Description of nuclear structures in light nuclei with Brueckner-AMD Tomoaki Togashi Yuhei Yamamoto Kiyoshi Katō Division of Physics, Hokkaido University P ｎ ｎ ｎ The 19th International IUPAP Conference on Few-Body Problems in Physics @Bonn, 31.Aug. - 5.Sep. 2009 ｎ P P ｎ

Introduction (1) One of the recent remarkable developments in theoretical nuclear physics; ab initio calculations based on the realistic nuclear force　　　　 3,4-body systems can be solved strictly by Faddeev, Gauss Expansion Method, and so on… R.B.Wiringa et al., PRC62 (2000) 014001 For heavier nuclei, Green’s function Monte Carlo (GFMC) no-core shell model (NCSM) coupled-cluster (CC) method unitary-model operator approach (UMOA) Fermionic molecular dynamics (FMD) + unitary correlation operator method (UCOM) tensor optimized shell model (TOSM) and so on… We can discuss cluster structures based on the realistic nuclear force.

Introduction (2) Recently, the studies with the Antisymmetrized Molecular Dynamics (AMD) have been developed. Y.Kanada-En’yo and H.Horiuchi, PTP Supple 142 (2001), 205 Advantages of AMD： － We can treat alpha-nuclei and non-alpha nuclei － The wave function is written in a Slater determinant form － Both states of shell model and cluster model can be described － no assumption for configurations AMD has succeeded in understanding and predicting many properties of both stable and unstable nuclei. However, the AMD calculations have been performed with phenomenological interactions.

T.Togashi, T.Murakami and K.Katō; Prog. Theor. Phys. 121 (2009) 299. Our strategy We develop the new AMD framework based on realistic nuclear forces. We apply the Brueckner theory to AMD and calculate G -matrix in AMD starting from realistic nuclear forces. Brueckner-AMD; the Brueckner theory + AMD T.Togashi and K.Katō; Prog. Theor. Phys. 117 (2007) 189. T.Togashi, T.Murakami and K.Katō; Prog. Theor. Phys. 121 (2009) 299. In this framework, we introduce no phenomenological corrections.

AMD wave function We calculate G-matrix in AMD. … A-body wave function Nucleon-nucleon (NN) correlations cannot be described sufficiently. (Slater determinant) single-particle wave functions We calculate G-matrix in AMD. Cooling method A.Ono, H.Horiuchi, T.Maruyama, A.Ohnishi, PTP 87 (1992) 1185 Energy Variation of A-body problem … The initial configuration is chosen randomly.

Brueckner-AMD The single-particle orbits in AMD Bethe-Goldstone equation H.Bando, Y.Yamamoto, S.Nagata, PTP 44 (1970) 646 AMD + Hartree-Fock method A.Dote, Y.Kanada-En’yo, H.Horiuchi, PRC56 (1997) 1844 single-particle w.f. of AMD starting energy : self-consistent Q -operator : the eigenstate of Hartree-Fock Hamiltonian occupied states single particle energy(s.p.e.) Cooling method G–matrices are changed because of dependence on s.p.e. The configuration is changed.

Correlation functions in Brueckner-AMD Bethe-Goldstone Equation correlation function Solution of the Bethe-Goldstone equation Model (AMD) pair wave function G -matrix element Correlation function Correlation functions are determined for every pair.

Decomposition of G -matrix G-matrix can be decomposed into central, LS, tensor by correlation functions. S.Nagata, H.Bando, Y.Akaishi, PTPS 65 (1979) 10 correlation function λ: rank of operators G-matrix with operators can be defined as . In the practical calculations, we use this potential with operators.

Application to light nuclei Interactions We use Argonne v8’ (AV8’) as a realistic nuclear interaction. ( cutting off electromagnetic interaction) AV8’; P.R.Wringa and S.C.Pieper, PRL89 (2002), 182501. Projection Parity : variation after projection (VAP) Spin (J ) : projection after variation (PAV) Parity eigenstate obtained with VAP (K -mixing) ⇒　We calculate the binding energy of the ground state and 　 the energy levels of the excited states. We present the Brueckner-AMD (B-AMD) results of some light nuclei (4He, 8Be, 7Li, 9Be, and 12C).

Result of 4He & 8Be level scheme Argonne v8' Result of 4He 4He ρ/2 4 0.1 0.08 0.06 Y 0.04 0.02 -4 4 [fm-3] [B.E.: -44.0 (MeV)] [B.E.: -56.5 (MeV)] Z binding energy(0+)‏ -24.6(MeV)‏ Few-body cal† EXP -25.9(MeV)‏ -28.3(MeV)‏ α+α cluster †H. Kamada et al. , PRC64 (2001) 044001

7Li level scheme Argonne v8' [B.E.: -29.6 (MeV)] [B.E.: -39.2 (MeV)]

9Be (parity -) level scheme Argonne v8' [B.E.: -41.9 (MeV)] [B.E.: -58.2 (MeV)]

9Be level scheme Argonne v8' [-41.9 (MeV)] [-58.2 (MeV)] α+α+n picture

12C level scheme Argonne v8' 3α cluster 3α-like [-70.4 (MeV)]

Huge shell-model model space is needed. Higher 0+ state of 12C The shell-model approaches have difficulty in describing the higher 0+ state of 12C. Huge shell-model model space is needed. P.Navratil et.al., PRL84 (2000) 5728 On the other hand, cluster models have been succeeded in describing the 0+2 state of 12C. Ex) E.Uegaki et.al., PTP 62 (1979) 1621 Cluster models indicate that a number of spatial configurations are needed to describe the 0+2 state of 12C.

Description of higher 0+ state of 12C In order to describe the higher 0+ state, it is necessary to superpose the intrinsic configurations different from 0+ ground state. We perform the energy variation with the orthogonality to the lowest state. Y.Kanada-En’yo, PTP117 (2007) 655 We perform the energy variation where the inertial axes are fitted in order to prevent us from obtaining the solution of rotated ground state. Intrinsic state for the excited state parity-projected states In order to create various spatial configurations, we perform the above variation under r.m.s.-radius and deformation parameter β constraint.

All states contribute coherently. 12C level scheme All states contribute coherently. Argonne v8' Argonne v8' [B.E.: -73.7 (MeV)] [B.E.: -92.2 (MeV)]

Summary & Proceeding works We developed the framework of AMD with realistic interactions based on the Brueckner theory. In the Brueckner-AMD, we can construct the G-matrix and correlation functions in AMD self-consistently. That means this method has no phenomenological corrections. We applied Brueckner-AMD to some light nuclei and succeeded to describe not only ground states but also excited states with cluster structures which shell model approaches cannot describe. Proceeding works Roles of tensor force for 8Be (discussed by Yuhei Yomamoto) The way how to introduce genuine 3-body forces

Appendix

d=1.0 [fm] 1E 3E 1O 3O G -matrix (central) renormalization of tensor force 1O 3O

G -matrix (spin-orbit, tensor) d=1.0 [fm] 3O LS 3O tensor reduction due to short range correlations 3E LS 3E tensor

Superposition of Slater determinants in Brueckner-AMD The G -matrix between the different configurations in bra and ket states is necessary. However, the G –matrix can be determined for only a single configuration. The G -matrix is calculated with the correlation functions. Correlation functions derived from bra and ket states T.Togashi, T.Murakami and K.Katō; Prog. Theor. Phys. 121 (2009) 299.

Features of G -matrix in Brueckner-AMD ①. The G-matrix can be solved in AMD framework because the single-particle orbits in AMD can be defined and applied to the Brueckner theory. starting energy: ②. The G-matrix in Brueckner-AMD is changed with the structures through the starting-energy dependence. The G-matrix in Brueckner-AMD is different from the case of nuclear matter and considers the property of finite nuclei.

Approximated β parameter : the moment of the inertia matrix Approximated β parameter

11B (parity-) level scheme Argonne v8' [B.E.: -54.5 (MeV)] [B.E.: -76.3 (MeV)] The inversion of 3/2- and 1/2- occurs.

What’s the origin of spin-orbit splitting ? No-core shell model calculations without genuine 3-body forces also indicate the inversion between 3/2- and 1/2- in 11B. C.Forssen et.al., PRC71 (2005) 044312 3-body forces may be essential to reproduce the energy levels of some nuclei.