Three-cluster states in the spectrum of : (1) Structure and β-decay of to 16.6 MeV state
Introduction
Recent works [1], [2], [3] have used the three-body cluster model to study the properties of low-lying states in Li, B. The calculated observables for Li, B 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 B. As a first step, we extend the three-cluster approach to the states in Be 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 Be 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, α––n) and binary (–p, –n) thresholds but unbound relatively to the α–α channel. Low width indicates that wave functions (WFs) of the 2+ states have a very small admixture of the α–α 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 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 α–α 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 reasonably well, we get the opportunity to study consistently transitions between three-cluster states in and , including the β-decay of to 2+ states in and electron capture to the 1+ state.
The unit system ℏ=c=1 is used in the work. The symbol “t” denotes T or cluster depending on context. Both T–p and –n 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 –n cluster arrangements, and for excited states in , α–T–p and α––n 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 –n scattering in a potential model
The first step in the studies of three-cluster states in 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 formPotential V1 for configurations with T=1 is fitted to T–n and –p scattering data and has already been used in our calculations [1], [2]. Potential V0 for T=0 is found using T–p and –n
Structure of the three-cluster 2+ and 1+ states in
As soon as we find the 3+1 potential, we can use it for the three-body calculations in . Like the 3+1 calculations, we should work in the representation of physical channels leading to the set of equationswhich 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 to 16.6, 16.9 MeV states in
The life-time of (770±3 s, =5.6 [23]) is governed by the β-decay to the first 2+ (3.04 MeV) state in . The β-decays to 16.6 MeV state have value 3.31 [12], [23], corresponding to a superallowed transition withHere 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 α–α structure of low-lying states in . It is reasonable to suggest that states around 16–20 MeV in 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 –n. Thus, to study three-cluster states, we first had to develop a potential formalism for T–p (–n) scattering. We were able to describe T–p and –n 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.
References (34)
- et al.
Nucl. Phys. A
(1996) Phys. Lett.
(1965)- et al.
Phys. Lett.
(1966) Nucl. Phys. A
(1978)- et al.
Phys. Lett.
(1964) Nucl. Phys.
(1966)- et al.
Nucl. Phys. A
(1979) Phys. Lett. B
(1990)- et al.
Phys. Lett. B
(1994) Nucl. Phys. A
(1988)