Isospin dependence of electromagnetic transition strengths among an isobaric triplet

Electric quadrupole matrix elements, M p , for the J π = 2 + → 0 + , (cid:3) T = 0, T = 1 transitions across the A = 46 isobaric multiplet 46 Cr- 46 V- 46 Ti have been measured at GSI with the FRS-LYCCA-AGATA setup. This allows direct insight into the isospin purity of the states of interest by testing the linearity of M p with respect to T z . Pairs of nuclei in the T = 1 triplet were studied using identical reaction mechanisms in order to control systematic errors. The M p values were obtained with two different methodologies: (i) a relativistic Coulomb excitation experiment was performed for 46 Cr and 46 Ti; (ii) a “stretched target” technique was adopted here, for the ﬁrst time, for lifetime measurements in 46 V and 46 Ti. A constant value of M p across the triplet has been observed. Shell-model calculations performed within the fp shell fail to reproduce this unexpected trend, pointing towards the need of a wider valence space. This result is conﬁrmed by the good agreement with experimental data achieved with an interaction which allows excitations from the underlying sd shell. A test of the linearity rule for all published data on complete T = 1 isospin triplets is presented.

This allows direct insight into the isospin purity of the states of interest by testing the linearity of M p with respect to T z . Pairs of nuclei in the T = 1t r i p l e t were studied using identical reaction mechanisms in order to control systematic errors. The M p values were obtained with two different methodologies: (i) a relativistic Coulomb excitation experiment was performed for 46 Cr and 46 Ti; (ii) a "stretched target" technique was adopted here, for the first time, for lifetime measurements in 46 V and 46 Ti. A constant value of M p across the triplet has been observed. Shell-model calculations performed within the fp shell fail to reproduce this unexpected trend, pointing towards the need of a wider valence space. This result is confirmed by the good agreement with experimental data achieved with an interaction which allows excitations from the underlying sd shell. A test of the linearity rule for all published data on complete T = 1 isospin triplets is presented. The exchange symmetry between the proton and the neutron is one of the most fundamental symmetries in modern physics, rooted in the near charge-symmetry and charge-independence of the nuclear force [1]. The concept led Heisenberg [2]t o introduce the isospin quantum number (T ) which can be assigned to any nuclear state such that, in the absence of charge-dependent forces, there will be an identical analogue state of the same isospin T in all isobars T z (= (N − Z )/2) in the range +T to −T .
In general, any interactions that depend on charge (the strongest of which is the electromagnetic interaction) are sufficiently weak that they do not disturb the symmetry of the underlying wave functions of these isobaric analogue states (IAS). Hence the assumption of identical wave functions among a set of IAS is usually considered to be safe. Much work has been undertaken recently studying differences in energy between excited states of mirror nuclei and T = 1t r i p l e t s (IAS with T = 1i n the three nuclei with T z = 0, ±1) -see, for example, references [3][4][5][6][7][8]. In these analyses, symmetry of the underlying wave functions is assumed and so the differences in excitation energy were interpreted in terms of nuclear structure phenomena. In a shell-model analysis, it is found that additional isospin non-conserving interactions (INC), beyond the usual two-body Coulomb force, were required to account for the data, e.g. [3,6]. In the case of T = 1t r i p l e t s , the INC interactions required are consistently large, and it is speculated that the charge dependence of the nuclear interaction itself plays a significant role [9,10]. This analysis of INC forces, however, does not yield any information on the purity of the isospin quantum number or deviations from the presumed identicality of the analogue wavefunctions.
Isospin mixing can, however, be studied through testing predictions that rely on the isospin purity of a set of analogue states. The established method is the isobaric multiplet mass equation (IMME). The IMME predicts that the total binding energy in a multiplet should be a quadratic in T z for a set of identical analogue states of pure isospin T in the presence of a two-body interaction with isoscalar, isovector and isotensor components. The IMME has been tested many times through high-precision mass measurements and, whilst there are a small number of notable deviations, the rule holds well -see, for example, Lam et al. [11]f o r a comprehensive review. Similar rules appear for electromagnetic transition strengths under the same assumptions as above for the IMMEsee reference [12]f o r details. The isospin dependence of the proton matrix element for a set of T → T analogue electromagnetic tran-sitions has a simple form, and for the analogue states of a T = 1 triplet may be written [13] where M 0 and M T Z =1 1 are the isoscalar matrix element and isovector matrix element (for the T z = 1 nucleus) respectively. Thus, in the limit of pure isospin, the analogue proton matrix elements should be exactly linear with T z . This rule is difficult to test precisely due to the experimental challenges in performing such measurements in proton-rich nuclei, since this requires either use of radioactive beams or production of nuclei from stable-beams but with very low cross sections. The most straightforward way to test this rule is through the study of a T = 1t r i p l e t of nuclei and measurement of the analogue B(E2) strengths between the T = 1, 0 + ground state and the T = 1, 2 + first excited state -in which case B(E2)(T z , 0 + → 2 + ) = (M p (T z )) 2 . Indeed, there is the potential for isospin mixing in the odd-odd N = Z system due to the close proximity of the T = 1 and T = 0s t a t e s . Prados-Estevez et al. [14]p e r f o r m e d a compilation of such B(E2) strengths for all T = 1t r i p l e t s for 22 ≤ A ≤ 42 and found, in general, good agreement with the rule within the error limits. There were some exceptions -most notably A = 38 [14] where deviation from linearity was indicated. However, for many of the cases shown, the error bars were too large to test linearity. Moreover, in general, the measurements for each triplet have come from different experiments, with different B(E2) measurement methods and different population/feeding routes to the state of interest -hence hidden systematic errors cannot be ruled out.
The A = 46 triplet, the topic of this work, is the heaviest for which all three B(E2)s are known, although the errors in the B(E2)s for 46 Cr and 46 V (at 20% [15] and 13% [16] respectively) are too large to test for any non-linearity. The A = 46 triplet is especially interesting since systematics of the experimental T = 0 and T = 1s t a t e s in 46 V [ 18] indicate that the lowest T = 1 and T = 02 + states are likely to be very close in energy and hence may result in mixing of isospin. The aim of the work presented here was to address these issues by performing B(E2) measurements for all three members of the A = 46 triplet. To reduce the impact of systematic errors, we also ensure that analogue B(E2)s are measured using the same technique, under identical experimental conditions, and that the states of interest were populated through an identical (analogue) mechanism. This enables a precise relative measurement of the transition strengths to test the linearity rule. We also introduce here a method for lifetime measurements, especially suited to studies with fragmentation beams, which we call the "stretched target" technique.
The experiment was performed at the GSI Helmholtzzentrum für Schwerionenforschung, Germany, with the FRS-AGATA-LYCCA setup -see [19,20]f o r further details. A 600 MeV/A 58 Ni beam, provided by the SIS18 synchrotron, impinged on a 2.5 g/cm 2 Be primary target. The isotopes resulting from the fragmentation reaction were separated and identified with the double-stage magnetic spectrometer FRS [21]b y means of the Bρ − E − Bρ method, which allows precise determination of the atomic number Z and the mass over charge-state ratio, A/q. The energy loss of the ions was measured with two Multi Sample Ionisation Chambers (MU-SICs) while the time of flight and the positions in the second and fourth focal planes were obtained from two plastic scintillators. Three different magnet settings were chosen to transmit the three secondary beams of interest, 46 Cr, 46 V and 46 Ti with beam energies of 180, 176 and 178 MeV/A respectively. The three beams impinged, separately, on two different secondary-target arrangements, described below, to measure the three analogue 0 + The secondary target (arrangement) was surrounded by the gamma ray tracking spectrometer AGATA [22], comprising, in this configuration, 22 HPGe detectors placed in the forward direction with respect to the beam line arranged in a combination of tripleand double-cryostats. For AGATA, the pulse-shape analysis methodology gives a position sensitivity of ≤ 5m m which, in turn, is crucial for γ -ray photo-peak resolution, which would otherwise be destroyed by Doppler broadening effects at β>0.5c, such as here. The outgoing ions were identified by the LYCCA calorimeter [23] which provides energy-loss, total energy and time-of-flight (ToF) measurements. The atomic number of the outgoing fragment was determined by E-E measurement given by a combination of DSSSD and CsI detectors in the LYCCA "wall" [23]. Determination of the mass of the outgoing fragment was not necessary since the fragment of interest is the same as the incoming secondary beam in both cases. The two LYCCA plastic scintillators placed just before and 3.4 m after the secondary target provide a precise event-by-event ToF measurement which is essential for the eventby-event Doppler correction. A single DSSSD detector is placed in close proximity to the secondary target in order to determine the interaction position, and the complementary position information given by the wall DSSSDs therefore provides the track of the outgoing ions. The position information from the LYCCA detectors was coupled to this information to provide full reconstruction of the trajectory of the incoming and outgoing ions for the purpose of Doppler correction and determination of scattering angle.
Two different methods were employed to determine the 0 + For each method two members of the triplet were measured -either 46 Cr/ 46 Ti or 46 V/ 46 Ti. Thus, in each case 46 Ti, for which the B(E2) is known very precisely, is measured, as a "reference" point to enable a relative measurement.
For the 46 Cr/ 46 Ti measurements, relativistic Coulomb excitation was used, for which the secondary-target comprised a 500 mg/cm 2 single gold foil. The cross section of the reaction is given by the relation: (2) where N B is the number of incoming ions detected in FRS, M TA is the number of target atoms per unit area and N γ is the number of counts in the γ -ray peak after background subtraction and efficiency corrections. In order to avoid any possible interference between the electromagnetic and the nuclear interactions, a cut  on the scattering angle of the ion was applied, such that the minimum impact parameter is given by The cross-section values obtained are (81.4 ± 6.6) mb for 46 Ti and (78.1 ± 14.0) mb for 46 Cr. The error budget is strongly dominated by the statistical contribution due to the number of counts in the γ -ray peaks. The electromagnetic transition probabilities have been obtained from the measured cross sections using the DWEIKO code [24], which provides calculations of elastic scattering differential cross sections, probabilities, and cross sections for inelastic scattering in nuclear collisions at intermediate and high energies in the framework of the Distorted Wave Born Approximation (DWBA), exploiting eikonal wave functions and solving coupled-channels equations [24]. The results are presented in Table 1 together with the previously reported values [15,16]. The 46 Ti result is consistent with the literature value, confirming the present analysis method. The 46 Cr measurement is also consistent with the literature value, with a slightly reduced error bar.
In the second method, for the 46 V/ 46 Ti measurements, lifetimes were determined using a new "stretched-target" method. A stack of three gold foils was used at the secondary target, to excite the ions of interest by electromagnetic interaction whilst measuring the lifetime using Doppler-shift effects. Each foil acts as both a target (to excite the 2 + 1 state through Coulomb excitation) and a degrader to vary the velocity of the incoming fragment. The result is a Doppler-shift profile that provides a distinctive line-shape with three peaks (for decays following the three targets), the shape of which depends on the lifetime. The target thicknesses were chosen to be (750 + 500 + 500) mg/cm 2 , separated by two 1m m gaps, all of which was optimised to provide maximum sensitivity Gamma-ray spectra for 46 V (top) and 46 Ti (bottom) using the triple-gold-foil "stretched-target" method. The Doppler correction has been optimised for the central "peak", and the correct energy of the transition is marked by the dashed line.
Only AGATA detectors at an angle ≤ 50 • to the beam direction are included, and an angle-dependent correction has been applied to spectrum to ensure that the peak centroids are at the same location for all angles. The solid (red) line is a full AGATA simulation produced under the same conditions as the experiment -see text for details. Inset: The χ 2 values (per degree of freedom) as a function of half life, and the resulting polynomial fit used to determine the half life and the statistical errors, also shown -see text for details. to half lives in the 5p s region. The three average velocities after the targets were, for 46 Ti (for example), v/c = 0.527, 0.509 and 0.488 respectively for decays after each of the three foils in turn. In this analysis, the Doppler correction was performed for the central "peak" which means that the position of the two other peak centroids will be dependent on angle. Since AGATA provides eventby-event angle information, a correction was applied to move the peak centroids to the same location in the spectrum for all angles. In this way the full range of angles can be included in a single spectral analysis for the lifetime. Events for all available laboratory angles ≤ 50 • were included (see below). The resulting spectra are shown in Fig. 2. In these spectra, and in the Coulomb excitation spectra in Fig. 1, the 2 + 1 → 0 + g.s. transition is the only peak observed in the whole spectrum.
In order to determine the lifetime, a full simulation was produced of the γ -ray spectral response. This involved a GEANT4 [25] simulation using an event generator that allows for Coulomb excitation of the single state (2 + 1 ) in each of the three targets. The secondary beam parameters (velocity, momentum spread, spatial distribution) were simulated from well-established ion-optical transport codes, and adjusted to match the observed experimental conditions. The γ -ray response was simulated through coupling this event generator to the full AGATA simulation code [26]. The experimental spectral resolution is dominated by the particle tracking capability of the system, specifically the effective LYCCA ToF resolution, which was adjusted in the simulation to reproduce the experimental width from the single-target spectra. The simulated data were then passed through the same analysis procedure as the experimental data. The final simulated spectrum was produced by adding the lineshape from the target to a polynomial background, the parameters of which were extracted from a fit  to the experimental spectrum between 700 and 1100 keV. For a specific simulated lifetime, the background and lineshape intensity were allowed to vary freely to obtain the best fit. This procedure was repeated, varying both the half life and the upper angle cut, to create a χ 2 surface. This was used to determine the optimum upper angle cut (i.e. narrowest χ 2 on the half-life axis). This yielded the upper angle cut of 50 • which was used for both 46 V and 46 Ti. The half life was determined from a fit of the resulting T 1/2 vs χ 2 plot and the statistical error determined from the evaluation of that fit at χ 2 min + 1. The analysis yields 4.23(31) and 5.26(30) ps (statistical errors) for 46 V and 46 Ti respectively. A systematic error was determined by varying, in the simulation, the two effects expected to have the strongest effect on the lineshape -the small energy dependence of the cross-section through the targets (calculated using DWEIKO for the simulation) and the gaps between the foils. For the former we introduced a 15% variation in the energy dependence and for the latter we allowed all three foils to move by ±100 μm. This yielded a 0.17 ps systematic error which was added in quadrature. The final results, and the corresponding 0 + g.s. → 2 + 1 B(E2) strengths are quoted in Table 2. Again, the agreement with the literature values is good, with the value for 46 V having a much reduced error bar. The absolute values of the proton transition matrix elements determined in this work are plotted in Fig. 3(a). The good agreement with the literature values for 46 Ti validates both methods. Since the aim was to determine relative B(E2)s across the triplet, the measured ratios M p (Cr)/M p (Ti) (from Coulex) and M p (V)/M p (Ti) (stretched target) are shown in Fig. 4. Since the conditions were identical for each measurement, only statistical errors have been included for the relative measurements. To reduce the error bars further, the values obtained in this work have been combined with the data from references [15][16][17]i n a weighted average, and shown in Figs. 3(b) and (c), along with a linear fit (dashed line). It is clear that the isospin rule (linearity of the matrix element with T z ) holds extremely well -implying no evidence for isospin mixing. Indeed, the M p values are remarkably constant across the triplet. This, in turn, implies a vanishingly small isovector proton matrix element -the linear fit (see Eq. (1)) yielding 0.0(15) and 62.2(15) efm 2 for the isovector and isoscalar matrix elements respectively. A recent complication of all available data by Morse et al. [42] indicates that measured isovector matrix elements are consistently small, relative to the isoscalar.
The linearity rule for transition matrix elements can be tested, generally, for isospin multiplets, by fitting a quadratic expression of the form M p = a + bT z + cT 2 z and extracting the c coefficientwhich should be zero in the limit of exact isospin symmetry. We have performed this analysis for the eight complete T = 1t r i p l e t s that exist in the literature, and the results are plotted in Fig. 5.
The data (i.e. M p ) for the even-even members of the A = 10 − 42 triplets are taken from the recent compilation of B(E2)s [ 27], with more recent published results [28][29][30]i n c l u d e d through adding them as a weighted average. Data for the odd-odd members of the triplet are taken from the most recent evaluations [31][32][33][34][35][36] and recent results on 10 B [ 37]. The data used for A = 46 are the weighted averages plotted in Fig. 3(b). Since the total matrix element varies significantly in this mass range, we have plotted the extracted c-coefficients normalised to the isoscalar matrix element, M 0 (extracted from a linear fit to the same data -see Eq. (1)). As expected the A = 46 triplet c-coefficient is consistent with zero, and it is clear that the A = 46 data now provide one of the most precise tests of the rule to date. It can also be seen that data on A = 30, 34 now deviate significantly from the prediction, once the most recent data are included.
The near equality of the experimental values for M p for A = 46 is, intuitively, unexpected since, assuming an inert 40 Ca core, the larger atomic number of 46 Cr should induce a higher transition probability. These data have been compared with shell-model calculations using the KB3G interaction [38]i n the fp valence space, as shown in Fig. 3(b). Calculations using the effective charges of reference [39], usually considered to be appropriate for this region, are shown by the solid line in Fig. 3(b) and the upper line in Fig. 4. The data are not well reproduced, neither in magnitude  Fig. 3(b)). The overall agreement, however, remains poor.
The above analysis clearly points towards the inadequacy of the fp valence space for these nuclei, suggesting the importance of core excitations from the underlying sd shell. To overcome this issue we used the ZBM2 interaction [41], with an inert core of 28 Si and the valence space consisting of the s 1/2 − d 3/2 − f 7/2 − p 3/2 orbits. This is shown by the solid line in Fig. 3(c) and the lower line in Fig. 4, where the effective charges of [39]a r e used as before. Now the magnitude and T z -dependence are much better reproduced. This clearly indicates that, for the A = 46 nuclei, the 40 Ca core is not closed, and the only way to explain the very flat dependence of the matrix elements with T z is by having a much larger (and hence more similar across the triplet) number of valence protons active for the 2 + 1 state through particle-hole excitations. This is consistent with the fact that the A = 46 mirror energy differences are unusually poorly reproduced by a shell-model analysis involving only the fp valences space -see, for example, reference [3]. The ZBM2 calculations (solid line in Fig. 3(c)) still do not reproduce the full magnitude of the B(E2). This is likely due to the lack of the f 5/2 , p 1/2 and d 5/2 orbitals. Since quadrupole excitations are mainly due to the coupling of L = 2o r b i t s , their absence will affect the B(E2). To demonstrate this, the energy gap between the f 7/2 and the p 3/2 orbits was artificially reduced by 500 keV, to increase the L = 2 interactions between the fp orbits. Indeed the agreement is now within error for two members of the tripletsee the dotted line in Fig. 3(c)).
In conclusion, the rule that (in the limit of pure isospin) the electromagnetic matrix elements should be exactly linear with T z has been tested with high precision for the heaviest T = 1 triplet for which data are currently available in the literature. We have compared our results with an analysis of the linearity rule for all known T = 1t r i p l e t s . An experimental method has been used that allows for a reliable relative measurement through using the same technique, experimental conditions and analogue population mechanism for different members of the triplet -hence reducing the impact of any systematic errors. The experiment was performed with the AGATA array where the superb position sensitivity and tracking capability of the new LYCCA set-up allowed for a high-resolution analysis. We have introduced a new method for lifetime analysis, the stretched target method, made possible by this high resolution. This method is especially well suited to spectroscopy using exotic fragmentation beams of high velocity (β ≥ 0.5), and so would be ideal for AGATA experiments at the future FAIR facility.