(3) empirical effective proton and neutron charges e π ¼ 1.30ð8Þe and e ν ¼ 0.452ð7Þe, respectively, that are in contrast to reports of e π ≈ 1.1-1.15e and e ν ≈ 0.6-0.8e for fp-shell nuclei near N ¼ Z. We demonstrate that these reports are erroneous and that, in fact, a universal set of effective charges can be used across the sd and fp shells. DOI: 10.1103/75ry-71sj

The nuclear shell model provides a fundamental view of nuclear structure [1,2]. It presumes independent particle motion in a spherical mean field with strong spin-orbit coupling, leading to significant energy gaps at "magic" proton and neutron numbers, Z; N ¼ 2, 8, 20, 28, 50, 82, and 126. Modeling nuclear structure then involves understanding the correlations or effective interactions between valence nucleons, which mix single-particle configurations, within a finite, truncated model space. However, exotic nuclei with increasing neutron excess relative to stable nuclei become sensitive to different aspects of the nuclear forces [3][4][5][6]. This leads to an evolution of the single-particle orbitals, causing standard magic numbers to disappear and new ones to emerge [7,8]. All effects from outside of the truncated model space are then treated with effective operators, which have been reported to have isospin and orbital dependencies beyond the sd shell [9,10].

Considerable interest has been directed toward the emergence of new subshell gaps at N ¼ 32 and 34 in neutron-rich nuclei within the fp shell. These gaps are currently understood to be driven by the elevation of the ν1f 5=2 orbital as protons are removed from the π1f 7=2 orbital. The existence of the N ¼ 32 subshell gap has been extensively confirmed near Z ¼ 20 by the measurement of high Eð2 þ 1 Þ values in 50 Ar [11], 52 Ca [12,13], 54 Ti [14,15], and 56 Cr [16][17][18]. These findings have been further supported by mass measurements of 51-54 Ca [19,20], 52;53 K [21], and 52-54 Sc [22,23]. However, mass measurements of Ti and V isotopes [24,25] have introduced some ambiguity.

The N ¼ 34 subshell closure is less clear. Initial evidence came from 54 Ca, where a high Eð2 þ 1 Þ of 2043 (19) keV was measured [26]. This was further supported by direct mass measurements of 55-57 Ca [27]. In 52 Ar, the 2 þ 1 state was measured at 1656 (18) keV [28]. Spectroscopic strengths from knockout reactions supported subshell closures at N ¼ 32 and N ¼ 34 [29,30]. However, the persistence of the N ¼ 34 subshell gap above Z ¼ 20 remains uncertain [15,25,[31][32][33][34][35].

Another point of interest for the region is the unexpectedly large charge radii of 49-52 Ca [36], where neutron filling shifts from the ν1f 7=2 to the ν2p 3=2 orbital. The sudden rise beyond N ¼ 28 is striking given the equivalent charge radii of 40 Ca and 48 Ca. In fact, the isotope shifts between 40 Ca and 48 Ca seem to be described solely by the E2 strength [37]. The change in charge radii beyond N ¼ 28 has been attributed to core polarization and a p-orbit halo, both of which can induce neutron skins [38]. The effective neutron charge, determined from a BðE2; 50 Ca, also showed a sudden change with the occupation of the ν2p 3=2 orbital [9]. A comparable result was reported from inelastic proton scattering on 50 Ca [39]. In this Letter, we investigate 54 Sc by decay spectroscopy at the Facility for Rare Isotope Beams (FRIB) using the FRIB Decay Station initiator (FDSi) [40,41]. 54 Sc is positioned one proton above the Z ¼ 20 shell closure and one neutron between the N ¼ 32, 34 subshell closures. This gives rise to relatively simple single proton-neutron (PN) coupled states at low excitation energies that are sensitive to the magnitudes of the N ¼ 32, 34 subshell gaps. A new nanosecond isomer is reported that corresponds to the most precise, unambiguous E2 transition strength in the neutron-rich fp region to date in a nucleus with valence protons above Z ¼ 20. The new B(E2) value and resulting effective charges provide a unique opportunity to probe core-polarization effects in neutron-rich fp-shell nuclei near N ¼ 32, 34.

The present results are from two experiments conducted at FRIB using the two-focal-plane configuration of the FDSi. In both experiments, a secondary cocktail beam of fully stripped ions produced in the fragmentation reaction between an 82 Se primary beam and a 9 Be target were delivered to the FDSi. At the first focal plane, γ rays and neutrons were detected using the DEcay Germanium Array initiator and the Versatile Array of Neutron Detectors at Low Energy [42,43], respectively. The Modular Total Absorption Spectrometer (MTAS) [44,45] was situated at the second focal plane. At both focal planes, cocktail beams centered around 52;54 K were implanted within a positionsensitive yttrium orthosilicate (YSO) detector [46,47]. See Refs. [48][49][50] for additional experimental details.

The γ rays observed following the β decay of 54 Ca and implantation of 54m Sc are presented in Figs. 1(a) and 1(b), respectively. The 247-keV β-delayed γ-ray transition, identified in this work with an absolute intensity of 62(8)%, was previously reported in Refs. [51,52]. The transition was attributed to a 247-keV state in 54 Sc with a spin parity of J π ¼ 1 þ based on the selectivity of the 0 þ parent decay. The statistics of the present work are higher by an order of magnitude than Ref. [52], but no additional discrete γ rays in 54 Sc were observed.

The fast-timing capabilities of the YSO and LaBr 3 scintillators at the first focal plane of the FDSi have PHYSICAL REVIEW LETTERS 135, 072501 (2025) enabled the identification of the 247-keV, 1 þ state in 54 Sc as a nanosecond isomer, as shown in Fig. 2. A maximumlikelihood fit to the β-γ time difference distribution yielded a 26.0ð22Þ stat ð3Þ sys ns half-life. The fit incorporated an exponentially modified Gaussian function with a background distribution derived from regions adjacent to the 247-keV peak.

The presence of a low-lying microsecond isomer in 54 Sc was initially reported in Ref. [53] with

, and confirmed [15,52] based on the detection of a 110-keV γ-ray peak following 54m Sc implantation. In this work, the microsecond 54m Sc isomer is confirmed as shown in Fig. 1(b) and no additional γ rays are observed in association with its decay. A fit to the implant-γ time-difference distributions yielded half-lives of 2.74(4), 2.80(6), and 2.72ð9Þ μs for the HPGe, LaBr 3 , and MTAS detectors, respectively. The weighted average, 2.75ð3Þ μs, agrees with previous values of 7ð5Þ μs [53] and 2.77ð2Þ μs [52].

The low-lying states of 54 Sc can be schematically interpreted with respect to a 52 Ca (Z ¼ 20, N ¼ 32) core, outside of which the odd 7=2 -proton from 53 Sc and odd 1=2 -neutron from 53 Ca spin couple to a 3 þ , 4 þ doublet. The population of the first 2 þ and 4 þ states in 54 Ti following the β decay of 54 Sc supports a 3 þ ground-state assignment for 54 Sc, consistent with discussions in Refs. [15,51,52]. Furthermore, this population is identical between the 54 Sc → 54 Ti and 54 Ca → 54 Sc → 54 Ti decay chains, suggesting the absence of a β-decaying isomer. The next lowest multiplet of states would then arise from a single-particle neutron excitation across N ¼ 34, corresponding to the excited 5=2 -state in 53 Ca coupled to the 7=2 -ground state in 53 Sc. The πf 7=2 ⊗ νf 5=2 coupling forms a sextet of states with

The 247-keV γ ray deexciting the nanosecond isomer corresponds to a 1 þ → 3 þ transition between the two multiplets with an experimental BðE2Þ of 1.93ð16Þ stat ð2Þ sys W.u. determined in this work. The 110-keV γ ray deexciting the microsecond isomer may represent either (a) a pure

transition within the πf 7=2 ⊗ νp 1=2 multiplet with BðM1Þ ≤ 5.92ð7Þ × 10 -6 W.u. and BðE2Þ ≤ 0.86ð1Þ W.u. We adopt the latter, which is consistent with discussions in Ref. [52], as the 2 þ and 6 þ states are expected to lie well above the antiparallel 1 þ state, and the in-multiplet M1 transition is expected to be strongly hindered; the strength to first order is given by BðM1;

Differences in neutron separation energies along the Ca isotopic chain, which emphasize the magnitude of the shell gaps, and level schemes for 54 Sc are shown in Figs. 3(a) and 3(b), respectively. Predictions by the KB3G [54] and GXPF1A [55] interactions, used in previous 54 Sc studies, are compared with recent UFP-CA (empirically adjusted) [56], SM Ã (perturbative ab initio) [57,58], and VS-IMSRG 1.8=2.0 EM (ab initio) [59][60][61] interactions. The UFP-CA Hamiltonian for the T ¼ 1 neutron-neutron interaction was obtained from constraints to data for the Ca isotopes [56]. For application to the Sc isotopes, the T ¼ 0 part of the GXPF1A Hamiltonian [55] was added. The T ¼ 1 part of the GXPF1A Hamiltonian was obtained from energy data for nuclei toward the middle of the fp shell. Thus, the T ¼ 1 parts of the UFP-CA and GXPF1A interactions are nucleus-dependent and the UFP-CA Hamiltonian should only be used for nuclei near Z ¼ 20 with N > 28. All calculations were performed with KSHELL [62] in the full fp model space. For the SM Ã interaction, the neutron g 9=2 orbit was also included. The KB3G and GXPF1A interactions predict N ¼ 34 gaps for Ca that are nonexistent and too strong, respectively, as shown in Fig. 3(a). The sensitivity of the low-energy structure of 54 Sc to the magnitude of the N ¼ 34 gap is PHYSICAL REVIEW LETTERS 135, 072501 (2025) 072501-3

reflected in the two corresponding level schemes, namely a 1 þ ; πf 7=2 ⊗ νf 5=2 state that is too low or too high, as illustrated in Fig. 3(b). The recent interactions (UFP-CA, SM Ã , and VS-IMSRG 1.8=2.0 EM) predict a weak N ¼ 34 gap relative to N ¼ 32 and better describe the experimental data and schematic interpretations. We note that the BðE2; 1 þ → 3 þ Þ strengths predicted by the empirically adjusted interactions (KB3G, UFP-CA, and GXPF1A) also increase with the magnitude of the N ¼ 34 gap and that the value for UFP-CA, which best describes the level energies, is larger than experiment.

The UFP-CA results are further investigated in Fig. 4 as a function of the PN interaction strength. All PN twobody matrix elements were multiplied by a scale parameter, V PN scale. In the weak coupling limit (V PN scale ¼ 0), the energies (and transitions) correspond to the excited neutron states in 53 Ca and the excited proton states in 53 Sc. As the PN strength increases, each multiplet splits and begins to mix with nearby configurations. At [63] is shown as a proxy for this limit, represented by the red band in Fig. 4(b). The PN interaction, at its nominal value (V PN scale ¼ 1), increases the E2 strength by a factor of 4. The large increase arises primarily from mixing with the second 1 þ configuration from the πp 3=2 ⊗ νp 1=2 multiplet, where proton p 3=2 → f 7=2 transitions across Z ¼ 28 have large intrinsic E2 strength. Note the large BðE2;

strength in the weak-coupling limit, and that the sum with BðE2;

remains relatively constant with the PN strength. The magnitudes of the Z ¼ 28 and N ¼ 34 gaps control the proximity of the two 1 þ configurations and therefore the mixing strength.

The BðM1; 4 þ → 3 þ Þ strength vanishes completely with either small adjustments to the PN interaction strength or 0.915 attenuation of the free neutron g s value; the total attenuation could be larger if the free proton g s were to be attenuated. However, the ground-state magnetic dipole moments of 49;51 Ca, which do not depend on the proton g s , fit best to a neutron g s attenuation of 0.957.

A survey of E2 data for neutron-rich Ca, Sc, and Ti isotopes is provided in Table I, including UFP-CA predictions with the "standard" isoscalar core-polarization effective charges of e π ¼ 1.5e, e ν ¼ 0.5e, and newly fitted effective charges of e π ¼ 1.30ð8Þe, e ν ¼ 0.452ð7Þe. Significantly larger effective charges would have been found if 1p -1h excitations across Z ¼ 28 or N ¼ 28 were not included, cf. Fig. 4(b) and Refs. [64,65]. The Blomqvist and Molinari oscillator parameter, ℏω ¼ 45A -1=3 -25A -2=3 [66], was used for all the (a) (b) (c) FIG. 4. UFP-CA calculations of 54 Sc as a function of PN strength for (a) the first four PN spin-coupled multiplets, and (b) the BðE2Þ values with newly fitted effective charges e π ¼ 1.30e, e ν ¼ 0.452e. (c) BðM1; 4 þ → 3 þ Þ values with free and quenched spin g factors. PHYSICAL REVIEW LETTERS 135, 072501 (2025) 072501-4

calculations, where BðE2Þ ∝ hr 2 ie 2 π;ν ∝ ℏ 2 ω 2 e 2 π;ν . Had ℏω 0 ¼ 41A -1=3 been used, the extracted effective charges would scale up by ℏω 0 =ℏω ¼ 1.068. The new effective charges improve the description of the E2 data in Table I from a reduced chi-squared, χ2 , of 18.1 to 1.1. The new effective neutron charge, e ν , is qualitatively consistent with ∼0.5e reported in Ref. [9] from the BðE2;

Ca but now includes fitting to the precise ground-state quadrupole moments of 49;51 Ca [67]. The new e π =e ν ratio of 2.88 (18), which is independent of ℏω, is also consistent with 3.5 (9) from inelastic proton scattering results on 50 Ca [39], but is now more precise by a factor of 5.

The new effective charges differ significantly from those previously established in the fp shell. For example, effective charges of e π ¼ 1.15e, e ν ¼ 0.8e (ratio of 1.44) were determined for 51 Fe= 51 Mn at N ≈ Z in the fp shell [68], and later updated to e π ¼ 1.12e, e ν ¼ 0.67e (ratio of 1.67) to best describe the stable N ¼ 28 and Z ¼ 28 chains [69]. The new values are also different from e π ¼ 1.1e, e ν ¼ 0.6e (ratio of 1.83) used recently to describe 49;50 Ti [70]. It has been argued that changes in the effective charges across the fp shell are due to an isospin or orbital dependence [9]. However, it is worth noting that the previous and new effective charges all have similar sums, namely e π þ e ν ≈ 1.7e-1.95e.

The E2 data and effective charges can be linearly systematized by plotting A n =M p versus A p =M p , as shown in Fig. 5. All data points should fall on a straight line with slope e π =e ν and intercept 1=e ν [64]. A p and A n represent the calculated transition amplitudes, which relate the E2 matrix element by hJ π f jjE2jjJ π i i ¼ A p e π þ A n e ν . M p is the experimental E2 matrix element, where BðE2Þ ¼ M 2 p =ð2J i þ 1Þ. J π f and J π i denote the spin parity of the final and initial states, respectively. Figure 5 shows that a large majority of fp-shell nuclei, including those at N ¼ 28, have A p ≈ A n meaning they are mostly sensitive to the sum of effective charges as opposed to the ratio. All of the previous cases are consistent with the new effective charges of e π ¼ 1.30ð8Þe, e ν ¼ 0.452ð7Þe. These values are equivalent to the universal effective charges in the sd shell, e π ¼ 1.36ð5Þe and e ν ¼ 0.45ð5Þe [71]. Further, they are equivalent to microscopic derivations by Dufour and Zuker, e π ¼ 1.31e and e ν ¼ 0.46e [72], which have been applied to sd [73], sd-pf [74], pf [75], fpg 9 d 5 [76], and pf-sdg [77] valence spaces. Therefore, universal effective charges of e π ¼ 1.33e, e ν ¼ 0.45e, based on empirical fits, can be used across the sd and fp shells. In summary, the structure of neutron-rich 54 Sc is reported from two experiments at FRIB using the FDSi. The 247-keV 1 þ state was identified as a nanosecond isomer while the previously known microsecond isomer was interpreted as a 4 þ → 3 þ transition between the πf 7=2 ⊗ νp 1=2 spin-coupled multiplet members. The new 1 þ lifetime and resulting effective charge analysis demonstrates that a universal set of effective charges can be used FIG. 5. Neutron versus proton transition amplitudes relative to the experimental E2 matrix elements for fp-shell nuclei listed in Table I [black line, e π ¼ 1.30ð8Þe, e ν ¼ 0.452ð7Þe]. Additional data points represent transitions in 50 Ti (N ¼ 28) and 51 Fe= 51 Mn (N ≈ Z) [68][69][70] with lines representing previously adopted effective charges. The proton amplitudes are zero for the Ca isotopes so the weighted average was adopted. See Table II in End Matter for more details. TABLE I. Experimental E2 strengths of neutron-rich fp-shell isotopes near N ¼ 32, 34 [9,33,63,67] compared with theoretical predictions using the UFP-CA interaction with the standard (e π ¼ 1.5e, e ν ¼ 0.5e) and fitted (e π ¼ 1.30e, e ν ¼ 0.452e) effective charges. BðE2Þ [W.u.] or Q [eb] Isotope J π i → J π f Experiment UFP-CA (1.5, 0.5) UFP-CA (1.3, 0.452) 49 Ca ð3=2Þ - g:s -0.036ð3Þ -0.043 -0.039 50 Ca 2 þ 1 → 0 þ 1 0.68(2) 0.82 0.68 51 Ca ð3=2Þ - g:s þ0.036ð12Þ þ0.042 þ0.038 55 Ca ½ - 1 → ð5=2Þ - 1 0.42(18) 0.26 0.21 51 Sc ð11=2Þ - 1 → ð7=2Þ - 1 1.9(5) 1.65 1.34 54 Sc 1 þ 1 → 3 þ 1 1.93(16) 2.44 1.91 54 Ti 2 þ 1 → 0 þ 1 6.0(12) 8.56 6.57 χ2 18.1 1.1 across the sd and fp shells, which should enable more consistent and accurate electric-quadrupole transition and moment calculations for a large number of atomic nuclei. No evidence for changes in the effective charges due to an isospin or orbital dependence is found. Sc 1 þ 1 → 3 þ 1 1.93(16) 8.38(35) 3.46 8.47 8.33 1.91 54 Ti 2 þ 1 → 0 þ 1 6.0(12) 19.1(19) 11.62 10.75 19.96 6.57 χ 2 norm 1.1 Theory (GXPF1A) 50 Ti 2 þ 1 → 0 þ 1 6.43(52) 18.76(76) 11.58 10.10 19.61 7.03 50 Ti 4 þ 1 → 2 þ 1 5.5(15) 23.27(317) 15.90 11.66 25.94 6.83 50 Ti 6 þ 1 → 4 þ 1 3.14(13) 21.13(44) 13.87 7.73 21.53 3.26 51 Mn ð27=2Þ - 1 → ð23=2Þ - 1 4.16(12) 36.18(52) 22.09 17.22 36.50 4.23 51 Fe ð27=2Þ - 1 → ð23=2Þ - 1 3.68(21) 34.02(97) 17.22 22.09 32.37 3.33 χ 2 norm 1.5 PHYSICAL REVIEW LETTERS 135, 072501 (2025) 072501-8