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

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 withhω ¯ = 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 ¯hω. 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 |Tz| values for the interaction. The second column refers to contribution to a single α particle. We use the JISP16 interaction withhω ¯ = 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 withhω ¯ = 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 ¯hω = 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 respective 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 ¯hω 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 withhω ¯ = 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 predictions 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 varioushω ¯ values in units of MeV...... 44

5.2 RGM calculations of the rotational band of 8Be for ¯hω = 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 ϕ [45] ϕ n1l1n2l2 Mn`NL n1l1n2l2 Coefficient for a transformation to the CM of equal mass particles (ϕ = 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. In this thesis, we put forward a new method for the study of nuclear clustering relying on the completely microscopic Configuration Interaction approach. We construct reaction cluster channels in a Harmonic Oscillator many- body basis that respect the symmetries of the Hamiltonian, are fully antisymmetrized, and carry a separable and controlled Center of Mass component. Such channels are then used to explore cluster signatures in Configuration Interaction many-body wavefunctions. The Resonating Group Method is then applied, utilizing the reaction channels as a basis to capture the essential cluster characteristics of the system. We investigate the emergence of nuclear clustering in 2α, 2α + n, 2α+2n and 3α systems using a No Core Shell Model approach from first principles, and traditional Shell Model studies of clustering in heavier nuclei.

xi CHAPTER 1

INTRODUCTION

One of the central questions that nuclear many-body physics seeks to answer revolves around the emergence of regularities such as nuclear shell structure, rotational bands and vibrational states to name a few, all observed in nuclear spectra [1, 2]. It has proven to be difficult to devise a model that describes the properties of atomic nuclei across the nuclear chart, and at various excitation energies, let alone from a completely ab initio view; namely starting from QCD interactions or at least interactions that describe the nucleon - nucleon phase shifts and properties of mass A = 3, 4 nuclei. While certain patterns, seen across the nulear chart, such as rotations and vibrations of the nucleus, are somewhat understood, nuclear clustering is yet another exotic phenomenon that is more difficult to study not only from a theoretical prespective, but also due to lack of experimental data. Nuclear clustering, the subject of this thesis, refers to the formation of substructures within the atomic nucleus. Light nuclei, with states that can be almost perfectly described as two or more substructures in relative motion, exhibit perhaps the most extreme degree of clusterization. Clusterization greatly affects not only the spectra of nuclei but also, and perhaps more importantly, the reaction rates for astrophysical processes. In this work we take a completely microscopic point of view in our approach. The main question we seek to answer concerns the emergence of clustering from nucleon-nucleon (NN) interactions, however, we also seek a general method that can probe the cluster features of nuclear systems and to study the role of coupling to the continuum of reaction states in altering spectroscopic characteristics of highly excited states.

The concept of a nucleus being formed as an assortment of α particles has existed since the beginning of nuclear science [3, 4, 5]. This idea was predominantly supported by both the fact that the α particle is very well bound and requires a rather large amount of energy to excite, and by experimental evidence showing α particle emission as the main decay channel for heavy elements.

1 This shows that the quartet arrangement involving the 2 protons and 2 neutrons that makes up the α particle is somehow energetically favorable. In support of clustering, early theories [3, 4, 6] emphasized that the binding energies of N = Z nuclei seem to follow a linear relation with the number of bonds that can be made between the α constituents of the nucleus. With the discovery of the neutron [7] these α models became disfavored, and attempts to describe the nucleus from its single particle constituents gained momentum. While geometric arrangements of α particles in multi-α nuclei were the first ones to be considered, different types of fragmentation were subsequently probed, identifying a vast number of excited nuclear states possessing cluster characteristics that contained non−α fragments [8, 9]. These excited clustered states were found to lie at energies near the respective cluster decay threshold, prompting Ikeda [10] to exemplify this property in the, now famous, Ikeda diagram (Figure 1.1). This idea ties in nicely with the Gammow picture of α decay, in which an α particle is formed in the nucleus and at some threshold energy can penetrate the Coulomb barrier. From an astrophysical prespective, the most important clustered state is the excited 0+ state in 12C. In 1954 Sir Fred Hoyle predicted [11, 12] the existence of this state as a neccessary doorway for the creation of 12C in the universe. Given that the 2α system 8Be is unbound, and there exists no bound A = 5 system, the formation of elements heavier than A = 4 would otherwise be hindered. From this argument the state is expected to have a strong 3α character, a result verified experimentally as it is seen to decay into a 3α final state, predominantly through the short-lived 8Be 0+ ground state resonance [13, 14, 15, 16, 17]. In the 1970s a large body of experimental data was collected aimed at elucidating the cluster aspects of light nuclei, see [18, 19, 20] for some specific examples, or [21] for a more complete survey of experimental data. The main tools were the α transfer reactions, such as (6Li, d), (7Li, t), and α knockout reactions such as (p, pα). These reactions were found to be selective to populating or depopulating states with some degree of clusterization. The nuclear shell model was seen as a conflicting view of nuclear structure at the time due to its single particle nature. Attempts to describe α clusters from within the shell model resulted in the powerful SU(3) symmetry based techniques [22, 23]. Despite some degree of success in select cases [24, 25], there were still significant deviations from experiment, making further theoretical work regarding quantitative agreement neccessary. This led to the rise of the Resonating Group Method (RGM) [26, 27], which aimed

2 α. α. α. α. α. α. α. α. α. α. α. α. α. α. α. α. α. α. α. α. 7.27 14.44 19.17 28.48 12.C 12.C α. 12.C α. α. 12.C α. α. α. 7.16 11.89 21.21

16.O 16.O α. 16.O α. α.

4.73 14.05 12.C 12.C Ex 13.93

Ikeda Diagram 20Ne. 20Ne. α.

9.32

24Mg.

Figure 1.1: Ikeda diagram showing the location of different modes of clusterization and the respective decay thresholds. at an explicit description of the nucleus as a multi-fragment clustered state, while maintaining a microscopic description for the cluster fragments and employing a fully antisymmetric wave function on the nucleonic level. Later, clustering was shown to emerge from a nucleon-nucleon (NN) interaction describing the NN phase shifts [28]; The Green’s Function Monte Carlo calculations showed the formation of two distinct centers in the density distribution of 8Be. At about the same time the method of Antisymmetrized Molecular Dynamics began to be extensively applied to the problem of molecular- type states in clustered nuclei [29]. Such states are known to exist in N > Z nuclei [30, 31], with the extra neutrons forming valence bonds between the α (or heavier) cores, similar to the role of electrons in molecules. Relying heavily on Group Theory, symmetry based approaches were also applied, treating the relative motion of the α particles as bosons, and, borrowing from molecular physics, exploiting triangular and tetrahedral spatial symmetries to describe the spectra, finding great success in both 12C and 16O [32, 33]. To this day, nuclear clustering is still under intense theoretical and experimental investigation. On the theory side, mean field approaches [34], lattice models [35, 36], large-scale ab initio shell

3 model calculations [37] and Bose-Einstein condensate wave functions [38, 39, 40] have all been employed to better understand how the clusters are formed, and which aspects of nuclear structure and reactions are most affected by the formation of substructures within the nucleus. Experimentally, the identification of clustered states becomes harder as the states are often broad (Γ ∼ 1 MeV) and quite high in excitation energy. The work presented here is organised as follows. In Chapter 2, we review the properties of the quantum Harmonic Oscillator (HO) and the related Configuration Interaction (CI) methods that are used throughout this thesis. Both the phenomenological traditional interacting shell model (SM) and the ab initio No Core Shell Model (NCSM) are introduced. The Resonating Group Method (RGM) is discussed as a way to merge the structure and reaction aspects of the nuclear system. We define the different types of Spectroscopic Factors (SF) and compare the methods of evaluating the degree of clustering in a nuclear state. Finally, a method for coupling the physics of the continuum to a finite quantum system is described. Chapter 3 serves to explore the various methods of creating explicitly clustered states within the HO many particle basis. We demonstrate our novel method that controls the Center-of-Mass (CM) coordinate of a many-body wave function in order to construct translationally invariant binary and ternary cluster channels. This approach overcomes various problems concerning the construction of α cluster channels and is general enough to be applicable to various clustering scenaria. Chapter 4 contains several explorations of clustering in various light nuclei using the Traditional Shell Model. Chapter 5 uses the RGM as a way to explore clustering from completely ab initio principles. Finally, Chapter 6 concludes the thesis by providing a brief review and summary of all the topics covered. Select theoretical topics, tables, and derivations can be found in the Appendices.

4 CHAPTER 2

METHODS

In this chapter we describe the methods used in this work. First we discuss the HO Hamiltonian and the properties of the wave functions. We briefly go over the basics of rotational bands in low- lying nuclear spectra and Elliott’s SU(3) model. Variational solutions to the many-body problem in terms of HO expansion of the wave function are reviewed, followed by discussion of the RGM. We discuss the concept of the spectroscopic factor and present different approaches to defining it. Finally, we show how the physics of the continuum can be incorporated in the shell model through the Feschbach projection formalism.

2.1 Harmonic Oscillator

The Harmonic Oscillator (HO) is perhaps the most widely encountered system in physics. It is deﬁned by a quadratic conﬁning potential and the Hamiltonian is given by p2 1 H = + mω2r2. (2.1) 2m 2 As the HO is analytically solvable, it becomes useful in approximating the solution of more complex systems near the stable point. In quantum mechanics, the wave function of a single particle trapped in a spherically symmetric HO potential is given by

s 2 3 n+l+2 l − −1/2 2 n! r −r2/2b2 l+1/2 r h~r|nlmi = ψnl (r) = ψnl (r, θ, φ) = b 2 π e L Yl (θ, φ), m m (2n + 2l + 1)!! b n b2 m (2.2) p where b = ¯h/mω is the so-called characteristic length of the oscillator. For alternative forms of the HO wave function see [41]. Here, the radial part of the wave function is positive near the origin. A phase convention used by some authors has an extra phase factor of (−1)n that assures that the radial wave function is positive as r → ∞. The energy corresponding to the wave function with quantum numbers |nlmi is E = ¯hω(N + 3/2), where N = 2n + l (2.3)

5 is the total number of HO quanta. Since the potential is a second power of r and the |nlmi are eigenstates we can invoke the Virial theorem and obtain for the expectation values of the kinetic T = p2/2m and potential V = mω2r2/2 energy ¯hω 3 hnlm|T |nlmi = hnlm|V |nlmi = 2n + l + . (2.4) 2 2 The oﬀ-diagonal matrix element for the kinetic energy reads

¯hω p hnlm|T |n + 1lmi = (n + 1)(n + l + 3/2), (2.5) 2 and since the Hamiltonian is diagonal in its eigenbasis |nlmi, the oﬀ diagonal matrix elements for the r2 operator are nearly identical

¯h p hnlm|r2|n + 1lmi = − (n + 1)(n + l + 3/2). (2.6) mω The range of the wave function can be evaluated by the rms value for r s p ¯h 3 hr2i = 2n + l + . (2.7) mω 2

The wave functions ψnlm (r) form a complete basis and can be used to expand another single-particle function. If the expansion is truncated by N, eq. (2.7) gives a measure of the range of validity of the expansion. While the HO possesses attractive properties, some of which are discussed later, it suffers from a gaussian fall-off of the wave function, a direct consequence of the potential being infinite as r → ∞. In real systems, the potential at large distances becomes 0, thus leading to a wave function decaying as exp (−r), not as exp −r2 which are the asymptotics of the HO wave functions. As a result systems that are loosely bound require a large number of excitations to be properly described in the HO basis [42].

2.1.1 Coordinate Transformations

Apart from the HO Hamiltonian being exactly solvable for a single particle, another advantage comes from its symmetry property under orthogonal transformations of coordinates in the many- body case. For example, when considering two non interacting particles in a HO potential, the Hamiltonian takes the form p2 1 p2 1 H = 1 + mω2r2 + 2 + mω2r2 , (2.8) 2m 2 1 2m 2 2

6 which allows us to write down the wave function of the system as a product of HO wave functions

Ψn1l1m1n2l2m2 (r1, r2) = ψn1l1m1 (r1)ψn2l2m2 (r2) (2.9)

Since the Hamiltonian is a rotational scalar, we can couple the two single particle wave functions into a total wave function with good angular momentum λ using the usual Clebsch Gordan coeﬃcients

X Ψλµ (r , r ) = [ψ (r ) × ψ (r )]λ = Cλµ ψ (r )ψ (r ). (2.10) n1l1n2l2 1 2 n1l1 1 n2l2 2 µ l1m1l2m2 n1l1m1 1 n2l2m2 2 m1m2

The Moshinsky or Talmi-Moshinsky recoupling scheme [43, 44] reﬂects the symmetry of the Hamiltonian to remain invariant under orthogonal coordinate transformations by an angle ϕ

r cos ϕ sin ϕ r + = 1 , (2.11) r− sin ϕ − cos ϕ r2

2 2 2 2 due to the orthogonality of the transformation, r1 + r2 = r+ + r−. Therefore the potential part of the Hamiltonian remains invariant under a coordinate canonical transformation. This is a subset of a more general set of canonical transformations that leave the Hamiltonian unchanged. The eigenstates of the multi-particle system are highly degenerate as the energy is given only by the total number of quanta; in the two particle case E = 2n1 + l1 + 2n2 + l2. The coordinate rotation leaves us within the same representation of the symmetry group, which in turn implies that the total wave function in the new coordinates (2.10) can be written as a linear sum of the wave functions in the old coordinates that are members of this irreducible representation

X n l n l ;λ Ψλµ (r , r ) = M + + − − Ψλµ (r , r ). (2.12) n+l+n−l− + − ϕ n1l1n2l2 n1l1n2l2 1 2 n1l1 n2l2

This requirement enforces HO quanta and parity conservation

2n+ + l+ + 2n− + l− = 2n1 + l1 + 2n2 + l2 (2.13)

(−1)l++l− = (−1)l1+l2 . (2.14)

The transformation coeﬃcients are known as the Moshinsky brackets and are matrix elements of the Wigner D-matrix. They depend on the angle of rotation between the two sets of coordinates

7 ϕ hn+l+n−l−|n1l1n2l2iλ ; see [45] for derivations of symmetries amongst diﬀerent angles and permu- tations of indices. An alternative shortened notation used here for the brackets is deﬁned as

Mn+l+n−l−;λ ≡ hn l n l |n l n l iϕ. (2.15) ϕ n1l1n2l2 + + − − 1 1 2 2 λ

The angular momenta coupled here, ~λ = l~1 +l~2, always obey the triangle conditions. The symmetry is limited to particles in oscillator potentials with the same frequency ω. Up until now, we have implicitly assumed that the particles are of the same mass. To treat 0 √ diﬀerent mass particles (m1, m2) we can ﬁrst use transformed coordinates (ri = miri), to eliminate an explicit dependence on mass from eq. (2.8) as

p02 p02 1 H = 1 + 2 + ω2 r02 + r02 (2.16) 2 2 2 1 2

0 p A particular transformation of the new coordinates (ri) is the one by an angle tan ϕ = m2/m1, which results in the new set of coordnates

√ 0 √ 0 m1r1 + m2r2 r+ = √ (2.17) m1 + m2 √ 0 √ 0 m1r1 − m2r2 r− = √ , (2.18) m1 + m2 √ with the Hamiltonian remaining invariant. A ﬁnal transformation of R = r+/ m1 + m2 and p r = r− m1m2/(m1 + m2) yields the following form for the Hamiltonian

P2 p2 1 1 H = + + Mω2R2 + µω2r2, (2.19) 2M 2µ 2 2 where we have deﬁned the total M = m1 + m2 and reduced µ = m1m2/M masses of the system and the new coordinates are given by

m1r1 + m2r2 R = , (2.20) M

r = r1 − r2. (2.21)

We recognize these as the CM and relative coordinates of the system. This exercise shows that the coeﬃcients for the transformation to the CM of the system are the ones corresponding to the angle p tan ϕ = m2/m1. The frequency of the HO for this transformation to be applicable needs to be the same for all constituents. There is a lot to gain from the availability of such a transformation; the HO basis is recognized as the only basis of expansion where the CM motion of a system of particles

8 can be separated exactly from the internal (Jacobi) coordinates [44, 46]. Since the recoupling can work between any HO states, we will use this to our advantage to construct states that have a well defined wave function in the relative coordinate between two clusters and an HO wave function for the total CM motion with N = 0 quanta. Varying definitions, derivations of the Moshinsky coefficients, sample values and symmetry properties can be found in references [43, 47, 48, 49, 50].

2.2 Conﬁguration Interaction Approach in a HO Basis

We begin this section by describing how to use the variational method to solve for single particle motion in a potential. We then discuss how the method can be generalized in the full Conﬁguration Interaction approach for the many-body problem, and conclude with discussion of the Traditional Shell Model (SM) and the newer No Core Shell Model (NCSM) approaches to nuclear structure.

2.2.1 HO Basis Expansion

As an introductory example and explanation of our RGM strategy in Section 2.5, we solve the problem of a single particle with its motion governed by a Hamiltonian H with rotational symmetry.

One way to go about this is to expand the unknown wave function φ`m (r) in terms of some known complete basis of functions and then obtain the expansion coefficients variationally. A special case of this is the HO expansion assuming a fixed ` and m ∞ ∞ X X φ`m (r) = Cnψn`m (r) = CnRn`(r)Y`m (θ, φ) (2.22) n=0 n=0 where we seek to determine the coefficients Cn. The infinite sum is truncated at some value nmax which converts the minimization problem to an eigenvalue equation

n Xmax Hnn0 Cn0 = ECn, (2.23) n0=0

0 where Hnn0 = hn`m|H|n `mi. The diagonalization yields not only an estimate for the ground state of the particle, but also the excited states. The Ritz-Rayleigh variational principle ensures that the ground state eigenvalue obtained this way is an upper bound to the actual solution to the problem. The upper bound for the ground state energy is dependent on the oscillator parameter ¯hω and has minimum at some optimal ¯hω value. As the size of the basis grows, the minimum becomes broader and thus thehω ¯ dependence of the ground state energy estimate diminishes. It is not a priori

9 guaranteed that the same oscillator parameter as the one for the ground state will yield the best estimate for all excited states. The method can be extended to the many-body problem, with the expansion now carried over many-body Slater determinants Φi, or other antisymmetrized conﬁgurations, thus guaranteeing full antisymmetrization of the solution

X hr1, r2 ... |Ψi = Ψ(r1, r2 ... ) = ciΦi(r1, r2 ... ). (2.24) i The single particle states used in the Slater determinant can again be HO wave functions and therefore all properties obtained will be ¯hω dependent. With a large enough basis all observables will converge to their exact values.

Depending on the orbitals included in the Slater determinant, we can define a number Ntot which is given by the sum of all HO quanta in the single particle wave functions of the determinant. At the same time, for an A fermion system, there exists a number Nmin which corresponds to the minimum number of HO quanta the system must have due to the Pauli exclusion principle. The truncation of the basis is performed based on the number of allowed excitations from this minimum number Nmax. For a space with a Nmax truncation of the basis, we use only Slater determinants that have Nmin ≤ Ntot ≤ Nmax. While the single particle case has matrix dimensions that scale linearly with Nmax, the many body basis size“explodes” combinatorially with the number of allowed quanta, reaching well in the millions even for a modest number of excitations Nmax = 6 in light nuclei. Although we discussed here the expansion in terms of HO wave functions, the method is quite general, allowing for any type of single particle functions to be used. If we choose to give up the symmetry properties associated with the HO, we can use wave functions defined as solutions to the mean-field Hartree-Fock equations as they are the ones that will bring us closer to the true ground state of the system expressed with a single Slater determinant, potentially reducing the size of space required to achieve convergence of results. In both cases the inclusion of the excited configurations and subsequent diagonalization of the Hamiltonian in the configuration space yields what is known as the Configuration Interaction (CI) variational method for solving the quantum many-body problem. In the next sections we will examine two variations of CI approaches, the Shell Model (SM) and No-Core Shell Model (NCSM).

10 2.3 Traditional Shell Model

The SM was one of the ﬁrst CI methods used to study nuclear structure, and still it remains one of the most popular methods to approach the nuclear many-body problem. The large jumps in particle separation energies as one moves across an isotopic (or isotonic) line are indicators of the so-called magic numbers 2, 8, 20, 28, 50, 82, 126 which in turn can be explained by the picture of independent particles ﬁlling up orbital shells [51, 52]. In contrast to the electronic structure of atoms (another system exhibiting such magic number structure) there is no externally created

ﬁeld like the Coulomb potential. Nevertheless, nucleons themselves create a mean ﬁeld (VMF) and one can use it for nuclear structure studies along with the residual interactions of the Hamiltonian

(Vres)

H = Trel + V = Trel + VMF + Vres. (2.25)

Gradual filling of the spherical mean field orbitals leads to the shell structure observed in nuclei. The model, just like atomic models, assumes that some of the nucleons in the core will be inert, at least for the low-lying states, and treats the remaining (valence) nucleons as active. This assumption works well, however, multiple effects, such as core polarization are treated effectively. The nucleons used in the shell model are now not bare nucleons, but dressed quasi particles whose interaction is expected to reproduce the spectra of nuclei. The SM is in some way a configuration space model, containing effective particles with effective properties and effective operators. All have appropriate structures and symmetries, but semi- phenomenological parameters are used (effective proton/neutron charges, etc). For initial estimates, Hartree Fock mean field or simple HO are commonly used. In particular, the residual part of the interaction can be obtained by either performing a series of unitary transformations on a realistic Hamiltonian [53, 54, 55, 56], selecting phenomenologically the coordinate form of the residual NN interaction and fitting the parameters to experiment[57, 58], or directly fitting the two body matrix elements of the interaction to reproduce a set of experimental data [59, 60, 61]. The last of these approaches has proven to be an accurate predicting tool for nuclear spectra, reduced transition rates and SF values, establishing the SM as one of the core techniques in nuclear physics.

11 2.4 No Core Shell Model

Building on the successes of the SM, the NCSM treats all A nucleons in a nucleus as active, point-like non relativistic fermions. In this non-perturbative approach, the NN interactions are used without any additional renormalization. The method is the many-body extension of the variational HO expansion. Due to the basis size explosion discussed earlier, various truncation and expansion schemes are employed in an attempt to circumvent the problem [62, 63, 64], with varying degrees of success. The many-body Schr¨odinger Hamiltonian in the NCSM has the form

2 1 X (pi − pj) X X H = + V (ri, rj) + V (ri, rj, r ) + ... (2.26) A 2m 2b 3b k i

2.5 Resonating Group Method

The Resonating Group Method (RGM) was introduced in 1937 by Wheeler [26] to deal with a problem of a nucleus being partitioned into multiple diﬀerent sub-structures. In the RGM, the wave function of a nucleus is projected onto the relative motion of sub-structures, which are the clusters of interest. For multiple reasons, it is an attractive model to study clustering as discussed in references [73, 74]. Here we present some key elements of the RGM which carry over into our approach

1. The wave function is totally antisymmetrized.

2. The non orthogonality of diﬀerent factorizations is treated exactly.

12 3. The interaction used can be a nucleon - nucleon interaction designed to ﬁt particle-particle scattering.

4. The structure and reaction parts of the problem are treated on equal footing.

5. The CM motion of the problem is also treated exactly.

While many RGM calculations have been performed since the 1960s [74, 27], the RGM combined with the NCSM in the NCSM/RGM method [75, 76] has recently re-emerged in ab initio approaches.

We outline here the basic principles of the RGM and defer the details of how it will be combined with the shell model to later chapters. In the RGM the binary cluster wave function can be written as a totally antisymmetrized product

0 0 0 0 hξ1ξ2ρ|Ψ i = Ψ (ξ1, ξ2, ρ) = A Ψ1(ξ1)Ψ2(ξ2)f(ρ) , (2.27) where we use primes to denote wave functions depending only on internal coordinates, we deﬁne 0 Ψ1,2 the wave functions describing the internal dynamics of the two clusters (ξ1, ξ2 are internal cluster coordinates) and f(ρ) is the wave function for the motion between the CM coordinates of the two clusters (ρ = RCM,1 − RCM,2). The total CM motion wave function is ommitted in all expressions in this section. We are interested in variationally calculating the relative wave function f(ρ). This can be done by expanding the function using a set of known functions. The δ functions were used originally, representing a channel where the two cluster fragments are a deﬁnite distance (here ρ0) apart

0 0 0 0 0 hξ1ξ2ρ|Φch(ρ )i = A Ψ1Ψ2δ(ρ − ρ ) (2.28) Z 0 0 0 0 hξ1ξ2ρ|Ψ i = f(ρ )|Φch(ρ )idρ . (2.29)

The wave function f(ρ) is to be determined variationally using the Hamiltonian of eq. (2.26). Using eq. (2.29), the variational condition transforms into the Hill-Wheeler equation [77, 78, 79] Z H(ρ, ρ0) − EN (ρ, ρ0) f(ρ0)dρ0 = 0. (2.30)

The functions H(ρ, ρ0) and N (ρ, ρ0) are known as the Hamiltonian and Norm kernels, respectively. They are found as matrix elements of the Hamiltonian and Norm operators in the channel

13 space

0 0 H(ρ, ρ ) = hΦch(ρ)|H|Φch(ρ )i (2.31)

0 0 N (ρ, ρ ) = hΦch(ρ)|1|Φch(ρ )i. (2.32)

The existence of the norm kernel enforces the Pauli principle at the channel level and controls normalization and orthogonality. Since the Norm kernel is positive semideﬁnite, all eigenvalues are non-negative, with the ones that are equal to zero representing the Pauli forbidden states. The positive semideﬁniteness can be demonstrated by expanding in some orthonormal basis function k set |Φchi

X k |Φch(ρ)i = bk(ρ)|Φchi (2.33) k 0 X ∗ 0 X ∗ 0 hΦch(ρ )|Φch(ρ)i = bk(ρ )bm(ρ)δkm = bk(ρ )bk(ρ), (2.34) km k which, if we pick an arbitrary function g(ρ) and evaluate the expectation value yields Z Z 2 X ∗ 0 ∗ 0 0 X g(ρ)bk(ρ)g (ρ )b (ρ )dρdρ = g(ρ)bk(ρ)dρ ≥ 0. (2.35) k k k When solving the RGM equations, our ﬁrst step is the diagonalization of the Norm kernel and the subsequent discarding of any eigenvector with a corresponding eigenvalue below some pre-set threshold governed by numerics.

The δ-function choice for the expansion is not unique; in our work we choose a basis consisting of HO wave functions to expand the relative motion, allowing for both the Hamiltonian and Norm kernels to be written down as matrices. For convenience we choose the same HO basis as the one used in the NCSM for the cluster and parent nucleus. In the HO expansion, the full channel wave function |χ(`)i for a given set of asymptotic quantum numbers (which in the following we abbreviate with a single label `) is given as an expansion of basis channel states (Φn`)

0 0 hξ1ξ2ρR|Φn`i = ψ0(R)A ψn`(ρ)Ψ1(ξ1)Ψ2(ξ2) (2.36)

(`) X (`) |χ i = χn |Φn`i. (2.37) n In the Slater determinant HO basis, the CM motion is unavoidable and thus we explicitly include it in the channel with a N = 0 HO quanta wave function. The expansion is similar to the one-body

14 variational CI treatment discussed in Section 2.2.1, however the bases used here are not orthogonal. The quantum number n here is used only for the number of nodes in the relative motion of the (`) clusters. The full function f(ρ) is now determined via the coeﬃcients χn . We re-write the RGM equation (2.30) in the HO channel basis

X (`) (`) X (`) (`) Hnn0 χn0 = E Nnn0 χn0 , (2.38) n n where the matrix elements are given by:

(`) Hnn0 = hΦn`|H|Φn0`i (2.39) (`) Nnn0 = hΦn`|Φn0`i. (2.40)

P (`)∗ (`) (`) The channel normalization requires n χn Nnn0 χn0 = 1. For large n, where the two fragments have a signiﬁcant probability of being found seperated by a distance that makes the eﬀects of the

Pauli principle negligible, the basis channels Φn` become approximately orthogonal. The removal of the Pauli blocked states is done by working in the image space of the Norm kernel. The channel (`) P (`) vectors |χ i, and the associated relative wave functions F`(ρ) = n χn Φn`, have proper orthog- (`) 0(`) onality and normalization properties hχ |χ i = δEE0 . With this reduction of the RGM equation to a matrix diagonalization problem, the only remaining task is the construction of the HO channels which will be the topic of the next chapter.

2.6 Spectroscopic Factors

In the study of clustering and reactions we are concerned with the role nuclear structure plays in rearrangement reactions or fragmentation into clusters; the Spectroscopic Factor (SF) is a relevant quantity [80, 81]. We limit our discussion to weak decays where the Fermi Golden rule and lowest order perturbation theory are valid, allowing for a formal deﬁnition of the SF. In this section we discuss various ways the SF have been historically deﬁned and their relations, as well as a sum rule for the strength distribution.

15 2.6.1 Traditional Spectroscopic Factors

The traditional SF is a measure of the overlap of a state with a particular cluster-core configuration. For single nucleon pickup reactions for example, the traditional SF is given by: 2 † 2 (A + 1Jf ||a`j||AJi) (old) X † S`j = hA + 1Jf Mf |a`jm|AJiMii = , (2.41) 2Ji + 1 Mf m where the initial |AJiMii and final |A + 1Jf Mf i states belong to nuclei with A and A + 1 particles, respectively. The double bar here refers to the reduced matrix element as defined in the Wigner- Eckart theorem [41]. The detailed balance relation for pickup, knockout, or decay processes is observed (old) S`j (i → f) 2Jf + 1 (old) = . (2.42) 2Ji + 1 S`j (f → i) In this definition, we ignored the recoil of the heavy nucleus in the decay process. This results in some of the strength being absorbed into CM-excited states in the calculation. To correct for this, one uses the so-called recoil coefficient which we will discuss in the next chapter. The SF can be generalized to cluster (many-particle) transfer/decay processes defined by the relative partial wave and total angular momentum of the process, and subsequently constructing a multi-particle operator in second quantized form that can take us from the initial (target) nuclear many-body state to states belonging to the final system. Many-body correlations play an important role in the calculation of SF, with the well-known example of significant enhancement in pair transfer by superconducting pairing correlations [82].

2.6.2 Normalized Spectroscopic Factors

Unlike single-particle operators that also provide basis states for the many-body problems, cluster operators and resulting channels are not following the same normalization and orthogonality conditions. The whole fractional parentage picture is obscured by multiple ways of fragmentation and by Pauli blocked components. Mathematically this is controlled by the Norm Kernel. Let us start with the traditional SF and discuss how they are calculated from the reaction channels with the HO expansion for the relative motion. The parent state wave function here will be denoted by ΨP and the channel follows the deﬁnitons of Section 2.5. The coeﬃcients

Fn` = hΨP|Φn`i, (2.43)

16 are fractional parentage amplitudes, leading to the following expression for the traditional spectroscopic factor (old) X 2 S` = |Fn`| (2.44) n Here, ` is an asymptotic observable but n is an internal channel label. Under the assumption that channels are orthonormal, eq. (2.44) is just a sum of probabilities of transition to each n.

As discussed in [83, 84], the implicit assumption is that the relative motion channels obey proper normalization conditions both for bound state and scattering boundary conditions. We see that 0 the channels defined in the form of |Φch(ρ)i or |Φnì do not obey such condition. However, channels obtained as eigenvectors of the norm kernel (in the image space) do obey the orthogonality condition. Therefore, without even considering the Hamiltonian kernel, we obtain such channels by diagonalizing the Norm kernel. The new channels |G˜kì are orthonormal, and therefore the binary cluster relative motion wave function and new SF are written as

−1/2 ˜ 1 ˜ |ψkì = Nk` |Gkì = √ |Gkì (2.45) Nk` 2 2 (new) X 2 X 1 ˜ X 1 X ˜ S` = |hΨP|ψkì| = hGk`|ΨPi = hGk`|ΦnìhΦn`|ΨPi . (2.46) Nk` Nk` k k k n

The index k is used to enumerate the different eigenvectors of N˜ and Nk` denotes the k-th eigenvalue of the Norm kernel. The new SF still has the same interpretation as a total sum over the probabilities of structurally separating the system into given clusters, but now with proper orthonormalization. This renormalized definition of the SF has a fixed sum rule for the total strength of a state [85]. If we now sum over all parent states with a given set of quantum numbers P , and use the fact that they form a complete set, we obtain the sum rule

X (new) X X X X S` = hψk`|ΨPihΨP|ψk`i = hψk`|ψk`i = 1, (2.47) P P k k k which is the number of channels included in the calculation.

2.6.3 Dynamic Spectroscopic Factors

In processes such as cluster decay, it is the dynamics, i.e. the Hamiltonian, that determine the weights of diﬀerent cluster basis conﬁgurations and thus summing all probabilities can be

17 inappropriate. The dynamic SFs provide an appropriate definition. We define them via the overlap of the parent nucleus many-body state with a RGM generated channel function. The advantage of this approach is that it allows the dynamics of the Hamiltonian to define the relative motion between the clusters. The channels obtained as RGM solutions are linear combinations of the channels used in the previous section and thus have the same normalization properties. The new channels now have some hierarchy defined from the RGM energy eigenvalues. The dynamic SF is defined ”per channel”, as the decay proceeds through a single channel at a given energy

2 (dyn) (`) X 2 2 S` = hΨP|χ i = χn|Fn`| . (2.48) n The SFs can vary signiﬁcantly when considering decays proceeding through diﬀerent channels; when considering a sum over all open channels the sum rule given in eq. (2.47) is still observed.

2.7 Coupling to Continuum

We now turn to the physics of the continuum and how it can be tied to the nuclear shell model. The discussion here follows closely the one found in [86]. The presence of open decay channels in a nucleus leads to a restructuring of the many-body states, separating them into two classes: broad short-lived states, known as superradiant, and long-lived, trapped states, that become nearly decoupled from the continuum. This phenomenon of superradiance has been discussed by many authors, whose efforts are summarized in the recent review [87]. Our interest lies in how the physics of clustering can affect, and in turn be affected, by the openness of the finite quantum system. While the phenomenon is well known, in the physics of nuclei it remains elusive; our inability to tune the continuum coupling, the short lifetimes of states of interest, the elaborate experimental tools required for this research, and the general complexity of the many-body physics make it difficult to unambiguously pinpoint the superradiance. Nevertheless, experimental advances towards exotic nuclei far from the valley of stability motivate the quest for superradiance in nuclei while hints of superradiant states are seen in states of light nuclei, shown in Figure 2.1 . The Fesh- bach projection formalism [90, 91] appears to provide a straightforward theoretical strategy for exploration of this phenomenon [87]. Under this formalism a full Hilbert space is divided into two subspaces, an intrinsic or internal subspace Q that describes the isolated system, while the remaining continuum of external reaction states belongs to the second subspace P . The full problem can

18 + ��

�� + + � + �+ � + �� + + � �+ �� (��) � ��+α ��+α ��+α �� + �+α �+ �� + �+α � + �� + + + + + + + ��

Figure 2.1: Broad alpha clustering states in light nuclei. The data for 10Be are taken from [88] and for 18O from [89]

be exactly projected, leading to the energy-dependent non-Hermitian eﬀective Hamiltonian H(E),

1 H(E) = HQQ + HQP HPQ. (2.49) E − HPP

Here, HQQ acts only in the intrinsic space Q, the operator HPP connects only states in the external

P space, and HQP is the coupling between the intrinsic and external spaces. The eﬀective Hamil- tonian operator (2.49) belongs to the intrinsic space and describes the evolution of states in the projected space. Therefore, the matrix elements of the transition matrix between reaction channels |Ac(E)i involved in the scattering process are

c 1 c0 T 0 (E) = hA (E)| |A (E)i. (2.50) cc E − H(E)

It is important that the same reaction channels deﬁne the eﬀective Hamiltonian itself. The second term in (2.49) involves virtual processes going through the continuum channels; using the Cauchy

19 principal value (PV), we can separate real and imaginary parts 1 i HQP HPQ = ∆(E) − W (E) (2.51) E − HPP 2 given as

X Z |Ac(E0)ihAc(E0)| X ∆(E) = PV dE0 and W (E) = 2π |Ac(E)ihAc(E)|. (2.52) E − E0 c c(open) The imaginary part here is of factorized form containing a summation over channels that are open at a given energy. This form assures unitarity of the scattering matrix [92, 93]. Indeed, taking the eﬀective Hamiltonian in the form where the imaginary part is explicitly factorized

H = H˜ − iπ AA† (2.53) allows us to write the S-matrix in the channel space as 1 1 − iπ K S = 1 − 2πT = 1 − 2πA† A = , (2.54) E − H 1 + iπ K which is explicitly unitary, being expressed using the Hermitian K-matrix that comes from the Hermitian part of the eﬀective Hamiltonian 1 K = A† A. (2.55) E − H˜ In the above formalism bold symbols correspond to matrices in the channel space spanned by vectors |Ac(E)i for channels that are opened at a given energy. For most decays, the number of channels that are open is relatively small; cases with just one dominating decay channel are very common. Thus, mathematically, the factorized term iπ AA† is of low rank; if this term is dominating the eﬀective Hamiltonian (2.53), then this would force segregation of eigenvectors into those orthogonal to channels |Ac(E)i and to eigenvectors that are nearly parallel to |Ac(E)i in the Hilbert space. The number of nearly parallel ones, those referred to as superradiant, equals to the number of open channels and these states accumulate almost all width (imaginary part) thus absorbing a large part of the decay strength [94, 95].

2.8 Rotational Bands

The existence of localized clusters in light nuclei results in rigid structures whose rotation gen- erates rotational bands observed in experiment. An outstanding example is 8Be, seen in Figure 2.1

20 with the 0+, 2+, 4+ states forming a rotational band. Rotational spectra are recognised as a part of nuclear structure, with clear experimental signatures, seen throughout the nuclear chart. In this section we brieﬂy present some of the basic components in the physics of rotating many-body systems neccessary for the description of cluster rotations [77, 96]. For rotations to be possible, there must be some relatively rigid deformed system. The rigid nature of some clustered systems allows for the rotational motion to be nearly decoupled from other types of excitations [97]. Therefore, the low-lying spectra can be described in terms of a single rotating intrinsic shape. This leads to the excitation energies of nuclear states described by the rotation of a rigid (non-vibrating) body to be given by

2 2 ¯h HR = AJ = J(J + 1). (2.56) 2I

While higher orders of J2 are possible, this is suﬃcient [77]. Here, I is the only parameter and is commonly known as the moment of inertia of the band in question in a semiclassical analogy. The other constant of motion is the projection K of the angular momentum on the deformation axis, which is used to label families of states belonging to the same rotational band. The case of K = 1/2 bands is special as it sometimes shows eﬀects of Coriolis decoupling. This can be understood in a particle-plus-rotor model where the odd particle system is considered as a single particle on top of a rigid rotor [96, 98]. The excitation energies for the band now further depend on the so-called Coriolis decoupling parameter a

2 ¯h h J+1/2 i Ex = J(J + 1) + a(−1) (J + 1/2) . (2.57) 2I

In cases where the Coriolis decoupling parameter is large, this results in a zig-zag shape for the spectra of the states as one goes up in J. Let us make a note on how rotational bands are identiﬁed, apart from the spacings of the states. Deformations are probed experimentally by measuring reduced electric quadrupole transition rates B(E2), as it is directly related to the quadrupole moment of the nucleus [77]

2 5 JK 2 B(E2,J + 2 → J) = Q C (2.58) 0 16π J+2K2K

Here, Q0 denotes the intrinsic quadrupole moment of the rotational band, labeled by K, considered in this case. Within a rotational band these transition rates are tens or even hundreds of times larger

21 than what one might expect from a single particle picture [77]. From an independent particle model prespective, the phenomenon of rotation and transitions within a rotational band are collective in nature where nucleons transition coherently from one state to the other. The large rates coupled with the spacing of diﬀerent J values given by eqs. (2.56,2.57) serve as signatures of a deformed rotating system.

2.8.1 Elliott’s SU(3) Model for Rotations

In Elliott’s seminal work [99, 100], dating back to 1958, the foundation was laid out on how rotational bands can be described within the nuclear shell model with a HO single particle basis. Traditionally there had been a separation between microscopic (shell model) and macroscopic (liquid drop) descriptions of nuclei. Elliott bridged the two concepts by presenting a microscopic formulation that displayed rotational band characteristics. By exploiting the U(3) symmetry of the harmonic oscillator potential to label the states according to their irreducible representation, the representation of collective rotational states within a microscopic model was uncovered. While Elliott worked within a single major HO shell, the results hold for spaces spanning multiple shells. The U(3) irreducible representations are labeled by three integer numbers, traditionally denoted by f1, f2, f3, each of them being the number of boxes in each of the rows of the Young pattern associated with the irreducible representation [101]. Since the U(3) representation of the HO states concerns only the spatial components, it follows that any symmetry specified here refers to the spatial permutational symmetry; the spin-isospin part is given by the conjugate Young pattern, so as to maintain total antisymmetry of the wave function [102]. The 9 generators are taken to be the HO single particle Hamiltonian, the five (algebraic) Quadrupole operators and the three angular momentum operators. The difference between the algebraic Quadrupole operators and the usual mass/electric quadrupole operators is the absence of matrix elements between shells that differ by two quanta. By further removing transformations that only yield an overall sign change, we obtain the SU(3) group for the description of nuclear states. The two integers labeling the SU(3) irreducible representation (λ, µ) are given with respect to their U(3) counterparts by λ = f1−f2 and µ = f2−f3. The stretched SU(3) irreducible representations (λ, 0) imply full permutational symmetry in the spatial part of the wave function and will be used later on for the construction of α particle wave functions.

22 CHAPTER 3

CONSTRUCTING CLUSTER CHANNELS

In this chapter, we present the main contribution of this work. We construct many-particle fully antisymmetric channel states for a system consisting of two fragments in relative motion described by a HO wave function. By construction, the channels have the same oscillator parameter ω, carry no HO quanta in their CM coordinate and respect rotational symmetry.

3.1 Antisymmetry

The Pauli exclusion principle is an important consideration when constructing many-body states. Let’s consider here, for example, the 8Be system clustering into two α fragments. Due to the compact nature of 4He, the ground state of the α particle can be approximated by a s4 configuration. Antisymmetrizing the total wave function for a channel with both α particles in the ground state configuration of the same HO well would result in a trivial 0 as the state is Pauli blocked. The lowest, in terms of HO quanta, allowed configuration for the 2α system would 4 4 then be the s p (Ntot = 4). Such issues were recognized in the past, requiring the relative wave function between the cluster and the nucleus to have a fixed number of quanta, so as to not be Pauli blocked [103, 104]. The four quanta in the above example are in the relative motion between the clusters, leaving their respective internal s4 structures unchanged. When utilizing CI methods with a HO single particle basis we assume non-spurious configurations with no quanta in the CM motion. We briefly comment on the state-operator polymorphism through which antisymmetrization is enforced in every step of the cluster channel construction. For more in depth discussion on the topic, Appendix A describes second quantized states and operators, and two examples are found in Appendix D. Since configuration interaction approaches utilize a second quantization occupation basis to expand the many-body states, the states we start with are by construction antisymmetrized. In order to enforce complete antisymmetry we write the many-body wave functions as second quantization many-body creation operators acting on the vaccum, with each operator weighed by

23 an appropriate coeﬃcient. For example, the state describing a trivial s4 α particle is a single Slater Determinant which can be written as 4 creation operators acting on the vaccum

† † † † † |Ψα(s4)i = aν+aν−aπ+aπ−|0i = Ψα|0i. (3.1)

Here, the subscript π, ν denotes protons and neutrons respectively, while the ± denotes the m = ±1/2 angular momentum projection of the total angular momentum. The rest of the quantum numbers are those of the s shell {n`j} = {0 0 1/2}. The (s4) notation is omitted in the rest of the text. We are working here in the isospin formalism so the proton and neutron labels refer to the two projections of isospin. Similarly, for the daughter state there exists an AD-body creation † operator ΨD that creates the state when acting on the vaccum. The channel state can then be constructed as a sequential action of many-body creation operators

† † |Ψi = ΨDΨα|0i (3.2)

hr|Ψi = A [Ψα(rα)ΨD(rD)] = Ψ(r), (3.3) where bold symbols denote all single particle coordinates for each wave function. We are not dealing here with Jacobi coordinates, so the wave function in eq. (3.1) carries a CM contribution. † † The comment we started from implies that, with the deﬁnition (3.1), ΨαΨα|0i = 0 because of Pauli blocking. Thus to obtain relative channels, we have to manipulate the CM part of the wave function.

3.2 The Gloeckner-Lawson Procedure

Before constructing the cluster channels we mention the method used to construct so-called non-spurious states, meaning here states that have zero HO quanta in the CM motion, i.e. the many-body wave function can be written as

0 hr|Ψi = ψ000(R)Ψ (ξ), (3.4) with ξ the internal nucleus coordinates and R the CM coordinate. We mentioned this subject for two particles in Section 2.1.1, but the Moshinsky bracket recoupling scheme is not eﬃcient for many particles.

24 The technique presented here is applicable in CI approaches where the HO is used as the single particle basis. Alternative approaches either eliminate the CM contaminants from the states after calculation [23], or construct the many-body states excluding the CM excitations [105]. A computationally eﬃcient procedure was proposed by Palumbo and Prosperi and further developed by Gloeckner and Lawson [106, 107]; we discuss it next. The method augments the many-body Hamiltonian in eq. (2.26) by explicitly adding a HO Hamiltonian for the CM coordinate, multiplied by an tunable parameter β

1 2 2 3 Hˆ = H + βHCM = H + β(TCM + AMω R − ¯hω), (3.5) 2 CM 2 and then proceed to solve the new Hamiltonian Hˆ . The existence of the explicit CM HO contribution to the Hamiltonian implies that now any CM excited states will have an additional energy related to their CM HO quanta ECM = β¯hωNCM. The parameter β is tuned so the CM-excited states are shifted higher in the spectrum and are absent in the low-lying spectrum of interest (a typical number is β = 10).

While in NCSM calculations with the Nmax truncation prescription the separation is exact, this does not hold for an arbitrary space truncation, including some traditional shell model valence spaces. Remarkably, in the non-exact case the separation of the CM motion of the system is good, with expectation values of the CM hamiltonian hβHCMi in the resulting states being in the order of ∼ 10 keV [85].

3.3 SU(3) Method

Earlier attempts dealing with α clustering in nuclei [22] utilized the SU(3) symmetry properties of the HO and dealt exclusively with cases where the valence-particle nuclear wave function lied completely in a single major oscillator shell. Furthermore they required the cluster structure to be described by a conﬁguration having no intrinsic HO quanta, so as the total ground state wave function of the cluster would be a (0,0) SU(3) irreducible representation. This approach is especially well suited to the α cluster, where one does not expect signiﬁcant additional contributions to the α ground state. The s4 structure for the α wave function in (3.1) has a full permutational symmetry on the spatial part of the wave function and all angular momenta and spin quantum numbers (L, S, J, T ) are coupled to 0. The spatial part is a product of gaussians so any exchange of particles

25 will leave the total wave function unchanged. This structure has no quanta, including the CM degree of freedom. Let us now consider an α particle moving with 4 quanta of motion relative to some p shell nucleus. We need to project the motion of the α particle with 4 quanta in the relative motion onto the valence space of interest. The answer to this problem is found to be the stretched SU(3) irreducible representation. As discussed in Section 2.8.1, stretched representations are of the form (N, 0) where N is the total number of HO quanta available in the relative cluster motion. In this case we need to find the (4,0) irreducible representation in the p4 configuration subspace. The reason behind the need for a stretched representation becomes apparent if one considers the Young diagram of the representation; a single line for all boxes, indicating full permutational symmetry in the spatial part. The conjugate pattern for the spin-isospin part will be fully antisymmetric [102] restoring the proper symmetry properties of the full wave function. Exactly how these stretched irreducible representations are constructed depends on the many- body scheme one chooses to represent the basis. A basis that uses SU(3) symmetry might seem like an ideal choice, however this would result in complications when trying to deal with interactions not written in the SU(3) scheme and require heavy use of SU(3) Clebsch-Gordan coefficients [108, 109]. We take a different approach and work in an M-scheme [41, 97] basis where we truncate the space of many-body states to our chosen configuration (particle distribution in the shells) and diagonalize an operator whose eigenstates are the configurations of interest and which lifts all possible degeneracies within the configuration [85, 110, 111]. The correct irreducible representation can then be chosen using the eigenvalues of some operator. Such an operator can be built as a sum of the following operators

2 2 2 2 Oi = {C2(SU3),J ,T ,L ,S ,MJ }, (3.6)

1 3 where C2(SU3) = 4 Q · Q + 4 L · L denotes the second order casimir operator of SU(3) and MJ is the Majorana exchange operator. The operator Q is the so-called algebraic quadrupole operator, with no matrix elements between major oscillator shells diﬀering by 2 quanta. Each of the Oi operators is multiplied by an irrational number to avoid any accidental degeneracies in the eigenvalues. It should be noted that C3(SU3) is not necessary as it only helps to tell the diﬀerence between the (N, 0) and (0,N) irreducible representations, though the latter do not possess the permutational

26 symmetry we seek. At the end of this process we have a four-body state for the conﬁguration labeled by η, written as an operator in the second quantization approach, that has the following structure

SU(3) |Ψαη:L i = |[4](N, 0) : L = J, S = T = 0i , (3.7) and as such it represents a component of an α-like state in the desired conﬁguration. While this method works ﬂawlessly, it relies on diagonalization of matrices that quickly expand in dimension. Furthermore, it is only applicable to clusters that can be modelled in a Ntot = 0 space (all particles in the s shell). The method was heavily used in [85]. These limits drive us to develop new methods for the construction of cluster channels that both scale better computationally and are applicable to heavier and more complex clusters.

3.4 Center of Mass Boosting Method

In the next two sections we present our newly developed method that can be used to construct cluster channels with the fragments in some deﬁnite relative HO motion. In Section 2.1.1 we saw how one can transform from single particle coordinates to CM-relative coordinates in the HO basis using a linear transformation. We will use such a transformation in Section 3.5, for the CM coordinates of the two fragments, transforming to the system of total CM and relative cluster coordinates. To proceed with the transformation, we devise a method that can construct the fragment CM excited (boosted) states to be recoupled. We do this via HO quantum raising operators acting directly on the CM coordinate of each cluster. The CM creation operators (and annihilation correspondingly) are deﬁned in the usual way as

† 1 B = √ (AMωRm − iPm ) (3.8) m 2AMω¯h where m denotes a speciﬁc magnetic projection of vectors . The nucleon mass is denoted by M to avoid confusion with the projection index. The operator is constructed using an isoscalar mass- density dipole E1 operator r r 4π ¯h D = (B† + B ), (3.9) m 3 2AMω m m and by taking the part that increases the number of quanta. It should be stressed that these operators are one-body operators that act on the many-body wave functions in second quantization of Section 3.1, see also Appendix A.

27 To raise the number of nodes in the CM coordinate, we need to construct a rotational scalar that contains two raising operators since each node contributes two quanta of energy. This can be achieved by taking a dot product of two creation operators

† † † † † † † † B · B ≡ B+1B−1 + B−1B+1 − B0B0 . (3.10)

The application of this operator on a harmonic oscillator wave function gives,

1p B† · B†ψ (R) = (2n + 2)(2n + 2` + 3) ψ (R). (3.11) n`m 4 n+1`m The normalization of the operator is derived in Appendix B. In a similar fashion, to control the † angular momentum part, we act with the B+1 operator to force the wave function up along the chain of m = ` states. s (` + 1)(2n + 2` + 3) B† ψ (R) = ψ (R). (3.12) +1 n`` 4(2` + 3) n`+1`+1

Finaly the m projection can be brought to the required value by repeated action of an angular momentum lowering operator. The angular momentum operator needs to be an axial vector and leave the total number of oscillator quanta unchanged. Thus the operator B† ×B is the only choice. The action of this operator is deﬁned as

√ † † p L+ψn`m (R) ≡ 4 2 B0B+1 − B+1B0 ψn`m (R) = (l − m)(l + m + 1)ψn`m+1(R) (3.13) √ † † p L−ψn`m (R) ≡ 4 2 B−1B0 − B0B−1 ψn`m (R) = (l + m)(l − m + 1)ψn`m−1(R). (3.14)

An alternative way of dealing with the angular momentum part was outlined in [23, 44]. The result of a sequential application of node and angular momentum operators is a many-body wave function with a definite structure in the CM coordinate. The relation between the SU(3) irreducible representations and the CM boosted states can now be elucidated. The CM boosted state extends over all configurations and, as the original state, has a norm of 1 by construction. On the other hand, the SU(3) irreducible representations also have a norm of 1 but there is one for every configuration that exists. The boosted wave function of an s4 α particle is a sum of all possible (N, 0) SU(3) irreducible representations going over all allowed distributions of particles with coefficients known analytically s 1 N! 4! Xη = . (3.15) N N Q αi Q 4 i(ni!) i αi!

28 They are commonly known as Cluster Coefficients [85, 23, 112], depending only on the distribution of the particles (αi) in the HO shells (ni). The label η is used as in the previous section to denote the configuration and N is the total number of quanta. A resummation of the wave functions of all irreducible representations weighted by their respective Cluster Coefficients would result in the CM boosted wave function. Table 3.1 lists the cluster coefficients for different configurations in CM boosted by 8 HO quanta α wave functions. The right-most column refers to an α particle wave function that is not purely s4 but contains contributions from excited configurations.

Table 3.1: Select conﬁguration content of NCSM wave functions for 4He withhω ¯ = 20 MeV boosted to a CM boosted state with n = 4 and ` = 0.

Conﬁguration Nmax = 0 Nmax = 4

(sd)4 0.038 0.035 (p)(sd)2(pf) 0.308 0.282 (p)2(pf)2 0.103 0.094 (p)2(sd)(sdg) 0.154 0.141 (p)(sd)(sdg)(pfh) 0.000 0.005 (p)(sd)(pf)(sdg) 0.000 0.009

This new method of controlling the number of CM quanta directly bypasses all problems discussed, and can raise any many-body wave function to higher quanta content. The boosted states with HO quantum numbers for the CM coordinate NLM are written in second quantization with their coordinate representation

† 0 X η SU(3) hr|ΨNLM i = hr|ΨNLM |0i = ψNLM (R)Ψ = XN Ψαη:L , (3.16) η completely separating the intrinsic and CM parts of motion. It can therefore act on heavier clusters that extend beyond the s shell and deal with realistic cluster wave functions that might extend into multiple shells.

3.5 Constructing the Channels

These CM boosted states for each cluster now need to be recoupled into a complete channel with some non-spurious wave function with deﬁnite relative motion. In Figure 3.1 we illustrate the construction of a cluster basis state which has the two clusters in relative motion described by the

29 D0

D0 nl(⇢)

0(RD) 0 ↵0 ↵

0(R) nl(R↵) O O

Figure 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.

wave function ψn` and the total CM by a gaussian wave function.. We can rewrite eq. (2.12) so that the coupling is to a ﬁnal total CM wave function with 0 quanta,

X Ψ` (R, ρ)Ψ0 Ψ0 = A [ψ (R) × ψ (ρ)]` Ψ0 Ψ0 = A Mn`00;` [ψ (R ) × ψ (R )]` Ψ0 Ψ0 , µ α D 0 n` µ α D ϕ n1l1n2l2 n1l1 α n2l2 D µ α D n1l1 n2l2 (3.17) p with tan ϕ = AD/Aα the angle required for this particular mass ratio recoupling, and we deﬁne

ψ0(R) ≡ ψ000(R) to shorten notation. Equation (3.17) demonstrates that one needs to actually create CM boosted states not only for the cluster, but also for the daughter nucleus. This requirement weakens as one moves towards larger mass asymmetries; the sum is weighted towards the case where only the light cluster has all CM quanta, due to the form of the Talmi-Moshinsky coefficients. To illustrate the skewness of the transformation for different masses we list the Moshinsky Coefficients for coupling to the same final state in Table 3.2. We see that at the extreme mass ratio d = 10 describing α clustering in the beginning of the fp shell (44Ti → α+40Ca ), a single configuration absorbs about 80% of the total sum.

30 Table 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.

n1, l1 n2, l2 d = 1 d = 5 d = 10 2,0 0,0 1/4 25/36 400/441 √ √ 1,1 0,1 −1/2 −5 5/18 −80 5/441 √ √ √ √ 0,2 0,2 1/ 6 5/9 6 40 2/441 3

Under the so-called no-recoil assumption, the daughter nucleus is heavier than the cluster and the only component of the transformation needed is the one where all the HO quanta are in the cluster CM. This transformation is depicted in Figure 3.1. The single coeﬃcient associated with the truncated transformation is known as the Recoil Coeﬃcient and is the inverse of a Moshinsky bracket. It has a relatively simple form [85, 113]

−1 N/2 n`00;` N MP RN = ϕM00n` = (−1) . (3.18) MD

Here, MP,MD represent the mass of the compound and daughter systems respectively, and N = 2n + ` is the number of HO quanta in the relative coordinate motion. For comparison, the relevant Moshinsky coefficients are the ones in the first row of Table 3.2. While this scheme works for SM structural (static) calculations, in which the channel wave function is projected onto a non-spurious parent state, ab initio RGM studies of clustering in light nuclei need to properly reconstruct the full translationally invariant CM wave function in order to capture the decay dynamics. This can be shown for the 2α system with an s4 approximation for the α particles. If we put an odd number of quanta in the relative motion, we would have two identical objects in an odd parity relative wave function. Such a wave function survives if we only consider the truncated form of the sum, as there is no reason the s4p3sd configuration would have a 0 norm a priori. Indeed, the symmetry of the Talmi Moshinsky brackets [45]

n`00;` ` n`00;` π M = (−1) π M (3.19) 4 n1l1n2l2 4 n2l2n1l1 implies that all components will cancel out when considering their symmetric distribution of quanta as ` is always odd. At the end of this procedure, we have obtained a many-body channel wave function that respects all symmetries imposed, has a non spurious CM component, and possesses the desired relative

31 motion. Such a wave function deﬁnes a channel with deﬁnite relative angular momentum and can be written as

JM n`JM h 0 0 I i Ψch = ψ0(R)A ψn`(ρ) × ΨC × ΨD . (3.20)

Here, the × symbol again denotes angular momentum coupling. When the cluster and/or daughter is a J = 0 particle, the situation simpliﬁes considerably, as there are no recoupling requirements.

32 CHAPTER 4

TRADITIONAL SHELL MODEL STUDIES

In this chapter results using the traditional shell model (SM) are presented. First we focus on the α particle wave function, in the s4 configuration. Light nuclei in the sd shell mass range are used to demonstrate the prevalence of α clustering in their low-lying states. We then turn to some specific considerations of cluster evolution as one moves from the N = Z line, and conclude with a Continuum Shell Model study of 13C that considers the effects of neutron decay of the system on its α characteristics.

4.1 α Particle Wave Function

Prior to discussing α clustering, we take a closer look at the α particle wave function. Even though this chapter presents results obtained for nuclei in the SM, the structure of the α particle is relevant for both this and the following chapters. Our goal is to ascertain whether the s4 approximation is a proper one. This investigation is carried out using the NCSM to obtain realistic wave functions for the α particle with 2-body interactions. Results of multiple calculations on the ground state wave function are shown in Figure 4.1 using the NCSM with the JISP16 interaction and multiple values forhω ¯ and Nmax. There are a few major points to be made here. First, we reaﬃrm that, in almost all cases, the s4 component comprises the bulk of the ground state wave function. To a large extent, the calculations appear to be converged, reaching the true ground state energy. Second, we observe that at least for some range ofhω ¯ (20-27 MeV), the s4 content of the wave function remains at a relatively high level (i.e. above 90%). The percentage is still high (> 75%) throughout a broad ¯hω range, pointing to a good approximation. Finally, we comment on the total binding energy of the α and how it converges with respect to the Nmax truncation. If we take for example the case of ¯hω = 22.5 MeV, 4 when moving from Nmax = 2 to Nmax = 4, the s component of the wave function changes by 1.6%. This change is accompanied with a drop in the total binding energy of 3.1 MeV, or about 10% of the total binding energy of 4He. This raises the possibility that large contributions to the

33 Figure 4.1: Ab initio calculations for the ground state energy (left) and the s4 component 4 of the He wave function (right) with the JISP16 interaction spanning multiple Nmax truncations and ¯hω values. energy come from non-dominant conﬁgurations [114] and therefore are not captured in this type of calculations. The SRG method appears to remedy this situation somewhat and such calculations will be discussed in a later section.

4.2 Clustering in the sd Shell

The sd shell is an attractive region to test methods describing clustering due to the abundance of experimental data and the compactness of the space. Furthermore, the sd space has been the target of intense theoretical interest throughout the years, leading to eﬀective interactions that describe and predict the spectroscopic characteristics remarkably well [60, 61, 116, 117]. Energy levels, reduced electromagnetric transition rates and particle transfer SF (which are extremely sensitive to the wave function content) are well reproduced, thus an extension to clustering is appealing. In this section we demonstrate that the method introduced in Section 3.4 is in exact agreement with the SU(3) based approach of the Cluster Nucleon Conﬁguration Interaction Model (CNCIM) [85, 118] and others [22] in the appropriate limit. We go beyond the s4 approximation which requires only the stretched SU(3) operators; this investigation is limited to the even-even N = Z and N = Z + 2 nuclei in the sd shell. The even-even N = Z nuclei are a good place to start as they can posses multi-α type structures as discussed in the introduction. The Nmax = 0 calcu-

34 Figure 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 withhω ¯ = 14 MeV and Nmax = 8. lation and SU(3) results previously reported [85] are in exact numerical agreement, as expected. The values for both the traditional and the new spectroscopic factors for ground state transitions are shown in Figure 4.2. A comparison between the dashed blue line and the solid/dotted lines shows the eﬀect of using the Fliessbach method for taking into account the channel normalization. The α SF no longer decrease with increasing mass number A, restoring the large values near the end of the shell. Furthermore, the need for extra scaling of the total strength to the 20Ne value is no longer neccessary to reproduce the data. The large α SF values in the sd shell are further supported by a strong level of quartet correlations, recently discussed in [119], although the SU(3) transfer operator used here is more speciﬁc than the general quartet coupling. The use of a realistic wave function for the α particle opens more channels for the relative motion. We chose ¯hω = 14 MeV to be in line with typical values used in the SM close to mid sd

35 Figure 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. shellhω ¯ = 41A−1/3 MeV [97]. The single SU(3) operator for 0+ → 0+ transitions was identified with having a cluster-daughter relative HO motion with n = 4 and ` = 0. As the realistic wave function contains configurations with Ntot =6 0, the channels with node number 4 ≥ n ≥ 4 − Nmax(α)/2 ≥ 0 are open. The channels are not orthogonal and some of their strength will be absorbed by the ground state as seen in Figure 4.2. The additional strength is countered by components of the channel that lie outside of it. We also look at the distribution of ` = 0 α strength for 20Ne 16 → O + α with three different α wave functions. The results for Nmax = 0, 4, 8 are shown in Figure 4.3. We see that the increased number of channels brings larger α SF values, a result of having different couplings available that can lead to a 0+ state. The norm kernel here is calculated after projection of the channel norm to the sd space. The projection is one of the reasons for the increase in SF and its effects will be investigated in the future. The role of non s4 components of the α wave function requires a full RGM treatment. These components would allow transitions forbidden in the SU(3) (8,0) transition operator. Possible experimental effects from the non-s4

36 components of the α particle are an interesting area fo future research.

4.2.1 The Case of 20Ne

The nucleus 20Ne is rather special in the sd shell due to its characterization as an α particle rotating around the closed 16O core [118, 120, 121, 122]. The ground state (new) SF is large (0.76) and consistent with observations for this and neighboring nuclei (22Ne, 24Mg) indicating that the high degree of clusterization is reproduced by the shell model calculation. All the members of the ground state rotational band can be described as an α particle orbiting the closed 16O core in a speciﬁc ` state. The α SF to the ground state of 16O, along with the reduced transition rates for the de-excitation of each state (B(E2)↓) are shown in Table 4.1. The yrast states show both a rotational behavior and α cluster characteristics. For each state there is only one channel, therefore the sum rule is exhausted at 1. From a theoretical prespective we understand the 16O +α rotational band

Table 4.1: Transition rates and α spectroscopic factors for the 20Ne rotational band members with the USDB interaction. J π B(E2)↓ (W.u.) B(E2)↓ [Exp] (W.u.) α SF (16O g.s.)

0+ - - 0.76 2+ 14.4 20.3 0.78 4+ 17.0 22 0.66 6+ 12.7 20 0.58 8+ 8.6 9.0 0.40 as being microscopically formed due to the collective motion described by the SU(3) operator. The correspondence between the rotational and the α character of the states is a direct consequence of the single SU(3) operator that describes α transfer and is an eigenstate of the Q · Q collective Hamiltonian enabling rotations [98]. The rotational band terminates at the 8+ state, in accordance to its description as an (8,0) SU(3) irreducible representation. Next we study how the formalism holds up in comparison with negative parity states generated from 1 particle-hole (ph) excitations in a psdpf space. The extension is neccessary as no negative parity states can be constructed in the N = 2 major oscillator shell. For this we use the PSDPF[124] interaction which was derived precisely in order to treat (0+1)ph states for sd shell nuclei. The interaction was fitted to experimental spectra, with the rms deviation of the final fit being about

37 (a) (b)

Figure 4.4: (a) Experimental population of states in 20Ne from (6Li,d) and (b) theoretical predictions 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].

200 keV. This can be compared with the 80 keV rms deviation obtained for USDB, and so we expect slightly larger deviations. We compare the results for the spectroscopic factors with results from a (6Li, d) reaction. What makes such reactions attractive in the study of α clustered systems is the high degree of selectivity with which they populate the spectrum of the final nucleus. This can be seen in Figure 4.4(a), where only a handful of the 60+ states that exist in 20Ne up to 12 MeV are populated with the majority of the states populated being natural parity states. For all states that are relatively well populated in the reaction, we observe large α strength in the theoretical calculation . There is still only one channel per angular momentum, and therefore the new SF sum rule is exhausted at a value of 1. The energies of the states are relatively close to the experimental counterparts, so this leaves little room for misinterpretation. The small population of the ground state and the first 0+ can be attributed to the angle of the detector in the experiment; for smaller angles the state should be further populated. The applicability of the shell model to clustering has been questioned for many years; limitations of valence space, problems arising from the use of the older definition of SF, the effective

38 Hamiltonian are all of concern. This example demonstrates that the shell model with the new deﬁnition for the SF can extract clustering information from the many-body wave functions. There are certainly limitations in the SM, that are being adressed [118], however, to the extent of this study we are not severely aﬀected.

4.2.2 Evolution of Clustering with Increasing Nucleons

Whether clustering survives when moving away from the N = Z line is also of interest [89]. In the following study, the nucleus 20Ne represents a core + α system which is known to be clustered. To investigate how the system responds to extra nucleons, we take states created to have a 16O + α structure with relative angular momentum ` = 0, 2, 4, 6 and add a neutron in the d5/2 orbital, recoupling to all possible angular momenta in each case. Alternatively, one can consider adding an α particle with a definite relative motion to the ground state of 17O. We end up with 18 channel states versus the 1935 many-body states of the full calculation. This idea of reducing the size of the problem using cluster configurations instead of all the many-body states possible in the space is at the center of interest in the CNCIM [85]. In the smaller subspace of these (non-orthogonal) basis channels, we now apply the Resonating Group Method approach and solve the generalized eigenvalue problem, with the matrix elements of the Hamiltonian Kernel calculated with the USDB interaction. The resulting low-lying spectrum is compared with both USDB and experimental values. The results are shown in Figure 4.5, with ground states showing binding energy and the rest of the states showing excitation energies from their respective ground states. While the gross features are reproduced there are two main points to make. First, the binding energy is 3.5 MeV less than the full USDB calculation, the latter being closer to experiment. Though not shown explicitly in Figure 4.5, the states coming from different ` couplings of the α+16O mix quite strongly in some cases. This implies that even though the system excitations are adequately described within this heavily truncated basis, the simple clustered picture of an α particle moving in some definite relative partial wave relative to 16O and the extra nucleon being a spectator is perturbed. This mixing can be experimentally probed by means of SF for different angular momenta obtained from transfer experiments [125]. The comparison of α SF is shown in Table 4.2, and is normalized to the ` = 2 wave of the ground state transfer to match the way

39 �/� + �� /� + �� /� + ��

+ � �/� �� /� + �� /� + �� + + �/� + �� /� �� /� ��

� �/� + �� /� + �� /� + �� (��) � + / �� + � � �/� + �� /� �� + - + - + - � �/� �� /� �� /� ��

Figure 4.5: Low-lying RGM, full USDB and experimental spectra of 21Ne.

experimental data is presented [125]. The experimental 9/2 values are grouped together with the ones for the 1/2 state because they are diﬃcult to resolve due to their small energy diﬀerence.

Table 4.2: Comparison between new and experimental SF for low-lying states in 21Ne.

Jπ S(new) S(exp) ` = 0 ` = 2 ` = 4 ` = 0 ` = 2 ` = 4 3/2+ 1.0 0.18 1.0 ± 0.05 0.42 ± 0.04 5/2+ 0.78 0.02 0.44 1.04 ± 0.41 ... 0.32 ± 0.18 7/2+ 0.9 0.14 0.91 ± 0.08 0.23± 0.04 9/2+,1/2+ 0.81 0.33 0.9 ± 0.05 0.29 ± 0.03

For reference, we also provide here the generalized eigenvalue problem kernels for the 5/2+ states constructed with ` = 0, 2, 4 (consistent with the order of matrix elements [` × 5/2](5/2).

−7.85 1.25 1.93  H =  1.25 −6.95 0.545 (4.1) 1.93 0.545 −8.9

 0.185 −0.03 −0.038 N =  −0.03 0.17 −0.015 . (4.2) −0.038 −0.015 0.219

40 The norm kernel demonstrates the degree of overlap between the diﬀerent channels. One should keep in mind that the non-orthogonality here comes from the extra neutron in the d5/2 orbital; the channels in 20Ne are completely orthogonal due to diﬀerent angular momenta.

4.3 Superradiance in 13C and α Clustering

We conclude this chapter with a final example of the 13C nucleus. This nucleus is interesting as it has two J π = 3/2+ states in close proximity, at 7.686 MeV and 8.2 MeV respectively. Both of these states are above the neutron separation energy and thus interact via coupling to a common neutron channel. The neutron transfer cross section from the 12C(d,p)13C reactions [126] clearly shows the two resonances with drastically different widths of 70 keV and 1.1 MeV respectively. Shell model calculations, for example using the same p − sd model Hamiltonian [127], roughly reproduce the energy of these states but predict nearly the same width for both. The experimental data and theoretical results concerning these two states are summarized in Tab. 4.3. The difference in the observed decay width by nearly 2 orders of magnitude and inability of the SM to describe this suggest superradiant behavior driven by strong continuum coupling. This situation is similar to a two-level model discussed in Ref. [128]. Furthermore, out of the two states, only one is seen in the alpha transfer experiment [129] thus raising a question whether the α transfer strength is also affected by the superradiant restructuring due to neutron continuum interaction. We investigate this using the p − sd valence space with interaction Hamiltonian from Ref. [127], we use J = 3/2+ eigenstates of the SM Hamiltonian as basis for continuum treatment. We first diagonalize the hamiltonian in the unrestricted psd space obtaining the 30 lowest J = 3/2+ states. In this truncated basis, the non Hermitean effective Hamiltonian takes the form

X H(E) = |αihα|H(E)|βihβ|, (4.3) α,β with the matrix elements, ignoring the self energy term, being in the form

i X c c hα|H(E)|βi = Eαδαβ − hα|A (E)ihA (E)|βi. (4.4) 2 c

c The decay channel |A (E)i = αc(E)|ci can be partitioned into a kinematic factor ( αc(E) )and a structural factor, both describing diﬀerent parts of the decay. The kinematic factor is responsible for the continnum density of states and channel normalization and is the only part

41 that explicitly depends on the energy. The factor |ci contains the many-body structure of the c channel. This allows us to write the overlap hA (E)|βi = αc(E)hc|βi. Since we are following a phenomenological approach, the only requirement for the kinematic factor is that it behaves correctly when approaching the threshold for the decay, i.e. it should identically drop to 0 when 1/2 E < Ethreshold. For this reason, we choose the form αc(E) = sPc (E), with Pc(E) the penetrability given by ! x dH(+) Pc(E) = Im (4.5) H(+)(x) dx x=kR and s a phenomenological scaling factor whose value is determined by the total sum of the widths; see [86] for more details.

Table 4.3: Energies, widths in MeV and α spectroscopic factors (structural) for the two lowest 3/2+ states in 13C

Exp. Exp. SM CSM SM CSM Energy Width Width Width alpha SF alpha SF 3/2+ (1) 7.686(6) 0.070(5) 0.809 0.094 0.026 0.054 3/2+ (2) 8.2(1) 1.1(3) 0.369 0.984 0.058 0.038

The results of this Continuum Shell Model study are shown in Tab. 4.3. For the neutron width, ﬁfth column, we indeed see a strong redistribution of widths which brings them to an agreement with experiment. There are changes to the alpha spectroscopic factors from this restructuring, the alpha strength is redistributed favoring the narrow state which appears to be consistent with observations [129]. This simple example demonstrates that the open channels will aﬀect the α spectroscopic characteristics of the system. The collectivization of decay characteristics, known as superradiance could provide a mechanism for the survival of highly clustered states well into the continuum.

42 CHAPTER 5

NO CORE SHELL MODEL STUDIES

In the previous chapter, we presented a study of nuclear clustering using the SM. While the insight gained from these problems is invaluable, we turn next to more fundamental questions about the emergence of nuclear clustering from ab initio 2-body forces.

5.1 The Nucleus 8Be

We begin by studying light α clustered systems from an ab initio prespective, focusing on 8Be which, due to numerous previous theoretical studies [130, 131, 132, 36], emerged as a benchmark for clustering methods. In addition, 8Be is a stark example of collectivity and rotations in the continuum [133, 134] where, as being well established experimentally in many light nuclei [31, 6, 89, 88], strongly clustered rotational bands survive the complexity of many-body dynamics. We start by constructing cluster basis channels to use in the RGM of Section 2.5. We perform calculations with up to N = 8 quanta in the relative motion coordinate, with N = 4 being the minimum in order to not have a Pauli blocked state. The full Moshinsky recoupling for the relative coordinate channel is employed

` 0 0 X n`00;` h † † i |Φn`m i = |ψ0(RCM )ψn`m (ρ)ΨαΨαi = Mn l n l Ψn l × Ψn l |0i, (5.1) 1 1 2 2 1 1 2 2 m n1l1n2l2

† where Ψnlm is an α ground state wave function boosted to CM quantum numbers nlm written in second quantized form. Employing the RGM with the JISP16 interaction, we obtain the results presented in Figure 5.1, demonstrating the gross features of the approach for a broad range ofhω ¯ values. A rotational band is formed by the 0+, 2+ and 4+ states emerges, and it is remarkable that this rotational band remains nearly undistorted in the full no-core shell model study of 8Be that has no a-priori assumption about any clustering structure. The minimum in the total energy lies in the region of ¯hω where the realistic α wave function is best described by the s4 conﬁguration. Since the states are unbound, these intermediate-range RGM solutions should be properly matched or combined in the Hilbert space with the asymptotic ones through other techniques such

43 Figure 5.1: Low-lying spectra of 8Be calculated with the JISP16 interaction at varioushω ¯ values in units of MeV. as R-matrix, CSM, or HORSE. For long-lived resonances the continuum coupling is weak and does not modify the structure, in this limit perturbation theory is applicable, therefore Fermi’s golden rule and the spectroscopic amplitudes characterize decay and reaction observables. In this approximation, the decay width is given by

2 Γ` = 2P` |F`(R)| , (5.2) where P is defined in eq. (4.5), F` is the RGM solution for the relative motion, and R is the so-called channel radius, i.e. an arbitrary radius where we assume the decay products no longer feel the effects of the nuclear potential. Table 5.1 lists the α decay widths for various ¯hω values along with the respective (dynamic) α SF for a fixed value of R = 3.6 fm. Overall the agreement with experiment seems satisfactory, however larger valence spaces are needed for the description of highly excited states. This is apparent from the decay width reduction as we move to higher values of ¯hω, where the range of validity (eq. 2.7) is smaller, thus the larger basis is required. We now take a closer look at the Hamiltonian and norm kernels and their contributions shown in Table 5.2. The s4 approximation for the α has all 4 particles in 0 quanta HO wave functions and thus the relative kinetic energy is Trel = 3 × 3¯hω/4 and the Coulomb energy between the two 2p protons is given by VCoul = e 2mω/¯hπ. For the 2α system, the norm of each cluster channel

44 Table 5.1: Alpha decay widths and spectroscopic factors for the first 3 states in 8Be using different oscillator parameters ¯hω. The S(dyn) are calculated with the lowest energy 8 solution of 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

+ + + (dyn) (dyn) (dyn) ¯hω (MeV) Γα(0 ) (eV) Γα(2 ) (MeV) Γα(4 ) (MeV) S0+ S2+ S4+ experiment 5.6 1.5 3.5 14.0 7.82 1.35 2.99 0.61 0.61 0.54 17.5 10.75 1.69 3.04 0.69 0.68 0.58 20.0 9.88 1.48 2.38 0.74 0.72 0.61 22.5 7.64 1.10 1.61 0.78 0.76 0.64 25.0 5.18 0.72 0.97 0.8 0.78 0.65 30.0 1.09 0.23 0.27 0.82 0.78 0.64 basis state N with increasing number of quanta N varies as [135]

2−N NN = 2(1 − 2 ), (5.3) asymptotically approaching N∞ = 2, the value for the normalization of two boson-like particles. The diagonal contribution to the relative kinetic energy (substracting the α particle internal contributions) is given by Trel = ¯hω(n + 3/4). Since both the norm and the α particle contributions are known, the diagonal kinetic energy matrix element can be written

¯hω 21 T (n, `) = N 2n + ` + . (5.4) rel 2n+` 2 2

The full Hamiltonian kernel for all values of ¯hω in Table 5.1 can be found in Appendix E for the partial waves ` = 0, 2, 4.

We further perform calculations where the α particle wave functions are obtained in the Nmax = 2, 4 truncations to probe the eﬀects of realistic cluster wave functions on the total binding energy. The value for ¯hω is ﬁxed at 20 MeV and the JISP16 NN interaction is again used. We observe, as seen in Figure 5.2, a reduction of the total binding energy. The correlations included via the use of Nmax = 4 α wave functions help lower the total energy by 12 MeV, which is approximately 4 equal to twice the amount the He binding energy is lowered when moving from an Nmax = 0 to an Nmax = 4 truncation (6.5 MeV). The spacings between the states also shift to the order of less than 1 MeV.

45 Table 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 |Tz| values for the interaction. The second column refers to contribution to a single α particle. We use the JISP16 interaction withhω ¯ = 25 MeV. Operator s4 α + α(2, 0) α + α(3, 0) α + α(4, 0) 1 1 1.5 1.875 1.96875 Trel 56.25 271.875 386.719 455.273 VCoul 0.892 6.48937 7.86807 8.02309 V0 -56.69 -211.267 -255.065 -259.621 V1 -11.31 -44.8899 -54.0692 -54.8474

5.2 The Nucleus 9Be

For the rest of the NCSM calculations we use the JISP16 interaction, selectinghω ¯ = 20 or 25 MeV. The next system depicting strong cluster characteristics is 9Be. Here, we assume a 5He + α clusterization for the system, with the 5He nucleus in the ground state. Since 5He is unbound, we will also consider the 1/2+ state as a channel component. Both clusters are taken to be in their 4 4 Nmax = 0 truncation, i.e. an s configuration for the α particle and an s + p3/2 configuration for 5He. As in 8Be, the minimum number of quanta available in the relative motion is N = 4, boosting the α particle to the p shell. We construct channels, both for positive and negative parity, up to ` = 5, and compare both with a full NCSM calculation and the states seen in experiment; the results are shown in Figure 5.3. There are three rotational bands of interest, identified in experiment For the ground state rotational band we observe small changes with respect to the number of available quanta in the relative motion, signifying that it can be well described within the p shell. On the other hand, the band members of the Kπ = 1/2− band exhibit a larger relative shift when increasing the number of relative quanta to N = 8, with the smaller Coriolis decoupling constant being recovered, consistent with experiment. The energy of the band relative to the ground state is about 5.5 MeV too high. The positive parity band displays a similar behavior, being 9 MeV higher than its experimental counterpart on average when considering the lowest relative quanta N = 5. The inclusion of channels with more quanta in the relative motion results in a lowering of about 5 MeV with respect to the ground state. Table 5.3 lists the squared overlaps (dynamic SF)

46 Figure 5.2: RGM calculations of the rotational band of 8Be for ¯hω = 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. and absolute energies for the NCSM states belonging to the rotational bands. Other states have SF less than 0.1 and are not listed. Apart from an overall lowering of energies, the ground state band remains relatively unchanged, even in the overlaps with the NCSM states. Since all calculations result in large spectroscopic factors, it is evident that we can construct the same state in multiple equivalent ways. Channels with a small number of HO quanta in the relative motion are more susceptible to large overlaps with one another, as they occupy a constricted space. As this number increases, the Pauli principle is felt less, and the channels slowly become orthogonal. Regardless, the SF are largest when considering the 8Be + n form for the channels pointing to it as the more relevant form of clusterization.

47 Table 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 ¯hω = 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. J π NCSM RGM (4) RGM (8) SF(4) SF(8) SF(8*) SF(8**) 3/2− -43.3 -16.0 -22.1 0.7 0.7 0.74 0.8 5/2− -39.8 -12.3 -18.6 0.67 0.68 0.73 0.79 7/2− -35.5 -7.59 -13.0 0.45 0.45 0.48 0.77 9/2− -28.0 -0.02 -6.48 0.51 0.52 0.56 0.78 1/2− -35.3 -6.61 -13.6 0.65 0.67 0.72 0.76 3/2− -29.8 -0.92 -9.91 0.6 0.63 0.69 0.77 5/2− -26.8 -3.96 -8.7 0.64 0.66 0.7 0.77

5.3 The Nucleus 10Be

The last Be isotope considered here is 10Be, which is also found experimentally to possess rotational bands and clustered configurations [136, 137, 138], fragmenting into 6He + α. There are two rotational bands of interest, both Kπ = 0+, with the first built on top of the ground state and + the second on top of the 02 state at 6.179 MeV. These cluster rotational bands can be identified in the NCSM calculation, with the second one at a high excitation energy (∼ 30 MeV).

Table 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).

π J NCSM Eexp (MeV) B(E2) (W.u.) SF(4) SF(6) SF(8) SF(8*) 0+ - - - 0.59 0.57 0.54 0.62 2+ 4.3 3.368 4.5 0.53 0.52 0.49 0.55 4+ 15.2 11.76 1.5 0.35 0.36 0.34 0.35 0+ 29.9 6.179 - 0.01 0.39 0.38 0.11 2+ 31.9 7.542 4.3 0.005 0.30 0.30 0.19 4+ 36.5 10.2 5.9 0.001 0.21 0.22 0.4 6+ 43.8 13.5([139]) 5.8 - 0.09 0.1 0.63

In Table 5.4 we list α SF for both rotational bands in five different RGM calculations, progres- sively including more quanta in the relative motion. The rotational band members are identified

48 Figure 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. in the NCSM from relatively large B(E2) transition rates. The the ground state rotational band and its cluster component lie predominantly in the p shell, while the excited rotational band is described as an excitation to upper shells involving two particles. We can further probe the band members of the second rotational band by constructing channels based on the first excited state of 6He with J π = 2+. The channels built with this configuration are not allowed to mix with the ones built on with ground state of 6He as they correspond to asymptotically orthogonal cases. The 6+ state has a low α SF to the 6He ground state, while the value is significantly higher for the 6He(2+) state. For both the higher spin members of the band (4+, 6+) we observe a preferential decay to the channels of the extended basis. While the theoretical calculation points to the dominance of an (8,0) rotational band in the spectrum, the band appears to mix with other SU(3) irreducible representations. There is also a shift in the energy of the state, deviating from the J(J + 1) expectation for the rotational band. The SU(3) decomposition of the NCSM states considered as rotational

49 band members is shown for the most important irreducible representations in Table 5.5, along with their particle-hole content.

Table 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 as in 5.4 J π (2,2) (0,3) (8,0) (6,1) 0p-0h 2p-2h 4p-4h 0+ 0.45 0.29 0 0 78.1% 12.2% 9.68% 2+ 0.40 0.29 0 0 77.9% 11.8% 10.3% 4+ 0.26 0.42 0 0 76.8% 11.2% 11.9% 0+ 0.02 0 0.57 0.09 4.25% 74.8% 20.1% 2+ 0.01 0.01 0.58 0.13 3.03% 75.5% 21.4% 4+ 0 0 0.51 0.19 2.35% 76.0% 21.6% 6+ 0 0 0.44 0.29 - 78.0% 22.0%

5.4 The 3α System

As a final calculation, we probe the cluster characteristics of the 3α system 12C. Various cluster geometries have been proposed for 12C states [140, 141, 35]. α clustering plays a major role in this nucleus and correspondingly for the formation of elements in nature. Here for the first excited 0+ (Hoyle) state the fraction of the direct decay, currently believed to be less then 10% of the total, and its competition with the sequential, via 8Be, one is of particular interest [14, 15]. The channels involving three clusters are more difficult to construct as there are now 2 internal Jacobi coordinates in the cluster system. In order to maintain translational invariance, the channels are built in 2 steps, first constructing a 2α system with some relative motion quantum numbers nρ, `ρ and then combining it with a third α cluster with the relative quantum numbers of the α+2α system denoted by nλ, `λ. We allow up to 12 quanta of relative motion to be shared amongst the 2 relative coordinates ρ, λ, and only look at states having J π = 0+, 2+, 4+. We employ the RGM procedure with three identical α particles, each in an s4 configuration using the N3LO Hamiltonian SRG evolved to λ = 1.5fm−1. Table 5.6 shows the results of the RGM and the NCSM compared to experiment. The unconstrained RGM wave function defines initial amplitudes for all types of asymptotic + 12 + three-α solutions. The overlaps of the NCSM 0 states with the RGM solution h C(01 )|RGMi =

50 Table 5.6: Ground state binding energy and excitation energies for rotational band members of 12C using the NCSM and RGM with ¯hω = 20 MeV.

J π NCSM Exp. RGM 0+ -108.99 -92.17 -94.08 2+ 3.4 4.4 4.8 4+ 12.5 14.1 11.9

12 + 0.841 and h C(02 )|RGMi = 0.229 are the structural contributions to the decay width of the states. The amplitude for the sequential decay process, proceeding via ground state of 8Be, can be evaluated by constructing the constrained 8Be + α channel separately, and projecting it out. For + the 02 state the magnitude of the sequential decay amplitude is 89% of the total, with remaining 11% corresponding to all other processes that do not proceed via the 8Be ground state. These amplitudes, their interference, ﬁnal state interactions, and phase space lead to observables, see ref. [142] and references therein.

51 CHAPTER 6

CONCLUSIONS

The main goal of this thesis was to provide a new method with which nuclear clustering can be described from a fully microscopic point of view. The boosting method has achieved this. We examined not only interacting systems of quasi-nucleons in the SM, but also showed how rotational bands arise completly from ab initio interactions in a clustered system. To investigate the clustered features in light nuclei, we first devised a method that leads to the construction of many-body states in an HO basis that describe a clustered system. The method relied on the generation of cluster fragment wave functions in a spurious CM state, and subsequently, through a careful application of the Moshinsky bracket prescription for the recoupling of HO wave functions to different coordinates, the creation of a fully microscopic channel with a definite relative motion for the fragments. Through the use of second quantization creation operators to describe the many-body Slater determinants full antisymmetrization was maintained through every step of the calculation. The cluster channels were first used for the calculation of α cluster SF in systems where the wave functions were obtained using the SM approach. This demonstrated how α clustering is prevalent in the low-lying spectra of light nuclei, and how it appears to be the favored mode of coupling between the protons and neutrons in nuclear ground states. The strong cluster component in the low-lying states of Ne isotopes was shown to survive both cross-shell excitations and the addition of an extra neutron, reproducting the gross features seen in experiment. A proof of principle study was conducted on the superradiant states of 13C, showing the effects of continuum coupling to nuclear state properties. The openness of the system can lead not only to changes in the decay widths of the channels open to decay, but also to significant restructuring of the many-body wave functions resulting in effects seen in transfer reactions. While the states considered did not possess any significant α cluster component, the calculation showed how the α SF can be affected by the openness of the system to a different type of decay, elevating the

52 continuum coupling to a possible mechanism for the existence of broad clustered states seen in experiment at energies above the decay threshold. Furthermore, ab initio studies of nuclear clustering were conducted to probe the clusterd nature of light Be isotopes. Using the nucleus 8Be, which naturally clusters into 2α particles, we investigated the robustness of the ground state rotational band for differenthω ¯ , even with small numbers of many-body states. The effects of realistic α fragment wave functions were also studied. The next 2 isotopes 9,10Be were also found to exhibit strong cluster characteristics. In the case of 9Be, we saw that not only the rotational bands emerge, but we recover the Coriolis decoupling of the K = 1/2 band. The different modes of clusterization 5He +α and 8Be +n where seen to overlap. In the case of 10Be, the excited states of the 6He cluster was shown to be neccessary in order to reproduce the excited rotational band, which was demonstrated to be predominantly a 2 particle excitation. There is a number of possibilities for extending the research presented in this thesis. The effects of continuum can furhter be elucidated with calculations having the α particle in the continuum. Clustering effects in neutron rich nuclei and how these can assist the stability close to the drip lines is perhaps another interesting question that can be answered with the traditional shell model. From an ab initio point of view, coupling the NCSM with the Continuum Shell model or with the HORSE method in order to obtain cross sections for (α, γ) reactions is one of the more ambitious goals.

53 APPENDIX A

SECOND QUANTIZATION

In this appendix we discuss the basics of second quantization both as it concerns the representation of many-particle states, the so-called occupation number representation, as well as that of the operators acting on said states. Even for small numbers of nucleons, keeping track of the wave function as a Slater determinant can be cumbersome. While the use of Slater determinants enforces antisymmetry of the wave function, it does not carry much more information given that the particles we are dealing with are identical. As a simpler way of writing the total wave function, we can opt for just keeping the labels of the states that are indeed occupied, hence the term occupation number representation. When dealing with fermions, as we do in this thesis, the maximum occupancy of a level (with all quantum numbers, spin, isospin, etc being speciﬁed) is 1. So for example a Slater determinant |Ψi for 4 particles occupying orbitals 1, 2, 3, 4 is written as

† † † † |Ψi = |1234i = a1a2a3a4|0i. (A.1)

We use |0i to denote the vaccum, ai|0i = 0 for any i within the space, the deﬁnition of which is case dependent; it could be a state with 0 particles, or a closed core depending on the model we are † using. Each of the operators ai creates a particle in orbital i and they all obey the anti-commutation relation † † {a1, a2} = 0. (A.2)

It follows immediately that any odd permutation of creation operators will result in an overall minus sign. To enforce uniqueness we deﬁne some arbitrary, ordering. We choose the so-called forward ordering, where the operator with the ”smallest” orbit goes to the left. In order to add an extra particle to an already made many-body state, we can simply act with one creation operator

† † † † † † † † † a4|123i = a4a1a2a3|0i = −a1a2a3a4|0i = −|1234i. (A.3)

Similarly a bra state is written down as annihillation operators to the right of the vaccum bra, with the order now reversed. This is straightforwad if we consider making the bra state by performing

54 a dagger operation on the ket state

† (|1234i) = h1234| = h0|a4a3a2a1. (A.4)

The enforcement of forward ordering is now responsible for the overlap of the two states being equal to 1 instead of -1, assuming the orbitals are orthogonal to one another, implying the commutation rule † {a1, a2} = δ12. (A.5)

Next, we comment on one-body operators, the most common of which is the single particle energy. The one-body operator in second quantization takes the form

ˆ(1) X (1) † O = Oij ai aj. (A.6) ij

From this form we see the meaning of such an operator, scattering a particle from orbital j to orbital i with a contribution of Oij. The so-called matrix elements Oij are evaluated from the traditional form of the interaction, usually depending or radial, spin and isospin coordinates

(1) ˆ(1) Oij = hi|O |ji. (A.7)

Similarly, the two-body operators can be written down

ˆ(2) X (2) † † O = Oijklai ajalak, (A.8) ijkl note the last 2 operators are in reverse order. Removing 2 particles from a state where both orbitals are occupied yields

a2a1|12i = |0i, (A.9) so no extra − sign is introduced. A more in-depth treatment of second quantization representation of states and operators can be found in Ref [77, 143, 144, 145].

55 APPENDIX B

CM BOOSTING OPERATOR NORMALIZATION COEFFICIENTS

Here, we discuss a form of the raising and lowering operators in spherical coordinates and the appropriate normalization for the action of their components. The spherical radial operators are given in terms of the rank-1 spherical harmonics r 4π rˆ = rY1 (B.1) m 3 m

The HO wave functions have the form deﬁned in Chapter 2, ψnlm (r) = hr|nlmi.

We start by looking at the action ofr ˆ1 on a stretched wave function |nlli. This action will generate a stretched state |n0l + 1l + 1i due to the additive properties of the m quantum number, leaving only the total normalization in question. Since there are both raising and lowering parts, we can expect N 0 = N ± 1. To derive the normalization of the operator, we can look at three separate parts of the total wave function, the normalization constant, the angular part and the radial part. We treat each of these separately, starting with the normalization constant s r 2n! 3 Nnl = 3 = Nnl+1 n + l + . (B.2) Γ(n + l + 2 ) 2

The angular part consists of a recoupling of the two Spherical Harmonics r s 4π 1p 2 2l + 1 1 l Y11Y = − 1 − cos θ P (cos θ) 3 ll 2 2π (2l)! l s 1 1 2l + 1 1 = P l+1(cos θ) 2 2l + 1 2π (2l)! l+1 s l + 1 = Y l+1. (B.3) (2l + 3) l+1

Finally, to derive an expression for the radial part, we use the Laguerre polynomial property

l+1/2 l+1+1/2 l+1+1/2 Ln (x) = Ln (x) − Ln−1 (x), (B.4)

56 while the factor of r coming from the operator deﬁnition is abosrbed in the rl part of the wave function. Putting everything together we arrive at the ﬁnal expression s (l + 1)(2n + 2l + 3) rˆ1|ψ i = |ψ i− nll 2(2l + 3) nl+1l+1 s (l + 1)n − |ψ i (B.5) (2l + 3) n−1l+1l+1

In order to construct an operator that directly raises the number of nodes in the wave function, it is clear that we need at least twor ˆm operators. Such an operator should be a rotational scalar. Here, the general dot product between two spherical tensor operators of rank k is deﬁned as

√ k (k) (k) X 00 (k) (k) T · S = 2k + 1 Ckmk−mTm S−m m=−k k X k−m (k) (k) = (−) Tm S−m, (B.6) m=−k which in our case simpliﬁes to

4π 2 2 rˆ · rˆ = r [Y11Y1−1 − Y10Y10 + Y1−1Y11] = −r . (B.7) 3

Ther ˆ · rˆ operator will change the number of quanta by either 0 or ±2 while leaving the angular momentum part intact. The normalization change is given by s 2(n + 1) N = N , (B.8) n+1l (2n + 2l + 3) nl s 2n N = N . (B.9) n−1l (2n + 2l + 1) nl

The radial part is governed only by the Laguerre polynomials for which the following relation holds

(α) (α−1) (α−1) xLn (x) = (n + α)Ln (x) − (n + 1)Ln+1 (x)

(α) (α) (α) = (2n + α + 1)Ln − (n + α)Ln−1 − (n + 1)Ln+1, (B.10)

The ﬁnal step is to construct the actual quanta raising operators. They are also spherical operators and, in accordance to their Cartesian counterparts are given by

rmω i b† = rˆ − pˆ , (B.12) m 2¯h m mω m wherep ˆm are the equivalent momentum operators, with their action easily calculated in a similar manner to the radial operators.

58 APPENDIX C

CALCULATION OF RADIAL INTEGRALS WITH HO WAVE FUNCTIONS

In this appendix we derive a formula to evaluate radial integrals of two HO wave functions and powers of r. Given that even highly oscillatory functions can be expanded as a sum of polynomials, this procedure can be used to evaluate the interaction potential form factors. The integral we wish to calculate has the generic form:

Z ∞ In2l2 = rkR (r)R (r)r2dr, (C.1) n1l1 n1l1 n2l2 0 with the radial Harmonic Oscillator wave functions having the following form s r 3 n+2l+3 l 2ν 2 n!ν 2 2 R (r) = Ll+1/2(2νr2)rle−νr = N Ll+1/2(2νr2)rle−νr . (C.2) nl π (2n + 2l + 1)!! n nl n

The phase is taken to be positive at the origin and we deﬁne ν = mω/2¯h. For the rest of this calculation we deﬁne the product N n2l2 = N N . (C.3) n1l1 n1l1 n2l2

Therefore, the integral takes the form

Z 2 I = N n2l2 rk+1+l1+l2 e−2νr Ll1+1/2(2νr2)Ll2+1/2(2νr2)rdr n1l1 n1 n2

r (k+l1+l2+3) 1 Z 2 = N n2l2 (2νr2)(k+1+l1+l2)/2e−2νr Ll1+1/2(2νr2)Ll2+1/2(2νr2)2νrdr. (C.4) n1l1 2ν n1 n2

2 Let’s deﬁne p = l1 + l2 + k and x = 2νr , yielding the integral in abbreviated form

1 (p+3)/2 1 Z I = N n2l2 x(p+1)/2e−xLl1+1/2(x)Ll2+1/2(x)dx. (C.5) n1l1 2ν 2 n1 n2

We proceed by expanding one of the two Laguerre polynomials in powers of x

n n X X (−1)m n + a La (x) = αa xm = xm. (C.6) n mn m! n − m m=0 m=0

59 Numerically this can be chosen to be the one with the least amount of nodes; here we expand n1. Restoring the expansion coeﬃcients for the Laguerre polynomials we obtain a formula that can be used to calculate the integral value as a sum of known functions.

(p+3)/2 n1 1 1 X Z I = N n2l2 αl1+1/2 x(p+1)/2+me−xLl2+1/2(x)dx n1l1 2ν 2 mn1 n2 m=0 (p+3)/2 n1 l1+l2+k 3 l1−l2+k 1 1 X Γ 2 + m + 2 Γ 2 + m + 1 = N n2l2 αl1+1/2(−1)n2 n1l1 mn1 2ν 2 l1−l2+k m=0 Γ 2 + m + 1 − n2 n2! n (p+3)/2 n2 1 m n2l2 1 (−1) X (−1) n1 + l1 + 1/2 = Nn l × 1 1 2ν 2 m! n1 − m m=0 l1+l2+k 3 l1−l2+k Γ 2 + m + 2 Γ 2 + m + 1 × . (C.7) l1−l2+k Γ 2 + m + 1 − n2 n2!

Given the bad numerical behavior of the Γ-functions and the increased time it takes to evaluate their respective logarithms, we can attempt to calculate each term in the sum based on the one before it so as to have relatively tame numbers to deal with. We can deﬁne

l +l +k 3 l −l +k m Γ 1 2 + m + Γ 1 2 + m + 1 (−1) n1 + l1 + 1/2 2 2 2 C(m) = m! n1 − m l1−l2+k Γ 2 + m + 1 − n2 n2! l +l +k 3 l −l +k m 3 Γ 1 2 + m + Γ 1 2 + m + 1 (−1) Γ(n1 + l1 + 2 ) 2 2 2 = 3 , (C.8) m! Γ(n1 − m + 1)Γ(l1 + m + ) l1−l2+k 2 Γ 2 + m + 1 − n2 n2! and now the next coeﬃcient is just   l1+l2+k + m + 3 l1−l2+k + m + 1 (n1 − m) 2 2 2 C(m + 1) = −  C(m), (C.9) (l1 + m + 3/2) l1−l2+k (m + 1) 2 + m + 1 − n2 leaving us with only C(0) to explicitly calculate.

60 APPENDIX D

CM BOOST AND RECOUPLING EXAMPLES

To demonstrate the approach outlined in Chapter 3, let’s consider two neutrons to be coupled to a final state in which their relative motion is in an ` = 1 wave, their spins coupled to S = 1 and total angular momentum of J = 0. We start by using general notation and will simplify at the end. The tradeoff in coupling single particles comes at the expense of having to deal with the internal spins of the clusters (here the spin of the neutrons). Since here only 1 HO quantum of energy is being added to the relative motion of the system, the nodal quantum number is supressed in the wave functions as it is always 0. The total final state we seek is written in the form

J h ` 0 0 Si Ψ`SJM = [ψ0(RCM ) × ψ`(ρ)] × ψs × ψs , (D.1) 2 2 M

† with s1,2 denoting the internal spin of the clusters. In the following, we denote with Bn`m the set of actions required to build a CM state with quantum numbers n`m. When recoupling single particles, the resulting wave function will have a deﬁnite total ` value and s = 1/2, and thus one can recouple to a good single particle total angular momentum j as

† 0 X jm+ms B01m ψ 1 = C 1 ψ1 1 j, (D.2) 2 ms 1m 2 ms 2 1 3 j= 2 , 2 obtaining the components for each orbital in the jj coupling scheme utilized by the SM. In the following, we supress the n, ` indices for B as they are understood to be 0 and 1 respectively. Expanding now eq. D.1, we can write down the full wave function as a weighted sum of CM excited conﬁgurations,

X X n`00;` JM `m` SmS † 0 † 0 Ψ`SJM = M C C C B ψ B ψ . n1`1n2`2 `m`SmS `1m1`2m2 s1ms1 s2ms2 n1`1m1 s1ms1 n2`2m2 s2ms2 m`mS n1`1m1 s s m 1 m 2 n2`2m2

If we now constrain ourselves to the 2 neutron example, we see that for any given term only 1 of the two neutrons can be excited in the CM coordinate to maintain the total number of quanta.

61 Evaluating the coeﬃcients, the result is

1 h i √ † 0 0 0 † 0 † 0 0 0 † 0 Ψ1100 = B1ψ− 1 ψ− 1 − ψ− 1 B1ψ− 1 + B−1ψ 1 ψ 1 − ψ 1 B−1ψ 1 6 2 2 2 2 2 2 2 2 1 h i √ † 0 0 0 † 0 † 0 0 0 † 0 − B0ψ 1 ψ− 1 − ψ 1 B0ψ− 1 + B0ψ− 1 ψ 1 − ψ− 1 B0ψ 1 , (D.3) 2 3 2 2 2 2 2 2 2 2 with the s = 1/2 index supressed for the intrinsic wave functions. It is important to remember that the ordering here matters as it represents the action of a state-operator on the vaccum. As the total M projection is 0, the streatched p3/2 orbitals with m = ±3/2 will not be occupied, and leave us with 6 availiable orbitals for each neutron; the ordering for the orbitals is shown in Table D.1. Transitioning to second quantization formalism, the state is now represented by a 2 body creation

Table D.1: Index ordering (k) of single particle orbitals. The ± superscirpt refers to the m = ±1/2 projection of each orbital.

k l j m k l j m 0 0 1/2 1/2 3 1 1/2 -1/2 1 0 1/2 -1/2 4 1 3/2 1/2 2 1 1/2 1/2 5 1 3/2 -1/2 operator r " r r r r # 2 1 2 1 2 Ψ† = − a†a† − a†a† − a†a† + a†a† 1100 3 3 1 4 3 1 2 3 0 5 3 0 3 "r r r r # 1 2 1 1 2 + √ a†a† + a†a† − a†a† + a†a† 3 3 0 5 3 0 3 3 1 2 3 1 4 † † † † = a0a3 − a1a2. (D.4)

Quite happily, the terms containing particles in the p 3 shell have cancelled out, as this would make 2 it impossible to couple the 2 neutrons to a ﬁnal state of 0 spin. Furhtermore, the phase of the states has come out to be exactly the same as if we were coulping the particles in the jj scheme, thus demonstrating the abscence of any arbitrary phase ambiguity that comes with obtaining the same state by diagonalization. The norm of the total wave function is 2 as the 2 particles are identical √ and the familiar factor of 1/ 1 + δ12 is necessary to obtain a normalized wave function. We can now move to a more interesting example, building a sp shell state comprised of two α particles. We choose to build the state that has the particles in n = 0, ` = 4, m = 4 relative motion.

62 We keep the notation of Bnlm from before. For second quantization notation see Appendix A. We deﬁne the indexing for the neutron states which in turn implies the α particle ground state

Table 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.

k l j m k l j m 1 0 1/2 -1/2 5 1 3/2 -3/2 2 0 1/2 1/2 6 1 3/2 -1/2 3 1 1/2 -1/2 7 1 3/2 1/2 4 1 1/2 1/2 8 1 3/2 3/2 conﬁguration is given by † † † † † |Ψαi = Ψα|0i = a1a2a9a10|0i. (D.5)

The wave function for the two α system can now be written

X Ψ† |0i = |Ψ i = C44 M0400;4 B† Ψ† B† Ψ† |0i, (D.6) 044 044 l1l1l2l2 0l10l2 0l1l1 α 0l2l2 α l1l2 where due to our choice of building a stretched m = ` state, only stretched CM excited states of the α particles are required. There are ﬁve combinations of quantum numbers that contribute

(l1, l2) = {(4, 0), (0, 4), (2, 2), (3, 1), (1, 3)}, yielding the form r 1 3 Ψ† = Ψ† B† Ψ† + B† Ψ† B† Ψ† − B† Ψ† B† Ψ† (D.7) 044 2 α 044 α 8 022 α 022 α 011 α 033 α

The consecuitive actions of the operator

† 1 † 1 † 1 † B = a a2 + √ a a1 + √ a a1, (D.8) 1 2 8 6 4 12 7

63 along with the proper ordering of the creation operators discussed in Appendix A yield for the three parts of eq. (D.7)

† † † 1 † † † † † † † † 1 † † † † † † † † ΨαB Ψα = √ a a a a a a a a + √ a a a a a a a a + 044 2 6 1 2 4 8 9 10 12 16 4 6 1 2 7 8 9 10 15 16 1 + √ a†a†a†a†a†a† a† a† + a†a†a†a†a†a† a† a† (D.9) 4 3 1 2 4 8 9 10 15 16 1 2 7 8 9 10 12 16 1 1 B† Ψ† B† Ψ† = a†a†a†a†a†a† a† a† + a†a†a†a†a†a† a† a† + 022 α 022 α 2 1 2 4 8 9 10 12 16 4 1 2 7 8 9 10 15 16 1 + √ a†a†a†a†a†a† a† a† + a†a†a†a†a†a† a† a† (D.10) 2 2 1 2 4 8 9 10 15 16 1 2 7 8 9 10 12 16 † † † † 1 † † † † † † † † 1 † † † † † † † † B Ψα B Ψα = −√ a a a a a a a a − √ a a a a a a a a − 011 033 6 1 2 4 8 9 10 12 16 2 6 1 2 7 8 9 10 15 16 1 − √ a†a†a†a†a†a† a† a† + a†a†a†a†a†a† a† a† . (D.11) 2 3 1 2 4 8 9 10 15 16 1 2 7 8 9 10 12 16

By direct substitution we obtain the wave function for the stretched state in second quantization r 2 1 Ψ† = a†a†a†a†a†a† a† a† + √ a†a†a†a†a†a† a† a† + 044 3 1 2 4 8 9 10 12 16 6 1 2 7 8 9 10 15 16 1 +√ a†a†a†a†a†a† a† a† + a†a†a†a†a†a† a† a† . (D.12) 3 1 2 4 8 9 10 15 16 1 2 7 8 9 10 12 16

We can evaluate the normalization of this function by computing the overlap

† 3 hΨ044|Ψ044i = h0|Ψ044Ψ |0i = , (D.13) 044 2 reaﬃrming previous results obtained through the RGM [78].

64 APPENDIX E

HAMILTONIAN KERNELS FOR THE 2α SYSTEM

(`) 8 Below we tabulate the Hamiltonian kernels H¯hω for Be RGM calculations with eight HO quanta in the relative motion. We approximate the α particle as an s4 conﬁguration, so the norm kernel is diagonal with values given by eq. (5.3).

 −21.002 12.8128 −5.72446 −15.3952 13.1604 −4.89252 (0) (2) H14 =  12.8128 9.89648 30.232  H14 =  13.1604 14.0454 30.2191  −5.72446 30.232 44.9108 −4.89252 30.2191 47.7248

−2.31252 12.6558 −3.10072 (4) H14 =  12.6558 23.2668 29.3868  −3.10072 29.3868 53.9659

−26.2072 17.8878 −4.58926 −20.1332 17.6485 −4.01868 (0) (2) H17.5 =  17.8878 11.5053 38.9481  H17.5 =  17.6485 16.0614 38.4008  −4.58926 38.9481 54.9993 −4.01868 38.4008 58.048

−5.96052 15.7803 −2.70874 (4) H17.5 =  15.7803 26.3265 36.3466  −2.70874 36.3466 64.9344

−27.2904 22.3532 −3.4803 −21.1413 21.555 −3.1318 (0) (2) H20 =  22.3532 15.5374 46.1311  H20 =  21.555 20.2172 45.0851  −3.4803 46.1311 65.2228 −3.1318 45.0851 68.3038

−6.79321 18.4189 −2.24897 (4) H20 =  18.4189 30.8422 41.9067  −2.24897 41.9067 75.3545

65 −26.1042 27.422 −2.28384 −20.0595 25.9845 −2.15899 (0) (2) H22.5 =  27.422 22.1413 54.1371  H22.5 =  25.9845 26.8226 52.5098  −2.28384 54.1371 78.1816 −2.15899 52.5098 81.189

−5.95531 21.4012 −1.71599 (4) H22.5 =  21.4012 37.522 48.0276  −1.71599 48.0276 88.1699

−22.6827 33.0095 −1.08043 −16.8915 30.8765 −1.16771 (0) (2) H25 =  33.0095 31.3835 62.9621  H25 =  30.8765 35.9527 60.6804  −1.08043 62.9621 93.9805 −1.16771 60.6804 96.8134

−3.37857 24.7128 −1.15027 (4) H25 =  24.7128 46.4615 54.7345  −1.15027 54.7345 103.499

−9.46902 45.4884 1.21365 −4.52577 41.8599 0.755614 (0) (2) H30 =  45.4884 57.8087 82.9668 H30 =  41.8599 61.8349 79.1951  1.21365 82.9668 134.179 0.755614 79.1951 136.391

 7.00845 32.2597 0.00593547 (4) H30 =  32.2597 71.228 69.9162  0.00593547 69.9162 141.867

66 APPENDIX F

TABLES OF SPECTROSCOPIC AMPLITUDES

Here we list spectroscopic amplitudes. The α operator is taken to have an s4 structure. The radial wave functions are positive near the origin, and l is coupled to s in that order. The phase convention for the channels after diagonalization of the Norm Kernel is ﬁxed by enforcing a positive diagonal of the matrix where the columns are the eigenvectors of the norm kernel. This was done so that in the limit of no overlap between channels they all remain unchanged (no arbitrary (−) phase introduced). The USDB interaction was used. First are the tables of even-even nuclei, then even-odd nuclei, and ﬁnally the odd-even and odd-odd nuclei. In the tables of even-even nuclei, the ground transitions are all to 0+ states, so the angular momentum ` = J.

20 16 + 22 18 + 24 20 + 26 22 + Ne → O(01 ) Ne → O(01 ) Ne → O(01 ) Ne → O(01 ) π π π π J Ex Sα J Ex Sα J Ex Sα J Ex Sα + + + + 01 0.00 -0.87 01 0.00 0.69 01 0.00 -0.59 01 0.00 0.72 + + + + 21 1.75 0.88 21 1.36 -0.60 21 2.11 -0.51 21 2.06 -0.73 + + + + 41 4.18 0.81 41 3.36 -0.43 22 3.75 -0.37 41 3.51 0.59 + + + + 02 6.70 0.38 22 4.28 0.27 41 3.99 0.51 22 3.77 0.25 + + + + 22 7.54 0.18 23 5.13 0.22 02 4.96 -0.40 02 4.64 0.37 + + + + 61 8.55 0.76 42 5.37 -0.23 23 5.55 -0.36 23 4.91 -0.26 + + + + 42 9.95 0.14 24 6.17 0.08 42 5.69 0.00 42 5.47 -0.27 + + + + 23 9.96 -0.00 61 6.24 0.14 03 6.10 -0.45 03 6.04 0.39 + + + + 24 10.49 -0.17 02 6.26 0.02 43 6.11 0.07 24 6.40 -0.08 + + + + 43 10.71 -0.36 43 6.30 0.35 24 6.23 -0.19 61 6.74 -0.47 + + + + 25 10.79 0.21 25 6.66 -0.24 25 7.37 -0.07 25 6.83 -0.05 + + + + 44 10.81 0.00 44 7.08 0.10 26 7.53 0.03 26 7.02 -0.09 + + + + 81 11.51 0.63 03 7.39 -0.57 44 7.71 0.26 27 7.26 -0.01 + + + 45 11.72 -0.16 26 7.69 0.23 43 7.46 0.13

67 28 24 + 30 26 + 24 20 + 26 22 + Ne → O(01 ) Ne → O(01 ) Mg → Ne(01 ) Mg → Ne(01 ) π π π π J Ex Sα J Ex Sα J Ex Sα J Ex Sα + + + + 01 0.00 0.72 01 0.00 0.70 01 0.00 0.64 01 0.00 0.66 + + + + 21 1.62 0.62 21 1.71 0.56 21 1.50 -0.60 21 1.90 0.53 + + + + 41 2.99 0.62 41 2.54 0.61 22 4.12 0.25 22 3.01 -0.19 + + + + 22 3.51 0.16 22 5.57 -0.67 41 4.37 0.04 02 3.63 0.33 + + + + 23 3.99 0.41 02 8.63 0.66 42 5.88 0.61 41 4.37 0.26 + + + + 42 4.87 0.06 42 8.81 0.80 02 7.34 -0.69 23 4.45 -0.28 + + + + 24 5.61 0.16 23 9.68 -0.21 23 7.48 0.42 24 4.88 0.24 + + + + 61 5.80 -0.57 24 13.34 0.41 61 8.27 -0.12 42 4.94 0.38 + + + + 02 6.09 -0.16 03 15.63 0.26 43 8.31 -0.12 03 5.03 0.50 + + + + 25 6.26 0.34 25 16.76 -0.17 24 8.97 0.20 25 5.39 -0.03 + + + 03 6.30 0.39 44 9.40 -0.00 43 5.52 0.02 + + + 43 6.40 -0.11 62 9.53 -0.52 44 5.89 -0.38 + + + 26 6.76 -0.05 25 9.57 -0.33 04 6.13 0.04 + + + 44 6.92 -0.23 45 9.58 0.07 26 6.68 0.11 + + 03 9.67 -0.09 45 6.73 0.03

28 24 + 30 26 + 32 28 + 28 24 + Mg → Ne(01 ) Mg → Ne(01 ) Mg → Ne(01 ) Si → Mg(01 ) π π π π J Ex Sα J Ex Sα J Ex Sα J Ex Sα + + + + 01 0.00 0.76 01 0.00 0.69 01 0.00 0.71 01 0.00 0.73 + + + + 21 1.52 0.59 21 1.59 0.52 21 1.50 -0.34 21 1.93 -0.57 + + + + 02 4.01 0.14 22 3.43 -0.04 41 2.71 0.44 41 4.61 0.46 + + + + 41 4.17 -0.10 41 3.89 0.18 22 5.75 0.73 02 4.84 0.40 + + + + 22 4.54 -0.24 23 4.79 -0.33 42 6.84 -0.20 42 7.00 -0.37 + + + + 23 4.79 -0.03 24 5.15 -0.34 23 7.13 -0.02 03 7.33 0.47 + + + + 42 5.21 -0.30 42 5.32 -0.23 43 8.57 0.82 22 7.52 -0.41 + + + + 24 5.57 -0.32 02 5.82 -0.15 24 8.87 0.15 23 7.90 -0.12 + + + + 25 6.07 -0.18 43 6.34 0.38 02 9.15 -0.65 24 8.38 0.13 + + + + 03 6.59 0.19 61 6.74 0.33 03 9.50 -0.11 61 8.41 0.20 + + + + 43 6.93 0.09 25 6.77 -0.07 61 10.03 1.00 25 8.49 0.42 + + + + 26 6.95 -0.12 26 6.85 0.19 25 10.18 0.07 43 9.39 -0.18 + + + + 04 7.13 -0.38 03 6.99 -0.55 44 11.11 -0.04 26 9.43 -0.00 + + + 44 7.06 0.02 26 11.12 0.08 04 9.50 0.01

68 30 26 + 32 28 + 34 30 + 32 28 + Si → Mg(01 ) Si → Mg(01 ) Si → Mg(01 ) S → Si(01 ) π π π π J Ex Sα J Ex Sα J Ex Sα J Ex Sα + + + + 01 0.00 0.74 01 0.00 0.77 01 0.00 -0.67 01 0.00 0.96 + + + + 21 2.27 -0.60 21 2.05 0.54 21 5.25 0.68 21 2.16 -0.87 + + + + 22 3.51 0.04 22 4.24 -0.15 41 8.28 0.88 02 3.46 0.04 + + + + 02 3.91 -0.02 02 4.99 0.04 22 8.72 0.02 22 4.38 -0.23 + + + + 23 4.87 0.07 23 5.75 0.43 02 8.76 -0.70 41 4.65 0.32 + + + + 41 5.33 -0.27 41 5.88 -0.25 23 10.32 0.17 23 5.45 0.10 + + + + 03 5.50 0.52 03 5.92 0.53 42 10.78 0.12 42 6.19 -0.20 + + + + 42 5.87 0.38 42 6.93 0.74 24 12.32 -0.59 24 6.67 0.01 + + + + 24 5.92 0.03 24 7.31 0.22 03 12.62 -0.11 43 6.90 -0.66 + + + + 25 6.51 -0.10 25 7.67 0.25 25 13.22 0.29 25 6.96 0.00 + + + + 04 6.74 -0.26 26 8.15 0.01 43 13.52 -0.11 03 7.33 -0.00 + + + + 26 7.03 0.45 04 8.40 -0.21 44 13.68 -0.32 26 7.40 0.14 + + + + 43 7.08 -0.27 27 8.45 0.36 04 13.69 0.18 04 7.46 0.18 + + 27 7.31 0.29 05 7.93 0.03

34 30 + 36 32 + 36 32 + 38 34 + S → Si(01 ) S → Si(01 ) Ar → S(01 ) Ar → S(01 ) π π π π J Ex Sα J Ex Sα J Ex Sα J Ex Sα + + + + 01 0.00 0.98 01 0.00 -0.97 01 0.00 0.99 01 0.00 -1.00 + + + + 21 2.13 -0.75 21 3.38 -0.96 21 1.82 0.90 21 1.84 0.92 + + + + 22 3.12 0.53 02 4.64 -0.20 22 4.25 -0.03 22 4.24 0.28 + + + + 02 3.75 0.11 22 6.26 -0.22 41 4.49 -0.05 02 6.03 -0.02 + + + + 23 4.12 -0.00 41 8.21 -0.91 02 4.52 0.07 41 8.76 1.00 + + + + 24 4.60 0.14 23 8.49 0.10 42 6.26 0.46 23 9.94 0.26 + + + + 41 4.83 0.13 24 10.07 0.05 23 6.38 -0.00 24 12.09 -0.03 + + + + 03 5.17 0.06 42 11.21 -0.37 24 6.79 0.19 03 16.46 -0.05 + + + + 25 6.04 0.00 03 11.72 -0.05 25 7.91 0.08 25 17.39 -0.03 + + + + 26 6.70 -0.06 43 12.48 -0.05 03 8.17 -0.01 42 17.96 0.09 + + + 42 6.75 -0.18 25 12.81 -0.11 26 8.29 0.11 + + + 43 6.88 -0.44 44 13.31 0.13 27 8.39 -0.00 + + 04 6.92 -0.06 43 8.62 0.75 + + 27 7.22 0.01 28 8.83 0.00

69 21 17 + 23 19 + Ne → O(5/21 ) Ne → O(5/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 3/21 0.00 -0.82 0.34 5/21 0.00 -0.52 0.44 0.09 + + 5/21 0.27 -0.72 0.12 0.54 1/21 1.06 -0.63 + + 7/21 1.76 -0.77 -0.31 0.33 7/21 1.81 0.64 -0.17 -0.10 + + 9/21 2.83 0.73 -0.47 -0.35 3/21 1.87 0.39 0.31 + + 1/21 2.86 -0.71 5/22 2.33 0.46 0.39 0.06 + + 5/22 3.72 -0.02 0.75 -0.22 9/21 2.48 -0.39 0.46 0.09 + + 11/21 4.38 -0.74 -0.01 0.33 3/22 3.33 0.25 -0.61 + + 5/23 4.63 0.49 0.01 0.52 1/22 3.51 -0.40 + + 3/22 4.91 -0.30 -0.72 5/23 3.72 -0.02 -0.16 0.59 + + 7/22 5.39 0.36 -0.59 0.19 7/22 3.78 0.25 0.25 0.29 + + 3/23 5.53 0.03 -0.24 11/21 3.79 -0.54 0.13 0.20 + + 1/22 5.96 0.55 3/23 3.84 0.56 0.17 + + 9/22 6.12 -0.08 0.25 -0.40 9/22 4.21 -0.42 -0.09 -0.06 + + 7/23 6.17 -0.05 0.05 -0.01 7/23 4.36 -0.08 -0.45 -0.08 + + 13/21 6.25 0.57 -0.45 -0.36 9/23 4.54 0.04 -0.24 -0.01 + + 9/23 6.26 -0.34 -0.44 0.04 5/24 4.97 0.13 -0.11 0.02

25 21 + 27 23 + Ne → O(5/21 ) Na → O(1/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 1/21 0.00 -0.68 3/21 0.00 -0.67 + + 5/21 1.76 0.59 0.28 0.30 1/21 1.38 -0.77 + + 3/21 2.04 -0.36 -0.73 7/21 1.90 -0.69 + + 5/22 3.17 -0.25 0.64 0.28 5/21 2.04 0.49 + + 3/22 3.17 0.11 -0.03 1/22 2.81 -0.19 + + 9/21 3.39 -0.45 -0.26 -0.33 3/22 2.89 -0.24 + + 7/21 3.77 -0.32 -0.41 -0.61 5/22 2.92 0.43 + + 7/22 4.45 -0.64 0.17 0.06 11/21 3.22 0.64 + + 5/23 4.53 0.34 0.14 0.08 5/23 3.23 -0.13 + + 3/23 4.63 -0.46 -0.12 3/23 3.79 -0.31 + + 5/24 4.90 0.26 0.05 0.23 9/21 3.98 -0.51 + + 9/22 4.99 0.32 -0.45 -0.19 7/22 4.35 0.19 + + 7/23 5.06 -0.25 0.16 0.27 9/22 4.51 0.32 + + 1/22 5.14 0.43 7/23 4.76 -0.33 + + 11/21 6.05 -0.53 -0.30 -0.41 5/24 5.08 -0.04 + + 7/24 6.05 0.11 0.03 -0.01 3/24 5.20 0.18

70 29 25 + 25 21 + Ne → O(5/21 ) Mg → Ne(3/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 3/21 0.00 0.69 -0.65 5/21 0.00 0.32 -0.07 + + 1/21 1.53 -0.43 1/21 0.61 0.02 + + 5/21 1.76 -0.64 0.64 3/21 1.10 0.29 0.52 + + 7/21 1.86 -0.58 0.58 7/21 1.72 -0.41 0.13 + + 9/21 2.43 -0.62 0.59 5/22 2.00 -0.34 -0.19 + + 3/22 2.92 0.05 -0.16 1/22 2.58 0.47 + + 7/22 3.48 -0.00 -0.10 3/22 2.81 -0.45 -0.48 + + 11/21 3.52 0.61 -0.60 7/22 2.90 -0.26 -0.66 + + 1/22 3.81 0.40 9/21 3.45 0.34 -0.07 + + 5/22 4.09 -0.32 0.20 9/22 3.90 -0.12 0.02 + + 5/23 4.95 0.25 -0.24 5/23 3.91 0.19 -0.30 + + 3/23 4.96 -0.32 0.39 3/23 4.33 0.48 -0.01 + + 7/23 5.55 -0.49 0.43 9/23 4.73 -0.23 -0.11 + + 5/24 5.74 -0.35 0.36 5/24 4.74 -0.30 -0.31 + + 9/22 5.83 -0.05 -0.00 7/23 4.89 0.28 0.20 + + 7/24 6.20 0.32 -0.32 11/21 5.13 -0.37 -0.01

27 23 + 29 25 + Mg → Ne(5/21 ) Mg → Ne(1/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 1/21 0.00 -0.67 3/21 0.00 -0.68 + + 3/21 0.99 -0.22 -0.59 1/21 0.04 -0.66 + + 5/21 1.67 0.68 -0.00 0.05 5/21 1.59 0.40 + + 5/22 1.90 -0.22 0.44 0.19 7/21 2.10 0.37 + + 7/21 3.08 0.65 0.16 0.01 3/22 2.33 0.01 + + 3/22 3.46 -0.42 0.03 1/22 2.63 -0.11 + + 7/22 3.46 -0.15 -0.12 0.24 5/22 3.15 0.07 + + 1/22 3.54 -0.10 3/23 3.50 -0.03 + + 3/23 3.60 -0.48 0.18 5/23 3.57 0.13 + + 9/21 3.99 0.38 0.25 0.16 9/21 3.98 0.12 + + 5/23 4.11 0.33 0.40 -0.14 7/22 3.99 0.34 + + 9/22 4.37 0.52 0.10 -0.12 5/24 4.26 0.25 + + 5/24 4.37 -0.08 -0.18 0.15 9/22 4.30 0.02 + + 7/23 4.71 -0.19 -0.19 -0.22 11/21 4.70 -0.47 + + 5/25 4.94 0.15 0.06 0.32 7/23 4.79 -0.20 + + 1/23 5.04 0.25 3/24 4.89 -0.36 + + 3/24 5.08 0.06 -0.33 7/24 5.09 -0.00

71 31 27 + 29 25 + Mg → Ne(3/21 ) Si → Mg(5/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 3/21 0.00 0.62 -0.64 1/21 0.00 -0.75 + + 5/21 1.39 -0.34 0.55 3/21 1.28 0.14 0.66 + + 7/21 2.11 -0.02 0.16 5/21 2.06 -0.68 -0.21 -0.01 + + 1/21 2.38 -0.09 3/22 2.52 -0.69 0.12 + + 5/22 3.04 0.21 -0.17 5/22 3.36 -0.17 -0.44 0.24 + + 3/22 3.31 -0.18 0.01 7/21 4.22 -0.18 0.04 -0.29 + + 11/21 3.59 -0.40 0.41 9/21 4.67 -0.55 -0.16 -0.01 + + 1/22 3.88 0.39 1/22 4.73 -0.32 + + 9/21 4.19 0.49 -0.37 5/23 4.90 -0.34 0.16 0.28 + + 5/23 4.36 0.07 -0.02 7/22 5.12 -0.24 -0.18 0.29 + + 7/22 4.47 0.45 -0.46 9/22 5.63 -0.20 -0.12 0.05 + + 3/23 4.84 -0.33 0.29 7/23 5.88 0.42 0.10 -0.07 + + 7/23 5.19 0.09 -0.31 3/23 6.05 0.03 -0.11 + + 9/22 5.20 -0.16 0.10 3/24 6.33 -0.22 -0.39 + + 5/24 5.78 0.49 -0.34 7/24 6.39 -0.07 0.05 -0.47 + + 3/24 5.79 0.14 -0.27 1/23 6.68 -0.21

31 27 + 33 29 + Si → Mg(1/21 ) Si → Mg(3/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 1/21 0.00 -0.67 3/21 0.00 -0.63 0.67 + + 3/21 0.00 -0.66 1/21 0.81 -0.48 + + 1/21 0.72 -0.65 7/21 4.19 -0.53 0.70 + + 5/21 1.57 -0.49 5/21 4.59 0.03 -0.01 + + 3/22 2.25 0.24 3/22 4.78 -0.35 0.45 + + 5/22 2.81 0.14 3/23 5.02 0.21 -0.09 + + 7/21 3.74 -0.35 1/22 5.10 0.62 + + 3/23 3.99 -0.15 9/21 5.73 -0.24 0.38 + + 1/22 4.66 0.14 5/22 6.11 -0.37 0.26 + + 7/22 4.68 -0.13 5/23 6.24 0.37 0.04 + + 9/21 5.06 -0.03 5/24 6.72 0.34 -0.18 + + 3/24 5.08 0.10 7/22 6.98 0.25 -0.19 + + 5/23 5.15 0.34 3/24 7.07 0.26 -0.34 + + 5/24 5.42 -0.13 1/23 7.18 -0.11 + + 3/25 5.48 0.24 7/23 7.40 0.38 -0.44 + + 9/22 5.59 -0.17 3/25 7.42 0.20 -0.21 + + 7/23 5.74 0.41 11/21 7.51 0.80 -0.94

72 33 29 + 35 31 + S → Si(1/21 ) S → Si(3/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 1/21 0.00 -0.67 3/21 0.00 0.96 -0.90 + + 3/21 0.00 -0.91 1/21 1.68 -0.96 + + 1/21 0.85 0.92 5/21 2.70 0.71 -0.77 + + 5/21 1.92 -0.67 3/22 2.80 0.06 -0.28 + + 3/22 2.24 0.28 5/22 3.37 -0.50 0.15 + + 5/22 2.95 -0.11 7/21 3.56 0.71 -0.78 + + 7/21 2.97 0.67 3/23 3.86 -0.15 0.25 + + 3/23 3.58 0.08 3/24 4.26 -0.10 0.06 + + 5/23 3.84 0.14 1/22 4.57 -0.04 + + 1/22 3.94 0.00 5/23 4.59 -0.28 0.06 + + 7/22 4.09 0.37 1/23 4.79 0.04 + + 9/21 4.21 0.11 3/25 5.19 0.02 0.00 + + 1/23 4.28 0.13 3/26 5.67 -0.02 -0.05 + + 3/24 4.36 0.00 7/22 5.81 0.09 -0.10 + + 5/24 4.61 -0.22 5/24 5.86 0.26 -0.46 + + 3/25 4.70 -0.01 1/24 6.09 0.07 + + 3/26 5.09 -0.03 5/25 6.18 -0.13 0.15

37 33 + Ar → S(3/21 ) π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + 3/21 0.00 -0.99 0.98 + 1/21 1.39 0.97 + 7/21 2.14 -0.88 0.87 + 5/21 2.77 0.45 -0.43 + 5/22 3.15 -0.42 0.05 + 3/22 3.65 0.00 -0.05 + 1/22 4.26 0.14 + 7/22 4.54 -0.17 0.34 + 3/23 4.84 0.00 0.00 + 3/24 4.96 -0.03 -0.01 + 1/23 5.33 -0.07 + 5/23 5.71 -0.17 0.02 + 5/24 6.12 0.60 -0.64 + 3/25 6.58 0.01 -0.11 + 1/24 6.73 0.00 + 1/25 6.77 0.11

73 22 18 + 23 19 + Na → F(11 ) Na → F(1/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 31 0.00 0.03 -0.04 3/21 0.00 -0.23 + + 11 0.34 -0.36 -0.54 5/21 0.40 -0.19 + + 41 0.96 0.04 7/21 2.17 0.36 + + 51 1.54 -0.07 0.08 1/21 2.17 0.56 + + 32 1.90 0.25 0.42 3/22 2.72 -0.15 + + 12 2.04 -0.68 0.43 9/21 2.76 -0.24 + + 21 2.23 0.00 5/22 3.75 -0.60 + + 33 2.95 0.30 -0.29 1/22 4.46 -0.30 + + 22 3.18 0.73 7/22 4.66 -0.38 + + 61 3.84 -0.10 7/23 5.24 -0.06 + + 13 4.09 0.30 0.12 5/23 5.42 -0.10 + + 23 4.15 0.23 11/21 5.59 0.19 + + 42 4.22 0.00 5/24 5.79 -0.21 + + 71 4.55 0.05 -0.12 3/23 5.83 -0.40 + + 52 4.62 0.02 0.01 11/22 6.02 0.02 + + 34 4.64 0.32 -0.37 9/22 6.03 0.03

24 20 + 25 21 + Na → F(21 ) Na → F(5/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 41 0.00 0.05 0.11 -0.03 5/21 0.00 -0.46 0.34 0.36 + + 11 0.54 -0.27 3/21 0.11 -0.58 0.29 + + 21 0.63 -0.10 -0.48 0.11 1/21 0.97 -0.72 + + 22 1.11 0.53 -0.23 0.09 3/22 1.98 0.37 0.37 + + 12 1.32 0.50 9/21 2.49 -0.37 0.49 -0.03 + + 31 1.34 -0.27 0.11 7/21 2.69 0.17 -0.11 0.09 + + 51 1.55 -0.14 0.00 5/22 2.81 -0.30 -0.46 0.01 + + 32 1.80 0.33 0.08 9/22 3.21 -0.18 0.15 -0.01 + + 23 1.81 0.18 0.22 -0.32 7/22 3.25 -0.66 -0.11 -0.09 + + 33 2.35 -0.24 0.28 3/23 3.52 -0.29 0.18 + + 34 2.63 0.16 0.00 5/23 3.80 0.33 -0.01 0.30 + + 42 2.65 0.34 0.26 -0.10 9/23 3.87 0.19 -0.13 0.34 + + 24 2.79 0.21 0.10 -0.14 3/24 3.99 -0.33 0.13 + + 43 3.00 -0.41 0.13 -0.26 1/22 4.12 0.02 + + 13 3.35 -0.01 7/23 4.13 -0.08 0.33 -0.16 + + 35 3.46 0.01 0.19 5/24 4.24 -0.37 0.28 0.26

74 26 22 + 27 23 + Na → F(41 ) Na → F(5/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 31 0.00 0.56 0.06 -0.04 5/21 0.00 -0.64 0.37 0.52 + + 11 0.00 -0.65 3/21 0.03 -0.77 0.49 + + 21 0.11 -0.00 0.25 -0.28 1/21 1.66 -0.54 + + 22 0.33 -0.47 0.19 0.31 9/21 2.19 0.58 -0.69 -0.01 + + 12 1.28 0.08 7/21 2.34 0.71 -0.06 -0.30 + + 41 1.63 0.48 -0.46 0.33 -0.06 -0.02 5/22 2.94 0.03 0.34 -0.35 + + 32 1.71 0.00 0.44 0.24 3/22 3.22 0.18 -0.00 + + 01 1.74 -0.69 9/22 3.77 0.06 -0.18 0.37 + + 42 1.99 0.11 -0.22 -0.31 0.31 0.38 1/22 3.86 -0.38 + + 23 2.06 0.41 0.03 0.23 5/23 3.87 -0.29 0.10 -0.07 + + 51 2.22 0.45 0.30 -0.19 -0.06 3/23 3.90 -0.23 -0.43 + + 24 2.24 -0.15 -0.42 0.17 3/24 3.94 0.00 -0.08 + + 13 2.45 0.17 5/24 4.14 0.17 0.27 0.11 + + 14 2.68 0.03 7/22 4.33 0.27 -0.25 0.27 + + 33 2.73 -0.09 0.14 0.10 5/25 4.51 -0.28 -0.11 0.01 + + 52 2.98 -0.36 0.29 0.03 0.02 7/23 4.70 0.09 0.15 -0.03

28 24 + 29 25 + Na → F(41 ) Na → F(5/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 21 0.00 -0.52 0.67 3/21 0.07 0.73 -0.51 + + 11 0.12 -0.38 0.49 9/21 2.04 -0.49 0.65 0.20 + + 31 0.62 -0.29 0.41 0.18 -0.50 1/21 2.15 -0.53 + + 41 0.79 -0.47 0.34 0.43 7/21 2.74 0.30 -0.06 -0.07 + + 32 1.13 -0.46 0.01 0.48 0.07 3/22 2.75 0.28 0.08 + + 22 1.39 -0.44 -0.27 7/22 2.94 -0.52 -0.14 0.38 + + 23 1.83 -0.22 -0.08 5/22 3.36 -0.10 -0.39 0.10 + + 12 2.13 -0.09 0.38 1/22 3.48 0.44 + + 33 2.51 -0.22 0.28 -0.30 0.14 1/23 3.98 0.24 + + 24 2.56 -0.09 -0.06 9/22 4.10 0.30 -0.27 0.23 + + 13 2.69 -0.20 0.13 5/23 4.11 -0.01 -0.21 0.16 + + 51 3.06 0.45 -0.31 -0.29 0.07 3/23 4.28 -0.26 -0.28 + + 14 3.14 -0.28 -0.38 5/24 4.68 -0.26 -0.16 -0.13 + + 42 3.25 0.20 -0.18 0.07 7/23 4.72 0.47 -0.05 0.13 + + 25 3.32 -0.07 -0.24 9/23 4.75 -0.06 -0.18 0.31 + + 61 3.35 0.42 -0.70 0.66 3/24 4.82 -0.21 -0.31

75 30 26 + 31 27 + Na → F(11 ) Na → F(5/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 21 0.00 0.39 5/21 0.00 0.62 -0.48 -0.51 + + 31 0.21 -0.24 0.03 3/21 0.37 -0.70 0.59 + + 11 0.25 0.64 0.26 9/21 1.82 -0.55 0.83 -0.75 + + 41 0.68 -0.31 1/21 3.89 0.78 + + 32 1.58 0.47 0.01 7/21 4.95 -0.89 0.32 -0.18 + + 22 2.08 0.20 3/22 5.20 0.59 0.19 + + 51 2.61 0.46 -0.59 5/22 5.77 -0.13 -0.55 0.26 + + 12 2.77 0.10 -0.47 9/22 6.83 -0.51 0.14 -0.16 + + 33 3.08 0.11 0.25 3/23 6.94 0.17 0.66 + + 61 3.18 -0.39 7/22 7.80 0.11 0.86 -0.87 + + 23 3.40 0.13 11/21 8.15 1.00 -1.00 -1.00 + + 42 3.55 0.51 5/23 8.16 -0.31 0.29 -0.73 + + 34 3.75 -0.31 0.37 1/22 8.86 -0.47 + + 24 3.80 -0.26 5/24 8.93 -0.64 0.20 -0.03 + + 13 3.80 0.29 -0.21 3/24 9.66 -0.06 0.25 + + 14 4.42 0.16 -0.15 9/23 9.78 -0.03 0.51 -0.62

26 22 + 27 23 + Al → Na(31 ) Al → Na(3/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 51 0.00 0.10 -0.14 -0.02 -0.03 5/21 0.00 0.39 -0.16 + + 31 0.52 -0.51 0.13 0.13 0.06 1/21 0.88 0.16 + + 11 1.03 0.09 0.13 3/21 1.06 -0.58 -0.78 + + 21 1.58 0.53 -0.08 7/21 2.33 -0.67 -0.20 + + 12 1.82 -0.44 0.07 5/22 2.70 -0.34 0.15 + + 13 2.11 -0.15 0.40 3/22 2.84 0.11 -0.16 + + 41 2.12 -0.66 -0.33 -0.02 9/21 2.99 -0.25 0.10 + + 22 2.12 0.00 0.00 1/22 3.83 -0.57 + + 32 2.15 -0.17 0.01 -0.34 -0.08 3/23 4.03 -0.08 -0.25 + + 33 2.40 0.35 0.60 0.27 0.01 5/23 4.40 -0.32 -0.54 + + 23 2.62 -0.09 -0.25 11/21 4.49 -0.39 0.01 + + 14 2.75 -0.03 0.32 7/22 4.71 -0.03 0.63 + + 24 2.92 -0.10 -0.00 5/24 4.73 0.24 0.11 + + 34 3.10 -0.03 -0.07 0.11 -0.39 5/25 5.33 -0.04 0.02 + + 25 3.23 0.00 0.00 9/22 5.40 -0.10 0.03 + + 52 3.40 -0.63 -0.17 0.11 0.02 5/26 5.42 -0.36 -0.09

76 28 24 + 29 25 + Al → Na(41 ) Al → Na(5/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 31 0.00 -0.57 -0.10 0.04 5/21 0.00 -0.59 0.49 0.41 + + 21 0.01 0.43 -0.30 -0.18 1/21 1.21 0.79 + + 01 0.98 0.18 7/21 1.86 -0.65 -0.11 0.18 + + 32 1.05 0.11 0.50 0.07 3/21 2.08 0.54 0.32 + + 11 1.20 -0.08 3/22 2.70 0.16 0.69 + + 12 1.50 -0.53 5/22 2.99 0.41 0.39 -0.22 + + 22 1.52 0.33 0.45 -0.00 5/23 3.10 0.18 0.19 0.36 + + 13 2.06 0.06 1/22 3.36 0.41 + + 23 2.07 0.21 -0.00 0.33 9/21 3.49 0.36 -0.29 -0.44 + + 41 2.21 0.47 0.25 -0.11 0.06 0.27 3/23 3.68 0.43 -0.12 + + 24 2.42 0.20 0.14 0.01 5/24 3.90 -0.03 0.01 0.16 + + 51 2.48 0.48 0.16 0.01 0.02 7/22 3.93 0.05 -0.42 0.03 + + 42 2.54 0.52 0.48 0.43 -0.05 -0.15 3/24 4.06 0.18 -0.05 + + 02 2.75 0.63 5/25 4.17 -0.03 -0.03 -0.10 + + 33 2.86 -0.10 -0.15 -0.11 1/23 4.33 -0.00 + + 14 3.00 0.32 5/26 4.45 0.05 -0.12 -0.07

30 26 + 31 27 + Al → Na(41 ) Al → Na(5/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 31 0.00 0.40 -0.54 0.11 0.26 5/21 0.00 -0.56 0.57 0.54 + + 21 0.14 -0.36 0.39 1/21 0.95 0.67 + + 11 0.56 0.36 -0.46 3/21 1.84 -0.50 0.33 + + 32 0.92 0.33 -0.14 -0.27 -0.21 9/21 2.51 0.21 -0.29 -0.43 + + 41 0.97 0.47 -0.24 -0.52 7/21 2.66 0.63 0.09 0.02 + + 22 1.62 -0.26 -0.37 5/22 3.35 0.50 0.33 -0.29 + + 23 1.87 0.56 0.10 1/22 3.44 -0.54 + + 12 2.09 -0.39 -0.43 3/22 3.74 -0.30 -0.29 + + 13 2.58 0.13 -0.51 5/23 3.93 -0.16 0.04 0.17 + + 42 2.65 -0.11 -0.16 -0.24 3/23 4.05 0.38 0.66 + + 33 2.68 -0.11 -0.03 0.24 -0.05 7/22 4.22 -0.37 -0.00 -0.08 + + 24 2.77 -0.11 0.20 9/22 4.66 -0.01 0.05 0.14 + + 51 2.99 -0.36 0.34 0.34 0.09 1/23 4.91 0.16 + + 14 3.12 -0.41 -0.07 3/24 4.94 0.02 -0.21 + + 43 3.25 -0.26 0.24 0.10 5/24 4.94 0.00 -0.08 -0.11 + + 25 3.29 0.00 0.36 3/25 5.29 -0.01 0.32

77 32 28 + 33 29 + Al → Na(11 ) Al → Na(3/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 11 0.00 -0.53 -0.28 5/21 0.00 -0.59 0.45 + + 41 0.54 -0.28 1/21 3.73 0.35 + + 21 0.66 0.52 3/21 4.53 0.30 0.92 + + 31 1.77 -0.23 -0.25 7/21 5.02 0.79 -0.07 + + 32 2.36 -0.57 -0.16 5/22 5.56 -0.47 -0.31 + + 22 2.39 0.25 3/22 6.33 0.21 -0.03 + + 12 3.00 -0.42 -0.19 9/21 6.59 0.50 0.31 + + 23 3.80 -0.05 7/22 7.32 -0.03 -0.95 + + 24 4.40 0.26 1/22 7.36 0.55 + + 33 4.75 -0.18 0.37 5/23 7.79 -0.13 0.60 + + 51 5.15 0.61 0.24 11/21 8.20 -0.89 0.87 + + 34 5.19 -0.22 -0.04 9/22 8.78 0.02 -0.53 + + 25 5.29 0.25 5/24 8.91 0.31 0.17 + + 13 5.47 0.19 0.32 3/23 9.27 0.11 -0.06 + + 42 5.74 -0.12 3/24 9.75 -0.84 -0.08 + + 35 5.80 -0.18 -0.38 7/23 9.98 0.06 0.03

30 26 + 31 27 + P → Al(51 ) P → Al(5/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 11 0.00 0.70 0.42 1/21 0.00 0.93 + + 12 0.65 -0.31 0.77 3/21 1.17 0.42 0.88 + + 21 1.60 0.35 0.59 5/21 2.31 -0.72 -0.19 -0.09 + + 31 1.99 -0.57 -0.02 0.12 0.54 1/22 3.24 -0.06 + + 32 2.40 -0.61 -0.23 0.16 -0.42 5/22 3.31 0.35 -0.65 -0.30 + + 22 2.58 0.77 -0.11 7/21 3.48 -0.34 -0.08 -0.22 + + 33 3.00 -0.04 -0.51 0.32 0.32 3/22 3.62 0.56 -0.03 + + 23 3.01 0.00 0.00 5/23 4.27 0.32 0.26 -0.44 + + 13 3.14 -0.46 0.03 3/23 4.37 0.27 0.05 + + 14 3.70 -0.13 -0.05 3/24 4.69 0.06 -0.07 + + 24 3.81 0.03 -0.46 7/22 4.77 -0.62 0.04 0.44 + + 34 4.13 0.13 0.40 0.55 0.04 5/24 4.82 -0.29 0.20 -0.05 + + 25 4.25 0.00 0.00 1/23 5.00 -0.01 + + 51 4.37 -0.69 -0.30 -0.16 -0.16 -0.15 5/25 5.20 0.10 -0.04 0.06 + + 26 4.39 0.15 0.05 9/21 5.39 -0.36 -0.30 -0.06 + + 41 4.43 0.42 0.33 -0.15 0.01 7/23 5.52 0.25 0.43 0.17

78 32 28 + 33 29 + P → Al(31 ) P → Al(5/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 11 0.00 -0.15 -0.76 1/21 0.00 0.93 + + 21 0.12 0.55 0.70 3/21 1.45 -0.16 -0.91 + + 12 1.12 0.66 0.29 5/21 1.96 0.61 0.10 0.23 + + 22 1.20 -0.13 0.08 3/22 2.63 -0.72 -0.06 + + 31 1.70 0.29 0.47 -0.15 -0.53 3/23 3.44 -0.12 -0.03 + + 13 1.98 0.34 -0.34 5/22 3.65 -0.33 -0.67 0.34 + + 32 2.23 -0.18 -0.55 -0.44 -0.42 7/21 3.85 0.32 -0.15 0.39 + + 23 2.24 0.10 -0.03 5/23 4.08 0.44 -0.04 -0.21 + + 24 2.59 0.05 -0.32 1/22 4.35 -0.02 + + 14 2.88 0.18 -0.10 3/24 4.99 -0.18 0.03 + + 33 2.95 0.32 -0.08 0.28 0.11 5/24 5.12 -0.45 0.01 0.23 + + 41 3.18 0.06 -0.30 -0.15 1/23 5.70 -0.05 + + 25 3.51 0.22 -0.06 9/21 5.77 0.29 0.05 -0.25 + + 34 3.75 0.14 -0.11 -0.36 0.06 7/22 5.90 -0.18 0.01 0.39 + + 26 3.76 -0.62 0.10 1/24 6.26 0.24 + + 15 3.81 -0.01 -0.13 3/25 6.54 -0.15 -0.05

34 30 + 35 31 + P → Al(31 ) P → Al(5/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 11 0.00 -0.52 0.80 1/21 0.00 -0.96 + + 21 0.40 -0.74 0.42 3/21 2.57 -0.43 -0.97 + + 12 1.51 0.38 0.37 5/21 4.82 -0.95 -0.53 0.33 + + 22 2.25 0.03 0.65 7/21 7.73 -0.59 -0.08 0.86 + + 31 2.79 0.11 -0.21 -0.49 0.90 5/22 7.74 0.08 0.76 0.05 + + 13 2.95 0.16 0.20 3/22 7.75 -0.82 -0.01 + + 23 3.17 0.15 0.35 9/21 8.46 -0.92 -0.85 -0.85 + + 41 4.18 0.63 0.02 0.22 5/23 8.83 -0.26 0.13 0.20 + + 14 4.50 -0.61 0.02 7/22 8.97 0.74 0.71 0.39 + + 24 4.84 0.47 -0.05 1/22 9.42 -0.23 + + 32 4.92 0.29 -0.28 0.13 -0.13 5/24 10.29 0.11 0.12 -0.04 + + 15 4.97 0.28 0.06 9/22 10.60 0.35 0.31 0.31 + + 25 5.95 0.05 -0.14 7/23 10.89 -0.26 -0.27 -0.28 + + 33 6.14 -0.86 -0.34 -0.20 -0.09 3/23 10.90 -0.33 0.02 + + 42 6.51 0.16 0.17 0.05 3/24 11.55 -0.04 0.19 + + 34 6.53 0.02 0.23 -0.10 0.02 1/23 12.19 0.05

79 34 30 + 35 31 + Cl → P(11 ) Cl → P(1/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 31 0.00 -0.90 0.93 3/21 0.00 0.94 + + 11 0.35 0.54 -0.63 1/21 1.23 0.97 + + 12 0.54 -0.73 0.68 5/21 1.68 0.82 + + 21 1.09 0.84 3/22 2.62 0.27 + + 22 1.73 0.24 7/21 2.75 0.47 + + 32 2.06 0.02 0.07 5/22 3.10 -0.27 + + 23 2.15 0.00 3/23 3.95 -0.06 + + 41 2.29 0.62 1/22 3.98 0.05 + + 13 2.40 -0.15 -0.14 9/21 4.16 0.05 + + 33 2.44 0.22 -0.12 7/22 4.21 0.20 + + 14 3.05 -0.07 0.15 5/23 4.61 0.02 + + 24 3.15 0.00 3/24 4.76 -0.00 + + 34 3.29 -0.06 0.04 1/23 4.93 -0.08 + + 51 3.67 -0.44 0.45 7/23 5.19 0.16 + + 35 3.71 0.07 -0.14 5/24 5.71 -0.14 + + 15 3.73 0.02 -0.09 3/25 5.80 -0.04

36 32 + 37 33 + Cl → P(11 ) Cl → P(1/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 21 0.00 0.92 3/21 0.00 0.97 + + 31 0.84 -0.71 0.79 1/21 1.88 0.99 + + 11 1.07 -0.77 0.56 5/21 3.20 -0.90 + + 12 1.73 0.56 -0.66 3/22 4.02 -0.22 + + 22 2.02 -0.32 3/23 5.40 -0.01 + + 23 2.46 0.06 5/22 6.84 -0.21 + + 13 2.55 -0.11 0.35 9/21 8.02 -0.88 + + 32 3.05 0.05 -0.14 7/21 8.12 0.93 + + 14 3.38 -0.01 0.10 1/22 9.06 0.13 + + 41 3.67 -0.71 3/24 9.65 -0.05 + + 33 4.57 -0.38 0.37 5/23 9.74 -0.36 + + 24 4.57 0.06 7/22 10.26 -0.35 + + 34 4.74 -0.27 -0.02 9/22 10.88 0.43 + + 25 5.00 -0.01 3/25 11.20 -0.05 + + 26 5.54 -0.05 1/23 11.51 -0.03 + + 15 5.59 -0.05 0.12 5/24 11.68 0.07

80 38 34 + 39 35 + K → Cl(11 ) K → Cl(3/21 ) π π J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 J Ex ` = 0 ` = 2 ` = 4 ` = 6 ` = 8 + + 01 0.46 -1.00 3/21 0.00 -1.00 1.00 + + 21 2.31 0.92 1/21 2.65 1.00 + + 22 3.21 -0.00 5/21 7.95 1.00 -1.00 + 23 4.70 0.28 + 02 6.49 0.02 + 41 6.81 0.00 + 24 7.16 0.00 + 42 9.22 -1.00 + 25 10.40 0.26 + 26 12.55 0.03

81 BIBLIOGRAPHY

[1] National Research Council. Nuclear Physics: Exploring the Heart of Matter. The National Academies Press, Washington, DC, 2013.

[2] Ani Aprahamian et al. Reaching for the horizon: The 2015 long range plan for nuclear science. 2015.

[3] George Gamow. Mass defect curve and nuclear constitution. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 126(803):632–644, 1930.

[4] L. R. Hafstad and E. Teller. The alpha-particle model of the nucleus. Phys. Rev., 54:681–692, Nov 1938.

[5] J.M. Blatt and V.F. Weisskopf. Theoretical Nuclear Physics. Dover Books on Physics. Dover Publications, 2012.

[6] Martin Freer. The clustered nucleus?α cluster structures in stable and unstable nuclei. Reports on Progress in Physics, 70(12):2149, 2007.

[7] James Chadwick. The existence of a neutron. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 136(830):692–708, 1932.

[8] N. Curtis, N. M. Clarke, B. R. Fulton, S. J. Hall, M. J. Leddy, A. St. J. Murphy, J. S. Pople, R. P. Ward, W. N. Catford, G. J. Gyapong, S. M. Singer, S. P. G. Chappell, S. P. Fox, C. D. Jones, D. L. Watson, W. D. M. Rae, and P. M. Simmons. Association of the 12C+12C breakup states in 24Mg with the quasimolecular resonances. Phys. Rev. C, 51:1554–1557, Mar 1995.

[9] N. Curtis, A. St. J. Murphy, N. M. Clarke, M. Freer, B. R. Fulton, S. J. Hall, M. J. Leddy, J. S. Pople, G. Tungate, R. P. Ward, W. N. Catford, G. J. Gyapong, S. M. Singer, S. P. G. Chappell, S. P. Fox, C. D. Jones, D. L. Watson, W. D. M. Rae, P. M. Simmons, and P. H. Regan. Evidence for a highly deformed band in 16O+16O breakup of 32S. Phys. Rev. C, 53:1804–1810, Apr 1996.

[10] Kiyomi Ikeda, Noboru Takigawa, and Hisashi Horiuchi. The systematic structure-change into the molecule-like structures in the self-conjugate 4n nuclei. Prog. Theor. Phys. Supp., E68:464–475, 1968.

[11] D. N. F. Dunbar, R. E. Pixley, W. A. Wenzel, and W. Whaling. The 7.68-mev state in C12. Phys. Rev., 92:649–650, Nov 1953.

[12] Fred Hoyle. On nuclear reactions occuring in very hot stars. i. the synthesis of elements from carbon to nickel. The Astrophysical Journal Supplement Series, 1:121, 1954.

82 [13] M. Freer, A. H. Wuosmaa, R. R. Betts, D. J. Henderson, P. Wilt, R. W. Zurm¨uhle,D. P. Balamuth, S. Barrow, D. Benton, Q. Li, Z. Liu, and Y. Miao. Limits for the 3α branching + 12 ratio of the decay of the 7.65 mev, 02 state in C. Phys. Rev. C, 49:R1751–R1754, Apr 1994.

[14] Ad.R. Raduta, B. Borderie, E. Geraci, N. Le Neindre, P. Napolitani, M.F. Rivet, R. Alba, F. Amorini, G. Cardella, M. Chatterjee, E. De Filippo, D. Guinet, P. Lautesse, E. La Guidara, G. Lanzalone, G. Lanzano, I. Lombardo, O. Lopez, C. Maiolino, A. Pagano, S. Pirrone, G. Politi, F. Porto, F. Rizzo, P. Russotto, and J.P. Wieleczko. Evidence for α-particle condensation in nuclei from the Hoyle state deexcitation. Phys. Lett. B, 705(12):65 – 70, 2011.

[15] M. Itoh, S. Ando, T. Aoki, H. Arikawa, S. Ezure, K. Harada, T. Hayamizu, T. Inoue, T. Ishikawa, K. Kato, H. Kawamura, Y. Sakemi, and A. Uchiyama. Further improvement of the upper limit on the direct 3α decay from the Hoyle state in 12C. Phys. Rev. Lett., 113:102501, Sep 2014.

[16] R. Smith, Tz. Kokalova, C. Wheldon, J. E. Bishop, M. Freer, N. Curtis, and D. J. Parker. New measurement of the direct 3α decay from the 12C Hoyle state. Phys. Rev. Lett., 119:132502, Sep 2017.

[17] D. Dell’Aquila, I. Lombardo, G. Verde, M. Vigilante, L. Acosta, C. Agodi, F. Cappuzzello, D. Carbone, M. Cavallaro, S. Cherubini, A. Cvetinovic, G. D’Agata, L. Francalanza, G. L. Guardo, M. Gulino, I. Indelicato, M. La Cognata, L. Lamia, A. Ordine, R. G. Pizzone, S. M. R. Puglia, G. G. Rapisarda, S. Romano, G. Santagati, R. Spart`a,G. Spadaccini, C. Spitaleri, and A. Tumino. High-precision probe of the fully sequential decay width of the Hoyle state in 12C. Phys. Rev. Lett., 119:132501, Sep 2017.

[18] N. Marquardt, W. Von Oertzen, and R.L. Walter. Evidence for a direct eight-nucleon transfer reaction and multiparticle excitation of 20Ne. Physics Letters B, 35(1):37 – 40, 1971.

[19] M. E. Cobern, D. J. Pisano, and P. D. Parker. Alpha-transfer reactions in light nuclei. iii. (7Li, t) stripping reaction. Phys. Rev. C, 14:491–505, Aug 1976.

[20] D. J. Pisano and P. D. Parker. Alpha-transfer reactions in light nuclei. ii. (3He, 7Be) pickup reactions. Phys. Rev. C, 14:475–490, Aug 1976.

[21] E. Hadjimichael and W. Oelert. Few-body problems. International review of nuclear physics: vol. 1 (1985). Singapore ; Philadelphia : World Scientiﬁc, c1986., 1986.

[22] J.P. Draayer. Alpha-particle spectroscopic amplitudes for sd shell nuclei. Nuclear Physics A, 237(1):157 – 181, 1975.

[23] Yu. F. Smirnov and Yu. M. Tchuvil’sky. Cluster spectroscopic factors for p-shell nuclei. Phys. Rev. C, 15(1):84, 1977.

83 [24] W. Chung, J. van Hienen, B.H. Wildenthal, and C.L. Bennett. Shell-model predictions of alpha-spectroscopic factors between ground states of 16 ≤ A ≤ 40 nuclei. Physics Letters B, 79(4):381 – 384, 1978.

[25] N. Anantaraman, C. L. Bennett, J. P. Draayer, H. W. Fulbright, H. E. Gove, and J. T¯oke. Systematics of ground-state α-particle spectroscopic strengths for sd- and fp-shell nuclei. Phys. Rev. Lett., 35:1131–1134, Oct 1975.

[26] John A. Wheeler. On the mathematical description of light nuclei by the method of resonating group structure. Phys. Rev., 52(11):1107–1122, 1937.

[27] K Wildermuth and E J Kanellopoulos. Clustering aspects in nuclei and their microscopic description. Reports on Progress in Physics, 42(10):1719, 1979.

[28] R. B. Wiringa, Steven C. Pieper, J. Carlson, and V. R. Pandharipande. Quantum monte carlo calculations of A = 8 nuclei. Phys. Rev. C, 62:014001, Jun 2000.

[29] Yoshiko Kanada-En’yo, Masaaki Kimura, and Akira Ono. Antisymmetrized molecular dynamics and its applications to cluster phenomena. Progress of Theoretical and Experimental Physics, 2012(1):01A202, 2012.

[30] W. von Oertzen. Nuclear spectroscopy, nuclear clustering and exotic shapes. Physica Scripta, 2000(T88):83, 2000.

[31] W. von Oertzen, Martin Freer, and Yoshiko Kanada-Enyo. Nuclear clusters and nuclear molecules. Physics Reports, 432(2):43 – 113, 2006.

[32] R. Bijker and F. Iachello. Cluster states in nuclei as representations of a U(ν + 1) group. Phys. Rev. C, 61:067305, May 2000.

[33] R. Bijker and F. Iachello. Evidence for tetrahedral symmetry in 16O. Phys. Rev. Lett., 112:152501, Apr 2014.

[34] J-P Ebran, Elias Khan, Tamara Nikˇsi´c,and Dario Vretenar. How atomic nuclei cluster. Nature, 487(7407):341, 2012.

[35] Evgeny Epelbaum, Hermann Krebs, Dean Lee, and Ulf-G. Meißner. Ab initio. Phys. Rev. Lett., 106:192501, May 2011.

[36] Serdar Elhatisari, Dean Lee, Gautam Rupak, Evgeny Epelbaum, Hermann Krebs, Timo A. L¨ahde, Thomas Luu, and Ulf-G. Meißner. Ab initio alpha–alpha scattering. Nature, 528(7580):111–114, 12 2015.

84 [37] A. C. Dreyfuss, K. D. Launey, T. Dytrych, J. P. Draayer, R. B. Baker, C. M. Deibel, and C. Bahri. Understanding emergent collectivity and clustering in nuclei from a symmetry-based no-core shell-model perspective. Phys. Rev. C, 95:044312, Apr 2017.

[38] A. Tohsaki, H. Horiuchi, P. Schuck, and G. R¨opke. Alpha cluster condensation in 12C and 16O. Phys. Rev. Lett., 87:192501, Oct 2001.

[39] Peter Schuck, Yasuro Funaki, Hisashi Horiuchi, Gerd Rpke, Akihiro Tohsaki, and Taiichi Yamada. Alpha particle clusters and their condensation in nuclear systems. Physica Scripta, 91(12):123001, 2016.

[40] Akihiro Tohsaki, Hisashi Horiuchi, Peter Schuck, and Gerd R¨opke. Colloquium: Status of α-particle condensate structure of the Hoyle state. Rev. Mod. Phys., 89:011002, Jan 2017.

[41] Amos de Shalit, Igal Talmi, and Igal Talmi. Nuclear shell theory. Pure and applied physics: v. 14. New York, Academic Press, 1963., 1963.

[42] Matteo Vorabbi, Angelo Calci, Petr Navr´atil,Michael K. G. Kruse, Soﬁa Quaglioni, and Guillaume Hupin. Structure of the exotic 9He nucleus from the no-core shell model with continuum. Phys. Rev. C, 97:034314, Mar 2018.

[43] M. Moshinsky. Transformation brackets for harmonic oscillator functions. Nucl. Phys., 13(1):104, 1959.

[44] Marcos Moshinsky. The harmonic oscillator in modern physics; : from atoms to quarks. Documents on modern physics. New York, Gordon and Breach [1969], 1969.

[45] V.D. Efros. Some properties of the moshinsky coeﬃcients. Nuclear Physics A, 202(1):180 – 190, 1973.

[46] M. A. Caprio, P. Maris, and J. P. Vary. Coulomb-sturmian basis for the nuclear many-body problem. Phys. Rev. C, 86:034312, Sep 2012.

[47] Yu. F. Smirnov and D. Chlebowska. Reduced widths for nucleon clusters in shell model. Nucl. Phys., 26(2):306, 1961.

[48] J Dobes. A new expression for harmonic oscillator brackets. Journal of Physics: A Mathe- matical and General, 10(12):2053, 1977.

[49] J.D. Talman and A. Lande. Computation of moshinsky brackets by direct diagonalization. Nuclear Physics A, 163(1):249 – 256, 1971.

[50] B. Buck and A. C. Merchant. A simple expression for the general oscillator bracket. Nucl. Phys. A, 600(3):387, 1996.

85 [51] Maria Goeppert Mayer. On closed shells in nuclei. Phys. Rev., 74:235–239, Aug 1948.

[52] Maria Goeppert Mayer. On closed shells in nuclei. ii. Phys. Rev., 75:1969–1970, Jun 1949.

[53] Morten Hjorth-Jensen, Thomas T.S. Kuo, and Eivind Osnes. Realistic eﬀective interactions for nuclear systems. Physics Reports, 261(3):125 – 270, 1995.

[54] G. R. Jansen, J. Engel, G. Hagen, P. Navratil, and A. Signoracci. Ab initio coupled-cluster eﬀective interactions for the shell model: Application to neutron-rich oxygen and carbon isotopes. Phys. Rev. Lett., 113:142502, Oct 2014.

[55] E. Dikmen, A. F. Lisetskiy, B. R. Barrett, P. Maris, A. M. Shirokov, and J. P. Vary. Ab initio eﬀective interactions for sd-shell valence nucleons. Phys. Rev. C, 91:064301, Jun 2015.

[56] H. Hergert, S.K. Bogner, T.D. Morris, A. Schwenk, and K. Tsukiyama. The in-medium similarity renormalization group: A novel ab initio method for nuclei. Physics Reports, 621:165 – 222, 2016. Memorial Volume in Honor of Gerald E. Brown.

[57] J. B. McGrory. The Two-Body Interaction in Nuclear Shell Model Calculations, pages 183– 203. Springer US, Boston, MA, 1972.

[58] EK Warburton and B Alex Brown. Eﬀective interactions for the 0p1s0d nuclear shell-model space. Physical Review C, 46(3):923, 1992.

[59] S. Cohen and D. Kurath. Eﬀective interactions for the 1p shell. Nuclear Physics, 73(1):1–24, 1965.

[60] B. A. Brown, W. A. Richter, R. E. Julies, and B. H. Wildenthal. Semi-empirical eﬀective interactions for the 1s-0d shell. Ann. Phys., 182:191, 1988.

[61] B. Alex Brown and W. A. Richter. New “USD” hamiltonians for the sd shell. Phys. Rev. C, 74:034315, Sep 2006.

[62] J P Draayer, T Dytrych, and K D Launey. Ab initio symmetry-adapted no-core shell model. Journal of Physics: Conference Series, 322(1):012001, 2011.

[63] R. Roth and P. Navr´atil.Ab initio study of 40Ca with an importance-truncated no-core shell model. Phys. Rev. Lett., 99:092501, Aug 2007.

[64] A. F. Lisetskiy, B. R. Barrett, M. K. G. Kruse, P. Navratil, I. Stetcu, and J. P. Vary. Ab-initio shell model with a core. Phys. Rev. C, 78:044302, Oct 2008.

[65] A. M. Shirokov, A. I. Mazur, S. A. Zaytsev, J. P. Vary, and T. A. Weber. Nucleon-nucleon interaction in the j-matrix inverse scattering approach and few-nucleon systems. Phys. Rev. C, 70:044005, Oct 2004.

86 [66] A M Shirokov, J P Vary, A I Mazur, S A Zaytsev, and T A Weber. Nn potentials from the j -matrix inverse scattering approach. Journal of Physics G: Nuclear and Particle Physics, 31(8):S1283, 2005.

[67] A. M. Shirokov, J. P. Vary, A. I. Mazur, and T. A. Weber. Realistic nuclear hamiltonian: Ab exitu approach. Physics Letters B, 644(1):33, 2007.

[68] Abdulaziz D Alhaidari, Eric J Heller, Hashim A Yamani, and Mohamed S Abdelmonem. The j-matrix method. Development and Applications (Springer, Berlin, 2008), 2008.

[69] D. R. Entem and R. Machleidt. Accurate charge-dependent nucleon-nucleon potential at fourth order of chiral perturbation theory. Phys. Rev. C, 68:041001, Oct 2003.

[70] R. Machleidt and D.R. Entem. Chiral eﬀective ﬁeld theory and nuclear forces. Physics Reports, 503(1):1 – 75, 2011.

[71] S. K. Bogner, R. J. Furnstahl, and R. J. Perry. Similarity renormalization group for nucleon- nucleon interactions. Phys. Rev. C, 75:061001, Jun 2007.

[72] Bruce R. Barrett, Petr Navr´atil, and James P. Vary. Ab initio no core shell model. Prog. Part. Nucl. Phys., 69:131, 2013.

[73] Y.C. Tang, M. LeMere, and D.R. Thompsom. Resonating-group method for nuclear many- body problems. Physics Reports, 47(3):167 – 223, 1978.

[74] Karl Wildermuth. A uniﬁed theory of the nucleus. Springer-Verlag, 2013.

[75] Soﬁa Quaglioni and Petr Navr´atil. Ab initio many-body calculations of nucleon-nucleus scattering. Phys. Rev. C, 79(4):044606, 2009.

[76] Soﬁa Quaglioni, Petr Navr´atil,Guillaume Hupin, Joachim Langhammer, Carolina Romero- Redondo, and Robert Roth. No-core shell model analysis of light nuclei. Few-Body Systems, 54(7):877–884, 2013.

[77] Peter Ring and Peter Schuck. The nuclear many-body problem. Texts and monographs in physics. New York : Springer-Verlag, c1980., 1980.

[78] P. Descouvemont and M. Dufour. Clusters in Nuclei, Vol 2, volume 848 of Lecture Notes in Physics, page 1. 2012.

[79] S. Yu. Igashov and Yu. M. Tchuvil’sky. Exchange eﬀects in composite-particle interaction. EPJ Web Conf., 38:16002, 2012.

[80] N.K. Glendenning. Direct Nuclear Reactions. World Scientiﬁc, 2004.

87 [81] Ian J Thompson and Filomena M Nunes. Nuclear reactions for astrophysics: principles, calculation and applications of low-energy reactions. Cambridge University Press, 2009.

[82] B.R. Americo and Z. Vladimir. Fifty Years Of Nuclear Bcs: Pairing In Finite Systems. World Scientiﬁc Publishing Company, 2013.

[83] T. Fliessbach and Hj Mang. Absolute values of alpha-decay rates. Nucl. Phys. A, 263(1):75, 1976.

[84] T. Fliessbach and P. Manakos. Alpha spectroscopic factors for light nuclei. J. Phys. G, 3(5):643, 1977.

[85] Alexander Volya and Yu M. Tchuvil’sky. Nuclear clustering using a modern shell model approach. Phys. Rev. C, 91(4):044319, 2015.

[86] Konstantinos Kravvaris and Alexander Volya. Quest for superradiance in atomic nuclei. AIP Conference Proceedings, 1912(1):020010, 2017.

[87] Naftali Auerbach and Vladimir Zelevinsky. Super-radiant dynamics, doorways and resonances in nuclei and other open mesoscopic systems. Rep. Prog. Phys., 74(10):106301, 2011.

[88] Anthony Nicholas Kuchera. 2013. Clustering Phenomena in the a=10 T=1 Isobaric Mul- tiplet, Florida State University, Electronic Theses, Treatises and Dissertations, Paper 8585, http://diginole.lib.fsu.edu/etd/8585/.

[89] M. L. Avila, G. V. Rogachev, V. Z. Goldberg, E. D. Johnson, K. W. Kemper, Yu M. Tchuvil’sky, and A. S. Volya. alpha-cluster structure of 18O. Phys. Rev. C, 90(2):024327, 2014.

[90] H. Feshbach. A uniﬁed theory of nuclear reactions. Ann. Phys., 5:357, 1958.

[91] C. Mahaux and H. A. Weidenm¨uller. Shell-model approach to nuclear reactions. North- Holland Pub. Co., Amsterdam, London, 1969.

[92] L. Durand. S-matrix treatment of many overlapping resonances. Phys. Rev. D, 14:3174, 1976.

[93] Alexander Volya. Time-dependent approach to the continuum shell model. Phys. Rev. C, 79(4):044308, 2009.

[94] V. V. Sokolov and V. G. Zelevinsky. On a statistical theory of overlapping resonances. Phys. Lett. B, 202:10, 1988.

[95] V. V. Sokolov and V. G. Zelevinsky. Dynamics and statistics of unstable quantum states. Nucl. Phys., A504:562, 1989.

88 [96] A. Bohr and B.R. Mottelson. NUCLEAR STRUCTURE (IN 2 VOLUMES): (In 2 Vol- umes)Volume I: Single-Particle Motion Volume II: Nuclear Deformations. 1998.

[97] V. Zelevinsky and A. Volya. Physics of Atomic Nuclei. Wiley, 2017.

[98] R. Casten. Nuclear structure from a simple perspective. Oxford studies in nuclear physics: 23. Oxford ; New York : Oxford University Press, 2000., 2000.

[99] JP Elliott. Collective motion in the nuclear shell model. i. classiﬁcation schemes for states of mixed conﬁgurations. In Proc. R. Soc. Lond. A, volume 245, pages 128–145. The Royal Society, 1958.

[100] James Philip Elliott. Collective motion in the nuclear shell model ii. the introduction of intrinsic wave-functions. In Proc. R. Soc. Lond. A, volume 245, pages 562–581. The Royal Society, 1958.

[101] M. Hamermesh. Group Theory and Its Application to Physical Problems. Dover Books on Physics. Dover Publications, 2012.

[102] A Castenholz. Algebraic Approaches to Nuclear Structure, volume 6. CRC Press, 1993.

[103] B. Buck, C. B. Dover, and J. P. Vary. Simple potential model for cluster states in light nuclei. Phys. Rev. C, 11:1803–1821, May 1975.

[104] Melina Avila. Clustering in 18O and ANC measurements using (6Li, d) reactions. PhD thesis, The Florida State University, 2013.

[105] F.Q. Luo, M.A. Caprio, and T. Dytrych. Construction of the center-of-mass free space for the SU(3) no-core shell model. Nuclear Physics A, 897:109 – 121, 2013.

[106] F. Palumbo and D. Prosperi. Eﬀects of translational invariance violation in particle-hole calculations. application to 208pb. Nuclear Physics A, 115(2):296 – 308, 1968.

[107] D.H. Gloeckner and R.D. Lawson. Spurious center-of-mass motion. Phys. Lett. B, 53(4):313 – 318, 1974.

[108] JP Draayer and Yoshimi Akiyama. Wigner and racah coeﬃcients for SU(3). Journal of Mathematical Physics, 14(12):1904–1912, 1973.

[109] C. Bahri, D.J. Rowe, and J.P. Draayer. Programs for generating clebschgordan coeﬃcients of SU(3) in SU(2) and SO(3) bases. Computer Physics Communications, 159(2):121 – 143, 2004.

89 [110] Alexander Volya and Yury M. Tchuvil’sky. Study of Nuclear Clustering Using the Modern Shell Model Approach, page 215. Exotic Nuclei, IASEN-2013. World Scientiﬁc, 09/10; 2014/12 2014. eds. E.Cherepanov, Yu. Penionzhkevich, D. Kamanin, R. Bark and J. Cornell.

[111] R. Wolsky, I. A. Gnilozub, S. D. Kurgalin, and Yu. M. Tchuvil’sky. Alpha-cluster and alpha- binucleon states of 1p-shell nuclei. Phys. At. Nucl., 73(8):1405, 2010.

[112] Munetake Ichimura, Akito Arima, E. C. Halbert, and Tokuo Terasawa. Alpha-particle spectroscopic amplitudes and the SU(3) model. Nucl. Phys. A, 204(2):225, 1973.

[113] P. Navr´atil.Cluster form factor calculation in the ab initio no-core shell model. Phys. Rev. C, 70(5):054324, 2004.

[114] Peter Kramer, Gero John, and Dieter Schenzle. Group theory and the interaction of composite nucleon systems. Number ARRAY(0x3d827a8) in Clustering phenomena in nuclei. Vieweg, Braunschweig, 1981.

[115] T.A. Carey, P.G. Roos, N.S. Chant, A. Nadasen, and H. L. Chen. Phys. Rev. C, 23(1):576(R), 1981.

[116] W. A. Richter, S. Mkhize, and B. Alex Brown. sd-shell observables for the USDA and USDB hamiltonians. Phys. Rev. C, 78:064302, Dec 2008.

[117] W. A. Richter and B. Alex Brown. Shell-model studies of the rp reaction 35Ar(p,γ)36K. Phys. Rev. C, 85:045806, Apr 2012.

[118] D. K. Nauruzbayev, V. Z. Goldberg, A. K. Nurmukhanbetova, M. S. Golovkov, A. Volya, G. V. Rogachev, and R. E. Tribble. Structure of 20Ne states in resonance 16O + α elastic scattering. Phys. Rev. C, 96:014322, Jul 2017.

[119] M. Sambataro and N. Sandulescu. Four-body correlations in nuclei. Phys. Rev. Lett., 115:112501, Sep 2015.

[120] Jun Hiura, Yasuhisa Abe, Sakae Sait, and Osamu End. Alpha-cluster plus 16O-core model for 20Ne and neighboring nuclei*). Progress of Theoretical Physics, 42(3):555–580, 1969.

[121] Takehiro Matsuse and Masayasu Kamimura. A study of α-widths of 20Ne based on 16O and α-cluster model. Progress of Theoretical Physics, 49(5):1765–1767, 1973.

[122] Takehiro Matsuse, Masayasu Kamimura, and Yoshihiro Fukushima. Study of the alpha- clustering structure of 20 ne based on the resonating group method for 16O + α analysis of alpha-decay widths and the exchange kernel. Progress of Theoretical Physics, 53(3):706–724, 1975.

90 [123] N. Anantaraman, H.E. Gove, R.A. Lindgren, J. Tke, J.P. Trentelman, J.P. Draayer, F.C. Jundt, and G. Guillaume. Alpha-particle strengths from the 16O(6Li, d)20Ne reaction. Nuclear Physics A, 313(3):445 – 466, 1979.

[124] M. Bouhelal, F. Haas, E. Caurier, F. Nowacki, and A. Bouldjedri. A psdpf interaction to describe the 1hω ¯ intruder states in sd shell nuclei. Nuclear Physics A, 864(1):113 – 127, 2011.

[125] N. Anantaraman, J. P. Draayer, H. E. Gove, J. T¯oke, and H. T. Fortune. Alpha-particle stripping to 21Ne. Phys. Rev. C, 18:815–819, Aug 1978.

[126] H. Ohnuma, B.A. Brown, D. Dehnhard, K. Furukawa, T. Hasegawa, S. Hayakawa, N. Hoshino, K. Ieki, M. Kabasawa, K. Maeda, K. Miura, K. Muto, T. Nakagawa, K. Nisimura, H. Orihara, T. Suehiro, T. Tohel, and M. Yasue. A comparative study of the 13C(p, pγ)13C and 13C(p, n)13N reactions at ep = 35 mev. Nucl. Phys. A, 456(1):61 – 74, 1986.

[127] Yutaka Utsuno and Satoshi Chiba. Multiparticle-multihole states around 16O and correlation- energy eﬀect on the shell gap. Phys. Rev. C, 83(2):021301, 2011.

[128] A. Volya and V. Zelevinsky. Non-hermitian eﬀective hamiltonian and continuum shell model. Phys. Rev. C, 67(5):054322, 2003.

[129] F. D. Snyder and M. A. Waggoner. Study of Be9(Li7, p), Be9(Li7, d), Be9(Li7, t), and Be9(Li7, α) reactions from 5.6 to 6.2 mev. Phys. Rev., 186:999–1012, Oct 1969.

[130] Fumiya Tanabe, Akihiro Tohsaki, and Ryozo Tamagaki. - scattering at intermediate en- ergiesapplicability of orthogonality condition model and upper limit of isoscalar meson- nucleon coupling constants inferred from potential tail. Prog. Theor. Phys., 53(3):677, 1975.

[131] R. G. Lovas, R. J. Liotta, A. Insolia, K. Varga, and D. S. Delion. Microscopic theory of cluster radioactivity. Phys. Rep., 294(5), 1998.

[132] R. B. Wiringa, Steven C. Pieper, J. Carlson, and V. R. Pandharipande. Quantum monte carlo calculations of A = 8 nuclei. Phys. Rev. C, 62:014001, Jun 2000.

[133] M.A. Caprio, P. Maris, and J.P. Vary. Emergence of rotational bands in ab initio no-core conﬁguration interaction calculations of light nuclei. Phys. Lett. B, 719(13):179 – 184, 2013.

[134] S. Pastore, R. B. Wiringa, Steven C. Pieper, and R. Schiavilla. Quantum monte carlo calculations of electromagnetic transitions in 8Be with meson-exchange currents derived from chiral eﬀective ﬁeld theory. Phys. Rev. C, 90:024321, Aug 2014.

[135] Hisashi Horiuchi. Chapter iii. kernels of gcm, rgm and ocm and their calculation methods. Prog. Theor. Phys. Supp., 62:90, 1977.

91 [136] S. Hamada, M. Yasue, S. Kubono, M. H. Tanaka, and R. J. Peterson. Cluster structures in 10Be from the 7Li(α,p)10Be reaction. Phys. Rev. C, 49(6):3192, 1994.

[137] N. Soi´c,S. Blagus, M. Bogovac, S. Fazini, M. Lattuada, M. Milin, D. Miljani, D. Rendi, C. Spitaleri, T. Tadi, and M. Zadro. 6He + α clustering in 10Be. EPL (Europhysics Letters), 34(1):7, 1996.

[138] D. Suzuki, A. Shore, W. Mittig, J. J. Kolata, D. Bazin, M. Ford, T. Ahn, F. D. Becchetti, S. Beceiro Novo, D. Ben Ali, B. Bucher, J. Browne, X. Fang, M. Febbraro, A. Fritsch, E. Galyaev, A. M. Howard, N. Keeley, W. G. Lynch, M. Ojaruega, A. L. Roberts, and X. D. Tang. Resonant α scattering of 6He: Limits of clustering in 10Be. Phys. Rev. C, 87:054301, May 2013.

[139] D. Dell’Aquila, I. Lombardo, L. Acosta, R. Andolina, L. Auditore, G. Cardella, M. B. Chat- terjiee, E. De Filippo, L. Francalanza, B. Gnoffo, G. Lanzalone, A. Pagano, E. V. Pagano, M. Papa, S. Pirrone, G. Politi, F. Porto, L. Quattrocchi, F. Rizzo, E. Rosato, P. Russotto, A. Trifirò,M. Trimarchi, G. Verde, and M. Vigilante. New experimental investigation of the structure of 10Be and 16C by means of intermediate-energy sequential breakup. Phys. Rev. C, 93:024611, Feb 2016.

[140] R. Bijker and F. Iachello. The algebraic cluster model: Three-body clusters. Ann. Phys., 298(2):334 – 360, 2002.

[141] M. Chernykh, H. Feldmeier, T. Neﬀ, P. von Neumann-Cosel, and A. Richter. Structure of the Hoyle state in 12C. Phys. Rev. Lett., 98:032501, Jan 2007.

[142] L. V. Grigorenko and M. V. Zhukov. Two-proton radioactivity and three-body decay. iii. integral formulas for decay widths in a simpliﬁed semianalytical approach. Phys. Rev. C, 76:014008, Jul 2007.

[143] R.D. Mattuck. A Guide to Feynman Diagrams in the Many-body Problem. Dover Books on Physics Series. Dover Publications, 1976.

[144] J. Suhonen. From Nucleons to Nucleus: Concepts of Microscopic Nuclear Theory. Theoretical and Mathematical Physics. Springer Berlin Heidelberg, 2007.

[145] G. Baym. Lectures On Quantum Mechanics. CRC Press, 2018.

[146] J. M. VonMoss, S. L. Tabor, Vandana Tripathi, A. Volya, B. Abromeit, P. C. Bender, D. D. Caussyn, R. Dungan, K. Kravvaris, M. P. Kuchera, R. Lubna, S. Miller, J. J. Parker, and P.-L. Tai. Higher-spin structures in 21F and 25Na. Phys. Rev. C, 92:034301, Sep 2015.

[147] R. Dungan, S. L. Tabor, R. S. Lubna, A. Volya, Vandana Tripathi, B. Abromeit, D. D. Caussyn, K. Kravvaris, and P.-L. Tai. High-spin states in 29Al and 27Mg. Phys. Rev. C, 94:064305, Dec 2016.

92 [148] R. Dungan, S. L. Tabor, Vandana Tripathi, A. Volya, K. Kravvaris, B. Abromeit, D. D. Caussyn, S. Morrow, J. J. Parker, P.-L. Tai, and J. M. VonMoss. Radiative decay of neutron- unbound intruder states in 19O. Phys. Rev. C, 93:021302, Feb 2016.

[149] P.-L. Tai, S. L. Tabor, R. S. Lubna, K. Kravvaris, P. C. Bender, Vandana Tripathi, A. Volya, M. P. Carpenter, R. V. F. Janssens, T. Lauritsen, E. A. McCutchan, S. Zhu, R. M. Clark, P. Fallon, S. Paschalis, M. Petri, A. O. Macchiavelli, W. Reviol, and D. G. Sarantites. Cross- shell excitations in 31Si. Phys. Rev. C, 96:014323, Jul 2017.

[150] Vandana Tripathi, R. S. Lubna, B. Abromeit, H. L. Crawford, S. N. Liddick, Y. Utsuno, P. C. Bender, B. P. Crider, R. Dungan, P. Fallon, K. Kravvaris, N. Larson, A. O. Macchiavelli, T. Otsuka, C. J. Prokop, A. L. Richard, N. Shimizu, S. L. Tabor, A. Volya, and S. Yoshida. 38,40 38,40 β decay of Si (Tz = +5, + 6) to low-lying core excited states in odd-odd P isotopes. Phys. Rev. C, 95:024308, Feb 2017.

[151] R. S. Lubna, Vandana Tripathi, S. L. Tabor, P.-L. Tai, K. Kravvaris, P. C. Bender, A. Volya, M. Bouhelal, C. J. Chiara, M. P. Carpenter, R. V. F. Janssens, T. Lauritsen, E. A. Mc- Cutchan, S. Zhu, R. M. Clark, P. Fallon, A. O. Macchiavelli, S. Paschalis, M. Petri, W. Reviol, and D. G. Sarantites. Intruder conﬁgurations of excited states in the neutron-rich isotopes 33P and 34P. Phys. Rev. C, 97:044312, Apr 2018.

[152] K Kravvaris and A Volya. Constructing realistic alpha cluster channels. Journal of Physics: Conference Series, 863(1):012016, 2017.

[153] Konstantinos Kravvaris and Alexander Volya. Study of nuclear clustering from an ab initio perspective. Phys. Rev. Lett., 119:062501, Aug 2017.

93 BIOGRAPHICAL SKETCH

Kostas raduated from the National Kapodestrian University of Athens in 2011 with a specialization in Condensed Matter Physics. He started working in Dr. Alexander Volya’s group at Florida State University in 2012, earning a Master’s Degree in 2014. He collaborated in experimental nuclear physics projects targeting high spin states in light nuclei [146, 147], radiative decays of neutron unbound states [148], shell evolution and intruder states [149, 150, 151]. Currently engaged in nuclear physics research targeting nuclear clustering and aspects of nuclear structure in light nuclei [152, 153, 86].