弱束縛核の動的性質と静的性質

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(1)九州大学学術情報リポジトリ Kyushu University Institutional Repository. 弱束縛核の動的性質と静的性質渡邉, 慎. https://doi.org/10.15017/1654646 出版情報：Kyushu University, 2015, 博士（理学）, 課程博士バージョン：権利関係：Fulltext available..

(2) Dynamic and Static Properties of Weakly-bound Nuclei. Shin Watanabe Theoretical Nuclear Physics, Department of Physics Graduate School of Science, Kyushu University 744, Motooka, Nishi-Ku, Fukuoka 819-0395, Japan February, 2016.

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(4) Abstract Many kinds of nuclear properties have been found and established thanks to the recent development in radioactive ion beams. Unstable nuclei have exotic properties never seen in stable nuclei, such as loss of magicity and large deformation. In particular, some nuclei near the neutron-drip line show “halo structure” in which a core nucleus is surrounded by one or a few nucleons. The halo nuclei have two important properties; one is the weakly-bound nature that one- or two-neutron separation energy is small (Sn or S2n . 1 MeV), and the other is the low angular-momentum nature that the extra neutron(s) are in s- and/or p-orbit. We can consider these properties as static properties of isolated nuclei. On the other hand, in the scattering of a weakly-bound nucleus, the projectile is easily broken up into constituent particles by a target nucleus. We can consider such properties as dynamic properties of projectile nuclei in the scattering. In order to understand nuclear physics comprehensively and deeply, we should study both the properties simultaneously. We then address the following three subjects. (I) How is projectile-breakup mechanism in 6 Li elastic scattering? (II) How are ground-state properties of neutron-rich Mg isotopes? (III) What is a measurable parameter quantifying halo nature of unstable nuclei? In subject (I), we investigate elastic scattering of 6 Li on targets (T) and clarify dynamic properties of 6 Li in the four-body scattering (n + p + α + T). Since the n + p subsystem of 6 Li has a bound state as deuteron (d), there exist not only four-body breakup processes (6 Li + T → n + p + α + T) but also three-body breakup processes (6 Li + T → d + α + T) in 6 Li scattering. These processes should be treated in a unified framework. This is nothing but the four-body version of continuum-discretized coupled-channels method (four-body CDCC). By using four-body CDCC, we find that 6 Li is mainly broken up into d + α in the scattering processes. The dα-dominance is an essential dynamic property of 6 Li. i.

(5) ii In study (II), we investigate recently-measured reaction cross sections (σR ) for neutron-rich Mg isotopes and deduce the static properties. We then reanalyze the so-called “island of inversion (IoI)”. The IoI is the region of Ne, Na, and Mg isotopes around the neutron number N = 20–22, and the nuclei in the IoI have exotic properties induced by the rapid shell evolution. Our analyses are based on the double folding model and antisymmetrized molecular dynamics (AMD). The framework reproduces the measured σR , and we then can deduce the ground-state properties (spin parity, total binding energy, and deformation) of Mg isotopes properly. By combing the present study on Mg isotopes with the previous study on Ne isotopes, we find that there exist large deformation beyond the IoI from N = 19 (31 Mg and 29 Ne) to N = 28 (40 Mg). We refer to this area as the peninsula of large deformation. This result indicates that the so-called island of inversion (once thought to be a region of N ≈ 20–22) might be enlarged. In study (III), we propose a parameter H quantifying the halo nature of oneneutron halo nuclei (a) composed of a core (c) and a neutron (n), and clarify the general properties of H. The parameter is defined by H = [σabs (a) − σabs (c)]/σabs (n) with the absorption cross sections σabs (x) of x (= a, c, and n) on the same target at the same incident energy per nucleon. We show that the H varies in 0 ≤ H ≤ 1, in particular, the halo nature is most developed when H = 1 and least developed when H = 0. The parameter H is thus expected to be a good indicator of halo nature. We deduce empirical values of H from the high-energy experimental data on σR where breakup effects are negligibly small. The empirical values of H are extrapolated to the weak-binding limit by using the model calculation based on the eikonal and adiabatic approximations. We find that H = 1 is realized only in the Sn = 0 limit for s-wave halo, and show that the point (Sn , H) = (0, 1) is universal for any s-wave halo in the limit. Through these analyses, we find some new properties such as dα-dominance in 6 Li scattering, the peninsula of large deformation beyond the IoI, and the universal point of H for s-wave halo nuclei. These new concepts help us to understand nuclei clearly and concisely..

(6) Acknowledgements This thesis, while all my responsibility, was built on support by many great people. With their warm guidance, advice, encouragement, and help, I was able to make my college life meaningful. I would like to express my heartfelt thanks to the people who supported me. First and foremost, I would like to express my sincere gratitude to my supervisor Masanobu Yahiro, who has supported me on all my Ph.D studies and research with his great insight. I learned not only various ideas on physics but also the ways of thinking in daily life; such as how to overcome difficulties, how to be a good leader and many more. Also, he has been a tremendous source of emotional support and then I was able to follow the path I believed in. Thanks to him, I can and will continue along my own way with confidence. Moreover, I would like to express my heartfelt gratitude to Assistant Prof. Takuma Matsumoto, who made direct contributions to this Ph.D thesis as my adviser. Discussions with him were so exciting that I learned a lot of practical techniques such as logical, computational, writing, and presentation skills. I was able to raise many humorous yet reasonable queries during the discussions because of his good nature. I would like to extend my special thanks to Associate Prof. Yoshifumi R Shimizu. His challenging questions, which came from his wide knowledge and sense, always offered a new insight from every perspective. I would like to thank Associate Prof. Kazuyuki Ogata for his thoughtfulness. When I was an undergraduate student, he gave seminars on nuclear reaction and I was fascinated in reaction dynamics. He surely led me to the field of nuclear reaction. It might have been that the title of this thesis was partly determined at that time. I would like to appreciate Associate Prof. Tomotsugu Wakasa for careful reading and giving useful comments as one of the juries. My gratefulness would go to Assistant Prof. Kosho Minomo. When he was a student of Kyushu University, he showed me the proper attitude on research as my senior. He is the model person I should follow. I wish to appreciate Associate Prof. Masaaki Kimura for teaching me how to consider and how to develop my researches with the recent trends. I would also like to show my profound appreciation to the SIGMA iii.

(7) iv collaboration group members, in particular, Associate Prof. Mitsunori Fukuda, Assistant Prof. Maya Takechi, Assistant Prof. Daiki Nishimura, and Prof. Takeshi Suzuki. Discussions with those experimentalists gave me different perspectives that I could have never obtained only through usual discussions at my laboratory. I also feel proud in having part in the developments of a study on reaction cross sections, which is actually the first study I did in my research life. I would like to appreciate the organizers of TALENT (Training in Advanced Low Energy Nuclear Theory) Course 5: Theory for exploring nuclear structure experiments, held in GANIL, Caen, France in August 2014, in particular, Prof. Richard F. Casten for giving me concrete suggestions on how to advance the “project” as my supervisor. I developed my understanding of nuclei through the educational lectures, difficult exercises, and thoughtful discussions. I am also grateful to the talented students for making the three weeks more valuable and memorable. I am willing and looking forward to study together again in the near future. I would like to thank all my fellow labmates. First, I would like to appreciate Shingo Tagami for offering professional suggestions every two hours, although they were sometimes beyond my understanding. I appreciate my friends Mitsuhiro Shimada and Junichi Takahashi for sharing joys and difficulties at the same time and place. I am extremely fortunate that you are my friends. I thank my fellow labmates Masakazu Toyokawa, Satoru Sasabe, Masahiro Ishii, and Junpei Sugano. I really enjoyed discussing everything, holding conversations, watching soccer games, driving every second day, drinking coffee, and so on. Of course, I will never forget the rice happening with my good memory! All of you have enriched my daily life. This work was supported by Grant-in-Aid from Japan Society for the Promotion of Science (JSPS) for JSPS Fellows (Grant No. 25·4319). Last but not least, I would like to express my deepest appreciation to my family: my parents, sisters, and grandparents for supporting me all my life..

(8) Contents Abstract. i. Acknowledgements. iii. 1 Introduction 1.1 Nuclear properties . . . . . . . . . . . . . . . . . . 1.2 Dynamic properties in nuclear reactions . . . . . . . 1.3 Static properties in nuclear structures . . . . . . . . 1.3.1 Island of inversion and nuclear deformation . 1.3.2 Halo nuclei . . . . . . . . . . . . . . . . . . 1.4 Overview of this thesis . . . . . . . . . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 2 Dynamic properties of 6 Li elastic scattering 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Formulation of three-body CDCC . . . . . . . . . . . . . . 2.2.1 Three-body Hamiltonian in the d + α + T model . . 2.2.2 Exact and CDCC wave functions . . . . . . . . . . 2.2.3 CDCC equation . . . . . . . . . . . . . . . . . . . . 2.2.4 Cross sections . . . . . . . . . . . . . . . . . . . . . 2.2.5 Discretization of breakup states . . . . . . . . . . . 2.2.6 Comparison between bin and pseudostate methods 2.2.7 Model setting in three-body CDCC . . . . . . . . . 2.3 Formulation of four-body CDCC . . . . . . . . . . . . . . . 2.3.1 Four-body Hamiltonian in the n + p + α + T model 2.3.2 Four-body CDCC wave function . . . . . . . . . . . 2.3.3 GEM in the n + p + α three-body model . . . . . . 2.3.4 Model setting in four-body CDCC . . . . . . . . . . 2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 6 Li + 209 Bi elastic scattering . . . . . . . . . . . . . v. . . . . . .. . . . . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . .. . . . . . .. 1 1 2 5 5 7 8. . . . . . . . . . . . . . . . .. 11 11 14 14 14 18 18 19 24 26 28 28 29 31 33 37 37.

(9) vi. CONTENTS. 2.5. 2.6. 2.4.2 6 Li + 208 Pb elastic scattering . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Four- and three-body channel-coupling effects 2.5.2 Effective d + α + T three-body model . . . . . Short summary . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 3 Static properties of neutron-rich Mg isotopes 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Double folding model . . . . . . . . . . . . . . . . . . . 3.2.2 AMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Test calculation of the DFM for stable nuclei . . . . . . 3.3.2 Application of the AMD+DFM to Mg isotopes . . . . 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Border of the island of inversion . . . . . . . . . . . . . 3.4.2 AMD-RGM calculation for 37 Mg . . . . . . . . . . . . . 3.5 Phenomenological analyses with def-WS . . . . . . . . . . . . 3.5.1 Formulation of def-WS model . . . . . . . . . . . . . . 3.5.2 Matter radii of Mg isotopes . . . . . . . . . . . . . . . 3.5.3 Weak-binding effects on σR of 37 Mg and deformed halo 3.6 Short summary . . . . . . . . . . . . . . . . . . . . . . . . . . 4 A measurable parameter quantifying halo nature and its universal properties 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 4.2 Theoretical analyses . . . . . . . . . . . . . . . . . 4.2.1 Absorption cross section . . . . . . . . . . . 4.3 Phenomenological analyses . . . . . . . . . . . . . . 4.3.1 Empirical values of halo parameters . . . . . 4.3.2 Halo parameter in the weak-binding limit . . 4.3.3 Properties of weakly-bound states . . . . . . 4.4 Short summary . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . .. . . . . . . . . . . . . . . .. . . . . . . . .. . . . . .. . . . . . . . . . . . . . . .. . . . . . . . .. . . . . .. . . . . . . . . . . . . . . .. . . . . . . . .. . . . . .. . . . . . . . . . . . . . . .. . . . . . . . .. . . . . .. 40 42 42 45 47. . . . . . . . . . . . . . . .. 49 49 50 51 54 56 56 57 60 60 62 66 66 68 70 72. . . . . . . . .. 75 75 77 77 80 80 83 85 87. 5 Summary and future perspective. 89. A Electric dipole transitions in the d + α two-cluster model. 93. B Matrix element of the complex-range Gaussian basis functions. 95.

(10) CONTENTS. vii. C Coulomb-breakup effects in four-body CDCC. 99. D Electric dipole transitions in the N + N + α three-cluster model. 101. E Transformation of deformation parameters. 103.

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(12) List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15. Chart of nuclides . . . . . . . . . . . . . . . . . . . . . Typical examples of three- and four-body scattering . . Breakup effects on d + 58 Ni scattering . . . . . . . . . Chart of nuclides around the IoI . . . . . . . . . . . . . Interaction radii of some light nuclei determined from interaction cross sections σI . . . . . . . . . . . . . . . Overview of this thesis . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the measured . . . . . . . . . . . . . . . .. . . . .. . 8 . 10. Breakup effects on 6 Li elastic scattering . . . . . . . . . . . . . . . . . . Coordinates in the d + α + T three-body model . . . . . . . . . . . . . Discretization in the momentum-bin method . . . . . . . . . . . . . . . The schematic k-distribution in the wave function . . . . . . . . . . . . Coordinates in the n + p + α + T four-body model . . . . . . . . . . . Three sets of Jacobi coordinates . . . . . . . . . . . . . . . . . . . . . . Energy spectrum of 6 Li with reference to the npα threshold . . . . . . . Rutherford ratio of elastic cross sections for 6 Li + 209 Bi scattering at Ein = 29.9 and 32.8 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . Rutherford ratio of elastic cross sections for 6 Li + 209 Bi scattering at Ein = 32.8–50 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rutherford ratio of elastic cross sections for 6 Li + 209 Bi scattering at Ein = 24, 28, and 29.9 MeV . . . . . . . . . . . . . . . . . . . . . . . . Total reaction cross section σR as a function of Ein . . . . . . . . . . . Elastic cross sections (normalized by the Rutherford cross section) for 6 Li + 208 Pb scattering at 29–210 MeV . . . . . . . . . . . . . . . . . . Elastic S-matrix elements for 6 Li + 208 Pb scattering at Ein = 39 and 210 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dα-probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic cross sections (normalized by the Rutherford cross section) for 6 Li + 208 Pb scattering at Ein = 39 and 210 MeV . . . . . . . . . . . . . ix. 3 3 4 7. 13 15 21 26 29 31 35 38 39 40 40 41 42 44 45.

(13) x. LIST OF FIGURES 2.16 Schematic picture of channel-coupling effects . . . . . . . . . . . . . . . 46 2.17 Same as Fig 2.8, but the results of three-body CDCC calculations with UdSF are plotted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14. 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8. Interaction cross sections for Ne isotopes on C targets at 240 MeV/nucleon Coordinates in the double folding model . . . . . . . . . . . . . . . . . Test calculations of the DFM for stable nuclei . . . . . . . . . . . . . . Total binding energy per nucleon and one-neutron separation energy as a function of mass number of Mg isotopes . . . . . . . . . . . . . . . . Deformation parameter |β2 | for Mg isotopes . . . . . . . . . . . . . . . Reaction cross sections σR for the scattering of Mg isotopes on a 12 C target at 240 MeV/nucleon . . . . . . . . . . . . . . . . . . . . . . . . . Reaction cross sections for the scattering of Mg and Ne isotopes on a 12 C target at 240 MeV/nucleon and Deformation parameter |β2 | . . . . A schematic picture of the peninsula of large deformation . . . . . . . . Energy spectra and Reaction cross sections with AMD and AMD-RGM Nilsson diagram of 37 Mg calculated by AMD . . . . . . . . . . . . . . . Comparison between AMD and def-WS calculations in reaction cross sections for Mg isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . Relation between the rms matter radius and the σR . . . . . . . . . . . The rms radii of Mg isotopes deduced from measured σR . . . . . . . . Nilsson diagram for 37 Mg calculated by the def-WS model and sensitivity of σR for Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51 53 57. Coordinates in the c + n + T three-body system . . . . . . . . . . . . . Schematic picture of the relation between H and the cross sections [σabs (a), σabs (c), σabs (n)] . . . . . . . . . . . . . . . . . . . . . . . . . . . The difference between σR (p) and σR (n) . . . . . . . . . . . . . . . . . Total reaction cross sections for p + 12 C scattering as a function of Ein Location of halo nuclei in the Sn -H plane . . . . . . . . . . . . . . . . . The rms matter radii of core and halo nuclei . . . . . . . . . . . . . . . Behavior of H as a function of Sn . . . . . . . . . . . . . . . . . . . . . z-integrated projectile densities ρ̄(b) of 11 Be, 17 C and 31 Ne . . . . . . .. 78. 59 60 61 62 63 65 65 69 70 71 72. 80 82 82 83 84 85 88. A.1 The d + α two-cluster model and its coordinates. . . . . . . . . . . . . 94 C.1 Coulomb-breakup effects on elastic cross sections of 6 Li on a 209 Bi target at Ein = 28–50 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 D.1 The N + N + α three-cluster model and its coordinates . . . . . . . . . 102.

(14) List of Tables The binding energy and the root-mean-square matter radius of the 6 Li ground state in the d + α two-body model . . . . . . . . . . . . . . . . 2.2 Parameter set of Vdα . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Optical potentials Ud and Uα used in CDCC calculations for scattering of 6 Li + 209 Bi at 29.9 and 32.8 MeV . . . . . . . . . . . . . . . . . . . . 2.4 Parameters of basis functions for I π = 1+ . . . . . . . . . . . . . . . . . 2.5 Parameters of basis functions for I π = 2+ . . . . . . . . . . . . . . . . . 2.6 Parameters of basis functions for I π = 3+ . . . . . . . . . . . . . . . . . 2.7 The binding energy and the root-mean-square matter radius of the 6 Li ground state in the n + p + α three-body model . . . . . . . . . . . . . 2.8 Parameter sets of optical potentials Un , Up , and Uα for scattering of 6 Li + 209 Bi at 24–50 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Parameter sets of optical potentials Un , Up , and Uα for scattering of 6 Li + 208 Pb at 29 and 39 MeV . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Parameter sets of optical potentials Un , Up , and Uα for scattering of 6 Li + 208 Pb at 73.7 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Parameter sets of optical potentials Un , Up , and Uα for scattering of 6 Li + 208 Pb at 210 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. 27 27 28 33 34 34 35 36 36 37 37. 3.4. Spin-parities I π , one-neutron separation energies Sn , and deformation parameters β̄ and γ̄ predicted by AMD . . . . . . . . . . . . . . . . . . The Woods-Saxon-potential parameters . . . . . . . . . . . . . . . . . . The deformation parameter set (β̄, γ̄) in AMD and the corresponding parameter set (β2 , γ) in the def-WS model for Mg isotopes . . . . . . . The rms radii of Mg isotopes deduced from measured σR . . . . . . . .. 4.1. Actual systems considered for the derivation of H . . . . . . . . . . . . 81. 3.1 3.2 3.3. 58 67 68 71. A.1 Recoil parameters in the 6 Li and 6 He two-cluster models. . . . . . . . . 94. xi.

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(16) Chapter 1 Introduction More than one century has passed since Rutherford found a nucleus as a massive core in an atom [1] through analyses of the scattering data measured by Geiger and Marsden [2]. During the last hundred years, a huge variety of nuclear properties has been discovered experimentally and theoretically. Nowadays, our interest is moving to unstable nuclei far from stable nuclei, which are great sources of new properties. Thanks to the recent development in production of radioactive ion beams, many exotic data have been accumulated for unstable nuclei rapidly. Now is the time to understand stable and unstable nuclei simultaneously.. 1.1. Nuclear properties. The nucleus is a quantum many-body system composed only of two kinds of fermions, i.e., neutrons and protons. Neutrons and protons are interacting each other through nuclear and Coulomb forces and form self-bound systems by themselves. The selfbound systems are nuclei. They can be classified with the neutron and proton numbers (N, Z), and each nucleus has its own property. Figure 1.1 is the chart of nuclides that specifies such nuclei by N in the horizontal axis and Z in the vertical axis. Each small cell corresponds to a nucleus, and the several theoretical and phenomenological analyses predict that there exist about 10,000 kinds of nuclei. Among them, black ones denote stable nuclei, while the remainders are all unstable nuclei. One can see from this nuclear chart that the number of stable nuclei is at most 300 and a huge number of unstable nuclei exist. The numbers 2, 8, 20, 28, 50, 82, and 126 are the so-called magic numbers. When a nucleus has the magic number for N and/or Z, it is stabilized more strongly than the neighbor nuclei. This property is called “magicity”. Also, nuclei have saturation properties for the density (ρ) at ρ ∼ 0.16 fm−3 and the biding energy (BE) at BE ∼ 8 MeV/nucleon. Until about the 1970s, nuclear physics 1.

(17) 2. Chap. 1 Introduction. were understood along the stability line (N ∼ Z), and the basic properties were simply believed to persist for unstable nuclei. A paradigm shift was brought about by new radioactive beam facilities such as RIKEN in Japan, GSI in Germany, and MSU in the US. Now, we are entering exotic regions far from the stability line. The experimental data obtained by the facilities showed that the magicity and the saturation properties are no longer valid and that unstable nuclei have novel properties. In particular, a “halo nucleus” in which one or two nucleons surround very far from a core nucleus is the most dramatic discovery. In the vicinity of the neutron-drip line, unstable nuclei have small one- or two-neutron separation energies such as Sn or S2n . 1 MeV. The nuclei turn out to have extremely dilute densities and much larger mean square radii compared with stable nuclei with the same mass. The weak-binding properties may lead to new nuclear properties we never expected. In order to understand nuclear physics comprehensively, we should study not only static properties of an isolated nucleus but also dynamic properties of a projectile nucleus in scattering processes. Here, static properties mean binding energies, saturation properties, magicities, and so on, whereas dynamic properties do projectile-breakup mechanism, dynamic core excitation in projectile, and so on. Typical examples of dynamic and static properties are shown in the following sections. Analyses of both the properties simultaneously allow us to understand a nucleus from different perspectives.. 1.2. Dynamic properties in nuclear reactions. Nuclear reaction is characterized by a projectile (P), a target (T), an incident energy (Ein ), and so on. Comprehensive understanding of scattering processes is a final goal of studies on nuclear reaction. This is nothing but an elucidation of dynamic properties. The many-body scattering problem is, however, quite difficult to solve directly. We then often pick up the important degrees of freedom from the many-body system. This reduces a many-body problem to a few-body one. In Fig. 1.2, deuteron scattering is described by the n + p + T three-body model, and 6 Li scattering is by the n + p + α + T four-body model. This procedure makes it possible to understand dynamic properties of nucleus-nucleus scattering. Projectile breakup is a typical dynamic property in scattering induced by weaklybound nuclei. A treatment of projectile-breakup effects is essential to describe the scattering. The continuum-discretized coupled-channels method (CDCC) is a fully quantum-mechanical method for treating projectile breakup, that is, dynamics among various kinds of channels including breakup (continuum) channels [4–6]. The theoret-.

(18) 1.2 Dynamic properties in nuclear reactions. 3. Figure 1.1: Chart of nuclides. The horizontal and vertical axes are the neutron and proton numbers, respectively. Each cell represents a single nucleus. Black (light-pink) cells denote stable (unstable) nuclei that are experimentally known. Deep-pink cells are isotopes created at RIKEN for the first time. The yellow area is the predicted region for existence of nucleus. Nuclei in the blue- and pink-shaded areas are expected to be produced at RIBF in nuclear spallation reactions of injected nuclei provided by a stable nucleus beam, and in nuclear fission reactions induced by the uranium beam, respectively. This figure is taken from Ref. [3].. p. 6Li. d. T n p. (a) Three-body scattering. T. n α. (b) Four-body scattering. Figure 1.2: Typical examples of three- and four-body scattering. (a) The d + T scattering is described by the n + p + T three-body model. (b) The 6 Li + T scattering is described by the n + p + α + T four-body model. For each scattering, a target nucleus (T) is assumed to be inert..

(19) 4. Chap. 1 Introduction. ical foundation of CDCC is shown by using the distorted Faddeev equation [5]. CDCC was first applied to d scattering by assuming the n + p + T three-body model; see Fig. 1.2(a). Figure 1.3 shows the elastic cross section for a 58 Ni target at Ein = 56 MeV. The solid line represents the result of CDCC calculations with d-breakup effects, whereas the dashed and dotted lines are the results of CDCC calculations with no d-breakup effect; more precisely, the D-state component of the deuteron ground state is included in the dashed line, but not in the dotted line. The details of CDCC calculations are shown in the caption of Fig. 1.3. The solid line is in excellent agreement with the experimental data up to backward angles. This suggests that d-breakup effects are essential for d scattering. Further analyses were made for d scattering in a wide range of incident energies, say Ein . 700 MeV [4]. The analyses showed that d-breakup effects are significant in the wide range of Ein . Nowadays, three-body dynamics in scattering of two-body projectiles is often analyzed by CDCC.. Figure 1.3: Breakup effects on d + 58 Ni elastic scattering at 56 MeV. The Rutherford ratio of elastic cross sections is plotted as a function of the scattering angle. The legends show channels included in CDCC calculations. Here, S0 (D0 ) means the deuteron ground-state channel with S-state (D-state) component only, and S ∗ (D∗ ) stands for d-breakup channels with S-state (D-state). This figure is taken from Ref. [4].. Our interest is now going to four-body dynamics in scattering of three-body projectiles; see Fig. 1.2(b) for 6 Li scattering as a typical case. CDCC for three- and fourbody scattering are now called three- and four-body CDCC, respectively. Four-body.

(20) 1.3 Static properties in nuclear structures. 5. CDCC is an extension of three-body CDCC, but the formulation is not straightforward because it is not easy to prepare the three-body continuum states of projectile before doing coupled-channel calculations. This problem was solved by two approaches; one is the combination [7] of the pseudostate discretization and the complex scaling method [8] and the other is the combination [9] of the momentum-bin discretization and the hyperspherical harmonics method [10]. Four-body CDCC is one of state-ofthe-art calculations in nuclear physics. In Chapter 2 of this thesis, we apply four-body CDCC to 6 Li elastic scattering. So far, 6 Li scattering was often described by the d + α + T three-body model. However, this is questionable, since d is a weakly-bound nucleus and the breakup effects are significant for d scattering as shown above. We then treat 6 Li scattering by assuming the n + p + α + T four-body model to take into account d-breakup explicitly. We focus on the four-body breakup process (6 Li + T → n + p + α + T) and the three-body breakup process (6 Li + T → d + α + T), and reveal breakup dynamics.. 1.3. Static properties in nuclear structures. 1.3.1. Island of inversion and nuclear deformation. In 1949, Mayer and Jensen made an important discovery about nuclear structure [11]. They introduced the strong spin-orbit coupling to the nuclear mean-field potential and excellently described the magic numbers. This work is the beginning of the nuclear shell model and provides the concept of single-particle motion in a complex nuclear many-body system. The concept is quite fundamental even in state-of-the-art nuclear many-body calculations, and makes it possible for us to understand nuclei simpler and clearer. The basic magicity is, however, no longer valid in some unstable nuclei. An example is the so-called “island of inversion (IoI)” [12]. Figure 1.4 shows the chart of nuclides around the IoI (N ∼ 20 and Z ∼ 12). 40 Ca is a doubly-magic stable nucleus with N = Z = 20. In 40 Ca, the large shell gap between sd- and pf -shells prevents neutrons (and protons) from getting excited easily. This makes 40 Ca stable and spherical. By contrast, 32 Mg is largely deformed in spite of having N = 20 [13,14]. The experimental result suggests that the N = 20 magicity disappears. Nowadays, the region of 10 Ne, 11 Na, 12 Mg isotopes around N ∼ 20 shown by the purple area is called the “island of inversion (IoI)”. However, the border of the IoI is still under consideration. We then discuss this point in this thesis. The history of the IoI dates back to 1969 when the first mass measurement was performed for neutron-rich Na isotopes by Klapisch and Thibault [15, 16]. It was pointed.

(21) 6. Chap. 1 Introduction. out that the observed two-neutron separation energy (S2n ) is anomalously larger than the Hartree-Fock prediction of the day. In 1990, Warburton et al. suggested that intruder (sd)−2 (pf )2 2~ω-configurations gain larger correlation energies than normal 0~ω-configurations [12]. This means that the N = 20 magicity disappears. The systematic measurements of neutron-rich nuclei near N = 20 [17] reinforced the prediction and indicated that the IoI extends the Z = 10–12 isotopic chains. Another exotic property seen in the IoI is strong deformation induced by the shell evolution. The large deformation was suggested by low excitation energies [18–22] and large B(E2) values [14, 23–25]. Recently, it was also suggested that the tensor component of nucleon-nucleon interaction plays an important role behind the shell evolution. Since nuclear deformation is closely related to the shell evolution, systematic determination of deformation is highly desired as a good indicator of the shell evolution. Traditionally, the deformation parameter β2 is often deduced from transition prob+ abilities between the ground and first 2+ excited states B(E2; 0+ gs → 21 ). For very neutron-rich nuclei, however, the measurements are not easy. Therefore, the excitation energies E(2+ ), E(4+ ), and the ratio R4/2 = E(4+ )/E(2+ ) via in-beam γ-ray spectroscopy are mainly used to deduce the deformation parameter. In fact, the measurements were carried out for Ne, Na, and Mg isotopes in and beyond the IoI [19–22] and suggested that there exist large deformation and shell quenching around the IoI. Furthermore, new measurements are scheduled for heavier nuclei such as 52 Ar, 62 Ti, and 78 Ni as the SEASTAR project (Shell Evolution And Search for Two-plus energies At RIBF) [26], and the systematic data on E(2+ ) are expected to be accumulated. However, in-beam γ-ray spectroscopy alone is not enough to deduce nuclear deformation precisely. In chapter 3 of this thesis, we also propose systematic measurements of reaction cross sections σR as a complementary approach to deduce deformation. The IoI is thus a typical example of the shell evolution and the elucidation is still quite important. The Radioactive Isotope Beam Factory (RIBF) in RIKEN has achieved a remarkable breakthrough on studies of the IoI, and produced novel data on the IoI. In fact, very recently, the σR were systematically measured for neutron-rich Mg isotopes. Mg isotopes cross the IoI and reach the drip-line nucleus 40 Mg with N = 28. Therefore, we can discuss (i) where the border of the IoI is and (ii) whether the N = 20 and 28 magicities still survive. In this thesis, we deduce ground-state properties (spin parity, total binding energy, and deformation) of Mg isotopes and then reanalyze the IoI by comparing the experimental data on σR with our microscopic model, and answer questions (i) and (ii) through the analysis..

(22) 1.3 Static properties in nuclear structures. 7. Figure 1.4: Chart of nuclide around the IoI. 40 Ca (N = 20) shows the magicity in which the large shell gap between sd- and pf -shells appears. This makes 40 Ca stable and spherical. By contrast, 32 Mg (N = 20) is largely deformed. This means that the N = 20 magicity is no longer valid. The region shown by the purple area is now called the “island of inversion”.. 1.3.2. Halo nuclei. In 1985, Tanihata et al. found a halo structure in 11 Li through systematic measurements of interaction cross sections (σI ) [27]. 11 Li is still the best example of a halo nuclei. 11 Li has a 9 Li core and two extra neutrons with very small two-neutron separation energy (S2n ∼ 300 keV). The nuclear radius is much larger than the empirical formula r0 A1/3 for stable nuclei, where r0 ∼ 1.2 fm and A is the mass number. The radius is even equal to that of 208 Pb. Figure 1.5 shows interaction radii for several nuclei determined from measured σI [28]. Prominent enhancements correspond to 11 Li, 11 Be, 14 Be, 17 B, and 17 Ne. Further systematic measurements confirmed that 6 He, 14 Be, 17 B, 19 B, and 22 C are twoneutron halo nuclei, and 11 Be, 15 C, and 19 C are one-neutron halo nuclei [29–31]. The experimental exploration of heavier halo nuclei is in progress. Nowadays, 31 Ne and 37 Mg are considered to be halo nuclei [32–36]. Thus, the experimental evidences on halo nuclei have been just accumulated. Now is the time to understand these halo nuclei systematically. In chapter 4 of this thesis, we propose a parameter quantifying halo nature of weakly-bound nuclei..

(23) 8. Chap. 1 Introduction. Figure 1.5: Interaction radii of some light nuclei determined from the measured interaction cross sections σI . The interaction radius (RI ) is defined by σI = π[RI (P) + RI (T)]2 . Prominent enhancements correspond to 11 Li, 11 Be, 14 Be, 17 B, and 17 Ne. This figure is taken from Ref. [28].. 1.4. Overview of this thesis. As mentioned above, many weakly-bound nuclei and the scattering of the nuclei can be well described by few-body models. In order to understand few-body dynamics systematically and deeply, we should analyze both dynamic and static properties. In this thesis, we then address the following three subjects: see Fig. 1.6 for the summary. (I) How is projectile-breakup mechanism in 6 Li elastic scattering? (II) How are ground-state properties of neutron-rich Mg isotopes? (III) What is a measurable parameter quantifying halo nature of unstable nuclei?.

(24) 1.4 Overview of this thesis. 9. As illustrated in Fig. 1.6, subject (I) is addressed in chapter 2, subject (II) in chapter 3, and subject (III) in chapter 4. In Chap. 2, we investigate dynamic properties of projectile-breakup processes in 6 Li elastic scattering by using four-body CDCC based on the n + p + α + T four-body model. In particular, we separate projectilebreakup processes into four- and three-body breakup ones, and clarify which process is important. In Chap. 3, we investigate static properties of neutron-rich Mg isotopes through analyses of recently-measured σR . Our fully-microscopic framework is based on Antisymmetrized molecular dynamics and the double folding model [37, 38]. The ground-state properties are deduced from the measured σR , and the border of the IoI is discussed. In Chap. 4, we propose a parameter H quantifying the halo nature. The empirical values of H are deduced from measured σR and extrapolated with the Eikonal and adiabatic approximation to the weak-binding limit. The universal property of H near the limit is discussed analytically and numerically. Chapter 5 is devoted to a summary and future perspective. This thesis is mainly based on the following three papers: 1. “Effects of four-body breakup on 6 Li elastic scattering near the Coulomb barrier” S. Watanabe, T. Matsumoto, K. Minomo, K. Ogata, and M. Yahiro, Physical Review C 86, 031601(R) (2012). 2. “Ground-state properties of neutron-rich Mg isotopes” S. Watanabe, K. Minomo, M. Shimada, S. Tagami, M. Kimura, M. Takechi, M. Fukuda, D. Nishimura, T. Suzuki, T. Matsumoto, Y. R. Shimizu, and M. Yahiro, Physical Review C 89, 044610 (2014). 3. “Four-body dynamics in 6 Li elastic scattering” S. Watanabe, T. Matsumoto, K. Ogata, and M. Yahiro, Physical Review C 92, 044611 (2015)..

(25) 10. Chap. 1 Introduction. Figure 1.6: Overview of this thesis. In Chap. 2, we investigate dynamic properties of breakup processes of 6 Li elastic scattering. In Chap. 3, we investigate static properties of neutron-rich Mg isotopes through analyses of the recently-measured σR . In Chap. 4, we propose a parameter H quantifying the halo nature and investigate the universal properties..

(26) Chapter 2 Dynamic properties of 6Li elastic scattering 2.1. Introduction. Systematic understanding of reaction mechanisms is an important subject in nuclear physics. This is nothing but understanding of dynamic properties of nuclei. In particular, systematic understanding of breakup dynamics of weakly-bound projectile is a long-standing subject. As three-body dynamics in the scattering of two-body projectiles such as d (= p + n), the importance of projectile-breakup processes was already confirmed. By contrast, as for four-body dynamics in the scattering of three-body projectiles such as 6 Li (= p + n + α), it is not understood well because the fully-quantum treatment of four-body scattering is extremely difficult. In four-body scattering, it is quite non-trivial which kind of breakup processes is favored. The continuum-discretized coupled-channels method (CDCC) is a fully-quantummechanical method to treat breakup processes not only for three-body scattering [4–6] but also for four-body scattering [40,41]. CDCC for three- and four-body scattering are called three- and four-body CDCC. Four-body CDCC was first proposed in Ref. [40] to describe 6 He scattering. The method was successful in reproducing measured elastic cross sections of the scattering. After that, four-body CDCC was further developed and applied to various kinds of reactions [7, 9, 40–50]. In this chapter, we consider 6 Li scattering. 6 Li is well described in the n + p + α three-body model, while 6 He is by the n + n + α model. In this sense, 6 Li is similar to 6 He. However, breakup dynamics of 6 Li is quite different from that of 6 He. 6 He is a well-known Borromean system in which the subsystems have no bound state. Therefore, the ground and breakup states of 6 He are constructed with only nnα threebody configurations. By contrast, the n + p subsystem has a bound state (d) in 6 Li. 11.

(27) 12. Chap. 2 Dynamic properties of 6 Li elastic scattering. Therefore, the ground and breakup states of 6 Li have dα two-body configurations as well as npα three-body configurations. This situation leads to the competition of two types of breakup processes in 6 Li scattering from a target (T): Four-body breakup process : Three-body breakup process :. 6. Li + T → n + p + α + T, 6 Li + T → d + α + T.. (2.1). Dynamics of 6 Li scattering is thus richer than that of 6 He scattering. This is the reason why 6 Li is analyzed by four-body CDCC. The importance of projectile breakup in 6 Li scattering was first pointed out by Satchler and Love [51, 52]. As a pioneering work, they performed systematic analyses on nucleus-nucleus elastic scattering with the optical potential constructed by the double folding (DF) model. Their calculations well describe measured elastic cross sections for any incident energy and target except for weakly-bound projectiles including 6 Li. For weakly-bound projectiles, the factor NR ≈ 0.6 for the real part of the DF potential is necessary to reproduce the measured cross sections, suggesting that the repulsive correction to the DF-potential is needed. The failure of the folding model for 6 Li scattering was solved by taking into account breakup effects with three-body CDCC based on the d + α + T three-body model [4]. Figure 2.1 shows elastic cross section for 6 Li + 208 Pb scattering at 99 MeV. The factor NI = 0.621 for the imaginary part of the DF-potential determined to minimize the difference between the model result and the experimental data; here NR is fixed at 1 in the model calculation. The dotted line represents the model calculation, but it cannot reproduce the experimental data. The solid line shows the result of three-body CDCC and it well describes the experimental data by virtue of breakup effects. As an interesting result, the DF-potential with NR = 0.633 and NI = 0.621 yields almost the same result as the CDCC result. The problem suggested in Refs. [51, 52] was thus resolved by 6 Li-breakup effects. However, this statement should be reinforced by four-body CDCC, since the d + α + T three-body model was assumed in the CDCC calculation. Keely et al. [53] analyzed 6 Li + 209 Bi elastic scattering near the Coulomb-barrier energy EbCoul ≈ 30 MeV with three-body CDCC based on the d + α + T model. However, the CDCC calculation could not reproduce the experimental data. They then introduced the normalization factor 0.8 for the phenomenological optical potentials of d-T and α-T systems to reproduce the experimental data. This problem is expected to be solved by four-body CDCC based on the n + p + α + T model. 6. Chapter 2 is devoted solely to dynamic properties of a weakly-bound stable nucleus Li, since the static properties, such as the total binding energy and the root-mean-.

(28) 2.1 Introduction. 13. Figure 2.1: Breakup effects on 6 Li +208 Pb elastic scattering at 99 MeV. The solid line shows the result of three-body CDCC calculation, and the dotted (dashed) line represents the result of the DF-potential with NR = 1 (NR = 0.633). This figure is taken from Ref. [4].. square radius, are known well. In order to achieve this purpose, we analyze 6 Li elastic scattering in a wide range of incident energies. 6 Li is constructed in the n + p + α three-body model. The three-body structure is constructed by the Gaussian expansion method (GEM) [54], and the scattering dynamics is then described with four-body CDCC based on the n+p+α+T model. Four-body CDCC reproduces the experimental data for 6 Li + 209 Bi scattering at Ein = 24–50 MeV and 6 Li + 208 Pb scattering at Ein = 29–210 MeV with no adjustable parameter. This allows us to investigate the four-body dynamics clearly. As an interesting result, 6 Li-breakup is mainly induced by d + α breakup and hardly caused by n + p + α breakup in the present range of Ein . We refer to this property as the dα-dominance, and clarify the reason why the dα-dominance is realized. Finally, we propose the effective three-body model that simulates the four-body calculation reasonably well..

(29) 14. 2.2. Chap. 2 Dynamic properties of 6 Li elastic scattering. Formulation of three-body CDCC. In this paper, we analyze 6 Li + T scattering with four-body CDCC based on the n + p + α + T four-body model and with three-body CDCC based on the d + α + T three-body model. We first formulate three-body CDCC.. 2.2.1. Three-body Hamiltonian in the d + α + T model. We consider 6 Li + T scattering by assuming the d + α + T three-body model. For later explanation, we first define the coordinate between d and α by r, and that between the center of mass of 6 Li and T by R (see Fig. 2.2). Three-body dynamics of 6 Li scattering is governed by the Schrödinger equation (H3 − E)Ψ(r, R) = 0,. (2.2). where Ψ represents the total wave function, and E denotes the total energy in the center-of-mass system. The three-body Hamiltonian H3 is given by H3 = KR + Ud + Uα +. e2 ZLi ZT + hdα , R. hdα = Kr + Vdα ,. (2.3) (2.4). where KR stands for the kinetic-energy operator regarding to R, and Ux (x = d, α) represents the optical potential between x and T. In the internal Hamiltonian hdα of 6 Li, Kr describes the kinetic-energy operator associated with r, and Vdα is the interaction between d and α. In Eq. (2.3), the Coulomb breakup is neglected and the Coulomb interactions of d-T and α-T systems are then approximated into e2 ZLi ZT /R, where ZA is the atomic number of nucleus A. This approximation works well in 6 Li scattering, because the electric dipole transition strength vanishes in the d + α twobody model [53, 55]; see Appendix A for more detail.. 2.2.2. Exact and CDCC wave functions. The total wave function Ψ for the three-body system can be expanded in terms of the total angular momentum J and its projection onto z-axis M : Ψ(r, R) =. X JM. CJM ΨJM (r, R),. (2.5).

(30) 2.2 Formulation of three-body CDCC. 15. 6Li. T. Rd. d r. R. α. Rα. Figure 2.2: Coordinates in the d + α + T three-body model.. where CJM is the expansion coefficient. The wave function ΨJM satisfies the Schrödinger equation (H3 − E)ΨJM (r, R) = 0. (2.6) In CDCC, 6 Li-breakup processes are described as transitions among the ground state 6 (Φgs 1m1 ) and the continuum states (ΦImI ) of Li, where the subscripts I and mI are the total spin of 6 Li and its z-component, respectively. 6 Li has no bound-excited state, and the ground state has an eigenenergy ε0 = −1.47 MeV with reference to the d + α threshold [56]. The internal wave functions of 6 Li are obtained by solving the Schrödinger equation [hdα − ε0 ]Φgs 1m1 (r) = 0,. (2.7). [hdα − εk ]ΦImI (k, r) = 0,. (2.8). where εk = ~2 k 2 /(2µr ) represents the eigenenergy of ΦImI (k, r), µr is the reduced mass of the d + α system, and k is the relative wave number conjugate to r. The explicit forms of the wave functions are given by (d). gs Φgs 1m1 (r) = φ1 (r)[Y0 (r̂) ⊗ η1 ]1m1 ,. ΦImI (k, r) = φI (k, r)i` [Y` (r̂) ⊗. (d) η1 ]ImI ,. (2.9) (2.10) (d). where φgs 1 and φI are the radial wave functions of ground and continuum states, η1 denotes the spin wave function of deuteron, and ` is the orbital angular momentum between d and α. In principle, the ΨJM can be expanded with the complete set {Φgs 1m1 , ΦImI }, 1=. X m1. gs |Φgs 1m1 i hΦ1m1 |. +. XZ ImI. ∞ 0. dk |ΦImI i hΦImI | ,. (2.11).

(31) Chap. 2 Dynamic properties of 6 Li elastic scattering. 16 as ΨJM (r, R) =. J+1 X. J [Φgs 1 (r) ⊗ χ1L (K0 , R)]JM. L=|J−1|. +. ∞ J+I Z X X. ∞ 0. I=0 L=|J−I|. dk [ΦI (k, r) ⊗ χJIL (K, R)]JM. (2.12). with J [Φgs 1 (r) ⊗ χ1L (K0 , R)]JM ≡. X. J h1m1 LML |JM i Φgs 1m1 (r)χ1LML (K0 , R), (2.13). m1 +ML =M. [ΦI (k, r) ⊗. χJIL (K, R)]JM. ≡. X. hImI LML |JM i ΦImI (k, r)χJILML (K, R),. mI +ML =M. (2.14) where the expansion coefficients χJ1LML (K0 , R) and χJILML (K, R) represent the relative motion between T and 6 Li in its ground and continuum states, K0 and K denote the relative wave number between 6 Li and T satisfying the energy conservation ~2 K02 =E 2µR ~2 K 2 εk + =E 2µR ε0 +. (ε0 = −1.47 MeV),. (2.15). (εk > 0 MeV),. (2.16). where µR is the reduced mass in the 6 Li + T system. For later discussion, we rewrite ΨJM in terms of the radial and angular parts as ΨJM (r, R) =. J+1 X. φgs 1 (r). L=|J−1|. +. χJ1L (K0 , R) 1L YJM (r̂, R̂) R. ∞ J+I Z X X I=0 L=|J−I|. ∞. dk φI (k, r) 0. χJIL (K, R) IL YJM (r̂, R̂), R. (2.17). with χJIL (K, R) L i YLML (R̂), R (d) IL YJM (r̂, R̂) ≡ i` [Y` (r̂) ⊗ η1 ]I ⊗ iL YL (R̂) JM .. χJILML (K, R) ≡. (2.18) (2.19). At this stage, ΨJM is the exact solution without any approximation, but it is quite difficult to get, because the right hand side of Eq. (2.17) has an infinite number of.

(32) 2.2 Formulation of three-body CDCC. 17. continuum states, each classified by I and k. In CDCC, therefore, the exact wave function ΨJM is approximated into the CDCC wave function ΨCDCC JM (r, R) =. J+1 X. φgs 1 (r). L=|J−1|. +. IX max. χJ1L (K0 , R) 1L YJM (r̂, R̂) R. J+I n max X X. I=0 L=|J−I| n=1. φ̂nI (r). χ̂nIL (Kn , R) IL YJM (r̂, R̂). R. (2.20). This means that the complete set {Φgs 1m1 , ΦImI } of Eq. (2.11) is replaced by the approximative one spanned by a finite number of discrete states {Φgs 1m1 , Φ̂nImI }. Namely, the model space 1≈. 1 X m1 =−1. gs |Φgs 1m1 i hΦ1m1 |. +. IX max. I n max X X. |Φ̂nImI i hΦ̂nImI | ≡ P,. (2.21). I=0 mI =−I n=1. where Φ̂nImI denotes the n-th discretized-breakup state, Imax and nmax are the maximum values of I and n. The procedure of discretization will be shown in Sec. 2.2.5. The CDCC method is thus based on the approximations composed of the discretization and the truncation of I and k. This enables us to obtain the reasonable three-body wave function. The CDCC solution has been justified as the leading-order solution of the distorted Faddeev equation [5].. The discretized-breakup state can be defined so as to satisfy the orthonormality: hΦ̂nImI |Φ̂n0 I 0 m0I i = δnn0 δII 0 δmI m0I ,. (2.22). hΦ̂nImI |hdα |Φ̂n0 I 0 m0I i = ε̂nI δnn0 δII 0 δmI m0I ,. (2.23). where ε̂nI = ~2 k̂n2 /(2µr ) and the discretized wave number k̂ satisfy the relation ~2 k̂n2 ~2 Kn2 + = E. 2µr 2µR. (2.24). Now, we redefine the first term of Eq. (2.20) as φ̂01 (r) ≡ φgs 1 (r),. (2.25). χ̂J01L (K0 , R) ≡ χJ1L (K0 , R).. (2.26).

(33) Chap. 2 Dynamic properties of 6 Li elastic scattering. 18. The CDCC solution ΨCDCC becomes JM ΨCDCC JM (r, R). =. IX max. J+I n max X X. I=0 L=|J−I| n=0. φ̂nI (r). χ̂JnIL (Kn , R) IL YJM (r̂, R̂), R. (2.27). where n = 0 (n ≥ 1) denotes the ground (discretized-breakup) state. As one can make out from Eq. (2.27), the ground and continuum states are treated in the same manner in CDCC.. 2.2.3. CDCC equation. ∗ IL By inserting Eq. (2.27) into the Schrödinger equation (2.2) and multiplying φ̂nI (r)YJM (r̂, R̂) from the left, and integrating over r and R̂, we can obtain a set of coupled-channels equation for χ̂Jγ (Kn , R): . ~2 L(L + 1) e2 ZLi ZBi ~2 d2 (J) + + Uγγ (R) + − (E − ε̂nI ) χ̂Jγ (R) − 2µR dR2 2µR R2 R X (J) =− Uγγ 0 (R)χ̂Jγ0 (R), (2.28) γ 0 6=γ. with the coupling potential D E (J) IL I 0 L0 Uγγ 0 (R) = φ̂nI (r)YJM (r̂, R̂) Ud + Uα φ̂n0 I 0 (r)YJM (r̂, R̂). r,R̂. ,. (2.29). where the combination of {nIL} is represented as γ for simplicity. By solving Eq. (2.28) under the outgoing-wave boundary condition: r χ̂Jγ (R). →. (−) HL (Kn R)δγγ0. −. K0 (J) (+) Ŝ H (Kn R) Kn γγ0 L. (R > RN ),. (2.30). (J). one can get the distorted wave χ̂γ and the S-matrix elements Ŝγγ0 , where RN is the (−) (+) range of nuclear coupling potentials, HL (HL ) is the incoming (outgoing) Coulomb function, and the elastic channel γ0 = {01L0 } is defined.. 2.2.4. Cross sections. Using the S matrix obtained above, we can calculate cross sections. The elastic scattering amplitude is given by (C). (N). fM10 M1 (θ, φ) = fM 0 M1 (θ) + fM 0 M1 (θ, φ). 1. 1. (2.31).

(34) 2.2 Formulation of three-body CDCC (C). 19. (N). where fM 0 M1 (θ) and fM 0 M1 (θ, φ) are the Coulomb and nuclear scattering amplitudes 1 1 given by θ η +2iσ0 −2iη ln sin 2 δM10 M1 , e =− 2 θ 2K0 sin 2 √ X π (N) fM 0 M1 (θ, φ) = hL01M1 |JM1 i hL0 M1 − M10 1M10 |JM1 i 1 iK0 J,L,L0 (C) fM 0 M1 (θ) 1. (2.32). 0. i(σL +σL ) J YL0 ,M1 −M10 (θ, φ), (2.33) × (2L + 1)1/2 (S01L,01L 0 − δ01L,01L0 )e. where η is the Sommerfeld parameter defined by η=. ZLi ZT e2 µR , ~2 K 0. (2.34). and σL denotes the Coulomb phase shift. From the scattering amplitude, we can get the elastic cross section as dσ(θ) 1 X = |fM10 M1 (θ, φ)|2 dΩ 3 0. (2.35). M1 ,M1. =. 2.2.5. 1 |f00 (θ, φ)|2 + 2 |f11 (θ, φ)|2 + |f10 (θ, φ)|2 + |f01 (θ, φ)|2 + |f1−1 (θ, φ)|2 . 3 (2.36). Discretization of breakup states. As shown in Sec. 2.2.2, an infinite number of breakup states {ΦImI (k, r)} are replaced by a finite number of discretized-continuum states {Φ̂nImI (r)}. In order to obtain the discrete states, the “momentum-bin method” and the “pseudostate method” have been widely used. Here, we explain both the method for better understanding. In this thesis, we apply the pseudostate method to all the calculations as shown later. Note that the choice of the basis functions {Φ̂nImI (r)} is not unique even in a certain method. Momentum-bin method In the momentum-bin method, the continuous breakup states {ΦImI (k, r)} at 0 < k < kmax is divided into a momentum bins [kn , kn+1 ] and the continuum states in each bin is integrated over k. The resultant state is denoted by Φ̂nImI (r). The momentum space is called a “momentum-bin”. Namely, this method is based on the following two procedures (assumptions): (i) Truncation.

(35) Chap. 2 Dynamic properties of 6 Li elastic scattering. 20. The model space is truncated up to kmax and Imax because the transitions to high-k and/or high-I states are negligibly small.. (ii) Discretization An infinite number of wave functions in the n-th momentum-bin [kn , kn+1 ] is replaced by one representative, because they change slowly in a bin.. Now, we define the second term of Eq. (2.17) as ΨBU . With the assumption (i), we can get ΨBU (r, R) ≡. ∞ J+I Z X X I=0 L=|J−I|. ≈. IX max. ∞. dk φI (k, r) 0. J+I Z X. I=0 L=|J−I|. kmax 0. χJIL (K, R) IL YJM (r̂, R̂) R. χJIL (K, R) IL YJM (r̂, R̂). dk φI (k, r) R. (2.37). (2.38). Next, the truncated momentum space [0, kmax ] is divided into momentum bins, each with a width ∆kn = kn+1 − kn , ΨBU (r, R) ≈. IX max. J+I n max Z X X. kn+1. dk φI (k, r) kn. I=0 L=|J−I| n=1. χJIL (K, R) IL YJM (r̂, R̂), R. (2.39). where k1 = 0 and knmax +1 = kmax . With the assumption (ii), we can suppose χJIL (K, R) ≈ χJIL (Kn , R),. (2.40). and can define χ̂JnIL (R) ≡. p ∆kn χJIL (Kn , R).. (2.41). Since χ̂JnIL is independent of k, ΨBU can be rewritten as ΨBU (r, R) ≈. IX max. J+I n max Z X X. I=0 L=|J−I| n=1. =. IX max. J+I n max X X. I=0 L=|J−I| n=1. ≡ ΨCDCC (r, R), BU. kn+1 kn. φ̂nI (r). 1 χ̂JnIL (R) IL √ dk φI (k, r) YJM (r̂, R̂) (2.42) R ∆kn. χ̂JnIL (Kn , R) IL YJM (r̂, R̂) R. (2.43) (2.44).

(36) 2.2 Formulation of three-body CDCC. 21. where the n-th discretized-breakup state is defined 1 φ̂nI (r) = √ ∆kn. Z. kn+1. dk φI (k, r).. (2.45). kn. The ΨCDCC is nothing but the second term of Eq. (2.20). The procedure of discretizaBU tion is schematically illustrated in Fig. 2.3.. k. k. Transitions to high-k states are small.. The wave functions in the same bin are almost the same.. =. (i) Truncation. (ii) discretization. ・・・. ∞. k. =. 0. =. g.s.. Figure 2.3: Discretization in the momentum-bin method. √ Thanks to the introduction of the factor 1/ ∆kn in Eq. (2.45), the Φ̂nImI satisfy the following orthonormality: hΦ̂nImI |Φ̂n0 I 0 m0I i = δnn0 δII 0 δmI m0I ,. (2.46). hΦ̂nImI |hdα |Φ̂n0 I 0 m0I i = ε̂nI δnn0 δII 0 δmI m0I ,. (2.47). where ε̂nI represents the energy average in a momentum bin [kn , kn+1 ], that is, ε̂nI. Z kn+1 ~2 k 2 1 dk = ∆kn kn 2µr 2 ~ 1 2 = (kn+1 + kn2 + kn+1 kn ) 2µr 3 q ~2 k̂n2 2 ≡ k̂n = (kn+1 + kn2 + kn+1 kn )/3 . 2µr. (2.48) (2.49) (2.50). The relative wave numbers K0 and Kn are taken to satisfy the energy conservation E = ε̂0 +. ~2 K02 ~2 Kn2 = ε̂nI + . 2µR 2µR. (2.51).

(37) Chap. 2 Dynamic properties of 6 Li elastic scattering. 22 Pseudostate method. In the pseudostate method, both bound and discretized-breakup states are obtained by diagonalizing hdα with L2 -type basis functions. The wave functions are expanded with a finite number of basis functions (ϕi : i = 1–imax ), Φ(r) =. imax X. ci ϕi (r),. (2.52). i=1. where imax is the number of basis functions and ci is the expansion coefficient. The coefficient ci is determined by the Rayleigh-Ritz variational principle δε[Φ] = 0,. (2.53). with the energy functional ε[Φ] =. hΦ|hdα |Φi , hΦ|Φi. (2.54). in which the energy expectation is minimized in the model space spanned by {ϕi }. For the trial function of Eq. (2.52), the expectation value can be written as P ∗ ij ci cj hϕi |hdα |ϕj i ε[Φ] = P ∗ . ij ci cj hϕi |ϕj i. (2.55). From the condition P ∗ P P cj ϕj i hϕi |hdα | j cj ϕj i ∂ ij ci cj hϕi |hdα |ϕj i hϕi | P ∗ j − P ∗ = 0, ε[Φ] = P ∗ ∗ ∂ci ij ci cj hϕi |ϕj i ij ci cj hϕi |ϕj i ij ci cj hϕi |ϕj i. (2.56). we can obtain the generalized-eigenvalue equation in the matrix expression; .   . Hij.  . .    − ε. Nij. .      cj  = 0,. (2.57). with the Hamiltonian- and norm-matrix elements, Hij = hϕi |hdα |ϕj i ,. (2.58). Nij = hϕi |ϕj i .. (2.59).

(38) 2.2 Formulation of three-body CDCC. 23. By solving Eq. (2.57), an infinite number of continuum states is automatically discretized into a finite number of positive-energy states (ε > 0). The states thus obtained are called “pseudostates”. This is one of the merits of the pseudostate method, i.e., we do not need the exact-continuum states to get the discretized-continuum states. This fact enables us to perform the four-body CDCC calculation in which discretized three-body continuum states should be prepared. The detailed formulation of fourbody CDCC based on the n + p + α + T model is shown in Sec. 2.3. Now, we show actual calculations. Let us go back to the two-body Schrödinger equation [Eqs. (2.7) and (2.8)]: [hdα − ε]ΦÌmI (r) = 0,. (2.60) (d). ΦÌmI (r) = φÌ (r)i` [Y` (r̂) ⊗ η1 ]ImI ,. (2.61). where the angular momentum ` is restored for later explanation. If the radial wave function is defined by φÌ (r) ≡. uÌ (r) , r. (2.62). Eq. (2.60) is reduced to the non-local Schrödinger equation: Z (r) [hdα. − ε]uÌ (r) = −. VOCM (r, r0 )uÌ (r0 )dr0 ,. (2.63). with ~2 d2 ~2 `(` + 1) + + Vdα (r), 2µ dr2 2µr r2 n o (FS)∗ (FS) (FS)∗ (FS) VOCM (r, r0 ) = lim λ u`=0 (r)u`=0 (r0 )δ`0 + u`=1 (r)u`=1 (r0 )δ`1 , (r). hdα = −. λ→∞. (2.64) (2.65). (FS). where VOCM is the pseudo-potential eliminating the forbidden states u`=0 for ` = 0 (FS) and u`=1 for ` = 1. In this study, we adopt the complex-range Gaussian functions [54]. The basis functions are defined by ϕi` (r) + ϕ∗i` (r) , 2 ϕ∗ (r) − ϕi` (r) 2 , gi` (r) = Ni`sin r`+1 e−λi r sin ωi r2 = Ni`sin i` 2i 2. fi` (r) = Ni`cos r`+1 e−λi r cos ωi r2 = Ni`cos. (2.66) (2.67).

(39) 24. Chap. 2 Dynamic properties of 6 Li elastic scattering. with the complex-range Gaussian function 2. ϕi` (r) = r`+1 e−(λi +iωi )r ,. (2.68). where λi and ωi are the range and oscillation parameters. Ni`cos and Ni`sin are the normalization factors determined from hfi` |fi` i = 1,. (2.69). hgi` |gi` i = 1,. (2.70). to stabilize the numerical calculation. With the basis, the radial wave function is expanded as uÌ (r) =. imax X. jmax. ci fi` (r) +. i=1. X. dj gj` (r),. (2.71). j=1. where ci and di are the expansion coefficients. The matrix elements are summarized in Appendix. B.. 2.2.6. Comparison between bin and pseudostate methods. It is instructive to compare the discretized-continuum states obtained by the bin (ps) (bin) method (Φ̂n ) and those by the pseudostate method (Φ̂n ). In order to understand what the discretized states are, we consider the following two overlap functions Φ̃(bin) (k) ≡ hΦ(k, r)|Φ̂(bin) (r)ir , n n Φ̃(ps) n (k). ≡. hΦ(k, r)|Φ̂(ps) n (r)ir. .. (2.72) (2.73). These functions show the distribution of the exact wave function Φ(k, r) in a certain discretized-continuum state. Similarly, we can define Φ̃(k, kc ) ≡ hΦ(k, r)|Φ(kc , r)ir Z = Φ∗ (k, r)Φ(kc , r)dr = δ(k − kc ),. (2.74) (2.75) (2.76). for the exact-continuum state Φ(kc , r). Needless to say, it is non-zero only at k = kc ; see Fig. 2.4 (a)..

(40) 2.2 Formulation of three-body CDCC (bin). The Φ̃n. 25. (k) can be analytically obtained as . Z Φ̃(bin) (k) n. =. ∗. Φ (k, r) Z. 1 √ ∆kn. Z. kn+1. dk Φ(k , r) dr 0. 0. kn. kn+1. 1 dk 0 δ(k − k 0 ) =√ ∆kn kn ( √ 1/ ∆kn (kn ≤ k ≤ kn+1 ), = 0 (k < kn , kn+1 < k).. (2.77). (bin). This shows that the Φ̂n (r) includes the exact wave function Φ(k, r) with the same √ weight 1/ ∆kn with a range of [kn , kn+1 ]; see Fig. 2.4 (b).. (ps). Since the Φ̃n (k) cannot be obtained analytically, we only show the schematic (ps) picture of the k-distribution. As illustrated in Fig. 2.4 (c), the Φ̃n (k) is distributed at around k ∼ k̂n in a certain weight. Note that the peak position is not necessarily (ps) at k̂n . The relation between k̂n and Φ̃n (k) is sZ k̂n =. (ps). dk k 2 |Φ̃n (k)|2 ,. (2.78). because ~2 k̂n2 2µ Z r ~2 2 (ps)∗ = dr Φ̂n (r) − ∇ + V (r) Φ̂(ps) n (r) 2µr ZZ ~2 2 0 (ps)∗ 0 = drdr Φ̂n (r) − ∇ + V (r) δ(r − r 0 )Φ̂(ps) n (r ) 2µr ZZ 0 0 (ps) = drdr 0 hΦ̂(ps) n |ri hr|hdα |r i hr |Φ̂n i ZZZZ 0 0 0 0 (ps) = drdr 0 dkdk 0 hΦ̂(ps) n |ri hr|ki hk|hdα |k i hk |r i hr |Φ̂n i ZZZZ ~2 k 2 ∗ 0 = drdr 0 dkdk 0 Φ̂(ps)∗ (r)Φ (k, r) δ(k − k 0 )Φ(k 0 , r 0 )Φ̂(ps) n n (r ) 2µr ZZ ~2 k 2 0 (k) = dkdk 0 Φ̃(ps)∗ δ(k − k 0 )Φ̃(ps) n n (k ) 2µr Z ~2 k 2 (ps) = dk |Φ̃ (k)|2 . 2µr n. ε̂n ≡. (2.79) (2.80) (2.81) (2.82) (2.83) (2.84) (2.85) (2.86).

(41) Chap. 2 Dynamic properties of 6 Li elastic scattering. 26. k. k Φ( ,. k. ). 0. Φ. ( ). Φ. 0. ( ). 0. g.s.. (a) Exact method. (b) Bin method. (c) Pseudostate method. Figure 2.4: The schematic k-distributions of the continuum states in the exact method (a), bin method (b), and pseudostate method (c).. 2.2.7. Model setting in three-body CDCC. The actual interactions and the model space of three-body CDCC are summarized in this subsection. Interaction between d and α (Vdα ) As for Vdα , we take the interaction of Ref. [57]: Vdα (r) = V CE (r) + V LS (r)` · S + V Coul (r),. (2.87). where V CE , V LS , and V Coul are the central, spin-orbit, and uniformly-charged Coulomb interactions, respectively. The explicit forms are defined as the Gaussian-type interaction given by 2. 2. V CE (r) = V1CE e−(r/rc1 ) + V2CE e−(r/rc2 ) , )2. )2. V LS (r) = V1LS re−(r/rs1 + V2LS re−(r/rs2 , ( 2 2e2 /(2rCoul )(3 − r2 /rCoul ), for r < rCoul , Coul V (r) = 2 2e /r, for r ≥ rCoul .. (2.88) (2.89) (2.90). The parameter set determined from the phase shift analysis reproduces the measured binding energy (ε0 ) but overestimates the root-mean-square matter radius (Rrms ). Hence, the interaction for the ` = 0 state is modified to reproduce both the measured ε0 and Rrms ; see Table 2.1 for the results. The actual parameter set used in this paper is summarized in Table 2.2. The ground and discretized-breakup states of (`, I π ) = (0, 1+ ), (2, 1+ ), (2, 2+ ),.

(42) 2.2 Formulation of three-body CDCC. 27. (2, 3+ ) with ε < εmax = 20 MeV (k < kmax = 1.14 fm−1 ) are taken as the model space P of three-body CDCC calculations. We have confirmed that the model space yields good convergence for the present 6 Li elastic scattering. Table 2.1: The binding energy (ε0 ) and the root-mean-square matter radius (Rrms ) of the 6 Li ground state. The calculation is done by the d + α two-body model. The experimental data are taken from Refs. [58, 59].. Calc. Exp.. Iπ 1+ 1+. m ε0 [MeV] Rrms [fm] −1.47 2.44 −1.4743 2.44±0.07. Table 2.2: Parameter set of Vdα taken from Ref. [57], but the interaction is slightly modified for the ` = 0 state to reproduce the measured Rrms and ε0 . V1CE [MeV] 0 -94.20 2 -82.98 `. V2CE rc1 rc2 [MeV] [fm] [fm] 0 1.960 31.00 2.377 1.852. V1LS [MeV/fm] -2.31. V2LS [MeV/fm] 1.42. rs1 [fm] 2.377. rs2 [fm] 1.852. rCoul [fm] 3.00 3.00. Optical potentials (Ud and Uα ) Since phenomenological Ud and Uα depend on the incident energy, we take them at the same incident energy per nucleon (same incident velocity) as in the 6 Li scattering of interest. As for 6 Li + 209 Bi scattering at 29.9 and 32.8 MeV, we take Ud and Uα at 5 MeV/nucleon for both the incident energies for simplicity. The potentials are taken from Ref. [60] and Ref. [62], respectively. They have the Woods-Saxon volume- plus surface-type form as follows. U (r) = V (r) + iW (r),. (2.91). 1 , (2.92) 1 + exp [(r − r0 A1/3 )/a0 ] 1 4 exp [(r − rD A1/3 )/aD ] W (r) = −W0 − W , (2.93) D 1 + exp [(r − rW A1/3 )/aW ] (1 + exp [(r − rD A1/3 )/aD ])2 V (r) = −V0. where A is the target mass, i.e., A = 209 for a summarized in Table 2.3.. 209. Bi target. The parameter sets are.

(43) Chap. 2 Dynamic properties of 6 Li elastic scattering. 28. Table 2.3: Optical potentials Ud and Uα used in CDCC calculations for scattering of 6 Li + 209 Bi at 29.9 and 32.8 MeV. The parameter sets are taken from Ref. [60] for Ud and Ref. [62] for Uα . V0 r0 a0 [MeV] [fm] [fm] 209 d + Bi 32.480 1.72 0.40 α + 209 Bi 96.44 1.376 0.625 System. 2.3. W0 rW [MeV] [fm] 9.75 1.72 -. aW [fm] 0.40 -. WD rD [MeV] [fm] 32.0 1.216. aD [fm] 0.42. Formulation of four-body CDCC. Four-body CDCC is recapitulated in this section. Four-body CDCC differs in the derivation of the internal wave function of projectile, but the procedure is common with three-body CDCC as shown below.. 2.3.1. Four-body Hamiltonian in the n + p + α + T model. In four-body CDCC, 6 Li + T scattering is described by assuming the n + p + α + T four-body model. First, we define the internal coordinate of 6 Li as ξ = {r, y}, and the relative coordinate between the center of mass of 6 Li and T as R (see Fig. 2.5). The Schrödinger equation for describing the four-body dynamics is written as (H4 − E)Ψ(ξ, R) = 0,. (2.94). where Ψ represents the total wave function, E denotes the total energy in the centerof-mass system. The four-body Hamiltonian H4 is given by e2 ZLi ZT + hnpα , R = Kr + Ky + Vnα + Vpα + Vnp ,. H4 = KR + Un + Up + Uα + hnpα. (2.95) (2.96). where KR stands for the kinetic energy operator regarding to R, Ux (x = n, p, α) represents the optical potential between x and T. Again, the Coulomb breakup is neglected. The approximation is confirmed to be good even in four-body CDCC; see Appendix C for the direct comparison. This is attributed to the fact that the E1 transition strength becomes zero even in the n + p + α three-body model; see Appendix D. In the internal Hamiltonian hnpα , Kr and Ky denote the kinetic energy operators associated with r and y, respectively, and Vab is the interaction between particles a and b..

(44) 2.3 Formulation of four-body CDCC. p. 6Li. n. Rp. 29 6Li. T. y. Rn n. R α. p. r. ξ = {r, y} α. Rα. Figure 2.5: Coordinates in the n + p + α + T four-body model.. 2.3.2. Four-body CDCC wave function. The exact four-body wave function ΨJM with the total angular momentum J and its projection onto z-axis M satisfies the Schrödinger equation (H4 − E)ΨJM (ξ, R) = 0.. (2.97). Also in four-body CDCC, the Eq. (2.97) is solved in the model space P : P =. 1 X. gs |Φgs 1m1 i hΦ1m1 | +. m1 =−1. =. IX max. IX max. I n max X X. |Φ̂nImI i hΦ̂nImI |. (2.98). I=0 mI =−I n=1. I n max X X. |Φ̂nImI i hΦ̂nImI | ,. (2.99). I=0 mI =−I n=0. where Φgs 1m1 is the ground state, and Φ̂nImI (n ≥ 1) is the n-th discretized continuum state with projectile spin I and its z-component mI . The quantum numbers n and I are truncated at nmax and Imax . Note that n = 0 represents the ground state, i.e., the ground state Φgs 1m1 is now written as Φ̂01m1 . The wave functions satisfy the following relations; hΦ̂nImI |Φ̂n0 I 0 m0I i = δnn0 δII 0 δmI m0I , hΦ̂nImI |hnpα |Φ̂n0 I 0 m0I i = ε̂nI δnn0 δII 0 δmI m0I ,. (2.100) (2.101). where ε̂nI (n ≥ 1) are the energies of Φ̂nImI , and ε̂01 means the ground-state energy (ε0 ). This model-space assumption reduces Eq. (2.97) to P (H4 − E)P ΨCDCC JM (ξ, R) = 0,. (2.102).

(45) Chap. 2 Dynamic properties of 6 Li elastic scattering. 30. for the four-body CDCC wave function ΨCDCC JM (ξ, R). =. IX max. J+I n max X X. [Φ̂nI (ξ) ⊗ χ̂JnIL (KnI , R)]JM ,. (2.103). I=0 L=|J−I| n=0. where χ̂JnIL describes the relative motion between T and 6 Li in Φ̂nI , and L denote the angular momentum regarding R. The relative wave number KnI between 6 Li and T satisfies the energy conservation: ~2 K02 = E, 2µR 2 ~2 KnI + = E, 2µR. (2.104). ε̂0 + ε̂nI. (2.105). where µR is the reduced mass of the 6 Li + T system.. Now, we redefine χ̂Jγ (KnI , R) L i YLML (R̂) ≡χ̂JnILML (KnI , R), R γ YJM (ξ, R̂) ≡ Φ̂nI (ξ) ⊗ iL YL (R̂) JM ,. (2.106) (2.107). where γ = {nIL} is defined for simplicity. Then, Eq. (2.103) is written as ΨCDCC JM (ξ, R) =. X χ̂γ (KnI , R) γ. R. γ YJM (ξ, R̂).. (2.108). Equation (2.108) exactly corresponds to Eq. (2.27) in three-body CDCC.. Once we have written Eq. (2.108), the CDCC equation for χ̂Jγ can be obtained straightforwardly by D. γ YJM (ξ, R̂). E. H4b − E. ΨCDCC JM (ξ, R). ‰,R̂. = 0,. (2.109). that is, . ~2 d2 ~2 L(L + 1) e2 ZLi ZBi − + + Uγγ (R) + − (E − ε̂γ ) χ̂Jγ (R) 2µR dR2 2µR R2 R X = Uγγ 0 (R)χ̂Jγ0 (R) (2.110) γ 0 6=γ.

(46) 2.3 Formulation of four-body CDCC. 31. with the coupling potential D E γ γ0 Uγγ 0 (R) = YJM (ξ, R̂) Un + Up + Uα YJM (ξ, R̂). ‰,R̂. .. (2.111). All we have to do here is to solve Eq. (2.110) under the standard boundary condition; see Eq. (2.30). This allows us to get the S-matrix elements and cross sections.. 2.3.3. GEM in the n + p + α three-body model. Three-body Hamiltonian The ground and discretized-breakup states are obtained with the Gaussian expansion method (GEM). In the GEM, three sets of Jacobi coordinates ξ c = {r c , y c } (c = 1, 2, 3) are taken as shown in Fig. 2.6. Thanks to this model setting, 5 He-p, 5 Li-n, d-α and np-α configurations are well incorporated, and a rapid convergence of the ground-state energy and cross sections can be obtained by increasing the model space P . n. p. n. p. r1. n. r2. y1. r3. y2. α. α c=1. p. y3. α c=2. c=3. Figure 2.6: Three sets of Jacobi coordinates ξ c = {r c , y c } in the n + p + α three-body model. Each set is identified with c (c = 1, 2, 3).. The Schrödinger equations for 6 Li wave functions are given by [hnpα − ε]Φε (ξ) = 0. (2.112). hnpα = Krc + Kyc + Vnα + Vpα + Vnp + VPF ,. (2.113). with the internal Hamiltonian. where VPF is the pseudo-potential based on the orthogonality condition model [63] to eliminate the Pauli forbidden states between α and nucleons φ0s (y i ) (i = 1, 2); VPF = lim λPF λPF →∞. 2 X i=1. |φ0s (y i )i hφ0s (y i )| ,. (2.114).