Existence of core excited 8 Be ∗ = α + α ∗ cluster structure in α + α scattering

...................................................................................................................Weshowtheexistenceofthe α + α ∗ cluster structure at the highly excited energy around E x = 20 MeV in 8 Be for the ﬁrst time in the coupled-channel calculations. An extended double-folding model derived using a realistic precise cluster wave function with a well-developed N + 3 N cluster structure for the ﬁrst excited state of 4 He was employed. The calculation reproduces the experimental phase shifts in α + α scattering up to E c.m. = 21 MeV well. The result shows that the well-developed core-excited α + α ∗ structure appears as resonances for L = 0 and 2 near the α + α ∗ threshold which correspond to the experimental states at E x = 20.20 MeV and E x = 22.24 MeV in 8 Be.

α+α scattering and its interaction potential have been most thoroughly studied both experimentally [29][30][31][32] and theoretically [8][9][10]33,34]. The phenomenological potential with a repulsive core at around R = 2 fm for the orbital angular momentum L = 0 and 2 waves was shown to originate from the Pauli principle [8][9][10]. On the other hand, the systematic and unified study of anomalous large angle scattering (ALAS), pre-rainbow and rainbow scattering and cluster structure in the bound and quasi-bound energy region for the typical α+ 16 O, α+ 40 Ca and 16 O+ 16 O systems has shown that the interaction potentials are deep [35][36][37][38][39][40][41][42][43][44][45] due to the Pauli principle [46]. A deep potential has been considered to be favored for the prototype α + α system [46,47], although ALAS and/or rainbows have not been observed clearly and typically because of the lightest double-closed identical boson system. The phenomenological potentials with a repulsive core for α + α, such as in Ref. [48], which has been widely used historically, is now understood as a mathematically supersymmetric partner [49] of the deep potential such as in Ref. [47].
Although the potential for α + α has been established, it is to be noted that the experimental phase shifts are reproduced successfully only at the low energy region below E L = 40 MeV where inelastic PTEP 2021, 113D02 Y. Hirabayashi and S. Ohkubo channels are closed [33]. Above E L = 42.4 MeV, where inelastic channels open, the experimental phase shifts with a non-smooth complicated structure have not been explained. The coupled-channel α + α * calculations, assuming a monopole vibration for the first excited state 0 + 2 of 4 He, were attempted as early as in the 1970s [50,51]. An experimental search for the cluster structure α + α * was also done [52,53]. However, the existence of the α + α * cluster structure has not been confirmed.
The structure of 8 Be at the highly excited energy region has also had attention paid to it from the viewpoint of the cosmological lithium problem [54,55]. One of the candidates to reduce the 7 Li abundance is the 7 Be(n,α)α reaction, and the 7 Be+n channel is opened as an highly excited state of 8 Be at E x = 18.899 MeV. Because the direct 7 Be(n,α)α breakup channel is strongly suppressed and the 7 Be(n,γ α)α reaction remains the dominant process, the highly excited states of 8 Be may affect the reaction corresponding the α-emitting cross section of these states [54].
The purpose of this paper is to show the existence of the α + α * structure as a prototype of α-core excited cluster structure for the first time, in which α is excited to the 0 + 2 state at 20.21MeV, analyzing α + α scattering using the coupled-channel method with an extended double-folding model derived from the precise wave functions for 4 He. We also show how the α cluster structure appears at a highly excited energy region near the α + α * threshold. The paper is organized as follows. In Sect. 2 the extended double-folding model is presented. Sect. 3 is devoted to the coupled-channel analysis of α + α scattering and the core excited α + α * cluster structure in 8 Be. In Sect. 4 the core excited α cluster structure in 8 Be is discussed. Our conclusion is given in Sect. 5.

The extended double-folding model
An extended double-folding model that describes all the diagonal and off-diagonal coupling potentials derived from the microscopic realistic precise wave functions for 4 He is given as follows: where ρ ( 4 He) ij (r) is the diagonal (i = j) or transition (i = j) nucleon density of 4 He calculated in the microscopic four-body calculations in Ref. [56]. In the coupled-channel calculations we take into account the 0 + 1 (0.0 MeV) and 0 + 2 (20.21 MeV) states of 4 He. For the effective two-body interaction v NN we use the M3Y interaction [57]. We introduce a normalization factor, N R . The Coulomb folding potential V C ij,kl (R) is similarly calculated in the folding model of Eq. (1) by replacing v NN by v Coul .

Analysis of α + α scattering and α + α * cluster structure
In Fig. 1(a) the calculated phase shifts for L = 0, 2 and 4 in α + α scattering calculated using the coupled-channel method (solid lines) with a double-folding model are compared with the experimental data [32]. Although N R = 0.999943 is used to reproduce the experimental ground state energy of 91.84 keV from the α + α threshold and the α decay width 5.57 eV of 8 Be [58] where calculated values are 91.84 keV and 5.5 eV, the calculated phase shifts, cross sections and the discussions that come up later are not very sensitive to this fine adjustment. The results calculated in the single channel by switching off the coupling (dashed lines) are also displayed for comparison. The behavior of the experimental phase shifts for L = 0, 2 and 4 at E c.m. < 20 MeV is reproduced well by the calculations. What is noticeable in Fig. 1(a) is that the kinks at around E c.m. = 20 MeV in the experimental data, which cannot be reproduced in the single-channel calculations [47], are also seen in the present coupled-channel calculations. Thus the kinks are found to arise from the coupling to the excited 0 The irregular behavior of the calculated phase shift at around E c.m. = 8 MeV is the redundant Pauli forbidden 0s state in the α + α * channel, as discussed later. Fig. 1(b) shows the resonance behavior of the cross section at around E c.m. = 20 MeV for L = 0 due to the s-wave coupling to the J = 0 + resonance with the α + α * cluster structure in 8 Be. It shows a closed channel resonance (Feshbach resonance) [59][60][61] below the α + α * threshold, in which the relative kinetic energy between two α particles is negative due to the loss of its energy by the excitation of α to the 0 + 2 state, and the wave function of the excited α + α * state shows an exact bound state nature without scattering amplitudes at large distance (see Fig. 5a). The calculated resonance behavior of the cross section can be well reproduced by the Fano resonance [62,63] formula (Fano profile), where k and δ b are the wave number of relative motion between two α particles and the background phase at the resonance energy, respectively. This formula is reduced to the Breit-Wigner formula [64] in a limit of δ b = 0. We take δ b = −50.0 • (see Fig. 1a  and the width parameter = 32 keV(dashed line). In Fig. 1(b), the calculated cross section shows a typical Fano resonance and the asymmetric shape comes from the interference between the elastic channel as a background continuum state and the 0 + 2 channel as a discrete excited state. Although the α width seems to be small compared with the experimental 0 + 2 state of 8 Be at E x = 20.20 MeV with the width about 350 keV [53,58], it may be due to the effect of the open channels such as p+ 7 Li and n+ 7 Be in 8 Be which are not taken into account in the present coupled-channel calculations. The coupling with states at higher energies such as the 1 − (T = 0) at 24.250 MeV, which has a well-developed N +3N structure as a parity doublet partner [65] of the 0 + 2 state at 20.210 MeV of 4 He, is also not included in the present framework. These states could be an origin of the wiggles in the highly excited states. A more involved coupled-channel calculation including higher excited states would be a future subject of study. On the other hand, the positions of the resonance energy are rather sensitive to the strength of the coupling potential.
In Fig. 2 the diagonal potentials for the α + α and α + α * channels are displayed. The diagonal nuclear potential for the α + α channel is similar to that in Ref. [47]. This potential accommodates the deeply bound redundant Pauli forbidden 0s and 1s states for L = 0 (Fig. 2a)   for L = 2. The Pauli forbidden states play a role in damping the amplitudes of the physical wave functions in the internal region, R < 2 fm, which corresponds to the core radius of the L-dependent shallow potential with a repulsive core in Refs. [8][9][10]. The gross behavior of the phase shifts is essentially reproduced by the diagonal α + α potential. The potential for α + α * is shallow, with its strength being about a quarter of the α + α potential at R = 0 fm (Fig. 2b) and is extended much to the outer region reflecting the N + 3N cluster structure of the 0 + 2 state of 4 He. This potential accommodates the redundant Pauli forbidden 0s state at −12.7 MeV, E c.m. = 8 MeV, from the α +α * threshold energy for L = 0 (Fig. 2b).
In order to make it easier to understand the physical meaning of the Pauli allowed physical states and their scattering state wave functions solved in the coupled-channel method, which will be discussed in Fig. 4 and Fig. 5, we show in Fig. 3 both the Pauli forbidden and allowed wave functions of the relative motion for L = 0, calculated in the bound state approximation in the single channel using the α + α potential and the α + α * potential. In the α + α channel the Pauli allowed physical states should satisfy the Wildermuth condition N = 2n + L = 4 with n being the number of the nodes in the relative wave function. For the α + α * channel the physical states should satisfy the Wildermuth condition N = 2. The number of the nodes in the Pauli allowed wave functions in Fig. 3 solved in the bound state approximation satisfies the Wildermuth condition. For the α + α channel the Pauli allowed wave function has two nodes with the outermost node at around 2 fm, which is the well-known energy independent node discovered in Refs. [8,9] and corresponds to the radius of the repulsive core potential for L = 0. For the α + α * channel the wave function of the Pauli allowed 0 + state has only one node at around 2 fm, which is also energy-independent due to the Pauli principle.
The shallow and extended properties of the potential for the α +α * channel in Fig. 2 are understood naturally from the wave function of the 4 He(0 + 2 ) state. This state at E x = 20.210 MeV with 100% p-decay width [58] is located 0.396 MeV above the p+ 3 H threshold (19.814 MeV) and 0.368 MeV below the n+ 3 He threshold (20.578 MeV) and has been shown to have the N + 3N cluster structure [56,66]. The calculated rms radius, 3.16 fm in Ref. [66] and 4.2 fm in Ref. [56], is much larger than that of the ground state of 4 He, 1.63 fm. This is very large even compared with the calculated size, 3.2-3.8 fm [67][68][69][70][71][72][73][74][75] of the well-developed gas-like dilute three α cluster 0 + 2 Hoyle state in 12 C at E x = 7.654 MeV, located 0.287 MeV above the α threshold energy.
To see the underlying structure in the kink of the phase shift at around E c.m. = 21 MeV for L = 2 seen in Fig. 1, the calculated elastic scattering cross section, the Argand diagram, and the calculated  inelastic cross section to the 0 + 2 channel are displayed in Fig. 4. We see a dip at E c.m. = 21.01 MeV in the elastic scattering cross section (Fig. 4a). This is related to the existence of the 2 + resonance in the α + α * channel. Although the elastic cross section does not show a Breit-Wigner type behavior of resonance because the phase shift does not pass through 90 • , the resonant nature can be confirmed by looking into the Argand diagram [76][77][78]. As seen in Fig. 4(b), the diagram of the elastic channel exhibits a rapid counterclockwise change of the phase as a function of energy at E c.m. = 21.01 MeV, which is the signature of a resonance in the coupled channels system. As a side note, the point where the Argand loop crosses the vertical axis from right to left with increasing energy corresponds to the phase shift of 90 • in the case without coupled-channel effects. It is to be noted, however, that the Argand loop does not generally imply the existence of a resonance unless a peak in the cross section is accompanied [76,77]. Our calculated cross section to the 0 + 2 channel for L = 2 in Fig. 4(c) clearly shows a sharp peak at this energy. Thus it is found that an L = 2 resonance occurs at E c.m. = 21.01 MeV above the α + α * threshold as a molecular resonance [27,28] with the centrifugal barrier. In this molecular 2 + state, the calculated mixing probabilities inside R = 4.5 fm are 99.76% for the α + α * component and 0.24% for the α + α component.
The MeV, the first node at around R = 1 fm moves outward, while the position of the second node at R = 2 fm is almost energy-independent, which is due to the orthogonality to the forbidden 0s and 1s states in the α + α channel indicated in Fig. 2(a). The Wildermuth condition N = 4 is satisfied for these wave functions. What is to be noticed is that at 19.44, 19.45 and 19.46 MeV no node appears at around R = 1 fm and the first node appears at R = 2 fm, which means that at these energies the wave functions no longer satisfy the Wildermuth condition N = 4 for the elastic channel; that is, the wave functions seem to disappear from the α + α channel. In fact, the calculated mixing probability inside R = 4.5 fm at resonance energy 19.45 MeV shows the dominant α + α * component (99.82%) while the α + α component is negligibly small, 0.18%. The wave functions at 19.44 and 19.45 MeV are out of phase at around R = 3 fm with those at the lower energies 19.35, 19.38 and 19.41 MeV as shown in Fig. 5(b). At 19.48 MeV, the first node appears again at R = 0.6 fm and the second node remains at the energy-independent R = 2 fm, thus again appears in the α + α channel. The energy interval between 19.435 MeV and 19.465 MeV, where the first node disappears, apparently violating the Wildermuth condition in the α + α channel, corresponds to the width of the resonance at E c.m. = 19.45 MeV. In Fig. 5(a) the wave function of the inelastic channel as well as that of the elastic channel on the resonance energy 19.45 MeV is shown. It should be noted that the number of the node satisfies N = 2 on this resonance energy although N = 4 is satisfied at other energies, that is, the number of node is decreased on the resonance. We see that the former has the first node at R = 2 fm, satisfying the Wildermuth condition in the inelastic channel, and the latter has the first node at R = 2 fm, violating the Wildermuth condition in the elastic channel. The inelastic channel wave function is much extended to 8 fm, reflecting that it has the α+ α * (3N + N ) structure.
In Fig. 5(a) the first node of the inelastic scattering wave function at 19. 45 MeV, which appears near R = 2 fm, is very close to that of the N = 2 wave function in Fig. 3(b) in the α + α * channel solved in the bound state approximation. This shows that in the resonance energies where the Wildermuth condition in the α + α channel does not apparently work, the inelastic wave function satisfies the N = 2 Wildermuth condition in the α + α * channel, being orthogonal to the redundant Pauli forbidden 0s state indicated in Fig. 2(b). This is understood by looking into the properties of the kink in the phase shift of L = 0 at E c.m. = 8.1 MeV in Fig. 1. From the changes of the wave functions with energy in the vicinity of E c.m. = 8.1 MeV, which is very similar to the behavior seen in Fig. 5, it is found that this kink is the redundant 0s Pauli forbidden state in the α + α * channel in Fig. 2(b). Thus the Paul forbidden states in both the α + α and α + α * channels are correctly incorporated in the present deep potential. The forbidden states in the α + α channel for L = 0 and 2 are in the deeply bound region, and the 0s forbidden state in the α + α * channel for L = 0 is also in the deeply bound region below the α + α * threshold. Since the solved physical L = 0 state with the α + α * structure is exactly orthogonal to this Pauli forbidden state, the present model calculations are a mimic of Saito's orthogonality condition model [79,80]. On the other hand, for L = 2 state there is no forbidden state in the α + α * channel.

Discussion
We discuss why a state with the α + α * structure in 8 Be was not obtained in the microscopic coupledchannel calculations using the resonating group method in Ref. [51] where none of the calculated phase shifts for L = 0, 2, and 4 as well as L = 1 and L = 3 pass through 90 • . This is probably because the wave function for the 4 He(0 + 2 ) state was not appropriate since the monopole vibration state was assumed. In fact, the precise wave function for 4 He(0 + 2 ) we used has a very extended 7/11 PTEP 2021, 113D02 Y. Hirabayashi and S. Ohkubo N + 3N cluster structure, which is essential in causing a strong coupling between the α + α and the α + α * channels. In Fig. 6 the energy levels with the α + α and the α + α * cluster structure obtained in the present calculations are compared with the experimental energy levels and the energy levels calculated using the tensor-optimized shell model (TOSM) with the AV8' force and the Minnesota force [81] and the Monte Carlo calculations [82]. The present calculation reproduces not only the experimental energy levels of the ground band but also the highly excited 0 + 2 and 2 + 2 states well. On the other hand, the shell model calculations locate the first excited 0 + state considerably lower than the experimental energy level. Although the structure of this 0 + state is not discussed in Ref. [81], it could correspond to the 0 + state with the α + α * cluster structure if the shell model space is enlarged sufficiently. In the ab initio calculations, Wiringa et al. [82] got no highly excited J = 0 + , 2 + states with T = 0 PTEP 2021, 113D02 Y. Hirabayashi and S. Ohkubo in the Green's function Monte Carlo calculations and a 0 + 2 at 28.62 MeV in the variational Monte Carlo calculations, which is far from the experimental 0 + 2 state. The rotational constant k = 2 /2J = 0.25 MeV, with J being the moment of inertia, of the α + α * cluster band estimated from the experimental states shows that the moment of inertia of the band is about twice as large as that of the ground band with k = 0.57 MeV. This large moment of inertia is reasonable considering that the α * has a very extended N + 3N cluster structure, as seen in Fig. 5(a), where the wave function of the 0 + 2 state is extremely extended up to more than 8 fm with the dilute density distribution compared with the α + α cluster structure of the ground state 0 + . We note that this band is very similar in extended nature to the K = 0 + 2 band with k = 0.28 built on the Hoyle state at 7.65 MeV in 12 C [73,83] and the K = 0 + band at 16.7 MeV in 16 O with very large moment of inertia, k = 0.064 MeV in Ref. [84] and k = 0.095 ± 0.020 MeV in Ref. [85] with the dilute α+ 12 C (0 + 2 ; 7.65 MeV) structure [22,86]. The present α + α * cluster picture of the second 0 + state seems to be supported in a microscopic three-cluster model calculation in Ref. [87], which locates a 0 + 2 state located at 1.44 MeV above the p+ 7 Li threshold (18.254 MeV), although its cluster character is not discussed in detail. The present results make an impact on the cluster structure study in nuclei by reinforcing the importance of the concept of core excitation in the lightest nuclei, which has been known in medium-weight and heavy nuclei. Also, as for the 7 Be(n, 4 He) 4 He reaction, it would be intriguing to investigate the incident energy dependence of the cross sections including the coupling to the α + α * channel at 20.2 MeV of 8 Be.

Conclusion
We have studied the α + α * cluster structure at the highly excited energy in 8 Be in the coupledchannel calculations using the double-folding model derived from the precise cluster wave function for the first excited state of 4 He(0 + 2 ). We found that the well-developed core-excited α + α * structure appears near the α + α * threshold as a closed channel resonance (Feshbach resonance) for L = 0 and a molecular resonance for L = 2 which correspond to the experimental states at E x = 20.20 MeV 9/11 PTEP 2021, 113D02 Y. Hirabayashi and S. Ohkubo and E x = 22.24 MeV in 8 Be, respectively. The present coupled-channel cluster model approach, which has been powerful for the cluster structure study with core excitation such as the α+ 40 Ca * (0 + 2 ) [24], α+ 16 O * (0 + 2 ) [23] and α+ 12 C * (0 + 2 ) [22] systems, is again found to be powerful for the lightest prototype α + α * cluster structure with α-core excitation in 8 Be.