Rotational bands in 11B and identification of diluted states

Differential cross-sections of the 11B + α inelastic scattering at E(α) = 65 leading to the most of the known 11B states at the excitation energies up to 14 MeV were measured. The data analysis was done by DWBA and in some cases by the modified diffraction model allowing determining the radii of the excited states. The radii of the states with excitation energies less than ∼ 7 MeV with the accuracy not less than 0.1-0.15 fm coincide with the radius of the ground state. This result is consistent with the traditional view of the shell structure of the low-lying states in 11B. Most of the observed high-energy excited states are distributed among four rotational bands. The moments of inertia of band states are close to the moment of inertia of the Hoyle state of 12C. The calculated radii, related to these bands, are 0.7 - 1.0 fm larger than the radius of the ground state, and are close to the radius of the Hoyle state. These results are in agreement with existing predictions about various cluster structure of 11B at high excitation energies. The state with the excitation energy 12.56 MeV, Iπ = 1/2+, T = 1/2 and the root mean square radius R ∼ 6 fm predicted in the frame of the alpha condensate hypothesis was not found.


Introduction
During long time 11 B nucleus was considered as good example of shell effects in light nuclei. Up to excitation energies ~ 7 MeV states of entire spectrum were described by different variants of shell models. Recently, however, a number of theoretical and experimental works appeared what resulted in increased interest to the 11 B structure. This interest was primarily connected with the predictions [1][2][3] about cluster configurations of various types co-existed in 11 B and the appearance of some experimental data indicating a greater probability of such a possibility [4].
Particular attention was drawn to the idea that there may be states in 11 B, analogues of the famous 0 + 2 state with excitation energy 7. 65 MeV in 12 C nucleus (the so-called Hoyle state). The Hoyle state properties, consisting of three weakly interacting alpha -clusters were crucial for verification and justification of the theory of alpha -particle condensation in nuclei [5], the main of which was the suggestion about its abnormally large radius. Accordingly, the Hoyle state analogues in 11 B (in which one proton is removed from alpha cluster, and having, therefore, the structure of α + α + t), must also have increased size.
Initially it was assumed [1] that the Hoyle state analogue in 11 B is the state 3/2¯ with excitation energy 8.56 MeV, which is not described by any variant of shell model. However, other possibilities were started to be discussed afterwards. For example, in [3] it was suggested that the true analogue of the Hoyle state is the state with excitation energy 12.56 MeV, despite contradictory data about its quantum numbers. Moreover, in [3] it was made even more ambitious assumption that this state has a "giant" radius R rms ≈ 6 fm, comparable to the radius of the uranium nucleus (!). The radius of 8.56 MeV state was still considered to be abnormally large, and it was predicted that this state is base for rotational band.
In [2,4] other unusual states in 11 B were predicted, such as quasimolecular states, with the structure of "two alpha -particle core plus three nucleons moving on their orbits". It is expected that the sequence of such states forms the positive parity rotational band, based on the 1/2 + , E* = 6.79 MeV state. It has been suggested [3] that this state also has an increased radius, 0.6 -0.8 fm larger than the radius of the ground state.
Thus, recently coexistence of several types of structures was predicted in 11 B, and this nucleus suddenly become in the focus of the entire problem of nucleon clustering in nuclei.
There are a lot of experimental studies of 11 B (see. e.g., [6] and references therein), but they did not affect the excitation energy region of interest for the problem. Due to the fact that many questions about 11 B states remained open, we have undertaken a new study of inelastic 11 B + α scattering at E(α) = 65 MeV. We measured the differential cross sections in a wide range of 11 B excitation energies and analyzed them together with the data available at other energies. Some preliminary results have been reported in [7].

Experiment and data analysis
The measurements were carried out at the University of Jyvaskyla K130 cyclotron (Finland) using the LSC (Large scattering chamber). The standard ΔE-E method was used. 4 telescopes were used simultaneously, each telescope made measurements for 2 degrees. The target was a self-supporting film of 11 B with 0.275 mg/cm 2 thickness. 12 C and 10 B were the only impurities in the target. The beam intensity was about 30 nA (for the particles). In the experiment, we used a system of beam monochromatisation, which allowed us to obtain a complete energy resolution about 150 keV.
The measured differential cross sections were analyzed by the Distorted Wave Born Approximation method (DWBA). To determine the radius of the 11 B nucleus it was supposed to use Modified Diffraction Model (MDM), described in details in [8], which allowed us to obtain selfconsistent data on the mean-square radii of a large number of excited states. MDM allows determining radii of the nucleus excited states through the difference between diffraction radii of excited and ground states using the expression (1): where R 0 -is the mean square radius of the ground state of the nucleus, R* dif and R dif (0) -diffraction radii calculated from positions of minima and maxima of experimental inelastic and elastic scattering angular distributions, respectively.

Low-lying states
The differential cross sections for low-lying 2.12, 4.44, 5.02 и 6.74 + 6.79 MeV states in 11 B were measured. It was shown that in the above unresolved doublet of excited states 6.74 MeV state dominates. The DWBA calculations were fulfilled.
Mean square radii of low-lying excited states of 11 В were received using MDM based on experimental data of current work and data [6] at 40 and 50 MeV. As it was expected, 11 B has equal non-enhanced radii in all low-lying states. The radii of low-lying (E * < 7 MeV) states practically coincide with the radius of the ground state.
The received results are consistent with traditional notions about shell structure of the low-lying states in the 11 B nucleus.

3/2 3¯, 8.56 MeV state and rotational band based on it
As it was mentioned in introduction, one of the main aims of this work was to get information about "abnormal" properties of 8.56 MeV state, and especially to determine its radius. Differential cross section with excitation of this state is shown of figure 1. Diffractive structure of angular distributions is shown clearly, however, the difference between DWBA calculations and experimental data takes place in the region of the second minimum. Compared with the elastic scattering, observed minima and maxima are shifted toward smaller angles, which is an indication of the increased radius of the 8.56 MeV state.
AMD -calculations [2] predict the existence of a rotational band, based on the 8.56 MeV state (see figure 2). There and in [4], it was suggested that the band is formed by sequence of states: 10.33 (5/2¯) -11.60 -13.14 (9/2¯) MeV. However, before these predictions, it was considered [9] that the spinparity of the 11.60 MeV state is 5/2 + and therefore it cannot be a member of this band.
As it was shown in [10], 10.33 and 13.14 MeV states are excited by angular momentum transfer L = 4. MDM calculations give R dif = 4.8 fm for these states. This diffraction radius is close to diffraction radius of recently observed state in 12 С with E* = 13.75 MeV (this state is excited by angular momentum transfer L = 4 in 12 C (α, α') scattering) and identified [11], as member of rotational band, based on the Hoyle state. It was shown [11] that the radius of the 13.75 MeV state is 0.8 fm larger than the radius of the 14.08 MeV state. Thus an estimation can be done, that the radii of above mentioned excited states of 11 В have radii 0.8 fm larger than radius of the ground state. So 10.33 (5/2¯) и 13.14 (9/2¯) MeV states have properties, similar to 8.56 MeV state and can be members of its rotational band. Separate discussion is required for question about spin-parity of the 3 rd predicted state of rotational band -the 11.60 MeV state, according to [2,4], J π = 7/2¯. But, in review [9] J π = 5/2 + .
In [10] we received using DWBA that this state is excited by angular momentum transfer L = 1 and L = 3, so J π = 5/2 + . If so, the question about the fact of the existence of this band arises. However, this result is not definitive. A significant contribution to the cross section the 11.44 MeV state can give, which has not been completely separated from the 11.60 MeV state, and for which there is no data on its spin and parity. In case J π = 7/2¯, this state should be excited by L = 2 and the use of MDM provides for the latter value R dif = 6.11 ± 0.35 fm.
These rotational bands are shown on figure 2 together with band in 12 C, based on the Hoyle state. Received from our experiment data about angular momentums transfer with excitation states, belonging to specified bands, are in agreement with known spin-parities of 11 B states. However they could not be determined unambiguously for the states 6.79, 9.88, 10.33, 13.14 and 13.16 MeV because of an insufficient energy resolution. Several special features in the J(J + 1) dependence of the excitation energies in figure 2 can be seen. Firstly, moments of inertia of the band states are very high and comparable. The largest of them (2I/ћ 2 ≈ 4.0, by the energy difference between the excitation energies 11.60 and 10.33 MeV) are observed for the higher members of the rotational band K = 3/2¯, for which cluster structure 2α + t is predicted. It is interesting that it is much larger than the moment of inertia of its analogue -the Hoyle state, for which 2I/ћ 2 = 2.7.  Secondly, there is a clear correlation between the moments of inertia and values of radii obtained using MDM from scattering data. Low-lying states of 11 B have "normal" radii and "reduced" moments of inertia about 2I/ћ 2 ≈ 1.1. These values are close to values for the first excited state of 12 C, 4.44 MeV. "Big" moments of inertia correspond to increased radii. The radii of the states, located at excitation energies above 7 MeV in 11 B, were obtained using MDM. The increased radii were found, at least, for one of the members of each band, and in most cases they are about 0.7 -1.0 fm larger than the radius of the ground state of 11 B. This leads to the conclusion that all states belonging to the bands under consideration have abnormal size. Theoretical works [1 -4] suggest a significant deformation of the rotational states of 11 B with E *> 7 MeV and it allows the increase of their radii. Radii and moments of inertia of these states are close to the corresponding values of the Hoyle state in 12 C nucleus.
Concerning the K=1/2 + band: this band produces the largest number of questions. In the measured spectra there were not observed any sign of contribution from the first member of the band, the 6.79 MeV state, to the unresolved group with the 6.74 MeV state. The second member, the 3/2 + , 9.88 MeV state was not observed at all. Spin-parity of the third member of the band, the 11.60 MeV state has not been established unambiguously. The estimated fourth band member, the 13.16 MeV state was not separated from the 13.14 MeV state. In the angular distribution the 13.16 state probably manifests itself, but probably with angular momentum transfer L = 1, which would mean that it cannot have spin 7/2 + , and it therefore doesn't belong to this band.

Radii of the states in 11 B
The levels of 11 B could be divided in 2 groups from the point of view of their radii values (see figure  3). The resulting radii of states with excitation energies up to ~ 7 MeV within the errors do not differ from the radius of the ground state, which is consistent with the traditional view of the 11 B shell structure.   At higher excitation energies formation of states belonging to rotational bands based on the 3/2¯ (8.56 MeV), 3/2 + (7.98 MeV), 5/2 + (7.29 MeV) and maybe 1/2 + (6.79 MeV) states was observed. All these bands are characterized by large moments of inertia (2I/ћ 2 = 3-4). The radii of rotational states, defined using MDM, were determined 0.7-1.0 fm larger than the radius of the 11 B ground state. The moments of inertia and the obtained values of the radii are close to the corresponding parameters of the Hoyle state in the 12 C and the rotational band, built on it, indicating cluster structure of these 11 B states. Particularly, the state with excitation energy 8.56 MeV can be considered as an analogue of the Hoyle state. The predictions of the theory of the coexistence of different structures in the 11 B, including cluster, are confirmed. Regarding the proposed band, based on the 6.79 MeV state, a number of uncertainties stay, including the fact of such band existence.

12.56/12.63 MeV state
Special attention should be provided to the problem of the 12.56 / 12.63 MeV state. Before the work [4], the only state at excitation energies 12.0 -12.9 MeV was the state at 12.56 MeV with spin -parity J π = 1/2 + , and isospin T = 3/2 [9], the isobar -analogue of the ground state of the 11 Be. In [3] it has been suggested that this state really has a spin -parity J π = 1/2 + , and isospin T = 1/2 and should be a "true" analogue of the Hoyle state in 12 C nucleus. The value of the isospin T = 1/2 is supported by the observation of the state decay with the emission of alpha -particles [12 -14].
In reaction 7 Li (α,α) [4] a strong resonance was indeed found, but with an excitation energy 12.63 ± 0.04 MeV. The state with excitation energy 12.56 MeV was not observed. In addition to the obvious value of the isospin T = 1/2, the 12.63 MeV state was received having the spin -parity J π = 3/2 + or 9/2 + . Possible value of the spin 9/2 + made it possible [4] to interpret the 12.63 MeV state, as the last member of the rotational band with K = 3/2 + .
In our experiment state with an excitation energy 12.6 MeV also was observed, so its isospin is T = 1/2. It is possible the state we observed is the state from [4] with excitation energy 12.63 MeV. Most probable value of the spin-parity according to our data is J π = 3/2 + . Analysis within MDM gave a "normal" value of the mean square radius (see figure 3), so prediction [3] has not been confirmed.

Conclusions
The 11 B nuclear has a number of excited states with the radii exceeding that of the ground state (size isomers). They are grouped in rotational bands whose moments of inertia correlate with enhanced radii values.
Radii and moments of inertia of the low-lying (E* < 7 MeV) states of 11 B are close to those of the ground state.
Radii and moments of inertia of rotational states with E*>7MeV are similar to those of the Hoyle state rotational band indicating their predicted cluster structure.
The 11 B state with E* = 12.6 MeV was observed, and thus its isospin is T = 1/2. However, the radius of this state, in contradiction with some predictions, is not significantly larger than the radius of the ground state, and thus the hypothesis about the presence of "giant" states in some light nuclei has not been confirmed.
Established numerous correlations between the values of moments of inertia of rotational band states and their radii are further confirmation of MDM ability to determine radii of excited states using experimental differential scattering cross sections.