Elsevier

Nuclear Physics A

Volume 665, Issues 1–2, 14 February 2000, Pages 105-121
Nuclear Physics A

Three-cluster states in the spectrum of 8Be: (1) Structure and β-decay of 8B to 16.6 MeV state

https://doi.org/10.1016/S0375-9474(99)00420-0Get rights and content

Abstract

The structure and spectrum of the three-cluster states in 8Be around 16–20 MeV was studied in the framework of the hyperspherical harmonic method. For that purpose we have developed a potential formalism for T–p and 3Hen (3+1) scattering and mechanism for the three-body mixed isospin calculations. We have studied the mechanism of the isospin mixing for 1+, 2+ doublets in 8Be and found a sharp sensitivity of isospin mixing to the experimentally not well defined parameters of the 3+1 intercluster interaction. The matrix elements of the β-decay of 8B to 2+ doublet 16.6, 16.9 MeV were calculated and estimates were made for the probability of electron capture to 17.64 MeV state. It is found that about 90% of the Gamow–Teller (GT) sum rule is concentrated in the three-cluster 2+ and 1+ states, situated below the three-body thresholds, within narrow boundaries (16.5–19.0 MeV).

Introduction

Recent works [1], [2], [3] have used the three-body cluster model to study the properties of low-lying states in 8Li, 8B. The calculated observables for 8Li, 8B were found to be in good agreement with experiment without any tuning of the model. While the issue of the applicability of three-cluster models is itself of interest, the structure of these nuclei is of importance for nucleosynthesis and the boron Solar neutrino problems as well. In [2] an estimate, based on the values of the asymptotic normalization constant, has been given for the astrophysical S-factor at zero energy S17(0)=19.2 eV·b, which agrees perfectly with recent experiments [4], [5]. In connection with the Solar boron neutrino problem we will consider the β-decay of 8B. As a first step, we extend the three-cluster approach to the states in 8Be which are Isobaric Analogue (or Gamow–Teller) states of those already studied. In that way we apply our model to the wider class of nuclei and phenomena and simultaneously get an opportunity to check it more thoroughly on the wider set of experimental data.

The other important issue is that the 2+ states in 8Be at 16.6, 16.9 MeV are experimentally known to have mixed isospin. They are bound with respect to all the three-body (α–T–p, α3Hen) and binary (7Lip, 7Ben) thresholds but unbound relatively to the αα channel. Low width indicates that wave functions (WFs) of the 2+ states have a very small admixture of the αα component. In this approximation we can consider these states as discrete spectrum states and, so far, the 2+ doublet is an example of isospin mixing in a bound (quasistationary) state. The nature of isospin mixing in the spectrum of 8Be was extensively studied, both experimentally and theoretically, since it was first emphasized in [6]. High quality experimental data exist concerning the properties of the high-lying 2+ states [7], [8] and β-decay to the lower of them [9], [10]. Extensive theoretical studies of the system were carried out by Barker [11], [12], [13]. Considerable efforts have been made to understand the interaction of the 2+ doublet with the αα continuum [14], [15], [16], [17], [18], [19], [20]. It was found difficult to reproduce isospin mixing matrix elements, extracted from experimental data, within the simple shell model. Detailed work was carried out by theorists to resolve this problem [21], [22]. We found that, compared with previous works, we can provide a new view of the isospin mixing phenomenon tracing it to the spectrum of the α-particle around binary thresholds and making a completely dynamic treatment of isospin mixing.

If we describe the spectrum of 8Be reasonably well, we get the opportunity to study consistently transitions between three-cluster states in 8Be and 8B, including the β-decay of 8B to 2+ states in 8Be and electron capture to the 1+ state.

The unit system ℏ=c=1 is used in the work. The symbol “t” denotes T or 3He cluster depending on context. Both T–p and 3Hen continuum of the α-particle are referred to as the “3+1 system”.

Section snippets

General notes

We found that the basis with definite isospin is not suited for calculations in the framework of cluster models. We use a basis, which we call “physical”, which does not have definite isospin but definite cluster partitions. For the low lying α-particle continuum the physical states have T–p and 3Hen cluster arrangements, and for excited states in 8Be, α–T–p and α3Hen arrangements. In both cases physical states are 50% isospin mixed. The term “physical” is used to emphasize the fact that

Studies of T–p and 3Hen scattering in a potential model

The first step in the studies of three-cluster states in 8Be is the development of the potential model for 3+1 scattering in the continuum of the α-particle. The “minimal” isospin-conserving model for this process considers the nuclear potential in the 3+1 channel to have the formV̂3+1=P̂T=0V̂0+P̂T=1V̂1.Potential V1 for configurations with T=1 is fitted to T–n and 3Hep scattering data and has already been used in our calculations [1], [2]. Potential V0 for T=0 is found using T–p and 3Hen

Structure of the three-cluster 2+ and 1+ states in 8Be

As soon as we find the 3+1 potential, we can use it for the three-body calculations in 8Be. Like the 3+1 calculations, we should work in the representation of physical channels leading to the set of equationsT̂−E+V̂αT+V̂αp+VCoul+(V̂1+V̂0)2ΨαTp+(V̂1V̂0)2ΨαHen=0,T̂−(E−0.764)+V̂αHe+V̂αn+(V̂1+V̂0)2ΨαHen+(V̂1V̂0)2ΨαTp=0,which are solved using the hyperspherical harmonics (HH) method. Here ΨαTp and ΨαHen are vectors of corresponding physical channel WFs in the hyperspherical representation (see

β-decay of 8B to 16.6, 16.9 MeV states in 8Be

The life-time of 8B (770±3 s, logft =5.6 [23]) is governed by the β-decay to the first 2+ (3.04 MeV) state in 8Be. The β-decays to 16.6 MeV state have logft value 3.31 [12], [23], corresponding to a superallowed transition withft1/2=6144.8±3.2 sBtotal=2ft(0+→0+)BF2BGT.Here λ=−1.268±0.002. The values BF,GT are reduced probabilities defined in an ordinary way. The phase volume factor f for β+ decay in the zero state width approximation can be estimated as 4.81×10−2 for 16.6 MeV 2+ state and

Conclusion

There is no doubt about the cluster αα structure of low-lying states in 8Be. It is reasonable to suggest that states around 16–20 MeV in 8Be are also “built” on the excitations of the α-cluster which are located between 20 and 23 MeV and has binary structure T–p or 3Hen. Thus, to study three-cluster states, we first had to develop a potential formalism for T–p (3Hen) scattering. We were able to describe T–p and 3Hen spectra well up to 3.5 MeV, fitting the T=0 part of the isospin-conserving

Acknowledgements

Numerous useful discussions with A.L. Barabahov and N.K. Timofeyuk are acknowledged. The authors thank J. Al-Khalili for carefully reading the manuscript and for useful comments. The work was partly supported by the Kurchatov Institute grant number 1 for young researches (L.V.G.) and RFBR grants 96-15-96548, 99-02-17610. L.V.G and N.B.S. are grateful to Royal Swedish Academy of Science for financial support.

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