Tracking changes in shell structure in neutron-rich nuclei as a function of spin

The structure of neutron-rich nuclei has recently become the focus of much theoretical and experimental effort. Central to the on-going investigations is the expectation that substantial modifications can occur to the intrinsic shell structure of nuclei with a sizable neutron excess [1]. Alterations to the energy spacings of the orbitals and/or to their ordering can have a considerable impact on global nuclear properties such as the nuclear shape or the type of excitations characterizing the level spectra.

As these proceedings amply demonstrate, a number of techniques have been developed to explore the properties of neutron-rich nuclei. Most take advantage of fragmentation of high-energy beams followed by selection of an isotope of interest through a separator. Measurements are then most often carried out either with the selected nucleus moving at high velocity [2] (Coulomb excitation, knockout reaction, etc) or after it has been brought to rest in the laboratory (mass measurements, β-decay studies, etc). This approach is particularly well suited for investigations of nuclear properties near the ground state. One of the purposes of the present contribution is to illustrate the power of a complementary approach. Using the resolving power of spectrometers such as Gammasphere [3] in combination with a variety of reactions ranging from fusion-evaporation with exotic beams or targets to multi-nucleon transfer reactions (so-called deep-inelastic reactions), it has been possible to investigate neutron-rich nuclei in a number of regions of the nuclear chart. In particular, over the last 10–15 years it has been shown that the yrast spectroscopy of hard-to-reach, neutron-rich nuclei, populated in deep-inelastic reactions, can be carried out successfully at energies 15–25% above the Coulomb barrier, when using γ–γ coincidence measurements [4]. From a broader perspective, γ-ray intensity measurements from these reactions have provided a mapping of the A and Z distributions of product yields. In all cases, multi-nucleon transfers were shown to lead mostly to less neutron-rich nuclei in the region close in mass to the heavy colliding partner. Conversely, species on the neutron-rich side of the light reaction partner were found to be favored as well. It is possible to understand these yield distribution patterns in terms of a general tendency toward N/Z equilibration of the dinuclear systems formed in deep-inelastic reactions. As stated above, this experimental approach is complementary to techniques with fast beams in that it provides a means to investigate neutron-rich nuclei to fairly high angular momentum, herewith providing the data required (i) to characterize further changes in shell structure while testing in detail large-scale shell-model calculations with the most modern interactions, and (ii) to characterize global properties, such as the evolution of the nuclear shape with spin.

This presentation will first discuss data gathered in nuclei close to the 'island of inversion' centered around ³²Mg and highlight challenges in describing the observations with modern effective interactions. It will subsequently review the nature of excitations in neutron-rich fpg-shell nuclei between Ca and Ni, where interactions incorporating the monopole tensor force are proposed to be responsible for a neutron sub-shell closure at N = 32. Also of particular interest in this region is the role of the g_9/2 orbital in driving the nuclear shape as has been demonstrated by the presence of collective excitations at moderate spin in neutron-rich Cr and Fe isotopes. This suggests that mixing between the 'deformed' and the 'shell-model' states may have a strong influence on their properties. This presentation will conclude with a discussion of recent results obtained in the direct vicinity of ⁶⁸Ni.

The 'island of inversion' in the neutron-rich sd shell is a region of the nuclear chart where the effects of shell evolution in nuclei far from stability have been studied for some time. Unexpectedly high binding energies, observed via mass measurements of ^31,32Na [5], were later reproduced in Hartree–Fock calculations when neutron f_7/2 intruder configurations were included [6]. The same effects were subsequently seen in Mg and Ne isotopes around N = 20 [7]. Systematic trends in E(2⁺₁) and B(E2;0⁺ → 2⁺) values have added weight to the arguments for large deformation in the ground- and low-lying states of N = 20 isotones below Z = 12, while data for Si, S, Ar, and Ca nuclei are characteristic of a large N = 20 shell gap. The large ground-state deformation of nuclei within the island of inversion is understood as resulting from deformed neutron states of f_7/2 and p_3/2 spherical parentage falling below similar states arising from the d_3/2 orbital as the neutron number increases. Recent theoretical efforts to account for the experimental observations have focused on the monopole component of the nucleon–nucleon interaction [8], revealing that both the central and the tensor forces dominate shell evolution. The central force is always attractive, with similar strength between two d orbitals as for a d-and-f pair of states. The additional effect of the tensor force between protons and neutrons, however, is attractive between the d_5/2 and the d_3/2 orbitals and repulsive between the d_5/2 and f_7/2 states. As the central force is stronger than the tensor force, the net effect is that of an attraction between both the d_5/2 and d_3/2 and the d_5/2 and f_7/2 orbitals, but less so in the latter case.

The nucleus ³⁰Mg is located at the boundary of the island of inversion. Recent neutron knockout experiments have confirmed this by contrasting the spectroscopic strength associated with the negative-parity orbitals in the ground states of ³⁰Mg and ³²Mg [9]: The strength in ³⁰Mg is a third of that found in ³²Mg, highlighting a dramatic change when entering the region of inversion. ³⁰Mg is, therefore, not expected to have significant ground-state deformation. However, the N = 20 gap should be smaller than for stable nuclei and, as such, configurations involving the f_7/2 and p_3/2 neutron orbitals should be observed at relatively low excitation energy. Furthermore, if intruder configurations involve f²_7/2 excitations, enhanced proton–neutron interactions may lead to collectivity in these excited states. These considerations motivated the quest for high-spin states in ³⁰Mg carried out by Deacon et al [10].

³⁰Mg was populated with the ¹⁴C(¹⁸O,2p) reaction which provides access to states of higher energy and spin than had been studied previously. The cross section for the reaction channel of interest being rather small, the measurement required the resolving power of the Gammasphere array [3] in combination with the mass and isotope selection provided by the Argonne fragment mass analyzer [11]. The ³⁰Mg level scheme can be found in figure 4 of [10]: of the 11 transitions identified, only two had been reported previously. Moreover, the coverage provided by Gammasphere enabled spin and parity assignments following an angular correlation analysis.

The experimental levels are compared to the results of the calculations in figure 1. An initial shell-model calculation was carried out with the universal sd (USD) effective interaction [12] within a full sd model space. The calculated states lie higher than those observed experimentally, and there also appear to be additional observed states compared to the calculation. In particular, states with negative parity are clearly outside the model space considered. Thus, calculations within the sd model space are insufficient to account properly for excitations in ³⁰Mg, even at relatively low spin. Extensions to the model space are needed to produce counterparts for all experimental states, and increase the overall level of mixing, compressing the lowest-lying states in energy, to give a better match in energies. Such calculations were performed using the Monte Carlo shell-model method (MCSM) with the SDPF-M interaction [13], to judge the influence of excitations into the fp-shell orbitals. This interaction has previously been used in calculations for a number of neighboring nuclei, including data on ³¹Mg [14]. To illustrate the role of cross-shell excitations, the positive-parity states from the MCSM calculations are shown on the right-hand side of figure 1, where states are labeled by spin-parity, excitation energy, and N_fp, the average number of neutron excitations into the fp shell. A pure 0p–0h state would have N_fp = 0, whereas a pure 2p–2h state has N_fp = 2, if the contribution of 4p–4h excitations is considered to be negligible. As discussed in [10], the MCSM calculations provide a way to disentangle cross-shell excitations from those involving the sd orbitals only. Specifically, they provide a means to account for the deexcitation patterns observed experimentally. For example, the first 2⁺ state is calculated to lie just 49 keV above the observed one, a closer level of agreement than for the USD result. Its structure is predominantly 0p–0h with N_fp = 0.79, as might be expected. The presence of a strong transition to this level from the observed 4⁺ state at 3379 keV suggests a similarity in structure. Therefore, the calculated 4⁺ state at 4450 keV would appear to be the theoretical partner, even though the energy is overestimated (see comment in what follows). In a similar fashion, the calculated states at 3000 and 3850 keV, both corresponding to cross-shell excitations with N_fp ∼ 1.5, would appear to correspond to the observed 2⁺ and 4⁺ states at 2465 and 3455 keV, respectively. The calculated 0⁺ level at 2120 keV may correspond to the excited 0⁺ state reported in [15], observed via an E0 transition to the ground state, and with an excitation energy of 1789 keV. The calculations indicate a significant 2p–2h configuration for this state, consistent with the conclusions drawn in [15]. Despite these successes, the MCSM calculations appear to have several specific problems associated with the 'intruder' excitations (N_fp ∼ 2) predicted at too high an excitation energy. This could be attributed to overprediction of the excitation energy of the 2p–2h configurations and/or overestimation of the mixing between 0p–0h and 2p–2h excitations. Comparison of the negative-parity (3⁻) state with the 1p–1h calculations also hints at the need to include a reassessment of the energy of cross-shell configurations. Thus, studies of higher spin states in nuclei such as ³⁰Mg suggest a path forward for future shell-model calculations, in terms of both improving single-particle energies and interactions and extending the model space available to cross-shell configurations. In this regard, it is worth pointing out that further information on 1p–1h excitations can be gathered from recent studies on less exotic nuclei such as ³⁰Si or ³⁰Al [16], for example. In these instances as well, comparisons between data and calculations suggest the need for improved single-particle energies.

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The neutron-rich Ca, Ti, and Cr isotopes have been the focus of many studies during recent years. This effort was prompted primarily by the observation of an N = 32 subshell closure in ⁵²Ca [17], ⁵⁴Ti [18], and ⁵⁶Cr [19], which has been inferred from mounting experimental evidence such as systematic variations in E(2⁺₁) energies and B(E2) values in the even–even Ca, Ti and Cr isotopes [20]. The two physical quantities are found to be anti-correlated; while the E(2⁺₁) energies increase significantly at N = 32, the B(E2) strengths are lowest. The situation mirrors that seen for the well-known neutron shell gap at N = 28 in nuclei of this region. This new aspect of shell structure in neutron-rich nuclei is understood as resulting from a significant shift in energy of the neutron νf_5/2 orbital caused by a strong proton–neutron monopole interaction [8]. When protons are removed from the πf_7/2 shell, the νf_5/2 orbital shifts up in energy relative to the νp_3/2 and νp_1/2 levels due to the weakening of the attractive monopole interaction. Many calculations within the full pf shell-model space have been carried out with different Hamiltonians and most reproduce the features of the yrast level structures fairly well. However, the magnitude of the shift in energy of the νf_5/2 orbital appears to be best reproduced by the GXPF1 family of Hamiltonians developed by Honma et al [21]. By comparing the level sequences of nuclei near the N = 32 gap with shell-model calculations, one is able to assess quantitatively the location of the single-particle states and their evolution as a function of Z and N. Examples abound and only a few selected cases are briefly discussed here.

Detailed comparisons of calculations with the yrast sequence in ⁵⁴Ti [18] illustrate the role of the νp_1/2 and νf_5/2 orbitals in the location of I^π = 7⁺–10⁺ levels in this nucleus. The ability of modern effective interactions to account properly for the monopole interaction is illustrated further by the study of the N = 31 isotones ⁵¹Ca, ⁵²Sc and ⁵³Ti [22], where comparisons between experiments and calculations have identified levels with dominant ν(p_3/2)²f_5/2 configurations coupled to πf_7/2 states; e.g. the 4320 keV, 9/2⁻ (ν(p_3/2)²f_5/2) level in ⁵¹Ca, the 3603 keV, 8⁺ state (πf_7/2)⊗(ν(p_3/2)²f_5/2) in ⁵²Sc, and the 21/2⁻ (πf_7/2)²⊗(ν(p_3/2)²f_5/2) level at 6056 keV in ⁵³Ti. Figure 2 illustrates the ability of the KB3G [23] and GXPF1A [21] interactions to reproduce the excitation energy of these states. The GXPF1A interaction appears to reproduce the data best: in the heaviest isotone, the difference between the experimental and the calculated values is only 51 keV, and the deviation between the data and the results of the calculation increases slightly to 138 keV in ⁵²Sc and is 476 keV for ⁵¹Ca. In all three isotones, the calculations also reproduce well the yrast states where a neutron occupies the νp_1/2 single-particle state (see [22]). Thus, while the interaction accounts well for the impact of the monopole interaction, the observed behavior of the states with predominant νp_1/2 and νf_5/2 configurations suggests that the p_1/2 − f_5/2 energy difference in Ca nuclei might be somewhat smaller than that predicted by the GXPF1A interaction.

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The same conclusion was reached in [24] from a comparison between experiment and theory in the case of ⁵⁵Ti (N = 33) [24]. Figure 3 provides the level scheme established by Zhu et al [24] and compares the data with calculations with the KG3B and GXPF1A interactions. Of importance in this case is the ordering of the two lowest states. Their quantum numbers have been firmly established in a recent knockout experiment [25]. The 1/2⁻ ground state can be viewed as the result of the coupling of a p_1/2 neutron to the ⁵⁴Ti core. The GXPF1A calculations interpret the 5/2⁻ first excited level in a straightforward manner as resulting from the ⁵⁴Ti⊗νf_5/2 coupling. Thus, contrary to the prediction of the KG3B interaction, the νf_5/2 state lies above the νp_1/2 orbital in the Ti isotopes, a result pointing to the preferred use of the GXPF1A interaction for the description of nuclei in the region. As discussed in [24], the couplings of a p_1/2 neutron to the ⁵⁴Ti 4⁺ and 6⁺ yrast excitations, accounts for the 9/2⁻ and 13/2⁻ states, and the calculations interpret the 17/2⁻ state as the ⁵⁴Ti(6⁺)⊗νf_5/2 coupling. In order to generate a 19/2⁻ state, the neutron core has to be broken and a p_3/2 neutron hole created. This is reflected experimentally in the 1881 keV energy of the 19/2⁻ → 17/2⁻ transition, and theoretically in the large probability (74%) of the [π(f²_7/2)⊗ν(f⁸_7/2p³_3/2p_1/2f_5/2)]_19/2⁻ configuration. While the agreement between experiment and theory for ⁵⁵Ti level structure can be viewed as satisfactory, it is worth pointing out that the energy difference between the lowest levels with predominant νp_1/2 and ν f_5/2 configurations (i.e. the 5/2⁻ − 1/2⁻ difference) is smaller experimentally than calculated by 307 keV. Thus, as in the case of the N = 31 isotones, it appears that the two single-particle orbitals may be somewhat closer in energy than the GXPF1A Hamiltonian would predict, although other effects may also contribute to this difference (such as the relative positions of the νp_1/2 and p_3/2 states, for example).

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Over the last decade, many other tests of modern effective interactions for the description of neutron-rich nuclei in the region just above doubly-magic ⁴⁸Ca have been carried out and the cases discussed above are only illustrative examples of the large experimental effort that has taken place. In this context, it is worth pointing to the work of [26], where the spectroscopy of ⁵²Ti was expanded significantly by exploring and focusing on high-lying, non-yrast states. The comparisons between data and calculations focus not only on energies, spins and parities, but also on lifetimes as the latter provide additional opportunities to probe the many-body wavefunctions. The experimental approach combined data from a fusion-evaporation reaction producing ⁵²Ti with information derived from β decay into the same nucleus following a deep-inelastic reaction [26]. Thirty-two states were observed, 14 of which had not been reported previously. In addition, lifetimes of nine states were measured using the Doppler shift attenuation method (DSAM). Comparisons between the data and shell-model calculations favor, to a degree, the results obtained with the GXPF1A effective interaction over those derived with the FPD6 [27] and KB3G Hamiltonians.

As indicated by the systematics of the 2⁺₁ energies of the even–even Cr isotopes, the effect of the N = 32 shell gap is less pronounced in this isotopic chain. This is expected since two additional protons occupy the f_7/2 shell and the π1f_7/2 − ν1f_5/2 proton–neutron monopole interaction is correspondingly stronger. Nevertheless, results from a study of the B(E2; 0⁺ → 2⁺₁) transition probabilities indicate that the collectivity of the 2⁺₁ state in ⁵⁶Cr (with N = 32) is significantly lower than that in the neighboring N = 30 and 34 isotopes, ⁵⁴Cr and ⁵⁸Cr, and is similar to that of N = 28, ⁵²Cr [28].

As already alluded to above, another experimental observation is that for neutron numbers above N = 32, the 2⁺₁ energy starts to drop considerably: from 1007 keV in ⁵⁶Cr to 880 keV in ⁵⁸Cr, 645 keV in ⁶⁰Cr, 446 keV in ⁶²Cr and 420 keV in ⁶⁴Cr. This trend of decreasing 2⁺₁ energies is associated with a small neutron gap at N = 40, which facilitates excitations to the νg_9/2 intruder orbital. Recent detailed studies of odd, neutron-rich Cr isotopes reveal that the yrast 9/2⁺ state, which is associated with this νg_9/2 intruder orbital, decreases in excitation energy with increasing neutron number: from 2087 keV in ⁵⁵Cr to 1507 keV in ⁵⁷Cr, and abruptly down to 503 keV in ⁵⁹Cr. In ^55,57Cr, decoupled rotational bands built on the ν g_9/2 state have been reported [29, 30], and these bands provide strong experimental evidence for the shape-driving potential of this intruder level, in these specific cases toward substantial prolate deformation. On the other hand, the long-lived 9/2⁺ isomeric state observed in ⁵⁹Cr appears consistent with an oblate core [31]. In any event, the evidence indicates an increasing involvement of the νg_9/2 orbital, when compared to the Ti isotopes. This point is illustrated further in figure 4, where the structure of ⁵⁶Cr is presented. In this case, the level scheme is characterized by two distinct sets of states: positive parity states of single-particle character and a collective negative-parity sequence. Indeed, the positive-parity yrast sequence of ⁵⁶Cr is reproduced satisfactorily by calculations with the GXPF1A Hamiltonian and the agreement between data and theory is comparable to that achieved in similar comparisons for the Ti isotopic chain. In fact, as discussed in [32], a similar agreement between low- and medium-spin levels and GXPF1A calculations is seen in ⁵⁸Cr, but not in ⁶⁰Cr where the experimental level structure is more compressed than that calculated. Thus, beyond N = 34, the fp model space appears to be too restrictive to account for the observations.

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As for the negative-parity sequence, figure 5 illustrates the striking similarity between the negative-parity, spin 7–17, sequence of ⁵⁶Cr and the two decoupled bands in the odd neighbors. The aligned angular momentum I_x experiences a similar increase with rotational frequency in all three bands, except perhaps at the highest spins where the beginning of a backbend is present in ⁵⁷Cr. Total Routhian surface (TRS) calculations described in [30] were carried out in order to track both the ground-state configuration and a negative-parity, two quasi-neutron configuration similar to those considered in the odd neighbors, i.e. with one particle occupying the same prolate-driving 1/2⁺[440] orbital of g_9/2 parentage and the other the f_5/2 state, emptying the p_3/2 and/or p_1/2 states correspondingly. These calculations are able to reproduce the observations in A = 55 − 57 Cr isotopes, as shown in figure 15 of [32]. However, thus far there is no experimental sign of a collective excitation in either the odd or the even A = 58–60 nuclei. Furthermore, a low-lying isomer associated with the g_9/2 state is present in ⁵⁹Cr. The TRS calculations with existing parameterizations encounter difficulties in reproducing the observations in detail although they indicate that the nuclear potential becomes rather soft and that a well-defined shape is absent in ⁵⁸⁻⁶⁰Cr.

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The picture that emerges from these comparisons is one where the intruder, shape-driving g_9/2 orbital needs to be taken into account to achieve a satisfactory overall description. In the A = 55–57 Cr nuclei, excitations involving this state appear in the experimental spectrum at sufficiently high excitation energy and angular momentum that they do not seem to disturb significantly the single-particle states associated with the fp space. The impact of the g_9/2 orbital is nevertheless significant as the presence of rotational bands indicates that the nuclear shape becomes prolate once this excitation is involved. As the neutron number increases, the g_9/2 state continues to come closer to the Fermi surface, and there are indications that the structure of ⁵⁹Cr is best explained in terms of excitations associated with an oblate shape with modest deformation. Shell-model calculations in the enlarged fpg space as well as TRS calculations with parameters suitable for neutron-rich nuclei of the region are badly needed to investigate the issues further. As a significant step in this direction, it is worth pointing to a recent development: the projected shell model (PSM) [33] has been applied to this mass region to investigate states in fp-shell systems at high spin. By exploiting the use of a deformed shell-model basis, the PSM can adopt a relatively large single-particle space that allows collective motion and cross-shell excitations to be taken into account. The PSM has successfully reproduced high-spin structural features in some fp-shell nuclei that involve νg_9/2 configurations (see, for example, [34]).

The role of the g_9/2 neutron orbital in driving the nuclear shape is illustrated further in the neutron-rich Fe nuclei which have also received much attention recently [35–38]. As seen in figure 5, the behavior of collective bands seen at moderate and high spin in ^59,60Fe mirrors that discussed above for ⁵⁵⁻⁵⁷Cr in terms of the alignment as a function of frequency. Calculations again account for the observations by invoking deformed configurations associated with the occupation of at least one νg_9/2 neutron state. Similar bands are also present in the N = 32 isotone ⁵⁸Fe and they are reproduced satisfactorily by PSM calculations [35]. Perhaps, more noteworthy in this instance is the presence of two band structures consisting of low-energy ΔI = 1ℏ transitions [35]. These are candidates for magnetic rotational bands. Unlike traditional deformed rotational bands, these structures are composed of strong magnetic dipole (M1) and weak E2 transitions. The orientation of these rotors is not specified by the deformation of the overall density, but rather by the current distribution induced by specific nucleons moving in high-j orbitals. In this interpretation, the angular momentum vectors of high-j protons and neutrons form two blades of a pair of shears that are almost perpendicular to each other at the bandhead. Along the bands, energy and angular momentum are increased by closing the blades of the shears; i.e. by aligning the proton and neutron angular momenta. Consequently, rotational bands can be formed in spite of the fact that the shapes of these nuclei stay nearly spherical. The name 'magnetic rotation' then alludes to the fact that the magnetic moment is the order parameter inducing a violation of rotational symmetry and, therefore, causing the presence of rotational-like structures in the spectrum. As discussed in [35], calculations reproducing the observed sequences indicate that the respective proton and neutron spins forming the blades are likely associated with two aligned proton holes in the πf_7/2 orbital (i.e. the π(f_7/2)⁻² configuration) and a ν[(f_5/2p_3/2p_1/2)³(g_9/2)] neutron excitation. To date, only one other case has been reported in this mass region where the g_9/2 orbital is involved in generating this collective mode: four candidate M1 bands have been identified in ⁶⁰Ni [39].

A recent measurement at the NSCL on ⁶⁴Cr [40], the Cr isotope with the most extreme proton–neutron asymmetry reachable today, has demonstrated that the shell gap at N = 40 appears to vanish in neutron-rich Cr. The energy of the first 2⁺ and 4⁺ states, compared to lighter isotopes in figure 6, were found to be 420(7) and 1131(11) keV, respectively. This behavior contrasts that of the Fe isotopes (also shown in figure 6), where the corresponding excitations of the N = 40 isotope, ⁶⁶Fe, are at 586(7) and 1417(11) keV. These 2⁺ excitations are by far the lowest observed in the N = 40 neutron-rich nuclei. Interesting systematics can also be seen in the ratio R between the 2⁺ and 4⁺ energies, which provides a convenient indicator of structural properties in terms of non-collective (⩽2.0), spherical–vibrational (∼2.0), transitional (∼2.5), or rigid-rotor (∼3.33) regimes. The data now available indicate that the ratio increases steeply toward a transitional character between N = 30 and 40 in the Cr isotopic chain (see figure 6). In contrast, the ratio remains roughly constant, and even decreases slightly, in the corresponding Fe chain. The relative excitation cross sections (deduced from the measurements of [40]) are also strikingly different for the Cr and Fe isotopes. The different behavior of both excitation energies and cross sections for these Cr and Fe isotopes point to a different intrinsic structure for the two N = 40 isotones. These observations represent a challenge for the most modern nuclear interactions. In fact, shell-model calculations reproduce the measured yrast level structures only by including both the g_9/2 orbital from above the N = 40 gap and d_5/2 state from the shell above N = 50 in the model space [41].

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As discussed in the preceding section, rotational bands have been delineated in ^58,59,60Fe, and ^55,56,57Cr which appear to be based on deformed structures that have at least one neutron g_9/2 orbital in their intrinsic configuration. The moments of inertia associated with these sequences in the even-Z systems are all very similar, but differ from those observed in the heavier Fe and Cr isotopes. This suggests that, whilst the heavier isotopes display collectivity, mixing between the 'deformed' and the 'shell-model' states might have a strong influence on their properties. Such effects are well documented in regions with shape coexistence, for example, in the neutron-deficient Hg and Pb isotopes [42]. In fact, the similarity between the Cr–Fe and Hg–Pb regions is rather striking, as shown in figure 7. The influence of the mixing between coexisting configurations has recently also been considered in the Cr–Fe region [43]. Simple two-band mixing calculations were carried out assuming that the yrast bands of these neutron-rich Cr and Fe even–even isotopes result from mixing between spherical and deformed states [43]. The excitation energies of the shell-model states are calculated using the GXPF1A interaction, which does not include g_9/2 or d_5/2 orbitals in the model space. The deformed states are calculated by a rotational model, defined by the 2⁺ excitation energy relative to the 0⁺ deformed ground states. Once the level structures of the shell model and deformed states are defined, the yrast states can be calculated using a two-band mixing model by determining the excitation energy of the two 0⁺ states and the interaction strength between the two structures, which is assumed to be constant for all levels. It was found that the excitation energies of the known levels in ^62,64,66,68Fe and ^60,62,64Cr can be reasonably reproduced by assuming a similar interaction strength V_int ∼ 100 keV for all isotopes. While all the states are mixed to some extent, the calculations indicate that the ground states of all the Fe isotopes are predominantly of spherical character, while the deformed ('intruder') 0⁺ level lies at 2.4 MeV in ⁶⁰Fe and decreases in energy with increasing N to ∼300 keV in the heavier isotopes, ^66,68Fe. In fact, these calculations suggest that the first excited state in ^64,66,68Fe is the deformed 0⁺ state, making it challenging to detect. For ^60,62,64Cr, the calculations result in a similar prediction that the excited 0⁺ state is the lowest excitation with an energy of ∼330 keV in ⁶⁰Cr and ∼200 keV in ^62,64Cr. However, unlike in the isotones ^64,66Fe, the model predicts that the ground states of ^62,64Cr are dominated by a deformed configuration. This result may well account for the difference in the relative behavior of the measured excitation cross sections in knockout between Cr and Fe isotopes discussed above [40]. Clearly, to test this hypothesis of shape coexistence further, additional measurements are required. Some of these are currently in the planning stage.

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In discussions of magic numbers, neutron number N = 40 has historically been a subject of debate, especially in the case of the Ni isotopes. Proton number Z = 28 is magic, and, for neutrons, a sizable energy gap at N = 40 is thought to separate the pf shell from the intruder g_9/2 state, potentially making Z = 28, N = 40 ⁶⁸Ni a doubly-magic nucleus. Experimentally, the occurrence of shell closures in ⁶⁸Ni was first suggested based on the observation of a 1770 keV 0⁺₂ level as the lowest excited state, followed by a 2⁺₁ state of relatively high excitation energy (2034 keV) [44, 45]. The discovery of several isomeric states in ⁶⁸Ni and in neighboring nuclei [46, 47] supported the case for its magic character further, as did the results from Coulomb excitation measurements indicating a B(E2,2⁺₁ → 0⁺) reduced transition probability roughly three times smaller than the corresponding value for ⁵⁶₂₈Ni₂₈ [48, 49]. However, based on recent high-precision mass measurements in the neutron-rich Ni isotopes (up to ⁷³Ni), the N = 40 shell closure appears to be more doubtful when inferred from changes in the two-neutron separation energies [50, 51]. It has been argued in the literature that the apparent contradiction between the B(E2) value and the separation energy is a consequence of the parity change across the N = 40 gap, with a sizable fraction of the low-lying B(E2) strength residing in excited states around 4 MeV, and the 2⁺₁ level being associated predominantly with a neutron-pair excitation [48, 52]. The size of the N = 40 gap is then of the order of 2 MeV only and the corresponding discontinuity in the sequence of orbitals corresponds at most to a sub-shell closure.

From these considerations, it is clear that a satisfactory description of nuclear structure in this mass region is still lacking. This is reflected in on-going theoretical efforts to determine the most appropriate interactions for use in the calculations, as well as in further experimental work. In this area, the most recent efforts have focussed on studies of single-particle excitations in the even ^64,66,68Ni isotopes [53], a search for collective excitations similar to those described above for Cr and Fe [54], a study of a possible new long-lived state in ⁶⁸Ni [55], and a new investigation of the neutron-hole nucleus ⁶⁷Ni. The data on the latter nucleus are presented below, as they provide an opportunity to test the most modern interactions while investigating the nature of yrast excitations up to moderate spin [56]. A level scheme built on the previously known 13 μs isomer in ⁶⁷Ni was established for the first time by exploring prompt and delayed coincidence relationships from deep-inelastic reaction products. The yrast and near-yrast sequences have been delineated up to an excitation energy of 5.3 MeV and a tentative spin and parity of (21/2⁻): see figure 8.

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In order to gain further insight into the nature of the observed ⁶⁷Ni states, large-scale calculations were carried out with the shell-model code ANTOINE [57] using both the jj44b [58]¹ and the JUN45 [59] effective interactions. Both Hamiltonians were restricted to the neutron f_5/2, p_3/2, p_1/2, and g_9/2 valence space and assume a ⁵⁶Ni core. However, the required two-body matrix elements and single-particle energies were obtained from fits to different sets of data. Specifically, the JUN45 interaction was developed by considering data in nuclei with Z ∼ 32 and N ∼ 50, and excludes explicitly the Ni and Cu isotopes as the ⁵⁶Ni core is viewed as being rather 'soft' [59]. In contrast, experimental data from Z = 28 − 30 isotopes and N = 48–50 isotones were incorporated in the fits in the case of the jj44b interaction [58].

The results of the calculations are compared with the experimental data in figure 8. With both interactions, the energy of the 9/2⁺ state is predicted lower than the measured value. This can be viewed as an indication that the adopted single-particle energy of the g_9/2 neutron orbital is too low in the two Hamiltonians. It is worth noting that the jj44b interaction calculates this state to lie within 192 keV of the data and indicates about a 25% admixture of the νg³_9/2 configuration into the 9/2⁺ wavefunction. With the JUN45 interaction, the level is predicted to lie 498 keV lower than in the data with roughly 33% of the wavefunction involving three neutrons in the g_9/2 orbital. This is possibly the result of the location of the g_9/2 orbital at a lower energy in the JUN45 Hamiltonian, as compared to that used in the jj44b case, which leads to larger configuration mixing in the wavefunction of the 9/2⁺ state. Overall, the calculated spectrum with both interactions appears somewhat compressed when compared to the data, as illustrated on the right side of figure 8. The correspondence between data and calculations is rather satisfactory when the computed excitation energies are expressed relative to the 9/2⁺ isomer as is done on the left-hand side of figure 8.

As discussed in detail in [56], the levels above the 9/2⁺ isomeric state can be understood as neutron excitations, with contributions of protons across the Z = 28 gap playing a minor role at best. Calculations with both interactions are in fair agreement with the data. They attribute a significant role to the g_9/2 neutron orbital for every state observed in this measurement. In fact, in most cases, significant νg²_9/2 and νg³_9/2 configurations are part of the wavefunctions. Similar observations have been made for other nuclei close to ⁶⁸Ni; see, for example, recent comparisons between calculations with the same jj44b and JUN45 interactions and data for ^65,67Cu in [60]. From these findings, it is concluded that even in a nucleus only one neutron removed from N = 40, the impact of a neutron shell closure is rather modest. As the g_9/2 neutron orbital is shape driving, multi particle-hole excitations involving this state may be expected to be associated with enhanced collectivity and it would be of interest to investigate the latter in future measurements.

Using the resolving power of spectrometers such as Gammasphere in combination with reactions ranging from fusion-evaporation with exotic beams or targets to deep-inelastic reactions, it has been possible to investigate to fairly high angular momentum neutron-rich nuclei in a number of regions of the nuclear chart. The approach provides tests of nuclear structure in these systems that are not available with other techniques.

The power of the techniques was illustrated through data from nuclei in the A ∼ 30 and ∼60 regions and challenges for current theoretical descriptions were discussed. Studies of high-spin states in nuclei near the N = 20 island of inversion suggest a path forward for future shell-model calculations in terms of both improving single-particle energies and interactions and extending the model space available to cross-shell configurations. The delineation to high angular momentum of yrast sequences in Ca, Ti and Cr nuclei has provided further evidence on the presence of a sub-shell gap at N = 32 as well as information on the interactions affecting the p_3/2, p_1/2 and g_9/2 orbitals. Collectivity manifests itself through the presence of rotational sequences at moderate spin in Cr and Fe isotopes near N = 40 and indications for mixing between spherical and collective excitations were presented. The data highlight the role of the g_9/2 orbital in driving the nuclear shape. Finally, recent results in the direct vicinity of ⁶⁸Ni indicate that the impact of the N = 40 neutron shell closure is rather modest. Generally speaking, the results discussed in this contribution highlight the challenge for theory to adequately account for cross-shell excitations.

The author would like to express his gratitude to the organizing committee of the Nobel Symposium for the opportunity to attend the meeting and to present this contribution. He is also indebted to his many collaborators: without their hard work and effort, the results presented above would not have materialized. He gratefully acknowledges constant fruitful interactions with A Gade, C Chiara, S Freeman, B Fornal, W Walters, as well as with his ANL collaborators M P Carpenter and S Zhu. This work was supported by the US Department of Energy, Office of Nuclear Physics, under contract no. DE-AC02-06CH11357.