E. Chacón

Group theory of the collective model of the nucleus

In the present paper we extend the group theoretical analysis of a previous publication to obtain explicitly, as a polynomical in sinγ, cosγ, the function φλμlk(γ) required in the discussion of the quadrupole vibrations of the nucleus. The states appearing in the collective model 〈νλμLV〉=F1λ(β) ΣKφλμLK(γ) DL*MK(φi), l= (ν−λ)/2, are then defined, as Fλl(β), DL*MK(φi) are well known. All matrix elements required in the collective model of the nucleus are related then with the expression (λμL;λ′μ′L′;λ″μ″L″= ∂π0ΣKK′K″ (LL′L″KK′K″) φλμLK(γ) φλ′μ′L′K′ (γ) φλ″μ″L″K″ (γ)sin 3γdγ, which is a reduced 3j‐symbol in the O(5) O(3) chain of groups.

U (5) ⊃ O (5 )⊃ o (3) and the exact solution for the problem of quadrupole vibrations of the nucleus

Over twenty years ago A. Bohr discussed the quantum mechanical problem of the quadrupole vibrations in the liquid drop model of the nucleus. States of definite angular momentum L could not be obtained exactly except when L=0,3. In the present paper we indicate how we can determine states for arbitrary angular momentum L and definite number of quanta ν in terms of polynomials of the creation operators characterized by irreducible representation (IR) of the chain of groups U(5) ⊆O(3). We furthermore characterize the states by a definite IR λ of O(5) by replacing the creation operators by traceless ones. These states are fully determined by an extra label μ that gives the number of triplets of traceless creation operators coupled to angular momentum zero. We show then how all the wavefunctions of the problem discussed by Bohr can be obtained in a recursive fashion and briefly discuss some of their applications.Over twenty years ago A. Bohr discussed the quantum mechanical problem of the quadrupole vibrations in the liquid drop model of the nucleus. States of definite angular momentum L could not be obtained exactly except when L=0,3. In the present paper we indicate how we can determine states for arbitrary angular momentum L and definite number of quanta ..nu.. in terms of polynomials of the creation operators characterized by irreducible representation (IR) of the chain of groups U(5) is contained inO(3). We furthermore characterize the states by a definite IR lambda of O(5) by replacing the creation operators by traceless ones. These states are fully determined by an extra label ..mu.. that gives the number of triplets of traceless creation operators coupled to angular momentum zero. We show then how all the wavefunctions of the problem discussed by Bohr can be obtained in a recursive fashion and briefly discuss some of their applications. (AIP)

GROUP THEORY OF THE INTERACTING BOSON MODEL OF THE NUCLEUS

Recently Arima and Iachello proposed an interacting boson model of the nucleus involving six bosons, five in a d and one in an s state. The most general interaction in this model can then be expressed in terms of Casimir operators of the following chains of subgroups of the fundamental group U(6): U(6) ⊆U(5) ⊆O(5) ⊆O(3) ⊆O(2), U(6) ⊆O(6) ⊆O(5) ⊆O(3) ⊆O(2), U(6) ⊆SU(3) ⊆O(3) ⊆O(2). To determine the matrix elements of this interaction in, for example, a basis characterized by the irreducible representations of the first chain of groups, then we only need to evaluate the matrix elements of the Casimir operators of O(6) and SU(3) in this basis as the others are already diagonal in it. Using results of a previous publication for the basis associated with U(5) ⊆O(5) ⊆O(3), we obtain the matrix elements of the Casimir operators of O(6) and SU(3). Furthermore, we obtain explicitly the transformation brackets between states characterized by irreducible representations of the first two chains of groups. Numerical p...

U(5) is contained inO(5) is contained inO(3) and the exact solution for the problem of quadrupole vibrations of the nucleus. [Liquid drop model]

Over twenty years ago A. Bohr discussed the quantum mechanical problem of the quadrupole vibrations in the liquid drop model of the nucleus. States of definite angular momentum L could not be obtained exactly except when L=0,3. In the present paper we indicate how we can determine states for arbitrary angular momentum L and definite number of quanta ν in terms of polynomials of the creation operators characterized by irreducible representation (IR) of the chain of groups U(5) ⊆O(3). We furthermore characterize the states by a definite IR λ of O(5) by replacing the creation operators by traceless ones. These states are fully determined by an extra label μ that gives the number of triplets of traceless creation operators coupled to angular momentum zero. We show then how all the wavefunctions of the problem discussed by Bohr can be obtained in a recursive fashion and briefly discuss some of their applications.Over twenty years ago A. Bohr discussed the quantum mechanical problem of the quadrupole vibrations in the liquid drop model of the nucleus. States of definite angular momentum L could not be obtained exactly except when L=0,3. In the present paper we indicate how we can determine states for arbitrary angular momentum L and definite number of quanta ..nu.. in terms of polynomials of the creation operators characterized by irreducible representation (IR) of the chain of groups U(5) is contained inO(3). We furthermore characterize the states by a definite IR lambda of O(5) by replacing the creation operators by traceless ones. These states are fully determined by an extra label ..mu.. that gives the number of triplets of traceless creation operators coupled to angular momentum zero. We show then how all the wavefunctions of the problem discussed by Bohr can be obtained in a recursive fashion and briefly discuss some of their applications. (AIP)

Complete set of states for microscopic nuclear collective models

For several years the authors have been interested in determining complete sets of states for macroscopic nuclear collective models, such as the Bohr–Mottelson one (BM) and the interacting boson approximation (IBA), as well as in their use in nuclear structure calculations. In the present paper we obtain a complete set of states for microscopic nuclear collective models such as those of Vanagas and of Filippov and Smirnov. For calculations in these models, one requires a set of states for the A nucleon system, in appropriate coordinates which include the ones related with collective degrees of freedom. As is customary in nuclear physics, the complete set of states is derived more conveniently if one assumes an oscillator interaction between the nucleons. We obtain explicitly this set of states when A≫1, showing that it can be expressed in terms of wavefunctions whose dependence on the collective coordinates is similar to those appearing in the BM model and in the IBA. We briefly indicate how this set of s...

Lie algebras in the Schrödinger picture and radial matrix elements

The main objective of this paper is to derive from a unified viewpoint particular one‐ and two‐body radial matrix elements with respect to oscillator and Coulomb states. All the results have been obtained previosly using either generating functions of Laguerre polynomials or group theoretical methods related to particular realizations of the Lie algebras associated with those states. We show, though, in this paper how some of the realizations proposed can be derived from physical considerations. The main idea is to translate well‐known realizations, in the Heisenberg picture, to the corresponding ones in the Schrodinger picture. The latter realizations allow us to define indecomposable (i.e., not completely reducible) and irreducible tensors of the Sp(2) [or equivalently the SU(1,1)] group for the one‐ and two‐body problem respectively. The evaluations of the radial matrix elements becomes then just a matter of applying the Wigner–Eckart theorem, giving rise to Wigner coefficients of SU(1,1) that have bee...

Analytic expressions for the matrix elements of generators of Sp(6) in an Sp(6)⊇U(3) basis

Work done by many authors indicates that important tools for calculations in microscopic collective models are the matrix elements of the generators of the symplectic group Sp(6) in an Sp(6)⊇U(3) basis. Rosensteel has derived recursion relations for these matrix elements while Filippov has determined them using generating function technique, but it would also be convenient to have explicit and analytic formulas for them. This is what we do in this paper for the case of closed shells, i.e., when the irreducible representation (irrep) of Sp(6) is characterized by equal values for the three weight generators in the lowest weight state. We also indicate how our results can be extended to the case of arbitrary irreps of Sp(6), i.e., when we have open shells.

Collectivity and geometry. II. The two‐dimensional case

In a previous paper of this series we showed that nuclear collective behavior can be related to the symplectic geometry of the many‐nucleon system, when we introduce in it appropriate constraints. We showed in that paper that a full discussion of collective behavior in the many‐body system requires—in a space of d dimensions—the basis for the irreducible representations of the Sp(2d) group in the chain Sp(2d)⊇Sp(2)×O(d), as well as the matrix elements of the generators of Sp(2d) in this basis. In the present paper we implement this program fully for d=2, both because of the mathematical interest of the chain Sp(4)⊇Sp(2)×O(2) and as a stepping stone to the chain Sp(6)⊇Sp(2)×O(3) which is central to the discussion of collective motions.

Confrontation of macroscopic and microscopic nuclear collective models

Since its start nuclear theory has lived with the dichotomy of viewing the nucleus microscopically, as a system of nucleons, or describing it macroscopically in terms of collective coordinates. In the last decade though, a point transformation has been introduced in which single particle coordinates can be expressed in terms of collective ones plus others, opening the possibility of deriving a microscopic collective model. In the present paper we confront the macroscopic and microscopic collective models, first in a space of two dimensions, in which we find explicitly the unitary representation in quantum mechanics of the canonical transformation that relates them. We then show how to extend every step of the analysis to the three‐dimensional problem, though there some of the states required are not yet available in analytic form. One of the fundamental problems in collective models of the nucleus is that of shape. We indicate what are the operators whose expectation values give a reasonable description o...

Collectivity and geometry. III. The three‐dimensional case in the Sp(6)⊇Sp(2)×O(3) chain for closed shells

As was indicated in previous papers, general Hamiltonians for systems of n particles in three‐dimensional space can be formulated in the enveloping algebra of the symplectic group Sp(6n). This group admits, among others, the subgroup Sp(6)×O(n) and, as has been noticed by many authors, collective Hamiltonians can be formulated in the enveloping algebra of Sp(6), so that their eigenstates can be characterized by a definite irreducible representation (irrep) of this group. The mathematical problem is then to determine the matrix elements of the generators of Sp(6) in a basis characterized by irreps of this group as well as of appropriate subgroups. In the present series of papers the subgroups chosen where Sp(2)×O(3) as the Casimir operator of Sp(2) when n → ∞ is formally related to the Bohr–Mottelson vibrational Hamiltonian (BMVH), while O(3) gives the angular momentum of the state. We give an algorithm for determining these states that closely parallels the procedure followed for the BMVH. Programs are be...