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Coupled-channel picture of the core nucleus for the expansion of the nuclear size in the drip-line region

We study the energy and radius of neutron rich nuclei toward to the drip-line region. We employ the cluster-orbital shell model approach to calculate the radius of the oxygen isotopes. The result shows that the calculated radius underestimates the observed one at 23O, though the binding energy and the drip-line of the oxygen isotopes are reproduced. Therefore, we propose a coupled-channel approach to explain the sudden increase of the radius of the oxygen isotopes at 23O. As a result, the sudden change of the radius can be descried in terms of the competition between the available valence neutron orbits and occupied orbits in the core nucleus. For the 23O case, we show that the excitation to the 1s1/2-orbit in the core nucleus is a key configuration to reproduce the radius.


Introduction
Nuclear size is one of an illustrative quantity for describing the nuclear systems. As the system comes close to the drip-line, and the binding energy becomes small, the observed radius shows a very large value. In general, such the phenomenon is called the "halo" structure of the nuclear system. The halo structure is described by a relatively stiff core nucleus and loosely bound valence nucleon(s) with a very small binding energy. The typical binding energy of the halo nuclei is a few hundred keV, which is small enough compared to the averaged separation energy of the normal nuclei such as ∼ 8 MeV.
An exceptional case for the relation between the binding energy and radius is observed on the oxygen isotopes. The drip-line of the oxygen isotopes is determined as 24 O and no bound state is observed for A ≥ 25. For 23 O and 24 O, an abrupt increase of the r.m.s.radii has been observed [1]. However, recent experiment [2] shows smaller radius for 23 O. In the theoretical study, it is not so straightforward to reproduce such the abrupt increase, since the one neutron separation energies of 23 O and 24 O are not so small compared to the typical "halo" nuclei such as 11 Be and 11 Li.
We have studied the oxygen isotopes by using the cluster-orbital shell model (COSM) approach [3,4]  above. Hence, we consider that the model space with the closed-core configuration plus valence neutrons is not enough to reproduce the general tendency of the radius of the oxygen isotopes from A = 16 to 24 within the consistency to the observed binding energy.
In this paper, we propose a coupled-channel picture for the 16 O core [5], and the increase of the radius can be explained as the competition between the occupied orbit in the core-excited configuration and the Pauli-blocking effect to the valence neutrons.

Formalism and the results
For studying weakly bound systems with the model space of the core and many valence nucleons, we employ the cluster-orbital shell model (COSM) approach [6]. All the coordinates for the valence nucleons are defined so that the origin is the center of mass of the core nucleus and vectors are spanned to each valence nucleon. For the oxygen isotopes, the core nucleus is large enough and the recoil part due to the subtraction of the center of mass motion becomes small. Therefore, we consider the COSM approach is suitable to study the oxygen isotopes and other heavier nuclei in the drip-line region.

M -scheme cluster-orbital shell model
In the COSM approach, we transform the total A-body Hamiltonian by using the coordinate systems defined from the center of mass of the core as follows: Here,t i andV i are one-body kinetic and potential terms for the ith valence nucleon.T ij is the recoil part, which comes from the subtraction of the center of mass motion,T G .Ĥ C is the Hamiltonian for the core nucleus. To solve the system described by the COSM Hamiltonian (1), we use the Gaussian basis function for the valence nucleons. For the case that the number of valence nucleons is N , the radial part of the basis function is defined as follows: where g i (r i ) is the Gaussian for the ith valence nucleon with a polynomial, which is proportional to the angular momentum l i , and its form is In order to simplify the treatment of the angular momentum part and the antisymmetrization, we employ an m-scheme approach for the angular momentum part of the basis function [4]. We only fix the z-components of the total spin and isospin described as |M M T and solve the system using the basis functions with the same fixed z-components as follows: ( If the basis size is sufficiently large, eigen vectors have a good quantum number for the total spin and isospin, and the coefficients c (m) play the role of the Clebsch-Gordan coefficients.
To discuss the structure of the core and the motion of the valence nucleons on the same footing, we use a semi-microscopic treatment for the interaction between the core and a valence nucleon. The interaction between the core and the ith valence nucleonV i is constructed by using the folding procedure of the wave function of the core nucleus Φ C aŝ  For the nucleon-nucleon interaction, we use the Volkov No.2 [8] interaction and modify the potential strength and parameters to reproduce the drip-line of the oxygen isotopes at 24 O.
Using the m-scheme COSM approach, we calculated the binding energy and r.m.s.radius of the oxygen isotopes. We perform calculations using two types of the core-size with respect to the change of the mass number and two different exchange parameters for the nucleon-nucleon interactions of the valence part.
For the core part, we use (α) the fixed core size and (β) the varied one, in which the size parameter has the mass number dependent as the b C ∼ A 1/6 . Here, b C is the width parameter of the wave function of the core, which is described by using the harmonic oscillator. For the valence part, two exchange parameters are applied to reproduce (i) the energy of 18 O ground state, and (ii) the drip-line of the oxygen isotopes at 24 O.
The calculated r.m.s.radii using the four different parameter sets for the core and valence nucleons are shown in Figs. 1(a) and (b). As seen from Fig. 1(b), for 16−22 O, the calculated radii using the fixed core size corresponds to the experimental one for both fixed-core cases, (α-i) and (α-ii). On the other hand, for 23 O and 24 O, the fixed core calculations do not show the sudden increase of the radius at 23 O even for the (α-ii) case, and the varied core-size case (β-ii) shows the better agreement compared to the other calculations.
This result suggests that a new mechanism is necessary to be introduced in order to describe the energy and radius of the oxygen isotopes consistently for 16−24 O. Further, it can be considered that the configuration of the core changes drastically from 22 O to 23 O as the consequence for the valence neutrons added.

A coupled-channel picture for 16 O
In order to reproduce the general tendency of the radius of the oxygen isotopes, and its sudden change at 23 O, we introduce a coupled-channel picture to the 16 O core [5]. We consider the essential point of this coupled-channel picture is to introduce an excitation of the neutron and proton (S = 1 and T = 0) in the core nucleus to the 1s 1/2 -orbit, see Fig. 2. The tensor induced core-excitation has been developed as the treatment of the tensor-optimized shell model (TOSM) approach [11]. With this condition, high-momentum components are taken into the model space as the size parameter of the orbit in the core, and the small size-parameter of the core is favored    [9,10]. Crosses with error bars are experiments [1]. Solid and open triangles with error bars are other experiments [2].
in the 2p2h-excitation channel.
In this work, we use the same king of mechanism of the TOSM to the 16 O-core. To reduce the model space, we restrict ourselves to a two-channel model for describing the core configuration. Here, we consider a neutron and a proton in the 0p 1/2 -orbit go up to 1s 1/2 -orbit, and other particle-hole excitations, for example the pairing-type one, are assumed to be renormalized into the 0p0h-or 2p2h-channel.
We parameterize the coupled-channel Hamiltonian as follows: The interactions in each channel and coupling parameter ∆ ij are adjusted to reproduce the energy and radius of the ground state of 17 O. We use the parameter for the diagonal part ∆E ≡ ∆ 22 − ∆ 11 = 10 (MeV). We calculate the energy and r.m.s.radius of the oxygen isotopes for 17−23 O using the coupledchannel approach within the core+n model space. Results are shown in Fig. 3. Here, we adjusted the strength of the core-n interaction (4) so as to reproduce the 17 O ground state. For 19−23 O, we use the same potential parameters. Nevertheless, the calculated energies agree with the experimental ones, which is shown in Fig. 3(a).
The mechanism to make different binding energies for heavier oxygen isotopes is as follows. In our model, the core size parameter in the first channel is taken to be proportional to the mass number as ∼ A 1/6 . The semi-microscopic potential (4) becomes weak for the large size parameter of the core wave function |Φ C . Hence, the binding energy becomes small with the large mass number. For the second channel, the 1s 1/2 -orbit of the valence neutron is occupied in the core configuration. The 1s 1/2 -component in the second channel of valence neutron is hindered due to the Pauli principle for 23 O, while the 1s 1/2 -orbit is unoccupied in the first channel. Therefore, the dominant component becomes only the first channel, and binding energy becomes small because of the smallness of the core+n potential with a large core-size parameter.
The calculated radii show a promising result in the view point for reproducing the sudden change of the radius at 23 O, see Fig. 3 (b). As mentioned in the calculated energies, due to the Pauli blocking effect for the second channel, the fist channel becomes dominant in 23 O. Since the first channel has the mass number dependence for the core size, the calculated radius with the first channel dominance show the large value for 23 O, while the second channel has the same amount of the contribution until 21 O.