UNDERSTANDING AND QUANTIFYING IMPACTS OF THE CONTINUUM ON NUCLEAR STRUCTURE By Joshua Wylie A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Physics — Doctor of Philosophy 2025 ABSTRACT With the opening of new facilities, such as the Facility for Rare Isotope Beams, exotic nuclei will be increasingly accessible. Many exotic nuclei are strongly coupled to the continuum re- sulting in interesting structure formations, such as halo nuclear states or nuclear resonances. In this thesis, the Gamow Shell Model framework is used to describe nuclei as Open Quantum Systems. This framework is a configuration-interaction shell model implemented in the Berggren basis which includes bound, resonant, and scattering states on equal foot- ing. Two case studies are presented to highlight the impact of continuum effects on nuclear structure. Spectroscopic Factors are calculated for 8,9C, 8B, 8,9Li, and 8He using a tradi- tional Shell Model approach (ignoring the continuum) and the Gamow Shell Model. The results from both methods are compared and demonstrate Spectroscopic Factors in these nuclei are dependent on the continuum. The newly discovered ephemeral nucleus 9N will be presented, which is unique with over half of its nucleons lying in the continuum. A projection method will be outlined to extract the continuum effects from the Gamow Shell Model to understand and directly quantify the effects of continuum coupling. Finally, a new outreach and community engagement demonstration, designed to educate the general public about nuclear structure and decays, will be discussed. This work is dedicated to all those who strive to make science and scientific education more accessible. Without your work, I may not be where I am today. iii ACKNOWLEDGMENTS There are countless people who have been crucial in guiding me to where I am today. That being said, I would like to thank those that have made this PhD possible. First, I would like to thank my advisor, Witold Nazarewicz, for his guidance, patience, and support. I am grateful for all the knowledge he has shared with me throughout this process, and he has instilled in me an appreciation for scientific rigor. Although initially intimidating, I have come to sincerely appreciate and value his challenges and questions, as they have pushed me to be a better scientist. I would also like to thank my committee members: Filomena Nunes, Paul Gu`eye, Metin Aktulga, and Johannes Pollanen for their support and encouragement throughout my PhD. I would also like to thank Paul in particular for his unending support and investment in my future career, especially during the end of my PhD. Although he was not an official advisor, I would like to thank Simin Wang (王思敏) for being an excellent secondary advisor and mentor to me throughout the past six years. From frequently coming into your office to emailing you in the middle of the night, I appreciate your willingness to answer my questions. Without your kindness and patience, this endeavor would have been much more difficult and overwhelming. I am also sincerely grateful to Nicolas Michel for both his technical support and his French language assistance, without which, I would have been in peril. From my time in Caen, I would like to thank Marek P(cid:32)loszajczak, David Cardona Ochoa, and Alan Cruz Dassie for welcoming me and all of the helpful discussions. Of course, I also appreciate everyone who has been a part of the research group and listened to my confused ramblings and helped me to refine and improve my work. Daniel, Eric, Kyle, and Pablo, I appreciate your willingness to offer help at a moment’s notice, espe- iv cially for this thesis. I would also like to thank Ante, Sudhanva, Yuanzhuo (远卓), Sylvester, Xingze (兴泽), and Josh for their support. Lastly, I would like to thank the students in the group (current and former): Aaron, An, Andrew, Bailey, Josh, Landon, Lauren, Maxwell, Mengzhi (孟之), Mookyong, and Tong (通) for their support and encouragement. I would also like to thank all of the other wonderful friends I have made along the way: Andy, Andy, Bob, Bryan, Cami, Cavan, David, Jordan, Kelly, Lexi, Mo, Nick, Patrick, Pierre, Richard, Senora, and Sheng. To all of the Mid Michigan Runners, thank you for the encouragement and welcome distractions from physics. A special thanks to Carissa, Hannah, and Julia for being the most fantastic friends and great people to weather the bad, sad, and pandemic times with. Without you, these six years would not have been nearly as joyful. To my family, thank you for your support and encouragement both before and during this time. I’m especially grateful to my parents, Carol and Eric, for pushing me to always be my best self. And to my Aunt Cathy and Uncle Brad, thank you for being available and supporting me throughout my entire university education. I am confident that this is the end of my educational journey; but if that were to change, I know you would be there to rely on. I would also like to thank my partners parents, Ed and Linda, who have welcomed me into their family and encouraged me throughout this process. I extend the biggest thanks to my partner, Renee, who has been kind, understanding, and patient with me, especially while I finished this thesis. You make the bad days better, you keep me focused and motivated, and make me want to be better every day. I look forward to what comes next! v TABLE OF CONTENTS Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Nuclear Structure Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Closed versus Open Quantum Systems Chapter 2. Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Berggren Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Gamow Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 3. Spectroscopic Factors and the Continuum . . . . . . . . . . . . . 3.1 Discrepancy Between Experiment and Theory . . . . . . . . . . . . . . . . . 3.2 Continuum Impact on Spectroscopic Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Parameters and Model Space 3.4 Impact of Continuum Coupling Calculations . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 4. Description of 9N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Particle-Emitting Nuclei 4.2 Nuclear Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Structure of 9N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 5. Projection of the Continuum . . . . . . . . . . . . . . . . . . . . . . 5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Shell Model Embedded in the Continuum . . . . . . . . . . . . . . . . . . . . 5.3 Quantifying the Continuum in GSM with Projection . . . . . . . . . . . . . 5.4 Toy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Realistic Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 6. Hands-on Nuclear Structure and Decays . . . . . . . . . . . . . . 6.1 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 3 6 6 13 17 18 21 22 23 28 30 30 31 32 40 42 42 42 46 57 63 67 69 70 75 79 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 vi Chapter 1. Introduction Nuclear physics is a broad discipline ranging from the use of radioisotopes in medical appli- cations to probing fundamental symmetries with atomic nuclei [1]. Underlying this broad range of topics is a dependence on nuclear structure information. While experimental data exists for approximately 3000 known isotopes [2, 3], there are many more unknown nuclei [4]. Predicting behaviors for such unknown nuclei, nuclear structure models are required. In this work, we are focused on an Open Quantum System (OQS) approach which is implemented via the Gamow Shell Model (GSM) [5]. First, we will focus on defining general nuclear attributes important to the discussions within this thesis. 1.1 Nuclear Structure Overview Nuclei are self-bound clusters of nucleons which interact with each other to produce emergent phenomena. Some of these phenomena are universal to all nuclei, such as binding energy, while others, such as magic numbers and nuclear halos, appear only for particular nuclei [6, 7]. Nuclear instability must be properly represented since the mechanisms of decay are intrinsically tied to their structures. A general quantity used to explain nuclear instability with respect to particle emission threshold is the Q-value. This value denotes the energy associated with removing the in- dicated particle(s); for example, removing one neutron Qn, two protons Q2p, or an alpha particle Qα. Q-values for one and two-nucleon decays are equal to the negative separation energy [8]. Fig. 1.1 depicts the various Qn,p and Q2n,2p values for the lithium isotopic chain; where positive (5Li) and negative (6Li) values correspond to prompt particle emission or a bound state, respectively. Nuclear states with energies near or above threshold will be ideal OQS candidates as these systems can be dominated by coupling to decay channels. 1 Figure 1.1: Illustration of openness in nuclear systems and the emergence of exotic phe- nomena at energies closer to threshold. We schematically show various isotopes of lithium ranging from a proton-resonance 5Li to neutron-unbound 10Li to two-neutron halo 11Li. Year discovered is presented for each isotope. Describing nuclear structure requires proper treatment of all interacting nucleons in a many-body formalism, but many systems cannot be exactly calculated with current com- putational resources and require approximations. Depending on the region of the nuclear chart one is working in, the proper choice of model is essential, as some start from a more microscopic approach while others make assumptions to reduce dimensionality. In the lighter region of the nuclear chart (Z ≤ 20) the widely-used theoretical approaches are ab initio or configuration-interaction Shell Model (SM)-based methods [9]. While each are useful in their own right, in this thesis we will focus on a variant of the SM as outlined in Sec. 2.2. To calculate nuclei within the shell model, we define a nuclear core to which valence nucleons are added. Observables can then be calculated relative to this inert core. For example, calculated nuclei with positive Q-values relative to the core, or a larger subsystem, would indicate the presence of spontaneous nucleon emission for a chain of nuclei. These 2 One and Two-Nucleon Separation Energies for Lithium IsotopesEnergy (MeV)Mass Number ACorrelation Dominated particle emitting systems for a variety of emissions η can be calculated as Qη = BE(Nresidue, Zresidue) BE(N, Z) 0 ≥ − (1.1) using the binding energy of the residual nucleus and decaying system respectively. Chains of nucleon emitters exist along specific isotopic and isotone lines, and are identifiable with respect to a residue. For example, there is a well-known chain of proton-emitting nuclei at the N = 2 isotone, relative to a 4He core, with Q thresholds ranging from Q1p to Q4p, for 5Li and 8C, respectively. Nuclei existing along these ridges are unstable and their lifetimes (T1/2) are associated with the decay width of a state Γ: T1/2 = ¯h ln 2 Γ . (1.2) Nuclei existing in these regions can be especially challenging for some nuclear structure mod- els which do not naturally include decays, such as a traditional SM. This dilemma motivates the need for a formalism which treats both structure and continuum states explicitly, and in our case, is accomplished in GSM. 1.2 Closed versus Open Quantum Systems Quantum-mechanical systems can be divided into two groups: Closed Quantum Systems (CQS) and OQS. CQS-type problems treat an object as a completely isolated system whose properties cannot be affected by external factors. OQS-type problems allow for a system to be affected by its environment and can produce new emergent behaviors, contrary to CQS. Generally, one may recognize an OQS by identifying an isolated quantum system that undergoes an irreversible process after coupling to the environment [10]. OQS are featured 3 across physics from macroscopic (molecular) systems to mesoscopic (nuclear) systems and have shared phenomena in the form of resonances and scattering states. A specific example of coupling with the environment that causes alterations in the system considered is the decoherence of qubits in quantum computers [11]. In the nuclear physics context, a nucleus (the system) can undergo decay by emitting a nucleon (the irreversible process) after coupling to the continuum (the environment). OQS phenomena vary from system to system and are dependent on the environment to different degrees. The development of any techniques describing OQS can be extended in a variety of fields, but nuclei are able to produce emergent behaviors seen in larger systems while re- taining far fewer degrees of freedom. One such example of interdisciplinary OQS phenomena accessible in nuclei is superradiance [12] which was first discovered in quantum optics [13]. Nuclei also are unique due to the presence of different species of fermions, and the interac- tions between these fermions is the interplay between strong and electro-weak forces. An example of probing unique physics with nuclei is tackling the question of non-exponential decay [14]. Theories that consistently treat nuclear OQS systems will use a formalism incorporat- ing bound and scattering (continuum) states simultaneously [5, 15]. Experimentally, these OQS will have a resonance structure or other collective features that appear near or above threshold. For a more approachable definition of a resonance, one could define it as a specific energy which produces a substantial increase in the cross section of a nucleus, or the proba- bility of an interaction occurring. Thus, a resonance is a local band in an energy spectrum where a projectile can alter the structure of a nucleus without being bound to or knocking another particle from the target. Essentially, a resonance is a feature in a nucleus where the lifetime is extended beyond what one might typically assume to be short-lived in the 4 scattering process. Additionally, understanding this continuum coupling would allow bet- ter understanding into the preference in decays of unbound states [16] as a decay could be considered the time inverse of a capturing reaction. Although nuclear resonances like 5Li are the systems most commonly associated with OQS, in reality the entire nuclear landscape should be describable with one coherent frame- work. If one has a Hamiltonian capable of describing one nucleus, from an OQS perspective, the same Hamiltonian should also be able to describe any other nucleus by successive chains of operations like particle capture and decay [17]. A key region where these dynamics are most clearly at play are in the correlation dominated section of Fig. 1.1 around threshold. Here, as indicated by the equation Sn λn ≈ − − ∆n, separation energies are on the same order of magnitude as the single-particle (s.p.) fields λn and many-body effects like pairing ∆n [17], leading to competition between these effects. Emergence of other features like nu- clear halos are also associated with the competition between coupling to the environment and coupling between particles inside the system [18]. For this reason, we will explore various exotic phenomena using a continuum-informed model, GSM. 5 Chapter 2. Formalism The formalism of GSM is fundamentally similar to the implementation of the traditional SM. Where GSM differs is most obviously in its choice of basis, the Berggren basis [19], whereas standard SM codes might use a harmonic-oscillator (HO) basis. First, we outline the motivation for the Berggren basis and the necessary formalism. 2.1 Berggren Basis As already highlighted, an OQS naturally allows the formation of metastable states that require decay by particle emission so an OQS must include outgoing scattering states. We shall assume that outgoing states will not be affected by short-range nuclear forces and they propagate as a plane wave when the particle is far from the residual nucleus. Ordinary quan- tum mechanics tends to be formulated in a Hilbert space where Hamiltonians are Hermitian, but it is not straightforward to implement decay processes in this formalism. To address this tension, Gamow extended the formalism to the complex plane by introducing an imaginary damping term (Γ) to the energy to address α-particle emission [20]. The states in this ex- tended framework belong to a Rigged Hilbert space [21–23] which includes states that obey the desired outgoing boundary conditions. Rigged quantum mechanics is a natural frame- work for intrinsically irreversible processes arising from the Hamiltonian itself, as opposed to extrinsic irreversible processes which occur due to environmental coupling and describable in a Lindbladian approach [22]. The development of this formalism is supported by the assumption of a bulk exponential decay property where the complex energy and half-life are related by: E = E0 − i Γ 2 6 (2.1) with energy E0 and decay width Γ as discussed surrounding Eq. 1.2. Berggren expanded on this framework [19] by including Siegert’s resonance definition [24] and used the asymptotics for a neutral particle: Ol(kr) ∼ ei(kr − lπ 2 ), Il(kr) ∼ e− i(kr lπ 2 ) − (2.2) for outgoing and incoming waves respectively. By imposing the boundary conditions: u(0) = 0; u(a) O′l(ka) u′(a) Ol(ka) = 0, − or ˜u(a) I′l(˜ka) − ˜u′(a) Il(˜ka) = 0, (2.3) where a corresponds to the radius at which the nuclear potential is effectively zero (for neutrons) or has long-range Coulomb behavior (for protons), the resonant nature of the system is respected [19]. Furthermore, this formulation modifies the general properties of our complex resonant states: ˜k = k∗, − ˜u(r) = u∗(r), and k = κ iγ − (2.4) with κ and γ being real and imaginary momentum terms, respectively. It should be noted that there are challenges when using this formalism, mainly that the states can increase exponentially and are not normalizable. Various approaches exist for normalizing such states, see Ref. [5] for details. From this point onward, our complex energy will be represented as E and complex momentum k, and the conjugation within the Berggren basis will be represented by a tilde. 7 2.1.1 One-body states For the current formalism, we will work in a spherically symmetric basis. This maintains many symmetry advantages, such as easy handling of the angular momentum and spin. Decomposing the one-body wave function is standard (including spin χs,ms) ψ(r, θ, ϕ) = R(r) Yℓ,m ⊗ χs,ms (cid:2) j m . (cid:3) (2.5) Taking advantage of the system symmetries for quantum numbers, the general one-body wave function can be rewritten as [8] ψ(n, ℓ, j, mj, ⃗r) = uψ(n, ℓ, j, r) r mℓ,ms (cid:88) ℓmℓsms jmj | ⟩ Yℓmℓ ⟨ (θ, φ)χs,ms; (2.6) where the angular terms are decomposed into the possible components with correct angular momentum coupling - e.g. those with ⃗j = ⃗ℓ + ⃗s and mj = mℓ + ms [8]. Due to the non-Hermitian nature of the Gamow resonant states (Eq. 2.4) in the com- plex plane, we illustrate possible solutions to the time-independent Schr¨odinger equation in Fig. 2.1. The parameters used can be found in Tab. 2.1 and follow with Ref. [5]. Bound wave functions are still normalized in the conventional sense as they are square integrable. An important distinction between different states is their behavior in the asymptotic region. For example, the narrow resonance in Fig. 2.1a appears to have a similar profile to the well- bound wave function, yet when extending to a larger radius, we see oscillations appear in its tail (seen in Fig. 2.1b). Conversely, the loosely bound wave function appears to not be well-localized, but we see its convergence at a larger radius. Broad resonances have extended localizations before being quickly dominated by their diverging behavior. Lastly, antibound 8 Figure 2.1: (a) Different classifications of one-body wave functions with real (solid lines) and imaginary (dashed lines) parts. (b) Radial extension of one-body wave functions (similar to (a)) for well bound, loosely bound, narrow resonance, and antibound states. (c) The locations in the complex plane of the presented wave functions in panel (a). Table 2.1: Parameters used to generate the corresponding s.p. states in Fig. 2.1. All wave functions are for a neutron in a one-body potential of A = 12, diffuseness d = 0.65 fm, and radius R0 = 3 fm along with the specified Woods-Saxon depth V0 and Spin-Orbit potential strength VSO. Additional information can be found in Chapter 2 of Ref. [5]. Type Well Bound Loosely Bound Narrow Resonance Broad Resonance Antibound V0 (MeV) VSO (MeV) State 0p3/2 1s1/2 0d3/2 0d3/2 1s1/2 7.5 0 7.5 7.5 0 50 43 63 50 40 1 k (fm)− (0, 0.7799) (0, 0.0149) (0.0918, -0.0003) (0.4202, -0.0693) (0, -0.0780) (also called virtual) states have no sense of localization and diverge immediately. These states naturally emerge in a potential well without a barrier, leading to an enhancement in the cross section [25, 26]. States can be more clearly separated by using the energy-momentum relation E k2. ∝ 9 05101520r(fm)−1.00−0.75−0.50−0.250.000.250.500.751.00WaveFunction(a)Exampleofs.p.wavefunctionswellboundantiboundbroadresonancelooselyboundnarrowresonance−0.20.00.20.40.60.8Re(k)(fm)−1−0.20.00.20.40.60.8Im(k)(fm)−1wellboundantiboundbroadresonancelooselyboundnarrowresonanceθ=−π4line(c)Exampleofs.p.poles050100r(fm)−0.20.00.20.40.6WaveFunction(b) Bound states (positive imaginary k) are real and exist on the negative energy axis while scattering states, which are also real, exist on the positive energy axis. These states lie on the first, physical, Riemann Sheet. Resonant states and antibound states are complex energies and lie on a second, sometimes called non-physical, sheet. These states can be accessed by a number of methods including a deformed contour. 2.1.2 Berggren Completeness Relation In order to define the completeness relation for GSM, we first demonstrate the minor differ- ences between a Berggren basis and the more traditional real-energy (Hermitian) complete- ness relation from basic quantum mechanics. Generally, states in a Hermitian Hamiltonian will be square integrable and normalizable. For non-square integrable states, a Dirac δ normalization may be imposed with the condition 2πC+(k)C−(k) = 1 where C±(k) are constants determined by Jost functions J ±(k) = 2ikC∓(k) ± which can be related to the Wronskian of u(k, r) [5, 27, 28] J ±(k) = u(k, r)u±′(k, r) u′(k, r)u±(k, r). − (2.7) (2.8) Here, u±(k, r) represent the solutions to the radial Schr¨odinger equation following incoming or outgoing boundary conditions. Details can be found in Ref. [5] for both neutral and charged particles. Since all states are orthonormal, a completeness relation can be defined. We focus on the 10 simpler case of an ℓ = 0 neutron where we consider the integral [5] I(K) = K 0 (cid:90) u(k, r)u(k, r′) 2πC+(k)C−(k) dk. Due to the momentum definition in Eq. 2.4, conjugation of states are invariant to k (2.9) k → − thus u(k, r) = u( k, r), − u±(k, r) = u∓( k, r), − and C±(k) = C∓( − k). (2.10) By taking advantage of these symmetries, integration bounds can be rewritten as [5, 27, 28] I(K) = K K (cid:90) − u+(k, r)u(k, r′) 2πC−(k) dk. (2.11) Knowing bound state poles only lie on the imaginary k axis in the complex k-plane, this integral can be simplified with Cauchy’s theorem. Introducing a contour integral along the upper half plane separates the residues from the non-resonant scattering continuum. Details can be found in Ref. [5, 27, 28], but the result is the Berggren completeness relation [5] un(r)un(r′) + + (cid:90)L n (cid:88) uk(r)uk(r′)dk = δ(r r′). − (2.12) The contour L + corresponds to the complex-energy scattering states, and the discrete un(r) are bound and resonant states inside L +. Fig. 2.1 presents the different components which make up the poles of the S-matrix and are present in Eq. 2.12. 11 2.1.3 Numerical Implementation Retaining the representation of the continuum in practical applications can be difficult as current computers cannot represent an infinite continuum of states. The contour integral contributions to the completeness relation in Eq. 2.12 is discretized as u(k, r)u(k, r′)dk + (cid:90)L Nd ≃ (cid:88)i=1 ui(r)ui(r′) (2.13) where ui(r) = ∆kiu(ki, r) and ∆ki is the discretization step weight at specific momenta [5]. By ensuring that these ui(r) are orthonormal, the discretized Berggren completeness relation (cid:113) can be written as Nd un(r)un(r′) + ui(r)ui(r′) n (cid:88) (cid:88)i=1 δ(r r′) − ≃ (2.14) In general, the analytic form of Eq. 2.12 would imply any arbitrary contour choice is sufficient as long as it encompasses the desired poles. With the discretized form, contours should be chosen such that they are not too close to the pole where numerical noise might lead to unwanted instability [5, 29–31]. However, the model space dimension d is directly tied to the number of discretization points chosen by N Nv s d ∝ (2.15) where Ns and Nv are the number of states and number of valence nucleons respectively [32]. A tension exists between the number of points required for good numerical completeness and the desire to keep model spaces within a calculable range. To optimize the computational resources required, one can reduce the number of discretization points Nd for the contour 12 or minimize the number of states considered Nv. By carefully choosing our contour for each partial wave (ℓ, j), we can minimize the number of states needed while maintaining computational stability and fidelity. 2.2 Gamow Shell Model For the Berggren basis to be useful in a SM or configuration-interaction implementation, the one-body completeness relation must be generalized to a many-body form. This is a result of SM wave functions being many-body vectors, not simple one-body wave functions. In order to make the familiar Dirac notation useful in the GSM context, we must make some modifications. 2.2.1 GSM Completeness Relation In order to be more amenable to SM formulations, we will work with N -body Slater deter- minants (SDs) Ψn | ⟩ = ψi1, ..., ψiA⟩ | = (2.16) ψi1(⃗r1) ... ψiA(⃗r1) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ... ... ψi1(⃗rA) . . . (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ... ψiA(⃗rA) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ψi built from the orthonormalized one-body states ( | ⟩ ) that make up our Berggren ensemble. Therefore, any SD built from a set of states ψk⟩ | and scattering s.p. states from the poles and contour. The generalization of the one-body may contain the bound, decaying resonance, completeness relation to a many-body representation is [5]: ˜Ψn ⟩ ⟨ | ≃ I, Ψn n | (cid:88) (2.17) and the derivation starting from the real-energy Newton completeness relation can be found in Ref [5, 27]. Different state conjugations ( ˜Ψ) are implemented than in the Hilbert space 13 formulation as discussed in Eq. 2.4. These tilde eigenstates represent effects from the time- reversal operator [33], and the physical connection is that particle-capturing states are rep- resented as bras and particle-emitting states as kets. Conjugate states belong to the dual space vectors, see Ref. [5, 34]. By using Eq. 2.17, we can check our system through the sum over all squared amplitudes of each configuration (not the squared absolute values like in the traditional SM) c2 n = 1. n (cid:88) This serves as a useful check to ensure any calculations are being handled properly. 2.2.2 GSM Hamiltonian The general form of the GSM many-body Hamiltonian [31] is H = Nval (cid:88)i (cid:34) ⃗p2 i 2µi Nval + Uc(i) + (cid:35) Vi,j + (cid:88)i=1,j>i (cid:20) ⃗pi⃗pj Mc (cid:21) (2.18) (2.19) with Nval being the number of valence nucleons and µi and Mc being the reduced mass of the nucleon and mass of the core respectively [5]. One-body effects are accounted by the core-nucleon potential Uc and two-body effects via valence nucleon interaction Vi,j. To elimi- nate center-of-mass motion energy contributions, the Cluster Orbital Shell Model framework (COSM) is used in Eq. 2.19 to provide a translationally invariant framework [5, 31]. The following Woods-Saxon (WS) form is used for the core-nucleon potential Uc(r) = V0f (r) 4Vℓs − 1 r df (r) dr ⃗ℓ · ⃗s + UCoul(r) (2.20) where f (r) = [1 + exp (r − R0)/a]− − 1. For neutrons, the one-body potential can be simpli- 14 fied by setting UCoul = 0 and obtaining Uc(r) = − 1 + exp 4Vℓs − 1 r V0 r (cid:16) R0 − a (cid:17) exp r R0 − a (cid:16) 1 + exp a r (cid:17) R0 − a (cid:104) (cid:16) (cid:17)(cid:105) ⃗ℓ 2 ⃗s · (2.21) with ⃗ℓ · ⃗s = j(j + 1) ℓ(ℓ + 1) − − 3 4 . Similarly, the same form can be used for protons along with the addition of the Coulomb term for a uniformly charged sphere of radius Rp r Rp 3 − (cid:18) (cid:16) 2 (cid:19) (cid:17) Ze2 2Rp Ze2 r Rp r ≤ . r > Rp (2.22) UCoul (r) =    There are many different choices for two-body potentials in GSM, but in the context of this thesis, we only use the valence nucleon-nucleon interaction V = Vc + VLS + VT + VCoulomb (2.23) with the central (c), spin-orbit (LS), tensor (T ), and Coulomb terms. The first three terms are based on the Furutani-Horiuchi-Tamagaki (FHT) force [35, 36]. These are based on spin-isospin projectors (ΠST ) Vc(r) = V 11 c f 11 c (r)Π11 + V 10 c f 10 c (r)Π10 + V 00 c f 00 c (r)Π00 + V 01 c f 01 c (r)Π01, VLS = VT (r) = Sij ⃗L · (cid:16) ⃗S LSf 11 V 11 LS(r)Π11, (cid:17) VT 11f 11 t (r)Π11 + V 10 T f 10 t (r)Π10 , (cid:105) (cid:104) where ⃗L is the relative orbital angular momentum, ⃗S = ( ⃗σi + ⃗σj)/2, and Sij = 3(⃗σi 15 (2.24) ˆr)(⃗σj · · ˆr) ⃗σi · − ⃗σj [37]. The distance between nucleons i and j is represented by rij and ˆr = ⃗rij/rij. Fundamentally, this interaction is based on a sum of Gaussians approach to include effective representation of a long-range term to represent pion exchange potentials and a short-range hard core term [37]. When using the FHT interaction, we note that three of the seven available terms (V 10 c , V 01 c , and V 10 T ) appear at leading order in effective field theory arguments with the remaining being higher order terms [31]. When attempting to adjust these parameters for optimiza- tion, including only leading order [31] or all [37] terms may provide better reproduction of spectra, depending on the application. Regardless of the inclusion of higher order terms, this interaction has been used to successfully calculate different nuclear systems [31, 37–40]. 16 Chapter 3. Spectroscopic Factors and the Con- tinuum Experimentally probing internal features of nuclei is a difficult process, but the data obtained can elucidate significant physical insight. Measured cross sections of different direct reactions (without forming any intermediate state) is one such tool that can be used to study nuclear phenomena. Some quantities discussed commonly by nuclear physicists are non-observables (or sometimes called unobservables) and are instead dependent on the chosen framework, see Ref. [41]. Single-particle energies are such quantities originating from a mean-field (shell model) level and dictate the ordering of energy levels for a specific calculation. In the context of compatible models, these non-observables can be incredibly useful especially if tied to an approximate experimental value. Spectroscopic factors (SFs) are another non-observable quantity which are inferred through ratios of cross section data. The SFs themselves can be derived via the number operator [8] under Racah [42] and Wigner’s convention, yet for the current discussion, it is more useful to discuss SFs in the context of radial overlap integrals. The radial overlap integral can be expressed as [5]: Ia bc;ℓj(r) = 1 √2Ja + 1 ΨJa a (cid:28) (cid:90)(cid:88)B (cid:103) ) B a†ℓj( (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) Jb Ψ b (cid:29) rℓj ⟨ | u B⟩ where we consider the overlap between a decaying nucleus a and its products a (3.1) b + c, → ultimately summing over all s.p. basis states . SFs are then defined as the squared norm B of Eq. 3.1 [5]: S = 1 2Ja + 1 JA A Ψ (cid:28) (cid:90)(cid:88)B (cid:103) 17 B a†ℓj( (cid:13) (cid:13) (cid:13) ) (cid:13) (cid:13) (cid:13) Ψ JA A − 1 − 1 2 . (cid:29) (3.2) An additional advantage to these definitions are that using a complete set of Berggren basis states results in the SF being independent to the choice of basis, whereas other forms have basis dependence [8, 43, 44]. These SFs can be used to calculate theoretical cross sections for direct reactions [5, 43–45] and for GSM specifically Snℓjmj σ nℓjmj s.p. . σ = (cid:88)nℓjmj (3.3) These formulations are tied to the notion of the spectator approximation or the approxima- tion that reactions are simply an A − 1 system with a lone nucleon outside. This chapter contains my contributions to a larger work focusing on the impact of the continuum on SFs [46]. 3.1 Discrepancy Between Experiment and Theory Despite being a non-observable quantity, SFs are inferred using the ratio of experimental to theoretical inclusive cross sections by [44, 47, 48] Rs = σexp σth . (3.4) Clearly, model dependence enters the calculation depending on how σth is obtained. Regard- less of the choice of the model, one could reasonably assume that these ratios would deviate about some mean value for all reaction probes across different nuclei for the chosen model. Any deviations in could be attributed to poor reproduction of specific physical behaviors within the chosen model. Indeed, this exact behavior is seen for the electron-induced knock- out reactions e, e′p for the stable nuclei considered in Fig. 3.1. However, in a sequence of papers [44, 47, 50] it was shown that, for knockout reactions in particular, there is a system- (cid:0) (cid:1) 18 Figure 3.1: Compiled data of different Rs using data in [44, 49]. We specify select nuclei (tri- angles) to highlight specific features, yet all available nuclear data can be found in Ref. [44]. 19 atic linear dependence in Rs(∆S), shown in Fig. 3.1. The quantity ∆S is calculated as the difference between the separation energy of the particle removed and its opposite species, a representation of isospin asymmetry [44, 51, 52]. For proton removal ∆S = Sp Sn and for − neutron removal ∆S = Sn Sp. − Focusing on either extreme of ∆S, one can notice commonalities between the highlighted nuclei in Fig. 3.1. For exotic nuclei, like 9C, the values of Rs for proton and neutron knockout span a large range unlike the stable nuclei like 12C. Additionally, when considering only a particular type of reaction, say neutron knockout, we can identify two isotopes 32Ar and 46Ar with vastly different Rs seemingly tied to whether the removed neutron was in the minority or majority nucleon species [51, 52]. Lastly, some nuclei such as the halo nucleus 15C [53] are not exotic nuclei with strong isospin asymmetry, yet 15C could be considered as a core-spectator system of 14C+n with the core being a magic nucleus of N = 8. The source of the trend in Rs is still under debate, but it is at least partially due to structural dependence on the nucleon removed. Specifically, the validity of the spectator ap- proximation for exotic states is the center of our focus. For many cases, this approximation is reasonable particularly when removing weakly-bound nucleons from their larger system [45]. One can recognize that a large overlap indicates that the parent and daughter maintain a largely consistent structure independent of the effects of the removed particle. Removing well-bound nucleons on the other hand may require strong alterations to the original parent configuration, making the daughter’s structure different and producing a small overlap. In each case, the nuclei with strong differences in the proton and neutron separation energies have Rs values either near one or suppressed strongly towards zero. Nuclei with large ∆S, tend to have a higher asymmetry of protons to neutrons and, in the case of light nuclei, these would be deemed “exotic” due to their shorter half-lives. At the extreme ends of the trends 20 identified in Fig. 3.1, many of the nuclei are located near dripline regions. 3.2 Continuum Impact on Spectroscopic Factors Considering nuclei near or beyond the driplines, an OQS formulation is essential due to con- tinuum dependence of such systems. In GSM, these SFs are determined via the spectroscopic amplitude [54, 55] Aℓj(kp) = ⟨ ΨA|| a+ ℓj(kp) ΨA || √2JA + 1 1⟩ − ; 2 ℓj = S 2 ℓj(kp), A (cid:90)(cid:88) (3.5) which describes similarity of a parent nucleus to its one-nucleon-removed daughter, just as the traditional SM formulation in Eq. 3.2. The formulation in Eq. 3.5 improves the structure representation from previous SM calculations since it includes physics associated with unbound states. To underscore why it is an improvement, we can contrast with known issues with well-localized bases that work for well-bound nuclear states, but not for weakly- bound systems without modification. An example focuses on the descriptions of 9 − 11Li with localized bases. For these nuclei, localized bases overbind the unbound 10Li and the 2n halo 11Li despite reproducing the structure of the well-bound 9Li subsystem [56, 57]. In this case, the importance of long-range dynamics has been demonstrated to resolve these issues [58]. These problems are addressed in GSM which can describe the unbound nature of the dripline systems like 10Li [31]. Although this is an oversimplified picture, the long distance wave function behavior can inadvertently be truncated using a localized basis, and this lack of long range components is thought to bind the system [56, 57]. In essence, we can better inform and represent states when choosing a basis that allows for natural occupations and avoids any possible discontinuities by treating all possible states on the same footing. Using the Berggren basis, this can be accomplished while including continuum degrees of freedom 21 in GSM which is contrary to the lack of a genuine continuum in a CQS representation. 3.3 Parameters and Model Space To probe the effects of the continuum on SFs, we perform a series of calculations for specific nuclei which are likely to be dependent on the continuum such as 8C, 8B, and 9C. The choice of these nuclei is based on recent experimental evidence from Ref. [49]. Although GSM does not have explicit control over continuum coupling strength, we enforce openness of the system by applying truncations on the number of valence particles allowed in the scattering continuum. For our calculations, we start with a HO basis (no continuum coupling) in a p-shell model space and ultimately open the system using a full Berggren basis calculation in a psd-model space with all four valence particles allowed to occupy the continuum for A = 8 nuclei. Berggren-basis calculations using a resonant 0p3/2 basis state were also performed and are denoted with the subscript “res” to better capture effects tied to width [31]. The purpose of using a resonant pole is to highlight basis independence of the calculations. To minimize the effect of core excitations, we selected 4He as the system core due to its tight binding. The GSM Hamiltonian is as defined in Sec. 2.2.2 using the standard WS with spin-orbit one-body interaction and the FHT two-body interaction. We used the calibrated parameters for ps-shell model space nuclei from Ref. [31] when performing the HO basis (SM analog) and ps Berggren basis calculations. We performed additional calculations with a psd- shell model space to explore the impact of higher ℓ states. For the ℓ = 2 states the, same one-body parameters were used as in Ref. [31], but we readjusted the only non-zero two-body term for homogeneous valence nucleon systems (V ST η = V 01 c ) until the ground state (g.s.) energy of 8C was within 0.2 MeV of the experimental value [2]. This is justified as the goal of this work is to best reproduce the 8C structure to adequately address continuum effects on 22 SFs. After determining our parameters for the two different model spaces in the proton-rich case, we applied the same parameters to the mirror nuclei 8He, 8Li, and 9Li. In all cases, the A = 9 nuclei are well-bound systems and were calculated allowing only two particles to occupy continuum states (Ncont = 2). Since we focused on the continuum impact on the daughter structure, the A = 8 systems were calculated with varying numbers of particles in the continuum (up to the maximum of Ncont = 4). The model space used in this work [46] included pole states for 0p3/2, 0p1/2, and (when applicable) 0d5/2. For Berggren basis calculations, we use the standard three segment con- tour form of Eq. 2.17 and each segment was divided into five Gauss-Legendre points. The segment geometry was chosen to be kpeak = 0.3 fm− 1, kmid = 0.4 fm− 1, and kmax = 4 fm− 1 with imaginary components being increased when needed for resonant systems. Fig. 3.2 shows the resulting spectra for each nucleus in the full possible model space, and the ener- gies appear to be reasonably well reproduced despite the Coulomb energy difference between mirror nuclei. 3.4 Impact of Continuum Coupling Calculations As discussed above, we study the effects of continuum coupling on wave function fragmen- tation for A = 9 nuclei through knockout of well-bound and weakly-bound nucleons. For isospin-asymmetric systems, the minority species nucleons will tend to be the well-bound nucleons while the majority species nucleons will be weakly-bound or even unbound. All SFs calculated are for parent g.s. to daughter g.s. transitions. 3.4.1 Minority Species Nucleon Removal First focusing on the minority species removal, we can compute the SF for 9C 3/2− 8C → 0+ + n using the four different model spaces outlined in Sec. 3.3. HO-SM calculations only in- (cid:0) (cid:1) (cid:0) (cid:1) 23 Figure 3.2: Energies of 8,9C, 8B, 8He, and 8,9Li calculated in GSM with the psdres-model space. Figure taken from [46]. cluding p-poles produced a SF of S2 = 0.86 whereas the GSM values range from S2 = 0.67 for ps-shell to S2 = 0.48 for psdres-shell calculations. Similarly, for the knockout in 9Li 3/2− 8He → 0+ + p the SF ranges from S2 = 0.85 for HO-SM to S2 = 0.48 for psdres- (cid:0) (cid:1) (cid:0) (cid:1) shell calculations. All SF values for the knockout of well-bound nucleons are presented in Table 3.1. To understand the reduction of SFs with increasing continuum coupling, we focus on the changing structure across each model space. By comparing the wave function decomposition between HO-SM and GSM-psd (Ncont = 4) in Fig. 3.3a, we can deduce the changes from continuum coupling. For 8C, we note the strong shift between HO-SM and GSM calcu- lations where the pole locations shift slightly in energy, but the squared amplitude of the GSM occupations drop significantly. Moreover, the decrease in squared amplitude for pole configurations is then redistributed among continuum states. This change is unsurprising as 8C is a system beyond the proton dripline, which decays by two steps of 2p emission, and 24 201510505E−Ec(MeV)8C0+18B2+19C3/2−11/2−18He0+18Li2+19Li3/2−11/2−1ExperimentTheory Figure 3.3: Squared amplitudes of configurations for the nuclei indicated using HO-SM and GSM. Shown are the split contributions from the GSM pole space and scattering continuum along with the occupations in the simpler HO-SM scheme. The energy presented is defined as the sum of s.p. energies for valence nucleons in each configuration relative to the lowest energy configuration. Figure taken from Ref. [46]. 25 0.00.20.40.60.8HO-SM0102030405060700.00.20.40.60.80102030400102030408C9C8He8Li9Li8B(a)(b)(c)(d)(e)(f)E (MeV)squared amplitude GSM (pole)GSM (cont) Table 3.1: Spectroscopic factors for the knockout of a p3/2 nucleon from the 3/2− g.s. of 9C and 9Li to the g.s. of 8C and 8He and knockout of a p1/2 nucleon to the g.s. of 8B and 8Li. The experimental neutron and proton separation energies [2] are shown (in MeV). The Model column indicates the particular model space for the calculation as described in Sec. 3.3. Ncont is the number of particles allowed in non-resonant continuum of A = 8 nuclei. The last row shows the contribution from the resonant 0p3/2 state. Table taken from Ref. [46]. Model Ncont 8C 9C 8Li Sn = 14.22 Sp = 13.94 Sp = 1.30 Sn = 4.06 8He 9Li 9Li 9C 8B → → → → HO-SM GSM-ps GSM-psd GSM-psd GSM-psdres 0 3 3 4 4 0.86 0.67 0.60 0.48 0.48 0.85 0.67 0.67 0.65 0.64 0.95 0.98 0.89 0.89 0.84 0.96 0.98 0.88 0.88 0.85 an ideal candidate for an OQS description. For the sake of illustration, we also provide a schematic view of how one may consider the partially-summed continuum configurations in Fig. 3.4. Partially-summed configurations are where all continuum states (n excitations) for each partial wave are summed together into one general scattering contribution. For example, in our case, 0p3/2 is a resonant pole, and the p3/2 contour is discretized by 15 points numerically, one will have 15 different configurations of π 0p3/2 3 pscat 3/2 from π 3 1p3/2 to π 0p3/2 3 16p3/2 . The corresponding (cid:16) (cid:17) squared amplitudes for each distinct configuration can be summed together to provide a (cid:16) (cid:17) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17) (cid:16) 0p3/2 (cid:16) total quantity representing discrete poles coupled with continuum states π 3 0p3/2 pscat 3/2 . (cid:17) Based on the schematic view, we illustrate that the change in the dominant HO-SM con- (cid:17) (cid:16) (cid:16) figuration π 0p3/2 4 = 95% dissipates into a distribution of continuum states. Similarly, we see the same dissipation occur in the mirror nucleus 8He in Fig. 3.3d. While changes (cid:16) (cid:17) in configurations between HO-SM and GSM may not be indicative on their own, when we 26 Figure 3.4: (Right) Example of the largest four partially-summed configurations contributing to the (Left) squared amplitude histogram for 8C [46]. We present a generic WS potential to illustrate bound versus unbound s.p. states within each configuration. The histogram bar color and meaning is the same as Fig. 3.3. consider the location and structure of the poles in the parent nuclei 9C and 9Li, we note a three dominant pole configuration structure (in the HO-SM and GSM poles) whereas 8C and 8He only have two dominant pole configurations. This alteration and reduction of dominant configurations thus is the driving factor for the suppression of the SFs in our study. 3.4.2 Majority Species Nucleon Removal We now consider the effects on SFs when removing majority species nucleons from 9C and 9Li to produce 8B and 8Li respectively. As shown in Table 3.1, increasing the available number of continuum states in our system does not produce a significant change to the SFs. In fact, the resulting SFs for weakly-bound nucleon removal saturate at a value near one, indicating that the assumption of a core-spectator system may be sufficient due to the similarities with an independent particle model approach. Focusing on the configurations, we see in Fig. 3.3 that parent and daughter share the 27 Figure 3.5: Example of largest 4 partially-summed configurations contributing to the squared amplitude histogram for 8B [46]. This figure follows the same labeling scheme as Fig. 3.4. same three dominant pole configurations, indicating less structural changes due to favoring pole-space occupations. Additionally, the dissipation of these pole states into continuum contributions is mitigated, as the daughter systems from majority nucleon removal will tend towards stability (where continuum contributions are unlikely). Again, if we consider a schematic example for 8B, shown in Fig. 3.5, we note that only one of the dominant four partially summed configurations contains direct continuum states. 3.5 Conclusions Using the GSM formalism, we have systematically explored the effects of continuum cou- pling on SFs. We have demonstrated [46] that continuum coupling contributions can pro- duce strong suppression, or quenching, of SFs when removing well-bound (minority species) nucleons. Sources of this difference can be attributed to the appreciable alterations to config- urations between parent and daughter nuclei. Specifically, when removing a minority species nucleon, the system moves in the direction of the dripline where continuum effects will sig- 28 nificantly alter the configurations. Simultaneously, we have shown that the effects of the continuum on SFs when removing weakly-bound (majority species) nucleons is rather weak and quickly saturates to a value not associated with SF suppression. Removal of majority species nucleons instead will push the remnant nucleus towards stability where continuum dominated configurations will be least likely to have substantial occupation. Continuum coupling effects on SFs were also shown to be qualitatively consistent with GSM using another model, Shell Modle Embedded in the Continuum (SMEC) [46]. The level of openness in SMEC was controlled by altering the number of decay channels avail- able, a different mechanism compared to GSM’s indirect basis truncation approach. Further investigations into the isospin dependence of nucleon-nucleon interaction strength is needed along with probes into larger angular momentum (ℓ > 3) systems which can also impact nucleon interactions. 29 Chapter 4. Description of 9N With the opening of the Facility for Rare Isotope Beams (FRIB) and improvements and innovations to existing radioactive ion beam facilities, nuclear experimentalists will be able to probe further into exotic regions of the nuclear chart [4, 59]. These exotic nuclei will have larger imbalances in the number of their constituent protons and neutrons and will be found closer to the nuclear driplines, as already demonstrated in recent work [60]. For lighter nuclei, the limits to the binding of these asymmetric nuclei are well defined experimentally since the driplines are known; however, beyond the dripline the limits to nuclear existence are not known experimentally and its definition is subject to debate [61]. This chapter contains my contributions to studying the structural and decaying properties of the extremely proton-rich nucleus 9N that has been observed recently [62]. 4.1 Particle-Emitting Nuclei For the purposes of this work, we define nuclear existence as the minimum time required to form a nuclear structure. An estimate of this time can be obtained by determining the period of a nucleon that traverses a nucleus of a given size, which for A 10 is roughly T1/2 ≈ 4.5 MeV [63]. If a nucleon can traverse a nucleus within this ≈ 10− 22 s or a decay width of Γ ≈ temporal boundary, then mean-field formation can occur which is an assumption for many nuclear structure models. Another benefit to this definition is that nuclear interactions which produce resonances or bound states should lead to some alteration of an incident particle. If the length of interaction time within the compound nucleus is too short, then it is virtually indistinguishable from a scattering state or a background continuum. Nuclear resonances exist between the driplines and this ephemeral (diffuse) region of the nuclear chart, and better probing these systems experimentally can give better insight into the edges of the 30 nuclear landscape. These ephemeral nuclei located beyond the dripline will primarily decay by (multi-)nucleon emission. The lightest section of the nuclear chart contains many nuclei that decay by nucleon emission and favor decay by steps of one- or two-proton (p or 2p) emission until ending at a particle-bound residue [64, 65]. Many of these nuclei, which decay by nucleon emission, have been observed, and more are likely to be discovered [66]. Currently, the list of nuclei which decay by emitting 3p from the ground state are: 7B, 13F, 17Na, and 31K [67–70] and 4p in total are: 8C and 18Mg [71, 72]. Although every nucleus is unique, these nucleon emitters share similar decay behavior. Due to nucleonic pairing, odd-Z proton emitters will favor single-proton emission while even-Z proton emitters tend to favor 2p emission [64, 65]. Focusing on a particular isotone chain (N = 2) we can better understand these general effects. Starting from a 4He core, one can build these nuclei by adding one proton (5Li), two protons (6Be), three protons (7B), and four protons (8C). This chain is of particular interest as 8C is a system with half of its nucleons in the continuum coupled to an α core. One may reasonably wonder where the boundary of proton emitters meets its ambiguous end and if there are any resonant nuclei further up this isotone chain? To address this question, we presented theoretical predictions of the structure of 9N, the next nucleus in the N = 2 chain and a 5p emitter, along with experimental observation in Ref. [62]. 9N is an interesting nucleus as over half of its constituent nucleons sit in the continuum and can be described as a 8C+p system. 4.2 Nuclear Features As we have highlighted, the formalism used in GSM naturally includes various complex- momentum poles of the S-matrix seen in Fig. 2.1. These resonant states are mathematical 31 constructs used to describe physical phenomena like resonances and for clarity we note the mathematical and physical distinction between the two terms. Moreover, careful defini- tions of features arising from nucleon interactions are paramount, as some nuclear structure features cannot be classified as resonances or bound states such as antibound states. Examples of these features can be seen even in simple systems like the dineutron and diproton. Although the dineutron (a system of two neutrons) is not bound, it is also not a resonant or scattering state (although closely related to the latter) [25, 73, 74]. Instead, the dineutron is classified as an antibound (or virtual) state or a state with exponentially increasing asymptotics as shown in Fig. 2.1 mathematically. Such states arise naturally in S-matrix formalism and manifest as a enhanced cross section and produce a large positive value of the phase shift derivative dδ/dk near zero energy [25]. Similarly, a diproton system is neither a bound state nor resonance, and is classified as a subthreshold resonance [73, 75]. Subthreshold resonances exist in the complex-k plane below the π/4 line and produce an enhanced scattering length like an antibound state. Both subthreshold resonances and antibound states can be associated as final state effects. We will focus on this distinction between these scattering features and resonances in 9N and 9He. 4.3 Structure of 9N 4.3.1 Prediction of 9N Structure As discussed in Sec. 4.1, 9N is three neutrons away from the lightest bound nitrogen isotope and is reasonably assumed to be a 8C+p resonance. To provide meaningful theoretical predictions, we built 9N as a 5p + α system and used the same interaction parameters and model space as Sec. 3.3 [46] to ensure good reproduction of the 8C g.s., without any 32 Figure 4.1: Calculated spectra using GSM. 8C and 9N(a) calculations use the same model space as Sec. 3.3 and serve as initial predictions. 9N(b) uses a modified model space to produce the 1/2+ state after comparison with experimental data. experimental information for 9N. We then applied these parameters to 9N and searched for a variety of possible J π states. Ultimately, we obtained converged predictions as shown in Fig. 4.1 which could be compared with experimental data. The experimental data did not contain enough statistics to provide constrained fits [62] as shown in Fig. 4.2. In particular, only a single-peak fit could be obtained with the available statistics (Fig. 4.2a), yet this required setting the magnitude of background event to 14% or lower [62] (an unreasonable assumption [76]). Consequently, we used GSM to provide new calculations with information from other nuclei to better constrain the experimental results. Specifically, we turned to the mirror partner 9He [77–82]. 9He belongs to the N = 7 isotone chain, which is known to contain low-lying 1/2+ and 1/2− doublets for nuclei with low proton numbers, such as 11Be [83]. In 11Be, an intruder s- wave state produces the doublet 1/2+ and 1/2− states which are bound and localized [83]. In 33 Figure 4.2: (a) Single-peak fit using an ℓ = 1 R-matrix lineshape and predicted spectra using GSM overlaid as blue vertical lines (from Fig. 4.1). (b) R-matrix lineshape using the refined GSM energies as vertical orange lines (discussed in Secs. 4.3.2 and 4.3.3) and the corresponding GSM refined energies. Modified from Ref. [62]. 34 246810121416(MeV)5pQ0102030Counts(b)00.10.2Efficiency0102030Counts(a)0.39N →8Cg.s.E(1/2-)E(3/2-)E(1/2-)E(1/2+)E(3/2-) 9He however, only the energy of the 1/2− state has been consistently measured, albeit with some disagreement on its width [77–79]. In contrast to 11Be, the presence of the 1/2+ feature in 9He does not have a uniform consensus and only some works indicate its existence [77, 79–82]. We focus on investigating the exotic features in 9He and the implications for the structure of 9N in the next section. 4.3.2 Refinement of 9N structure using 9He Despite using a model space which included continuum s-wave states, initial searches for a 1/2+ state were unsuccessful for both 9N and 9He. In the case of 9He, previous Berggren basis calculations were able to produce a 1/2+ state, yet they did so by including an additional 1s1/2 basis pole. By using this additional pole in their calculations, the 1/2+ state manifested as an artificially bound state [37, 63] and precluded these works from determining if 9He was a genuine state or antibound feature. It should be noted that it is theoretically possible to obtain the same results without including the 1s1/2 basis pole, but it requires a very dense discretization of the s1/2 contour to meet the unitarity condition [40]. Moreover, it has been shown that including the bound 1s1/2 pole in the GSM basis can help convergence rates and accuracy for bound many-body states, yet many-body states with antibound behavior will not uniformly benefit from this speedup [30]. Great care is warranted when focusing on the unbound 1/2+ states to avoid improper handling of fragile features such as antibound states. Since reproducing antibound states without including a 1s1/2 basis pole requires a significant number of non-resonant scattering continuum states (contour discretization points), we will forgo this approach due to model space size constraints in favor of including the 1s1/2 basis pole. We checked the effects of this basis pole on the 1/2+ state by varying the depth of the WS one-body basis potential. Comparing the calculation results to experimental values for 9He [77], we find that the best 35 reproduction of the 1/2+ feature coincides with an antibound 1s1/2 basis pole just before crossing threshold. It is worth noting that this antibound state is sensitive to the placement of the 1s1/2 basis pole. This basis dependence, while not ideal, is unsurprising as the use of the 1s1/2 basis pole supplants the finer contour discretization which would otherwise be required. After determining that the best 1/2+ reproduction occurs with a barely-antibound 1s1/2 basis pole, we extended this search procedure to GSM of 9N. Since protons cannot be anti- bound due to the presence of the Coulomb barrier [84], we cannot use the exact same basis as 9He in a proton-rich version. The behavior which best reproduced the 1/2+ state 9He was an antibound 1s1/2 pole nearest threshold; thus, we chose an analogous version for the proton basis where a narrow resonant 1s1/2 pole was selected. We can study the effects of this basis choice by comparing these results with that of a bound basis pole and find that the energy will vary by roughly one MeV but the decay width may be strongly suppressed (see Fig. 4.3). The effect the choice of basis can have on the convergence of decay widths of many-body GSM states is well documented [29, 31, 37, 85]. As discussed in the references therein, depending on the goals of the practitioner, one may favor better energy reproduction over calculating an exact decay width as the later may require more computational resources. Since the completeness relation holds, if one chooses enough non-resonant continuum states, the resulting energies will remain albeit with different occupations than if using an artificial basis pole [85]. For a majority of many-body states, which are bound states or narrow resonances, the tension of convergence rate to reproducibility of states does not usually pose a significant dilemma (see the bound states in Fig. 4.3). However, we see in the case of near- or above- threshold states, the choice of using resonant basis poles can lead to variation in the energy 36 Figure 4.3: Calculation of isobaric analog states for the A = 9 chain scaled by a factor of aZ2 where a = 0.21 to compensate for Coulomb energy corrections. Experimental results are shown in black and two versions of GSM results are shown in blue with an antibound (a) basis pole and bound (b) basis pole. States with a gold star above them are those with well 3) and proper calculation of the expected converged energies ( T . Previous experimental results are shown from Refs. [2, 77], but we do not isospin ⟩ ⟨ include the experimentally estimated 1/2+ energy for 9N as it was determined in this work (see Fig. 4.5) [62]. ˜Ψ Ψ ⟩ | ⟨ (cid:12) (cid:12) (cid:12) ˜Ψ | ⟨ 10− Ψ | ⟩ − H ≤ (cid:12) (cid:12) (cid:12) E 37 and decay width. This holds true if the contour discretization is insufficient, and computing the decay width can be unstable [31]. Despite the possible instability associated with using unbound basis poles, the widths of the 1/2− state in 9He and 9N are well under control for both bound and resonant basis poles. Using an unbound basis pole causes the 9N 1/2+ (cid:1) state to be much broader than the bound basis pole calculation (which has an unusually (cid:0) narrow width). 4.3.3 Implications of 9N Structure After developing the method to reproduce the structure of 9He and applying it to 9N, we are able to investigate the dependence of the energies on the choice of basis. As discussed in the section above, we systematically vary the depth V0(ℓ = 0) of the auxiliary potential to generate the Berggren basis and track the trajectory of the resulting many-body states as shown in Fig. 4.4. If we rely on the method developed for 9He, looking for a barely narrow-resonant 1s1/2 basis pole, we find the value for 1/2+ which best matches this criteria is around V0(ℓ = 0) = 100.4 MeV. We also note that this basis variation does not impact the trajectory of the 1/2− state in a meaningful way. Due to the demonstrated stability of the 1/2− state regardless of basis in Fig. 4.4, the experimentalists utilized this GSM complex energy to constrain their anti-correlated two- peak fit (see Fig. 4.2b). By using this GSM 1/2− energy as a constraint, the two-peak fit produced a significance beyond the capabilities of Monte Carlo [62]. The two-peak-plus- background hypothesis concludes that the data is highly likely to originate from 9N structures at a 5σ threshold. Resulting experimentally determined energies and the GSM results using the refined basis are illustrated in Fig. 4.5. 38 Figure 4.4: Trajectory of the many-body eigenvectors for the GSM calculation of 9N with varying WS basis depth V0. We show the π/4 subthreshold boundary as a black dashed line. The experimental data point is determined by using the GSM 1/2− state to constrain the two-peak fit (shown in Fig. 4.2). 39 0.20.00.20.40.60.81.01.2Re(k)0.40.30.20.10.0Im(k)V0=100.4V0=86.4V0=92.4V0=94.4V0=102.4V0=104.4V0=105.4V0=106.9V0=100.4V0=95.4V0=105.49N Many-body state locations with respect to8C+pThreshold1/2+1/2−Exp1/2+ Figure 4.5: Energy spectrum for the N = 2 isotone decay chain relative to the 4He core. GSM calculations are shown alongside experimental values. Figure taken from Ref. [62]. 4.4 Conclusions Rare isotopes existing near or beyond the driplines are challenging systems for both ex- periment and theory. Of this class of nuclei, the 5p-emitting nucleus 9N is a particularly interesting as over half of its constituent nucleons are unbound as its intermediate state 8C+p contains another unbound nucleus. We provided theoretical predictions for the structure of 9N and this data assisted in the experimental analysis which determined the presence of 9N within a ≈ 5σ threshold [62]. Predicted energies for the low-lying doublet structure in 9N are in good agreement with the experimental energy location of these states. Furthermore, this system is interesting for mirror (isospin) symmetry as the level inversion in the N = 7 chain continues for 9N’s mirror, 9He. Interestingly, 9He consists of an antibound 1/2+ and a real resonance 1/2−, and this doublet is present in 9N. However, the 1/2+ state in the proton-rich nucleus appears to be transformed into a genuine resonance rather than a scattering feature under the current 40 0246810C8+pBe6+p3α+p5Li5+p4B7+p2p2p2+0+0+0+2ppGSMC8N9GSM)-(1/2)+(1/2Energy (MeV)Energy (MeV)0+1/2GSMExpHe9-1/2-1-3-2 Hamiltonian and GSM framework. Unfortunately, the experimental statistics were low in this experiment and the two-peak character could not be constrained simultaneously, relying on a statistical analysis of different theoretically-informed peak profiles. Future experiments can potentially resolve this issue and more OQS model development is needed to probe these highly exotic systems where even the current GSM implementation struggles. 41 Chapter 5. Projection of the Continuum 5.1 Motivation Within GSM, the continuum coupling in resonances is not well quantified or understood. Other nuclear OQS models however incorporate continuum effects via explicit coupling terms. This class of continuum SM is thus complementary to GSM calculations, as was discussed for SMEC in Sec. 3.5. Essentially, one model class (GSM) incorporates the continuum in a “black box” perspective where continuum effects are introduced at a basis level; while the other model class (SMEC) has tunable parameters for direct manipulation and exploration of the continuum. Currently, there is no direct quantitative comparison between the two methods and their relation has been assumed to be valid at a theoretical level. We aim to provide quantitative evidence to support or dispute these assumptions. Ordinarily, one can explore the effects of the continuum in GSM only from the many-body structure perspective, as finer mechanisms associated with direct control of continuum coupling strength are not accessible in GSM [5, 29] without modification [38, 86]. Of course, this property of GSM is beneficial as it limits the dimensionality of the same many-body problems which would otherwise be intractable with other OQS models such as SMEC [29]. 5.2 Shell Model Embedded in the Continuum Before we can formulate any methods to extract quantitative continuum coupling from GSM, we must first outline the formalism for SMEC to identify possible starting points. SMEC incorporates OQS nature by dividing the system into two subspaces, a localized space (Q) with A − 1 bound particles and a continuum space (P ) with the scattering nucleon(s) in the continuum without coupling to the Q space [18, 87]. This subspace picture was first outlined 42 by Feshbach in Refs. [88, 89] as a method to unify nuclear structure and reaction theories. Since the development of SMEC [90], it has had a long history of highlighting single-particle continuum effects [18, 90–92], including decay channel effects on exotic states [46, 93, 94], and highlighting potential effects of two-particle systems [87]. To perform SMEC calculations in the one valence nucleon case, one first needs to generate wave functions for the total function space. The Q space sets are L2 functions from the standard SM while the P space sets are s.p. scattering states. Decays for the A-nucleon system will occur when the scattering states are coupled to the SM states. Both wave functions can be obtained by solving the Schr¨odinger equation: HSM | E = E Φi ⟩ − Hcc′ (cid:1) (cid:88)c′ (cid:0) (SM) i Φi | c′(+) ξ E for localized space Q , ⟩ = 0, for continuum space P (5.1a) (5.1b) where Eq. 5.1a corresponds to the discrete SM calculations (CQS picture) and Eq. 5.1b to the scattering states [87]. The SM Hamiltonian (CQS) is HSM and Hcc′ the coupled-channel (CC) Hamiltonian. Since the decay mechanisms are directly tied to = H0 + Vcc′ is the scattering nucleon, the channels are controlled by the nucleon’s relative motion with the 1 residue states A − ΦA j | 1 − . One can then define projection operators for each space over ⟩ their set of states ˆQ = N (cid:88)i=1 ΦA | i ⟩ ⟨ ΦA i | , ˆP = ∞ dE 0 (cid:90) ξE⟩ ⟨ | ξE| such that each projection on a vector from its complement space gives ˆQ = = 0, ξE⟩ | ˆP ΦA i ⟩ | = 0. 43 (5.2) (5.3) Operators, such as the Hamiltonian, may also be projected such as ˆQH ˆQ = HQQ = HSM and ˆP H ˆP = HP P = Hcc. One will notice that at the current step in the formulation, we still have two distinct non-coupled spaces. Assuming that the different spaces follow ˆQ + ˆP = Id, we can calculate a mixed projection ωi | ⟩ = HP Q | Φi ⟩ which produces a third wave function (+) P (E) = G (+) ω i | ⟩ (+) P (E) = ˆP (E G , ωi | ⟩ HP P ) ˆP . − (5.4) (5.5a) (5.5b) Eq. 5.5b is the Green’s function of a nucleon’s motion in the scattering space. We can now write the solution using all three function sets ΨE⟩ | = Id | ΨE⟩ = ˆQ ΨE⟩ | + ˆP ΨE⟩ | ΦA i ⟩ | (cid:111) (cid:110) . This wave function can be ξE⟩} {| , and , by expanding (+) ω i ⟩ | (cid:110) (cid:111) operated on by ΨE⟩ | ˆQ ˆP ΨE⟩ | ΨE⟩ | = = which can be expanded to E − (cid:0) ξE⟩ | HQQ(E) − 1 HQP | (cid:1) (+) P (E)HP Q ˆQ ξE⟩ ΨE⟩ | + G ΨE⟩ | = + ξE⟩ | ΦA i ⟩ | + (+) ω i | (E) ΦA i | ⟨ E (cid:0) ⟩ (cid:17) − HQQ(E) (cid:1) 1 − (cid:88)i,k (cid:16) 44 , (5.6a) ΦA | k ⟩ ⟨ ΦA k | HQP | ξE⟩ . (5.7) We note that in this formalism, an energy dependent effective Hamiltonian is obtained HQQ(E) = HQQ + HQP G (+) P (E)HP Q. (5.8) which captures the effects of continuum coupling on the CQS Hamiltonian HQQ. Therefore, HQQ is the OQS Hamiltonian within the Q space, and from this effective, energy-dependent Hamiltonian we can calculate its eigenvalues to obtain the energies ˜Ei(E) and resonance widths ˜Γi(E): Ei = ˜Ei(E = Ei); Γi = ˜Γi(E = Ei). (5.9) Formally, we assign the second term of Eq. 5.2 to be the continuum coupling term WQQ(E) = HQP G (+) P (E)HP Q [46]. The continuum coupling term can have different forms, but in the case of simple interactions the continuum coupling is dependent on some over- all strength parameter [46, 93]. As a result, the continuum coupling strength is explicitly dependent on this parameter. After obtaining the eigenstates of our effective Hamiltonian ΨJπ α ⟩ | , which include coupling to a specific decay channel A − 1Z(Kπ′) , a continuum-coupling correlation energy Jπ ℓj ⊗ (cid:105) (cid:104) can be calculated to quantify the continuum-induced mixing [93, 95] EJπ corr.,α(E) = ΨJπ WQQ(E) α (E) | ⟨ ΨJπ α (E) ⟩ | = ΨJπ α (E) ⟨ |HQQ(E) − ΨJπ α (E) HQQ| . (5.10) ⟩ 45 5.3 Quantifying the Continuum in GSM with Projec- tion Aside from the work in Chs. 3 and 4, GSM has been used to study many OQS phenomena in nuclear physics from threshold effects like Wigner cusps [54] to the structure of exotic nu- clei [30, 39, 72]. The differences in the formalism of coupling two spaces compared to using a holistic Rigged Hilbert Space are highlighted in Fig. 5.1. A limitation to GSM is the inability to control specific phenomena (such as the characterization of individual decay channels) and the correlation between these phenomena and the strength of continuum coupling, as both are implicitly handled in GSM. In other words, interesting many-body structural phenomena can be modeled, but we cannot directly probe or quantify the mechanisms that drive the desired behavior. This dilemma is the motivation for our work, and our goal is to follow the Feshbach projection approach of SMEC to isolate the CQS and pure-scattering continuum terms from the GSM continuum couplings. 5.3.1 Formalism For well-bound systems, GSM’s results can be indistinguishable from results calculated in a traditional SM as these nuclei will have little continuum dependence (recall Fig. 3.3). This conclusion is unsurprising as bound states defined in any similar basis will likely share the same spatial localization. We take advantage of this property and assume that within a GSM calculation, there must exist a subspace which is representative of the SM solutions for well- bound nuclei, see Fig. 5.2. The main question is then, how might we use this assumption to build operators to project-out the SM-like components of a full GSM calculation? There are a few possible methods we can practically do this, each with their own particular challenges, 46 Figure 5.1: (Top) Representation of different approaches depending on choice of Hilbert Space (SMEC) or Rigged Hilbert Space (GSM) formalisms. Resonances manifest in both (Bottom) We show the functional cases, yet the mechanisms differ by the space choice. “zones” of both approaches. SMEC requires coupling between to separate spaces to produce the OQS effects. GSM’s Rigged Hilbert Space contains all components for OQS, but it is difficult to disentangle the continuum coupling effects. but we will first describe the assumptions we will use to make these operators. To create a CQS projection operator, we must use some set of localized basis states to construct a space we believe to be entirely closed, with Dirichlet boundary conditions imposed. These boundary conditions require that outside of some domain, the wave function must be zero, such as u(r = 0) = 0 and u(r = Rbox) = 0, and could be considered as imposing an infinite square well boundary condition where particles cannot enter or escape (decay). 47 Im(k)Re(k)ScatteringStatesBoundStatesQ spaceP spaceResonantStatesScatteringContinuumIm(k)Re(k)ScatteringStatesBoundStatesResonantStatesNonresonantScatteringContinuumDiscrete States(Bound and Resonant)Hilbert SpaceSMECRigged Hilbert SpaceGSMQ spaceP spaceNonresonant Scattering Continuum space (blue) in GSM by accounting for states gener- Figure 5.2: Depiction of isolating a ated in a well-localized basis. The complement space is shown in violet and contains any features not captured by the -space. This figure is an illustration of applying the Feshbach projection technique onto GSM as presented in Fig. 5.1. Q Q Using these CQS states Φ ⟩ | , we build a general CQS projector = Q N (cid:88)i Φi | Φi ⟩ ⟨ | (5.11) or more generally A = |  | Φ1 Φ2 | ... ΦN | | |       and = A Q 1 − AT AT A (cid:17) (cid:16) (5.12)        m N × with vectors Φ having length m. Although we would expect higher energy components of our system to be associated with continuum effects, it is certainly possible to have high energy components in Q which happen to coincide with discretized scattering states we are attempting to isolate. This would be cases where pseudo-continuum states generated from a CQS basis are similar to select, genuine, scattering states as their boundaries coincide with 48 Q Well-Bound StatesP Complement spaceNon-Resonant ScatteringContinuumDiscrete StatesRadially-Extended Bound States a specific radial point of a node in a scattering state rbox = rnode. Since OQS by definition have features which cannot be naturally described in a CQS framework, Q will only act on a subspace of the OQS space; thus, there must be some complementary space where I + , P ≈ Q → P ≈ I . − Q (5.13) Here we assume that the working Berggren basis is large enough to encompass long range effects. In the above equation, I represents the A-body Fock space of the Berggren ensemble employed in GSM. This I is not a true unity operator, like one finds in a Hilbert space, as the system is built from a reference core rather than the vacuum. From the constructed CQS and scattering-space projection operators, any GSM operator can be decomposed via O OGSM = ( P OGSM = + ) Q OGSM ( P + + ) Q QOGSMQ POGSMQ + QOGSMP + POGSMP . (5.14) Continuum coupling for specific observable quantities can be obtained by projecting their corresponding operators, such as the Hamiltonian: HGSM = ( + ) HGSM ( + ) , HGSM = P Q HGSMQ Q CQS P Q HGSMQ + P + Q Coupling HGSMP + . HGSMP P Scattering (5.15) (cid:124) As shown, projected energies can be associated as E (cid:123)(cid:122) (cid:123)(cid:122) (cid:124) (cid:125) QQ (cid:125) (cid:124) (cid:123)(cid:122) HGSMQ Q = (cid:125) being the CQS-like 49 energy, E PP = HGSMP P being the pure-scattering energy, and E PQQP = HGSMQ P + being the continuum-continuum coupling term. For more manageable notation HGSMP Q we will adopt the following convention: HGSMQ HGSMQ Q P = H QQ = H PQ , , HGSMP HGSMP P Q = H PP = H QP , . Considering the typical GSM (Rigged Hilbert space) picture, one would expect the de- composed version to be of the form of Fig. 5.3. Focusing on the H QQ component, one would anticipate that the first Riemann energy sheet would only contain few discrete continuum contributions but have many of the bound states, assuming the localized states used in are Q a good representation. Additionally, in the second Riemann sheet of H QQ , some low-energy Gamow states may appear depending on the system and will be shifted onto the real axis as shown in Fig. 5.3. CQS projection essentially will capture the localized components of the resonant states with the remainder being left in the -space. P On the other hand, the H PP component would be expected to contain all pure scatter- ing terms and perhaps a weakly-bound, or severely radially extended, state. More likely, we would expect this term to instead collect more Gamow states and retain the coupling terms that are larger, towards pure continuum states. In either case, we can more easily see in the second Riemann sheets that these projected systems might still retain some res- onant (Gamow) states but will only highlight the real-energy and continuum contributions respectively. The final part, which is the goal of this work, is to then extract the continuum coupling H QP + H PQ . Before continuing, we explicitly define the corresponding SDs and bases to avoid any 50 Figure 5.3: Illustration of the CQS and pure scattering components of the Hamiltonian after applying the projection operators in Eq. 5.15. The complex k-plane has been divided into its two energy Riemann sheets in order to highlight the behaviors we would expect to capture operators. Riemann Sheet I shows the projection of the physical sheet using the and Riemann Sheet II shows the projection onto the non-physical sheet. Resonant states are on the non-physical sheet but can be accessed via contour deformations. and Q P confusion. Single-particle states will be referred to using a lower-case Greek letter ( and ϕ ⟩ | ψ | ) and any many-body states will be upper-case Greek letters - ⟩ Φ ⟩ | and Ψ ⟩ | for CQS and OQS respectively. Generally, to build a nucleus we apply creation operators to add n valence nucleons onto a closed shell c | ⟩ to form the arbitrary wave function In our case we define the SDs as g†1g†2...g†n = Ξi | ⟩ (cid:104) . ⟩ c i | (cid:105) CQS OQS (GSM) = Φi | ⟩ = Ψi | ⟩ a†1a†2...a†n c i | ⟩ (cid:105) (cid:104) , i = 1, ...d, b†1b†2...b†n (cid:104) (cid:105) c i | ⟩ , i = 1, ...d, (5.16) where a† and b† are the creation operators for the chosen CQS and Berggren bases respec- tively. Here, d refers to the corresponding quantum numbers for each state. Many-body basis SDs ΨB i ⟩ | eigenstate of HGSM) as used in GSM can naturally expand a particular resonant state ci ΨB i ⟩ | . Ψν | ⟩ = (cid:88)i 51 Ψν | ⟩ (an (5.17) Im(k)Re(k)DiscreteScatteringStatesDiscreteScattering StatesRiemann Sheet IBoundStatesResonantStatesNonresonantScatteringContinuumRigged Hilbert SpaceGSMIm(E)Re(E)Im(E)Re(E)Bound StatesQWeakly-BoundStateResonantStatesAntibound StateScattering StatesPIm(E)Re(E)Im(E)Re(E)ResonantStatesRiemann Sheet II Note that from now on, for clarity, any state with a B superscript refers to a basis state and states with Greek letter subscripts ( Ψν | ) are specific Gamow states (eigenvectors of the ⟩ Hamiltonian). We will discuss some additional notation for clarity since we will be working with two different bases. If any state or operator is surrounded by parenthesis accompanied by a subscript or just accompanied by a subscript, this indicates the operator is in that corresponding basis e.g. IΨ = (I)Ψ is in the GSM basis Ψ. This is helpful when an explicit basis state is not written like ΨB . i ⟩ | First, we consider the operation of on a single GSM basis state Q ΨB j ⟩ Q | = (cid:88)i ΦB | i ⟩ ⟨ ΦB i | ΨB j ⟩ ΦB i ⟩ | Mij. = (cid:88)i (5.18) A transformation coefficient Mij arises which will allow transformations between Φ and Ψ basis vectors. This can be expanded into matrix using a complete set of GSM basis vectors (in other words, the identity) IΨ = Q (cid:32) (cid:88)i ΦB | i ⟩ ⟨ ΦB i |(cid:33)   (cid:88)j ΨB | j ⟩ ⟨ ˜ΨB j | = ΦB | i ⟩ ⟨  ΦB i | ΨB j ⟩ ⟨ ˜ΨB j | ΦB i ⟩ | Mij ˜ΨB j | ⟨ = M. (5.19) = (cid:88)ij (cid:88)ij The Q operator can be applied in the opposite direction and has a slightly different form IΨQ =   (cid:88)j ΨB | j ⟩ ⟨ ˜ΨB j |  ΦB | i ⟩ ⟨ ΦB i |(cid:33) = (cid:32) (cid:88)i ΨB | j ⟩ ⟨ ˜ΨB j | ΦB i ⟩ ⟨ ΦB i | ΨB j ⟩ | ˜Mij ΦB i | ⟨ = ˜M . (5.20) = (cid:88)ij (cid:88)ij 52 Here M and ˜M each denote the conjugation status of our GSM state ( ΨB | ⟩ and ˜ΨB ⟨ ). Since | the proper GSM conjugation is simply time reversal (no complex conjugate in Eq. 2.10), we need this distinction as the CQS state could require a complex conjugation. In the case where we have only real wave functions, this reduces to a case where M = ˜M T , but the most general definition requires this distinction. The matrix M is NCQS × dimensions are swapped. From these expressions, we can write the projection operator NGSM and its partner’s in Q the GSM basis QΨ = ˜M M ˜M 1 − M (5.21) (cid:16) to create a projection operator with NGSM × (cid:17) NGSM dimensions. Because M and ˜M represent the overlaps between SDs in the CQS and GSM bases, we can express the overlaps as [96–98] Mi,j = Φi ⟨ Ψj | ⟩ = c | ⟨ ainain − 1...ai1b†j1 ...b†j2 c b†jn | ⟩ ϕi1| ψj1⟩ ⟨ ψj2⟩ ϕi1| ⟨ ... ϕi2| ψj1⟩ ⟨ ψj2⟩ ϕi2| ⟨ ... ϕi1| ψjn⟩ ⟨ ϕi2| ψjn⟩ ⟨ ... ... . . . ... = (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ⟨ ⟨ ϕin| ψj1⟩ ψj2⟩ ϕin| ... (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ψjn⟩ (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ϕin| ⟨ (5.22) with the second line showing the overlaps between each one-body state making up the SDs. 53 Likewise, we can calculate ˜Mi,j = = = ⟩ c ⟨ binbin 1...bi1a†j1 − a†jn | c ⟩ ˜Ψi ⟨ Φj | ˜ψi1| ⟨ ˜ψi1| ... ⟨ ϕj1⟩ ϕj2⟩ | ˜ψi2| ⟨ ˜ψi2| ⟨ ... ϕj1⟩ ϕj2⟩ ˜ψi1| ⟨ ϕjn⟩ ˜ψi2| ϕjn⟩ ⟨ ... ... . . . ... (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ...a†j2 ˜ψin| ⟨ ˜ψin| ⟨ ... ˜ψin| ⟨ ϕj1⟩ ϕj2⟩ (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ϕjn⟩ (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (5.23) which is crucial to get our basis-transformed IΨQ QΨ. notation, any reference to should be taken as Q IΨ = ˜M M . In the following generalized 5.3.2 Implementation Since both GSM and CQS wave functions are spherically symmetric, they have the general form of Eq. 2.6 and the primary difference is the radial wave function. Use of these occupation representation M and ˜M operators then provides a convenient feature whereby the overlaps can be pre-computed on the s.p. level and the matrices can be built by taking determinants of these scalar values. Calculating these overlaps can be reduced further as each s.p. wave function overlap would contain ϕi ⟨ ψj | ⟩ = rmax 2π 1 rmin (cid:90) 0 (cid:90) 1 (cid:90) − u∗ϕ(n, ℓ, j, r) r uψ(n′, ℓ′, j′, r) r ×   mℓ,ms (cid:88) C jm ℓmℓsms Y ∗ℓmℓ (θ, φ) s, ms ⟨ |  ×    ,s′z (cid:88)m′ℓ C j′m′ ℓ′m′ℓ Yℓ′m′ℓ s′s′z (θ, φ) s, ms | ⟩   r2d(cos θ)dφdr, (5.24) 54 and generally for any spherical harmonic overlap 2π 1 0 (cid:90) 1 − (cid:90) Y ∗ℓmℓ (θ, φ)Yℓ′m′ℓ (θ, φ)d(cos θ)dφ = δℓℓ′ δmℓm′ℓ which leads to the reduced form of s.p. overlaps (noting that s, ms ⟨ s′, m′s⟩ | = δss′ δmsm′s ) ϕi ⟨ ψj | ⟩ = rmax rmin (cid:90) u∗ϕ(n, ℓ, j, r)uψ(n′, ℓ′, j′, r)dr × (cid:88)mℓ,ms;m′ℓ ,m′s C ℓsj mℓmsmj C ℓ′s′j′ m′ℓ m′sm′j δℓℓ′ δmℓm′ℓ δss′ δmsm′s = ϕ ψ u j ⟩ i | u ⟨ (cid:88)mℓ,ms;m′ℓ ,m′s C ℓsj mℓmsmj C ℓ′s′j′ m′ℓ m′sm′j δℓℓ′ δmℓm′ℓ δss′ δmsm′s . (5.25) For many of the precomputed s.p. overlaps the different values of the δ functions will be zero and this binary check can prevent any need to compute the radial integrals. Up until now, we have also been assuming nucleons of the same species but inclusion of isospin Tz as an extra input quantum number produces the criteria δTzT ′z s.p. states. between the two overlapped After the s.p. overlaps are calculated and the M and ˜M operators are computed, GSM results can be projected. An important feature of projection operators is idempotence, or that successive operations of a projection operator produce the same result as a single operation, = Q QQ . Due to potential numerical instabilities, it will be important to check amongst different operations that idempotence holds by IΨ = ⟨ Ψν Ψν |QQ| Ψν Ψν |Q| ⟩ ⟩ − ⟨ = 0 or IH = Ψν ⟨ H |QQ Ψν QQ| Ψν H |Q ⟩ − ⟨ Ψν Q| ⟩ = 0. (5.26) 55 The complement operator P has the same idempotent criteria. The remaining choice one must make is which CQS basis to use to generate the projection operators. 5.3.3 Box Basis Our basis of choice for the purposes of this work is a so-called “box basis” or a basis whereby we impose Dirichlet boundary conditions on our system. The primary effect of this choice is that any solutions to the radial wave equation u must follow the boundary conditions u(r = 0) = u(r = rbox) = 0 at the origin and the edge of the box. From the perspective of an introductory quantum mechanics course, this would be the same as working from an infinite square well in spherical coordinates where the potential is zero inside the box and infinite beyond the rbox. Infinite wells have a well-known property that the energy excitation spectrum follows a general relation E ∝ 2 r− box [99, 100]. A convenient result of this feature is the level of openness in the system can be directly controlled by the box radius, and in the limit as rbox → ∞ , the energy level spacing becomes infinitesimally small, a continuum limit. Another effect of changing box radius is that the number of low-lying energy states will increase and move down from the high-energy pseudo-continuum. A limit to this basis choice is the stability of the low-lying wave functions which coin- cides with the energy of the states. To maintain consistency with the GSM contour, which represents the maximum energy cutoff, we only use states with energies of 400 MeV or lower (a typical cutoff value). While this limitation does not allow for a full representation from the box basis, we have found that the use of a few states is sufficient for projection. 56 Figure 5.4: Energies for GSM and HO-SM basis calculations with respect to the WS depth V0 associated with each calculation crossing threshold. Solid lines indicate real energy while dashed lines are for imaginary components (if applicable). Note the black dotted lines highlight where each calculation crosses threshold for the 9He(1/2−) state. We highlight the lack of a discontinuity in the SF for HO-SM calculations. 5.4 Toy Problem Before applying the projection code to realistic problems, the projection operator perfor- mance must first be benchmarked. To avoid cumbersome calculations, or potentially diffi- cult to interpret results, we will consider a textbook example of the emergence of Wigner cusps (discontinuities in the SF) for a 9He+n 10He system as the WS depth parameter → (V0) is varied [5]. We show in Fig. 5.4 such effects where the SF becomes discontinuous across threshold only for the GSM (OQS) calculations. We note in Fig. 5.4 that there is a bend in the energy of 9He as a result of crossing threshold. This example was selected as it guarantees a system that exhibits threshold dependence. Additionally, this example is ideal for benchmarking since it uses a simplified model space consisting of only a single 0p1/2 basis pole, a p1/2 discretized contour with 15 points per segment, and a 8He core. 57 −50−48−46−44−42−40V0(‘=1)(MeV)−8−6−4−20E−Ecore(MeV)CalculatedEnergiesHOE[9He(1/2−)]HOE[10He(0+)]GSME[9He(1/2−)]GSME[10He(0+)]−50−48−46−44−42−40V0(‘=1)(MeV)1.01.21.41.61.82.0SpectroscopicFactorsS29He(1/2−)+νp1/2→10He(0+)HOGSM Figure 5.5: Projected energies for the considered nuclei. Solid lines indicate real energy while dashed lines are imaginary components. We show the idempotence check of Eq. 5.26 in the rightmost panel to verify the projection operators are idempotent. The projection operator was constructed using only the 0p1/2 box state at a box radius of rbox = 3.9 fm. Q While this model space is not a realistic representation for the structure for 9He or 10He, its simplicity will quickly identify if the projection operators capture threshold effects and remain idempotent. Following the outline in Eq. 5.15, the projection operators can be applied to the Hamilto- nian to obtain the projected energy components along with a check to ensure the operators are idempotent (Eq. 5.26). To start benchmarking with the simplest case, we will use only the lowest box state (0p1/2) and a box radius of 3.9 fm to compute the projection opera- tor. Previously, in Fig. 5.4, we noted that the GSM energies did not appear to have strong changes when crossing threshold, yet in Fig. 5.5 we can clearly see via the projected energies that effects from the continuum produce discontinuities. Moreover, we find that the contin- uum coupling energy is less than that of the SM or pure-scattering terms and serves as a binding effect when only considering this bound basis pole. Taking inspiration from Eq. 5.2, a similar continuum correlation energy can be obtained for each of the projected components 58 −0.6−0.4−0.20.00.20.4Re(cid:2)˜E(9He)(cid:3)−10−8−6−4−2024E−Ecore(MeV)Projectedenergiesfor(GSM)9He(1/2−)EQQEQP+EPQEPPEGSMEsum−0.6−0.4−0.20.00.20.4Re(cid:2)˜E(9He)(cid:3)−15−10−505E−Ecore(MeV)Projectedenergiesfor(GSM)10He(0+)ESMQQESMQP+ESMPQESMPPESMGSMESMsum−0.6−0.4−0.20.00.20.4Re(cid:2)˜E(9He)(cid:3)−15−10−5051015log10(|IH|)Idempotencechecksforprojectedenergies9He10He Figure 5.6: Projected continuum correlation energies for the considered nuclei. Color scheme, line styles, and box parameters are the same as in Fig. 5.5. We show the derivative of the correlation energy to highlight a discontinuity (inset) for the first order derivative in the coupling term for 10He. (represented generically as XX) by Ecorr XX = Ψ(J π) HGSM | ⟨ XX (E) − HCQS XX | Ψ(J π) ⟩ . (5.27) In this form, the OQS and CQS components are projected onto the same subspace, allowing for a consistent description between the two quantities. This is analogous to taking the difference between the energy dependent Hamiltonian HQQ(E) and the SM Hamiltonian HQQ in the SMEC formalism. In cases where the CQS effects across threshold are effectively constant, we see only slight changes in the projected continuum correlation energies as in Fig. 5.6. As discussed in Sec. 5.3.3, the box basis has two adjustable parameters that control the representation of the Q space, the box radius rbox and the number of excitations included n. Before applying our method to realistic systems, the effects of changing these two parameters 59 −0.6−0.4−0.20.00.20.4Reh˜E(9He)i−8−6−4−2024E−Ecore(MeV)ContinuumCorrelationEnergiesfor9He(1/2−)EcorrQQEcorrPPEcorrPQQP−0.6−0.4−0.20.00.20.4Reh˜E(9He)i−15−10−50510E−Ecore(MeV)ContinuumCorrelationEnergiesfor10He(0+)EcorrQQEcorrPPEcorrPQQPddEEcorrPQQP Figure 5.7: Projected energies using only the 0p1/2 bound s.p. box state at different values of box radius. Each set of lines belong to a plane corresponding to an rbox value. The color scheme is the same as Fig. 5.5 where blue is E . We see a general convergence of the values at larger box sizes. , and orange is E , green is E PQQP QQ PP must be benchmarked systematically. We will only focus on the impacts of the projection of the Hamiltonian only, as projecting SFs requires additional formulation of three different projection spaces corresponding to the parent, daughter, and free nucleon. 5.4.1 Benchmarking Box Radius Effects Starting with the box radius, it was demonstrated in Sec. 5.3.3 that the behavior of the box basis states should follow an infinite square well. Since the box basis is generated using a WS plus Spin-Orbit (plus Coulomb for protons) potential with Dirichlet boundary conditions imposed, we should recover any bound states of the WS potential as long as rbox extends far enough to allow the bound state(s) to form. For this toy system, there is one bound 0p1/2 pole; therefore, increasing rbox should force the projected energies to converge to a specific value once the bound state is properly accounted for. Indeed, this is what is seen in Fig. 5.7 where the real and imaginary projected components stabilize. While the convergence of the projection operators when using only bound states is a 60 Figure 5.8: Projected energies using the lowest two box states (0p1/2 and 1p1/2) at different values of box radius. We see that there is no convergence with increasing box radius. The color scheme is the same as Fig. 5.7. helpful check, the question of how including higher-order box state excitations (pseudo- continuum states) will impact the projection operators still remains. In this case, one would not expect to have perfect convergence to a specific projected value as the pseudo-continuum (excited box) state will constantly change with box radius, recalling the behavior of an infinite well. Fig. 5.8 demonstrates this behavior as the inclusion of the next excited state causes fluctuations in the projected energies above threshold. For the n = 1 box wave function, the overall behavior does not vary significantly with rbox since it is a pseudo-continuum state where the number of nodes of the wave function are directly tied to the infinite well representation. Despite the general line shape similarity, the fluctuations in the projected energies are a result of how well the box wave function for a given radius happens to coincide with the GSM s.p. scattering states. This is an obvious model dependence, but it is shared for many CQS models which use a fixed box boundary to represent a pseudo-continuum. 61 5.4.2 Benchmarking Box Excitation Effects Since Sec. 5.4.1 has demonstrated that the projection operators return sensible results, so long as box radii are large enough to encompass the WS bound states, impacts and limitations on the number of box excitation states included can be investigated. Theoretically, there is no limit to the number of box state excitations one can include so long as the dimensionality of the box space does not exceed the GSM space. The same holds true for any projections onto a HO-SM calculation to obtain a continuum correlation energy from Eq. 5.27. As one can see in Eq. 5.21, if the dimensions change from NGSM > Nbox to NGSM < Nbox then QΨ will project onto an overcomplete basis since the projection space exceeds the system space. In this extreme case, the calculation of Eq. 5.21 will likely result in a singular matrix or, at best, a collapse of idempotence. In practice, numerical noise will exist and could lead to premature loss of idempotence of the projection operator. We checked this effect by performing a series of calculations of the toy model using a range of included maximum excitations. Fig. 5.9 illustrates the degree to which numerical noise impacts the projection operators. We see in this case that after including nmax = 5, the idempotence check increases orders of magnitude and it will become useless after a few more excitations are included. This problem is also present for the 10He case where the changes in the idempotence check magnitude are even sharper. Constant checking of the idempotence is imperative as Fig. 5.9 illustrates the system behavior may not provide obvious indications to the increased loss of idempotence. Thankfully, these higher excited states at the current box radius coincide with energies far above the typical GSM energy cutoff. Moreover, the goal of using box states in our projection operator were to reliably capture any CQS behavior, not have the space reproduce the genuine continuum Q 62 Figure 5.9: (Left) Projected imaginary energies including an increasing number of excitations from the box basis at a consistent rbox = 4 fm. (Right) We note that after including more excited states, idempotence will fail. states. 5.5 Realistic Nuclei For an initial test of GSM projection in a realistic case, we selected the nuclei 5He and 6He which are constructed with valence neutron(s) on a 4He core. These two nuclei have been studied frequently with methods using the Berggren basis [27–29, 31, 37, 101, 102] and one-body WS and two-body FHT interactions have been calibrated for these nuclei [31, 37]. For this work, the model space will be reduced to include only p1/2 and p3/2 states which are necessary to reproduce the states in 5He and are sufficient for calculations of 6He. The contours will be discretized with 27 points (9 per segment) with the points kpeak = (0.3, − 0.3), kmid = (0.9, 0.0), and kmax = (4.0, 0.0). One-body and two-body parameters are taken from Ref. [31], as we will use a WS core-nucleon and FHT nucleon-nucleon interaction. To determine the effects different thresholds have on 5He and 6He, we again vary the WS depth V0 to transition from artificially bound to unbound cases as shown in Fig. 5.10. This realistic case provides a new stress-test of the projection operators as it includes different 63 Figure 5.10: Left: Calculated energies for the ground states of 5He and 6He. We show the real (solid) and imaginary (dashed) parts as well as the two thresholds for 5He and 6He respectively as vertical dotted lines. Right: SFs calculated in GSM with varying WS depth. j basis states which compete in the structure of the A = 6 system. We highlight that this case also provides two thresholds to study, the transition from bound to unbound for 5He and for 6He. Furthermore, the behavior of 6He is that of Borromean system which decays by 2n emission since 5He is neutron unbound [40, 54, 103, 104]. At each threshold, we expect the projected components to reveal qualitative energy changes in the system as the wave functions will transition from being effectively describable with Dirichlet conditions to outgoing asymptotics. Fig. 5.11 depicts the effects of projection using only the lowest energy basis poles for the considered partial waves. Again, specific behaviors appear that were also seen in the toy problem when including only bound states in , such as a coalescence of Q two projected components ( and P Q ) at threshold and the binding effect of the coupling term E PQQP . This behavior continues for the excited 6He(2+) state which is shown in Fig. 5.12. The excited state of 5He is unbound for the entire domain of the considered V0 parameter, yet the excited state in 6He retains the desired threshold crossing feature. For 64 −50−45−40−35−30V0(‘=1)(MeV)−10−8−6−4−2024E−Ecore(MeV)GSMEnergiesGSME[5He(3/2−)]GSME[6He(0+)]HOE[5He(3/2−)]HOE[6He(0+)]−50−45−40−35−30V0(‘=1)(MeV)−1.0−0.50.00.51.01.52.02.53.0SpectroscopicFactorsS26He(0+)→5He(3/2−)+nGSMHO Figure 5.11: Projected ground state (g.s.) energies using the lowest box states (0p1/2 and 0p3/2) at rbox = 4 fm. We show the two thresholds for 5He and 6He respectively as vertical dotted black lines. The center and left plots use the same legend. Figure 5.12: Projected first excited state energies using the lowest box states (0p1/2 and 0p3/2) at rbox = 4 fm with respect to the 5He g.s. energy. We show the two thresholds for 5He and 6He respectively. 65 −2−101Re(cid:2)˜E(5Heg.s.)(cid:3)−8−6−4−202468E−Ecore(MeV)Projectedenergiesfor(GSM)5He(3/2−)EQQEQP+EPQEPPEGSMEsum−2−101Re(cid:2)˜E(5Heg.s.)(cid:3)−12.5−10.0−7.5−5.0−2.50.02.55.0E−Ecore(MeV)Projectedenergiesfor(GSM)6He(0+)−2−101Re(cid:2)˜E(5Heg.s.)(cid:3)−15−10−5051015log10(|IH|)Idempotencechecksforprojectedenergies5He6He1.92.02.12.22.32.4Re(cid:2)˜E(5Heg.s.)(cid:3)−10−50510E−Ecore(MeV)Projectedenergiesfor(GSM)5He(1/2−)1.92.02.12.22.32.4Re(cid:2)˜E(5Heg.s.)(cid:3)−12−10−8−6−4−2024E−Ecore(MeV)Projectedenergiesfor(GSM)6He(2+)EQQEQP+EPQEPPEGSMEsum1.92.02.12.22.32.4Re(cid:2)˜E(5Heg.s.)(cid:3)−15−10−5051015log10(|IH|)Idempotencechecksforprojectedenergies5He6He Figure 5.13: Variation of box radius from rbox = 1 to rbox = 20 fm. Projected energies for the ground state of 5He were obtained using the lowest box states (0p1/2 and 0p3/2). the unbound 6He(2+), we note that at shallow values of V0 the state was not well-converged E Ψν | H ( | 5Heg.s.. Despite the lack of convergence in the GSM calculations, the projection operators > 0.5 MeV) and this is represented in Fig. 5.12 at larger values of Ψν | ⟩ − ⟩ | still function well and have little loss in idempotence. Moreover, ignoring the convergence issues, the resonant state clearly dissolves into the continuum due to the large imaginary term, yet the projected components still handle these sharp changes well. We follow the toy model checks to diagnose the stability of the projection operators in this realistic case. The results of varying the box radius are plotted in Fig. 5.13 and the impact of changing the number of box excitations included in the projection operator are shown in Fig. 5.14 for the ground states only. These checks are consistent with what was observed in the toy model. Specifically, there is convergence of projected components to certain values when using only bound states in with large box radius, and a loss of Q idempotence is observed when including higher-lying excited box states in the projection operator. Both checks illustrate that the projection operators, even in realistic cases, can 66 Figure 5.14: Variation of box excitations included from n = 0 to n = 7. Projected energies for the ground states were obtained using a box radius of rbox = 4 fm. To maintain readability, only the imaginary part is shown. reliably provide quantified continuum coupling using localized states and the inclusion of pseudo-continuum states (E > 0 in the CQS case) is feasible but requires constant checking of the idempotence. Additionally, with any variation in either box radius or excitations included, the resulting effects on projected energies appear to pivot about the corresponding threshold of the system. This indicates that these projection operators respect the thresholds in GSM. 5.6 Conclusions Using a set of states generated with Dirichlet boundary conditions, we have demonstrated that projection of continuum effects in GSM using a CQS basis is possible and determined possible limitations to this approach. In principle the choice of projection basis is arbitrary, but in practice the set of states used to generate the projection operator should vary con- sistently in a parameter space. Further investigations into other realistic systems should be done across different shells such as 17,18O or 41,42Ca. Investigating these systems will reveal if the behaviors noted in this work appear across the nuclear landscape, and these general 67 features may provide more insight into generic continuum effects. Since these systems will require different model spaces of higher ℓ states, one can probe other threshold effects. For example, Wigner cusps do not manifest for ℓ > 1, but discontinuities will appear at higher order derivatives of the SFs. The order of derivative follows the trend of ℓ 1 so cusps − appear in ℓ = 2 states at first order derivatives and at second order for ℓ = 2 [54, 105]. Other methods to extract continuum energy correlation factors, like in Ref. [106] which allow for direct probing of effects such as aligned states, would be complementary to this work. Direct comparison between such approaches and this projection method should be done to qualitatively understand how continuum coupling impacts nuclear structures. Ad- ditionally, combinations of OQS models can be used to provide insight into specific features. We have shown in this work that topics including SF suppression (Ch. 3) and the structure of ephemeral nuclei (Ch. 4) are strongly dependent on the continuum. Using the suite of spe- cialized methods to extract specific aspects of continuum coupling should be used to better quantify the nature of the continuum. In the case of exotic resonant states, like antibound states, the use of these projection techniques could also be used to highlight deficiencies in a CQS basis by generating a projection operator from a chosen basis. 68 Chapter 6. Hands-on Nuclear Structure and Decays Nuclear science is an ever-evolving field which greatly benefits from new perspectives, and im- proving accessibility and inclusivity is crucial its continued development [1]. Simultaneously, pushes to improve the public’s scientific literacy and increase engagement with students will be crucial for both maintaining public support of scientific programs and for developing fu- ture workforces. To this end, we have developed a new demonstration [107] that highlights nuclear structure and decays while aiming to be accessible to a wide audience. Nuclear phe- nomena are illustrated in this demonstration by layering wooden blocks with a fixed set of rules emulating nuclear shells. Providing audiences with opportunities to engage with nuclear science can be difficult due to the microscopic size of nuclei. Constan developed [108] a method for explaining reaction mechanisms in accelerator facilities using magnetic marbles, and highlighted the possible products that can be made from the impact of two nuclei. Whittaker created a demonstration using magnetic disks to describe static nuclear properties from a quark level [109]. While both highlight crucial aspects of nuclear formation, building from the quark level and illustrating statistical reaction processes, there is a gap in the literature focusing on time-dependent processes like decay. Our demonstration seeks to address this gap by allowing nuclei to be built from a nucleon level and introducing mechanisms of decay such as γ-decay, β±-decay, α-emission, and nucleon emission. Fission is not explored in this demonstration as it occurs in heavy nuclei which have large numbers of nucleons, a scale that is not conducive for this demonstration. This chapter contains work submitted to The Physics Teacher and is currently under 69 Figure 6.1: Diagram of dimensions for the required blocks and cardboard support pieces. Nuclear magic numbers are represented by shape A (for the 2 shell closure) and shape B (the 8 and 20 shell closures). Sides referenced for each block are as shown in the given label. review [107]. Authors of that manuscript are Joshua Wylie, Pablo Giuliani, Kyle Godbey, and Sylvester Agbemava. 6.1 Rules First, we define the basic properties of the demonstration before discussing the physics concepts covered. The materials required for the demonstration are 1.5cm 2.5cm 7.5cm wooden blocks, with at minimum 20 painted red (protons) and 28 painted blue (neutrons), and cardboard shapes cut to the specifications in Fig. 6.1. The demonstration can be done individually or in a competitive format. In either case, a set of base rules exists that one must follow, and these are represented in Fig. 6.2 but are also listed (block sides are labeled in Fig. 6.1): 1. Protons: Must lay on side 1, be spaced 2 cm apart, and be parallel to each other. ≈ 70 Shape AShape B8.25cm8.25cm15.75cm15.75cm7.75 cm4 cm4 cmH=2.5cmL=7.5cmW=1.5cm234 cm Figure 6.2: Rules of the demonstration accompanied by pictorial representations. We provide a few starting scenarios to illustrate allowed and forbidden actions within the rules. 71 Example of rules 1-6Start!Rule 5Rule 2cRule 1, 3Rule 3-5Protons must be on side 1 and spaced 2.5cm apart EXCEPT at their ends where they can pairRule 1SpacingPairingNo SpacingNeutrons can be in any orientation on side 2Rule 2Shared proton and neutron level (Bad)Neutron perpendicular to other neutron (Bad)1) Protons are on side 1, parallel, spaced 2cm unless paired2) Neutrons are on side 2, perpendicular to protons, and can touch3) Protons start on starting platform4) Build protons or neutrons on top of starting proton(s). Structure must be free-standing5) Levels only contain one nucleon type (protons or neutrons)6) Magic number pieces added when 2, 8, 20, and 28 are reached for one nucleon typeShell closure (cardboard added)!H=2.5cmL=7.5cmW=1.5cm23B1B2B3B2continued Pairing between two protons is allowed by having sides 3 join together. 2. Neutrons: Must lay on side 2 and can touch in any configuration, but they must be parallel to one another and perpendicular to protons. 3. The demonstration must begin with placing one proton on the starting platform, shape A. 4. Subsequent nucleons may be added atop this platform or the starting proton respecting rules 1-3. 5. Each level can contain only one nucleon species, and no more than two sequential levels of nucleons of the same species can be on top of each other. 6. When enough nucleons have been added, a player may add a magic number cardboard stabilizer to the structure. 7. (Optional) Each level of protons or neutrons has a maximum number of blocks allowed. Isospin symmetry and the presence, or lack, of Coulomb effects due to charge are represented in the first two rules. Rules 3 and 5 ensure that stable neutron matter or extremely neutron- rich matter is not a possible outcome of the demonstration. While rule 7 best mimics the general SM scheme, see Fig. 6.3, it can be complex for younger participants and left optional. We allow participants to vary the level order (regardless of adherence to rule 7) so they may represent other nuclear features like islands of inversion. Finally, after a nucleus has been constructed, and assuming that the nucleus has not yet collapsed, one can introduce vibrations to the table the nucleus sits on to simulate the effects of quantum mechanics. The introduction of these effects might induce a decay, where a nucleon changes orientation, falls, or the entire structure collapses. It is recommended that participants use something which vibrates the table as a constant frequency and varying amplitudes to illustrate nucleon motion over time at different energies. If such a tools is 72 Figure 6.3: Example of 13C structure in relation to a schematic SM picture. unavailable, participants can carefully shake or hit the table to accomplish the same purpose. After introducing quantum effects (vibrations) participants will notice some structure survive the process while others decay. The goals are to either (1) build the largest, most stable, nucleus or (2) build the most exotic nucleus. For the purposes of this demonstration, “exotic” is defined as nuclei with strong isospin asymmetry such as 8C or 28O. Excited states can be produced by promoting one nucleon to a higher level as depicted in the side branches B1 and B3 of Fig. 6.3. Additional rules for a competitive version between multiple participants can also be implemented, and these rules are: 1. Using only one hand, and not touching any additional blocks, players may (a) place a new nucleon onto the nucleus or (b) excite an existing nucleon by moving it to a higher level than its current position. 73 2 blocks on 0s1/2 Shell2 blocks on 0s1/2 Shell4 blocks on 0p3/2 Shell4 blocks on 0p3/2 Shell1 block on 0p1/2 ShellLevel 1 (0𝑠1/2): 2 BlocksLevel 2 (0𝑝3/2): 4 BlocksLevel 3 (0𝑝1/2): 2 Blocks2813C Figure 6.4: Depiction of additional competitive rules including an illustration of scoring a round. 2. Nucleons cannot be de-excited (moved to a lower level). 3. Once the proper nucleon number requirement is met, a player may use their turn to place a magic number board instead of a nucleon. 4. (Optional) Nucleons cannot be added to levels below a magic number board unless it is to replace a vacancy left by an excited nucleon. Examples of these additional rules are provided in Fig. 6.4. A scoring mechanism is also included to encourage participants to play multiple rounds and to cleverly create unstable nuclei. Points are awarded to the player which made the last stable move. Points are assigned 74 Scoring:Point Values for decay types:•Nucleon emission (N): +1 point•β or γ-decay (β, γ): +2 points•Magic Numbers (M): +1 point per board on nucleusAdd all points from decayed nucleons and any magic numbers in play. Total goes to the player who played the last stable move. GOAL: First to 25 wins!Ex: It decayed on Orange’s turn, so Blue will get the points. We have two protons which fell onto their sides, β-decaying into two neutrons. Blue gets 2(β)+2(β)+1(M)=5 points!1. Using one hand only, players may:3. Once the minimum number of protons and neutrons have been added, a player may add a Magic Number boardAdd a nucleonExcite an existing nucleonOR2. Nucleons cannot be de-excitedExample: 8 protons and 8 neutronsPair a nucleon by sliding itORExtra Rulesi. All players have placed 1 block down making 4He. Blue placed magic numberii. Orange places proton on closed 4Heiv. Grey adds proton on 6Be making 7Bv. Blue excites proton on 7Bvi. Orange excites proton on 7Biii. Green pairs proton on 5Li making 6BeExample gamevii. Nucleus decays after Orange plays!Blue wins the round! as follows and ends once a player has reached 10 points: • Nucleon emission: One or more nucleons fall completely off of the nucleus (1 point). • β-decay: A nucleon falls down to a lower level of the opposite nucleon type or onto the opposite side it started on, i.e. a proton falls landing on side 2 changes to a neutron (2 points). • γ-decay: A nucleon falls down to a lower level of the same nucleon type or falls onto the same side it started on, i.e. a proton falls landing on side 1 remains a proton (2 points). • Additional points are added onto the decay score for each Magic Number board on the nucleus. Extra points are assigned to decays which are less likely to occur based on the nature of the demonstration, as (multi-)nucleon emission is the most likely. 6.2 Concepts As outlined in Sec. 6.1, we retain simple rules associated with protons and neutrons. This is done to avoid any misconceptions that nuclei form in a sterile environment, contrary to the dynamic processes in which they actually form, like in stars. Furthermore, capturing emer- gent behavior from simple rules highlights the ideal balance of maintaining model simplicity and predictability. Nucleonic properties are the first focus of this demonstration. We assert that there are two species of nucleons, one of which can only be near others of the same species as pairs. Participants may easily recognize this importance as it is associated with electric charge, which can be compared to the repulsive effects when placing two magnets of the same align- ment near each other. Starting from these Coulomb and general fermionic properties, one 75 can explore a variety of phenomena related to the nuclear chart. Participants frequently question why the nuclear chart bends to favor an excess of neutrons rather than a constant N = Z trend. By imposing or neglecting the proton spacing rule (Coulomb effects), partic- ipants determine that the repulsive nature leads to favoring neutron excess when building larger nuclei (40Ca or larger). We developed and provide a companion website [110] which includes a feature to highlight this deviation. Aside from nucleon properties, nuclear shells are another incredibly important concept to nuclear structure. Since the introduction of the SM by Maria Goeppert Mayer [111] to address the presence of magic numbers (also highlighted in the companion website [110]), many other phenomena have been associated with the presence of shells. This demonstration allows participants to follow general shell structure to understand how specific shell order- ings change the accessible nuclei. For example, following the exact level scheme provided, participants may be able to build 48Ca with ease, yet other exotic nuclei may be difficult to construct. When participants alter the shell ordering, creating islands of inversion, they find some nuclei are easier to build. By utilizing the shell structure in this demonstration, describing excited states becomes a natural extension by allowing lower-lying blocks to be moved to a higher level. The competitive version in particular encourages such promotion and can produce some nuclear structures that will not normally be constructed in an isolated (single-player) view. Promotion of nucleons, or other unstable moves, by other participants provides an ideal example of external environmental impacts on nuclear structure (OQS). Lastly, decays are a natural feature of this demonstration. In the cases where the tower remains stable, requiring additional energy to break it, participants will identify that some configurations are stronger than others. Even for towers containing the same number of nucleons, different configurations (ground states or excited states) will be associated with 76 Figure 6.5: Depictions of two decay modes in the context of this demonstration and relation to the corresponding decay mode. different levels of stability. For cases where the nuclear towers do undergo a decay, partici- pants can learn more about the systems. By leaving holes in lower levels, participants may learn that some decays become more favorable, such as β± or γ-decays. They can recognize that different configurations are more likely to produce desired decay properties. These de- cays are illustrated in the context of the demonstration in Fig. 6.5. A full view of all decay modes in this demonstration are illustrated in Fig. 6.6 and found on the companion web- site [46]. γ-decays occur in the demonstration by a nucleon falling from an elevated level onto its original orientation, whereas a β±-decay arises from a change in orientation. Both require 77 Ground state of 11BeThe neutron is now in the same orientation and level as the protons, so it becomes a proton too.You made 11B!Quantum Effects (vibrations)β decayThe neutron underwent a conversion into a proton, electron, and neutrino (not shown) to make 11B!8C (2 neutrons to 6 protons)Decays with 4 proton emission!Nucleon emission8C (2 neutrons to 6 protons)Decays with 4 proton emission using two pairs of 2 protons!Quantum Effects (vibrations) Figure 6.6: Screenshot of companion website with all decay modes higlighted for example nuclei. 78 that the nucleon remain on the nuclear tower. Conversely, (multi-)nucleon and α-emission explicitly require the nucleons to fall completely off of the nucleus. We find these decay pro- cesses occur within the demonstration. A final mechanism which participants can consider is partial or total disintegration, whereby part or the entire nuclear tower collapses. This can be related to catastrophic events which will destroy much of the original nucleus such as photodisintegration. Typically such events will require significant energy from vibrating the surface the tower rests on. 6.3 Conclusions As we have demonstrated in this work, nuclear shell effects and decay properties dramatically impact nuclear structure. This demonstration was inspired to describe both of these aspects, similar to how GSM describes structure and reaction properties simultaneously. 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