Course 5: Theory for exploring nuclear structure experiments

The first course 5 was held from August 11 to August 29 in 2014 at GANIL, Caen, France and was organized in collaboration with the University of Basse- Normandie. The course materials are available at http://talent.ganil.fr/2014/.

Motivation, background and aims

The study of nuclear structure and the models used to describe atomic nuclei are experiencing a renaissance. This is driven by three technological revolutions: accelerators capable of producing and accelerating exotic nuclei far from stability; instrumentation capable of detecting the resulting reaction products and gamma radiation, often on an event-by-event basis, in situations where data rates may be many orders of magnitude less than has been traditional; and computing power adequate to analyze the resulting data, often on-line, and to carry out sophisticated theoretical calculations to understand these nuclei at the limits of stability and to unravel what they tell us about nuclei and their structural evolution. The newly gathered data are revolutionizing our understanding of nuclei and leading to great strides in the models used to describe them. The models of the last half-century are now being seen as merely projections onto a subset of nuclei (those near stability and onto the proton-rich side of the line of stability) of more general theoretical approaches.

As noted above, we cannot expect the same quantity of data on exotic nuclei that we have traditionally had. Therefore we need to develop better signatures of structure, better models, with fewer parameters and with the richness needed to describe a much larger variety of structures. From the student’s perspective, understanding the underlying theory, the differences in competing interpretations and how to distinguish them becomes all the more important. Finally, one of the best opportunities provided by the study of such a wider swath of nuclei is the ability to follow chains of isotopes (or isotones) over wide ranges of nucleon number, often spanning two or more major shells, to see how structure changes and how the occupation of very different nucleonic orbits affects the interplay of residual interactions and configuration mixing. It is the goal and motivation of this course to introduce and develop many of the tools needed to carry out forefront research in this new era.

Overview, format and objectives

Synopsis: The course was held from August 11 to August 29 in 2014 at GANIL, Caen, France and was organized in collaboration with the University of Basse- Normandie. The course ran full-time for three weeks and consisted of 45h of lectures, 45h of exercises and a project-based assignment of 2 weeks work. The total workload was approximately 170 hours, corresponding to 10 ECTS in Europe. The final assignment was graded with marks A, B, C, D, E and failed for Master students and passed/not passed for PhD students.

Overview: This three-week TALENT course on nuclear theory focused on the interpretation of data on the structure of nuclei. The aims were to familiarize the student with the phenomenology of nuclear structure and to discuss a wide variety of models and techniques to interpret such data. The theoretical bases of these approaches was discussed and practical exercise sessions were organized requiring the use of available computer codes as well as hand calculations. The set of lectures provided students with the tools necessary to perform calculations, and to use these calculations to interpret data from experiments on nuclei near and far from stability.

Format: We had approximately forty-five hours of lectures and a comparable amount of practical exercise sessions, including the setting of individual problems and the organization of mini-projects. The mornings will consist of lectures and the afternoons will be devoted to exercises and student projects. These two 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. As with previous TALENT courses, we envision the following features for the exercise 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 exercises and the lecturers will be available to help students and work with them during the exercise sessions.
Often, 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: At the end of the course students should have greatly expanded familiarity with a variety of nuclear models, should be able to carry out explicit calculations with these models and to think about new data on atomic nuclei in order to understand their structural implications, and should have a grasp of the evolution of structure with proton and neutron number and the drivers of such evolution. Efficient methods to estimate structure from a minimum of data and to predict the structure of unknown nuclei in the absence of data will be developed. The students will also have an enhanced ability to decide which observables and which experiments are key to further progress in a variety of structural domains (magic nuclei, near-magic nuclei, nuclei with emergent collectivity, well-deformed nuclei, both even- and odd-mass nuclei). More specifically, at the end of the course students should:

Be able to look up and interpret data on the evolution of nuclear structure.
Have a working knowledge of angular momentum or SO(3) theory in quantum mechanics and have a qualitative idea of its generalization to arbitrary Lie algebras.
Understand the uses and limitations of algebraic methods in many-body physics.
Be familiar with the basic models of nuclei (spherical shell model, Nilsson model, pairing models, geometric and algebraic collective models).
Be able to carry out simple calculations (hand and computer, for both even-mass and odd-mass nuclei) with a variety of the above models.
Be able to use all of the above models and techniques to understand drivers of shell structure and structural evolution.
Be able to exploit a number of phenomenological schemes to interpret data on nuclear structure and to estimate the properties of unknown nuclei.
Be familiar with a number of extensions of the above basic models and concepts such as, for example, symmetries and supersymmetries in Bose-Fermi systems, mixed-symmetry states in nuclei, quantum phase transitions, etc.

Detailed course content

Overview: Given that several sub-topics are involved and that students will have a variety of backgrounds and experience, it is important to give students the language needed to communicate. After this background is introduced we will begin by looking at the data on atomic nuclei. It has been said often in recent years that there are two general perspectives with which to view nuclei: a "microscopic" (literally "femtoscopic") approach focusing on nucleonic motion, quantum numbers and interactions, and a “macroscopic” approach that looks at the nucleus as a whole, as a coherent many-body system with its own properties. The course will look at both avenues to understanding nuclei, to their inter-relationships, and will illustrate how advances in one can illuminate the other. The development of nuclear models divorced from the context of the data that they need to interpret is often confusing and bewildering. Therefore, early lectures will provide an overview of the data on atomic nuclei spanning from light nuclei to superheavy species, looking at data ranging from integral observables such as masses and charge radii to specific spectroscopic data that provide insights into structure either from the nucleonic perspective or from the many-body perspective of nuclear symmetries, many-body quantum numbers, selection rules and the like.

Introduction to the course (1 hour-PVI and RFC)
Introduction to nuclear data and phenomenology (2 hours-RFC)
These early lectures will survey of the behaviour of atomic nuclei as a function of neutron and proton numbers and related valence nucleon numbers. Observables include integral observables such as masses and charge radii, and spectroscopic data on energy levels, patterns of level spins, R42 and other combinations of energy levels, B(E2) values, moments, E0 transitions. Patterns of these observables in sequences of nuclei lead to evidence for shell structure, shape/phase transitional regions, and emergent collectivity. The use of differentials of observables will be discussed. The beauty of simple patterns and regularities will be highlighted. Emphasis will be on the most efficient observables for various structural scenarios. This survey will be accompanied by a first interpretation in terms of simple models and phenomenological schemes, including schemes based on understanding the microscopic drivers of structural evolution. The neutron-proton interaction and the evolution of collectivity. Simple interpretations leading to schemes for the prediction of the structure of unknown nuclei and/or for evidence for shell structure. Empirical extraction of neutron-proton interaction strengths using nuclear masses. Discussion of how to evaluate different models, introducing the idea of “theoretical uncertainty”.
Exercises included: inspecting the data on specific nuclei and nuclear regions to deduce shell and collective structure, carrying out phenomenological analyses to correlate such data, and using these data and using ideas about the competition between pairing and the proton-neutron interaction to estimate the properties of unknown nuclei.
Simple collective models (3 hours-RFC)
A more detailed introduction to collective models as a guide to structure and to understanding what nuclei are doing. Relation to microscopic models and why nuclei do what they do: nucleon versus many-body degrees of freedom. The emergence of collectivity in nuclei. Simple collective models: vibrator (harmonic and anharmonic), models for transitional nuclei, axially symmetric rotor, tri-axial rotor, gamma-soft nuclei, Alaga rules and band mixing.
Exercises included: interpreting the data on nuclei with weak and strong collectivity, and on the transition regions between them: Identifying and interpreting phonon and multi-phonon spectra in “spherical” nuclei, using coefficients of fractional parentage to understand energies and B(E2) values in anharmonic vibrational nuclei, rotational and vibrational structures in deformed nuclei, calculations of rotation-vibration coupling. Parameters versus fit quality.
Basic theoretical tools for nuclear structure (3 hours-PVI)
Angular momentum in quantum mechanics: Clebsch-Gordan coefficients, 6j and 9j symbols. Wigner-Eckart theorem. Occupation-number formalism. Symmetry, anti-symmetry and mixed symmetry; coefficients of fractional parentage. The independent-particle model, harmonic oscillator, square well, basic behaviour of nucleon orbits, shell structure, simple residual interactions (delta, pairing).
Exercises included: (1) computation of 3nj coefficients, (2) construction of two-fermion and two-boson states in an m-scheme basis, (3) construction of many-particle states in a coupled basis, (4) interpretation of two-nucleon spectra such as 210Pb.
Algebraic methods for many-body physics (3 hours-PVI)
Algebraic formulation of the quantal many-body problem. Symmetries, quantum numbers, dynamical symmetries, partial and quasi dynamical symmetries. Basic notions of group theory with a qualitative discussion of generalized Clebsch-Gordan coefficients and their relation to coefficients of fractional parentage, of the generalized Wigner-Eckart theorem and Racah’s factorization lemma.
Exercises included: (1) U(3) dynamical symmetry of the harmonic oscillator, (2) introduction to the code ArbModel and its use in simple fermion and/or boson calculations.
The solvable nuclear shell model (3 hours-PVI)
Hartree-Fock approximation and the notion of mean field. A review of solvable correlated shell-model hamiltonians: Racah’s SU(2) model of pairing (seniority), Wigner supermultiplet SU(4) theory, Elliott’s SU(3) model of rotation.
Exercises included: (1) pairing and delta interactions between identical nucleons in a single-j shell (seniority), (2) partial conservation of seniority in a j=9/2 shell, (3) HO shell with neutrons and protons interacting through a quadrupole force [SU(4) and SU(3) symmetries]. All exercises will be done numerically with ArbModel; some will also be done analytically.
The shell model for deformed nuclear shapes – the Nilsson model (3 hours-RFC)
Discussion and derivation of the Nilsson model: intuitive “derivation” of the complex Nilsson diagram without calculation. Understanding the Nilsson quantum numbers. The Nilsson model for arbitrary shapes. Identifying specific Nilsson orbits from direct reaction data on deformed nuclei (fingerprint patterns). Role of pairing in the systematics of Nilsson states in deformed nuclei. Rotational motion in odd-mass nuclei and the coupling of single-particle and collective angular momenta. Coriolis coupling.
Exercises included: deducing the Nilsson diagram without calculation and estimating Nilsson wave functions by inspection, introduction to a code for Nilsson model calculations, interpreting illustrative deformed nuclei (with examples from the rare-earth region) in terms of specific Nilsson excitations and rotational spectra built on top of them, Coriolis calculations, developing Nilsson diagrams for nuclei with hexadecapole shape components by hand and by explicit calculation.
Geometric collective models (GCM) (3 hours-MC)
Collective coordinates. The quadrupole variables alpha and beta. Symmetries of the collective wave function in the intrinsic frame. Exactly solvable collective models: the harmonic quadrupole oscillator, the infinite square-well potential, the Davidson potential, rigid-rotor and soft-rotor models. An algebraic approach to geometric collective models.
Exercises included: interpreting the spectra on even-even nuclei with the GCM, dealing with the various parameters, explicit calculations with the GCM code.
The interacting boson model (3 hours-JJ)
Dynamical symmetries of the IBM, their quantum numbers, selection rules, and wave function, the symmetry triangle, symmetry breaking. Introduction to the group theory of the IBM and to relevant bases for IBM wave functions. The Consistent Q formalism (CQF) and Extended CQF (ECQF). Boson number dependence of IBM predictions. Shape phase transitions in the IBM and their connection to Landau theory. Order and chaos in the symmetry triangle. The Arc of Regularity and its role as an internal border in the triangle. Partial and quasi dynamical symmetries. Description of intruder (2p-2h) states.
Practical calculations with the IBM (1 hour-RFC)
Techniques for calculations of arbitrary nuclei using the ECQF: contours in the symmetry triangle. Understanding how changes in parameters affect the model predictions. Expected accuracy of the model for different observables. Different approaches to IBM calculations. Calculations, comparison with data, including, for deformed nuclei, the relation of IBM predictions to the Alaga rules. IBM calculations in alternate bases.
Exercises included: (1) IBM and dynamical symmetries (2) partial dynamical symmetry. All exercises will be done numerically with ArbModel; some will also be done analytically.
Pairing models (3 hours-AOM)
Review of basic nuclear structure concepts that are needed to understand the effects of a pairing force. Study of the main ingredients of the effective nucleon-nucleon interaction, construct a simple model of pairing and discuss some methods to solve the model Hamiltonian. In particular, the BCS solution and the concept of quasi-particles will be used to explain the pairing gaps and moments of inertia. Possible signatures of pairing phase transitions driven by angular momentum, temperature and mass number. Use of two nucleon transfer reactions as specific probes of pairing correlations, with particular emphasis on the enhancement of pair-transfers. Microscopic origin of the pairing force in nuclei.
Exercises included: (1) A simple model, consisting of two j-shells will be introduced and solved numerically to help understand the competition between the single-particle separation energy and the pairing strength and the concepts of pairing vibrations and rotations, in analogy to the more familiar case of quadrupole deformations. (2) Solution of the BCS equations and applications to some realistic cases. Fortran programs for the above will be provided.

Extensions of the basic models (15 hours)
The aim is to present a number of specialized topics to show how the concepts and models discussed previously can be applied to problems of current interest in nuclear physics. We propose five specialized sessions of 2-to-3 hours each, leaving some flexibility to organize short lectures on topics of particular interest to the students. The program of the last week is therefore still subject to change and iteration. Individual discussions are organized related to the students’ projects and in the last days the final project assignments are made.
Quantum phase transitions in nuclei (2-3 hours-NP)
Signatures of criticality in the evolution of the nuclear shapes across the neutron-proton plane. Overview of specific data indicating sudden structural changes in various isotopic and isotonic chains of nuclei. Examples from the interacting boson model and the geometric collective model. The X(5) model and the confined beta-soft model. Phase transitional behaviour in other models including power law potentials.
Exercises included: comparison of X(5) with data in nuclear transition regions, etc
Symmetries in Bose-Fermi systems and supersymmetries (2-3 hours-JJ)
Application of symmetry techniques to systems of interacting bosons and fermions. Bose-Fermi symmetry classifications. Nuclear supersymmetries U(6/4) and U(6/12). Discussion of examples of odd-mass and odd-odd nuclei in the platinum and gold region.
Exercises included: Bose-Fermi symmetries and partial Bose-Fermi symmetries. All exercises will be done numerically with ArbModel; some will also be done analytically.
Mixed-symmetry states in nuclei (2-3 hours-NP)
Neutrons and protons in the IBM. F-spin multiplets of nuclei. Mixed-symmetry and scissors states. F-spin and isospin.
Emergence of collective behaviour from ab-initio methods (2-3 hours-MC)
No-core configuration interaction methods. Convergence and extrapolation. Clustering in ab initio approaches. Emergence of rotational signatures. Multi-shell SU(3) correlations. Symplectic dynamics.
Exercises included: analysis of results from ab initio calculations.
Pairing in exotic nuclei (2-3 hours-AOM)
We will conclude the subject of pairing, by outlining some new phenomena expected in exotic nuclei, with particular emphasis on neutron-proton pairing along the N=Z line and pairing in weakly bound systems.
Exercises included: in-depth discussions of two recent experiments addressing these topics.

The organization of the day was as follows:

  Time 	           Activity
9am-12pm 	Lectures, directed exercises
12pm-2pm 	Lunch
2pm-6pm 	Hands-on sessions, computational projects
6pm-7pm 	Wrap-up of the day

Teachers and organizers

The teachers were

Mark A. Caprio (University of Notre Dame), mcaprio@nd.edu
Richard F. Casten (Yale University), richard.casten@yale.edu
Jan Jolie (University of Cologne)
Augusto O. Macchiavelli (Lawrence Berkeley National Laboratory), aomacchiavelli@lbl.gov
Norbert Pietralla (Technische Universität Darmstadt), pietralla@ikp.tu-darmstadt.de
Piet Van Isacker (GANIL), isacker@ganil.fr

The organizers were

Richard F. Casten (Yale University), richard.casten@yale.edu
Piet Van Isacker (GANIL), isacker@ganil.fr
Francesca Gulminelli (University of Basse-Normandie), gulminelli@lpccaen.in2p3.fr
Marek Ploszajczak (GANIL) and contact person, ploszajczak@ganil.fr

Audience and prerequisites

Audience: Students and post-doctoral fellows interested in models for nuclear structure, phenomenological techniques for interpreting and predicting the structure of stable as well as exotic nuclei. The material will be of interest, and accessible, to both theorists and theoretically inclined experimentalists, and will include learning the practical use of a wide variety of single-particle and collective models.

Prerequisites: Prospective student participants are expected to be familiar with standard mathematical methods for scientists and to have a practitioner’s knowledge of intermediate level quantum mechanics. Students who are not familiar with the above will be expected to study some selected material in advance of the course.