% for 74 Kr and 20% for 74 Rb. The ratio of reactions happening in the first or second degrader was assumed to be the ratio of the degrader thicknesses.
The TRIPLEX was installed upstream of the S800 spectrograph [32]. Reaction residues observed in the focal plane of the S800 [33] were identified using the energy-loss vs time-offlight method, with data taken for 74 Rb and 74 Kr simultaneously. The incoming beam was characterized by measuring its energy, y-position, and direction of travel at the target position as reconstructed by the S800 particle-tracking by taking data with neither target nor degraders installed. The energy loss of the beam through the target was measured with no degraders present, and then the degraders were installed to determine the energy loss through all three foils. The velocity v between the degraders (Eq. (4)) was calculated by taking the average energy measured in the S800 and correcting for the calculated energy loss in the second degrader.  74 Rb recoils. For 74 Kr, several states are clearly populated, including the 2 + 1 , 4 + 1 , and 6 + 1 states. In contrast, only the 2 + 1 state is strongly populated in 74 Rb, with a small population of the 4 + 1 state visible. The insets show a closer view of the 2 + 1 → 0 + 1 transition for each nucleus, with the fast, reduced, and slow components labeled. The data in both panels were taken with 1 mm separations between the target/first degrader and between the first degrader/second degrader, and the Doppler correction was performed using the velocity after the first degrader.
De-excitation γ rays were detected in the Gamma Ray Energy Tracking In-beam Nuclear Array (GRETINA) [34]. For this experiment, GRETINA consisted of seven modules, each containing four 36-fold segmented high-purity germanium crystals. The TRIPLEX was mounted inside of GRETINA and shifted about 12 cm upstream of the center of the array. Accounting for the change in the target position, the array was configured with four detectors between 20 • -50 • and three detectors positioned around 70 • . GRETINA was crucial to the success of this experiment due to its ability to localize γ -ray interaction points to within a few millimeters. The γ -ray position resolution of GRETINA, coupled with the particle-tracking of the S800, allows excellent Doppler-corrected energy resolution as described in Ref. [35]. Resolving three peaks for each transition has been a major challenge to the DRDDS [29], which is overcome by the combination of GRETINA and the S800. The spectra detected in coincidence with 74 Rb and 74 Kr recoils are shown in Fig. 2, with the transitions labeled by the initial and final states. The insets show the three components of the 2 + 1 states each labeled according to the velocity at which the photons are emitted. The velocity used for Doppler-reconstruction was the velocity after the first degrader, which was corrected event-by-event based on the energy measured in the S800. Therefore, the middle peak for each decay corresponds to the rest-frame γ -ray energy. Fig. 2 shows spectra for which the separation between the target/first degrader and first degrader/second degrader was 1 mm, which as mentioned earlier corresponds to a flight time of about 10 ps at a velocity of β = 0.35. Inspection of the spectra in Fig. 2 makes it clear that the slow component of the 2 + state is most strongly populated, indicating a lifetime significantly longer than 10 ps for both 74 Kr and 74 Rb.
The analysis of the lifetimes of the 2 + 1 states in 74 Kr and 74 Rb was performed with the aid of a Monte Carlo simulation program. Simulating the data addresses several issues, including the detector efficiency for each of the Doppler-shifted components, emission of photons within the foils, and evaluating the importance of feeding effects by including or excluding γ -ray cascades. The simulation package [36] is based on the Geant4 framework [37] and has been updated to include the GRETINA+TRIPLEX geometry. The software accepts several inputs which are used to reproduce the physical conditions measured during the experiment, including the beam properties described earlier, the momentum transfer during the reaction on target, and the acceptance of the S800 spectrograph. The simulation can also assign a fraction of the reactions to occur in the degraders instead of the target as determined in the analysis. Simulated γ -ray spectra were generated for the transitions observed in the data, with the lifetime of the 2 + 1 state varied systematically. The simulated spectra were then fit to the data in order to determine the 2 + 1 -state lifetimes from a χ 2 minimization.

Analysis
The 2 + 1 state lifetime of 74 Kr was determined first to verify the performance of the DRDDS technique. Fig. 3(a) shows the results of fitting simulated decays from the 2 + 1 state to the ground state. Feeding from excited states was also included in this fit, which was fixed from the intensities of the higher-lying transitions as shown in Fig. 2. The shaded regions in this figure indicate the range used to determine the 2 + 1 -state lifetime. It was found that about 30% of the 2 + 1 → 0 + 1 decays come from direct population of the 2 + 1 state, with the remaining population originating with feeding from the 4 + 1 , 6 + 1 , and 2 + 2 states. Simulations varying the lifetime of the 2 + 1 state were generated and fitted to the data, and the reduced χ 2 values from these fits were used to generate the χ 2 distribution shown in the inset of Fig. 3(a). The minimum of this distribution gives a lifetime of τ 2 + 1 = 31(3) ps. The uncertainty includes both statistical and systematic contributions, with the main sources of systematic uncertainty being the positioning of the target and second degrader foils relative to the first degrader, as well as the position of the TRIPLEX relative to GRETINA. Each of these uncertainties contributes at about the 3% level. The sensitivity of the fit to the functional form used to describe the background was tested and found to be negligible. This result is in good agreement with the most precise published value of 33.8(6) ps, obtained with a differential-decay curve and recoil distance method [38]. It should be noted that these data were previously analyzed in Ref. [39] using a method based on a more standard RDDS technique, and was independently reanalyzed here with consistent results.
To evaluate the sensitivity of the DRDDS to feeding from shortlived states, the fitting procedure was repeated for 74 Kr without any feeding included in the simulations. The lifetime of the 4 + 1 state in 74 Kr is about 6 ps [39] and the higher-lying states are considerably shorter still [26,38], which satisfies the condition that the feeding lifetime should be shorter than the target-degrader flighttime. The results are shown in Fig. 3(b). The best-fit lifetime in this condition is τ 2 + 1 = 33(3) ps, which includes systematic effects similar to the result which includes feeding. The 2 ps difference between the two fits to the 74 Kr data may be attributable to feeding; nevertheless, this result is consistent with that obtained when feeding is considered and also with the previously determined lifetime [38]. It should be noted that the fast peak is not reproduced well by the simulations in this case, which is a consequence of Fig. 3. Doppler-corrected γ -ray energy spectra for 74 Kr and 74 Rb along with the best-fit results of the 2 + 1 -state lifetime for each nucleus. The data and uncertainties are indicated by the crosses, and the simulated spectra are shown by the solid lines. The shaded areas indicate the range used in the fits to determine the 2 + 1 -state lifetime, and the exponential background is shown by the dashed line. Panel (a) shows the fit to the 74 Kr data including all observed feeding levels, while panel (b) shows the same fit when feeding is neglected (see text for details). Panel (c) shows the fit to the 74 Rb data. The insets show the reduced χ 2 distribution, the minimum of which was used to determine the lifetime. the lack of feeding transitions that would otherwise delay γ -ray emission and thereby reduce the population of this peak. However, since the fast peak is not explicitly considered in the DRDDS method, this does not alter the lifetime result.
The analysis of 74 Rb was performed in the same manner as 74 Kr. Feeding was neglected in this analysis, due both to the apparently little feeding from higher-lying states as well as lack of information on their lifetimes. As shown in Fig. 2, about 70% of the population goes to the 2 + state in this reaction, with the remainder going to the 4 + state. Although this state decays through a γ -ray with an energy of 575.1(4) keV [20], it apparently has a short lifetime and mostly populates the fast peak at approximately 600 keV. Since this lifetime is unknown, the insensitivity of the DRDDS technique to feeding from short-lived states is particularly important in this analysis. As with 74 Kr, simulations were generated for the 2 + 1 → 0 + 1 transition of 74 Rb, with the lifetime of the 2 + 1 state varied. These simulations were fitted to the data, with the resulting best fit shown in Fig. 3(c). The reduced χ 2 distribution is shown in the inset to this panel, and from the minimum of this distribution the lifetime was determined to be τ 2 + = 27(6) ps, which includes a small increase in the uncertainty to account for the fact that the reduced χ 2 does not go to one [40]. The uncer- tainty includes systematic effects similar to those in 74 Kr, and in addition includes a 6% contribution deduced from the difference between the two fits to the 74 Kr data to account for possible feeding. Assuming a lifetime for the 4 + state similar to that in 74 Kr and including feeding in the simulations changes the lifetime by about 1 ps, which indicates that this is a reasonable estimate of the uncertainty from feeding from this state. An attempt was made to fit the 4 + 1 lifetime using the technique of Ref. [39], but the statistics are too low to achieve a reliable result. There is no evidence of decay from the J π = 3 + , T = 0 state at 1005.4(3) keV, which would feed the 2 + 1 state via a 527.6(2)-keV γ ray [20]. Including simulations for decay from this state to the 2 + 1 state with lifetimes in the range 0-30 ps had no effect on the measured 2 + 1 -state lifetime.

Discussion
From the lifetime, the reduced transition probability for the decay of the 2 + 1 state of 74 Rb can be calculated to be B(E2; 2 +  [22]. The relevant relation given in Ref. [22] can be written The systematic behavior of the isoscalar and isovector contributions to the transition matrix elements is shown in Fig. 4 for a wide range of nuclei along the N = Z line. For the T = 1 triplets, the isoscalar and isovector components of the matrix elements are plotted as black circles and red triangles, respectively. Although only three measurements are available for nuclei beyond mass A = 42, the isoscalar components of the matrix elements display a marked increase in the nuclei above the sd shell, in keeping with the observed increase in collectivity in this region. Notably, the isoscalar components of the matrix elements in the odd-odd nuclei follow very closely the evolution of the matrix elements derived from even-even N = Z nuclei, shown by the solid line and error band in Fig. 4, which suggests that the collectivity in the T = 0 and T = 1 states has a common origin. This observation mirrors the findings of Ref. [54] based on an analysis of the energy systematics of N = Z nuclei, in which it is found that T = 1 pairing is primarily responsible for collective excitations while T = 0 pairing results in single-particle characteristics. Further, the authors of that work propose that the ground states of even-even nuclei and the T = 1 states of neighboring odd-odd nuclei are both built on T = 1 coupling of nucleon pairs and thus have essentially the same correlation properties. The near identity of M 0 in neighboring species in Fig. 4 supports this interpretation.
In contrast to the isoscalar components of the matrix elements shown in Fig. 4, the isovector components show essentially no correlation with mass number. Furthermore, the isoscalar components of the matrix elements account for almost the entire transition strength, while the isovector components are consistent with zero. This suggests that the dominance of the isoscalar matrix elements in sd-shell nuclei may be a general property of nuclei on the N = Z line, with no enhancement of the M 1 components up to A = 74. Specifically for A = 74, the fact that M 1 is consistent with zero serves as an indirect indicator of shape coexistence in 74 Rb. As mentioned in the introduction, the analogue of the 0 + 2 state that mixes with the ground state in 74 Kr has not been seen in 74 Rb, which makes direct determination of shape coexistence in this nucleus difficult. The small M 1 value indicates that there is no significant evolution between the T z = 1 and T z = 0 members of the A = 74 multiplet, such that 74 Rb and 74 Kr should exhibit the same degree of shape coexistence.
Finally, theoretical calculations continue to make progress in our understanding of this region, although they remain challenging [17,55]. The only theoretical prediction of the B(E2) in 74 Rb of which the authors are aware is Ref. [56], in which shell model calculations of N = Z nuclei between A = 68-76 have been performed. Although the authors of that study note that their calculated transition rates are too small due to truncations on their model space, they predict a rapid increase in collectivity starting at A = 72. They attribute this to occupation of the g 9/2 d 5/2 orbitals, which drives a shape phase-transition between nuclei with A ≤ 70 and A > 70. This predicted increase in collectivity is qualitatively in agreement with the trend observed in Fig. 4, and the overall trend in the predicted B(E2) values matches the experimental data.
In summary, the lifetime of the 2 +