This paper shows a brief review on CDCC and the microscopic reaction theory as a fundamental theory of CDCC. The Kerman-McManus-Thaler theory for nucleon-nucleus scattering is extended to nucleus-nucleus scattering. New development of four-body CDCC is presented. An accurate method of treating inclusive reactions is presented as an extension of CDCC and the Glauber model. §1. IntroductionThe construction of microscopic reaction theory is one of the most important subjects in nuclear physics. It is a goal of the nuclear reaction theory. Furthermore, the construction is essential for many applications. Particularly for the scattering of unstable nuclei, there is no reliable phenomenological optical potential, since measurements of the elastic scattering are not easy. An important theoretical tool of analyzing inclusive reactions is the Glauber model. 1) The theoretical foundation of the model is shown in Ref.2). The model is based on the eikonal and the adiabatic approximation. It is well known that the adiabatic approximation makes the removal cross section diverge when the Coulomb interaction is included. The Glauber model has thus been applied mainly for lighter targets in which the Coulomb interaction is negligible; see for example Refs. 3)-9) and Refs. 10), 11) for Coulomb corrections to the Glauber model.Meanwhile, the method of continuum discretized coupled channels (CDCC) 12), 13) is an accurate method of treating exclusive reactions such as the elastic scattering and the elastic breakup reaction in which the target is not excited. The theoretical foundation of CDCC is shown in Refs. 14)-16). Actually, CDCC has succeeded in reproducing data on the scattering of not only stable nuclei but also unstable nuclei; see for example Refs. 17)-28) and references therein. The dynamical eikonal approximation 29) is also an accurate method of treating exclusive reactions at intermediate and high incident energies where the eikonal approximation is reliable. The nucleon removal reaction is composed of the exclusive elastic-breakup component and the inclusive nucleon-stripping component. CDCC and the dynamical eikonal approximation can evaluate the elastic-breakup cross section, but not the stripping cross section.The experimental exploration of halo nuclei is moving from lighter nuclei such as He and C isotopes to relatively heavier nuclei such as Ne isotopes. Very recently, Takechi et al. measured the interaction cross section σ I for the scattering of 28−32 Ne at 240 MeV/nucleon and found that σ I is quite large particularly for 31 Ne. 30) A halo structure of 31 Ne was reported with the experiment on the one-neutron removal reaction. 31) This is the heaviest halo nucleus in the present stage suggested experi-

We propose a fully quantum-mechanical method of treating four-body nuclear breakup processes in scattering of a projectile consisting of three constituents, by extending the continuum-discretized coupled-channels method. The three-body continuum states of the projectile are discretized by diagonalizing the internal Hamiltonian of the projectile with the Gaussian basis functions. For 6 He+ 12 C scattering at 18 and 229.8 MeV, the validity of the method is tested by convergence of the elastic and breakup cross sections with respect to increasing the number of the basis functions. Effects of the four-body breakup and the Borromean structure of 6 He on the elastic and total reaction cross sections are discussed. PACS numbers: 21.45.+v, 21.60.Gx, 24.10.Eq, The study on neutron-halo nuclei has become one of the central subjects in the unstable nuclear physics since the discovery of such nuclei [1]. In scattering of a two-neutron-halo nucleus such as 6 He and 11 Li, the projectile easily breaks up into its three constituents (n+n+core), indicating that the scattering should be described as a four-body (n+n+core+target) reaction. Then an accurate theory for treating such a fourbody breakup is highly desirable.So far the eikonal and adiabatic calculations were proposed and applied to 6 He and 11 Li scattering around 50 MeV/nucleon [2,3,4,5]. Since these calculations are based on semi-classical approaches, they work well at higher incident energies. In fact, the elastic cross section of 6 He+ 12 C scattering at 229.8 MeV has recently been measured [6] and successfully analyzed by the eikonal calculation with the sixnucleon wave function of 6 He [7]. However, these approaches seem not to be applicable for low-energy scattering such as 12 C( 6 He, 6 He) 12 C at 3 MeV/nucleon [8] measured very recently.In this rapid communication, we present a fully quantummechanical method of treating four-body nuclear breakup. The method is constructed by extending the continuumdiscretized coupled-channels method (CDCC) [9] that treats three-body breakup processes in scattering of the two-body projectile. In CDCC, the total scattering wave function is expanded in terms of bound and continuum states of the projectile. The continuum states are classified by the linear (k) and angular momenta, and they are truncated by setting an upper limit to each quantum number. The k-continuum is then divided into small bins and the continuum states in each bin are averaged into a single state. This procedure of discretization is called the average (Av) method. The S-matrix elements calculated with CDCC converge as the modelspace is extended [9]. The converged CDCC solution is the unperturbed solution of the distorted Faddeev equations, and corrections to the solu- * Electronic address: taku2scp@mbox.nc.kyushu-u.ac.jp tion are negligible within the region of space in which the reaction takes place [10].Also for four-body breakup processes in scattering of the three-body projectile, CDCC has to prepare three-body bound and discretized-continuum states of t...

A new method of pseudostate discretization is proposed for the method of continuum discretized coupled channels to deal with three-body breakup processes. In the method, discrete S-matrix elements to the pseudo (discretized) continuum states are transformed into smooth ones to the exact continuum states of the projectile. As for the basis functions for describing pseudostate wave functions, we take real-and complex-range Gaussian functions, which form in good approximation a complete set in a finite configuration space being important for breakup processes. This "approximate-completeness" property is essential to make transformed S-matrix elements accurate. Moreover, the use of these Gaussian bases is expected to be very useful to describe four-body breakup processes. Accuracy of the method is tested quantitatively for two realistic examples: elastic and projectile-breakup processes in d+ 58 Ni scattering at 80 MeV and those in 6 Li+ 40 Ca at 156 MeV.

The deformation of Ne isotopes in the island-of-inversion region is determined by the doublefolding model with the Melbourne g-matrix and the density calculated by the antisymmetrized molecular dynamics (AMD). The double-folding model reproduces, with no adjustable parameter, the measured reaction cross sections for the scattering of 28−32 Ne from 12 C at 240MeV/nucleon. The quadrupole deformation thus determined is around 0.4 in the island-of-inversion region and 31 Ne is a halo nuclei with large deformation. We propose the Woods-Saxon model with a suitably chosen parameterization set and the deformation given by the AMD calculation as a convenient way of simulating the density calculated directly by the AMD. The deformed Woods-Saxon model provides the density with the proper asymptotic form. The pairing effect is investigated, and the importance of the angular momentum projection for obtaining the large deformation in the island-of-inversion region is pointed out.

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We accurately analyze the 6 He+ 209 Bi scattering at 19 and 22.5 MeV near the Coulomb barrier energy, using the continuum-discretized coupled-channels method (CDCC) based on the n+n+ 4 He+ 209 Bi four-body model. The three-body breakup continuum of 6 He is discretized by diagonalizing the internal Hamiltonian of 6 He in a space spanned by the Gaussian basis functions. The calculated elastic and total reaction cross sections are in good agreement with the experimental data, while the CDCC calculation based on the di-neutron model of 6 He, i.e., the 2 n+ 4 He+ 209 Bi three-body model, does not reproduce the data. PACS numbers: 21.45.+v, 24.10.Eq, 25.70.De

We present a novel method of smoothing discrete breakup cross sections calculated by the method of continuumdiscretized coupled channels. The smoothing method based on the complex scaling method is tested with success for a 58 Ni(d,pn) Exploring unstable nuclei far from the stable line is one of the most important subjects in nuclear physics. The unstable nuclei have exotic properties such as the halo structure [1][2][3] and the island of inversion [4]. As a feature of reactions induced by unstable nuclei, the projectile easily breaks up into its constituents. One of the most reliable methods for treating the projectile breakup processes over a wide range of incident energies is the continuum-discretized coupled channels (CDCC) method [5,6]. In CDCC, the scattering wave function of the total system is expanded with a finite number of bound and discretized continuum states of the projectile. The space spanned by these states is called the model space. The S-matrix elements calculated with CDCC converge as the model space is extended [7,8]. The converged CDCC solution is the unperturbed solution of the distorted Faddeev equations, and corrections to the solution are negligible within the spatial region in which the breakup processes take place [9,10].For scattering of a two-body projectile, the continuum states are classified by linear and angular momenta k and l, respectively, between the two constituents. In CDCC, these momenta are taken up to upper limits, the k continuum is divided into small bins, and the continuum states in each bin are averaged into a single state. This discretization procedure is called the average (Av) method. The Av method has been widely used, but its application has been limited to three-body breakup reactions as we will show. An alternative to the Av method is the pseudostate (PS) method [11][12][13][14][15][16], in which the continuum states {ψ (−) (k)} are replaced by pseudostates { n } obtained by diagonalizing the internal Hamiltonian of the projectile in a space spanned by L 2 -type basis functions. One can adopt the transformed harmonic oscillator (THO) [11] or the Gaussian [12,13] as the L 2 -type basis functions. The validity of the PS method was confirmed for scattering of twobody projectiles by the agreement between CDCC solutions calculated with the two discretization methods [12][13][14][15]. * tmatsumoto@nucl.sci.hokudai.ac.jp

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