Enhanced collectivity along the N = Z line: lifetime measurements in 44 Ti, 48 Cr, and 52 Fe

. Lifetimes of the 2 +1 states in 44 Ti, 48 , 50 Cr, and 52 Fe were determined with high accuracy exploiting the recoil distance Doppler-shift method. The reduced E2 transition strengths of 44 Ti and 52 Fe diﬀer considerably from previously known values. A systematic increase in collectivity is found for the N = Z nuclei compared to neighboring isotopes. The B (E2) values along the Ti, Cr, and Fe isotopic chains are compared to shell-model calculations employing established interactions for the 0 f 1 p shell, as well as a novel eﬀective shell-model Hamiltonian starting from a realistic nucleon-nucleon potential. The theoretical approaches underestimate the B (E2) values for the lower-mass Ti isotopes. Strong indication is found for particle-hole cross-shell conﬁgurations, recently corroborated by similar results for the neighboring isotone 42 Ca. A detailed manuscript has meanwhile been published in Physics Letters B [1].


Motivation
The 0f 1p shell above the 40 Ca core is a fertile region for shell-model studies. Several interactions like KB3G [2], FPD6 [3], and the GXPF1A [4] are available and well established, yielding good agreement with various experimental observables, such as energy spectra or -with the inclusion of Coulomb contributions -isospin-dependent effects like mirror-energy differences. In fact, a considerable number of nuclei near the N = Z line is accessible via stable-beam fusionevaporation reactions with reasonable production yields. This opens the door for high-precision measurements of lifetimes and transition strengths to be compared to theory. This work focuses on the N = Z nuclei 44 Ti, 48 Cr, and 52 Fe. Reduced transition strengths are sensitive signatures to describe collective excitations in this region of the Segrè chart. Along the isotopic chains, (sub-)shell closures can be identified by a drop of the B(E2; 2 + 1 → 0 + g.s. ) value which increases towards midshell. In the chain of titanium isotopes such a drop was previously observed at  [5,6]. Data points of the same element are connected and the Ti and Fe chains are slightly shifted horizontally with respect to the Cr data to guide the eyes. (b) Adopted B 4/2 ratios for even-even N = Z nuclei between 40 Ca and 56 Ni. Limits for an ideal vibrator (long dashed purple line), an ideal rotor (dotted green line), and the limit for a non-collective behavior (B 4/2 = 1) (black line) are indicated. See text for details. Fig. 1 (a)) which is not expected with regard to the shell closure at N = 20. For 48 Cr and 52 Fe the B(E2) values are only known with considerable errors.
Furthermore, reduced transition strengths in these nuclei are of high interest to investigate the systematics of B 4/2 ratios, as recently discussed in Ref. [7]. The B 4/2 ratio (sometimes also referred to as RE4) is defined as follows: In rotational and vibrational nuclear models as well as in shell-model calculations the value of B 4/2 is always larger than one. B 4/2 ratios deduced from previously adopted B(E2) values evolve dramatically in the 0f 7/2 shell along N = Z from a vibrational behavior for 44 Ti and 52 Fe to a single-particle value at the 0f 7/2 mid-shell for 48 Cr (cf. Fig. 1(b)).

Experiment
Lifetime measurements of the 2 + 1 states in 44 Ti, 48,50 Cr, and 52 Fe were performed at the Institute for Nuclear Physics, University of Cologne employing the Cologne plunger device [8]. Ion beams were provided by the FN tandem accelerator at 62, 26, and 86 MeV inducing nat. Mg( 23 Na, xnp) 44 Ti, 40 Ca( 10 B, np) 48 Cr, 27 Al( 28 Si, αp) 50 Cr, and 27 Al( 28 Si, 2np) 52 Fe fusion-evaporation reactions, respectively. Excited recoil nuclei left the target with velocities of 1 to 3% of the speed of light and were finally stopped within a 9.6-mg/cm 2 thick Au foil. Emitted γ rays were detected by a setup of twelve high-purity germanium (HPGe) detectors. The HPGe detectors were placed in rings centered at polar angles of θ 0 = 0 • (1 detector), θ 1 = 45 • (6 detectors), and θ 2 = 143 • (5 detectors) with respect to the beam axis. During the experiments, data were recorded for different target-to-stopper distances. For each target-to-stopper distance coincident γ rays were sorted into γγ matrices correlating groups of the different detector rings.

Results
In the analysis lifetimes were determined using the recoil distance Doppler-shift (RDDS) technique in combination with the differential decay-curve method (DDCM) [8]. To reduce systematic uncertainties caused by unobserved side feeding, the DDCM was applied for γγcoincidence data. Lifetimes of the 2 + 1 states were obtained from the intensity ratios of shifted (SH) and unshifted (US) components (i) of the depopulating transition for gates on the shifted part of feeding transitions and (ii) of the direct feeding transitions for coincidence requirements on the unshifted component of the 2 + 1 → 0 + g.s. transition. Exemplary gated γ-ray spectra of the 2 + 1 → 0 + g.s. transition in 50 Cr are shown in Figs. 2 (a, b). The lifetime τ is deduced from the weighted mean of each lifetime τ i obtained at a distance i in the sensitive range. The resulting plots for the 2 + 1 lifetime of 50 Cr are given in Figs. 2 (c, d). Corresponding spectra and lifetime curves of 44 Ti, 48 Cr, and 52 Fe are published in Ref. [1]. Results of the determined lifetimes and corresponding B(E2) values are summarized in Table 1.  40 Ca target [9,10]. Challenging aspects of these experiments were the lack of precise knowledge of the stopping power, unknown feeding contributions, and the decreasing sensitivity of the DSAM for lifetimes longer than 1 ps. The present value provides a solution of the puzzling question of the robustness of the N = 20 shell closure (see Fig. 3(a)). The B(E2; 2 + 1 → 0 + g.s. ) = 26.9 +1.7 −1.4 W.u. of 48 Cr and B(E2; 2 + 1 → 0 + g.s. ) = 19.6(6) W.u. of 50 Cr are in good agreement with previously known values. In the case of 48 Cr the relative uncertainty was further reduced and amounts to 5.8%. Both values follow the overall trend along the Z = 24 isotopes (cf. Fig. 3(b)). Similar to 44 Ti, also the B(E2; 2 + 1 → 0 + g.s. ) value of 23.0 +1.3 −1.1 W.u. in 52 Fe exceeds the previously known value of 14.2(18) W.u. [11] and suggests a stronger collectivity at N = Z = 26. Both B(E2) values, the previously known as well as the present one, are consistent with the expectation of an increased collectivity with respect to the N = 28 shell closure, however, the present slope is more pronounced (see Fig.3(c)). To sum up, the B(E2; 2 + 1 → 0 + g.s. ) values in 44 Ti and 52 Fe are considerably larger than the previously measured ones. Thus, an enhanced collectivity close to the doubly-magic nuclei 40 Ca and 56 Ni at N = Z is found.

Discussion
Furthermore, shell-model calculations were performed to compare the experimental results with theory. Shell-model calculations were performed using the NuShellX@MSU code [12]. For comparison three established interactions were employed, namely the KB3G [2], FPD6 [3], and GXPF1A [4]. Single-particle excitations are limited to the full 0f 1p valence space. For the calculation of the matrix elements, effective charges e π = 1.5 e and e ν = 0.5 e were employed. Furthermore, realistic shell-model calculations, using the shell-model code Antoine [13], have been performed [14]. The calculated B(E2; 2 + 1 → 0 + g.s. ) values are reported in Table 1. An overview on the evolution of the B(E2) values for Ti, Cr, and Fe isotopes is shown in Fig. 3.
For each interaction the theoretical B(E2) values along the chain of titanium isotopes are in poor agreement with the experimental ones close to N = Z (see Fig. 3(a)). A large deviation is observed for N = 20 where the theoretical values are considerably smaller than the experimentally determined values. This discrepancy decreases with an increasing number of valence neutrons toward N = 26. However, the sub-shell closure at N = 28 is also not reproduced by the calculations. In the neighboring even-even isotone 42 Ca a recent investigation revealed a similar result [15]. Shell-model results underpredict the enhanced experimental B(E2) values. It was suggested that the reduced theoretical values arise from 2hω sd-shell excitations, which are not included in the 0f 1p model space [16]. Evidence for the importance of core excitations is also found in the study of the schematic α-cluster model where excitation energies of all rotational bands in 44 Ti are reproduced using an α + 40 Ca(I π ) system [17]. Moreover, their calculated B(E2; 2 + 1 → 0 + g.s. ) = 18 W.u. value [17] is in a better agreement with the present experimental results.  In the chain of chromium isotopes the different shell-model calculations reproduce the trend of the measured B(E2) values for N < 28, whereas the measured values are overestimated for N ≥ 28 (cf. Fig. 3(b)).
Only partial agreement between experiment and theory is found for the iron isotopes (see Fig. 3(c)). For light isotopes at N = 24, 26 results from the realistic calculation and the FPD6 interaction are consistent with the present experimental data. However, nice agreement is achieved between the KB3G interaction and experiment for N ≥ 28. It is noteworthy, that no calculation describes the huge experimentally observed drop in B(E2) values between N = Z = 26 and the shell closure at N = 28; it is underestimated by each interaction.
Considering  Fig. 4. The present evaluation of the B 4/2 ratios shows considerable differences for 44 Ti and 52 Fe with respect to the previous results. The observed strong variations of the B 4/2 ratios along the chain of isodiapheres, which were discussed in Ref. [7], are not confirmed. Moreover, the evaluated vibrator-like behavior for 44 Ti and 52 Fe is now suggested to be more rotor-like. The puzzle of the minimum value for mid-shell 48 Cr close to unity could not be resolved and remains subject of future studies.

Summary
These proceedings report on lifetime measurements in N = Z nuclei 44 Ti, 48 Figure 4. B 4/2 ratios for even-even N = Z nuclei between 40 Ca and 56 Ni. Results from this work are given in red squares and the adopted values are shown in black triangles. Same color code as in Fig. 1 (b) is used. See text for details.