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Clustering in Light Nuclei with Configuration Interaction Approaches Konstantinos Kravvaris

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Clustering in Light Nuclei with Configuration Interaction Approaches Konstantinos Kravvaris Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2018 Clustering in Light Nuclei with Configuration Interaction Approaches Konstantinos Kravvaris Follow this and additional works at the DigiNole: FSU's Digital Repository. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES CLUSTERING IN LIGHT NUCLEI WITH CONFIGURATION INTERACTION APPROACHES By KONSTANTINOS KRAVVARIS A Dissertation submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2018 Konstantinos Kravvaris defended this dissertation on May 11, 2018. The members of the supervisory committee were: Alexander Volya Professor Directing Dissertation David Kopriva University Representative Ingo Weidenhoever Committee Member Simon Capstick Committee Member Laura Reina Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the dissertation has been approved in accordance with university requirements. ii ACKNOWLEDGMENTS This thesis would not have been possible without the constant encouragement of Professor Alexan- der Volya. His patience was a critical ingredient throughout the research and writing process. It was always a pleasure discussing research, administrative and general topics. I am also grateful for the committee members' guidance when necessary. I would also like to thank Professor Jorge Piekarewicz for being a great teacher and exposing me to more problems in nuclear physics. Finally, special thanks goes to Professor Efstratios Manousakis for all his help before and during my PhD studies. I would also like to thank everyone who made the non-research part of my studies enjoyable and interesting, Maria Anastasiou, Konstantinos Kountouriotis, Katerina Chatziioannou, David Fernandez, Hailey Nicol (soon to be Fernandez), and Daniel Lanford. Finally, for navigating the bureaucratic maze, I would like to thank Jonathan Henry, Brian Wilcoxon and Felicia Youngblood. iii TABLE OF CONTENTS List of Tables . vi List of Figures . viii List of Symbols . ix List of Abbreviations . .x Abstract . xi 1 Introduction 1 2 Methods 5 2.1 Harmonic Oscillator . .5 2.1.1 Coordinate Transformations . .6 2.2 Configuration Interaction Approach in a HO Basis . .9 2.2.1 HO Basis Expansion . .9 2.3 Traditional Shell Model . 11 2.4 No Core Shell Model . 12 2.5 Resonating Group Method . 12 2.6 Spectroscopic Factors . 15 2.6.1 Traditional Spectroscopic Factors . 16 2.6.2 Normalized Spectroscopic Factors . 16 2.6.3 Dynamic Spectroscopic Factors . 17 2.7 Coupling to Continuum . 18 2.8 Rotational Bands . 20 2.8.1 Elliott's SU(3) Model for Rotations . 22 3 Constructing Cluster Channels 23 3.1 Antisymmetry . 23 3.2 The Gloeckner-Lawson Procedure . 24 3.3 SU(3) Method . 25 3.4 Center of Mass Boosting Method . 27 3.5 Constructing the Channels . 29 4 Traditional Shell Model Studies 33 4.1 α Particle Wave Function . 33 4.2 Clustering in the sd Shell . 34 4.2.1 The Case of 20Ne.................................. 37 4.2.2 Evolution of Clustering with Increasing Nucleons . 39 4.3 Superradiance in 13C and α Clustering . 41 iv 5 No Core Shell Model Studies 43 5.1 The Nucleus 8Be ...................................... 43 5.2 The Nucleus 9Be ...................................... 46 5.3 The Nucleus 10Be...................................... 48 5.4 The 3α System . 50 6 Conclusions 52 Appendix A Second Quantization 54 B CM Boosting Operator Normalization Coefficients 56 C Calculation of Radial Integrals with HO Wave Functions 59 D CM Boost and Recoupling Examples 61 E Hamiltonian Kernels for the 2α System 65 F Tables of Spectroscopic Amplitudes 67 Bibliography . 82 Biographical Sketch . 94 v LIST OF TABLES 3.1 Select configuration content of NCSM wave functions for 4He with ¯h! = 20 MeV boosted to a CM boosted state with n = 4 and ` =0. .................. 29 3.2 Moshinsky coefficients for different mass ratios d = AD=Aα coupling to a final HO wave function with CM quantum numbers N = L = 0 and relative quantum numbers n = 2; ` =0. ......................................... 31 4.1 Transition rates and α spectroscopic factors for the 20Ne rotational band members with the USDB interaction. 37 4.2 Comparison between new and experimental SF for low-lying states in 21Ne. 40 4.3 Energies, widths in MeV and α spectroscopic factors (structural) for the two lowest 3=2+ states in 13C...................................... 42 5.1 Alpha decay widths and spectroscopic factors for the first 3 states in 8Be using different oscillator parameters ¯h!. The S(dyn) are calculated with the lowest energy solution of 8 the RGM in each case projecting on the Nmax = 4 corresponding Be wave function. Widths are calculated with a fixed channel radius of R = 3:6 fm . 45 5.2 Contributions to the Hamiltonian from different interaction components in MeV. Then numbers in parentheses indicate number of nodes n and angular momentum ` in the relative coordinate of the 2α channel. V0;1 correspond to the different jTzj values for the interaction. The second column refers to contribution to a single α particle. We use the JISP16 interaction with ¯h! = 25 MeV. 46 5.3 Negative parity band energies and SF for 9Be. The numbers in parentheses denote the number of available quanta in the relative motion between the 2 clusters. The SF are calculated with ¯h! = 20 MeV. The sixth column marked with a single star refers to dynamic SF calculated with additional channels built on the 5He(1/2−) state. The final column marked with two stars refers to a calculation with channels restricted to having an 8Be +n form. 48 5.4 Dynamic SF for rotational band members in 10Be. The last columns labeled with a (*) refers to calculations with channels constructed with the excited 2+ state of 6He. The states belonging to the two different rotational bands are separated by a line. The SF is evaluated with the dominant channel(s). 48 5.5 Leading SU(3) irreducible representation components and particle-hole content of states recognized as belonging to rotational bands in 10Be. The ordering is the same asin5.4............................................ 50 vi 5.6 Ground state binding energy and excitation energies for rotational band members of 12C using the NCSM and RGM with ¯h! = 20 MeV. 51 D.1 Index ordering (k) of single particle orbitals. The ± superscirpt refers to the m = ±1=2 projection of each orbital. 62 D.2 Index ordering (k) of neutron single particle orbitals for the sp space. The proton indexing follows the same order but offset by 8. 63 vii LIST OF FIGURES 1.1 Ikeda diagram showing the location of different modes of clusterization and the re- spective decay thresholds. .3 2.1 Broad alpha clustering states in light nuclei. The data for 10Be are taken from [88] and for 18O from [89] . 19 3.1 A schematic depiction of the process of transfering the quanta of motion from the CM of an α cluster to the relative motion between the α and a daughter nucleus. The primes indicate the (invariant) internal wave functions of both nuclei. 30 4.1 Ab initio calculations for the ground state energy (left) and the s4 component of 4 the He wave function (right) with the JISP16 interaction spanning multiple Nmax truncations and ¯h! values. 34 4.2 Old (dashed) and New (solid) spectroscopic factors comparison with experimental data from [115, 25] using the USDB [61] interaction. Red (blue) poinst correspond to α knockout (pickup). The dotted line corresponds to a calculation with a realistic α wave function, obtained using the NCSM withh! ¯ = 14 MeV and Nmax = 8. 35 4.3 Comparison between (new) α SF calculated with different wave functions for the α particle. For the Nmax = 0; 4; 8 there are 1; 3; 5 channels open respectively. 36 4.4 (a) Experimental population of states in 20Ne from (6Li,d) and (b) theoretical pre- dictions for SF in a (0+1)ph calculation of 20Ne. To match the experimental figure, energy is increasing to the left. Details of the calculation are given in main text. Figure (a) is reproduced with permission from [123]. 38 4.5 Low-lying RGM, full USDB and experimental spectra of 21Ne. 40 5.1 Low-lying spectra of 8Be calculated with the JISP16 interaction at varioush! ¯ values in units of MeV. 44 5.2 RGM calculations of the rotational band of 8Be for ¯h! = 20 MeV with the JISP16 interaction and with the α particle wave function taken from a NCSM calculation with the corresponding value of Nmax. The value N in the plot reflects the total number of HO quanta available to the relative motion between the two clusters. 47 5.3 Cluster rotational bands in 9Be. N refers to the total number of quanta available in the relative motion; even (odd) N corresponds to negative (positive) parity bands. Top (bottom) plot corresponds to positive (negative) parity states. 49 viii LIST OF SYMBOLS π 3:1415926 ::: ¯hc 197:327 ::: MeV fm n The number of nodes of in the radial coordinate, n = 0; 1; 2 ::: l The orbital angular momentum of a single particle orbital. m The z - axis projection of the angular momentum. ` The relative orbital angular momentum of two clusters. N The total number of oscillator quanta, N = 2n + l. Nmax Number of allowed excitation quanta in No Core Shell Model. s; p; d; f : : : Spectroscopic notation for orbitals with l = 0; 1; 2; 3 ::: α Ln(x) Generalized Laguerre polynomial. Ylm(θ; φ) Usual Spherical Harmonic. JM Cj1m1j2m2 Clebsch Gordan Coefficient coupling j1m1 and j2m2 to JM [41]. Mn`NL Moshinsky Coefficient for a transformation with angle ' [45] ' n1l1n2l2 Mn`NL n1l1n2l2 Coefficient for a transformation to the CM of equal mass particles (' = π=4) ix LIST OF ABBREVIATIONS HO Harmonic Oscillator CI Configuration Interaction SM Traditional Interacting Shell Model NCSM No Core Shell Model RGM Resonating Group Method CSM Continuum Shell Model Harmonic Oscillator Representation HORSE of Scattering Equations SF Spectroscopic Factor NN Nucleon-Nucleon JISP J-Inverse Scattering Potential x ABSTRACT The formation of sub-structures within an atomic nucleus, appropriately termed nuclear clustering, is one of the core problems of nuclear many-body physics.
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