ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 3, с. 409-413
= ЯДРА ^^
A CLUSTER CALCULATION FOR 6He SPECTRUM
©2014 I. Filikhin*, V. M. Suslov**, B. Vlahovic***
North Carolina Central University, Durham, USA Received January 29, 2013
The 6He nucleus is considered as the cluster a + n + n system. The excitation energies of the low-lying levels are calculated using the configuration-space Faddeev equations. The analytical continuation method in a coupling constant is applied for calculation of resonance parameters. The an interaction is constructed to reproduce the results of Д-matrix analysis for an-scattering data. A realistic AV14 potential describes the nn interaction. Additional three-body potential adjusted by the ground state energy of6He is used. The energies of the low-lying resonances of 6He(0+, 0+, 2+) are reasonably reproduced by the calculations.
DOI: 10.7868/S0044002714030088
We cordially dedicate this work to Vladimir B. Belyaev on the occasion of his 80th birthday
1. INTRODUCTION
The cluster phenomena in light nuclei are intensively studied within three-body models in the last decade [1, 2]. In particular, the 6He and 9Be nuclei may be described as the mirror a + n + n and a + + a + n cluster systems (see [3] for another example of such "mirror" systems). Spectral properties of the 6He nucleus were studied within the cluster model in a number of works [4—11], taking into account different aspects of the model to obtain reliable description for 6He. This description includes the most important orbital momentum configurations; three-body force which is needed; types of clusterization in the system, problem of implementation of the Pauli principle; and others.
In the present work we focus on the potential of the an inter-cluster interaction, which has to reproduce the set of the experimental data as well as the an scattering data and low-lying spectrum of 6He and 9Be. In [12] we have proposed the an potential constructed to reproduce the results of R-matrix analysis for an-scattering data [13]. This potential allows to describe the low-lying spectrum of 9Be to be close to the experimental data. The same an-potential is applied to calculate the low-lying spectrum of the 6He nucleus.
E-mail: ifilikhin@nccu.edu
E-mail: vsuslov@nccu.edu
E-mail: vlahovic@nccu.edu
Note that three-body force has to be included into the model to obtain the experimental value for the energy of the 6He ground state. The potential forms essentially the low-lying spectrum of the a + n + n system and may depend on total orbital momentum for different states. The three-body potential is motivated by recognizing that at short distances the spatial structure of this nucleus does not necessarily have three-body character [4]. The problem, which we consider in this regard, is the necessity of the three-body potential having strong orbital momentum dependence that has to be included into the model. It is worth to note that in the mirror n + a + a system a three-body potential is not needed as was shown in [12]. Strong three-body force creates difficulties for predictive potential of three-body model due to uncertainty of parameters of the potential which have to be adjusted by use of experimental data.
2. FORMALISM
The 6He nucleus is described within three-body a + n + n model as two-neutron halo nucleus. The nucleus has weak bound state since the a + n and n + n subsystems of a + n + n are unbound (the Borromean type of nucleus [2, 5]). The bound state is the 0+ state. Due to the Pauli principle for the system of five nucleons, the s-wave configuration of the an subsystem is suppressed for the system, while the p-shell configuration dominates.
Our calculations are based on the Faddeev equations in configuration space [14]. The total wave function of the a + n + n system is decomposed in the sum of the Faddeev components U and W
corresponding to the (nn)a and (an)n types of rearrangements, respectively:
^ = U + (I + P)W,
P is permutation operator for identical particles (the pair of neutrons). In this notation the differential Faddeev equations are given as follows
(Ho + Vnn - E)U = Vnn(W + PW), (1) (Ho + V«n - E)W = Van(U + PW).
The Argonne V14 (AV14) potential [15] was used for the nn interaction. Comparing the potentials of singlet—triplet (s = 0, t = 1) and triplet—triplet (s = = 1, t = 1) spin—isospin states it is assumed that the triplet (s = 1) components are essentially weaker. The LS-coupling scheme is used for partial wave analysis of the equations 1 [2, 16]. The LS basis allows to restrict the model space to the states with the total spin S = 0 (when the spin projections of nucleons are antiparallel). The possible spin—orbital momentum configurations with S = 1 are not taken into account in our calculations. According to evaluations of different authors the total contribution of the S = 1 configuration is in the range from 5% to 14% for the a + n + n ground state [7].
The an potential Van acts in s, p, and d states of the pair. The p- and d-wave components include central and spin—orbit parts: V^n(r) = V£(r) + + (s, l)Vso(r). The coordinate dependencies of the components have the form of one- or two-range Gaussians functions. The s-wave component is repulsive [6, 9, 17] to simulate the Pauli exception for an in the s state. This an potential proposed in our work [12] is a modification of the potential from [5].
An effective three-body force simulating the effects of violation of the strong cluster structure of 6He [4, 6] is taken into account. Two reasons for such violation may be noted: the first one is that the 3H + 3H threshold is close to the energies considered (12 MeV) (see [6] for the details) and the second is the use of the cluster model with "frozen core" [18] when the distortion of the alpha-particle is ignored.
The three-body potential is defined by one-range Gaussian function: V3bf (p) = V0 exp(-ap2). Here p is hyper-radius of the three-body system: p2 = = x2 + y2, where x, y are the mass-scaled Jacobi coordinates [2]. Adding this potential into left-hand side of the Faddeev equations (1) one can reproduce the experimental value of the 6He binding energy. The adjustment may be done by using the two free parameters Vo and a.
The three-body potential may have different parameters for different L states of cluster systems [7— 11]. This L dependence is phenomenologically defined
so that the results of calculations for resonances could be closer to experimental data.
Bound-state problem formulated by Eq. (1) is numerically solved applying the finite difference approximation with the spline collocation method [19]. The eigenvalues are calculated by the inverse iteration method.
The method of analytical continuation in a coupling constant is used to calculate parameters of the resonances [12]. The coupling constant is the depth of an additional non-physical three-body potential. The potential has the form: V3(p) = —5 exp(-ap2) with parameters a, 5 > 0 that can be varied. For each resonance there exists a region |5| = |501 where a resonance becomes a bound state. In this region we calculated 2N bound-state energies corresponding to 2N values of 5. The continuation of this energy (set as a function of 5) onto complex plane was carried out by means of the Pade approximant:
-VE = J2thc/ (i + X>O),
i=l ' \ i=l J
where ( = — S. The complex value of the Pade approximant for 5 = 0 gives the energy and width of resonance: E(5 = 0) = Er + ir/2. Accuracy of the Pade approximation for the resonance energy and width depends on the distance from the three-body threshold, and on order N of the Pa de approximant.
3. RESULTS
Using the three-body potential defined by the parameters: V0 = —1.661 fm"2 and a = 0.2 fm"2 we obtained the value —0.9726 MeV for the a + + n + n ground-state energy, that is close to the binding energy of 6He (—0.973 MeV) [20]. The orbital momentum configurations taken into account for this calculation are shown in Table 1. The numerical convergence for the binding energies relative to increasing model space is fast. The main contribution, that provides the bound state of the system, comes from the p wave of the an interaction, due to attractive p-wave component of the an potential. Note that the calculations without the three-body potential do not result in a bound system.
The calculations for the 6He resonances are performed for the 0+, 2+, 1", 2" states. In Table 2 we present the orbital momentum configurations taken into account for the (nn)a and (an)n rearrangement channels for these calculations. The results for the parameters of the resonances are given in Table 3, where we present also the experimental data [20]. Our calculation predicts two levels for each 0+ and 2+ states. The 2+ resonance, being the first excited state
Table 1. Convergence of the calculations for binding energy (E) with the model space extending by the partial waves of subsystems (The (nn)a and (an)n are rearrangement channels, l is orbital momentum pair of particles, A is orbital momentum of relative motion of the third particle to the center of mass of the pair.)
(nn)a (an)n E, MeV
I 024 0 1 2
A 024 0 1 2 -0.973
I 02 0 1 2
A 02 0 1 2 -0.972
I 0 0 1 2
A 0 0 1 2 -0.968
I 02 0 1
A 02 0 1 -0.957
I 024 0 1
A 024 0 1 -0.957
I 0 0 1
A 0 0 1 -0.953
I 024 0
A 024 0 Not bound
of 6He, has been well studied experimentally [20]. Other resonances are not reliably defined by experiments. In particular, the existing the 1- resonance is not clear in this time as was discussed in [9]. One can see reasonable agreement between our results and the experimental data from Table 3.
The best fit to measured energy was obtained for the 0+ levels. Thus, we may say that two free parameters of the three-body potential (strength and range parameters) are fixed to reproduce the 0+ and 0+ energy simultaneously.
Calculation for the 2+ and 2- energy levels did not match experimental energy with the same accuracy. The f
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