Felix Villars

Adiabatic time-dependent Hartree-Fock theory in nuclear physics

The adiabatic limit of time-dependent Hartree-Fock theory (ATDHF) is developed in a systematic way so as to provide a basis for the description of large amplitude, slow, collective motion in nuclei. A variational principle is used to find the best approximation to a solution of the time-dependent Hartree-Fock equation in the form of a parametrized wave packet φ(p, q), p, q being time-dependent c-numbers. The mean energy for the wave packet φ(p, q) defines a classical collective Hamiltonian H(p, q). The evolution of the parameters p and q is described by the canonical equations of motion associated with H. A set of self-consistency conditions, of a formally stationary type, determines possible optimal wave packets. An iteration scheme for the solution of the self-consistency conditions is proposed. The quantal interpretation of the classical description of collective motion embodied in the Hamiltonian H(p, q) is discussed. Finally, two simple, well understood illustrative examples are presented to illustrate the application of the general formalism developed in this paper.

Unified theory of nuclear rotations

Abstract The nuclear Hamiltonian is expressed in terms of the total angular momentum variables, and of intrinsic particle coordinates by means of a contact transformation. The intrinsic structure is approximated by a Hartree-Fock solution, and the Coriolis coupling diagonalized by admitting particle-hole excitations. It is shown that in this approximation, the rotational energy takes the conventional form, as given by the “unified” model, the moment of inertia being determined by a self-consistent “cranking” formula. This, and corresponding results for matrix elements of multipole operators substantiate the validity of the more intuitive procedures based on the unified model and on the use of single particle orbitals in deformed potential wells.

A note on rotational energy levels in nuclei

Abstract It is shown that the kinetic energy T of any system of interacting particles may be written in a “canonical form”, which displays the dependence of T on the total angular momentum. A formula is given for the zero order moments of inertia associated with collective rotation of the system. The formalism is applied to nuclei, and for purposes of comparison, to molecules. In addition, it is shown that an unsymmetric formulation of the many particle problem, in which only the nucleons of a “core” participate in the definition of the body fixed axes, leads to a Hamiltonian which is related by a simple averaging process with the “unified” model of Bohr and Mottelson. Finally, a connection is established between the “unified” model of Bohr-Mottelson and the “cranking model” of Inglis.

Collective energies from momentum- and angular momentum-projected determinantal wavefunctions☆

Abstract Determinantal many-particle wavefunctions, as used in Hartree-Fock theory of finite nuclei, are not eigenfunctions of the total linear momentum, nor, generally, of the total angular momentum. A simple derivation is given for the expression for the energy in terms of momentum- or angular momentum projected determinants. A variational principle based on this expression is used to determine the collective energy (center of mass kinetic and rotational energy). This method gives the correct energy of center of mass motion, and in the case of strong deformation, the Thouless-Valatin self-consistent cranking result for the rotational energy.

ELEMENTARY QUANTUM THEORY OF NUCLEAR COLLECTIVE ROTATION

Abstract A new, entirely quantal description of nuclear collective rotation is developed. The basic element is the explicit introduction of the pair (or pairs) of operators which describe the collective motion; in the case of rotation this is the total angular momentum J and a phase angle ϕ. A representation is then constructed, in which either J or ϕ is diagonal. The non-collective degrees of freedom are described in a subspace of the set of antisymmetrized independent particle wave functions φn; this subspace is obtained by using suitable projection operators. The formalism is developed for “nuclei” in two-dimensional space only. It is shown that the ϕ-representation is particularly useful and leads to the construction of an operator H J = H ( i ) + JH ( c ) + 1 2 J 2 Q , whose eigenvalues are the set of all levels with angular momentum J. These levels may be found by the usual HF and RPA procedures. It is shown that the moment of inertia of the ground state rotational band is given by the Thouless-Valatin formula.

RPA equations for 2-particle--1-hole states

Abstract Within the Dyson equation approach to higher correlation functions we derive RPA-like equations for the 2-particle-1-hole problem. The equations can be entirely expressed by the RPA sub-solutions of the particle-hole and particle-particle problem. In this way our equations are very well suited for weak or intermediate coupling situations.

The coupling of a large amplitude collective motion to RPA excitations

Abstract Adiabatic large amplitude motion along a path of minimal potential energy is shown to be coupled to residual modes of excitation. These are described by RPA modes denned at every point q of the adiabatic path. A semi-classical Hamiltonian for this system is constructed, which exhibits the coupling terms between the two types of modes.

The linked-cluster expansion for the deuteron optical potential☆

Abstract The Goldstone-Hugenholtz linked-cluster expansion for the transition amplitude is derived for elastic deuteron scattering reactions. From this expression, the deuteron optical potential is derived, and the relationship between this potential and the single-nucleon optical potential is established.

Collective rotations in nuclei

Abstract A systematic approach is made to the problem of determining the parameters of collective motion in nuclei. Formulas are given for the moment of inertia of the rotational energy and for the collective g -factor of the magnetic moment. These expressions are not based on the use of a model of the nucleus. Such a model will be needed, however, for the numerical evaluation of the expressions derived. The essential elements of a usable model are briefly discussed.

Nuclear collective motion and the Born-Oppenheimer approximation

Abstract This paper attempts to implement the analogy between nuclear collective dynamics and the dynamics of molecular rotation and vibration. The procedure used is illustrated for the case of a single canonical pair Q , K of collective operators. It contains two essential steps. First, the construction of a factorable trial function modelled after the well-known Born-Oppenheimer ansatz. Redundant degrees of freedom are eliminated by projection. Second, a decomposition of the hamiltonian into a series, each term being the product of a power of the collective momentum K and of a cofactor which commutes with the collective position operator Q . This decomposition is unique. These two steps, used in conjunction, lead to an expression for the mean energy in terms of a collective hamiltonian H (p, q) . Two optimization conditions are applied, which jointly determine both the intrinsic wave function and the exact form of the collective operator Q (given its general nature). As a result of several well-defined approximations in the evaluation of H (p, q) , a self-consistent cranking-type expression energes for the inertial parameter. Small amplitude oscillations near the unconstrained equilibrium are described by RPA-type equations.