Shape coexistence and the role of axial asymmetry in $^{72}$Ge

The quadrupole collectivity of low-lying states and the anomalous behavior of the $0^+_2$ and $2^+_3$ levels in $^{72}$Ge are investigated via projectile multi-step Coulomb excitation with GRETINA and CHICO-2. A total of forty six $E2$ and $M1$ matrix elements connecting fourteen low-lying levels were determined using the least-squares search code, gosia. Evidence for triaxiality and shape coexistence, based on the model-independent shape invariants deduced from the Kumar-Cline sum rule, is presented. These are interpreted using a simple two-state mixing model as well as multistate mixing calculations carried out within the framework of the triaxial rotor model. The results represent a significant milestone towards the understanding of the unusual structure of this nucleus.

The quadrupole collectivity of low-lying states and the anomalous behavior of the 0 + 2 and 2 + 3 levels in 72 Ge are investigated via projectile multi-step Coulomb excitation with GRETINA and CHICO-2. A total of forty six E2 and M 1 matrix elements connecting fourteen low-lying levels were determined using the least-squares search code, gosia. Evidence for triaxiality and shape coexistence, based on the model-independent shape invariants deduced from the Kumar-Cline sum rule, is presented. These are interpreted using a simple two-state mixing model as well as multistate mixing calculations carried out within the framework of the triaxial rotor model. The results represent a significant milestone towards the understanding of the unusual structure of this nucleus.
The structure of low-lying states in even-even Ge isotopes has been the subject of intense scrutiny for many years due to the inherent challenge of interpreting their systematics as a function of mass A. These nuclei possess at least one excited 0 + state in their low-energy spectrum that differs from the ground state in its properties. Systematically, the energy of the 0 + 2 level varies parabolically with A and reaches a minimum in 72 Ge, where it becomes the first excited state. The existence of even-mass nuclei with a J π = 0 + first excited state is an uncommon phenomenon which, to date, has been observed in only a few nuclei located near or at closed shells: 16 O [1], 40 Ca [2,3], 68 Ni [4,5], 90 Zr [6], 180,182 Hg [7][8][9], 184,186,188,190,192,194 Pb [10][11][12][13][14][15]. There are also examples of such nuclei where a subshell appears to play a role similar to a closed shell such as 96,98 Zr [16,17]. These cases have all been explained as resulting from shape coexistence due to the presence of intruder configurations; i.e., configurations involving the excitation of at least one pair of nucleons across a shell or subshell energy gap [18].
The structure of 72 Ge is highly unusual in that this nucleus is far from closed shells and, yet, possesses a 0 + first excited state. It shares this distinction with only two other known nuclei: 72 Kr [19] and 98 Mo [20]. It should be noted that while 68 Ni might be doubly magic [4,21] with a presumed subshell closure at N = 40 [22], there is no evidence to date of spherical ground-state configurations in any of the other N = 40 isotones [23]. There are, however, strong experimental indications of enhanced collective behavior in or near the ground states of the N = 40 neutron-rich Fe and Cr isotones [24]. In the Ge isotopes, the absence of a subshell closure at N = 40 can be inferred from the fact that the 2 + 1 level with the highest energy appears in 70 Ge rather than 72 Ge. Thus, understanding the nature and origin of this anomalous 0 + 2 state in 72 Ge has been a major challenge for collective model descriptions.
The theory of collectivity in nuclei is predominantly focused on models with quadrupole degrees of freedom. The simplest of these models, based on the quantization of a liquid drop [25,26], describes quadrupole collectivity as either due to harmonic quadrupole vibrations of a spherical shape or to rotations and vibrations of a deformed quadrupole shape: a prolate or oblate spheroid or an axially asymmetric ellipsoid. None of these variants of nuclear collectivity possesses 0 + first excited states. More sophisticated models abound, for example models based on boson degrees of freedom [27], and most notably the interacting boson model [28], but to produce a first excited 0 + state usually requires extreme parameter choices fitted to the nucleus under investigation. Other models with quadrupole collectivity introduce "pair-excitations" in an ad hoc way. These approaches are motivated by the shape coexistence phenomena observed at and near closed shells. These models contain parameters that are fitted to the properties of low-lying excited 0 + states, most notably to a large pair excitation energy parameter which is overcome by enhanced quadrupole correlations. The latter lower the energy of the pair configuration, even to the extent that the intruder configuration becomes the ground state [29][30][31][32]. It should be noted that microscopic collective models are now beginning to provide first insights into such structures [33].
In this letter, a detailed study of quadrupole collectivity in 72 Ge by multi-step Coulomb excitation is reported. This nucleus has been studied by Coulomb excitation in the earlier work of Kotliński et al. [34]. The extended set of E2 matrix elements obtained in the present study now permits a model-independent view of the shapes exhibited by 72 Ge up to moderate spin (8 ). This, to the best of our knowledge, is the most extensive set of shape invariants achieved for any nucleus with a first-excited 0 + state. The deformation and asymmetry of the groundstate band are found to be remarkably constant. The shape invariants from the yrast and non-yrast states support an interpretation based on shape coexistence, with the ground-state configuration exhibiting a triaxial deformation of γ ∼ 30 • . These results are compared with a recent version of the triaxial rotor model. It is worth noting that the nucleus 72 Ge and its neighbors have also been the focus of detailed and varied spectroscopic investigations using multi-nucleon transfer reactions: these are summarized with an attempt at a simple pairing occupancy picture in Ref. [18], but this picture has not been connected to the quadrupole collective properties of the nuclei involved.
Multi-step Coulomb excitation of the 72 Ge nucleus was carried out by bombarding a 0.5 mg/cm 2 -thick 208 Pb target, sandwiched between a 6 µg/cm 2 Al front layer and a 40 µg/cm 2 C backing, with a 301-MeV 72 Ge beam delivered by the ATLAS facility at Argonne National Laboratory. The γ rays emitted in the deexcitation were detected with the γ-ray tracking array, GRETINA [35]. At the time of the experiment, this array consisted of 28 highly-segmented coaxial high-purity germanium (HPGe) crystals grouped into 7 modules. The scattered projectile and recoiling target nuclei were detected in kinematic coincidence with the γ rays by CHICO2, an array of 20 position-sensitive parallel-plate avalanche counters arranged symmetrically around the beam axis [36]. CHICO2 covers 68% of the solid angle around the target and provides a position resolution (FWHM) better than 1.6 • in θ (polar angle) and 2.5 • in φ (azimuthal angle) relative to the beam axis. In addition, it achieves a time resolution of 1.2 ns (FWHM), sufficient for a measurement of the time-of-flight difference, ∆T tof , between the reaction products as a function of the scattering angle, θ. This enables an event-by-event reconstruction of the reaction kinematics and a precise Doppler-shift correction of the γ rays. Figure 1 provides the resultant γray spectrum measured in coincidence with the scattered 72 Ge projectiles. The insert depicts a two-dimensional histogram of ∆T tof vs. θ demonstrating the clear separation (with a mass resolution of ∆m/m ≈ 5%) between the reaction participants. A partial scheme, incorporating all the relevant levels populated in the present measurement, (Color online) Doppler-corrected γ-ray spectrum obtained in kinematic coincidence with 72 Ge ions. Note that the 1114-keV transition belongs to 72 Ge, but has not been placed and, thus, was not included in the analysis. The insert illustrates the performance of the CHICO2 array in discriminating between the projectile and target nuclei. is provided in Fig. 2, where the black-colored transitions are those observed in the prior Coulomb-excitation measurement [34]. The resolving power and efficiency of the present experiment is illustrated by the many additional transitions given in red.
The Coulomb-excitation analysis was performed using the coupled-channel least-squares search code, gosia [38,39], which constructs a standard χ 2 function based on comparisons between the measured γ-ray yields and theoretical ones calculated from an initial set of transition and static matrix elements. Since the Coulomb cross sections depend on the relative phases as well as on the sign and magnitude of the Eλ matrix elements (ME), a random set of the MEs used as starting values in the χ 2 search were chosen to sample all possible signs of the interference term. Furthermore, all intra-band transitions within the ground-state and quasi-γ bands as well as the 2 + 3 → 0 + 2 transition were assigned positive MEs. The signs of all other MEs for the inter-band transitions were determined relative to these, and varied during the fitting process to avoid being trapped in local χ 2 minima. It should be noted that the signs of these MEs, i.e., intraand inter-band transition matrix elements are not experimental observables. In contrast, the signs of the static or diagonal matrix elements determine whether the deformation is prolate (negative ME for K = 0) or oblate (positive ME for K = 0).
In order to enhance the sensitivity to the MEs and to exploit the dependence of the excitation probability on the particle scattering angle, the data were partitioned into seven angular subsets: This resulted in a total of about 70 data points for the χ 2 analysis. In addition, known spectroscopic data such as lifetimes, branching and E2/M 1 mixing ratios were included as constraints of the relevant parameters during the fitting process. The final MEs and their associated errors are displayed in Table I. For the purpose of this discussion, only the relevant E2 MEs are tabulated, while the E3 ME for the 3 − 1 state is given here; i.e., I i | |M (E3)| |I f = 0.199(4) eb 3/2 . The quoted errors for all MEs were derived in the standard way by constructing a probability distribution in the space of fitted parameters and requesting the total probability to be equal to the chosen confidence limit (in this case 68.3%). These errors include the statistical and systematic contributions as well as those arising from cross-correlation effects.
The quadrupole collectivity of the low-lying states in 72 Ge and the associated shapes can be analyzed quantitatively using the model-independent invariant sum rules of Kumar [42] and Cline [43], which construct the deformation parameters [ Q 2 , cos 3δ ] from a complete set of E2 MEs determined via Coulomb excitation. The quadrupole invariants, Q 2 and cos 3δ , describe the nuclear charge distribution via rotationally-invariant scalar products of the quadrupole operators which relate the reduced E2 MEs to the quadrupole deformation parameters [43]. In this formalism, the parameters, Q and δ, have correspondence to the collective model variables, β and γ, defining the overall quadrupole deformation and axial asymmetry, respectively. The experimentally deter- −0.14 mined expectation values of the sum of products of E2 MEs, Q 2 , and the axial asymmetry quantity, cos 3δ , for the relevant states within the ground-state (gsb) and the quasi-γ bands are presented in Fig. 3. The almost constant and nonzero value of Q 2 [ Fig. 3(a)] indicates that 72 Ge is deformed in its ground-state band. In addition, the behavior of the asymmetry parameter [ Fig. 3 (c)] is consistent with a triaxially-deformed shape for this band. Figs. 3(b) and 3(d) provide evidence that a similar conclusion can be drawn for the quasi-γ band as well. Figure 4 presents the invariant sum rules analysis for the two lowest 0 + states of 72 Ge, in comparison with those of neighboring Ge isotopes. The present and the previous results are given with filled and open symbols, respectively. While it is evident from Fig. 4 that a welldefined shape transition occurs when going from mass 70 to 72; i.e., the ground-state configuration for 70 Ge becomes the 0 + 2 excitation in 74,76 Ge and vice versa for the 0 + 1 levels in the latter two nuclei, the results of the present investigation (filled symbols) indicate a more subtle transition and show that the 0 + 1 and 0 + 2 states in 72 Ge have essentially the same Q 2 value. This is in contrast to previous investigations (see open symbols for 72 Ge), partly because of the 0 + 2 → 2 + 3 ME, which was determined here for the first time, and found to contribute  significantly to the sum rule. It should be noted that, while the mean value of the shape parameter is almost the same for these two 0 + states, it does not necessarily follow that they are characterized by the same shape. In fact, the identical magnitude of the Q 2 values can be interpreted as a consequence of mixing of two distinct (or unperturbed) configurations characterizing these states, each being associated with a different shape. Within a simple two-state mixing model, the 0 + 1 and 0 + 2 states can be expressed as linear combinations of two unmixed 0 + u1,u2 basis levels with different mixing amplitudes and unique deformations [44]. Assuming no interaction between the intrinsic states; i.e., I u1 | |M (E2)| |I u2 = 0, and using the four MEs connecting the 0 + 1,2 to the 2 + 1,3 mixed states, the intrinsic, unmixed basis states, initially separated by ∆ int = 32 keV, are shifted by 330 keV with respect to one another by a mixing ME of 345 keV. The wavefunction of the mixed 0 + 1 state contains an amplitude cosθ 0 = 0.72(3) of the unmixed groundstate wavefunction, which indicates maximum mixing [cos 2 θ 0 = 0.52(4)], and is consistent with values derived in two-neutron and alpha transfer measurements [45]. In principle, the mixing strength can also be determined on the basis of the known E0 monopole strength between the two 0 + states [ρ 2 (E0; 0 + 2 → 0 + 1 ) × 10 3 = 9.18(2) in 72 Ge] [46]. This would however, require knowledge of the difference in the mean-square charge radii of the two configurations which is at present unavailable. Thus, while the simple two-state mixing calculations describe the mixing between configurations of the 0 + states reasonably well, ambiguities associated with the influence of the highly-mixed 2 + "subspace" (comprising the 2 + 1 , 2 + 2 , and 2 + 3 states), as evidenced by the various decay branches and magnitudes of the linking MEs (see Tab. I), remain to be addressed. Hence, a complete description requires, at the minimum, a three-band mixing calculation that takes into account the contributions of the 2 + states with, in addition, the inclusion of triaxiality, as indicated by the cos 3δ analysis of Fig. 4.
In order to quantify the role of triaxiality and present a better understanding of shape coexistence in the lowlying states, a revised version of the triaxial rotor model with independent inertia and electric quadrupole tensors [47][48][49][50][51] was applied to the newly deduced E2 MEs of 72 Ge. This version is a departure from the standard use of irrotational flow moments of inertia; e.g., the Davydov-Filippov rotor model [52]. Using this model, the E2 MEs for states within the ground and quasi-γ bands were calculated [53] with a minimum set of assumptions and compared to the experimental results. The three model parameters required to describe the E2 MEs of a triaxial rotor include the deformation, Q 0 , the asymmetry of the electric quadrupole tensor, γ, and the mixing angle of the inertia tensor, Γ. These parameters are determined analytically in this study from the 0 1 ||M (E2)||2 1 , 0 1 ||M (E2)||2 2 , and 2 1 ||M (E2)||2 1 experimental MEs (cf. Ref. [49]) and result in Q 0 = 1.45 eb, γ = 27.0 • , and Γ = −23.2 • .
The results of triaxial rotor model calculations [53], designated as (TRM), are compared with the data in Figs. 5 and 6. For completeness, calculations for a symmetric rotor (Symm) and triaxial rotor with irrotational flow moments of inertia (DF) [52] are also included. All three versions of the rotor model equally reproduce the ground [ Fig. 5(a)] and quasi-γ intra-band transitions [ Fig. 5(b)]. Beyond this point, however, the symmetric rotor (Symm) has little to no value in describing the data. For the quasi-γ to ground inter-band transitions, the TRM and DF calculations appear to perform equally well, predicting large values when the experimental values are large and likewise when they are small (see Fig. 6). However, there is a consistent failure to reproduce the 6 1 ||M (E2)||4 2 and 4 1 ||M (E2)||6 2 MEs (see indices 8 and 11 in Fig 6); these ∆I = 2, ∆K = 2 transitions have been shown to be very sensitive to interference effects [49,51]. With the exception of the 4 2 state, perhaps the most important outcome is the ability of the TRM calculations to reproduce the static quadrupole moments; i.e., the diagonal E2 MEs I i ||M (E2)||I i . This is presented in Figs. 5(c) and 5(d), where both the experimental I i ||M (E2)||I i values and the general trend with spin are in good agreement with expectations of a triaxial rotor. Configuration mixing between two triaxial rotor models was utilized [54] to investigate further the nature of the first excited 0 + 2 level and the 2 + 3 state possibly built upon it, and to determine their impact on the E2 MEs of the ground and quasi-γ bands. A simple approach was adopted to extract the model parameters without any effort to further adjust the parameters for better agreement. This approach was to factorize the solution into two 2 × 2 subspaces in order to solve the ∆K = 2 triaxial mixing (see above) and ∆K = 0 configuration mixing. Then, using the parameters from these solutions, the E2 MEs for the full 4 × 4 space were calculated. The ∆K = 0 configuration mixing was solved using the 0 1 ||M (E2)||2 1 , 2 1 ||M (E2)||0 2 , 0 1 ||M (E2)||2 3 , and 0 2 ||M (E2)||2 3 experimental MEs. The adopted parameters for the configuration mixing are θ I=2,∆K=0 = 21.1 • . From the ∆K = 0 configuration mixing alone, the MEs 2 1 ||M (E2)||2 1 , 2 3 ||M (E2)||2 3 , and 2 1 ||M (E2)||2 3 are predicted to be 0.66, -0.28, and -0.17 eb, respectively, but the corresponding experimental values are -0.16, -0.02, and 0.24 eb. Hence, ∆K = 0 configuration mixing alone is not sufficient to describe the data and suggest including ∆K = 2 triaxial mixing.
Finally, the sum of 2 + quadrupole moments provides another interesting perspective on triaxiality and shape coexistence. This sum over a complete basis or subspace should be zero [55]. For a triaxial-rotor model, there are only two 2 + states, whose quadrupole moments must be equal and opposite. For 72 Ge, 2 1 ||M (E2)||2 1 = −0.16( +7 −2 ) and 2 2 ||M (E2)||2 2 = 0.179( +3 −6 ), yielding a sum of 0.02( +8 −6 ). If the sum is made over the three 2 + states observed in the present study, where 2 3 ||M (E2)||2 3 = −0.02( +2 −14 ), the result is 0.00( +8 −15 ). Assuming that the 2 + 3 state is part of a second triaxial rotor configuration, there must then be another 2 + level with a diagonal E2 ME equal to 0.00( +8 −15 ), consistent with γ 2 = 30 • . It can thus be concluded that the quadrupole moment sum is consistent with shape coexistence between two triaxial structures, but it is also consistent with other scenarios; e.g., shape coexistence between triaxial and spherical structures. It can also be argued, based on the magnitude and the general pattern of the MEs linking the 2 + 3 level to the lower-lying states, that this state may not be directly associated with the 0 + 2 level, as has often been assumed. In fact, the TRM calculations account for only ∼ 32% of the measured B(E2) value for this state. The introduction of gamma softness, perhaps resulting from a two-phonon character, may be required to account for the remaining strength. Further work will be required to clarify this issue.
In summary, the collective properties of the groundstate and quasi-γ bands as well as the first excited 0 + and 2 + 3 states in 72 Ge have been investigated in a modelindependent way using sub-barrier multi-step Coulomb excitation. The model-independent shape invariants obtained from the Kumar-Cline sum rule analysis of the low-spin structure provide compelling evidence for the coexistence of two triaxially-deformed configurations associated with the 0 + 1 and 0 + 2 states. Within a two-state mixing model, the extracted matrix elements agree with this shape coexistence interpretation, but also require maximum mixing between the wavefunctions of the first two 0 + states. The results of multi-state mixing calculations performed within the framework of the triaxial rotor model demonstrate the importance of the triaxial degree of freedom and indicate that the low-spin structure of 72 Ge can be adequately described by mixing of two triaxial rotors. This result reveals that a nucleus, apparently unlike any other known collective one, possesses a simple underlying structure. This interpretation, by way of coexistence of different shapes, supports structural simplicity when the phenomenology appears complex.
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Of-