Boson Space Research Articles

Microscopic approach to magnetic excitations in IBM-2

Starting from the collective SD subspace of the shell model, we construct boson images of the hamiltonian and the M1 and M3 operators by using the OAI method. In the magnetic operators one-boson as well as two-boson terms are considered. Energy spectra and g-factors of 130,134Ba are well reproduced by this method. Relatively strong magnetic dipole and octupole transitions at E x ∼ 2 MeV are predicted. The M1 and M3 electron scattering form factors are calculated. It turns out that the magnetic dipole operator is mainly a one-body operator in boson space, whereas the magnetic octupole operator contains important two-boson terms, which give sizeable contributions to the excitation strengths. These two-boson terms tend to diminish the F-spin selectivity of M3 excitations.

Physical boson basis states in the boson expansion theories.

The use of physical boson basis states is stressed for the calculations in the boson space. The explicit form of physical boson basis states in terms of bosons is derived for the nonunitary boson mapping of Dyson. The ambiguity in the normalization introduced due to the use of bi-orthonormal basis states is satisfactorily resolved, resulting in a Hermitian matrix. This Hermitian matrix is found to coincide with the Hamiltonian matrix in the fermion space. The model cases where the use of a boson basis is justified are shown to be consistent with our view of using physical boson basis states.

Dyson boson mapping in S-D subspace

The validity of a truncated boson space invented by Zirnbauer and Brink in the Dyson boson mapping is examined by using a one-j (= 23 2 ) model. The pairing operator is well reproduced without regard to deformation. When the deformation of a nucleus is smaller than 0.3 the quadrupole interaction is also well approximated by this method.

On the boson mapping of fermion collective pairs

We study the problem of the mapping of fermion collective pairs onto particle-particle bosons and of different fermion operators (hamiltonian, one- and two-particle transfer operators) onto corresponding boson ones and we test the consequences of the truncation to lowest orders of these boson operators. We find that, although the lowest-order terms in the expansion of the operators in boson space lead to matrix elements between boson states which display the qualitative behaviour of the corresponding ones between fermion states, higher-order terms are required to get a quantitative agreement when a large number of particles are involved, as a direct consequence of the increased role of the Pauli principle.

IBA mapping of microscopic hamiltonians

Using the seniority-conserving mapping of a realistic fermion hamiltonian into a boson space it is shown that this leads to serious problems for quadrupole forces in a single j-shell. The lack of agreement of the exact spectrum with the mapped one persists if realistic forces are used. This observation invalidates many “microscopic justifications” of the interacting boson model by lowest-order boson mapping.

Cranked Hartree approximation for systems with many interacting bosons

An extension to bosons of a cranked Hartree variational method is developed. This method is not limited to s and d bosons and can be extended to more complex boson spaces. An application to well deformed systems including g ( l = 4) bosons is compared with the exact solution of the hamiltonian H = − KQ 2· Q 2.

Boson expansion based on the extended commutator method in the Tamm-Dancoff representation

Formal aspects of boson expansions in the Tamm-Dancoff representation are investigated in detail. This is carried out in the framework of the extended commutator method by solving in complete generality the coefficient equations, searching for Hermitian as well as non-Hermitian boson expansions. The solutions for the expansion coefficients are obtained in a new form, called the square root realization, which is then applied to carry out an analysis of the relationship between the type of expansion and the boson space in which the expansion is defined. It is shown that this new realization is reduced to various well-known boson theories when the boson space is chosen in an appropriate manner. Further discussed, still on the basis of the square root realization, is the equivalence, on a practical level, of a few boson expansion approaches when the Tamm-Dancoff space is truncated to a single quadrupole collective component.NUCLEAR STRUCTURE Boson expansion, commutator method, Tamm-Dancoff representation, particle-pair representation, Hermitian and non-Hermitian boson expansions, square root realization, truncation of the fermion and boson spaces.

Boson-Fermion Expansions

Boson expansion theory is extended so as to allow for the coupling of several fermion-like valence particles or holes to the collective vibrations of a closed shell, described in terms of boson degrees of freedom. The result is a faithful mapping of the original many-fermion problem into a certain boson-fermion subspace without redundant variables. Generalizations of the Dyson and Holstein-Primakoff representations are derived with the aid of al­ gebraic techniques. § l. Introduction The boson expansion method exploits the remarkable fact that fermion behavior can be perfectly replicated within a subspace of a suitable boson Hilbert space.ll It provides an ideal microscopic avatar for phenomenological theories of collective motion, from the venerable Bohr-Mottelson model to that current cynosure, the IBA, all of which are formulated in terms of bosons.2> Following the pioneering work of Beliaev and Zelevinsky 3> and of Marumori, Yamamura and coworkers, 4> the subject of boson expansions has had a slow and, at times, precarious evolution/> but is now in the midst of a renaissance. It has even been applied recently to particle physics.5> The renewed popularity of boson expansion theory is certainly due in part to its utility and the interest in the IBA; but the opportunities for rederivation by the most fashion­ able methods, such as path-integration, 6> generalized coherent-state representa­ tions,n etc., have probably also played a role. In the usual boson expansion method, the entire fermion space is mapped injectively into the physical subspace of the so-called ideal space. The ortho­ gonal complement of the physical subspace is called the unphysical subspace, since it has nothing to do with the fermion problem. If the number of particles is even, the physical subspace is purely bosonic with a distinct boson for each fermion-pair excitation.8>,9> If the number of particles is odd, the physical subspace lies in the tensor product of the boson space with the space of an ideal odd fermion. Therefore, all fermion pairs are bosonized, and at most one nucleon may persist as fermion-like.*>,lo>.m

Microscopic investigations of the structure of the bosons and the hamiltonian of the interacting boson model

In this paper we aim at a formal understanding of the IBA hamiltonian. We start from the point of view that the s- and d-bosons are the L = 0 and L = 2 members of two correlated protons or neutrons. The IBA bosons are therefore considered to be particle-particle bosons. These can conveniently be calculated within the broken pair approximation which is explained in some detail in the text. We then map the correlated two-particle states thus obtained onto a boson space; this is performed using either Marumori's or Dyson's mapping. We also shortly discuss how through a renormalization procedure we can eliminate the less collective members of the bosons to finally arrive at the IBA hamiltonian.

Boson mappings for schematic nuclear models with the symmetry of SO(5)

In the past, several schematic nuclear models have been proposed with the symmetry of the Lie algebra SO(5). We derive a set of mappings from the shell model spaces in which these models are defined to suitable boson spaces. The derivations are carried out by two distinct algebraic methods, which are customarily associated with the names Holstein-Primakoff and Dyson, respectively. The relative utility of the two is seen to depend on the particular mapping. The mappings are useful for the study of the collective properties of the various models and of generalizations of each of them. This is illustrated with one of the models. Some algebraic details, some ideas concerning schematic model building, and an extension of the results of this paper are presented in appendices.

A unification of boson expansion theories: (II). Boson expansions as provided by the functional representation method

It is shown that the boson expansions hitherto known, as well as an infinite number of the new ones, can be derived in a unified way in terms of the functional representation technique. Each boson expansion (boson representation) is valid for the carrier space of an irreducible representation of a semisimple compact Lie subgroup of SO(2 N + 1) and can be presented as Dyson-type, Holstein-Primakoff-type or Garbaczewski (Marumori)-type expansion with the corresponding properties concerning finiteness, convergence and hermitian conjugation. The physical boson space can be determined for every expansion and the projection operator is explicitly found. The methods for solving the Schrödinger equation are discussed and the generator coordinate method is shown to be equivalent to the boson expansion approach.

The spectral-distribution method in boson space

The spectral-distribution method is extended to a system of monopole (L=0) and quadrupole (L=2) bosons. For noninteractingL=0 bosons, analytical expressions for the first four-moments are given. The main conclusion of this paper is that, in the case of bosons, the presence of many single-particle levels seems to be essential in generating a normal-level density, the number of particles playing a minor role.

A test of boson mappings of fermion systems

The Otsuka-Arima-Iachello Method, the Belyaev-Zelevinsky-Marshalek boson expansion method, and the boson expansion theory are each used to map a solvable fermion hamiltonian onto a boson space. Comparison of the spectra and transition rates obtained by these three boson mapping methods are compared to the exact values.

New method for studying the microscopic foundations of the interacting boson model

We describe (i) A mapping, using a multishell seniority basis, from a prescribed subspace of a shell model space to an associated boson space. (ii) A new dynamical procedure for selecting the collective variables within the boson space, based on the invariance of the trace. (iii) A comparison with exact calculations for a multi-level pairing model, to demonstrate that the method works.

Examples of the relationship between the shell model and the Bohr collective hamiltonian: The multilevel extension of the Lipkin-Meshkov-Glick model

The multilevel extension of the Lipkin-Meshkov-Glick model, a polynomial in the generators of the Lie algebra SU( n), is mapped onto a boson space with n − 1 bosons (generalized Holstein-Primakoff transformation). By expansion in powers of N −1, where N, the total number of fermions, completely characterizes the transformation, one generates the associated collective hamiltonian. Previous studies of the potential energy surface and of the semi-classical limit are regained as special cases.

Many-body theory of core holes

An electronic Hamiltonian appropriate for describing core-hole processes is derived and its properties are discussed. Based on this Hamiltonian a many-body theory is presented, and a useful linked-cluster theorem is proven. A distinction between relaxation and correlation terms can be made in each order of the perturbation expansion. Under certain well-defined conditions the core-ionization process can be visualized as a Franck-Condon transition between two multidimensional potential-energy surfaces in an abstract bosonic space. The corresponding approximation, the boson approximation, is discussed and applied to the water molecule.

Shape of eigenvalue distribution for bosons in scalar space

Analytic expressions for obtaining first four moments of noninteracting as well as interacting bosons have been given in terms of matrix elements. These expressions have been utilized to obtain general conditions under which the eigenvalue distributions for bosons tend towards the Gaussian distribution. Results for pairing and $Q\ifmmode\cdot\else\textperiodcentered\fi{}Q$ operators in an $s\ensuremath{-}d$ boson shell have also been given.NUCLEAR STRUCTURE Analytic expressions, first four moments, noninteracting and interacting bosons, eigenvalue distribution, conditions leading to Gaussian distribution, pairing and $Q\ensuremath{-}Q$ operators, $s\ensuremath{-}d$ boson space.

On the correspondence between fermion and boson states and operators

Mapping of shell-model (fermion) Hamiltonians onto boson Hamiltonians which underly the interaction boson model 1–5) is investigated. A simple correspondence is defined and a sufficient condition given for shell-model Hamiltonians to simply correspond to finite hermitian boson Hamiltonians. A special case is discussed where diagonalization of a shell-model Hamiltonian for valence protons and neutrons can be exactly carried out in an equivalent (but different) boson space. If, however, the proton Hamiltonian and neutron Hamiltonian are diagonal in the seniority scheme, mapping of fermion states onto orthogonal boson states cannot be a simple correspondence. In that case the boson quadrupole operators equivalent to fermion guadrupole operators cannot be single-boson operators but must be more complicated, ones.

Features of Nuclear Deformations Produced by the Alignment of Individual Particles or Pairs

The interacting boson model is first analysed at the level of "real" bosons with a view to establishing the conditions for the occurrence of rotational band structure. The conditions on the effective interactions are much less restrictive than those required for the occurrence of SU (3) symmetry. The wave functions for the aligned bosons are analysed from the microscopic point of view in order to compare with the coupling scheme based on the alignment of individual particles in the deformed potential. In general, these two coupling schemes, both of which yield rotational band structure, lead to major differences in the nuclear properties. The region of validity of the boson coupling scheme is studied in terms of the effective interactions between the nucleons. Only for rather small deformations, is it justified to truncate the boson space to J = 0 and 2.

Scalar averages in the boson space

Scalar averages in the boson space