Theoretical nuclear spectroscopy of highly excited states of 8Be nucleus

The results of ab initio calculations of the spectroscopic characteristics of the excited states of 8Be nucleus are presented. A satisfactory description of a variety of the spectroscopic data known from experiments has been achieved. A number of levels have been predicted that have not yet been observed experimentally. The results obtained demonstrate broad prospects for using ab initio approaches to study the spectra of light nuclei.


Introduction
High-precision methods for describing light nuclei properties and reactions induced by light nuclei collisions are advancing nowadays.One of the most successful approaches to compute nuclear binding energy, excitation energies of nuclear levels, nuclear moments and radii is the No-Core Shell Model (NCSM) [1][2][3][4][5][6].This method is based on realistic in particular,from Chiral Effective Field Theory [7,8].To adopt ab initio approaches for a description of the decay properties of nuclear resonance states it is necessary to describe various cluster channels.The most advanced methods combine NCSM and Resonating Group Model (RGM).They are called the (NCSM/RGM) [9] and the NCSM with continuum (NCSMC) [10,11].To describe resonances using calculations of scattering phase shifts, the SS-HORSE method was proposed in [12].These methods make it possible to describe the total and, in rare cases, partial decay widths.
To broaden the energy range of decaying nuclear states and the list of fragmentation channels, as well as the list of characteristics available for ab initio studies, including the partial widths, we developed another approach [13][14][15][16].The developed procedure includes the following steps.The calculation of the NCSM Hamiltonian eigenvalues (EVs) and eigenfunctions (EFs) is carried out.For a more precise determination of the energies of the levels the extrapolation procedure based on the calculated EVs is used.Next, projecting of the EFs onto the wave functions (WFs) of the cluster channels is performed.The resulting projections -cluster form factors (CFFs) describe the relative motion of decay products in A-nucleon configuration space in each state.The CFF () lSJ  is defined as the overlap In this study as a whole, we calculated the positions of levels of 8 Be up to 24 MeV of the excitation energy for all states and 28 MeV for low spin ones as well as the partial widths of the decay of all these levels into various channels.The Daejeon16 potential [8], which is based on ChEFT and is well proved for calculating the spectra of nuclei with mass A≤16, their sizes, and other characteristics, including the decay widths studied in our previous papers, was used in the calculations.The NCSM computations were carried out with the use of Bigstick code [17].The basis is limited by the value Nmax=10.The extrapolation procedure presented in [18] is used.

Results of the calculations and discussion
Some of the results obtained, are tabulated in Table 1 together with the evaluated data presented in the spectroscopic database [19].It contains the information about all levels lying in the energy region E < 21 MeV.For the resonances with reliably identified values J π ,T presented in [19] this region is expanded up to 24 MeV.The entire list of levels predicted by us is too wide to present them in this paper.
Let us compare, first of all, the measured and calculated values of the excitation energy E*.Only in two cases: 4 + ,0, E*(exp.)= 19.86MeV and 0 + , E*(exp.) = 20.2MeV, the energy difference exceeds 0.7 MeV.If, in addition, we take into account that the total binding energy 8 Beg.s.(reference point) is reproduced with an accuracy of 0.005 MeV, the quality of the results can be qualified as high.
In the second column of the Table the RMSVs of the isospin Ť are presented.This value illustrates the isospin impurities of each state.The calculation results are in good agreement with the qualitative data from the tables [19], but provide much more detailed information.So, they point to a rather large impurities of states 0 + ,0, E*(exp.)= 20.2MeV and 1¯,1, E*(exp.)= 20.9MeV: 0.315 and 0.673.They also indicate zero values of the isospin T of states 1¯, E*(exp.)= 19.40MeV and 4¯, E*(exp.)= 20.9MeV.Indeed, RMSVs of them are 0.071 and 0.049 respectively.
The other columns of Table 1 show the calculated partial decay widths of the 8 Be states into alpha (Гα), proton (Гp), and neutron (Гn) channels, which are characterized by the angular momentum of relative motion l and the channel spin S together with the measured total widths (Гt(exp)).The symbol < denotes a small (< 100 eV) width value.The typical difference between the experimental and calculated results is 1.5 ÷ 2 times.In our opinion, for such highly excited states, the origin of such a deviation may be both an inexactness of the theoretical approach, and various inaccuracies generated by the difficulty of experimental studies and their interpretation (see the example below).Regardless of the origin of the discrepancy, given the extremely wide range of decay widths, it can be argued that the decay widths calculated with such accuracy can serve as a fairly reliable means of classifying nuclear levels.
So let us focus on the examples in which the discrepancy is more pronounced.For state 2¯, E*(exp.)= 18.91 MeV the calculated total width is about six times greater than the one evaluated from different measurements [19].However, the available experimental data are generally contradictory.Thus, the processing of high-precision measurements of the cross section of 7 Li(p,n) 7 Be reaction led to the result Гn =55 ±20 keV, which practically coincides with our calculation.The analysis of reaction 7 Be(n,p) 7 Li, which was carried out in [21] demonstrate quite different values of partial decay widths: Гp= 1409 keV,   Гn= 225 keV, i. e. the proton decay channel is dominating.At the same time, these input data describe the cross section of the neutron capture at the thermal point with high precision.So, the proton decay width obtained by us is in more or less satisfactory agreement with the just presented.Most likely the origin of such strong contradictions between the experimental data is the proximity of the discussed state to the neutron decay threshold.This makes the values of the cross sections of reactions with neutrons in the entrance and exit channels very sensitive to variations in the energies and widths of the s-resonances.For the same reason (proximity to the neuron threshold), the results of calculations of the widths of the decay of levels into s-wave channels also become unstable.In addition, for such cases, the characteristics of resonances of this type can be seriously affected by their strong coupling with the swave continuum.An alternative way to eliminate the discrepancy can be the identification of the observed state 2¯ E = 18.91 MeV with the state 2¯,1 E = 20.2MeV predicted by us in case that level 2¯,1 lies significantly lower and is not observed.Also noteworthy are three overlapped broad resonances 4 + , 2 + , 0 + in a narrow energy range 19.86 ÷ 20.2 MeV.In addition to the above significant discrepancies between the calculated and measured energies of the first and the third of them, the calculation gives a very large alpha decay width Гα of the former one.It is quite possible that the source of these discrepancies is the difficulty in interpreting the results of measurements of reaction cross sections under conditions of strong resonance overlap.
Note that due to the calculations of the decay widths, we managed to quite reliably reinterpret the level presented in the spectroscopic tables as 3 (+) , 21.5 MeV (marked with *), identifying it as 2 + ,1.Also, the spin of the state (1,2) ¯,1, 24.0 MeV (marked with **), is identified as J=2.
Table 1 demonstrates several levels predicted by us, but not found in experiments which are marked with -----.The reasons for this are, for the most part, clear.The 0¯,0 19.07 MeV state is difficult to observe, since it is close to the neutron threshold (see above).The other levels have either very large or very small widths and are strongly overlapped between themselves and other levels, that is why they are hardly to be identified.It should be noted that some of performed earlier NCSM calculations which use various versions of NN-interaction also predict some of these states approximately in the same energy region (see, for example, [22]).
In summary it may be said that calculations in the framework of our approach provide almost complete spectroscopic information.Ability to successfully describe the set of partial decay widths of nuclear states in a wide range of excitation energies has been demonstrated.The predictive power of the discussed approaches seems to be sufficient to assert that the levels obtained in the calculations actually exist in the vicinity of the predicted energy and have properties that are close, in a qualitative sense, to those obtained theoretically.For exotic nuclei and for highly excited states of habituated nuclei, the characteristics computed in this way can, in part, serve as a substitute for the experimental ones.
Based on the results of the analysis of other ab initio approaches, we believe that at present the foundations of a new line of research are being laid -the theoretical spectroscopy of light nuclei.
WFs of the initial nucleus and fragments respectively, Â is the antisymmetrizer, and N is the integral operator providing normalization of the ket WF.Next, matching of the CFFs with the corresponding asymptotic wave functions at the points where the logarithmic derivatives coincide, is applied to compute the decay widths of the levels.Root mean square values (RMSVs) of the isospin 2

Table 1
Excitation energies (MeV) and decay widths of the 8 Be states (keV).