Развитие алгебраической теории коллективных движений атомных ядер тема диссертации и автореферата по ВАК РФ 01.04.16, доктор наук Ганев Хубен Ганев

• Ганев Хубен Ганев
• доктор наук
• 2020

Оглавление диссертации доктор наук Ганев Хубен Ганев

Contents

1 Introduction

1.1 Actuality of the problem

1.2 The aim and tasks

1.3 Scientific novelty

1.4 Scientific and practical significance of the obtained results

1.5 Method

1.6 Approbation

1.7 Publications

1.8 Structure of the dissertation

1 The phenomenological approach

2 The Interacting Vector Boson Model

2.1 The building blocks

2.2 The algebraic structure

2.2.1 The Up(3) 0 Un(3) chain

2.2.2 The SU(3) 0 UT(2) chain

2.2.3 The O±(6) chain

2.2.4 The U(3, 3) chain

2.2.5 The model Hamiltonian

2.2.6 The dynamical symmetries of the IVBM

3 The unitary dynamical symmetry limit

3.1 Dynamical symmetry

3.1.1 The Basis

3.1.2 The Hamiltonian

3.1.3 Tensor operators

3.1.4 Matrix elements

3.2 Application

3.2.1 Energy spectra

3.2.2 Energy staggering

3.2.3 Transition probabilities

4 Geometrical structure of the IVBM

4.1 Shape structure

4.1.1 The Up(3) 0 Un(3) limit

4.1.2 The 0(6) limit

4.1.3 The SU(3) 0 Ut(2) limit

4.2 The generalized IVBM Hamiltonian and its phase diagram

5 Triaxial shapes in the IVBM

5.1 The shape of the SU*(3) limit

5.2 Perturbation of the SU*(3) dynamical symmetry

5.2.1 The Majorana perturbation

5.2.2 Phase transition between O(6) and SU*(3) limits

5.3 The energy surfaces of real nuclei

5.4 Angular momentum projected energy surfaces

6 The U(3, 3) dynamical symmetry limit

6.1 Dynamical symmetry

6.1.1 The Basis

6.1.2 The Hamiltonian

6.1.3 Tensor operators

6.1.4 Matrix elements

6.2 Application

6.2.1 Energy spectra

6.2.2 The energy staggering

6.2.3 Transition probabilities

7 The orthosymplectic extension of the IVBM

7.1 Fermion degrees of freedom

7.1.1 Fermion dynamical symmetries

7.1.2 Bose-Fermi symmetry

7.2 Dynamical supersymmetry

7.2.1 The Basis

7.2.2 The Hamiltonian

7.3 Odd mass nuclei

7.3.1 The even-even core

7.3.2 Energy spectra

7.3.3 Transition probabilities

7.4 Simultaneous description of even-even, odd-mass and doubly odd nuclei

8 Chiral doublet bands

8.1 The even-even core

8.2 The energy spectra of odd-mass and odd-odd nuclei

8.3 Electromagnetic transitions

8.3.1 E2 transitions

8.3.2 M1 transitions

8.3.3 Numerical results

8.4 The B(M1) staggering

II The microscopic shell-model approach

9 The Proton-Neutron Symplectic Model

9.1 The symplectic geometry

9.2 The dynamical content. Motion groups

9.3 The collective and intrinsic coordinates

9.4 The physical meaning of the collective variables

9.5 The wave functions

9.5.1 The collective wave functions

9.5.2 The intrinsic wave functions

9.5.3 The permutational symmetry

9.5.4 The spin part

9.6 The GCM(6) model

9.7 Shell-model representations of the proton-neutron symplectic model

9.7.1 The shell-model classification of nuclear collective states

9.7.2 Determination of the symplectic bandhead

9.8 Structure of the model Hamiltonian and many-particle Hilbert space

10 Many-particle limits of the PNSM

10.1 Contraction of the Sp(12,R) Algebra

10.2 The U(6)-Phonon Model

10.3 The Coupled Two-Rotor Model

11 The PNSM matrix elements

11.1 Some U(u1 + u2) D U(u\) ® U(u2) isoscalar factors

11.1.1 Two-component boson system

11.1.2 The method of calculation

11.2 Tensorial properties of the Sp(12,R) generators

11.3 Matrix elements of the Sp(12,R) generators

11.4 Matrix elements of the collective potential

12 Structure of the low-lying positive-parity states

12.1 The model Hamiltonian

12.2 Application to 166Er

12.3 Application to 154Sm

12.4 Application to 238U

13 Structure of the low-lying negative-parity states

13.1 The model Hamiltonian

13.2 Application to 154Sm....................................................................23б

13.2.1 The energy spectra

13.2.2 The energy staggering

13.2.3 Transition probabilities

13.3 Application to 238U

13.3.1 The energy spectra

13.3.2 The energy staggering

13.4 The E1 transitions in the extended proton-neutron symplectic model

13.4.1 The PNSM collective observables

13.4.2 The wsp(12,R) algebra and its representations

13.4.3 Matrix elements of the dipole operator

13.4.4 Comparison of the B(E1) transition strengths with experiment for 154Sm

and 238U

14 Summary and conclusions 2бб

14.1 Main results of the dissertation

14.2 List of scientific publications on which the dissertation is based ......................2б0

Введение диссертации (часть автореферата) на тему «Развитие алгебраической теории коллективных движений атомных ядер»

Chapter 1 Introduction

1.1 Actuality of the problem

The nuclear structure physics is devoted to the study of the properties of nuclei at low excitation energies (typically up to 2—5 MeV). Excitations of the atomic nucleus manifest themselves in a large variety of phenomena, ranging from single-particle character, collective behavior to nuclear quasimolecular states. Nuclear physicists attempted to formulate these diverse excitation mechanisms in terms of nuclear models usually focusing on specific ways of excitations, often neglecting the links to other models developed to describe different types of phenomena. A long-standing goal of nuclear theory is to determine the properties of atomic nuclei from the interactions among the protons and neutrons. The atomic nucleus is a quantum many-body system which contains from few to couple of hundreds strongly interacting nucleons. The latter, in turn, are composed from s and d quarks. Taking into account the hierarchy in the physical world, the big hope of the theoreticians is to derive and explain the observed properties of nuclear structure from the fundamental lows of the nature and strongly interacting subparticles, constituting the protons and neutrons — namely, the quarks and gluons. Despite of the recent progress of the so called ab initio approaches (e.g., no-core shell model, coupled-cluster channel, Monte-Carlo shell model) in describing the properties of light nuclei using different high precision realistic interactions, inspired from the Quantum Chromodynamis — the theory of interacting quarks and gluons — and meson exchanged theory, the realization of this hope is still a huge challenge for particle and nuclear physicists. Instead, the modest aim of nuclear theory is to explain the experimentally observed nuclear phenomena at its own nuclear level, i.e. by considering the protons and neutrons as elementary particles. Indeed, for the energies of interest (1 — 10 MeV) in nuclear structure studies the subnucleon degrees of freedom can be suppressed.

The non-relativistic nuclear many-body problem requires the solving of the many-body Schrodinger equation

H ...,rA) = E ...,rA), (1.1)

which is formidable task. In attempt to obtain a tractable approximation to the many-body nuclear problem, various approaches based on different physical methodology have been proposed and used in nuclear structure physics: semiclassical, mean-filed, geometric, cluster, boson expansion,

algebraic, etc. Due to the enormous complexity of the nuclear systems and the lack of the complete and exhaustive knowledge of the strong nucleon-nucleon potential, there are variety of methods and models developed during the years for the successful description of various features of the atomic nuclei (see, e.g., [1, 2, 3]). These models differ in their assumptions and approaches and are characterized by varying degrees of microscopy: some are essentially macroscopic, describing primarily collective excitations of atomic nuclei in terms of nuclear shapes (e.g., rotations of deformed ellipsoids, vibrations), while others focus on the specific orbits of individual nucleons and the interactions among these nucleons. Generally speaking, these models are not contradictory, but rather complementary approaches to describing different manifestations of the behavior of nuclei. It is unrealistic to expect that any given model can explain all or even a dominant part of the observed nuclear phenomenology. Thus, we need many approaches to build up a complete picture because we will never have a single theory that simultaneously describes all aspects of collective motions in atomic nuclei.

Thus, not able to solve the many-particle Schrodinger equation (1.1) because of its complexity, we replace it by the following one

Hc^o = E (1.2)

for the restricted Hamiltonian H0, which we hope to solve. The Hamiltonian H0 is an appropriately chosen part of H that corresponds to some, e.g. collective, effects of nuclear dynamics using some particular criteria. For instance, as such a criterium can be used the irreducible decomposition of H with respect to some group G. Then, an additional integral of motion (or in quantum-mechanical terms, exact quantum number) appears. The physical picture of the nuclear structure that is described by the wave function thus provides a dynamical model of the nucleus, generated by the Hamiltonian H0. Depending on the nuclear potential, the energy can take discrete, continuous or combined discrete-continuous values. In the present work, we will consider only discrete values of the energy, for which case the space of solutions for the Schrodinger equation (1.2) (or (1.1)) is spanned by the square-integrable wave functions and the state space is the Hilbert space.

An alternative to the exact solution of the many-particle Schrodinger equation is the algebraic approach to nuclear structure, based on the symmetry. Symmetry is an important concept in physics. In finite many-body systems, such as atomic nuclei, it appears as time reversal, parity, and rotational invariance, but also in the form of dynamical symmetries [3, 4, 5, 6, 7]. Dynamical symmetry is arguably the most fundamental concept in physics. In nuclear physics, dynamical symmetry was already used implicitly in the early works of Heisenberg [8] and Wigner [9] and more explicitly exploited by Elliott [10] in his SU(3) shell model of nuclear rotations and in the seniority scheme for coupling nucleons in pairs [11, 12, 13, 14, 15]. However, the most useful and extensive application of the concept of dynamical symmetry was within the framework of the popular Interacting Boson Model (IBM) [16]. It turns out that a wide range of dynamical models are solvable by virtue of underlying dynamical symmetries. Indeed, almost all models of nuclear structure have an algebraic structure. In this respect, we give some definitions which are used in what follows.

By symmetry we will mean any set of transformations of the states of a nucleus that leave some important attributes of the nucleus invariant.

Definition: A symmetry group of a system is a group of transformations of the system that leave its Hamiltonian invariant.

There are several variations in the literature on the definition of a dynamical group. The following definition is most useful for present purposes.

Definition: A dynamical group for a Hamiltonian H is a Lie group of unitary transformations of the Hilbert space H of a system with Hamiltonian H such that the subspaces of H that are invariant under the transformations of the dynamical group are spanned by eigenstates of H.

A dynamical symmetry for a classical system is a group of transformations of the phase space for the system having the property that all possible dynamical motions of the system are given by sequences of transformations of an initial starting point. Moreover, starting from any initial state, it must be possible to reach all other states of the phase space by transformations that are contained in the dynamical group (mathematically expressed in terms of group orbits; see, e.g., Ref.[17]). The group orbits are of importance in the nuclear structure theory because they allow to interpret the collective motions in nuclei in classical terms. The collective motions are then defined as motions on a certain orbit hypersurface of a given dynamical collective group in the configuration space. Given a dynamical group and a suitable Hamiltonian, the classical dynamics of a system is expressible in terms of the Lie algebra of infinitesimal generators of the group. Further, as we will demonstrate throughout the present work, the classical model of collective motions in nuclei is quantized simply by constructing the unitary representations of the dynamical group. Then the Hilbert space for the quantized system becomes a carrier space of such representations. The infinitesimal generators of a given dynamical group comprise a so called spectrum generating algebra.

Definition: A Lie algebra g is said to be a spectrum generating algebra (SGA) (or a dynamical algebra) for a given Hamiltonian H if H can be expressed as a polynomial in the elements of g.

In quantum mechanics, the elements of the SGA are identified with the physical observables of the model, represented by Hermitian operators. Thus, a useful SGA for a model is one for which all the observables of physical interest are either elements of the SGA or simple functions of these elements. In this regard, we point out that a given symmetry may have many interpretations and the success of a certain model does not necessary justify any particular interpretation. Indeed, a given model has two complementary aspects — a mathematical structure and a physical interpretation. In confronting the prediction of a model with experimental data, one may or may not be testing both aspects of the model. For example, two models can share the same mathematical structure but describe different physics. Moreover, the two models may lead to similar observable properties. Then, the observation of such properties cannot distinguish between the two models. For example, the SU(3) algebraic structure consisting of quadrupole and angular momentum generators is differently realized in the Elliott model [10] and in the Interacting Boson Model [16]. Moreover, the two models may lead to identical energy and angular momentum spectra and have a complete set of quadrupole matrix elements in common, but nevertheless the SU(3) SGA's of the two models have different microscopic interpretation. This is an example when an abstract collective algebra can have more than one microscopic interpretation. Conversely, there are also

examples in which more than one algebraic structure can be associated with a given set of physical observables. Many other examples of spectrum generating algebras will be given in the text, both on the phenomenological and microscopic levels.

In the fully algebraic approach, in which the interaction between nucleons as well as the physical operators of interest are expressed in terms of the elements of a given SGA, the eigenvalue problem of the nuclear Hamiltonian is transformed from the solving a set of differential equations to the problem of its matrix diagonalization. Moreover, in the limits in which the interaction and physical operators can be expressed solely in terms of the invariant operators of a single group-subgroup chain, according to which the basis states are classified, one obtains analytical solutions for the eigenvalues and eigenfunctions. Such limiting cases are referred to as dynamical symmetry limits or simply dynamical symmetries.

Symmetry approach opens the possibility to study the properties of atomic nuclei phenomeno-logically. Usually, in the phenomenological models only few collective degrees of freedom are taken into account in the description of nuclear data. Thus, in attempts to describe a finite set of data for a system which exhibits some recognizable patterns, it is natural to pick a dynamical group whose irreps exhibit similar patterns and to construct a corresponding model description of the data. It is worth noting that the construction of the first successful models of nuclear structure was due to the observation of patterns in nuclear data and the emergence of simplicity in their interpretation. For instance, the observation of spins of odd-mass nuclei and single-particle separation energies led to their interpretation in terms of the nuclear shell model [18, 19], whereas the patterns in the quadrupole moments of odd-mass nuclei [20, 21] and their interpretation [22] similarly led to a simple collective model [23] and then to a unified model [24, 25] in which nucleons move in a deformable shell-model potential with vibrational and rotational degrees of freedom. Patterns are signatures of underlying symmetries and their recognition can be of enormous help in understanding the properties of atomic nuclei. Several symmetries of the nucleus as a system of nonrelativistic interacting nucleons that give rise to the total angular momentum, parity, third projection of the isospin quantum number, and permutational symmetry of nuclear states are in common use in nuclear structure physics. These are the exact integrals of motion which the nuclear wave function must satisfy. This make it possible to construct models of nuclear structure that preserve the exact integrals of motion. Such models are called kinematically correct models [26, 27]. The commutation relations that underlie the basic algebraic structures of sets of symmetry-related operators in quantum mechanics are also unavoidable, as for example, the operators of position and momentum coordinates which close on the Heisenberg-Weyl algebra.

Having chosen the dynamical group, one then has the flexibility to choose the irrep and the parameters of the Hamiltonian and other physical operators of the model. Often in different phenomenological models, their building blocks can be considered simply as auxiliary tools which allow to generate given algebraic structures with certain symmetry, regardless of their microscopic composite character. In this respect, many properties of atomic nuclei have been investigated using algebraic models, in which one obtains bands of collective states which span irreducible representations of the corresponding dynamical groups [3, 4, 5, 6, 7].

Besides in its simplest, phenomenological form, the algebraic approach exists also in a micro-

scopic version. A model is called fully microscopic (or simply microscopic) if the antisymmetriza-tion is completely involved in the construction of the many-fermion Hilbert space of nucleus. The Lie algebra of observables of the model, i.e. its SGA, is then expressible in terms of many-particle coordinates (position, momentum and spin). Moreover, in the microscopic models, the collective effects for example are derived from all the single-particle degrees of freedom. As will see further, this can be done in a very elegant way, using the group theory, by restricting the model many-body Hamiltonian to the Hilbert state space with a definite O (A — 1) symmetry, where A is the number of protons and neutrons; or which is the same, by projecting its O(A — 1)-scalar part [28, 29, 30, 31, 32]. In this way, in contrast to the phenomenological collective models in which the collective (rotational and vibrational) modes are postulated, in the microscopic collective models they are derived from the Schrodinger equation for the many-particle nuclear Hamiltonian.

According to the restricted dynamics approach [28, 29, 30, 31, 32], the restricted Hamiltonian H0 can be introduced by using the operator decomposition

H = Ho + Hi + H2 + ... (1.3)

with the first term acting within some subspace H(Ar) of the space , presented as a direct sum

H(A) = H(Ar,) © H(Ar") © .... (1.4)

In (1.4), A denotes all integrals of motion of the original many-body Hamiltonian H, whereas r denotes the characteristics of the subspace H(Ar). In the restricted dynamics approach both the Hamiltonians and spaces in which they act play equally important role. One may say that H0 is restriction of the Hamiltonian H on the space

The Hamiltonians H0 conserve additional quantum numbers, i.e. they possess additional integrals of motion that characterize the spaces involved. Based on the algebraic technique, there are two ways to do such a restriction. The first one is given by an irreducible decomposition of H with respect to some group G:

H = H(0) + H(1.5)

where H0 =

H (°)

is the G-scalar part of H, and H(m) with ¡ = (0) denotes the rest terms. The second method of restriction is based on the multilinear decomposition of H in terms of infinitesimal operators {Xk} of the group G, i.e. using SGA. Since the infinitesimal operators of the group G in the G-irreducible basis are diagonal with respect to G irreducible representations, this guarantees the restriction of H to

From this, it follows that the second kind of restriction within the restricted dynamics approach [28, 29, 30, 31, 32] is identical to the fully algebraic approach [3, 4, 5, 6, 7, 16] to nuclear structure. In particular, the restricted collective spaces H(Aw) that will be considered throughout the dissertation are characterized by a definite O(A — 1) (or equivalent to it symplectic bandhead) symmetry (u), which appears as an additional integral of motion. We will demonstrate also that for the phenomenological collective models their Hamiltonians act on the nonphysical Hilbert spaces H(Aw) with (u) = (0), obtaining in this way a very simplified collective dynamics. For the microscopic collective models, the collective irreducible

spaces in which the model Hamiltonians act are characterized by a definite O(A — 1) symmetry (w) = (0), thus providing the state spaces to be Pauli allowed subspaces of the many-particle nuclear Hilbert space.

There are many reasons [33, 34, 35, 36, 37] for expressing models of nuclear structure in terms of dynamical groups and spectrum generating algebras. This can be viewed by the many roles that the symmetry plays at each stage in the process of understanding the nuclear phenomena in terms of interacting protons and neutrons. In the present work, we will use the following operational sequence in understanding the nuclear structure [35]:

1) Observe the phenomena by recognizing patterns in the experimental data;

2) Construct phenomenological models to describe the patterns;

3) Define the limitations and the domain of validity (refine the models);

4) Understand a given phenomenological model in microscopic terms by embedding it in the microscopic nuclear theory (shell model) which can also be refined;

5) Use the symmetry of obtained microscopic model to define an appropriate shell-model coupling scheme and relevant model spaces, as well as to identify the kinds of shell-model configurations needed to describe the collective excitations in nuclei.

So, first, one can use the symmetry to interpret nuclear data. Indeed, algebraic models have proven to be powerful in correlating a large body of data on excited states in nuclei (cf. IBM [16]). The success of algebraic models is based on the fact that any pattern recognized in nuclear experimental data is associated with a certain pattern of symmetry. Indeed, the symmetry of a given system is what give it structure.

Second, the symmetry makes the model calculations easier to do: it helps one to classify basis states and calculate required matrix elements. However, this is only a part of what dynamical symmetry is about. The expression of a certain physical model in the precise language of group theory is an excellent test of whether or not the model is well defined. If, for example, the dynamics contained in the model is incomplete, this will become evident because the set of proposed dynamical motions will not form a group. An example of this is presented by the irrotational flows. One finds that a sequence of two irrotational flows may not be irrotational, i.e. the set of irrotational-flow rotations does not form a group.

Third, the algebraic structure reveals the symmetries of a given model which provide us with a connection between different models of nuclear structure. This is the case when two (or more) models share a common algebraic structure, or when one model has a dynamical symmetry algebra that contains another dynamical algebra of another model as subalgebra. Algebraic structures of collective models thus provide valuable techniques for probing the relationships between them and their microscopic structures. In this respect, dynamical symmetry appears as a unifying concept in terms of which different (phenomenological or/and microscopic) models of nuclear structure can be expressed.

It is well known that the shell model provides the basic formal framework for understanding nuclei in terms of interacting protons and neutrons. It makes two essential assumptions: first, that the building blocks of nuclei are interacting nucleons and second, that the effective interactions between them are not so strong as to completely destroy the shell structure of the underlying

independent-particle motion. Although other shell models can be constructed, based on more realistic single-particle potentials, such as the Woods-Saxon or Hartree-Fock potentials, the harmonic oscillator shell model has huge practical advantages and through the natural mechanisms it provides for identifying the relevant shell-model spaces, incorporates the microscopic collective models much more directly. The harmonic oscillator shell model is an exceedingly rich algebraic model with a wealth of dynamical subgroup chains and solvable submodels. Thus, fourth, one can give a collective model a microscopic interpretation by embedding it in the shell model; i.e., by expressing it as a submodel of the shell model. This is possible provided the collective model is compatible with the shell model. An example of a complete algebraic model that is a submodel of the shell model is provided by the Elliott's SU(3) model [10].

When a collective model is embedded in the shell model, its wave functions become expressible in microscopic shell-model terms. We can then make use of all the microscopic observables of the shell model to ask more detailed questions of the dynamics than the model alone can answer. For example, the nuclear rotor model does not contain current operators. Thus, it cannot say if the current flows of a rotating nucleus are of a rigid- or irrotational-flow character. They could be some combination of the two. One can only make inferences about the current flows from the values of the moments of inertia. However, if rotor model states are embedded in the shell model, one can use the microscopic current operators to probe the currents and any other dynamical properties of interest. Furthermore, it becomes possible to learn what is missing from a model and what is needed to correct it.

Expressing phenomenological models as submodels of a more complete microscopic theory enhances both the models and the theory in which they are embedded. An example of this is the connection of the phenomenological Goldhaber-Teller model [38] of giant dipole resonance to the shell model carried out by Brink [39], who had showed that giant dipole excitations of the GT model appear as a coherent superposition of shell-model excitations. Thus, a GT model was given a microscopic foundation. Conversely, the physical content of the more detailed shell-model description of the dipole resonance was exposed as being driven by the simple dynamics of its phenomenological submodel.

The embedding problem becomes straightforward once it is recognized that both the shell model and the collective models are algebraic models with dynamical groups. Thus, a collective model becomes a submodel of the shell model if its dynamical group is expressed as a subgroup of a dynamical group for the shell model. An algebraic model is one whose fundamental observables are elements of a Lie algebra. The full Lie algebra of observables of the shell model is huge (infinite) which is why the shell model (with major-shell mixing) is an unsolvable problem and why we have to make simpler models. Fortunately, it has a subalgebra which is easier to manage; i.e., the Lie algebra of all one-body operators. The corresponding dynamical group is then the group of one-body unitary transformations. The essential point is that the whole shell-model Hilbert space spans a single irrep of this Lie group (and of its Lie algebra).

By embedding different models in a common formal theory, as is the shell model in nuclear structure physics, one can hope to understand the extent to which models complement each other and together provide an interpretation of a wide range of phenomena. If a model does not really

fit in such a framework, it may be appropriate to replace the model with a better or more realistic one that does.

Having a given algebraic model embedded in the shell model, one can further use the symmetry to define a relevant shell-model coupling scheme for detail microscopic calculations of nuclear properties and to provide a natural mechanisms for identifying the relevant shell-model subspaces. There might be different optimal coupling schemes for different classes of nuclei: closed-shell, singly closed-shell, and doubly open-shell nuclei. For instance, the spectra of single closed-shell nuclei indicate a pair coupling scheme in a spherical harmonic oscillator shell-model basis, whereas the deformation and rotational bands indicate collective model and, e.g., the SU(3) D SO(3) coupling scheme. In this way, the dynamical groups and SGA's provide us with a symmetry-adapted basis in which different kinds of nuclear dynamics are most naturally expressed; i.e. a basis with the required many-nucleon correlations. In the present work we are interested in heavy strongly deformed nuclei, so the most physically relevant many-body basis that naturally incorporates their collective properties (deformations and rotations), as we will try to convince later, is provided by the (one- and two-component) symplectic scheme basis in which the Elliott's SU(3) dynamical subgroup plays a central role.

One of the advantages of formulating a given model algebraically, in terms of a dynamical algebra, is that this also allows the model to be expressed clearly and succinctly so that its physical content becomes transparent. This exposes the limitations and possible extensions of the model. In Part I of the present dissertation we will see, e.g., how to extend the Interacting Vector Boson Model (IVBM) [40] by inclusion the fermion degrees of freedom so that it becomes able to describe the collective properties of odd-A and doubly odd nuclei. Further, in Part II we will see that the IVBM dynamics is of irrotational-flow type, just like in the Bohr-Mottelson (BM) collective model

[41].

Additionally, it turns out that in many situations in which the presence of large symmetry-braking interactions, invalidating the use of dynamical symmetry limits, disturb/distort the exact dynamical symmetry but remain many of its algebraic properties, albeit in an average quasi-dynamical sense. For example, the persistence of rotational structure in spite of symmetry-breaking interactions is well understood in phenomenological terms. Recall that, if it were not for the centrifugal and Coriolis forces, there would be a rotational analogue to Galilean invariance. This means that for nuclei with large moments of inertia, that are perturbed little by the centrifugal and Coriolis forces in states of relatively low angular momentum, there should be an effective separation of the intrinsic and rotational variables. Thus, while spin-orbit and pairing forces are known to mix rotational bands, it is expected that their primary effect is to induce vibrational fluctuations in the intrinsic shapes of well-deformed rotational nuclei but allow the rotational dynamics to survive. An expression of this concept of adiabatic decoupling of different degrees of freedom in the language of group representation theory has led to the concepts of embedded representations [42] and quasi-dynamical symmetry [43, 44]. In simple terms, quasi-dynamical symmetry is explained as a possibly strong coherent mixing of near-to-equivalent irreps of the dynamical group. In this way, the introduction of a concept of quasi-dynamical symmetry based on the precise mathematical concept of adiabatic or embedded representations enrich the realm

of group-theoretical approach to nuclear structure. The embedded (adiabatic) representations of a group or Lie algebra can be defined as the weighted average of a number of irreps. In these situations, rotational dynamics (presented by adiabatic mixing of different irreps) is adiabatically decoupled from the other (high-energy) degrees of freedom and the observed rotational structure of nuclei is preserved. As we will see in Chapters 12 and 13, the concept of an adiabatic representations is undoubtedly an essential ingredient of many kinds of realistic collective motions. We will see also that in model calculations for nuclei in which rotational dynamics is dominant, the collective states are highly coherent mixtures of relatively few irreducible representations of the appropriate dynamical groups.

Another advantage of expressing a model algebraically is that new physical interpretation may emerge. In other words, they may be more than one way of identifying the observables of an algebraic model with physical quantities. In this regard, the algebraic (group-theoretical) structures of a given collective model are particularly useful in finding all the possible realizations (unitary irreducible representations) of a model in quantum mechanics. For example, as Wigner has showed in his classical paper [45], the quantization of the space-time geometry of a free relativistic particle leads to wave functions for a particle with well-defined four-momentum but with no intrinsic spin. The extra spin degrees of freedom appear in the theory as non-abelian gauge degrees of freedom. A similar example appears in the one-component microscopic theory of nuclear motion. The configuration space for quadrupole vibrations and rotations is given by the five-dimensional space of nuclear shapes whose coordinates are the quadrupole moments of the nucleus. This space can be generated by the general linear group GL(3,R), which generators together with quadrupole moments {Q2^} span a Lie algebra of the general collective motion group in three dimensions GCM(3). The group GCM(3) has a structure very similar to that of the Poincare group and like it the GCM(3) has representations carried by the collective wave functions with intrinsic spin [46]. However, in the collective model, the intrinsic spin has the physical significance of vortex angular momentum [47, 48] (also called pseudo-momentum [49]). Moreover, the addition of vorticity to the collective model as an intrinsic gauge degree of freedom results in a complete range of possible collective flows from irrotational-flow (zero vorticity) to rigid rotations. In this way it was shown [33, 34, 35, 36, 37] that the collective model of Bohr and Mottelson admits microscopic realization first by augmenting it by vorticity degrees of freedom, important for the appearance of low-lying collective states, and second by making it compatible with the composite many-fermion structure of the nucleus. The result is the one-component SSp(6,R)1 symplectic model [50] of nuclear collective motion, called also a microscopic collective model, which is a submodel of the nuclear shell model. Similarly, in the present work we will see how the addition of an intrinsic structure, related to the vortex degrees of freedom, to the phenomenological IVBM leads to formulation of a theory of collective motions in the two-component proton-neutron nuclear systems compatible with the composite many-nucleon structure of the nucleus. This latter case is an example when two physical models, subject to Part I and II respectively, share a common mathematical structure but have different quantum-

1 Throughout the present work, we will use the notation Sp(2n,fi) for the group of linear canonical transformations in 2n-dimensional phase space. Some authors denote the Sp(2n,fi) group by Sp(n,fi).

mechanical realizations.

The one-component symplectic model, as will see further, is important in the theory of nuclear structure because of its clear relationships both to phenomenological models and microscopic theory. But its real value lies in its ability to interpret the achievements and defects of phenomenological collective models in microscopic terms because the symplectic model provides a very general microscopic framework that embraces both the phenomenological collective models and microscopic shell model theory. The whole shell-model Hilbert space can be expressed as a sum of irreducible symplectic subspaces. Thus, in principle, any shell-model state can be expressed in a basis of symplectic model states which thereby exposes the collective model content of microscopic wave functions.

The present work explores the properties of atomic nuclei that are indicators of the emergence of simple dynamics associated with the symmetries that are available to a nucleus as a many-body system. In particular, it focuses on the dynamical symmetries associated with the nuclear collective motions. Symmetries that provide good quantum numbers, or even approximately good quantum numbers, are obviously useful. However, symmetry can be used to decompose the nuclear Hilbert state space into a sequence of invariant subspaces with respect to a given dynamical algebra, ordered such that their contributions to the observed states of nuclei are of decreasing importance. A standard example of this is given by the symmetry of the three-dimensional spherical harmonic oscillator shell model which proves to be particularly relevant for many states of near closed-shell and singly closed-shell nuclei. An alternative, based on the dynamical symmetry of the six-dimensional harmonic oscillator which proves to be much more appropriate for the rotational states of heavy deformed nuclei, is presented in this work on both phenomenological and microscopic shell-model levels.

1.2 The aim and tasks

The ultimate aim of the present dissertation is the development of a microscopic theory of nuclear collective motions aimed at understanding the nuclear collective dynamics in terms of interacting protons and neutrons. In realizing this aim we use the following powerful algebraic strategy of constructing such a microscopic theory. It starts with a phenomenological model in terms of a Lie algebra of observables, which is able to capture many of the collective properties of atomic nuclei. Further we seek a microscopic, many-particle realization of this algebra in terms of the position and momentum coordinates of the particles of the system. It then remains to identify the relevant shell-model irreducible representations of this algebra (known as a spectrum generating algebra or simply dynamical algebra) to obtain a microscopic version of the model.

In pursuing this strategy, it is often necessary to adjust the considered phenomenological model so that its algebraic structure becomes compatible with that of the microscopic system it represents. However, often in combining different degrees of freedom, in the attempt to understand a given complex system, new intrinsic gauge degrees of freedom appear. For instance, it was found that in order to combine the rotational and vibrational dynamics in the Bohr-Mottelson collective model [16] one needs the extra non-abelian SO(3) gauge degrees of freedom, corresponding to

the vorticity [33, 34, 35, 36, 37]. By adding the vorticity, one obtains a more realistic version of the model. Further, by doing this and expressing the model in terms of a dynamical group, the BM collective model becomes a submodel of nuclear shell model. In this way, the application of the algebraic strategy yielded a microscopic version of the BM collective model, augmented by the intrinsic vortex spin degrees of freedom and compatible with the nucleon structure of nucleus, known as Sp(6, R) symplectic model or sometimes called a microscopic collective model [33, 34, 35, 36, 37]. To see that the latter is a submodel of the shell model, one needs only to observe that the Sp(6, R) Lie algebra consists of all one-body operators, expressed as a bilinear combination of the position and momentum nucleon coordinates.

Following the above strategy in Part I we first develop and apply the symplectic and or-thosymplectic extensions of the phenomenological IVBM for the description of various nuclear phenomena in heavy strongly deformed nuclei. Various applications of the IVBM in this part can be considered as an application of the symplectic-based effective theory of nuclear collective motions in the two-component nuclear systems by using effective interactions and effective transition operators in appropriately chosen truncated collective spaces. Further, in Part II, we propose and develop a fully microscopic Proton-Neutron Symplectic Model (PNSM) of collective excitations that aims the microscopic shell-model description of collective properties of heavy nuclei, consisting of protons and neutrons.

In pursuing our strategy, the following tasks are formulated in achieving the aim of present dissertation:

• Clarifying the role of the symplectic Sp(12, R) dynamical group in the IVBM by applying the new dynamical symmetries, which arise as a result of the symplectic extension of the model.

• Study of the first few positive- and negative-parity collective rotational bands, for which new experimental data are obtained up to very high angular momenta, in the heavy even-even nuclei from the rare-earth and actinide mass regions, including some of the fine structure energy level effects.

• Study of the possible shapes in the IVBM. Construction of the phase diagram of IVBM.

• Examining the possibility of obtaining triaxial shapes in the phase structure of the IVBM.

• Study of the triaxiality in atomic nuclei within the framework of the IVBM within its irreducible symplectic collective space.

• Introduction and application of the orthosymplectic extension of the IVBM, which incorporate the fermion degrees of freedom, for studying the collective properties of heavy odd-mass and odd-odd nuclei. Simultaneous description of the low-energy collective spectra in the even-even, odd-A and doubly odd nuclei within the supersymmetric (orthosymplectic) extension.

• Study of the chiral rotation in doubly odd nuclei within the orthosymplectic extension of the IVBM.

• Developing of an algebraic microscopic theory of nuclear collective motions in the two-component many-particle proton-neutron nuclear systems.

• Examining the type and number of collective and intrinsic degrees of freedom in the many-particle two-component proton-neutron nuclear systems.

• Examining the possible collective flows in the PNSM and revealing of its dynamical content.

Construction of the simplest kinematically correct nuclear wave functions.

• Understanding the collective dynamics within the proton-neutron symplectic model in microscopic and macroscopic terms.

• Constructing of the shell-model representations of the PNSM in the many-particle Hilbert space.

• Study of the macroscopic, hydrodynamical limits of the PNSM.

• Development of the computational techniques of PNSM, including the explicit calculation of the required isoscalar factors and matrix elements of physically interesting operators.

• Application of the newly proposed PNSM for obtaining the microscopic shell-model structure of the low-lying positive-parity states in the strongly deformed heavy even-even nuclei.

• Obtaining of the microscopic shell-model structure of the low-lying negative-parity states in strongly deformed heavy even-even nuclei within the framework of the PNSM.

• Study of the low-lying collective E1 dipole strengths in the extended PNSM.

• Study of adiabatic decoupling of the rotational dynamics from other degrees of freedom. Examining the role of emerging quasi-dynamical symmetries.

1.3 Scientific novelty

In the dissertation new results are obtained for the first time, the main of which are:

• The symplectic extension of the IVBM is developed. The new dynamical symmetries which appear as a result of this extension are studied and applied for description of different nuclear phenomena in heavy even-even nuclei.

• The phase structure of the IVBM is obtained for the first time. The possibility of obtaining triaxial shapes is considered.

• The supersymmetric (orthosymplectic) extension of the IVBM is developed, which allow the collective properties of heavy odd-mass and doubly odd nuclei to be described.

• The orthosymplectic extension is applied for the first time for the description of chiral doublet bands in doubly odd nuclei.

• A fully microscopic PNSM of nuclear collective motions is formulated for the first time by considering the possible collective flows and symplectic geometry of the two-component nuclear system.

• The macroscopic limits of the PNSM, which appear for large dimensional representations, are further obtained. As a result, two simplified models of nuclear collective motion, expressed in simple geometrical terms, arise as macroscopic limiting cases of the PNSM.

• Computational techniques for the practical application of the PNSM at U(6) level are developed, including the explicit calculation of the required isoscalar factors and matrix elements of physically interesting operators.

• The PNSM is applied for the first time in obtaining the microscopic shell-model structure of the positive-parity states in well deformed heavy even-even nuclei, namely 166Er, 154Sm and

238 U

• Revealing of the dynamical and quasi-dynamical symmetries of the underlying protonneutron collective dynamics in the microscopic structure of positive-parity states in well deformed heavy even-even nuclei.

• The first application of the symplectic-based shell-model approach to the structure of negative-parity states in heavy nuclei, in particular to 154Sm and 238U.

• The central extension of the proton-neutron symplectic model with the semi-direct structure WSp(12, R) = [HW(6)]Sp(12,R) is proposed, which allows to include explicitly in the theory various many-particle correlations which lie outside the enveloping algebra of Sp(12,R).

• Study of the low-lying electric dipole strengths in heavy even-even nuclei in the extended PNSM.

1.4 Scientific and practical significance of the obtained results

By applying different DS limits of the IVBM in Part I, we are able to clarify the role of Sp(12, R) as a dynamical group of the possible collective excitations in the two-component nuclear systems. The latter, in turn, allows us to be more precise in the constructing of relevant model Hamiltonians, adequate for the description of a more complete proton-neutron collective dynamics.

The obtained results in the Part II are an important step towards the development of a practical, computationally tractable microscopic theory of nuclear collective motion in the heavy nuclei. From the conceptional point of view, the proposed symplectic-based approach provides a general shell-model framework for studying microscopically the observed collective excitations in the two-component many-particle nuclear systems and represents a further step towards the development of a more general and comprehensive microscopic theory of collective motion in atomic nuclei consisting of protons and neutrons.

The practical significance of the obtained results consists of the fact that the algebraic microscopic theory of proton-neutron collective excitations, which is formulated in the present dissertation, opens the path for studying the microscopic structure of strongly deformed heavy nuclei, for which the conventional shell-shell model techniques are very formidable even for the modern computing facilities. It allows to identify effectively the kinds of shell-model configurations needed to describe the rotational states of strongly deformed heavy nuclei and provides a practical way to involve different many-particle correlations (e.g., dipole, quadrupole, octupole, etc.) of interest in the theory.

Symplectic representation theory makes it possible to diagonalize a given collective model Hamiltonian in a basis of shell-model states.

1.5 Method

In the present work, we use the elegant part of mathematics known as a representation theory of Lie algebras and Lie groups or simply a group theory. The group theory is the precise way of

expressing mathematically different symmetry patterns, but for the present purposes it can just be thought as a multi-dimensional extension of the familiar three-dimensional angular momentum techniques.

The group representation approach allows the construction of a Hamiltonian of a system which is, or nearly so, invariant under a certain group of symmetry transformations. It then allows one to construct basis of states realizing the symmetry and to calculate explicitly the matrix elements of different physically interesting operators, themselves classified by the symmetry. This further allows many properties of atomic nuclei to be investigated using algebraic models, in which one obtains bands of collective states which span irreducible representations of the corresponding dynamical group.

Nucleus is a quantum many-body system of strongly interacting protons and neutrons. To obtain some information about the structure of atomic nuclei one needs to solve the many-body Schrodinger equation. However, with the increase of the number of particles and the number of available states one very rapidly arrives to the explosion dimensionality problem within the microscopic many-particle shell model. Thus, a truncation of a model space to the manageable size is required. In this regards, the algebraic approach provides us with different efficient truncation schemes on both the microscopic and macroscopic, phenomenological levels which lead to various fermion or boson models of nuclear structure, respectively. By doing this, the algebraic structure of a given model of nuclear motions provides also practical means for selecting appropriate finite dimensional Hilbert subspaces in which one can isolate a certain type of nuclear dynamics by using renormalized or effective interactions and renormalized transition operators (by exploiting effective charges) in the sense of Lee-Suzuki [51].

As we will demonstrate in what follows, the algebraic methods provide important insights and powerful tools for investigating nuclear structure of heavy nuclei.

1.6 Approbation

The results obtained in the present work were reported on many international conferences (see the list below) and were also discussed at different seminars, given at the Laboratory of Theoretical Physics (Dubna, Russia) and the Institute of Nuclear Research and Nuclear Energy (Sofia, Bulgaria).

THE INTERNATIONAL CONFERENCES (talks given by the author):

• 28th International Workshop on Nuclear Theory, Bulgaria, Rila Mountains, Bulgaria, June 21-26, 2009.

• 29th International Workshop on Nuclear Theory, Bulgaria, Rila Mountains, Bulgaria, June 20-26, 2010.

• Current Problems in Nuclear Physics and Atomic Energy, Kyiv, Ukraine, September 3-7,

• 31th International Workshop on Nuclear Theory, Bulgaria, Rila Mountains, Bulgaria, June

24-30, 2012.

• XX International School on Nuclear Physics, Neutron Physics and Applications, Varna, Bulgaria, September 16-22, 2013.

• 32th International Workshop on Nuclear Theory, Bulgaria, Rila Mountains, Bulgaria, June 23-29, 2013.

• XVI International Conference on Symmetry Methods in Physics, Dubna, Russia, October

13-18, 2014.

• XXI International School on Nuclear Physics and Applications, & International Symposium on Exotic Nuclei, Varna, Bulgaria, September 6-12, 2015.

• International Conference "Nuclear Structure and Related Topics", Dubna, Russia, July

14-18, 2015.

• LXV International Conference on Nuclear Physics "Nucleus 2015", Saint Petersburg, Russia, June 29-July 3, 2015.

• XXII International School on Nuclear Physics and Applications, Varna, Bulgaria, September 10-16, 2017.

• The International Conference "Nuclear Structure and Related Topics", Burgas, Bulgaria, June 3-9, 2018.

1.7 Publications

The main results of dissertation are published in the referred journal articles [A1-A19] and in the full text conference proceedings [B1-B11] (see the List of publications on which the dissertation is based on). These papers are published as follows: Phys. Rev. C - 11, Eur. Phys. J. A - 3, J. Phys. G: Nucl. Part. Phys. - 1, Nucl. Phys. A - 1, Int. J. Mod. Phys. E - 3, J. Phys.: Conf. Ser. - 3, EPJ Web of Conferences - 2, AIP Conf. Proc. - 1, conference book's proceedings - 5.

1.8 Structure of the dissertation

The dissertation consists of an Introduction, 12 Chapters, and a Conclusion with a list of the main results obtained by the author, references and a list of the publications on which the dissertation is based. The volume consists of two parts and is presented on 282 pages, contains 96 figures, 21 tables and a list of cited references consisting of 414 items.

The Part I represents the phenomenological approach to nuclear structure within the framework of the phenomenological algebraic Interacting Vector Boson Model [40] in its symplectic and orthosymplectic extensions. It contains 7 Chapters in which the new extensions of the IVBM are developed and exploited in a full account. Symplectic and orthosymplectic dynamical symmetries allow the change of the number of excitation quanta or phonons building the collective states providing for larger representation spaces and richer subalgebraic structures to incorporate more complex nuclear spectra.

The Part II represents the microscopic approach to nuclear collective motion. In this part it is shown how the phenomenological IVBM can be generalized in a way to be compatible with the

proton-neutron composite structure of the nucleus. Along this line, by considering the possible collective flows and the symplectic geometry of the two-component nuclear systems, a fully microscopic Proton-Neutron Symplectic Model is formulated as a generalization of both the IVBM and the one-component Sp(6, R) symplectic model. The latter is often referred to as a microscopic collective model of nucleus. In Part II it is proved that the IVBM is a very particular case of the PNSM and corresponds to the two-fluid irrotational-flow collective model of Bohr-Mottelson type, which contains only two irreducible Sp(12,R) subspaces — the even (scalar) and the odd (one-particle) nuclear Hilbert spaces that correspond to the case of even and odd-mass nuclei, respectively.

• In Chapter 2, the IVBM is presented: its building blocks, the general rotationally invariant Hamiltonian and the role of Sp(12, R) as a dynamical group of the model. The algebraic structure of the IVBM is given by a lattice of Sp(12, R) subgroups and the consideration of four dynamical symmetry limits, when the Hamiltonian can be written as a linear combination of the Casimir operators of a single reduction chain only.

• In Chapter 3, the unitary dynamical symmetry (DS), defined by the reduction

Sp(12, R) d U(6) d SU(3) 0 U(2) d S0(3) 0 U(1)

is considered aiming the description of the first positive- and negative-parity bands up to high angular momenta for many strongly deformed even-even nuclei from the rare-earth and actinide mass regions [A1,A10,B7]. The algebraic notion of the "yrast" states is introduced as states with a given L, built up of minimal number of vector excitations N — the eigenvalue of the total number of bosons. The correspondence between the observed collective states and the symplectic basis states, based on this notion, leads to the appearance of a vibrational term in the eigenvalues of the Hamiltonian, which affects the "yrast" energies. This term plays the role of an interaction between the different bands under consideration, and in particular is responsible for the correct reproduction of the odd-even staggering of the lowest positive- and negative-parity band energies.

For the practical application of this DS the symplectic basis in the Sp(12, R) irreducible space is constructed and the role of symplectic generators as transition operators between different basis states is clarified [A2]. Further, the tensorial properties of the Sp(12, R) generators are considered with respect to the the unitary DS chain, which allow the matrix elements of Sp(12, R) operators and any function of them to be calculated [A2] in a pure algebraic way by exploiting the generalized Wigner-Eckart theorem. With the help of the obtained matrix elements, the reduced B(E2) and B(E 1) transition probabilities between the collective states of the ground and first Kn = 0^ negative-parity band are compared with experiment, described earlier in this chapter.

• In Chapter 4, the geometrical structure [A6,B3,B6] of the IVBM which corresponds to a specific ground state configuration is obtained by means of the IVBM coherent states. The basis nuclear shapes — namely, the spherical, axially deformed prolate and y-unstable — are obtained in the Up(3) 0 U„(3), SU(3) 0 Ut(2) and 0(6) DS limits of the IVBM, correspondingly. The phase diagram of the generalized IVBM Hamiltonian is constructed and studied [A6,B3].

• In Chapter 5, it is shown how the triaxial shapes can be obtained in the IVBM by different perturbations of the SU*(3) phase structure [A7,B4]. In the end of this chapter, it is shown that

a more accurate angular momentum projection procedure [B6] changes the topology of the pure SU(3) and SU*(3) PES, leading to oblate and maximum triaxial shapes, respectively.

• In Chapter 6, the U(3, 3) dynamical symmetry [A8,A9,B5] of the IVBM which is defined by the reduction

Sp(12, R) D U(3, 3) D Up(3) 0 Un(3) D U*(3) D SO(3)

is introduced for studying the axial asymmetry in heavy even-even nuclei. This DS is appropriate for nuclei in which the one type of particles is particle-like and the other is hole-like, as for example, in the Os-Pt region. The symplectic basis is constructed along this DS chain [A8]. The effect of a Majorana interaction on the energy of the non-perturbed U(3, 3) DS Hamiltonian is examined [A7,B4,B5]. The inclusion of a Majorana term to the model Hamiltonian allows the range from a 7-rigid to 7-unstable structures of the 7-band to be covered by the considered perturbation of the U(3, 3) DS Hamiltonian. Further, the tensor properties of the symplectic generators with respect to the U(3, 3) DS are considered, which allow to calculate the matrix elements [A8] of the basic Sp(12,R) operators. With the help of the latter, the B(E2) and B(M1) transition probabilities [A8,A9] between the states of the ground and 7 bands are compared with experiment for 190Os and 192Os. The excitation energies of the ground and 7 bands in 190Os, 192Os and 112Ru are also well reproduced within the considered DS. Moreover, the perturbed U(3, 3) DS allows the odd-even energy staggering between the states of the 7 band, exhibiting a 7-rigid or 7-unstable characteristic structure, to be correctly reproduced.

• In Chapter 7, a supersymmetric (orthosymplectic) extension [A3] of the IVBM is introduced by incorporating the single-particle fermion degrees of freedom in a simple and straightforward way, involving in addition the fermion pair Lie algebra. Fermion, the Bose-Fermi and supersymmetry dynamical symmetries are shortly considered which still lead to exact analytical solutions for the odd-A and doubly odd nuclei. By considering the simplest particle-core coupled-type physical picture, e.g., the states of the odd-A nuclei are obtained as a result of the coupling of a particle with intrinsic spin taking a single j-value to a boson core whose excitation states belong to an Sp(12, R) irrep. By using an algebraic notion of yrast states (i.e. the states with given L, which minimize the energy with respect to the number of bosons N) the experimentally observed collective states of odd mass nuclei are mapped onto the SU(3) stretched states of the symplectic basis, which allows the energy levels of the ground and first excited bands to be reproduced very well up to very high angular momenta. The important role of the symplectic structure of the model for the proper reproduction of the intraband B(E2) behavior for odd-A nuclei is revealed [A3]. The success of the supersymmetric extension of the IVBM is based on the (ortho)symplectic structures which allow the mixing of the basic collective modes — rotational and vibrational ones — arising from the algebraic yrast conditions. In this way, because of the mixing of rotational and vibrational-like modes, it is demonstrated that various collective properties of the even-even underlying cores and the extended Bose-Fermi systems could be, respectively, obtained even within the framework of the same dynamical symmetry limit, in contrast to the other algebraic approaches (e.g. IBM) in which different dynamical symmetries or their mixtures are required. In the end of this chapter, the orthosymplectic extension of the IVBM is further fully exploited for the simultaneous description of the collective states in even-even, odd-mass and doubly odd nuclei for two sets of neighboring

nuclei with various collective properties [B2].

• In Chapter 8, the chiral rotation in some doubly odd nuclei from the A ~ 130 region is studied in the supersymmetric extension of the IVBM [A4,A5,B1], which still leads to exactly solvable limit that yields a simple and straightforward application to real nuclear systems. In the calculations, a consistent procedure that includes the analysis of the even-even and odd-even neighbors is employed, which leads to a purely collective interpretation of the chiral doublet bands. It is shown that the good agreement between theoretical and experimental band structures is a result of the mixing of the basic rotational and vibrational collective modes which is traced back to the level of the even-even cores. This allows for the correct reproduction of the high-spin states of the collective bands and the correct placement of the different bandheads. The important role of the symplectic terms entering in the corresponding transition operators is revealed for the correct reproduction of the behavior of both B(E2) and B(M1) strengths, which are crucial for establishing the nature of the twin bands. Additionally, for the case of 128Cs, it is demonstrated that the observed odd-even staggering of both B(E2) and B (M1) values could be reproduced by the introduction of an appropriate interaction term of quadrupole type [A5], which produces such a staggering effect in the transition strengths.

• In Chapter 9, a fully microscopic proton-neutron symplectic model [A11,B9] of collective motions is formulated by considering the symplectic geometry and possible collective flows in the two-component many-particle nuclear systems. The full dynamical group of the whole many-particle system allows the separation of the nuclear variables into kinematical (internal) and dynamical (collective) ones, which in turn allows to determine the number and type of collective degrees of freedom in the two-component many-particle nuclear systems purely by a group-theoretical consideration of the relevant coordinate transformation of the microscopic many-particle configuration space. The simplest kinematically-correct nuclear wave functions (i.e., microscopically translationally-invariant, which preserve the observed integrals of motion) are constructed [A11,B9] in terms of collective and their complementary intrinsic coordinates and are represented correspondingly as a product of collective and intrinsic components. Dynamical content of the PNSM is revealed by considering different motion subgroups of Sp(12, R), and in particular by constructing its GCM(6) submodel [A11]. It is also shown that the GCM(6) and Sp(12,R) models appear as hydrodynamic irrotational-flow collective models of the two-component nuclear system that include 21 collective irrotational-flow degrees of freedom, augmented by a SO (6) and U(6) intrinsic structure, respectively, associated with the vortex degrees of freedom.

Further, the representation theory of the PNSM in the many-particle shell-model Hilbert space is considered [A13]. The relation of Sp(12,R) irreducible representations of the PNSM to the shell-model classification of the basis states is considered by extending of the state space to the direct product space of SUp(3) 0 SUn(3) irreps, generalizing in this way the Elliott's SU(3) model for the case of two-component system. The Sp(12,R) model then appears as a natural multimajor-shell extension of the generalized proton-neutron SU(3) scheme which takes into account the core collective excitations of monopole and quadrupole, as well as dipole type associated with the giant resonance vibrational degrees of freedom. It is shown that each Sp(12,R) irreducible representation is determined by a symplectic bandhead or an intrinsic U(6) space which can be

fixed by the underlying proton-neutron shell-model structure, so the theory becomes completely compatible with the Pauli principle. It is demonstrated that this intrinsic U(6) structure is of vital importance for the appearance of the low-lying collective bands without involving a mixing of different symplectic irreps [A13,B9]. Finally, it is shown that the full range of low-lying collective states could, in principle, be described by a microscopically based intrinsic U(6) structure that is renormalized due to the coupling to the giant resonance vibrations. Finally, the structure of model Hamiltonian and many-particle Hilbert space is shortly discussed.

• In Chapter 10 are considered the many-particle (also referred to as macroscopic or hy-drodynamic) limits [A12,B8] of the PNSM which show how a given microscopic discrete system starts to behave like a continuous fluid and reveal further its physical content. The macroscopic limits, which take place at large dimentional representations, are obtained by purely algebraic way using the formal expansion-contraction group-theoretical procedure. The algebraic approach thus allows to interpret a given microscopic algebra of collective observables at macroscopic level in simple geometrical terms. Consequently, it is shown that as a result of the contraction, two new simplified macroscopic models of nuclear collective motion appear. The first one is the U(6)-phonon model [A12,B8] with the semi-direct product structure [HW(21)]U(6), which is shown to be actually an alternative formulation of the original proton-neutron symplectic model in the familiar IBM-terms. The second model which appears in double contraction limit is the two-rotor model with the ROTp(3) ® ROTn(3) D ROT(3) algebraic structure [B8]. The latter, in contrast to the original two-rotor model, is shown to be not restricted to the case of two coupled axial rotors. In this way, the second contraction limit of the PNSM is shown to provide the phenomenological two-rotor model with a simple microscopic foundation.

• In Chapter 11, the computational technique [A14,A15] required for practical application of the PNSM at U(6) level is developed. The aim is to obtain the PNSM matrix elements, which are obtained by purely algebraic way by using a generalized Wigner-Eckart theorem with respect to the symmetry-adapted basis of the PNSM. The application of this theorem depends upon the knowledge of the corresponding isoscalar factors (IFs) which were not available. Thus, as a first step it is shown what kinds of IFs appear in the diagonalization of the model Hamiltonian of the two-component many-body systems [A14]. Some of relevant isoscalar factors, needed for the calculation of PNSM matrix elements, are further obtained using a building-up procedure [A14]. With the help of obtained IFs, the matrix elements [A15] of the Sp(12,R) generators of the PNSM are next obtained in a U(6)-coupled basis in the space of fully symmetric representations. This allows further the matrix elements of any physical operator of interest, such as the relevant transition operators or the collective potential, to be calculated. As an illustration, the matrix elements of the basic irreducible tensor terms which appear in the U(6) decomposition of the long-range full major-shell mixing proton-neutron quadrupole-quadrupole interaction Qp • Qn are presented [A15].

• Using the obtained matrix elements for the collective potential, in Chapter 12 the PNSM is firstly applied [A16,A17,B10,B11] to the simultaneous description the low-lying states of the lowest ground, 3 and 7 bands in three strongly deformed heavy nuclei, namely 166Er [B10], 154Sm [A16] and 238U [A17]. For this purpose, the algebraic model Hamiltonian is diagonalized

in a SUp(3) ® SUn(3) symmetry-adapted basis for 166Er, and a U(6)-coupled basis, respectively, for 154Sm and 238U which is restricted to the state space spanned by the fully symmetric U(6) irreps. As a result, the microscopic shell-model structure of the collective states of the ground, 3 and y bands for the three nuclei under consideration is obtained. It is shown that a good description of the energy levels of these bands for the all three nuclei, as well as the intraband B(E2) transition strengths between the states of the ground band (and of y band for 166Er) is obtained without the use of an effective charge [A16,A17,B10,B11]. For 166Er, the results for the microscopic structure show the presence of a good SU(3) dynamical symmetry [B10]. For the other two nuclei 154Sm and 238U, we show the relevant SU(3) and U(6) decomposition of their wave functions. The calculations show that when the collective quadrupole dynamics is covered already by the symplectic bandhead structure, as in the case of 154Sm, the results show the presence of a very good U(6) dynamical symmetry [A16,B11]. In the case of 238U, when we have an observed enhancement of the intraband B(E2) transition strengths, then the results show small admixtures from the higher major shells and a highly coherent mixing of different irreps which is manifested by the presence of a good U(6) quasi-dynamical symmetry [A17,B11] in the microscopic structure of the collective states under consideration. The (parameter-free) results for B(E2) collectivity, obtained for the three nuclei 166Er, 154Sm and 238U, are shown to be very close to those of the phenomenological one-parameter rigid rotor model. The close agreement between the rigid rotor and the PNSM is a strong implication that they effectively describe the same rotational dynamics, albeit in the sense of quasi-dynamical symmetry. But, the most important point is that the PNSM calculations allow to identify the kinds of shell-model configurations needed to describe the rotational states of strongly deformed nuclei.

• In Chapter 13, the new microscopic theory is further applied for the first time for obtaining the microscopic structure of the low-lying negative parity states [A18,A19] of the Kn = 0]" and Kn = 1i bands in 154Sm and 238U without the introduction of additional degrees of freedom, inherent to other approaches to odd-parity nuclear states. For this purpose, the model Hamiltonian is diagonalized in a U(6)-coupled basis, restricted to state space spanned by the fully symmetric U(6) irreps of the lowest odd irreducible representation of Sp(12,R). It is demonstrated that the positive- and negative-parity collective bands can be treated on equal footing within the framework of the microscopic symplectic-based shell-model scheme. It is shown that a good description of the energy levels of the two bands under consideration, as well as the reproduction of some energy splitting quantities which are usually introduced in the literature as a measure of the octupole correlations, is obtained for these two nuclei. It is further shown that practically there are no admixtures from the higher shells in the microscopic structure of low-lying collective states with negative-parity in 154Sm and that this points to the presence of a very good U(6) dynamical symmetry [A18]. Additionally, it is shown that the structure of the collective states under consideration for this nucleus shows also the presence of a good SU(3) quasi-dynamical symmetry [A18]. For 238U, likewise the positive-parity states, the microscopic structure of the low-lying negative-parity states [A19] show small admixtures from the higher major shells and a highly coherent mixing of different irreps (a good U(6) quasi-dynamical symmetry).

Further, the low-energy E1 transitions are tackled within the framework of PNSM. In order

to study the electric dipole strengths, the explicit matrix elements of dipole operator are obtained [A19]. To achieve this, we make the central extension [A19] of the proton-neutron symplectic model that has the semi-direct structure wsp(12,R) = [hw(6)]sp(12,R), which in contrast to the sp(12,R) algebra among its generators contains the electric dipole operator, which allows to calculate the reduced E1 transition strengths. It is demonstrated that this extension introduces 1hw 1p-1h (one proton or one neutron Jacobi particle raised by one shell) excitations [A19] to the PNSM 2hw like particle 1p-1h (one proton or neutron Jacobi particle raised by two major shells) and proton-neutron 2hw 2p-2h (one proton and one neutron Jacobi particles raised by one shell) core excitations. In this way, all kinds of np-nh shell-model configurations with any even or odd number of harmonic-oscillator quanta are incorporated in the theory. It is demonstrated also that any collective operator of physical interest that can be brought in the form of an arbitrary function of either even or odd power in the many-particle position and momentum coordinates will lie in the enveloping wsp(12,R) algebra. Next, we calculate the required matrix elements of the dipole transition operator and compare theoretical values for the low-energy B (E1) transition strengths between the states of the ground band and Kn = 0]~ band in 154Sm and 238U with experiment. It is shown that the results obtained for the B (E1) values reproduce well the experimental data for the two nuclei under consideration without the use of an effective charge [A19], which could be considered as a significant achievement of the present approach.

Part I

The phenomenological approach

With the intensive development of the experimental facilities, a lot of new data on the collective spectra of nuclei throughout the nuclear chart is accumulated. It reveals the complicated character of the nuclear motions, yielding different degrees of mixing of the basic rotational and vibrational collective modes. In turn, this requires a corresponding degree of complication of the nuclear structure models, but still they should remain analytically solvable and so providing the tool for the interpretation of the observed data. This is most easily achieved within the framework of the phenomenological algebraic models of nuclear structure. Indeed, a large body of experimental data have been analyzed and structured using different algebraic models. The most popular among them is the Interacting Boson Model [16] which has been widely used to describe collective properties of vibrational, 7-unstable and rotational nuclei.

The phenomenological approach to nuclear structure, based on the revealing of hidden symmetries in nuclei, has been extensively used. In this approach it is assumed that as a model Hamiltonian one can consider

H0 = H0 + V (Xk), (1.6)

where H0 is the Hamiltonian of some simple system, like the harmonic oscillator or rigid rotor, having a symmetry group G. The potential V(Xk), expressed as a function of the generators {Xk} of some SGA g, beaks the symmetry group G of the Hamiltonian (1.6) to the rotational group SO(3) and in this way shifting the degeneracy of the G-multiplet of states generates the spectrum of the nuclear system. Usually, the potential is expressed solely as a function of Casimir operators (invariants) of relevant group(s), i.e. V = V(Ck). In this case, one immediately obtains analytic solution for the eigenvalues and eigenfunctions.

In many early works, it was assumed that the rotational states in many heavy nuclei can be considered as members of broken multiplets of different groups. For instance, in [52, 53] and [54] as such a group, to which multiplets of the experimentally observed states correspond to, was considered SL(3, R) and SU(3), respectively. What can be the reason for such an assumption? It should be pointed that the practical success of a certain algebraic model depends upon the choice of the relevant model observables. If only the few degrees of freedom essential to the collective phenomena are contained in the algebra, then the model will be a considerable simplification of the full many-body problem with 3A degrees of freedom. In this regard, we shortly review some of the possible spectrum generating algebras of physical observables, based on some general theoretical considerations, which proved to be appropriate in the description of nuclear collective motion.

It is well known that the heavy nuclei from the rare-earth and actinide mass regions have large quadrupole moments pointing to their nonspherical equilibrium shapes. Additionally, it is known also that the reduced probability for E2 transitions is proportional to the non-diagonal matrix elements of the charged quadrupole moment operator between initial and final states, whereas the diagonal matrix elements of this operator define the static quadrupole moments of nuclei in the corresponding states. The calculation of the reduced matrix elements of interest requires the knowledge of detailed structure of nuclear wave functions. If, however, the wave functions of nucleus form a basis for representation of some group G, which Lie algebra contains the (charge) quadrupole moment operators, then the required reduced matrix elements can be calculated in a pure algebraic, group-theoretical way.

The key role played by deformation in nuclei thus makes the quadrupole moment a dominant property of the nucleus. In this way, the Lie algebra appropriate for description of the nuclear collective states should contain among its generators, at least, the five components of

the quadrupole momentum operators and the three components {Lk} of the angular momentum operators, respectively. If these are the only observables it contains, the model can have one of the three possible structures — the rot(3) = [R5]so(3) of the rigid-rotor model of Ui [55], the su(3) of Elliott's model [10], or the sl(3, R) structure of the model of Weaver and Biedenharn [53]. For all three algebras, which are able to produce bands of collective states with rotational properties, one has the common commutation relations (in coupled form):

[L,L] = —V2L,

[L, Q] = —V6Q. (1.7)

The three algebras are distinguished by the [Q,Q] commutator:

Í0, for rot(3)

3VÎÔL, for su(3) (1.8)

—3VÏÔL, for sl(3,R).

Dynamical groups ROT(3) and SL(3,R) (and their SGA's) are noncompact and hence they are appropriate for the description of infinite rotational band, whereas the compact dynamical group SU(3) and its Lie algebra are appropriate for the description of finite rotational bands.

In [54, 56, 57] some arguments have been exposed that for nuclear systems, Sp(6,R) turns out to be the most appropriate dynamical group, which allows to describe both the rotational and vibrational collective motions in nuclei. The role of Sp(6, R) as the full dynamical group of the three-dimensional harmonic oscillator was clarified for the first time in the pionering work of Goshen and Lipkin [58]. We note that Sp(6,R) contains all three dynamical groups mentioned as subgroups and, in addition, it contains the harmonic oscillator Hamiltonian operator as its generator. This can easily be seen by considering the following two chains of subgroups

Sp(6, R) D U(3) D SU(3) D SO(3), (1.9)

relating the Sp, (6, R) model [50] to the shell model, and

Sp(6, R) D SL(3, R) D ROT(3) D SO(3), (1.10)

revealing its dynamical content.

If Sp(6, R) or another group G is the true dynamical or spectrum generating group for a given nucleus, then all states of this nucleus should be unified in a single multiplet. The interaction between the nucleons breaks this G symmetry and the degeneracy in energy of the states of this multiplet is lifted. In order to answer the question how the G symmetry is broken let us recall that in the spectra of deformed nuclei different rotational bands have been observed. From another side, it is known that the states of finite rotational bands can be unified in a single multiplet of

SU(3), which is broken, e.g., due to the quadrupole-quadrupole interaction. In this way, if we believe that the rotational bands observed in nuclear spectra are finite, then it seems natural to consider observed collective states as belonging to several broken SU(3) multiplets, embedded in a single Sp(6,R) irreducible representation [54, 56, 57]. The interaction between the nucleons therefore breaks the Sp(6, R) symmetry into different SU(3) multiplets, which in turn decompose into states of different angular momenta.

There are also other ways to broke the Sp(6,R) symmetry. A criterium for preferring one or another way of breaking of dynamical symmetry, of course, should be the comparison of the calculated B(E2) transition probabilities with the experimentally measured values. At this point we want to point out that the breaking of the assumed symmetry in a given nuclear system can practically be obtained in different ways. In particular, the breaking can be carried out via the Hamiltonian, via the used transition operator, or by using simultaneously a combination of the two. Examples of these ways of breaking the symmetry G are given throughout the dissertation.

For the description of collective spectra of spherical nuclei, an appropriate spectrum generating group prove to be SU(5) (see, e.g., [59]). But, the most successful works turned out to be those that use U(6) as a dynamical group. This group appears in various approaches, based on different physical pictures of nuclei. In Ref.[60], the group U(6) appears as an algebraic realization of Truncated Quadrupole Phonon Model, whereas in Refs.[61, 62] — as an algebraic realization of the Interacting Boson Model. The two models have been proven to be isomorphic to each other (cf., e.g., [63]). The equivalence has been established in respect to the matrix elements of their physical observables and their corresponding operators [64]. Something more, the strict mathematical proof of the unitary equivalence of the physical boson-spaces of the basis states of the two models also exists [65], which is the reason for considering the two models as different realizations of the same phenomenological SU(6)-boson model. It turned out that the U(6) group encompassed both the quadrupole vibrations (U(5) dynamics) and the SU(3) group of rotational motion, plus an unexpected 0(6) group structure [66] that is related to the description of y-unstable motion. These three dynamical symmetries, U(5), SU(3), and 0(6), were extensively explored in three major papers in the period 1976-1979 [61, 62, 66]. These papers sparked a rapidly increasing activity in the community of nuclear physics. It was soon realized that the IBM could be considered as an algebraic approximation to the Bohr-Mottelson collective model [67]. Many relationships (see, e.g., recent work [68]) between the IBM [16] and BM collective model [41] have been established using the known fact that two finite Hilbert spaces of equal dimension are isomorphic to each other. It is worth noting to mention also that the three subgroups, U(5), SU(3), and 0(6), embedded in U(6) have appeared in the late 1950s in the classification scheme of sd-shell nuclei [10], based on the Racah [69, 70, 71] group-theoretical methods and ideas of jj-coupling scheme in the atomic shell theory in 1940s and adapted by Jahn, Flowers and others in the beginning of 1950s to nuclear structure [72, 73, 74].

The U(6) group appears also in the microscopic approach of Vanagas [75, 76], where it was given a generalization of the kinetic energy operator without the involving the semiclassical picture of the structure of nucleus. In these papers, it was shown also that the algebraic structure of U(6) is appropriate for the description of spectra with both rotational and vibrational properties.

In the approach given in Refs.[75, 76], the group U(6) C Sp(12,R) was associated with the 6 quadrupole collective degrees of freedom which are microscopically expressed in terms of many-particle coordinates. We want to stress also that the three algebraic structures (U(5), SU(3) and 0(6)) of the IBM-1 [16] were obtained even earlier in Refs.[75, 76] (cf. [61, 62, 66]) where they were embedded in the Sp(12, R) group via U(6) C Sp(12, R). Further, using the Sp(12, R) D U(6) algebraic structure and a symplectic-type decomposition of a rather general microscopic collective Hamiltonian in U(6) irreducible terms, a microscopic version of the generalized (including the monopole degree of freedom) Bohr-Bottelson collective model and the IBM Hamiltonians was obtained in Ref.[77], based on the "deep collectivity" concept. According to the later, all A — 1 Jacobi quasiparticles equally contribute to the collective effects and as a result a new interpretation of the IBM was obtained, without referring to active fermions coupled to s and d bosons. There, the interacting boson approach is shown to be a very simplified version of the generalized BohrModel collective model when both models are treated microscopically. The works of Vanagas [75, 76, 77] strengthens the role of U(6) in describing the collective effects, although assuming different interpretation, and are based on the general microscopic theory of collective motion within the framework of the restricted dynamics approach [26, 27, 29, 32].

Besides the Elliott's [10] early breakthrough and the subsequent idea of considering the sd-boson U(6) group with its dynamical symmetries, many efforts have been made in trying to connect the observed nuclear collective dynamics (vibrations, rotations, etc.) to the microscopic structure making use of algebraic methods (see,e.g., Refs.[33, 34, 35, 36, 37, 26, 27, 75, 76, 29, 77, 32] for overviews of these methods). It turned out, as we will prove in more detail in the beginning of Part II of the present work, that the Sp(6, R) symplectic group emerges as an appropriate dynamical group for the one-component many-body theory of collective motion [33, 34, 35, 36, 37] and is at the same time a dynamical group for the three-dimensional harmonic oscillator [58, 78], thus guiding us towards the shell model [33, 34, 35, 36, 37]. A connection with the IBM has been explored [34], leading more recently to a computationally tractable version of the Bohr-Mottelson collective model [79, 80].

Finally, the U(6) group appear in the phenomenological Interacting Vector Boson Model [40], which is a generalization of the broken-SU(3) model [81, 82], used for description of the collective properties of heavy even-even nuclei, by introducing an additional quantum number which was initially called "pseudospin" (later replaced by the more appropriate term "T-spin" by analogy to the F-spin of IBM-2 [16]). This additional degrees of freedom extends the SU(3) dynamical symmetry to U(6) C Sp(12, R), which structure is generated by two types of vector quasiparticles (or phonons) that have been used to build up collective spectra in heavy even-even nuclei. As a result of different building blocks, the U(6) substructures (cf., e.g., Eq.(2.9)) of the IVBM differ from those of the IBM, as demonstrated in the next chapters. In the IVBM, the fully algebraic interactions between the two vector bosons are used that were expressed by the U(6) SGA. The U(6)-conserving version of the IVBM was applied successfully [83] to a description of the low-lying collective rotational spectra of the even-even medium and heavy mass nuclei.

With the advent of ever improving experimental nuclear physics facilities, a large repository of data on the structure of atomic nuclei is being amassed [84]. This data reveals the complex nature

of nuclear degrees of freedom, requiring different mixings of the basic rotational and vibrational collective modes. This in turn calls for better and often more sophisticated nuclear structure models, that should remain tractable and easy to apply while yielding very reasonable theoretical interpretations of the data. With the aim of extending the earlier applications of the IVBM to incorporate the new experimental data on states with higher spins and to incorporate new excited bands, in first part of the present work, we explore the symplectic extension of the IVBM, for which the dynamical symmetry group is Sp(12,R), and some of its practical applications. This extension is realized from, and has its physical interpretation over basis states of its maximal compact subgroup U(6) C Sp(12,R). As we will prove throughout this part of the dissertation, this leads naturally to the description of not just energies, but to finer structural effects like a staggering of levels between the ground and octupole bands up to states of very high spins. In general, the advantages of exploiting symplectic structures, in addition to gaining a direct geometrical interpretation (cf. Chapters 3 and 4), are quite transparent when a change in the number of phonons needed to build collective states is included in the theory as this results in larger model spaces that can accommodate the more complex structural effects as realized in nuclei with nucleon numbers that lie far from the magic numbers of closed shells.

The symplectic extension of the IVBM provided an useful tool to obtain its relation [85] to a version of the IBM that contains the s and d bosons with F-spin F =1 and a p-boson with F = 0, by means of a mapping procedure that uses pairs of vector bosons as constituents of the IBM ones. In addition the IVBM with Sp(12, R) dynamical group, which algebraic structure will be presented in detail in next chapters, contains all the phenomenological models based on the dynamical groups ROT(3), SL(3,R), SU(3) and Sp(6,R) respectively, which role was mentioned above. All this motivate the development and application of the phenomenological algebraic IVBM and its symplectic and orthosymplectic extensions for description of the collective excitations in heavy nuclei.

Заключение диссертации по теме «Физика атомного ядра и элементарных частиц», Ганев Хубен Ганев

14.1 Main results of the dissertation

The main results of the present dissertation which are raised to the defence are:

• The symplectic extension of the IVBM is developed. The new dynamical symmetries which appear as a result of this extension are studied and applied for description of different nuclear phenomena in heavy even-even nuclei.

• The phase structure of the IVBM is obtained for the first time. The possibility of obtaining triaxial shapes is considered.

• The supersymmetric (orthosymplectic) extension of the IVBM is developed, which allow the collective properties of heavy odd-mass and doubly odd nuclei to be described.

• The orthosymplectic extension is applied for the first time for the description of chiral

doublet bands in doubly odd nuclei.

• A fully microscopic PNSM of nuclear collective motions is formulated for the first time by considering the possible collective flows and symplectic geometry of the two-component nuclear system.

• The macroscopic limits of the PNSM, which appear for large dimensional representations, are further obtained. As a result, two simplified models of nuclear collective motion, expressed in simple geometrical terms, arise as macroscopic limiting cases of the PNSM.

• Computational techniques for the practical application of the PNSM at U(6) level are developed, including the explicit calculation of the required isoscalar factors and matrix elements of physically interesting operators.

• The PNSM is applied for the first time in obtaining the microscopic shell-model structure of the positive-parity states in well deformed heavy even-even nuclei, namely 166 Er, 154Sm and

238 и

• The first application of the symplectic-based shell-model approach to the structure of negative-parity states in heavy nuclei, in particular to 154Sm and 238U.

• Study of the dynamical and quasi-dynamical symmetries of the underlying proton-neutron collective dynamics in the microscopic structure of positive- and negative-parity states in strongly deformed heavy even-even nuclei.

• The central extension of the proton-neutron symplectic model with the semi-direct structure WSp(12, R) = [HW(6)]Sp(12,R) is proposed, which allows to include explicitly in the theory various many-particle correlations which lie outside the enveloping algebra of Sp(12,R).

• Study of the low-lying collective E1 dipole strengths in strongly deformed heavy even-even nuclei within the extended PNSM shell-model framework.

Список литературы диссертационного исследования доктор наук Ганев Хубен Ганев, 2020 год

14.2 List of scientific publications on which the dissertation is based

Publications in refereed journals

[A1] H. Ganev, V. P. Garistov, and A. I. Georgieva, Description of the ground and octupole bands in the symplectic extension of the interacting vector boson model, Phys. Rev. C 69, 014305 (2004).

[A2] H. G. Ganev and A. I. Georgieva, Transition probabilities in the U(6) limit of the Symplectic Interacting Vector Boson Model, Phys. Rev. C 76, 054322 (2007).

[A3] H. G. Ganev, Collective states of the odd-mass nuclei vjithin the framework of the interacting vector boson model, J. Phys. G: Nucl. Part. Phys. 35, 125101 (2008).

[A4] H. G. Ganev, A. I. Georgieva, S. Brant, and A. Ventura, New description of the doublet

bands m doubly odd nuclei, Phys. Rev. C 79, 044322 (2009).

[A5] H. G. Ganev and S. Brant, Structure of the doublet bands in doubly odd nuclei: The case of 128Cs, Phys. Rev. C 82, 034328 (2010).

[A6] H. G. Ganev, Phase Structure of the Interacting Vector Boson Model, Phys. Rev. C 83, 034307 (2011).

[A7] H. G. Ganev, Triaxial shapes in the interacting vector boson model, Phys. Rev. C 84, 054318 (2011).

[A8] H. G. Ganev, Transition probabilities in the U(3,3) limit of the symplectic IVBM, Phys. Rev. C 86, 054311 (2012).

[A9] H. G. Ganev, Axial asymmetry in the IVBM, Eur. Phys. J. A 49, 55 (2013).

[A10] H. G. Ganev, Simultaneous description of low-lying positive and negative parity bands in heavy even-even nuclei, Phys. Rev. C 89, 054311 (2014).

[A11] H. G. Ganev, Collective degrees of freedom of the two-component nuclear system, Eur. Phys. J. A 50, 183 (2014).

[A12] H. G. Ganev, U(6)-phonon model of nuclear collective motion, Int. J. Mod. Phys. E 24, 1550039 (2015).

[A13] H. G. Ganev, Shell-model representations of the proton-neutron symplectic model, Eur. Phys. J. A 51, 84 (2015).

[A14] H. G. Ganev, Some U(n1 + n2) D U(n1) ® U(n2) isoscalar factors, Int. J. Mod. Phys. E 26, 1750057 (2017).

[A15] H. G. Ganev, Matrix elements of the proton-neutro symplectic model, Int. J. Mod. Phys. E 27, 1850021 (2018).

[A16] H. G. Ganev, Structure of the low-lying positive-parity states in 154Sm, Phys. Rev. C 98, 034314 (2018).

[A17] H. G. Ganev, U(6) quasi-dynamical symmetry in 238 U, Nucl. Phys. A 987,112 (2019).

[A18] H. G. Ganev, Microscopic structure of the low-lying negative-parity states in 154Sm, Phys. Rev. C 99, 054305 (2019).

[A19] H. G. Ganev, E1 transitions in the extended proton-neutron symplectic model, Phys. Rev. C 99, 054304 (2019).

Full-texts in conference proceedings

[B1] H. G. Ganev, A. I. Georgieva, S. Brant, and A. Ventura, Structure of the doublet bands in doubly odd nuclei with mass around 130, Proceedings of the XXVIII International Workshop on Nuclear Theory (June 22-27, 2009, Rila Mountains, Bulgaria), ed. S. Dimitrova, Printed by BM Trade Ltd., Sofia, Bulgaria 2010, pp. 177.

[B2] H. G. Ganev and A. I. Georgieva, Simultaneous Description of Even-Even, Odd-Mass and Odd-Odd Nuclear Spectra, AIP Conf. Proc. 1203, 17 (2010).

[B3] H. G. Ganev, Phase Structure of the Interacting Vector Boson Model, Proceedings of the XXIX International Workshop on Nuclear Theory (June 20-26, 2010, Rila Mountains, Bulgaria), ed. A. Georgieva and N. Minkov, (Published by Heron Press, Sofia, 2010), pp. 119.

[B4] H. G. Ganev, Triaxiality in the IVBM, Proceedings of the XXXI International Workshop on Nuclear Theory (June 24-30, 2012, Rila Mountains, Bulgaria), ed. A. Georgieva and N. Minkov, (Published by Heron Press, Sofia, 2012), pp. 204.

[B5] H. G. Ganev, On the structure of triaxial nuclei, Proceedings of the 4-th International Conference on Current Problems in Nuclear Physics and Atomic Energy (Institute for Nuclear Research, Kyiv, 2013), pp. 390.

[B6] H. G. Ganev, Nuclear shapes in the Interacting Vector Boson Model, Proceedings of the XXXII International Workshop on Nuclear Theory (June 23-29, 2013, Rila Mountains, Bulgaria), ed. A. Georgieva and N. Minkov, (Published by Heron Press, Sofia, 2013), pp. 141-150.

[B7] H. G. Ganev, Negative parity states in the IVBM, J. Phys. Conf. Ser. 533, 012015 (2014).

[B8] H. G. Ganev, Contraction limits of the proton-neutron symplectic model, EPJ Web of Conferences 107, 03012 (2016).

[B9] H. G. Ganev, The proton-neutron symplectic model of nuclear collective motions, J. Phys. Conf. Ser. 724, 012016 (2016).

[B10] H. G. Ganev, Structure of the low-lying positive parity states in the proton-neutron

symplectic model, J. Phys. Conf. Ser. 1023, 012013 (2018).

[B11] H. G. Ganev, U(6) dynamical and quasi-dynamical symmetry in strongly deformed heavy nuclei, EPJ Web of Conferences 194, 05002 (2018).

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