Intruder band mixing in an ab initio description of 12Be

The spectrum of 12Be exhibits exotic features, e.g., an intruder ground state and shape coexistence, normally associated with the breakdown of a shell closure. While previous phenomenological treatments indicated the ground state has substantial contributions from intruder configurations, it is only with advances in computational abilities and improved interactions that this intruder mixing is observed in ab initio no-core shell model (NCSM) predictions. In this work, we extract electromagnetic observables and symmetry decompositions from the NCSM wave functions to demonstrate that the low-lying positive parity spectrum can be explained in terms of mixing of rotational bands with very different intrinsic structure coexisting within the low-lying spectrum. These observed bands exhibit an approximate SU(3) symmetry and are qualitatively consistent with Elliott model predictions.


Introduction
A breakdown of the N = 8 shell closure in neutron-rich 12 Be is supported by both experimental  and theoretical  evidence.The compressed 0 + 1 to 2 + 1 level spacing [1,45], larger proton radius [19], and higher B(E2; 2 + 1 → 0 + 1 ) value [16,22], in comparison with neighboring 10 Be, all point to a significantly deformed intrinsic state, not the spherical shape expected at a shell closure.Shell model and cluster molecular orbital descriptions, as well as spectroscopic factors, all indicate the 12 Be ground state is an admixture of 0ℏω (normal) configurations with a filled neutron p shell and 2ℏω (intruder) configurations with two neutrons promoted to the sd shell [8][9][10][11][26][27][28].Although the specific admixture is model dependent, the largest contribution is consistently identified to be 2ℏω, i.e., the ground state is an intruder state.
Efforts to understand the structure of these low-lying states in 12 Be have largely been restricted to phenomenological models.In this work, we focus on understanding the low-lying spectrum of 12 Be from an ab initio perspective.Specifically, we use the no-core shell model (NCSM) framework [46] in which energies and wave functions are obtained by solving the non-relativistic Schrödinger equation in a basis of antisymmetrized products of harmonic oscillator states 1 .Early ab initio calculations of 12 Be were computationally limited to model spaces insufficient to reproduce the intruder nature of the ground state [40,41].However, here, by combining computational advances with use of the Daejeon16 internucleon interaction [50] based on chiral Intruder states are thought to be a result of competition between shell structure and particle correlations [51][52][53][54].Near a shell closure, normal configurations have little correlation energy, while intruder configurations can achieve a much larger correlation energy through deformation.Thus intruder states are expected to have highly deformed intrinsic states relative to other nearby normal states (shape coexistence), which often results in intruder rotational bands with large moments of inertia.In the low-lying NCSM-calculated spectrum presented in this work, such intruder bands are observed along with a normal band built on the excited 0 + state.Similar bands were observed in previous theoretical investigations [39,41].In this work, the intrinsic shapes of these bands are probed by calculating proton and neutron radii and quadrupole moments.
However, the simple description of shape-coexistent rotational bands is insufficient to describe the low-lying spectrum of 12 Be.Transitions between bands with markedly different intrinsic shape are expected to vanish [55,56].Thus the measured 0 + 2 → 0 + 1 E0 transition [14] can be taken as an indication of mixing [55].To gain insight into the mixing of the states as well as extract properties of the pure rotational bands, we apply a two-state mixing analysis to the calculated spectrum.We   demonstrate that two-state mixing combined with the rotational picture well describes the low-lying spectrum of 12 Be.In light nuclei, where intrinsic shape is often not sharply defined, the assumption of vanishing transitions between states with different shape is not as well motivated as in heavier systems.However, as we demonstrate in this work, the vanishing interband transitions can alternatively be understood in the context of an emergent approximate symmetry, specifically Elliott's SU(3) symmetry [57][58][59][60][61][62][63][64][65][66][67][68] which is tied to both nuclear rotation and deformation as well as microscopic correlations.We will demonstrate that the rotational bands exhibit this approximate symmetry and discuss the consequences for transition strengths.
In this work, we first present the NCSM calculated spectrum and identify emergent rotational bands (Sec.2).We then apply the two-state mixing model to "un-mix" the rotational bands and extract information about the intrinsic states of the pure rotational bands (Sec.3).Finally, we interpret the NCSM results in the context of Elliott's SU(3) framework (Sec.4).

Intruder band in 12 Be
Rotational states are characterized by a deformed intrinsic state rotating in the lab frame.An intrinsic state which is rotationally symmetric about one of the principal axes is labeled by K, the projection of angular momentum J onto the symmetry axis in the body-fixed frame.Rotational band members, i.e., states with the same intrinsic state but different angular momenta, are identifiable as states connected by E2 transitions enhanced relative to single particle estimates.Band members have characteristic energies given by  where I is the moment of inertia and E 0 is the energy intercept.
Rotational bands emerge in the low-lying spectrum of 12 Be, as shown in Fig. 1.Excitation energies (symbols) are plotted versus J on an axis that is scaled by J(J + 1) so that band members lie along a straight line.Gray lines between states denote E2 transitions.Thickness and shading are proportional to the transition strength.Three rotational bands are identified: two K = 0 bands (red diamonds and blue circles) and a K = 2 band (gold hexagons).Fits to the rotational energy formula are indicated by the dashed lines.Excitation energies of the yrast K = 0 band are in reasonable agreement with experimental values (green lines) [1] for the 0 + 1 , 2 + 1 and (probable) 4 + 1 . 2The calculated bandhead of the yrare K = 0 band lies just above the experimental 0 + 2 state.However, the calculated 2 + member of the yrare band over-predicts the possible 2 + state at 4.590 (5) MeV by about 1 MeV. 3 The bandhead of the K = 2 band is in good agreement with a probable 2 + state observed at 7.2(1) MeV.
With access to the underlying calculated wave functions, we can probe the structure of the band members.By decomposing the wave functions by number of excitation quanta N ex , we demonstrate that all of the members of the yrast band (red diamonds) are intruder states.In the NCSM, wave functions are expanded in terms of configurations, i.e., distribution of particles over oscillator shells, with N ex up to some cutoff N max .We classify a state as "normal" if the largest single contribution to the wave function is N ex = 0 and as "intruder" otherwise.To classify the bands in Fig. 1, wave functions of the band members are decomposed by N ex (Fig. 2).Each of the states in the K = 0 yrast band members have a largest contribution from N ex = 2 configurations [Fig.2(a)].Thus the states form an intruder band.In contrast, the two states forming the yrare K = 0 band (blue circles) are normal states, i.e., with largest contribu-tion from N ex = 0 configurations [Fig.2(b)].For the remainder of this paper, we will label the K = 0 bands as intruder (i.e., K = 0 + intr ) and normal (i.e., K = 0 + norm ).The K = 2 band (not shown in Fig. 2) is also an intruder band.
Under the adiabatic assumption, the energy scale for rotational excitations is small compared to the energy scale for intrinsic excitations.Thus the intrinsic structure is the same for all members of a band. 4That the members of each band have similar N ex decompositions (Fig. 2) is approximately consistent with this assumption.
The intrinsic state is also assumed to have a well-defined quadrupole shape.Here we consider the proton and neutron radii and intrinsic quadrupole moments Q 0 , which characterize the quadrupole shape of the intrinsic state; these properties are related to the quadrupole deformation by β ∝ Q 0 / ⟨r 2 ⟩.Since the r 2 operator is scalar, a radius calculated in the lab frame can immediately be identified with a radius in the body-fixed frame.However, the intrinsic quadrupole moment can only be obtained indirectly from quadrupole moments and B(E2) values.For a symmetric rotor both these observables are proportional to the same intrinsic quadrupole moment, where the proportionality factors are given in terms of Clebsch-Gordan coefficients and angular momentum dimension factors [69]: For an ideal rotor, the radii of members of a rotational band are expected to be constant.As shown in Fig. 3, radii within each band are indeed similar in size, but values are not constant.Although the values are not converged with respect to N max , proton and neutron radii in the intruder band members [Fig.3(a,b)] have a clear dependence on angular momentum J.In the rotational framework, the increase in radius with J could be attributed to centrifugal stretching.In the normal band [Fig.3(c,d)], there is again an angular momentum dependence.However, here, both the proton and neutron radii decrease with increasing J.
Similarly, the Q 0 extracted from the spectroscopic quadrupole moments and intraband E2 transitions should also be constant.In Fig. 4, the (a) proton and (b) neutron intrinsic quadrupole moments of the intruder band are extracted from the quadrupole moments of the 2 + intr and 4 + intr states as well as the 2 + intr → 0 + intr and 4 + intr → 2 + intr B(E2) values.For the normal band, (c) Q 0,p and (d) Q 0,n are obtained from the quadrupole moment of the 2 + norm state and the 2 + norm → 0 + norm B(E2) value.Though there are some small discrepancies in the values obtained for Q 0,p in either band, the values overall appear to be comparable and thus consistent with rotor model expectations.There is, however, a larger spread in the values extracted for Q 0,n within each band, but the values do appear to be converging towards a more similar value by N max = 12.While an approximate rotational picture emerges in the 12 Be spectrum, the spread in radii and intrinsic quadrupole moments within each band suggests that the simple picture provided by the rotational model is incomplete.An additional discrepancy is noted in the energies of the yrast band.As shown in Fig. 1, the 0 + intr lies almost 1 MeV below the rotational energy (red dashed line) obtained by fitting (1) to the 4 + and 6 + members of the intruder band. 5However, as we demonstrate in the following section, deviations from rotational expectations can largely be understood as a consequence of mixing [70] of same-J members of the two rotational bands.

Band mixing
Deviations from rotational expectations can largely be understood as a consequence of mixing [70,71] between same-J members of the two K = 0 bands.To gain insight into the mixing of these states, we apply a two-state mixing model in which we assume that the proton E0 transitions between pure intruder and normal states vanish.
Our assumption that the E0 transitions between the pure states vanish is motivated by the observation that the radii and quadrupole moments of the two bands differ.As shown in Fig. 3(a,c), the proton and neutron radii of the 0 + and 2 + intruder band members are larger than those of the corresponding normal band members at each N max .Moreover, the Q 0,n of the intruder band is significantly larger than that of the normal band, implying a significant difference in neutron quadrupole deformation and thus in the intrinsic shape.In the limit where the intrinsic state is assumed to have definite shape (and is thus an eigenstate of r 2 ), matrix elements of the E0 operator between bands with different intrinsic shapes must vanish.
Requiring the E0 transition between the pure states to vanish allows us to extract a mixing angle θ, which can be used to extract values for energies, radii, and electromagnetic transitions and moments for the pure rotational bands.The mixing angle between two states |ψ 1 ⟩ and |ψ 2 ⟩ is then forced to be At N max = 12, the mixing angles between the two 0 + states and two 2 + states are θ 0 + = 26.3• and θ 2 + = 11.2 • , respectively.As shown in Fig. 5, the degree of mixing is highly N maxdependent.The fractions P(pure) of the (left) 0 + intr and (right) 2 + intr state, coming from the pure intruder (red diamonds) and pure normal (blue circles) states, respectively, are shown in Fig. 5(a).At N max = 6, the 0 + intr state is approximately 70%  1 ) [1] given by green square in panel (a).Estimate for r p (0 + 2 ) inferred from known experimental quantities (Sec.3) given by purple square in panel (c).Note error bar on estimated r p (0 + 2 ) includes only experimental errors and does not account for error arising from assumptions made in estimation.pure intruder and 30% pure normal.Then at N max = 8, the state is nearly 50% pure intruder and 50% pure normal.The fraction of the state which is pure normal then decreases with increasing N max .Note that, since the mixing of the states is symmetric, when the 0 + intr state is 70% pure intruder and 30% pure normal, the 0 + norm is 70% pure normal and 30% pure intruder.The 2 + states follow a similar evolution with N max .By N max = 12, the 2 + states are almost entirely pure states.
Much of the N max dependence of the mixing is an artifact of levels crossing as energies of the band members converge with N max .Though excitation energies within a band, and thus the moment of inertia, appear to be well-converged even at low N max [66], different bands converge at different rates.The energies of the 0 + and 2 + members of the K = 0 bands are shown in  Although both the E0 moment of the ground state (proportional to [r p (0 + 1 )] 2 ) and the 0 + 2 → 0 + 1 E0 transition strength are measured [14,19], an experimental value for the mixing angle θ exp 0 + cannot be obtained as above, since the proton E0 moment of the 0 + 2 state is not measured.However, if we take the 2 + states to be essentially unmixed (which the NCSM calculation suggests is reasonable), we can deduce an approximate value for θ exp 0 + from the known E2 transition strengths [1].Specifically, tan , and thus θ exp 0 + ≈ 18 • ± 2 • .The corresponding fraction of the physical states which come from the pure intruder state are shown in Fig. 5(a) (purple symbols labeled as Exp).Although the calculated θ 0 + [and thus P(pure) for the 0 + intr state] is not converged with respect to N max , it appears to be converging towards a value that is reasonably consistent with θ exp 0 + .Using the approximate experimental mixing angle, we can then extract an approximate value for r p (0 + 2 ) by inverting (4).Combining θ exp 0 + with the measured ground state proton radius r p (0 + 1 ) and measured E0 transition between the 0 + states, we obtain the approximate value r p (0 + 2 ) = 2.26 fm, which is shown in Fig. 3(c) (purple circle).This approximate r p (0 + 2 ) is slightly smaller than the calculated r p (0 + norm ).Like the E0 transition, E2 transitions between pure band members with different intrinsic shapes are expected to vanish.Calculated E2 transitions strengths are shown in Fig. 6 between the (filled) mixed and (open) pure states.Though not converged, the calculated 2 + intr → 0 + intr and 2 + intr → 0 + norm transition strengths are reasonably consistent with experimental values (green squares).The impact of the mixing on the interband 2 + → 0 + transition for both the intruder and normal bands is small, in part because the matrix elements of M p (E2) are similar in value at N max = 6 and 8 where the mixing is largest.The effect on the interband transitions (purple x's) is more notable.Transitions between the pure states with different intrinsic shape are expected to vanish.As shown in Fig. 6, the transitions between the pure bands (open purple crosses) are highly suppressed relative to the transitions between the mixed states (filled symbols).Neutron interband transitions between pure states (not shown) are also highly suppressed.Note that the 2 + 2 → 2 + 1 transition strength is converging towards that between the pure states, providing further evidence that the 2 + states are nearly pure states by N max = 12.
With the two-state mixing picture established, we turn our attention to the impacts of mixing on the observables discussed above (Figs.1-5).Most of the discrepancies from rotational expectations noted in the previous section can be understood as resulting from mixing of the pure bands.
As noted in Sec. 2, the energy of the 0 + intr state [Fig. 1 (red filled symbols)] is pushed down relative to the rotational energies (red dashed line) obtained by fitting the rotational energy formula (1) to the 4 + and 6 + states.This difference in energy between the 0 + intr state and the rotational prediction is largely a result of two-state mixing; level repulsion pushes the 0 + intr down in energy relative to the pure state, while pushing the 0 + norm state up.The energy of the pure 0 + intr state (red open symbol) is much closer to the rotational energy.
Notice that the level repulsion between the 0 + states creates an illusion regarding the moments of inertia of the bands.If one naively interprets the energy difference between the (mixed) 0 + and 2 + states as a measure of the rotational moment of inertia, then it appears as though the the bands have near identical moments of inertia.In contrast, the slope of the rotational energy fit to the pure normal states (blue dashed line) is more than 1.5 times larger than the slope of the rotational energy fit for the intruder band (red dashed line fitted to the 4 + and 6 + members), which translates into a moment of inertia which is 1.5 times smaller for the normal band than for the intruder band.
Both the radii and intrinsic quadrupole moments within each band are more congruous, for the pure states, with the expectation that these properties be constant for all rotational band members.As shown in Fig. 3(e,g), the proton radii of both bands are nearly constant within a band.Similarly, while the neutron radii of both bands [Fig.3(f,h)] do still have a J dependence, the effect is much smaller.The intrinsic quadrupole moments of the pure states [Fig.4(e-h)] are also more consistent with the rotational expectations.In particular, there is very little difference among the Q 0,n values within each pure band.The exception to this pattern is the Q 0,p of the pure normal band.As shown in Fig. 4(g), the variation in Q 0,p values extracted from the quadrupole moment Q(2) and B(E2) of the pure states is larger.The difference in extracted Q 0,p may be an indication that the two pure normal states do not form a well-defined rotational band, even though the E2 transition between the states is  [16,22] and 2 + intr → 0 + norm [14] transitions. enhanced.
The differences in radii between the pure intruder and pure normal states are also much more pronounced.Most notable is the difference in neutron radii of the 0 + states: whereas the neutron radii of the mixed 0 + states [Fig.3(b,d)] are very similar in value, the neutron radius of the pure 0 + intr state [Fig.3(f)] is significantly larger than that of the pure 0 + norm state [Fig.3(h)].By N max = 12 both the proton and neutron radii of the pure intruder band are more than 0.1 fm larger than the corresponding radii of the pure normal band.Note that, because the E0 transitions are assumed to vanish between pure states, the radii of the mixed states can be interpreted as weighted averages of the radii of the pure states, with weights determined by the mixing angle.
The large difference in Q 0,n of the pure bands mirrors the difference in the moments of inertia of the two pure bands and thus suggests that the difference in moment of inertia reflects and underlying change in neutron intrinsic structure.The Q 0n of the pure normal band [Fig.4(h)] is close to zero, as one would expect for a nucleus with a closed neutron shell.This small Q 0,n translates to a much smaller moment of inertia than that of the pure intruder band, as shown in Fig. 1.
The pure states are also more distinctly intruder and normal in their N ex content.Decompositions by N ex of the pure states are obtained by first un-mixing the calculated wave functions of the 0 + and 2 + states.The wave functions of the pure states can then be decomposed by N ex in the same manner as the wave functions of the mixed states.The N ex decompositions are shown in Fig. 2 for the (c) pure intruder and (d) pure normal band members.The pure 0 + intr and 2 + intr states look more "intruder-like."The N ex = 0 contributions to the mixed intruder states [Fig.2(a)] nearly vanish in the pure states.In addition, the decompositions of the members within a band are all nearly identical.
Combining the rotational picture with two-state mixing provides a reasonable description of the low-lying positive parity spectrum of 12 Be.In this section, the assumption that the E0 transition vanishes between pure states is motivated by the assumption that the intrinsic state of the rotational bands is an approximate eigenstate of the r 2 operator.In the following section, we demonstrate that vanishing transitions can also be motivated by the emergence of an approximate SU(3) symmetry, without assuming rotational structure or well defined intrinsic shape.Though the SU(3) symmetry is only approximate, it does provide selection rules on the transition operators which support enforcing vanishing interband transitions.

Emergence of Elliott's SU(3) picture
Elliott's SU(3) rotational framework provides a link between microscopic correlations and nuclear rotation and deformation.In this framework, there is a rotational intrinsic state which has definite SU(3) symmetry with quantum numbers (λµ) [57,58].In the limit of large λ and µ, (λµ) can be associated with the nuclear deformation parameters β and γ, with larger values of λ and µ corresponding to a more deformed intrinsic shape [73,74].Microscopically, each particle in a harmonic oscillator configuration has SU(3) quantum numbers (λµ) = (N, 0), where N is the oscillator shell number for the given particle.A many-body state with definite total (λµ) can be obtained by coupling together SU(3) quantum numbers of each particle according to SU(3) coupling rules.This SU(3) state also has definite total spin S obtained by coupling the spins of each particle together according to angular momentum coupling rules.
Elliott's framework captures the competition between shell structure and correlation energy.In this framework, the model Hamiltonian is typically given by Ĥ = Ĥ0 − χ Q • Q, where H 0 is the harmonic oscillator Hamiltonian, χ is a strength parameter, and Q is an SU(3) generator which is closely related to the physical (mass) quadrupole operator but cannot move a nucleon between oscillator shells [51,59,75].The first term, H 0 , gives rise to shell structure, while the (negative) quadrupole-quadrupole term gives preference to more deformed nuclear states.
Traditionally, Elliott's model was restricted to the shell model valence space, which maps implicitly onto the N ex = 0 subspace.However, such a picture cannot describe intruder states.Thus, to describe 12 Be, we extend the framework to include SU(3) states in either the N ex = 0 or the N ex = 2 subspaces.Within each of these subspaces, the largest deformations correspond to SU(3) many-body states with quantum numbers N ex (λµ)S = 0(2, 0)0 and 2(6, 2)0, respectively.Qualitatively, the very large deformation associated with a 2(6, 2)0 many-body state, as compared with that of a 0(2, 0)0 state, overcomes the (positive) harmonic oscillator energy required to excite particles out of the valence space, bringing the J = 0, 2 intruder band members below the normal states.
The Elliott model Hamiltonian gives rise to rotational bands.The Q • Q term in the Hamiltonian can be re-expressed in terms of the SU(3) Casimir operator ĈSU(3) and the orbital angular momentum operator L2 .The Hamiltonian then becomes Ĥ = [ Ĥ0 −χ ĈSU(3) ]+3χ L2 , where the eigenvalue of H 0 −χ ĈSU(3) corresponds to E 0 in (1) 6 and the eigenvalue of 3χL 2 corresponds (for S = 0) to the J(J + 1) term in (1).Unlike the simplified rotational picture presented in Sec. 2, the SU(3) intrinsic states are not presumed to be symmetric about any of the principal axes.Thus more than one band, with different K quantum numbers, can be projected out from the same SU(3) intrinsic state.
To identify the SU(3) content of the intrinsic states of the bands, we decompose the wave functions of the band members into contributions from subspaces with definite N ex (λµ)S . 7ecompositions of the 0 + and 2 + band members into different N ex (λµ)S contributions are shown in Fig. 7.Only subspaces contributing ≥ 5% to at least one state are shown.As expected, for both of the intruder states [Fig.7(a,b)], the largest contribution is from the subspace labeled by N ex (λ, µ)S = 2(6, 2)0.(This is also true for the K = 2 band, not shown.)Also as expected, the largest N ex (λ, µ)S contribution to the normal K = 0 band members [Fig.7(c,d)] comes from the 0(2, 0)0 subspace.These decompositions confirm that the bands shown in Fig. 1 are consistent with those expected in the Elliott rotational framework.
Of course, Elliott's SU(3) symmetry is only approximate [61,62,[64][65][66][67][68].The largest single N ex (λ, µ)S contribution in both bands is less than 50%, with the remaining probability fragmented over many other subspaces with different number of N ex .However, a substantial part of the fragmentation is due to the two-state mixing; Fig. 7(e-h) shows the decompositions for the pure states.In the pure intruder band [Fig.7(e,f)], the decompositions of the 0 + intr and 2 + intr states are nearly identical.For the normal band [Fig.7(g,h)], contributions arise with the same quantum numbers, but their relative magnitudes vary.Notably, the 2 + norm state [Fig.7(g)] has a more significant S = 1 contribution from states labeled by 0(0, 1)1, which may indicate a weakening of an underlying 2α cluster structure.This difference in decompositions also provides context for the differences in the Q 0,p values extracted for the pure normal band in Fig. 4(g).We note that, in both the pure normal and pure intruder states, much of the remaining fragmentation over N ex comes from subspaces labeled by quantum numbers which would be consistent with symplectic excitations [65,67,[80][81][82][83][84].
Despite SU(3) symmetry breaking, there is no significant overlap in the distributions for the pure intruder band and for the pure normal band.The stark difference in SU(3) content of the two bands provides a more microscopically based interpretation of why interband transitions between the pure bands vanish.Both the E0 and E2 transition operators can be expressed as a linear combination of SU [64,85], where b † and b are the boson creation and annihilation operators.From SU(3) and spin selection rules on each tensor term, it can be shown that the matrix elements of either operator vanish between any subspace that contributes ≥ 5% to the pure intruder band and any that contributes ≥ 5% to the pure normal states.

Conclusion
In this work we have investigated the underlying structure of the low-lying positive parity states of 12 Be.As we demonstrate, the intruder nature of the lowest lying state emerges ab initio in the NCSM framework, without any assumptions of, e.g., shell structure, clustering, or symmetry.With the calculated energies, radii and electromagnetic transitions in reasonable agreement with experiment, the calculated wave functions can now be used to probe the underlying structure of the low lying spectrum of 12 Be.
Within the ab initio calculated spectrum for 12 Be, signatures of nuclear rotations emerge.Rotational bands are identified as states connected by enhanced E2 transitions with energies approximately consistent with characteristic rotational energies.In particular, a K = 0 intruder band built on the ground state and a normal K = 0 band built on the first excited 0 + appear.However, a simple symmetric rotor model is insufficient to describe the 12 Be spectrum.The 0 + intr energy deviates from the rotational J(J + 1) energy relation, while radii and intrinsic quadrupole moments are inconsistent within each band.Decompositions of band members by N ex and SU(3) symmetry also highlight inconsistencies in the intrinsic structure within each band.
Band mixing can explain the discrepancies between the NCSM-calculated observables and the rotational picture.The low-lying spectrum can thus be described in terms of mixing between pure bands with very different intrinsic structure, namely an intruder band and a normal band.By assuming that the proton E0 transitions between the pure bands vanish, we deduce a mixing angle and use it to extract properties of the pure bands from the NCSM-calculated observables.The (extracted) observables, e.g., radii and intrinsic quadrupole moments, associated with the pure bands, as well as the energy of the pure 0 + intr state, are significantly more consistent with rotational model expectations.Moreover, N ex and SU(3) symmetry decompositions are more constant within each band.
Both of the K = 0 bands as well as the K = 2 intruder band exhibit an approximate SU(3) symmetry.Within each band, the largest SU(3) contribution comes from the same SU(3) subspace, notably the SU(3) subspace associated with the largest deformation in the corresponding N ex subspace.Moreover, the angular momenta of the band members are exactly those expected in Elliott's rotational model for an intrinsic state with quantum numbers corresponding to that largest contributing symmetry subspace.Although the SU(3) symmetry is only approximate, the pure states have notable contributions from only a few SU(3) subspaces.Much of the apparent fragmentation of SU( 3) is instead a result of the two-state mixing.
The mixing framework applied in this work assumes E0 transitions between the pure bands vanish.This assumption is typically motivated by the argument that the E0 operator cannot connect states with very different intrinsic shape.In light nuclei intrinsic shape is often not well defined.However, the stark difference in the SU(3) content of the pure bands provides a microscopic explanation for vanishing E0 interband transitions: selection rules forbid E0 transitions between any of the SU(3) subspaces contributing significantly to the intruder band members and any of the SU(3) subspaces contributing significantly to the normal band members.
Thus, a remarkably simple picture emerges from the ab initio calculated spectrum, for which the only input was the internucleon interaction.The low lying spectrum of 12 Be can be described as mixing of a K = 0 intruder band and a K = 0 normal band with very different intrinsic structure.

Figure 1 :
Figure1: Calculated spectrum of12 Be with ℏω = 15 MeV at N max = 12.The J axis is scaled by J(J + 1) but labeled by J. Lines connecting states denote E2 transitions.Thickness and shading correspond to transition strength.Horizontal green lines indicate experimental energies[1].Parentheses indicate J π assignment is only tentative.Open symbols indicate excitation energies of pure states (see Sec. 3).Dashed lines indicate rotational energy fit to pure band members.

Figure 2 :
Figure 2: Decomposition by N ex of wave functions of representative members of the (a) K = 0 yrast band and (b) K = 0 yrare band identified in Fig. 1 (Sec.2).(c,d) Decomposition by N ex of pure wave functions (Sec.3) of representative members of the same bands, respectively.Decomposition by N ex is trivially obtained by summing probability of harmonic oscillator configurations with given N ex .

Figure 3 :
Figure 3: Proton and neutron radii with respect to N max of representative members of the (a,b) K = 0 intruder, (c,d) normal, (e,f) pure intruder and (g,h) pure normal bands.All values shown for ℏω = 15 MeV.Experimental value for r p (0 +1 )[1] given by green square in panel (a).Estimate for r p (0 + 2 ) inferred from known experimental quantities (Sec.3) given by purple square in panel (c).Note error bar on estimated r p (0 +2 ) includes only experimental errors and does not account for error arising from assumptions made in estimation.

Figure 4 :
Figure 4: Proton and neutron intrinsic quadrupole moments of the (a,b) K = 0 intruder, (c,d) normal, (e,f) pure intruder, and (g,h) pure normal bands with respect to N max as extracted from the calculated quadrupole moments and B(E2) values.All values shown for ℏω = 15 MeV.

Figure 5 :
Figure 5: (a) Fraction of the (left) 0 +intr and (right) 2 + intr wave function coming from the pure intruder state (red diamonds) and pure normal state (blue circles), given by cos 2 θ and sin 2 θ, respectively.Fractional contributions corresponding to an approximately deduced experimental mixing angle are indicated by purple symbols in the Exp.column.See Sec. 3 for details.(b) Convergence of NCSM calculated energies of the 0 + and 2 + band members (filled) and energies of the pure band members (open) with respect to N max .Experimental values given by horizontal green line[72].

Fig. 5 (
Fig. 5(b).At N max = 6 the members of the normal band (blue circles) are lower in energy than those of the intruder band (red diamonds).At N max = 8, where the states are maximally mixed, the pure states (open symbols) are nearly degenerate.Within increasing N max the pure states move further apart in energy and the mixing correspondingly decreases.Although both the E0 moment of the ground state (proportional to [r p (0 + 1 )] 2 ) and the 0 + 2 → 0 + 1 E0 transition strength are measured[14,19], an experimental value for the mixing angle θ

Figure 7 :
Figure 7: Decompositions of wave functions into contributions labeled by N ex (λµ)S for the (a,b) intruder, (c,d) normal, (e,f) pure intruder, and (g,h) pure normal 0 + and 2 + states.Only those subspaces contributing ≥ 5% to at least one of the states are shown. 12 fm 2 intrFigure 6: Convergence with respect to N max of intraband (top) and interband (bottom) E2 transition strengths, shown as the magnitude of the reduced matrix element.Experimental values (green squares) are given for 2 + intr → 0 + intr