We start by showing a decays-eye view of the lower portion of the nuclear chart in Figure 1B. The zig-zaggy drip lines are defined in this part of the chart and these lines display easily understood pairing features. Namely, even atomic number (Z) elements have proton drip lines more removed from stability and the neutron-drip line is scalloped with even neutron number (N) isotopes inside and odd N isotopes outside the drip line. As required by the energetics, the number of nucleons emitted is that required to land inside the drip line. The N = 6 isotones, note upper orange arrow, extend from 1 to 4 protons emitted from ¹⁵F to ¹⁸Mg [14] (with the residue in each being ¹⁴O). The N = 2 isotones extend from ⁵Li up to ⁹N with the latter (see star) exhibiting the record length decay chain of 5 protons [15]. By examination of the subevents, it is often possible to reconstruct the kinetic decay chain, see, for example, [5]. If Z = odd, the first decay step is always emission of a single proton and long decay sequences are concatenations of 1p and prompt 2p decay steps. The latter principally, but not exclusively, occurs when there is no energy and isospin allowed 1p decay path.

Using a secondary beam of ³⁷Ca impinging on a ⁹Be target, resonances corresponding to the ground states of ³⁴K and ^37,38Sc were found, see Figure 1C. Using the IMS determined decay energies and the known mass excesses of the daughters, three new masses were determined [9]. These mass measurements allow for an extended look at neutron and proton separation-energy trends, which are shown in the upper panels of Figure 2 (The new masses allowed for calculation of the data represented by stars.) The lower panels in this figure show the separation energy differences defined by Δ S n ( N , Z ) = S n ( N , Z ) − S n ( N + 1 , Z ) = [ Δ M ( N + 1 , Z ) + Δ M ( N − 1 , Z ) ] − 2 Δ M ( N , Z ) and an equivalent expression for protons.

First, take note of the expected behavior. The jumps in Δ S n at N = 20 and N = 28 illustrate the classic neutron shell closures. The reduced increase in Δ S n for 21 41 Sc 20 (red, top data sequence) should be noted and we shall return to this observation. Next, note that at N = 16 , the raw data (points connected with dotted lines) suggest a neutron shell closure for ³⁶Ca (blue). (The word “suggest” is used as one expects a general increase in neutron separation energy with decreasing neutron number.) This had previously been noted [16]. However, the new data point, for Z = 19 (orange star), indicates that the enhanced binding for N = 16 has largely diminished. Again, we shall return to this observation. Finishing on what is, more-or-less, expected; note that the change in proton separation energies exhibit a clear peak for ⁴⁰Ca (Figure 2D, blue points and dotted line). One observes a diminution of Δ S p and the apparent loss of the enhancement of the proton removal energy when N recedes below 20.

Before proceeding to the explore the not-so-obvious trends, for which some inklings were provided above, we have to appreciate that there are three structure issues at play in these mass derived quantities. Two of these are the standard issues of nuclear shells and pairing of like nucleons. The remaining issue, unimportant for heavier nuclei or neutron-rich nuclei, is the so-called Wigner or n-p congruence energy [17, 18]. The latter, included early on in macroscopic mass models, results in extra stabilization near N = Z and arises from T = 0 (but not necessarily J = 1 ) neutron-proton pairing correlations [19]. The real separation energies are enhanced if the parent N / Z asymmetry is smaller, suppressing its mass, than that of the daughter. If one desires to focus only the impact of nuclear shells, pseudo separation energies should be constructed which remove congruence-energy effects. Such Wigner-“corrected” separation energies, are not observables as they remove, in a model-dependent way, one structure effect. As the Wigner energy rapidly reduces away from N = Z , Wigner-removed pseudo values of Δ S n ( N , Z ) are strongly reduced if the central nucleus has N = Z (as the actual separation energies are inflated by n/p congruence) and will increase this quantity if either of the nuclei corresponding to one nucleon added or removed has N = Z . To generate the Wigner-removed pseudo-separation energies we employ the procedure suggested by Goriely e t a l [20]. The results are shown as solid lines, without points, in Figure 2. The shading between the solid and dashed lines highlights the Wigner-energy contribution.

We are now ready to return to the not-so-obvious trends. The Wigner-energy-removed pseudo-separation energies confirm the suggestion of a N = 16 subshell closure as one observes that Δ S n ( N , Z ) increases from N = 18 to N = 16 for potassium ( Z = 19 ) isotopes similar to the trend observed for calcium isotopes, compare orange and blue solid lines without dots in Figure 2B. (Removing the Wigner energy suppresses the pseudo-separation energy for 19 37 K 18 more than for 19 35 K 16 , as the former is closer to N = Z .)

A Z = 14 subshell closure is most clearly seen as a peak in Δ S p between N = 20 and N = 17 , see Figure 2D. At N = 16 , there is no evidence for this feature. With 16 neutrons, the ν 0 d 5 / 2 and ν 1 s 1 / 2 orbitals are nominally filled, so adding another neutron starts filling the ν 0 d 3 / 2 orbital. Through the tensor interaction [21], neutrons occupying the ν 0 d 3 / 2 will stabilize the π 0 d 5 / 2 , increasing the energy gap between it and the higher lying π 1 s 1 / 2 . This effect explains the observed low proton occupation of the π 1 s 1 / 2 orbit in 14 34 Si 20 , which lead to the suggestion that this nucleus is doubly magic [22]. More insight into this topic can be found in the paper by J. Chen found in the present issue [23]. Finding the mirror of this effect in 20 34 Ca 14 is a future research opportunity.

Neither the real nor the Wigner-removed pseudo-proton-separation energy differences show an increase at Z = 20 for N < 19 (The recent invariant-mass work added data allowing for the calculation of the values for N = 17 and N = 16 , stars in Figure 2D.) In these cases, the Wigner modification is of little consequence. This analysis confirms that Z = 20 has lost its “magicity” for N < 19 . This conclusion had previously been reached through the two-nucleon removal cross section for ³⁸Ca [24] and measurement of the B (E2) for ³⁶Ca [25]. This enfeebling of the Z = 20 shell for neutron deficient isotopes has also been mentioned in a recent global examination of shell gaps over the whole chart of nuclides [26]. However, with some introspection, data from ⁴⁰Ca (e,e’p) [27] told us three decades ago that even ⁴⁰Ca had a somewhat open proton sd shell and an appreciable cross-shell f 7 / 2 spectroscopic factor of about 1/3, (results confirmed by (d,³He) proton knockout studies [28].) Another point of heuristic value deduced from panels (B) and (D) of Figure 2, is that congruence is a non-negligible contributor to the stability of ⁴⁰Ca.

Finally, we return to an observation made above from Figure 2B - the reduced increase in Δ S n for 21 41 Sc 20 (red) compared to the two other isotones plotted ( either 20 40 Ca 20 or 19 39 K 20 ). The Wigner-energy-removal modification only amplifies this observation and therefore we must also conclude that the N = 20 shell is significantly weakened for Z > N .

One example of isospin symmetry found in the continuum is mated pairs of 2p emitters. Figure 3 shows two such cases [29, 30]. The schemes on the top show the ground-state 2p decay of Z = even, T = 2 nuclei. These decays are characterized by each proton removing 1/2 of the total available decay energy, a characteristic of decays unperturbed by intermediates and thus indicating “direct” 2p decay. (Experience has taught that if a potential intermediate is broad, it leaves no “finger print” on the decay.) The lower decay schemes show the same T = 2 to T = 1 decays rotated in isospace into the T z = T − 1 nuclei, i.e., the decays of the analogs. In these cases, while there are single-proton energetically-allowed narrow intermediates, there are no energetically and isospin allowed intermediates. (These potential intermediate states are T = 1/2.) As in the T z = 2 cases (top), the two protons share the decay energy equally. In the A = 8 analog decay, the charged-particle IMS was coupled with the gamma detection to confirm that the 2p decay populated the isobaric analog state in ⁶Li [31] (In the other case, the addition of excitation energy of the 2p daughter’s T = 1 gamma-decaying analog state to the measured 2p decay energy yielded the energy of the previously unobserved T = 2 state in ¹²N [30].) One would also expect another mated pair for A = 16, i.e. 16 Ne g s and its T = 2 analog in ¹⁶F. Despite considerable effort, no clear evidence for the second of this pair has been found. We suspect that the resolution of the riddle lies in the failure of isospin allowed 2p decay to effectively compete (at Z = 9) with isospin violating 1p decay.

Another beautiful example of isospin symmetry in the continuum is the mated rotational bands in the A = 10 nuclei ¹⁰Be and ¹⁰C. These nuclei become unbound (to n and 2p emission) at 6.812 and 3.821 MeV, respectively. The ground and 2 1 + states are particle bound in both cases and have been known for decades. Other than the 0 2 + state in ¹⁰Be, all other states in either the ground rotational band or those built on the second 0 + state are in the continuum. Tentative, but highly plausible, reconstructions of the ground and excited rotational bands in these two nuclei, as well as the analog of the excited (T = 1) band in the intermediate odd-odd ¹⁰B nucleus, are shown in Figure 4. All of the states for ¹⁰Be shown in this standard rotational (excitation energy vs. spin) plot have been known for years. Only the spin of what is now assigned as 4 1 + was uncertain, although it was known to be T = 1 [32]. (Note that in the assignments made in Figure 4, 4 1 + belongs to the excited, but much lower moment-of-inertia, excited band while 4 2 + belongs to the ground-state band.). The spin assignments made for ¹⁰C only became possible when a highly plausible assignment could be made for 0 2 + , the search for which was rather tortuous but for which the final chapters were IMS studies, one with an incorrect assignment [33] which prompted another study which lead to the assignment used in Figure 4 [34]. The correct assignment was made based on the similarity of the 3-body correlations for this state with those for other 0 + 2p decays. Using similar logic, the higher spin states could be given tentative assignments [34]. While this spin assignment method is novel and should be viewed with some measure of skepticism, confidence in the assignments is generated by the fact that the apparent moments-of-inertia are constant and the same in the two bands independent of isospin projection. (In three cases for the excited band.) While these assignments must be considered tentative, the results, taken at face value, show that the rotational structures in these clustered nuclei show remarkable insensitivity to decay thresholds.

Isospin symmetry can be broken by asymmetric coupling to the continuum. The classic case, considered by both Ehrman [35] and Thomas [36] is for the A = 13 pair ¹³C and ¹³N, see Figure 5C, where the ground and first three excited states of the former are bound to neutron decay while for the latter all but the ground state are unbound to proton decay. The excitation energies of the 3 / 2 − and 5 / 2 + states are similar in the two nuclei while the excitation energy of the unbound 1 / 2 + state in ¹³N is downshifted by 0.73 MeV relative to its mirror state. The base explanation is simply that, for states unconfined by an angular momentum barrier, the Coulomb energy for the proton-rich case is less, i.e., the wave functions are slightly expanded, for states coupled to the continuum.

While several examples of what has been come to be known as “Thomas-Ehrman” (TE) shifts have been known for decades, the study of proton-rich nuclei by IMS has extended the list of known examples several of which are shown in the other panels of Figure 5. The A = 11 and A = 17 cases, 5 (B) and (F) are similar to the A = 13 case (C) in that the 1 / 2 + state is down shifted relative to the 1 / 2 − , and the 5 / 2 − state in (F). In the A = 17 case (F), the ground states are used as references and the ordinate is again (as in (C)) the excitation energy. However, to display the shift in the A = 11 case (B), we have chosen to fix the energy ordinate zero to the p + ¹⁰C decay threshold, the decay products of ¹¹N, and reference the mirror schemes to one another using levels with finite ℓ composition, blue dotted lines. For A = 16 (D), all the levels in the p-rich ¹⁶F are unbound while none of those in the mirror are. Using the graphical tool employed in (B), one notes that, if the two high-spin levels are used to align the level schemes, the two levels that can decay by s-wave emission are down shifted.

The ground and first excited states for both ¹⁰Be and ¹⁰C (A) are bound, however there are two levels in ¹⁰C well below the third excited state in ¹⁰Be but above both the 1p and 2p decay thresholds (The mirror 1n and 2n thresholds for ¹⁰Be are above all levels in question.) One of these levels is 0 2 + , the band head of the second rotational band, see Figure 4, and the other, which decays to p + ⁹B, has been assigned J π = 2 + , see [37] and references cited therein, an assignment consistent with direct reaction data. (This state is not part of either of the rotational bands shown in Figure 4).

The remaining panel of Figure 5E represents a research opportunity. All the levels shown are known [32] except the 4 + in ¹⁶Ne. While states with J π ≤ 3 can be reached with contributions from the second proton s orbit, J π = 4 + states cannot. Finding this state, allows for an assessment of the actual downshifts of the lower levels, including the ground state, and thus estimates of the contribution from the second s orbit.