This three-week TALENT course on nuclear theory will focus on the interpretation of data on the structure of nuclei using the Nuclear shell model as main tool.

Format:

We propose approximately forty-five hours of lectures over three weeks and a comparable amount of practical computer and exercise sessions, including the setting of individual problems and the organization of various individual projects.

The mornings will consist of lectures and the afternoons will be devoted to exercises meant to shed light on the exposed theory, the computational projects and individual student projects. These components will be coordinated to foster student engagement, maximize learning and create lasting value for the students. For the benefit of the TALENT series and of the community, material (courses, slides, problems and solutions, reports on students' projects) will be made publicly available using version control software like git and posted electronically on github.

As with previous TALENT courses, we envision the following features for the afternoon sessions:

• We will use both individual and group work to carry out tasks that are very specific in technical instructions, but leave freedom for creativity.
• Groups will be carefully put together to maximize diversity of backgrounds.
• Results will be presented in a conference-like setting to create accountability.
• We will organize events where individuals and groups exchange their experiences, difficulties and successes to foster interaction.
• During the school, on-line and lecture-based training tailored to technical issues will be provided. Students will learn to use and interpret the results of computer-based and hand calculations of nuclear models. The lectures will be aligned with the practical computational projects and exercises and the lecturers will be available to help students and work with them during the exercise sessions.
• These interactions will raise topics not originally envisioned for the course but which are recognized to be valuable for the students. There will be flexibility to organize mini-lectures and discussion sessions on an ad-hoc basis in such cases.
• Each group of students will maintain an online logbook of their activities and results.
• Training modules, codes, lectures, practical exercise instructions, online logbooks, instructions and information created by participants will be merged into a comprehensive website that will be available to the community and the public for self-guided training or for use in various educational settings (for example, a graduate course at a university could assign some of the projects as homework or an extra credit project, etc).

Objectives and learning outcomes:

At the end of the course the students should have a basic understanding of

• Configuration interaction methods (nuclear shell-model here) as a central tool to interpret nuclear structure experiment
• Have an understanding of single-particle basis functions and the construction of many-body basis states built thereupon. Examples are basis states from a Woods-Saxon potential, harmonic oscillator states and mean-field based states from a Hartree-Fock calculation. The single-particle basis states are orthonormal and are used to construct a corresponding orthonormal basis set of Slater determinants.
• Develop an understanding of what defines an observable.
• Understand how theory can be used to interpret experimental quantities (separation energies and shell gaps for example).
• Understand how experiment is used to extract transition probabilities and information about ground and exited states.
• Understand how second-quantization is used to represent states and compute expectation values and transition probabilities of operators
• Understand how the Hamiltonian matrix is constructed from this orthonormal basis set of many-body states (linear expansion of Slater determinants)
• For nuclear systems like the \( sd \)-shell, essentially all nuclei can be studied using direct diagonalization methods. In the construction of the shell-model code during the first two weeks, the students will learn to compute the ground state and the excited states of selected \( sd \)-shell nuclei. This project applies to all students. During the last week, students can pursue more individually defined projects
• The students will also learn to understand the basic elements of effective shell-model Hamiltonians and how to interpret the calculated properties in terms of various components of the nuclear forces (spin-orbit force, tensor force, central force etc). We will provide the students with the necessary tools to perform such analyses.
• Understand how to use shell-model calculations to calculate decay rates and transition probabilities and relate these to various electromagnetic transition operators and operators for beta-decays and double-beta decays.
• Develop a critical understanding the limits of shell-model studies and how these can be related to interpretations of data such as results from in-beam and decay experiments.
• Understand how to use second-quantization to construct one-body and two-body transfer operators, overlap functions, spectroscopic factors and experiments related to spectroscopic factors.
• For the students which wish to follow a more computational path during the last week, iterative eigenvalue solvers will discussed. Similarly, efficient representations of many-body states and computations of Hamiltonian matrices will also be discussed.
• We will also discuss modern shell-model codes like NushellX. This suite of programs can be used by students to pursue their own projects. Applications of NushellX to the calculations of various observables will discussed and students who wish to use NushellX, can define individual projects during the last week.