$\beta$ Decay and Isomeric Properties of Neutron-Rich Ca and Sc Isotopes

A correlation time of 5 s was used to analyze the $\beta$ decay of ⁵³Ca to ⁵³Sc, where the expected half-life of ⁵³Ca was on the order of several hundred milliseconds. With a relatively long correlation window, the analysis was restricted to pixels located towards the edge of the implantation DSSD. The $\beta$-delayed $\gamma$-ray spectrum for ⁵³Ca in the range 0 to 3 MeV is presented in Fig. 3(a). One transition at 2109.0$\pm$0.3 keV has been assigned to the $\beta$ decay of ⁵³Ca. The newly-assigned 2109-keV transition was initially reported by us in Ref. Craw09 , and has been recently confirmed in one-proton knockout from ⁵⁴Ti McDa09 . An absolute $\gamma$ intensity of 56$\pm$12% was determined for this 2109-keV transition from a Gaussian fit of the peak area, the absolute efficiency of SeGA, and the number of $\beta$ rays correlated with ⁵³Ca parent decays. The absolute intensity of this transition suggests that the majority of $\beta$ intensity from the decay of ⁵³Ca, excluding the previously established 40$\pm$10% $\beta$$n$ branch Lang83 , proceeds through the 2109-keV excited state. This level has been tentatively assigned spin and parity 3/2^-, on the basis of the apparent allowed $\beta$ decay from a (1/2)^{- 53}Ca ground state. The only other possibility for a state populated in allowed $\beta$ decay is a 1/2^- assignment, which is excluded due to the observed direct transition to the 7/2^{- 53}Sc ground state.

The decay curve for ⁵³Ca gated on the 2109-keV transition is presented in Fig. 3(b). Here, all pixels of the DSSD were considered, since the additional $\gamma$-coincidence requirement significantly reduced background events. The data were fitted with a single exponential decay, and a half-life value of 461$\pm$90 ms was extracted. This value is higher than the value (230$\pm$60 ms) obtained previously by Mantica ${\it et~al.}$ Mant08 , which was deduced from a half-life curve with no additional $\gamma$-ray requirement. The required $\gamma$ coincidence in the present result eliminates uncertainties arising from daughter decay and background contributions, and provides a more accurate half-life determination. However, as noted in Ref. Mant08 , the possibility of a second $\beta$-decaying state in ⁵³Ca cannot be excluded.

Previous measurement of the ⁵⁴Ca half-life suggested a value of order 100 ms Mant08 . With such an expectation, a correlation time of 1 s was used for this short-lived nucleus, and data over the entire surface of the DSSD were considered. The $\beta$-delayed $\gamma$-ray spectrum of ⁵⁴Ca in the range 0 to 1.8 MeV is shown in Fig. 4(a). One transition at 247.3$\pm$0.3 keV, with an absolute intensity of 65$\pm$9%, has been assigned to the $\beta$ decay of ⁵⁴Ca. This transition corresponds to the $\gamma$ ray observed by Mantica ${\it et~al.}$ Mant08 , and placed as a ground state transition depopulating a 247-keV state in ⁵⁴Sc, assigned as $J^{\pi}$ = 1⁺. Missing $\beta$ intensity can be attributed to a possible $\beta$$n$ branch in the ⁵⁴Ca decay.

Fig. 4(b) provides the decay curve obtained by requiring a coincidence with the 247-keV $\gamma$ ray. This decay curve yielded a half-life value of 107$\pm$14 ms, within 1$\sigma$ of the previous value Mant08 .

Information regarding the $\beta$ decay of ⁵³Sc is sparse. Sorlin ${\it et~al.}$ Sorl98 reported the half-life of ⁵³Sc as $>$ 3 s, and did not assign $\gamma$-ray transitions to the decay. Given the anticipated long half-life, a correlation time of 10 s was used for analysis of this isotope, and only pixels on the edge of the DSSD were considered for the half-life and absolute $\gamma$-intensity determinations. On the other hand, the $\gamma$$\gamma$ coincidence analysis, which is less sensitive to random background, made use of data from the full DSSD surface. The $\beta$-delayed $\gamma$-ray spectrum for ⁵³Sc of Fig. 5(a) contains six transitions, one corresponding to a known $\gamma$ ray in the decay of the daughter ⁵³Ti Park77 , and five assigned to the $\beta$ decay of ⁵³Sc. Four additional transitions, not apparent in the spectrum of Fig. 5(a), were identified in the $\gamma$$\gamma$ coincidence analysis, and confirmed to be present in the $\gamma$-singles spectrum considering data over the full DSSD surface. In the same $\gamma$-singles spectrum, two other weak transitions were identified, and assigned to the decay. Transitions observed with energies of 292, 340, 629, 1237, and 1576 keV agree with those observed in deep-inelastic work on ⁵³Ti Forn05 .

The decay curve for $\beta$-decay events correlated with ⁵³Sc implantations, with the requirement of a high-energy decay $\gamma$ ray in coincidence with the decay event, is shown in Fig. 5(b). The use of high-energy $\gamma$ rays above 1 MeV limited the background contribution. The fit of a single exponential decay to these data yielded a half-life of 2.4$\pm$0.6 s.

The proposed decay scheme for levels populated in ⁵³Ti by the decay of ⁵³Sc is presented in Fig. 6. The levels at 1237, 1576, 2205 and 2498 keV, and the associated $\gamma$-ray transitions were positioned in accord with Ref. Forn05 and confirmed by $\gamma$$\gamma$ coincidence data. The spectrum of $\gamma$-ray transitions coincident with the 292-keV transition revealed a new $\gamma$ ray at $\sim$413 keV [see Fig. 7(a)], which was placed in cascade with the 292-keV line, depopulating a new state at 2910 keV. A 701-keV transition was observed in coincidence with the 629-keV line [Fig. 7(d)], which was placed as connecting another new state at 2906 keV with the level at 2205 keV. Both of these transitions, and the corresponding doublet of levels at $\sim$2910 keV have been confirmed by Cieplicka ${\it et~al.}$ Ciep09 using Gammasphere data from the reaction ⁴⁸Ca + ²³⁸U, which populates states in ⁵³Ti up to nearly 4 MeV. The Gammasphere work also confirms the placement of the 483-keV transition connecting the 1719- and 1237-keV levels, and is the source of the precise energy values for the 483- and 1576-keV transitions. The absence of $\gamma$ rays coincident with the 1720-keV and 1970-keV transitions lead to placement of these transitions feeding the ground state directly. A 267.6$\pm$0.3-keV $\gamma$ ray (with an absolute intensity of 6$\pm$3%) remains unplaced in the ⁵³Ti level scheme.

Spin and parity assignments proposed in Fig. 6 were adopted from the deep inelastic work of Fornal ${\it et~al.}$ Forn05 , and from comparison of the Gammasphere results with shell-model calculations Ciep09 . All assignments are consistent with the observed $\beta$ feeding pattern from the (7/2)^{- 53}Sc ground state. The apparent ground-state feeding is consistent with zero.

The analysis of the ⁵⁴Sc $\beta$ decay, with a half-life of order $\sim$400 ms Lidd04 , considered a correlation time of 5 s, again restricted to include pixels towards the edge of the central implantation DSSD. The $\beta$-delayed $\gamma$-ray spectrum for ⁵⁴Sc in the range 0 to 2 MeV is presented in Fig. 5(c). Eleven transitions were identified in this spectrum, and the four lowest in energy were eliminated as candidates for transitions in the ⁵⁴Sc decay, based on their apparent half-lives, and likely belong to the decay of the daughter nucleus, ⁵⁴Ti. $\gamma\gamma$ analysis indicates that these four transitions are in coincidence with one another, and the lowest energy transition at 108 keV corresponds to a known transition in ⁵⁴V Grzy98 . The remaining seven $\gamma$ rays were assigned to the $\beta$ decay of ⁵⁴Sc. One additional transition was observed in the $\gamma$-spectrum considering data over the entire DSSD surface, and assigned to the decay of ⁵⁴Sc. The transitions observed at 1002, 1021 and 1495 keV were previously observed in $\beta$ decay Lidd04 and deep-inelastic studies Forn04 .

The decay curve of ⁵⁴Sc-correlated decay events with the additional requirement of a coincident 1002-, 1021-, 1495-, 1504- or 1966-keV $\gamma$-ray transition is given in Fig. 5(d). A fit to these data including a single exponential decay and constant background yielded a half-life for ⁵⁴Sc of 526$\pm$15 ms, longer than the value of 360$\pm$60 ms determined by Liddick ${\it et~al.}$ Lidd04 . The discrepancy between the present value and previous half-life measurements is attributed to the complex deconvolution of the $\beta$-decay curve. A new half-life of 2.1$\pm$1.0 s was determined here for the daughter ⁵⁴Ti, which is slightly longer than the literature value Dorf96 . Additionally, missing $\beta$ intensity is suggestive of a neutron-branching contribution of 16$\pm$9%. Inclusion of these factors in fitting a $\beta$-decay curve impacts the fit to data. The additional $\gamma$-ray coincidence condition taken here permits a half-life determination that requires only two parameters, and extraction of a more accurate value.

The proposed decay scheme for ⁵⁴Sc to levels in ⁵⁴Ti is shown in Fig. 8. Placement of the 1002-, 1021- and 1495-keV transitions follows assignments made in previous works Lidd04 ; Forn04 . The new transitions at 485, 840, 1504 and 1966 keV were placed based on $\gamma$$\gamma$ coincidences. Energy-sum relationships were used to place the 1504-keV cross-over transition connecting the states at 3000 and 1495 keV, and the 2517-keV cross-over transition connecting the state at 2517 keV with the ground state.

Apparent $\beta$ feedings and log $ft$ values deduced from the absolute $\gamma$ intensities are included in Fig. 8. The $J^{\pi}$ of the ⁵⁴Sc ground state was previously limited to (3,4)⁺ Lidd04 . Apparent $\beta$ feeding is observed to both the 2${}_{1}^{+}$ and 4${}_{1}^{+}$ states previously identified in Ref. Forn04 , at 1495 and 2497 keV respectively. The suggested allowed decay to both of these states further restricts the spin and parity for the ⁵⁴Sc ground state to 3⁺. No feeding is expected to the 0^{+ 54}Ti ground state according to the selection rules for allowed $\beta$ decay, assuming a $J^{\pi}$ = 3⁺ ground state for ⁵⁴Sc. $J^{\pi}$ values for the states with energy 1495, 2497 and 2517 keV were adopted from previous works Lidd04 ; Forn04 . Apparent allowed $\beta$ feeding from the (3)^{+ 54}Sc ground state limits the spin and parity of the newly placed levels at 3000, 3338 and 3461 keV to $J^{\pi}$ = 2⁺, 3⁺, or 4⁺.

The isomeric $\gamma$-ray spectrum collected within 20 $\mu$s of a ⁵⁴Sc implantation event is presented in Fig. 9(a). The 110-keV isomeric transition was initially reported by Grzywacz ${\it et~al.}$ Grzy98 , and assigned E2 multipolarity. The transition was observed here with high statistics, and an improved half-life of 2.77$\pm$0.02 $\mu$s was deduced [Fig. 9(b)]. This more precise value is consistent with the result of Grzywacz ${\it et~al.}$, and an the E2 multipolarity assignment based on comparison to Weisskopf lifetime estimates. Based on the (3)⁺ spin and parity of the ⁵⁴Sc ground state and an E2 isomeric transition, the limits on $J^{\pi}$ for the 110-keV level are $J^{\pi}$ = 1⁺ or 5⁺. $J$ = 1 is excluded as this state was not directly populated following the $\beta$ decay of ⁵⁴Ca. Thus, under the assumption of an E2 multipolarity for the transition, the 110-keV isomeric state in ⁵⁴Sc is tentatively assigned a spin and parity of 5⁺, as originally suggested by Grzywacz ${\it et~al.}$ Grzy98 . However, this assignment assumes a single-particle nature for the final and initial states. More complicated wavefunctions, involving configuration mixing, open the window for a 4⁺ spin and parity assignment for the 110-keV isomeric state. This possibility is explored further in Section IV.B.

A correlation time of 1 s was used for the analysis of events associated with ⁵⁵Sc, based on previous half-life measurements Sorl98 ; Lidd04 ; Mant08 indicative of a $\sim$100 ms half-life. Analysis of the $\beta$-gated $\gamma$-ray spectrum was again restricted to pixels towards the edge of the central implantation DSSD. The spectrum in the range of 0 to 2 MeV is given in Fig. 5(e). A total of 13 $\gamma$-ray transitions were identified in this spectrum; of these, 9 correspond to known $\gamma$ rays in the decay of the daughter isotope, ⁵⁵Ti. The remaining 4 transitions have been assigned to the $\beta$ decay of ⁵⁵Sc. One peak in the $\gamma$-singles spectrum, at 349.6$\pm$0.7 keV, is very broad, and is likely a doublet. A known transition in the $\beta$ decay of ⁵⁵Ti, with energy 349.3$\pm$0.6 keV, may contribute to this peak. The transition with energy 592 keV corresponds to the $\gamma$ ray previously observed in the $\beta$ decay of ⁵⁵Sc by Liddick ${\it et~al.}$ Lidd04 . States in ⁵⁵Ti have been studied by deep inelastic scattering with Gammasphere at the ATLAS facility Zhu07 . The $\gamma$ rays reported here with energies 350, 592, 1204 and 1554 keV agree with transitions observed in that work.

A summed decay curve from gates on the $\gamma$ rays in ⁵⁵Ti with energies 592, 712, 1204 and 1554 keV is given in Fig. 5(f). This curve was fitted with a single exponential decay and constant background, and the resulting half-life of 96$\pm$2 ms is consistent with previous measurements Lidd04 ; Mant08 .

The level scheme for ⁵⁵Ti populated following $\beta$ decay of ⁵⁵Sc is illustrated in Fig. 10. Placement of the 350-, 592-, 1204- and 1554-keV transitions is based on previous deep inelastic results Zhu07 , which established the yrast level structure for ⁵⁵Ti up to $J^{\pi}$=(19/2)^-. One additional transition, not seen in Ref. Zhu07 , with energy 712 keV was observed in coincidence with the 1204- and 592-keV transitions. This 712-keV transition was placed depopulating a new state with energy 2508 keV.

Tentative $J^{\pi}$ values were assigned for the ground state and excited states at 592, 1796 and 2146 keV in Ref. Zhu07 , based on comparisons to shell model calculations with the GXPF1A interaction Honm05 . The assignments proposed in that work included $J^{\pi}$ = 1/2^- for the ⁵⁵Ti ground state, which has recently been confirmed by a one-neutron knockout measurement from ⁵⁶Ti to states in ⁵⁵Ti performed at GSI Maie09 . Such a $J^{\pi}$ assignment would exclude allowed $\beta$ decay from the ⁵⁵Sc ground state populating the ⁵⁵Ti ground state directly. Thus, the missing $\beta$ intensity of 17$\pm$7% suggests the possible presence of a $\beta$n branch in the ⁵⁵Sc decay. Absence of transitions in ⁵⁴Ti (see Fig. 8) in the $\beta$-delayed $\gamma$-ray spectrum may indicate that the $\beta$n branch mainly populates the ⁵⁴Ti ground state directly. The spin and parity assignments proposed by Zhu ${\it et~al.}$ Zhu07 are adopted here. Apparent allowed $\beta$ decay to the newly placed state at 2508 keV limits this state to $J^{\pi}$ = 5/2^-, 7/2^- or 9/2^-.

Analysis of the $\beta$ decay of ⁵⁶Sc was completed using a correlation time of 1 s, and the full DSSD surface was considered. The $\beta$-delayed $\gamma$-ray spectrum for ⁵⁶Sc in the range of 0 to 2 MeV can be found in Fig. 5(g), where nine transitions have been assigned to the decay of ⁵⁶Sc, and are summarized in Table 1. Two additional transitions were assigned to the decay of the granddaughter nuclide, ⁵⁶V. The transitions with energies 592, 690, 751, 1129 and 1161 keV were previously observed in both $\beta$ decay Lidd04 and deep-inelastic experiments Forn04 . It was noted in Ref. Lidd04 that the transition with energy 592.3$\pm$0.5 keV was within the error of a known transition in ⁵⁵Ti, and the possibility of $\beta$-delayed neutron emission was suggested. Another known transition in ⁵⁵Ti, at 1204 keV Zhu07 is also present in the delayed $\gamma$-ray spectrum for ⁵⁶Sc, confirming $\beta$-delayed neutron emission in this decay.

Liddick ${\it et~al.}$ Lidd04 proposed two $\beta$-decaying states in ⁵⁶Sc: a low-spin state with $T_{1/2}$ = 35$\pm$5 ms, and a higher-spin level with slightly longer $T_{1/2}$ = 60$\pm$7 ms. Given the different spin values, the two $\beta$-decaying states were observed to populate different levels in the ⁵⁶Ti daughter. Gated decay curves were generated for each $\gamma$-ray transition identified in Fig. 5(g), and fitted with a single exponential decay and constant background. The resultant half-life values are summarized in Table 1. Values deduced from the decay curves gated on the 690- and 1161-keV $\gamma$ rays, which depopulate levels at 2979 and 2289 keV respectively, are in agreement with the previous half-life determination for the higher-spin isomer Lidd04 . The transitions assigned to the $\beta$n decay also have half-lives consistent with decay from the higher-spin state, as do the transitions with energies 1467 and 1495 keV. The weighted average of the half-lives for transitions populated solely by the higher-spin isomer is 75$\pm$6 ms.

The half-life of the lower-spin, $\beta$-decaying state Lidd04 was previously deduced from a two-component fit to the decay curve gated on the 1129-keV $\gamma$ transition, which depopulates a state fed by both $\beta$-decaying states in ⁵⁶Sc. Two newly-identified $\gamma$-ray transitions at 751 and 1712 keV decay with half-lives similar to that extracted for the lower-spin isomer. These $\gamma$ rays apparently depopulate states fed exclusively by the lower-spin isomer. The weighted average of the half-lives for the 751- and 1712-keV transitions yields a half-life for the lower-spin isomer of 26$\pm$6 ms.

Levels in ⁵⁶Ti and ⁵⁵Ti populated in the decay of the two $\beta$-decaying states in ⁵⁶Sc are shown in Fig. 11. States in ⁵⁵Ti are known from the present ⁵⁵Sc $\beta$-decay results, previous $\beta$-decay studies Lidd04 , and deep-inelastic work Zhu07 . The three states at 1129, 2289 and 2979 keV in ⁵⁶Ti were previously identified and assigned tentative spin and parity values Lidd04 ; Forn04 , which were adopted in Fig. 11. Observed $\gamma$$\gamma$ coincidences between the 1129- and 751-keV transitions suggest the placement of an additional state populated by the lower-spin $\beta$-decaying state at 1880 keV. In addition to populating the two states at 2289 and 2979 keV directly, observed $\gamma$$\gamma$ coincidences between the 1495- and 690-keV transitions suggest that the higher-spin $\beta$-decaying state populates a level at 4474 keV. Only the 1467- and 1712-keV $\gamma$ transitions remain unplaced in the present level scheme for ⁵⁶Ti.

Absolute $\gamma$-ray intensities were determined for the $\beta$-delayed $\gamma$-ray transitions in the ⁵⁶Sc decay by comparison of the number of observed $\gamma$ rays, adjusted for the absolute efficiency of SeGA, with the number of observed ⁵⁶Sc decays. The number of parent decays in this case was determined using the total number of implantations, taken from the particle identification, and the average $\beta$-detection efficiency of 11.4$\pm$0.4%. The absolute $\gamma$-ray intensities are included in Table 1, and in the decay scheme of Fig. 11. Apparent $\beta$ branches were deduced from the absolute $\gamma$-ray intensities. Decay from the higher-spin isomer apparently populates both the 4⁺ and 6⁺ states in ⁵⁶Ti. This suggests that the higher-spin isomer has $J^{\pi}$=5⁺, in contradiction to the previous assignment of (6,7)⁺ by Liddick et al. Lidd04 . However, the large $Q_{\beta}$ value for the decay allows the (4)⁺ state at 2289 keV to be populated by cascades from above, resulting in the observed intensity difference between the 690- and 1161-keV transitions. Given this possibility, the higher-spin $\beta$-decaying state has been tentatively assigned as (5,6)⁺. The state with an energy of 4474 keV, also populated by the higher-spin $\beta$-decaying state, has $J^{\pi}$ limited to 4⁺, 5⁺, 6⁺ or 7⁺, depending on the spin and parity of the higher-spin $\beta$-decaying state in ⁵⁶Sc.

Decay from the lower-spin $\beta$-decaying state in ⁵⁶Sc apparently directly feeds the first excited 2⁺ state in ⁵⁶Ti, which limits the spin and parity of this $\beta$-decaying state to 1⁺, 2⁺ or 3⁺. The $J^{\pi}$ for the ⁵⁶Sc ground state can be further restricted if there is direct $\beta$ feeding to the ⁵⁶Ti ground state. The absolute intensity of the 592-keV transition in the $\beta$n daughter ⁵⁵Ti suggests a lower-limit for the neutron branching ratio of 14$\pm$2%. Even under the assumption that the two unplaced $\gamma$ transitions directly populate the ⁵⁶Ti ground state, there is 29$\pm$7% of the $\beta$ intensity unaccounted for. Two scenarios can account for the missing intensity: either direct feeding to the ⁵⁶Ti ground state or $\beta$n decay populating the ground state of ⁵⁵Ti directly. These possibilities were investigated by reanalyzing the $\beta$ decay of ⁵⁶Sc with the longer correlation time of 5 s. With a longer correlation time, it was possible to compare the intensities of $\gamma$-ray transitions in the decay of the $\beta$n daughter ⁵⁵Ti (672.5 keV, $I_{\gamma}^{absolute}=44\pm 4$% Mant03 ), and the $\beta\beta$ granddaughter ⁵⁶V (668.4 keV, $I_{\gamma}^{absolute}=26\pm 2$% Mant03_2 ). The ratio of $\beta$ to $\beta$n decays in ⁵⁶Sc can be determined by the ratio of the intensities of the 668- and 673-keV transitions, which are detected with nearly the same efficiency in SeGA. Approximately 70% of the decays of ⁵⁶Sc populate states in ⁵⁶Ti. Thus, the missing $\beta$ intensity cannot be fully accounted for by $\beta$n decay to the ground state of ⁵⁵Ti, and there is apparent direct population of the 0^{+ 56}Ti ground state, which must be from the lower-spin $\beta$-decaying state. This suggests a spin and parity for the lower-spin $\beta$-decaying state of $J^{\pi}$=1⁺. Direct feeding from the (1)⁺ $\beta$-decaying state limits $J^{\pi}$ of the state at 1880 keV to (0,1,2)⁺.

The isomeric $\gamma$-ray spectrum collected within a 20-$\mu$s time window following a ⁵⁶Sc implantation is presented in Fig. 12(a). Five transitions were observed. Those with energies 140, 188 and 587 keV were previously reported by Liddick ${\it et~al.}$ Lidd04 .

The observed $\gamma$-ray energy as a function of the time elapsed between a ⁵⁶Sc implantation event and prompt $\gamma$ emission is presented in Fig. 12(b). The five transitions appear to decay with the same half-life. The projection of Fig. 12(b) onto the time axis, gated on the five isomeric transitions, is included in Fig. 12(c). The resultant decay curve was fitted with a single exponential decay and a constant background, resulting in a half-life of 290$\pm$30 ns.

The low-energy structure of ⁵⁶Sc, populated by isomeric decay (as presented in Fig. 11), was based on observed $\gamma$$\gamma$ coincidences (see Fig. 13), as well as relative intensity and energy-sum relationships.

A novel approach was taken to determine which of the two $\beta$-decaying states in ⁵⁶Sc is populated in the isomeric decay from the 775-keV level. A half-life curve was constructed considering only $\beta$-decay events correlated with ⁵⁶Sc implantation events that were in turn correlated with one of the five prompt $\gamma$ rays. The resultant decay curve is presented in Fig. 14. The deduced short half-life of 30$\pm$5 ms provides evidence that the isomeric transitions populate the lower-spin $\beta$-decaying state.

The tentative spin and parity assignment of the lower-spin $\beta$-decay isomer in ⁵⁶Sc as 1⁺ permits tentative $J^{\pi}$ assignments to the remaining levels populated in decay of the isomeric state in ⁵⁶Sc. The 188-keV transition, which directly depopulates the isomeric level, is proposed to be of E2 character, based on the comparison to Weisskopf half-life estimates, and the competing 48-keV transition to be of the M1 type. It follows that the 140-keV transition must also have M1 multipolarity. The observed intensities of the 727-keV and 140-keV transitions then suggest the former $\gamma$ ray to be an E2 transition. Finally, the remaining 587-keV transition must have M1 multipolarity. These assignments lead to the tentatively assigned $J^{\pi}$ = 2⁺, 3⁺ and 4⁺ values for the states in ⁵⁶Sc with energies 587, 727 and 775 keV above the lower-spin $\beta$-decaying state respectively, again assuming that this state in ⁵⁶Sc has $J^{\pi}$=1⁺ quantum numbers.

⁵⁷Sc was studied in a separate experiment from that used to deduce the results for the isotopes previously discussed. However, the decay of ⁵⁷Sc to states in ⁵⁷Ti was analyzed in a similar manner. A correlation time of 500 ms was used for the analysis of this decay, which had a previous half-life measurement of order tens of milliseconds. Pixels over the entire DSSD surface were considered in the analysis.

The $\beta$-delayed $\gamma$-ray spectrum for the decay of ⁵⁷Sc is presented in Fig. 5(i). Four transitions in this spectrum were attributed to the $\beta$ decay of the ⁵⁷Ti daughter nucleus, and one transition was known from the $\beta$ decay of the ⁵⁷V granddaughter. The remaining five $\gamma$ rays in Fig. 5(i) were attributed to the $\beta$ decay of ⁵⁷Sc, and are listed in Table 2. No previous information on $\beta$-delayed $\gamma$ rays in ⁵⁷Sc is available from the literature. The transition at 1127 keV is close in energy to a known transition in ⁵⁶Ti, at 1128.7$\pm$0.3 keV Lidd04 ; Forn04 , and may suggest a $\beta$-delayed neutron branch from ⁵⁷Sc to levels in ⁵⁶Ti.

The decay curve for ⁵⁷Sc-correlated $\beta$ decays with the additional requirement of a decay $\gamma$-ray coincidence is given in Fig. 5(j). The decay curve was fitted with a single exponential decay and a constant background, and the deduced half-life was 22$\pm$2 ms. The new half-life value is slightly longer than the previous ⁵⁷Sc half-life measurement of 13$\pm$4 ms Sorl03 , which did not include a $\gamma$-ray tag.

No $\gamma$$\gamma$ coincidence data were available due to low statistics. The absolute intensity for the 364-keV $\gamma$ ray suggests that it directly populates the ⁵⁷Ti ground state, and it has been placed as a ground-state transition. No other conclusions regarding the low energy structure of ⁵⁷Ti could be drawn from the present data.

The low-energy structure of ⁵³Sc can be interpreted as a weak coupling of the valence $1f_{7/2}$ proton to states in ⁵²Ca, which, assuming a robust $N$=32 subshell closure, can be viewed as a doubly magic core Gade06 ; Craw09 . Coupling of the valence proton to the first 2⁺ state in ⁵²Ca should produce a quintet of states in ⁵³Sc with $J^{\pi}$ values ranging from 3/2^- to 11/2^-, at energies centered around $\sim$2.6 MeV. Studies of ⁵³Sc via deep inelastic reactions Bhat09 have populated 9/2^- and 11/2^- states near this energy, which are likely members of the $\pi$$1f_{7/2}$$\otimes$2⁺ multiplet. The $\beta$ decay of ⁵³Ca was observed to populate only one state in ⁵³Sc. The population of a single state in the decay of ⁵³Ca decay limits the spin and parity of the ⁵³Ca ground state to $J^{\pi}$=1/2^-, as allowed $\beta$ decay from a $J^{\pi}$ = 3/2^- or 5/2^{- 53}Ca ground state would be expected to populate several levels of the $\pi$$1f_{7/2}$$\otimes$2⁺ multiplet. The single new level at 2109 keV populated in the decay from a $J^{\pi}$=1/2^{- 53}Ca ground state is a likely candidate for the 3/2^- member of the $\pi$$1f_{7/2}$$\otimes$2⁺ multiplet. The three known levels in ⁵³Sc (see Fig. 15) reside at just over 2 MeV, nearly the energy expected within the weak coupling framework. The success of this scheme in describing the structure of ⁵³Sc provides further support for the robust nature of the $N$=32 subshell closure in the ₂₀Ca isotopes.

The structure of the even-$A$ ₂₁Sc isotopes can be described in terms of coupling of the valence $1f_{7/2}$ proton particle with $2p_{3/2}$, $2p_{1/2}$, $1f_{5/2}$, and at higher energies, $1g_{9/2}$, valence neutron structures to states in a corresponding ₂₀Ca core. This is shown most directly by considering the cases of ⁵⁰Sc and ⁵²Sc, where this simple description explains many aspects of the known low-energy structures.

⁵⁰Sc has one proton and one neutron particle outside of the doubly magic ⁴⁸Ca core, and the lowest energy levels can be described in terms of the coupling of a valence $1f_{7/2}$ proton and $2p_{3/2}$ neutron. The low-energy structure, known from $\beta$ decay Albu84 , transfer reactions Fist94 , and charge exchange reactions Ajze85 can be seen in Fig. 16(a). The four states lowest in energy arise from the configuration $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$2p_{3/2})^{1}$, which produces four states ranging from $J^{\pi}$ = 2⁺ to 5⁺. The particle-particle coupling rules Paar79 suggest that the states in this multiplet are arranged in a downward-opening parabola, in good agreement with the experimental energy levels. Promotion of the valence neutron in ⁵⁰Sc to the $\nu$$2p_{1/2}$ level will give the configuration $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$2p_{1/2})^{1}$, producing two states with $J^{\pi}$=3⁺, 4⁺. Two 3⁺ states are identified in the level scheme of ⁵⁰Sc above 2 MeV, and either may be a candidate for the 3⁺ member of this doublet. The energy spacing between multiplets arising from configurations involving the $\nu$$2p_{3/2}$ and $\nu$$2p_{1/2}$ states is directly related to the $\nu$$2p_{3/2}$-$\nu$$2p_{1/2}$ single-particle spacing, and thus the $\nu$$2p_{3/2}$-$\nu$$2p_{1/2}$ spin-orbit splitting. The apparent separation of these two multiplets in ⁵⁰Sc is of order 2 MeV, a gap sufficient to account for the established $N$=32 subshell closure in ₂₁Sc.

The low-energy structure of ⁵²Sc can be similarly described in terms of coupling of a valence $1f_{7/2}$ proton particle and $2p_{3/2}$ neutron hole, taking ⁵²Ca as the inert core. States in this odd-odd nucleus have been identified following $\beta$ decay Huck85 , in-beam $\gamma$-ray spectroscopy following secondary fragmentation Gade06_2 and deep-inelastic work Forn08 ; Bhat09 . The known levels are presented in Fig. 16(b). As discussed in Refs. Forn08 ; Bhat09 , the lowest-lying quartet of states in ⁵²Sc can be associated with the configuration $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$2p_{3/2})^{-1}$. The four states with $J^{\pi}$=2⁺-5⁺ should form an upward-opening parabola, since the coupling involves a proton particle and neutron hole. Shell-model calculations based on the effective interactions GXPF1 Gade06 and GXPF1A Forn08 , which assume an inert ⁴⁸Ca core, both result in the near-degeneracy of the 3⁺ and 4⁺ states of this multiplet. The energy separation between these states has not been established experimentally, but is likely small. The success of the Pandya transformation Pand56 in relating the particle-particle $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$2p_{3/2})^{1}$ states of ⁵⁰Sc to the particle-hole $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$2p_{3/2})^{-1}$ states in ⁵²Sc also suggests that the energy separation between the 3⁺ and 4⁺ states is small in ⁵²Sc, and that the configurations of these low-lying levels in both nuclei is fairly pure. Calculations using the GXPF1 effective interaction Gade06 confirm the configurational purity of the four lowest-energy levels. The coupling of the $1f_{7/2}$ proton to an excited $2p_{1/2}$ neutron to give a pair of states with $J^{\pi}$=3⁺, 4⁺ accounts for the observed 4⁺ state in ⁵²Sc at $\sim$1.7 MeV. The 6⁺ state at $\sim$2.3 MeV in ⁵²Sc has been shown in shell-model calculations to have parentage including this configuration, though the extent of this contribution varies between shell-model interactions. Calculations using the GXPF1A interaction Forn08 suggest that the primary parentage of the 2.3-MeV 6⁺ state is $(\pi$$1f_{7/2})^{1}$$\otimes$$[\nu$$(2p_{3/2}^{2}2p_{1/2}^{1})]$, while the $(\pi$$1f_{7/2})^{1}$$\otimes$$[\nu$$(2p_{3/2}^{2}1f_{5/2}^{1})]$ configuration is predicted to dominate the first 8⁺ state in ⁵²Sc, experimentally observed at $\sim$3.6 MeV Forn08 . The energy separation between states arising from the $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$2p_{3/2})^{-1}$ and $(\pi$$1f_{7/2})^{1}$$\otimes$$[\nu$$(2p_{3/2}^{2}2p_{1/2}^{1})]$ configurations was $\sim$2 MeV, corresponding to the spin-orbit splitting between the $\nu$$2p_{3/2}$ and $\nu$$2p_{1/2}$ single-particle states, in agreement with the situation in ⁵⁰Sc. As noted by Fornal ${\it et~al.}$ Forn08 , the apparent separation between the $\nu$$2p_{1/2}$ and $\nu$$1f_{5/2}$ single-particle states, as inferred from the energy spacing between the 6⁺ and 8⁺ states in ⁵²Sc, is slightly smaller than that predicted in the GXPF1A interaction.

The low-energy structure of ⁵⁴Sc, with two additional neutrons, should exhibit significant change. The $\nu$$2p_{3/2}$ orbital is now fully occupied, and the ⁵⁴Sc structure at low energy should reflect coupling of the valence $1f_{7/2}$ proton particle to either a $2p_{1/2}$ or $1f_{5/2}$ neutron, taking ⁵²Ca as the inert core. The $\nu$$2p_{1/2}$ single-particle level is expected to lie below the $\nu$$1f_{5/2}$ state for the ₂₁Sc isotopes, based on the observations in ⁵²Sc. Assuming such an ordering, the $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$2p_{1/2})^{1}$ coupling yields a doublet of states with $J^{\pi}$=3⁺, 4⁺, which is expected to be lowest in energy in ⁵⁴Sc. Shell model results Lidd04 predict the 3⁺ and 4⁺ states of this configuration to be close in energy, and thus the spin and parity of the ⁵⁴Sc ground state from a theory perspective is not clear. The ground state is tentatively assigned $J^{\pi}$=3⁺ in the present work. Considering the situation in terms of the coupling, it may be natural to assume that the isomeric 110-keV level corresponds to the 4⁺ state arising from the $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$2p_{1/2})^{1}$ configuration. However, within the simple coupling framework it is difficult to accept an in-multiplet M1 transition hindered so strongly as to yield the experimental isomer half-life of 2.8 $\mu$s. In this case, the two excited states identified in ⁵⁴Sc, the isomeric (5)⁺ level and higher-lying 1⁺ state, may be explained as arising from the configuration $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$1f_{5/2})^{1}$. This coupling results in a sextet of states ranging from $J^{\pi}$=1⁺ to 6⁺. The particle-particle coupling would produce a downward-facing parabola. There are, however, inconsistencies in the experimental level ordering in ⁵⁴Sc when compared with shell model calculations. The 1⁺ state lies above the 5⁺ state in energy in the level scheme as presented in Fig. 16(c), a result which is inconsistent with calculations using the GXPF1 effective interaction. An alternative $J^{\pi}$ assignment for the states in ⁵⁴Sc with a 4⁺ ground state and 6⁺ 110-keV isomeric level is a possibility, but the placement of the 6⁺ level below the 1⁺ level is also unexpected. The apparent experimental level ordering would be explained if the $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$2p_{3/2})^{-1}$ configuration was the origin of the 3⁺ and 5⁺ states. However, as the $\nu$$2p_{3/2}$-$\nu$$2p_{1/2}$ spin-orbit splitting is a fairly constant $\sim$2 MeV, this multiplet is expected at much higher excitation energy in ⁵⁴Sc, making this possibility unlikely.

An alternative to the 5⁺ assignment for the 110-keV isomeric state is a 4⁺ assignment. While not expected based on comparison of the deduced isomer lifetime with Weisskopf single-particle estimates, which favor an E2 multipolarity for the 110-keV transition, a 4⁺ spin and parity cannot be excluded for the 110-keV state. Calculations using the GXPF1 Honm02 , GXPF1A Honm05 and KB3G Pove01 interactions all predict a doublet of states near the ⁵⁴Sc ground state with $J^{\pi}$=3⁺ and 4⁺. The transition probabilities, $B(M1:3^{+}\to 4^{+})$ and $B(E2:3^{+}\to 4^{+})$ calculated with these three interactions are shown in Table 3, along with the calculated partial half-lives. In all cases, the M1 component would dominate the transition. However, the $B(M1:3^{+}\to 4^{+})$ values are small, giving rise to half-lives of order nanoseconds or longer – much slower than the picoseconds expectation from the single-particle Weisskopf estimate. The $B(M1:3^{+}\to 4^{+})$ value is very sensitive to the degree of mixing in the wavefunctions for the 3⁺ and 4⁺ states. With increased configuration mixing, there is increased suppression of the $B(M1:3^{+}\to 4^{+})$ value, hindering the transition. A 4⁺ spin and parity assignment is in better agreement with the expectation for the level ordering in ⁵⁴Sc. However, more work is required to make a firm spin assignment for the 110-keV isomeric state in ⁵⁴Sc.

The 1⁺ state at energy 247 keV in ⁵⁴Sc can only be explained by the $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$1f_{5/2})^{1}$ configuration. Even if this is the only member of the multiplet identified, the energy separation between the ground state with $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$2p_{1/2})^{1}$ and the excited states from $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$1f_{5/2})^{1}$ appears to be small. Shell model results using the GXPF1 effective interaction Lidd04 predict a $\nu$$2p_{1/2}$-$\nu$$1f_{5/2}$ separation of more than 1 MeV, which manifests itself in ⁵⁴Sc as an expanded low-energy level structure [see Fig. 16(c)]. The compression observed here between multiplets arising from coupling with the $\nu$$2p_{1/2}$ and $\nu$$1f_{5/2}$ neutron states suggests a $\nu$$2p_{1/2}$-$\nu$$1f_{5/2}$ single-particle energy separation smaller than that assumed in the GXPF1 Hamiltonian, and is possibly inconsistent with a robust $N$=34 subshell closure.

Another rapid structure change should be evident in ⁵⁶Sc. The addition of two neutrons to ⁵⁴Sc should fill the $\nu$$2p_{1/2}$ single-particle orbital. The unpaired valence neutron will then occupy the $1f_{5/2}$ state. Under this assumption, the lowest energy states in ⁵⁶Sc are expected to arise from the $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$1f_{5/2})^{1}$ configuration. The resulting sextet of states with $J^{\pi}$=1⁺ to 6⁺ should be arranged in a downward facing parabola, with the 4⁺ state at the vertex. The challenge in comparing expectations in ⁵⁶Sc to the new data is the unknown position of the higher-spin $\beta$-decaying state, relative to the low-energy structure established by the isomer decay. However, the states at low energy in ⁵⁶Sc with $J^{\pi}$=1⁺, 2⁺ and (5,6)⁺ are difficult to explain in a simple way outside of the $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$1f_{5/2})^{1}$ configuration. The 3⁺ and 4⁺ states cannot be trivially assigned the same configuration, as they may also arise from the $(\pi$$1f_{7/2})^{1}$$\otimes$$(\nu$$2p_{1/2})^{1}$ coupling, depending on the relative position of the $\nu$$2p_{1/2}$ and $\nu$$1f_{5/2}$ single-particle orbitals. In fact, significant mixing would be expected between the 3⁺/4⁺ states in the $\pi$$1f_{7/2}$-$\nu$$1f_{5/2}$ and $\pi$$1f_{7/2}$-$\nu$$2p_{1/2}$ multiplets. This mixing is already reflected in the GXPF1 calculated levels, even with a large $\nu$$2p_{1/2}$-$\nu$$1f_{5/2}$ gap.

The experimentally-known levels below 4 MeV in the odd-$A$ neutron-rich ${}^{53,55,57}_{\phantom{53,55,}22}$Ti isotopes are represented in Fig. 17. The levels of the ₂₂Ti isotopes have been well characterized Jans02 ; Dinc05 ; Forn05 ; Forn04 ; Zhu07 by the GXPF1A Honm05 shell-model effective interaction.

Within the GXPF1A interaction, the ordering of the neutron single-particle states places the $2p_{1/2}$ orbital below the $1f_{5/2}$ orbital for the Ti isotopes. Given such an ordering, ⁵⁵Ti₃₃ is expected to have a 1/2^- ground state, as the first neutron above the $N$=32 subshell closure should occupy the $2p_{1/2}$ orbital. The first excited state in ⁵⁵Ti then corresponds to excitation of the valence neutron into the $1f_{5/2}$ orbital. The recent direct determination of the ground state spin and parity in ⁵⁵Ti Maie09 provides confirmation for the ordering of the $\nu$$2p_{1/2}$ and $\nu$$1f_{5/2}$ orbitals, with the $\nu$$2p_{1/2}$ orbital lying lower in energy, similar to the ordering in the ₂₁Sc isotopes. Thus, it appears that the $\nu$$2p_{1/2}$-$\nu$$1f_{5/2}$ level ordering changes already in going from ₂₃V to ₂₂Ti, though a significant energy gap between these orbitals is not present in the Ti isotopes Zhu07 .

Based on the assumed neutron-level ordering, ⁵⁷Ti₃₅ is expected to have a 5/2^- ground state, with a neutron configuration of $2p_{3/2}^{4}2p_{1/2}^{2}1f_{5/2}^{1}$, and a first excited state with $J^{\pi}$=1/2^-, corresponding to excitation of a $2p_{1/2}$ neutron into the $1f_{5/2}$ orbital. Thus, this is essentially an inversion of the two lowest states in ⁵⁵Ti. Calculations with the GXPF1A interaction have placed this first excited state at an energy of 422 keV Lidd05 . In the present results, one state has been placed in the level scheme of ⁵⁷Ti at an energy of 364 keV, which may correspond to the 1/2^- first excited state. If this is the case, $\beta$ decay from the ⁵⁷Sc 7/2^- ground state should not directly populate this state, and it must be fed indirectly from higher-lying states. However, the 364-keV transition is much stronger than the other transitions observed in the decay, and direct feeding cannot be excluded. Allowed $\beta$ decay from ⁵⁷Sc, with a 7/2^- ground state, populating the first-excited state in ⁵⁷Ti would suggest a reversal of the level ordering shown in Fig. 17, and would present a significant challenge for theory. However, before the origin of the 364-keV state can be confirmed, a more complete picture of the low-energy states in ⁵⁷Ti is required.