Electric dipole excitation of the halo nucleus 11Be in a discrete basis

Electric dipole excitation of the halo nucleus 11Be in a discrete basis Trabajo de Fin de Grado Doble Grado en Física y Matemáticas María Pérez García Tutorizado por D. Jesús Casal Berbel abstract 1 Abstract Halo nuclei have become a major focus in current nuclear physics due to their distinct characteristics which challenge conventional nuclear models. This work focuses on the study of an archetypical halo nucleus, 11Be. The nucleus is modelled as a 10Be-𝑛two-body system with two different potentials: a Woods-Saxon potential and a Gaussian potential. Using a matrix formalism, we will numerically solve the Schrödinger equation by diagonalizing the Hamiltonian in a known transformed harmonic oscillator basis, using the MATLAB software. In particular, we will try to reproduce different observables of interest that illustrate the halo characteristics of 11Be, focusing primarily on its energy levels, root-mean-square radius and electric dipole transition. For the root-mean-square radius, values of 7.1613 fm (Woods-Saxon) and 6.6733 fm (Gaussian) are obtained. Similarly, regarding the electric dipole transition, the computed results for (𝐸1,1 2 + →1 2 −)are equal to 0.2788 e2fm2 and 0.3661 e2fm2, respectively. These values are consistent with the halo nature of the nucleus, which leads to large spacial extension and electromagnetic transition probabilities. 2 Resumen Los núcleos halo actualmente son un foco de interés en el campo de la física nuclear debido a sus características distintivas, que desafían los modelos nucleares tradicionales. Este trabajo se centra en el estudio de un núcleo halo arquetípico, el 11Be. Para ello, modelamos el núcleo como un sistema de dos cuerpos (10Be-𝑛) usando dos potenciales distintos: un potencial de Woods-Saxon y uno Gaussiano. Empleando un formalismo matricial, resolvemos la ecuación de Schrödinger de forma numérica en el software MATLAB, diagonalizando el Hamiltoniano en una base de oscilador armónico transformada. En particular, vamos a intentar reproducir distintos observables de interés que ilustran en el 11Be las características del halo, centrándonos principalmente en su espectro de energías, el radio cuadrático medio y la transición dipolar eléctrica. Para el radio cuadrático medio, se obtienen los valores de 7.1613 fm (Woods-Saxon) y 6.6733 fm (Gaussiano). Así mismo, con respecto a la transición dipolar eléctrica, los valores calculados para (𝐸1,1 2 + →1 2 −)son iguales a 0.2788 e2fm2and 0.3661 e2fm2, respectivamente. Estos elevados valores son consistentes con la naturaleza de halo de este núcleo, que da lugar a una gran extensión espacial y altas probabilidades de transición electromagnética. Contents Abstract.............................................. 1 Resumen ............................................. 2 1 Introduction 5 1.1 Nuclearphysics ...................................... 5 1.2 Thenuclearshellmodel.................................. 8 1.3 Halonuclei......................................... 11 1.4 The 11Benucleus ..................................... 14 1.5 TheSchrödingerequation ................................ 15 1.5.1 The Schrödinger equation for two-body systems in a central potential . . . 15 1.5.2 Solving the Schrödinger equation with a matrix formalism . . . . . . . . . 17 1.6 Electric transition probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6.1 Transitionsumrules............................... 21 2 Characterization of 11Be 23 2.1 Basisfunctions....................................... 23 2.1.1 TheHObasis ................................... 23 2.1.2 TheTHObasis .................................. 24 2.2 The bound states of 11Be ................................. 25 2.3 The unbound states of 11Be................................ 29 3 Electric dipole excitation 33 3.1 Permitedtransitions.................................... 33 3.2 Boundstatetransitions .................................. 34 3.3 Transitions into the continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Sumrule .......................................... 39 3.4.1 Root-mean-square radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4.2 Transition to a forbidden state . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.3 Verification of the sum rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4 Comparison between two potentials 45 4.1 Energylevels........................................ 47 4.2 Radialfunctions...................................... 48 4.3 Electricdipoletransition ................................. 49 5 Summary and Conclusions 53 Chapter 1 Introduction Describing the structure of atomic nuclei from its fundamental interactions remains one of the central challenges in Nuclear Physics. The strong nuclear force originates from the residual interaction between quarks, which is described by Quantum Chromodynamics. The complexity of this approach makes direct calculations unfeasible, and thus requires nuclear interactions to be modelled with effective theories based on phenomenological observations. The predictive power of these models can be put to the test with exotic nuclei, such as the 11Be halo nucleus. This introductory chapter serves to contextualize the 11Be nucleus within Nuclear Physics and to outline the relevant theoretical framework. In the following chapter, the energy spectrum of the nucleus will be characterized using a Woods-Saxon (WS) potential; obtaining both the bound and unbound energy levels of relevant states. This provides a basis for describing the electric dipole transition in Chapter 3. Finally, in the fourth chapter, the previous model will be compared to a alternative framework employing a Gaussian potential. 1.1 Nuclear physics Nuclear Physics stands as one of the pillars of contemporary physics, playing a central role in our understanding of the world. It is often known by the wider public by events that have shaped our history, such as nuclear power and weaponry. Its importance, however, extends further than this, providing substantial contributions in medicine and technological advances. In addition to investigating the fundamental properties of nuclei, Nuclear Physics makes significant contributions across various areas of physics, encompassing phenomena from the microscopic to the cosmic scale. The development of this field of physics traces its origins to the study of matter, which dates back to ancient Greek philosophers. The term nucleus was first introduced by Rutherford in 1911, after studying the scattering of 𝛼particles on a gold foil target [1]. The experimental observations were inconsistent with the plum pudding atomic model proposed by J.J. Thomson, which expected slightly deflected 𝛼particles by the target. Instead, most 𝛼particles were observed exiting the gold foil with little to no deflection, while only a small fraction were scattered at large angles. These 6 inconsistencies led Rutherford to propose a new atomic model. An atom is a region of mostly empty space, with a small concentrated area in the center that contains most of the mass and all of the positive charge, denoted as the nucleus. Electrons orbit the nucleus, which is several orders of magnitude smaller than the atom. This same scientist was also responsible for the discovery of the proton in 1918, after identifying Hydrogen nuclei when bombarding Nitrogen gas with 𝛼particles, and the proposal of the existence of the neutron, a particle that increased the mass of the nucleus without altering its charge. The experimental confirmation of this new proposed particle came in 1932. Chadwick concluded that the penetrating radiation emitted when bombarding 9Be with 𝛼particles agreed with the proposed neutrons and could not be explained by photons. Thus, a nucleus is formed by 𝑁neutrons and 𝑍protons. The total number of nucleons, is know as the mass number and is denoted as 𝐴. The 1940s and 1950s were marked not only by significant theoretical developments, like the nuclear shell model, but also by major technological advancements. The former earned Maria Goeppert-Mayer and J. Hans D. Jensen the Physics Nobel Prize in 1963, for their discoveries concerning nuclear shell structure. Contemporaneously, the first controlled self-sustaining nuclear chain reaction was achieved in a project known as Chicago Pile-1, directed by Fermi. Soon after, the infamous Project Manhattan led to the creation of nuclear weapons, and the first Nuclear Power Plant began to provide electricity to the civil population. The description of nuclear structure and behavior begins with a set of fundamental definitions. An atomic nucleus is the central core of an atom, where protons and neutrons are tightly bound together by the strong nuclear force. Nuclei with an equal number of protons are know as isotopes, whereas those with the same number of neutrons are named isotones. Nuclei that share the total number of nucleons, 𝐴, are isobars. The mass of neutral atoms is approximately proportional to the mass number and can be expressed as: 𝑀(𝑍, 𝐴)𝑐2=𝐴𝑢𝑐2+ Δ(𝑍, 𝐴), where 𝑢is the atomic mass unit and is defined as 𝑢=𝑀(12𝐶)∕12 ≈ 931,494 MeV/c2. The discrepancy of the proportionality of the mass with 𝐴is accounted for in Δ(𝑍, 𝐴), the mass defect. The binding energy of any system is the rest energy of the constituents minus the rest energy of the composite system. When the binding energy is positive, the system is said to be bound. For a nucleide, its binding energy can be expressed as the sum of the atomic binding energy (∼eV) plus the nuclear binding energy of protons and neutrons in the nucleus (∼MeV). The energy necessary to extract a proton or a neutron from a bound nucleus is known as the separation energy and are denoted as 𝑆𝑝and 𝑆𝑛respectively. The periodic table classifies elements and is useful for describing their chemical properties. For a better understanding of the different isotopes, the Segrè Chart was introduced (Figure 1.1). It plots bound nucleides in terms of the number of neutrons, on the x axis, and the number of protons in the y axis. It is a clear way of mapping nuclear behaviors such as the decay type and lifetime. The limits of the chart are determined by the proton and neutron driplines. These are lines where adding 1. introduction 7 one more nucleon makes the system no longer bound: i.e., they are marked by 𝑆𝑝= 0 and 𝑆𝑛= 0 respectively. Any nucleus outside the area delimited by the driplines in the chart is unbound or can only exist as a resonance. Neutron-rich nuclei decay by 𝛽−, while an excess in protons leads to both 𝛽+and electron capture. The valley of stability is the region of the Segrè Chart where stable nuclei are found. For 𝑍 < 20, stable nuclei have a similar number of protons and neutrons, 𝑍≈𝑁. For heavier nuclei, 𝑍 > 20, the number of neutrons exceeds that of protons to counter the proton repulsion. Nuclei that are further from the valley of stability often display properties that differ significantly from those of stable nuclei, so the models that work for stable nuclei generally fail to accurately describe them. These exotic nuclei have a ratio of nucleons significantly different from the stable nucleus closest in mass [2]. Due to their short lifetimes, exotic nuclei are challenging to study and must be investigated shortly after their production. The improvement of accelerator facilities in the 1980s provided the technology necessary for experimental investigation, which was needed to achieve a deeper understanding. Their distinct properties test the limits of nuclear models and provide insight into nuclear interactions and reaction mechanisms. In particular, light exotic nuclei are weakly bound systems which present stability against particle emission, but decay by 𝛽processes into more stable nuclei with short lifetimes. An example of a light exotic nuclei is the main focus os this work, 11Be. Figure 1.1: A representation of the Segrè Chart [3]. Each square represents an isotope characterized by a specific number of protons on the vertical axis and neutrons on the horizontal axis. The color of each square indicates the main decay mode of the corresponding isotope. 14 1.4 The 11Be nucleus The main focus of the work is the study of the 11Be nucleus; a nucleus with 4 protons and 7 neutrons. It is an unstable Beryllium isotope with a half-life of 13.76 s [14]. It decays by 𝛽−radiation to 11B. The significant interest in this nucleus stems from its status as one of the most well-known examples of a one-neutron halo nucleus. Notably, it shows a large transition strength for the dipole electric transition between its ground state and the first excited state; the strongest amongst light nuclei [15]. A particularly intriguing feature of 11Be is the inversion of parity in its ground state, which deviates from the expected shell model ordering and has prompted extensive theoretical and experimental investigation into its configuration. Figure 1.6 shows the neutron levels’ parity inversion. For a traditional shell model, the last neutron should occupy the 1𝑝1 2 level, as displayed for the proton levels. However, this level switches positions with the 2𝑠1 2 level for the neutrons. The inversion of the 1𝑝1 2 and 2𝑠1 2 level was first predicted by Talmi and Unna [16]. This leads to the ground state being 1 2 +, instead of the expected 1 2 −with the conventional shell model. The 1 2 +ground state was confirmed experimentally by Alburger et al. [17]. They found that beta-ray transitions in 11Be occur to the excited states of 11B at 6.80 MeV and 7.99 MeV, both which had been determined to have even parities. According to the selection rules of beta decay, the parity must remain unaltered in permitted transitions, thus the ground state of 11Be must have a positive parity. The nucleon configuration for the ground state is: 𝜋∶ (1𝑠1 2 )2(1𝑝3 2 )2 𝜈∶ (1𝑠1 2 )2(1𝑝3 2 )4(1𝑠1 2 )1. Consequently, the first excited state is 1 2 −, with a corresponding nucleon configuration: 𝜋∶ (1𝑠1 2 )2(1𝑝3 2 )2 𝜈∶ (1𝑠1 2 )2(1𝑝3 2 )4(1𝑠1 2 )0(1𝑝1 2 )1. It is surprising that the 11Be nucleus has two bound states, as halo nuclei tend to only have one bound state [18]. The ground state has a neutron separation energy of 501.64 ± 0.25 keV [14] and the excited state has an excitation energy of 320.04±0.10 keV [14]. Higher lying states are unbound resonances, such as 5 2 +and 3 2 −. The low binding energy of the valence neutron is responsible for the large matter radii, characteristic of halo nuclei. The 11Be nucleus can be modelled as a two-body system composed of a 10Be core and a valence neutron, as displayed in Figure 1.7. The 10Be core has a 0+ground state and a 2+first excited state at 3368.03±0.3keV [14]. This excitation energy is enough to understand 11Be as a neutron orbiting the 10Be core [2]. The last neutron extends various femtometres outside the range of interaction that binds it to the core. A great part of the behavoir of 11Be can be explained without considering the core excitations, if the parity inversion is taken into consideration. 1. introduction 15 Figure 1.6: Level scheme for 11Be. It shows how, for a traditional shell model, the last neutron should occupy the 1𝑝1 2 level, as displayed for the proton levels. However, this level switches positions with the 2𝑠1 2 level for the neutrons [2]. Figure 1.7: Schematic representation of the halo structure of 11Be, with the 10Be core and a valence neutron. 1.5 The Schrödinger equation 1.5.1 The Schrödinger equation for two-body systems in a central potential The Schrödinger equation is the fundamental equation of quantum mechanics, describing the evolution of the quantum state of a system. We will model the nucleus as a two-body system; therefore, we are interested in the Schrödinger equation for a two-body problem in a central potential: 𝑖ℏ 𝜕 𝜕𝑡Ψ( 𝑟1, 𝑟2;𝑡) =  𝐻Ψ( 𝑟1, 𝑟2;𝑡).(1.5) As known from classical mechanics, with an appropriate change of coordinates, a two-body problem can be separated into two independent one-body problems. One corresponds to the evolution of the centre of mass, while the other consists of a single-body problem of reduced mass 𝜇subjected to a 16 potential 𝑉(𝑟), where: 𝜇=𝑚1𝑚2 𝑚1+𝑚2 , and 𝑟 =𝑟1−𝑟2is the relative cordinate. The position of the centre of mass is denoted as  𝑅and 𝑀=𝑚1+𝑚2. In the new set of coordinates { 𝑟,  𝑅} the Schrödinger equation takes the following form: 𝑖ℏ 𝜕 𝜕𝑡Ψ(𝑟,  𝑅, 𝑡) = [−ℏ2 2𝜇∇2 𝑟−ℏ2 2𝑀∇2 𝑅+𝑉(𝑟)]Ψ(𝑟,  𝑅, 𝑡).(1.6) As the previous equation does not have terms which combine 𝑟,  𝑅, 𝑡, the wave function can be factorized into Ψ(𝑟,  𝑅, 𝑡) = 𝜑(𝑟)𝜒( 𝑅)𝜏(𝑡). Additionally, given that the central potential 𝑉(𝑟)does not explicitly depend on time, the solution to the temporal part of the wave function is known: 𝑇(𝑡) = 𝑁𝑒−𝑖𝐸𝑇 ℏ𝑡,(1.7) and only the time-independent Schrödinger equation remains to be solved: [−ℏ2 2𝜇∇2 𝑟−ℏ2 2𝑀∇2 𝑅+𝑉(𝑟)]𝜑(𝑟)𝜒( 𝑅) = 𝐸𝑇𝜑(𝑟)𝜒( 𝑅),(1.8) where 𝐸𝑇is the sum of the energy contributions due to the movement of the centre of mass, 𝐸𝑅, and the relative movement, 𝜀. The Equation 1.8 can be further separated into two equations; one for the movement of the centre of mass, while the other depends on the relative coordinate. We are only interested in the latter, as if no external forces are applied, the movement of the centre of mass will be uniform, which is irrelevant when describing the nucleus. The equation which concerns us, written in spherical coordinates, is: [−ℏ2 2𝜇∇2 𝑟+𝑉(𝑟)]𝜑(𝑟)=−ℏ2 2𝜇[1 𝑟 𝜕 𝜕𝑟 +1 𝑟2(𝑟𝜕 𝜕𝑟)2 −𝐿2 ℏ2𝑟2]𝜑(𝑟) + 𝑉(𝑟)𝜑(𝑟) = 𝜀𝜑(𝑟).(1.9) Analogously, this equation can be further separated: (−ℏ2[1 𝑟 𝜕 𝜕𝑟 +1 𝑟2(𝑟𝜕 𝜕𝑟)2]+ 2𝜇𝑟2(𝑉(𝑟) − 𝜀))𝑅(𝑟)=−𝜆𝑅(𝑟),(1.10) 𝐿2𝑌(𝜃, 𝜙) = 𝜆𝑌 (𝜃, 𝜙).(1.11) The Equation (1.11) is the eigenvalue equation for the angular coordinates, which has as eigenvalues 𝑙(𝑙+ 1)ℏ2and as eigenfunctions the spherical harmonics. For the radial part, Eq. (1.10), we can introduce a new radial function 𝑢(𝑟) = 𝑟𝑅(𝑟). Together, subtituting the eigenvalues of Equation (1.11), the resulting expression is: 1. introduction 17 [−ℏ2 2𝜇 𝑑2 𝑑𝑟2+𝑉(𝑟) + 𝑙(𝑙+ 1)ℏ2 2𝜇𝑟2]𝑢(𝑟) = 𝜀 𝑢(𝑟).(1.12) This is a time-independent Schrödinger equation in one dimension, where 𝑙is the angular momentum and 𝜀is the energy associated with the relative movement of the two bodies. The potential now considered is: 𝑉(𝑟) + 𝑙(𝑙+ 1)ℏ2 2𝜇𝑟2, an effective potential which includes a centrifugal barrier. If the potential is taken as null for 𝑟→∞, then the bound solutions will have 𝜀 < 0, while the unbound solutions, which form a continuum, will have 𝜀 > 0. The eigenstates of the system will be expressed as: 𝜓(𝑟) = 1 𝑟𝑢(𝑟)(𝑟),(1.13) where results from the coupling on the spherical harmonics 𝑌(𝜃, 𝜙)with the relevant spins. If one of the particles has a zero spin, we can express the coupled basis as: 𝑙𝑠𝐽 𝑀 (𝑟)=[𝑌𝑙(𝑟)⊗ 𝜅𝑠]𝐽𝑀 ,(1.14) where 𝑠is the spin of the remaining particle and 𝐽is the total angular momentum. 1.5.2 Solving the Schrödinger equation with a matrix formalism Solving the Shröndinger equation is the equivalent to diagonalizing the Hamiltonian in a basis. We want to solve Eq. (1.12), a time-independent Shröndinger equation, which is no other than an eigenvalue equation for the Hamiltonian operator: 𝐻𝜑𝑘=𝐸𝑘𝜑𝑘.(1.15) The equivalent problem consists in finding the eigenfunctions 𝜑𝑘as well as the eigenvalues 𝐸𝑘for 𝑘= 1,2.... To do so, we begin by considering a basis of the Hilbert space of the problem, denoted as {𝜙𝑛}, with 𝑛= 1,2,3.... The eigenstates solution to the equation can be expressed in terms of the basis: 𝜑𝑘= ∞ ∑ 𝑛=1 𝑐(𝑘) 𝑛𝜙𝑛.(1.16) This expansion can then be substituted into the eigenvalue equation, Eq. (1.15): 𝐻 ∞ ∑ 𝑛=1 𝑐(𝑘) 𝑛𝜙𝑛=𝐸𝑘 ∞ ∑ 𝑛=1 𝑐(𝑘) 𝑛𝜙𝑛.(1.17) 18 Obtaining the eigenstates {𝜑𝑘}is equivalent to calculating the coefficients of the expansion, {𝑐(𝑘) 𝑛}. To achieve this, we multiply the equation by 𝜙𝑚on the left and integrate in the whole Hilbert space. Employing Dirac notation: ∞ ∑ 𝑛=1 𝑐(𝑘) 𝑛⟨𝑚|𝐻|𝑛⟩=𝐸𝑘 ∞ ∑ 𝑛=1 𝑐(𝑘) 𝑛⟨𝑚|𝑛⟩=𝐸𝑘𝑐(𝑘) 𝑚.(1.18) For the deduction, the orthogonality property of the basis has been taken into account, ⟨𝑚|𝑛⟩=𝛿𝑚,𝑛. The equation (1.18) can be written in a more transparent manner with a matrix formalism: ⎛⎜⎜⎜⎜⎜⎝ 𝐻1,1𝐻1,2𝐻1,3… 𝐻2,1𝐻2,2𝐻2,3… 𝐻3,1𝐻3,2𝐻3,3… ⋮ ⋮ ⋮ ⋱ ⎞⎟⎟⎟⎟⎟⎠ ⎛⎜⎜⎜⎜⎜⎝ 𝑐(𝑘) 1 𝑐(𝑘) 2 𝑐(𝑘) 3 ⋮ ⎞⎟⎟⎟⎟⎟⎠ =𝐸𝑘⎛⎜⎜⎜⎜⎜⎝ 𝑐(𝑘) 1 𝑐(𝑘) 2 𝑐(𝑘) 3 ⋮ ⎞⎟⎟⎟⎟⎟⎠ ,(1.19) where 𝐻𝑛,𝑚 stands for ⟨𝑚|𝐻|𝑛⟩. The expression reached, Eq. (1.19), is an equivalent problem to solving the time-independent Schrödinger equation. To obtain the eigenvalues and eigenstates of the problem of interest, we merely need to diagonalize the matrix: ⎛⎜⎜⎜⎜⎜⎝ 𝐻1,1𝐻1,2𝐻1,3… 𝐻2,1𝐻2,2𝐻2,3… 𝐻3,1𝐻3,2𝐻3,3… ⋮ ⋮ ⋮ ⋱ ⎞⎟⎟⎟⎟⎟⎠ .(1.20) This matrix can be interpreted as a representation of the Hamiltonian operator on the {𝜙𝑛}basis. A full representation of the system requires an infinite matrix, in correspondence with the infinite basis needed. In practice, however, it is necessary to resort to a finite basis. This is sufficient for the description of the physical properties of the problem and leads to convergence studies, which will be presented in the following chapters. The negative eigenvalues of 𝐻represent bound solutions of the system’s Schrödinger equation, and the solutions associated with positive eigenvalues are known as pseudostates and represent the continuum states for a discrete basis. They are the Hamiltonian’s eigenstates for a truncated square-integrable basis [19]. 1.6 Electric transition probability Stemming from the system wave functions many observables can be calculated. Among them, in this work, we will be studying electromagnetic transitions; in particular, the dipole electric transition. An excited nucleus can be de-excited from its initial configuration to a lower energy state. Typically, nuclear excited states decay to their ground state (g.s.) emitting energetic photons, i.e., gamma radiation. These transitions occur within the same nucleus 𝐴𝑋, contrary to what occurs in beta and alpha decay. It encompasses a re-ordering of nucleons, in addition to a lowering of mass from an exited 𝑚∗𝑐2to the lowest mass value 𝑚𝑐2(the g.s.) [5]. 1. introduction 19 The excited states of a nucleus are eigenstates of the Hamiltonian constituted by the nuclear Hamiltonian and the radiation Hamiltonian. In principle, one should expect no transitions to occur, as they are eigenstates of the operator. However, in reality, transitions are observed. The nucleus decays from an initial state |𝑖⟩with an 𝐸𝑖energy and 𝐽𝜋𝑖 𝑖, to a final state |𝑓⟩with a lower energy 𝐸𝑓and 𝐽𝜋𝑓 𝑓. Here 𝐽stands for the total angular momentum of the nucleus and 𝜋for the parity of the state. This occurs due to the coupling of the nuclear charge to the radiation, represented by the electromagnetic interaction Hamiltonian, 𝐻′. From time-dependent perturbation theory, an expression for the transition probability, 𝜆, can be obtained: 𝜆𝑖𝑓 =2𝜋 ℏ|⟨𝑖|𝐻′|𝑓⟩|2𝜌𝑓(𝐸𝑓),(1.21) know as the Fermi’s Golden Rule [4]. The quantity 𝜌𝑓(𝐸𝑓)is known as the density of final states. The electromagnetic interaction Hamiltonian is responsible for both electric and magnetic transitions. The classification of these transitions depends on the multipole order, with each type obeying different selection rules. In the case of electric transitions, they follow a set of transition rules: 𝜋𝑖=𝜋𝑓(−1)𝜆 𝐽𝑖= 𝐽𝑓+ 𝜆. (1.22) The angular momentum selection implies that the difference in angular momentum between the initial and final states is of 𝜆units of ℏfor an 𝐸𝜆 transition, and the electric transitions involve a natural parity change. Particularly, for the transition which concerns us, the electric dipole transition (𝐸1), the selection rules are as follows: •There is a change in parity between the initial and final nuclear wave function. •The angular momentums must satisfy  𝐽𝑖= 𝐽𝑓+ 1. The reduced probability of electric transition between two bound states is defined as: (𝐸𝜆, 𝐽𝑖→𝐽𝑓) = ∑ 𝜇 𝑀𝑓|⟨𝐽𝑓𝑀𝑓|𝜇(𝐸𝜆)|𝐽𝑖𝑀𝑖⟩|2 =2𝐽𝑓+ 1 2𝐽𝑖+ 1 |⟨𝐽𝑓||𝜇(𝐸𝜆)||𝐽𝑖⟩|2, (1.23) where ⟨𝐽𝑓||(𝐸𝜆)||𝐽𝑖⟩is a reduced matrix element following the definition of [20]. The electric multipole operator for an 𝐴-body system, 𝜇(𝐸𝜆), is defined as: 𝜇(𝐸𝜆) = 𝑒 𝐴 ∑ 𝑖=1 𝑍𝑖𝑟𝜆 𝑖𝑌𝜆,𝜇(𝑟𝑖).(1.24) Here, 𝑒𝑍𝑖is the charge of each of the bodies and 𝑟𝑖the coordinate with respect to the centre of mass of the whole system. When studying the 11Be nucleus, we are considering a two-body model. Consequently, we can simplify the previous expression to: 20 𝜇(𝐸𝜆) = 𝑒𝑍1𝑟𝜆 1𝑌𝜆,𝜇(𝑟1) + 𝑒𝑍2𝑟𝜆 2𝑌𝜆,𝜇(𝑟2).(1.25) The indices 1 and 2 refer to each of the bodies. As in classical mechanics, in a two-body problem, the position vectors can be expressed in terms of the relative distance, in the following way: 𝑟1= − 𝑚2 𝑚1+𝑚2 𝑟;𝑟2=𝑚1 𝑚1+𝑚2 𝑟. (1.26) If we also introduce an effective charge: 𝑍(𝜆) 𝑒𝑓𝑓 =𝑍2(𝑚1 𝑚1+𝑚2)𝜆 +𝑍1(−𝑚2 𝑚1+𝑚2)𝜆 ,(1.27) then the electric multipole operator can be presented as: 𝜇(𝐸𝜆) = 𝑒𝑍(𝜆) 𝑒𝑓𝑓 𝑟𝜆𝑌𝜆𝜇(𝑟).(1.28) Thus, the reduced matrix element can be expressed as: ⟨𝐽𝑓||(𝐸𝜆)||𝐽𝑖⟩=𝑒𝑍(𝜆) 𝑒𝑓𝑓 ⟨𝑓|𝑟𝜆|𝑖⟩⟨(𝑙𝑓𝑠𝑓)𝐽𝑓||𝑌𝜆||(𝑙𝑖𝑠𝑖)𝐽𝑖⟩,(1.29) where ⟨𝑓|𝑟𝜆|𝑖⟩is the integral of 𝑟𝜆with the radial parts of the initial and final wave functions, ⟨𝑓|𝑟𝜆|𝑖⟩=∫∞ 0 𝑟𝜆𝑢∗ 𝑓(𝑟)𝑢𝑖(𝑟)𝑑𝑟. (1.30) On the other hand, the reduced element concerning the harmonic spherical functions can be written in terms of the spin and angular momenta [2]: ⟨(𝑙𝑓𝑠𝑓)𝐽𝑓||𝑌𝜆||(𝑙𝑖𝑠𝑖)𝐽𝑖⟩=𝛿𝑠𝑖𝑠𝑓(2𝐽𝑖+ 1)1∕2 {𝐽𝑓𝐽𝑖𝜆 𝑙𝑖𝑙𝑓𝑠𝑖} × (−1)𝜆+𝑙𝑓+𝐽𝑖+𝑠𝑖(2𝑙𝑓+ 1)1∕2⟨𝑙𝑓||𝑌𝜆||𝑙𝑖⟩, (1.31) where: ⟨𝑙𝑓||𝑌𝜆||𝑙𝑖⟩=[(2𝑙𝑖+ 1)(2𝜆+ 1) 4𝜋(2𝑙𝑓+ 1) ]1∕2 ⟨𝜆0𝑙𝑖0|𝑙𝑓0⟩.(1.32) When dealing with a transition between a bound state and a continuum of states |𝑘 𝐽𝑓⟩, the transition probability differs from Eq. (1.23) [2]: 𝑑(𝐸𝜆) 𝑑𝜀 =2𝐽𝑓+ 1 2𝐽𝑖+ 1 𝜇 𝑘 (2𝜋)3ℏ2|⟨𝑘 𝐽𝑓|(𝐸𝜆)||𝐽𝑖⟩|2.(1.33) In the case of calculating this transition employing a finite pseudostate basis, the transition probability (𝐸𝜆)is given by: 1. introduction 21 (𝑁)(𝐸𝜆;𝑖→𝑓) = 2𝐽𝑓+ 1 2𝐽𝑖+ 1 |⟨Ψ(𝑁) 𝑖,𝐽𝑓||(𝐸𝜆)||𝐽𝑖⟩|2,(1.34) where N is the dimension of the basis. These discrete values can be related to the continuous distribution with different approaches. A simple way is by the following approximation: 𝑑(𝐸𝜆) 𝑑𝜀 |||||𝜀=𝜀𝑘 ≈1 Δ𝑘(𝑁)(𝐸𝜆;𝑖→𝑓),(1.35) where Δ𝑘= (𝜀𝑘+1 −𝜀𝑘)∕2 is an estimation of the width of the energy for each pseudostate. An alternative way of reaching a continuous distribution when working with pseudostates is by using a smearing function: 𝑑(𝐸𝜆) 𝑑𝜀 =∑ 𝑘 𝑔(𝜀, 𝜀𝑘)(𝐸𝜆;𝑖→𝑓),(1.36) where 𝑔(𝜀, 𝜀𝑘)is commonly taken as a Gaussian or Lorentzian function, with a width parameter 𝜎 [21]. The determination of the width parameter when working with pseudostates is unclear and is frequently taken as a free parameter. During this work, we will utilize Gaussian functions. In our calculations, we consider that, for each of the 𝑘Gaussian functions, the mean value of the distribution is 𝜀𝑘and the FWHM is: FWHM𝑘=𝜀𝑘+1 −𝜀𝑘−1.(1.37) As the relationship between the FWHM and the standard deviation 𝜎of normal distribution is FWHM ≈ 2.3548 𝜎[22], the width parameter will be taken as: 𝜎𝑘=𝜀𝑘+1 −𝜀𝑘−1 2.3548 .(1.38) 1.6.1 Transition sum rules Electromagnetic transitions can give substantial information when studying the structure of nuclei. Nonetheless, modelling transitions is not a simple process as they require the calculation of excited states [23]. Sum rules present an alternative way of studying these transitions. Although the formulas previously discussed pertain to the distribution of (𝐸𝜆)as a function of energy, the total sum of transition strengths is the sum of the values from Eq. (1.34) for all of the pseudostates. So, the non-energy weighted sum rule is [2]: (𝐸𝜆)𝐽𝑓= 𝑁 ∑ 𝑛=1 (𝐸𝜆) = 2𝐽𝑓+ 1 2𝐽𝑖+ 1 𝑁 ∑ 𝑛=1 |⟨Ψ(𝑁) 𝑖,𝐽𝑓||(𝐸𝜆)||𝐽𝑖⟩|2 =2𝐽𝑓+ 1 2𝐽𝑖+ 1 𝑁 ∑ 𝑛=1⟨𝐽𝑖||(𝐸𝜆)||Ψ(𝑁) 𝑖,𝐽𝑓⟩⟨Ψ(𝑁) 𝑖,𝐽𝑓||(𝐸𝜆)||𝐽𝑖⟩. (1.39) This may be expressed considering the radial and angular parts separately: 22 (𝐸𝜆)𝐽𝑓= (𝑍(𝜆) 𝑒𝑓𝑓 𝑒)22𝐽𝑓+ 1 2𝐽𝑖+ 1 𝑁 ∑ 𝑛=1 ⟨𝑖|𝑟𝜆|𝑛⟩⟨𝑛|𝑟𝜆|𝑖⟩ ×(⟨(𝑙𝑓𝑠𝑓)𝐽𝑓||𝑌𝜆||(𝑙𝑖𝑠𝑖)𝐽𝑖⟩)2. (1.40) If the completeness relation is applied to the radial part, this expression can be further simplified. The completeness relationship is only entirely valid if the basis employed is infinite. However, if the dimension of basis is sufficient, this is a reasonable approximation and 𝐼≈∑𝑁 𝑛=1 |𝑛⟩⟨𝑛|. For this reason, the sum rule allows us to assess the suitability of a basis with 𝑁pseudostates. The resulting expression is assuming the completeness relation is fulfilled is: (𝐸𝜆)𝐽𝑓= (𝑍(𝜆) 𝑒𝑓𝑓 𝑒)22𝐽𝑓+ 1 2𝐽𝑖+ 1 ⟨𝑟2𝜆⟩𝑖× (⟨(𝑙𝑓𝑠𝑓)𝐽𝑓||𝑌𝜆||(𝑙𝑖𝑠𝑖)𝐽𝑖⟩)2,(1.41) where ⟨𝑟2𝜆⟩𝑖is the mean value of 𝑟2𝜆for the ground state. Consequently, the sum rule for the electric dipole transition (𝐸1) can be used to calculate the root-mean-square radius from experimental data. To do so, it is key for the sum rule to include not only the sum of the continuum states, but the transitions to other bound states. Chapter 2 Characterization of 11Be 2.1 Basis functions The radial functions that describe the nuclear wave function may be obtained by solving the Schrödinger equation using a matrix formalism, as discussed in Section 1.5.2. This requires a complete basis of functions, typically constructed from the eigenstates of a known Hamiltonian. 2.1.1 The HO basis A well-studied potential is the isotropic harmonic oscillator in three dimensions: 𝑉(𝑟) = 1 2𝜇𝜔2𝑟2.(2.1) As it is a central potential, the radial and angular parts of the wave function can, therefore, be treated independently. The radial part of the solutions of the HO are: 𝑢𝐻𝑂 𝑛𝑙 (𝑟) = 𝑛𝑙𝑟𝑙+1𝑒−𝑟2 2𝑏2𝐿𝑙+1 2 𝑛(𝑟2 𝑏2)𝑛= 0,1,2... (2.2) where 𝐿𝑙+1 2 𝑛are the generalized Laguerre polynomials, 𝑏=√ℏ 𝜇𝜔 the characteristic length of the oscillator and 𝑛𝑙 a normalization constant: 𝑛𝑙 =√1 𝑏3√𝜋 2𝑛+𝑙+2 𝑛! 𝑏2𝑙(2𝑛+ 2𝑙+ 1)!!.(2.3) Although analytical, this is not the most suitable basis for describing the eigenstates of 11Be, or, in general, a system subject to finite-range interactions. The bound wave functions of a finite potential decrease asymptotically as 𝑒−𝛼𝑟, while the radial solutions of the harmonic oscillator have a 𝑒−𝛼𝑟2 Gaussian-like behavior. Consequently, a large number of basis functions are needed to appropriately reproduce the asymptotic behavior of bound states. 30 5 10 15 20 25 N 0 2 4 6 8 10 Energy (MeV) Figure 2.7: The energy spectrum for 1 2 +for 𝛾= 1.5fm1 2in terms of the size of the basis employed. 5 10 15 20 25 N 0 2 4 6 8 10 Energy (MeV) Figure 2.8: The energy spectrum for 1 2 −for 𝛾= 1.5fm1 2in terms of the size of the basis employed. Figure 2.9: Energy spectrum for 1 2 +for N = 26 in terms of 𝛾. Figure 2.10: Energy spectrum for 1 2 −for N = 26 in terms of 𝛾. Nevertheless, describing the region of the continuum of interest is not the only criterion when determining the transformation parameter. This must be chosen taking into account the observables wanted to be described. To exemplify, even though the low-lying states of the continuum are best represented for 𝛾= 1 fm1 2, a larger basis is needed to reach convergence for the bound levels. In Figures 2.11 and 2.12, the wave function of pseudostates corresponding to the first few unbound states of 1 2 +and 1 2 −, respectively, are shown for fixed parametres, 𝑁= 26 and 𝛾= 1.5fm1 2. It is important to note that as a result of these pseudostates not representing true continuum wave functions, they exhibit an oscillatory behavior only up to large distances—not to infinity—since the basis functions themselves decay. True continuum states, in contrast, would extend to infinity. In addition, the instabilities observed in both figures are also typical features for the continuum states described with a discrete basis. 2. characterization of 11be 31 Figure 2.11: Radial functions for the first unbound states of 1 2 +for 𝑁= 26 and 𝛾= 1.5fm1 2. Figure 2.12: Radial functions for the first unbound states of 1 2 −for 𝑁= 26 and 𝛾= 1.5fm1 2. Chapter 3 Electric dipole excitation We shall now proceed to study the electric dipole transition in the 11Be nucleus. From the 𝐸1transitions between bound nuclear levels, the 1 2 + →1 2 −in 11Be is the strongest one measured amongst light nuclei [15]. This is due to its halo structure; the significant extension of the neutron wave function enhances the transition strength in comparison to typical nuclear transitions. 3.1 Permited transitions When studying (𝐸1) transitions, the levels involved in the transition are required to fulfil the selection rules; they must have opposite parities and their total angular momentums must be able to couple to  1. The transitions which concern us are those which connect with the ground state, 1 2 +. For this reason, there are only two transitions we ought to consider, with 1 2 −or 3 2 −states. Prior to calculating the probability of each of the transitions, it is essential to express Eq. (1.23) for each of the particular cases. Firstly, the effective charge, Eq. (1.27), may be expressed as: 𝑍1) 𝑒𝑓𝑓 = −4 𝑚2 𝑚1+𝑚2 ,(3.1) since for the two-body system we are considering is composed of a 10Be core, which has 4 protons (𝑍1= 4) and a valence neutron, (𝑍2= 0). Secondly, we proceed with the calculation of the reduced matrix elements ⟨𝐽𝑓||(𝐸𝜆)||𝐽𝑖⟩ (Eq. 1.29)). To do so, we will now discuss each transition independently. For the transition between 1 2 +and 1 2 −the relevant quantum numbers for each state are (𝑗𝑖, 𝑙𝑖, 𝑠𝑖) = (1 2,0,1 2)and (𝑗𝑓, 𝑙𝑓, 𝑠𝑓) = (1 2,1,1 2), respectively. Therefore: 34 {1 2 1 21 0 1 1 2}=1 √6 ⟨1000|1 0⟩= 1. Hence, the reduced element concerning the harmonic spherical functions ⟨(1 1 2)1 2||𝑌1||(0 1 2)1 2⟩, Eq. (1.31), is equal to −1 √4𝜋; as the (2𝐽𝑖+ 1)1 2(2𝑙𝑓+ 1)1 2=√6terms cancel out the 6𝑗symbol. Consequently, for the 1 2 + →1 2 −transition, the resulting expression is: (𝐸1) = 𝑒21 4𝜋(−4 𝑚2 𝑚1+𝑚2)2 ⟨𝑓|𝑟|𝑖⟩2.(3.2) Analogously, for the transition between 1 2 +and 1 2 −the relevant quantum numbers for each state are (𝑗𝑖, 𝑙𝑖, 𝑠𝑖)=(1 2,0,1 2)and (𝑗𝑓, 𝑙𝑓, 𝑠𝑓)=(3 2,1,1 2), respectively. Proceeding in the same manner, the reduced element concerning the harmonic spherical functions ⟨(1 1 2)3 2||𝑌1||(0 1 2)1 2⟩, Eq. (1.31), is equal to 1 √4𝜋. For this reason, the resulting expression for the 1 2 + →3 2 −transition is: (𝐸1) = 2 𝑒21 4𝜋(−4 𝑚2 𝑚1+𝑚2)2 ⟨𝑓|𝑟|𝑖⟩2.(3.3) Note that the condition that only transitions that satisfy the selection rules will have a non-null probability is encompassed in Equation (1.31) by the Wigner 6𝑗symbol. For a symbol: {𝑗1𝑗2𝑗3 𝑙1𝑙2𝑙3},(3.4) to have a non-zero value, the following triplets must satisfy the triangular conditions for angular momentum coupling: (𝑗1, 𝑗2, 𝑗3),(𝑙1, 𝑗2, 𝑙3),(𝑗1, 𝑙2, 𝑗3)and (𝑙1, 𝑙2, 𝑗3). The triangular conditions for a triplet (𝑗𝑎, 𝑗𝑏, 𝑗𝑐)are: •|𝑗𝑎−𝑗𝑏|≤𝑗𝑐≤𝑗𝑎+𝑗𝑏. •𝑗𝑎+𝑗𝑏+𝑗𝑐is an integer. Knowing the quantum numbers belonging to the ground state, it can be easily deduced that the only plausible transitions are those previously considered. 3.2 Bound state transitions The 11Be nucleus only has one excited state 1 2 −, which happens to permit a dipole electric transition from the ground state, 1 2 +. With the expression previously deduced, Eq. (3.2), the probability of the transition can be calculated for a fixed basis. In Figure 3.1, the values of (𝐸1) are portrayed in terms of the size of the basis and the value of the transformation parameter 𝛾. The probability of 3. electric dipole excitation 35 the transition has an incremental tendency and converges to (𝐸1) = 0.2728 e2fm2. Although, the value of the transition probability is first reached for the basis with 𝛾= 1.5fm1 2, convergence is reached for all the cases presented for 𝑁∼ 23. Figure 3.1: Convergence of the dipole electric 1 2 + →1 2 −transition for different values of 𝛾. The single particle model used, which ignores the structure of the 10Be core, gives a value for the probability of electric dipole transition between the two bound energy levels of the nucleus of (𝐸1) = 0.2728 e2fm2. This is comparatively larger than the experimental values; in [29], they obtain the strength of the transition to be (𝐸1) = 0.102 e2fm2. The discrepancy between the theoretical model and the experimental value is due to the single particle model not considering excitations of the core. The electric dipole transition is only permitted when the 10Be core is in its ground state, 0+. Selection rules prohibit transitions from the core’s first excited state (2+), where the valence neutron is forced to occupy the 1𝑑5 2 energy level (  𝐽𝑖≠ 𝐽𝑓+ 1). Therefore, considering a model which assumes the core to always be in its lowest energy state overestimates the probability of a transition occurring. In [30] when comparing a single particle model with a particle-plus-core (CEX) model which considers the 0+and 2+states of the 10Be core, the CEX model reduces (𝐸1) by 0.14 e2fm2compared to their single particle model. 3.3 Transitions into the continuum To complete the description of the electric dipole transition, the transitions from the ground state to permitted continuum states must also be studied. The only states that satisfy the selection rules are 1 2 −and 3 2 −. Equation (1.34) provides a method for calculating discrete values of the transition probability, (𝐸1). However, to obtain the desired continuous distribution, it is necessary to resort to one of the approaches presented in the first chapter (Eq. (1.35) and (1.36)). The first method is presented in Figure 3.2 for the transition 1 2 + →1 2 −(𝑐𝑜𝑛𝑡.). Eq. (1.35) is applied to the discrete 36 values of (𝐸1), providing the values of the continuous distribution 𝑑𝐵(𝐸1) 𝑑𝜀 evaluated at the same energies. To represent the 𝑑𝐵(𝐸1) 𝑑𝜀 distribution, the discrete points have been smoothed using a spline. This is shown in Figure 3.2 for bases with 𝛾= 1.5fm1 2and diverse sizes, 𝑁. It is clear that as the dimension of the basis increases, the shape of the distribution converges. The main differences can be appreciated at the peak of the curve, which only stabilizes for 𝑁∼ 30. In addition, for clarity purposes, the discrete values calculated with Eq. (1.35) have been plotted for 𝑁= 41, showing how these values are adjusted with the corresponding spline curve. Figure 3.2: The 𝐸1transition probability, 𝑑𝐵(𝐸1) 𝑑𝜀 , for 1 2 + →1 2 −(𝑐𝑜𝑛𝑡.), in terms of the size of the THO basis. The distributions have been calculated using Eq. (1.35) and adjusted with a spline function. The basis parametre is fixed to 𝛾= 1.5fm1 2. The alternative method, using smearing functions, is presented for the 1 2 + →3 2 −(𝑐𝑜𝑛𝑡.)transitions. The smearing functions employed are Gaussian functions. For each of the functions used, the mean value of the distribution is taken as the associated energy 𝜀𝑘and the corresponding standard deviation, 𝜎𝑘, is taken as shown in Eq. (1.38). This method provides a smooth curve for the transition probability directly from the discrete values of (𝐸1) calculated with Eq. (1.34), instead of having to then apply an interpolation method. In the same manner, to be able to compare both methods, Figure 3.3 shows the resulting 𝑑𝐵(𝐸1) 𝑑𝜀 distributions for several THO 𝛾= 1.5fm1 2basis, in terms of the dimension of the basis. Once again, there are slight differences near the maxima of the curves. This is strictly due to the method employed to obtain the discrete values of the transition probability (Eq. (1.34)). As we are using a pseudostate basis to describe the continuum, we only have a discrete representation of the positive energy spectrum. Consequently, the energy values surrounding the maximum will vary for each basis size 𝑁. Therefore, (𝑁)(𝐸𝜆;𝑖→𝑓)will be available for different values of 𝐸for each basis, slightly altering the shape of the peak. In summary, both methods provide feasible ways of calculating 𝑑𝐵(𝐸1) 𝑑𝜀 . 3. electric dipole excitation 37 Figure 3.3: The 𝐸1transition probability, 𝑑𝐵(𝐸1) 𝑑𝜀 , for 1 2 + →3 2 −(𝑐𝑜𝑛𝑡.), in terms of the size of the THO basis. The distributions have been calculated using a Gaussian smearing function, Eq. (1.36). The basis parametre is fixed to 𝛾= 1.5fm1 2. In order of making a more meaningful comparison, Figure 3.4 shows the converged transition probability distributions (𝑁= 41). In both cases, 1 2 −and 3 2 −, the transition probabilities calculated with both methods are indistinguishable for most of the energy interval considered. Only for 𝐸 ≲ 0.5MeV, in the vicinity of the maximum, there is a minor deviation between the approaches; the peak is more pronounced when using the spline. In addition to the contributions discussed, the total transition probability is also shown; i.e., the sum of the 1 2 −and 3 2 −distributions. To be able to calculate the total transition probability it is necessary to have the continuous distribution for both contributions, as the discrete representation of the continuum differs between 1 2 −and 3 2 −. Therefore, if we were to apply merely Eq. (1.34) (first method), we could obtain the strength of each of the transitions evaluated at certain energies, but it would be unfeasible to combine the strength of both contributions. In contrast, the smearing function (second method) directly provides a smooth distributions, hence this difficulty is not encountered. In the following discussion, the second method is used when comparing with the experimental data, as it appears to reproduce them more accurately. To further asses the validity of the theoretical analysis, it is confronted with experimental data. In Figure 3.5, the total transition probability calculated with the Gaussian smearing function is plotted against the results extracted from the Coulomb dissociation experiments from Ref. [31]. The model employed provides realistic results for the most part of the continuum. The single-particle model depicts a satisfactory description of the transition probability for the continuum without the inclusion of core excitations. Note that, for the transition between the bound states, the single particle model largely overestimated the data, so this model fails to describe simultaneously both parts of the electric dipole transitions. 38 Figure 3.4: The figure shows 𝐸1transition probabilities, 𝑑𝐵(𝐸1) 𝑑𝜀 , for the 1 2 + →1 2 −,1 2 + →3 2 −and total transitions. Comparing, the distributions obtained using spline and smearing functions. The basis parametres are fixed to 𝑁= 41 and 𝛾= 1.5fm1 2. Figure 3.5: Experimental and calculated 𝑑𝐵(𝐸1) 𝑑𝜀 strength distributions. The experimental data from Ref. [31] is shown against the total strength distribution. The theoretical distributions correspond to the smearing function method. The THO basis has parametres 𝑁= 41 and 𝛾= 1.5fm1 2. 3. electric dipole excitation 39 3.4 Sum rule In this section, we are going to verify the fulfilment of the transition sum rule for the electric dipole transition. Equation (1.41) gives the transition strength between the ground state and the set of final states with a fixed total angular momentum 𝐽𝑓. As studied in during this chapter, the only states that permit 𝐸1transitions are 1 2 −and 3 2 −. Therefore, we must particularize Eq. (1.41) for both cases, and the total value of the sum rule will be the addition of the two contributions mentioned. The terms in Eq. (1.41) dependent on the final state are 2𝐽𝑓+1 2𝐽𝑖+1 and the reduced matrix element, ⟨(𝑙𝑓𝑠𝑓)𝐽𝑓||𝑌𝜆||(𝑙𝑖𝑠𝑖)𝐽𝑖⟩. For the transitions involving the 1 2 −states, these terms are evaluated as: 2𝐽𝑓+ 1 2𝐽𝑖+ 1 = 1 ⟨(1 1 2)1 2||𝑌1||(0 1 2)1 2⟩= − 1 √4𝜋 . Similarly, for those involving the 3 2 −states: 2𝐽𝑓+ 1 2𝐽𝑖+ 1 = 2 ⟨(1 1 2)3 2||𝑌1||(0 1 2)1 2⟩=1 √4𝜋 . Consequently, considering both contributions, the total sum rule can be expressed as: 𝑆𝑇=3 4𝜋(4𝑚2 𝑚1+𝑚2)2 𝑒2⟨𝑟2⟩𝑖.(3.5) The expression reached solely depends on the root-mean-square radius of the ground state. Experimentally, this property is used to extract the value of the rms radius from the strength of the 𝐸1 transition. We are going to compare the strength of the transition obtained through two different approaches. Firstly, we consider the sum of all of the contributions (discussed in Sections 3.2 and 3.3), as well as a transition to a bound Pauli state that will be studied in the subsection 3.4.2. Secondly, we will calculate the rms radius employing the wave functions constructed with the THO basis. Applying Equation (3.5), this yields a value for 𝑆𝑇. The transition sum rule will be satisfied if both results coincide. 3.4.1 Root-mean-square radius Building upon the previous discussion, this section addresses one of the methods employed to calculate the total transition strength. To do so, one must calculate the root-mean-square radius of the nucleus, which can be expressed as: 46 contributions for the 𝑝waves, the Gaussian forms are repulsive. Even so, the spin-orbit term may still be positive or negative depending on the value of 𝐽of the corresponding state, depending on the value of  𝑙⋅𝑠𝑛=1 2(𝐽2−𝑙2−𝑠2 𝑛). Table 4.1: The parameters (𝑆𝑐and 𝑆𝑠𝑜 in MeV; 𝑏𝑐and 𝑏𝑠𝑜 in fm) for the Gaussian potential taken from [34]. Waves 𝑠 𝑝 Parameters 𝑆𝑙=0 𝑐𝑏𝑙=0 𝑐𝑆𝑙=1 𝑐𝑏𝑙=1 𝑐𝑆𝑙=1 𝑠𝑜 𝑏𝑙=1 𝑠𝑜 -8.40 3.5 40.0 3.5 63.52 3.5 The Gaussian and Woods-Saxon potentials exhibit distinct characteristics, each offering advantages for describing different aspects of the nuclear structure. The Gaussian potential stems from the effective interaction between two nucleons and is particularly useful when describing few-body systems. In contrast, the Woods-Saxon potential is a mean field potential that describes the average interaction experienced by a nucleon within a nucleus and is more realistic for heavy nuclei. This potential exhibits a Fermi profile and has a long-range tail due to the exponential fall-off. This characteristic is advantageous for the description of a halo, as the long tail depicts the diffuse matter distribution. In contrast, the Gaussian form decays faster, poorly describing the long tail. Figures 4.1 and 4.2 show the form of the potentials for different states, for 𝑠and 𝑝waves, respectively. When examining the potential corresponding to the ground state (Figure 4.1), it becomes evident that the shape assumed by each potential differs significantly. The Gaussian one is substantially shallower and weaker. For the excited state, (Figure 4.2), our model incorporates both the central and spinorbit terms of the potential. Once again, the Woods-Saxon potential has a greater depth. In addition, it also diverges at the origin. 02468 r (fm) -60 -40 -20 0 V (MeV) Woods-Saxon Gaussian Figure 4.1: Woods-Saxons and Gaussian potential for 1 2 +with parametres from Tables 2.1 and 4.1. 02468 r (fm) -40 -30 -20 -10 0 V (MeV) Woods-Saxon Gaussian Figure 4.2: Woods-Saxons and Gaussian potential including the spin-orbit term, for 1 2 − with parametres from Tables 2.1 and 4.1. 4. comparison between two potentials 47 4.1 Energy levels Having established the potential in the previous section, we now proceed to analyze the resulting energy spectra. As we aim to reproduce the electric dipole transition, we are interested in the spectra of the 1 2 +,1 2 −and 3 2 −states. In the same manner as with the Woods-Saxon potential, we are going to employ the matrix formalism to expand the Hamiltonian on a finite THO basis. During this chapter, for simplicity the basis parametres will be fixed as 𝛾= 1.5fm1 2and 𝑏= 1.4fm. For the bound states, we shall study the 1 2 +,1 2 −states. For 1 2 +, we now only obtain one bound energy level, at 𝐸= −0.5025 MeV. This is a consequence of the well being adjusted to intentionally exclude the Pauli state for 1𝑠1 2 . For 1 2 −, the negative eigenvalue appears at 𝐸= −0.1828 MeV. Both energy levels agree in great measure with the experimental values found for 11Be; 𝐸= −0.502 MeV and 𝐸= −0.182 MeV respectively. In Particular, the value obtained for 1 2 −more accurately reflects the experimental observations than the Woods-Saxon potential ( 𝐸= −0.1916 MeV). This is due to the parametres in [34] being more carefully chosen to accurately represent the energy spectrum. Furthermore, for these basis parametres the convergence of the bound state energy levels is rapid, stabilizing for 𝑁∼ 15. When studying the 𝐸1transition, we are also concerned about the representation of the continuum of the 1 2 −and 3 2 −. Firstly, represented in Figure 4.3 are the energy levels up to 6MeV obtained for 1 2 −with a basis of size 𝑁= 25 for the Woods-Saxon and Gaussian potential. Below the 0MeV threshold we find the corresponding bound states. As for the discrete representation of the continuum, the spectra provide the same number of pseudostates, as they both only have one bound state. Moreover, they both portray the same behavior: closer to 0MeV the eigenvalues are closer together and begin to separate as the energy increases. The description, however, differs. The energy levels do not appear at the same values and this will affect the calculation of the 𝐸1transition. Figure 4.3: Energy spectrum of 1 2 −for 𝑁= 25 for the Woods-Saxon and Gaussian potential for 𝛾= 1.5fm1 2. 48 Secondly, in Figure 4.4 are the energy levels up to 6MeV obtained for 3 2 −with a basis of size 𝑁= 25 for the Woods-Saxon and Gaussian potential. It is worth noting that, for the Gaussian potential, we lack the Pauli state that appears for the Woods-Saxon potential. Consequently, the representation of the continuum for the Gaussian potential will be represented with one more pseudostate for an equally sized basis and there will be no need of calculating the value of (𝐸1;1 2 + →3 2 −)for the total transition sum rule, discussed in the previous chapter. Figure 4.4: Energy spectrum of 3 2 −for 𝑁= 25 for the Woods-Saxon and Gaussian potential for 𝛾= 1.5fm1 2. 4.2 Radial functions Having defined the potentials, an important consequence of the choice is its impact on the spatial extension of the wave function. An observable sensitive to the form of the potential is the root-meansquare radius, which provides insight into the spatial distribution of the nucleons. In particular, we are concerned by the rms of 11Be in its ground state as it determines the value of the sum rule for the electric dipole transitions. When comparing the rms radius for both potentials, we find that the matter distribution is more confined for the Gaussian potential. The rms for this potential is ⟨𝑟2⟩1 2 𝐺= 6.6733 fm, in comparison to the Woods-Saxon potential for which the rms is ⟨𝑟2⟩1 2 𝑊 𝑆 = 7.1613 fm. The absolute value of the radial functions, |𝑢(𝑟)|, obtained for the corresponding bases of dimension 𝑁= 41 are represented in Figure 4.5. For the Woods-Saxon potential, the radial function has an additional zero due to the Pauli state that appears for 1 2 +using this potential. Consequently, this forces the matter distribution to be centred further away from the origin, in order to preserve the normalization of the function. The obtained rms is larger and represents how the valence neutron extends further away from the core. In contrast, for the Gaussian potential, the radial function lacks zeros; it only vanishes at the 4. comparison between two potentials 49 origin and when the radial coordinate tends to infinity. The decay of distribution is more rapid, suppressing the wave function at large distances. The matter distribution is centered closer to the origin. This may result in an underestimation of the spatial extension of the halo, leading to a less realistic behavior of the valence neutron. To place our results in context, it is essential to contrast them with the experimental data. Previously, the experimental data calculated the rms to be ⟨𝑟2⟩1 2 𝑔.𝑠 = 5.77 ± 0.16 fm [31]. However, this value was deduced from the value of the total sum rule, where the contribution between the two bound states was not taken into account. Therefore, this value probably underestimates the rms of 11Be. Regardless, for both potentials, the theoretical calculations give us a larger value for the rms in the ground state. In fact, they both present significantly larger values of the matter radius than the surrounding nuclei in the nuclear chart; portraying the halo characteristic of the 11Be nucleus. Figure 4.5: Absolute value of the radial function |𝑢(𝑟)|for the Gaussian and Woods-Saxon potential for bases with 𝑁= 41 and 𝛾= 1.5. 4.3 Electric dipole transition In this section, we are going to examine the electric dipole transition when utilizing a Gaussian potential. To begin with, we have calculated the transition probability between the two bound states making use of Eq. (3.2) and obtained a transition probability of (𝐸1; 1 2 + →1 2 −) = 0.3661 e2fm2, using a THO basis of parametre 𝛾= 1.5fm1 2. Convergence was reached for 𝑁∼ 16. The value obtained is greater than with the Woods-Saxon potential, where (𝐸1; 1 2 + →1 2 −)=0.2728 e2fm2; a calculation that already overestimated the experimental data. Here, for the Gaussian potential, 50 we can also account for a portion of the overestimation to the simplified model, which does not take into account the structure of the 10Be core. However, comparatively, this potential more inadequately models the dipole transition between the bound states. For this potential, there is no need to calculate (𝐸1; 1 2 + →3 2 −), as it lacks a Pauli state for 3 2 −. Nevertheless, it is worth noting that if we add both of the contributions to the 𝐸1transition between bound states for the WS potential, they add up to 0.3697 e2fm2. This magnitude has a similar magnitude than (𝐸1; 1 2 + →1 2 −)for the Gaussian potential. To continue, we are going to study the electric dipole transitions from the ground state into the continuum. This continuous distribution has two contributions; transitions that end either in 1 2 −or 3 2 −unbound states. For bases with 𝑁= 41 the transition probabilities have been calculated making use of Eq. (1.34) and the continuous distribution has been obtained with the convolution method (Eq. (1.36)). This is represented in Figure 4.6. It is evident from the figure that 𝑑𝐵 𝑑𝐸 (𝐸1) distribution obtained with the Gaussian potential fails to reproduce the experimental data from the RIKEN accelerator [31]. It substantially underestimates the transition probability for all of the energy range considered. This deviates from the results given by the Wood-Saxon potential, where the experimental data adjusts to the theoretical curve obtained. From this comparison, it becomes apparent how one of the potentials is better suited for describing halos and their distinctive properties. Moreover, the figure also depicts the transition probability for 1 2 + →1 2 −(𝑏𝑜𝑢𝑛𝑑); comparing the theoretical calculations to the experimental value from [29]. Figure 4.6: (𝐸1) strength function for both the Gaussian (red dashed) and WS (solid blue) potentials. These are compared to experimental values, both for the transition between bound states (𝜀 < 0) [29] and the continuous distribution (𝜀 > 0) [31]. 4. comparison between two potentials 51 When studying the electric transition probability, it is necessary to verify the transition sum rule. It not only gives us information about the transition itself, but also validates that the basis used satisfies the completeness relationship; a feature of the basis that has been used when deducing the expressions employed to calculate the transition probabilities. The corroboration of the sum rule is shown in Figure 4.7. On one hand, strength of the transition is calculated in terms of the rms in the ground state using Eq. (3.5). It converges for 𝑁∼ 13 to a value of 𝑇(𝐸1) = 1.4247 e2fm2. On the other hand, the right-hand side of the figure portrays the significance of each of the contributions, which add up to the same value for the total transition probability 𝑇(𝐸1). Comparing these results with the experimental data, (𝐸1) = 1.05 ± 0.06 2e2fm2[31], considering as before only the contributions due to transitions into the continuum, this potential provides an almost identical value, 1.0586 e2fm2. It seems to be in better agreement with experimental measurements than the Woods-Saxon form. This is a coincidence, as it has been shown that this potential poorly describes the different electric transition probabilities for the 11Be nucleus; both the (𝐸1,1 2 −𝑏𝑜𝑢𝑛𝑑)and continuous distribution are not suitably reproduced. Figure 4.7: The two methods of obtaining the sum rule are portrayed. The plotted points show the convergence of 𝑆𝑇(𝐸1)deduced from the rms in terms of the size of the basis 𝑁, for 𝛾= 1.5 fm1 2. While on the right-hand side of the figure, the strength of each type of transition to the total transition probability is displayed. Chapter 5 Summary and Conclusions This work is framed within the Nuclear Physics field. This area of physics stands out for its phenomenological character, as a result of the complicated nature of nuclear forces. The strong nuclear force fundamentally stems from the interaction between quarks, which is described by Quantum Chromodynamics. Working with these interaction is unfathomable at the nuclear level. Instead, the nuclear interactions are modelled with effective theories that capture the essential behavior of nuclei, with potentials whose parameters may be adjusted to the experimental data. Exotic nuclei test the limits of these models, providing further insight into nuclear interactions. In particular, the 11Be nucleus presents a parity inversion of the 1𝑝1 2 and 2𝑠1 2 levels with respect to the traditional nuclear shell model. It also displays a notably extense matter distribution and strong electric dipole transition. The main objective is to solve the time-independent Schrödinger equation for the 11Be nucleus and compute the aforementioned obsevables. This equation has analytical solutions only for certain potentials, but, in general, the solutions must be obtained using numerical methods. The calculations presented in this work were conducted using MATLAB. With this software the Schrödinger equation was solved using a matrix formalism, using a finite THO basis to describe the eigenstates of the Hamiltonian. The convergence of the results were studied in terms of the basis parametres in order to perform more efficient calculations. The nucleus is modelled as a two-body system (10Be −𝑛) which interacts with a phenomenological potential. We have compared the results obtained using two different potentials: the Woods-Saxon and Gaussian potentials. The 11Be nucleus presents an exotic halo structure and is characterized by having its two bound states in close proximity to the breakup threshold. Consequently, exciting the nucleus into continuum states requires little energy and the probability of electric dipole transition at low energies is large. To characterize this nucleus and its halo structure, we have calculated the most relevant observables, which are summarized in Table 5.1. To begin with, we obtained the energy levels for both bound states, which properly adjust to the experimental value. Then, we obtained the rms radius and as a result, the matter radius, which clearly portrays the diffuse matter distribution of the nucleus; the magnitude is significantly larger than surrounding nuclei in the nuclear chart. With 54 regard to the electric dipole transition, we first obtained the strength of the transition between the two bound states. The strength of this transition was overestimated and a further description is necessary to fully reproduce its behavior. In contrast, this same single-particle model accurately reproduces the electric dipole transition into the continuum for the Woods-Saxon potential. The Gaussian potential, however, fails to correctly describe this transition, making it evident how one of the potentials is more adequate when describing the particular features of the (𝐸1) strength of this halo nucleus. Finally, we also satisfactorily verified the transition sum rule for both potentials; not only providing further knowledge on the transition and the rms radius, but validating the completeness relationship of the finite bases used. Table 5.1: Results obtained for the 11Be nucleus. Observable Woods-Saxon Gaussian Experimental 𝐸𝑔.𝑠 (MeV) −0.5033 −0.5025 −0.50164 ± 0.00025 [14] 𝐸1𝑠𝑡 (MeV) −0.1916 −0.1828 −0.1816 ± 0.0010 [14] ⟨𝑟2⟩1 2 𝑔.𝑠 (fm) 7.1613 6.6733 5.77 ± 0.16 [31] (𝐸1,1 2 + →1 2 −)(e2fm2)0.2788 0.3661 0.102 [29] 𝑇(𝐸1) (e2fm2)1.6423 1.4247 1.05 ± 0.06 [31] In conclusion, the results obtained are satisfactory and allow for a robust physical interpretation of the one neutron halo. It is worth highlighting, upon comparing our results with the experimental data, that the sum rule obtained from the Coulomb breakup does not take into account the (𝐸1) between bound states, as well as only considers transitions up to 4MeV. Therefore, 𝑇(𝐸1) and the extracted rms radius may be underestimated. Notably, when considering only the contributions included in the experimental procedure, our models show better agreement with the measurements. The discrepancies of the models with the experimental data are coherent and can be justified by their limitations. 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