Collective Motion In Nuclei Research Articles

Collective motions in nuclei and the spectrum generating algebras T5 × SO(3), GL(3,R), and CM(3)

Cusson's classical treatment of the collective rotations of a discrete system of N particles is extended to the full quantum mechanical system by means of a straightforward generalization of Villars' canonical transformation. In this manner, Bohr's collective Hamiltonian, with various values for the rotational mass, is microscopically derived. The nature and criteria for the existence of various collective flows in a many-body system are also given. The collective parts of the Hamiltonian are then separately expressed in original particle coordinates and momenta and in this manner the possibility of microscopic calculations for the collective motions is suggested. Finally appropriate microscopic Hamiltonians for the S.G.A.'s T5 × SO(3), GL(3,R), and CM(3) are determined.

Particle motion in guide potentials and the collective nuclear motion

The motion of a particle which is constrained by a guide potential to move on a curve is studied in the framework of the Generator Coordinate Method (GCM). In the limit of narrow guide potentials a differential equation for the wave function of the constrained motion is obtained which differs from the corresponding Schrodinger equation by an additional potential. This additional potential is due to the embedding of the curve in the space and depends on the form of the guide potential and on the curvature of the curve. Nonadiabatic transitions in the constrained motion are possible for finite widths of the guide potential. The coupling terms are given explicitly and it is shown that an adiabatic limit exists. Since the GCM can equally well describe the collective motion of nuclei, some insight into the more complicated problem of collective motion is obtained from its analogies to the studied problem of constrained particle motion: The collective motion of a nucleus can be considered as the motion of a particle with variable mass along a curve in a guide potential which is given by the interaction potential between the nucleons. It is shown that Schrodinger's quantized kinetic energy is correctly used in the cranking model and that the additional potential terms mentioned above are included there by the definition of the collective potential energy. Approximations to the idealized GCM used here are discussed and the connection with the method of Born, Oppenheimer and Villars is indicated.

Centrifugal stretching of a classical rotator and collective motions in nuclei

The energy of a classical rotator with the inclusion of the effect of centrifugal forces is calculated. The resulting set of parametric equations is found to yield an excellent description of rotational states in deformed nuclei. The formulation within the framework of the hydrodynamical model also predicts β-vibrational energies in good agreement with the observed levels. Our results are practically identical with those obtained from the Davydov–Chaban model with axial symmetry. Comparison with other models, which include the rotation–vibration interaction corrections, establishes the relative merit of our simple approach.

The decay of 17673Ta

Measurements have been performed on the direct gamma spectrum and the spectra of the gamma radiations coincident with conversion electrons emitted by 176Ta. Combining these results with the data given by Harmatz et al. it was possible to set up a level scheme in which nearly all the transitions found by the experiments could be placed. This level scheme has been compared with those of neighbouring nuclei and with the results of computations starting from some models for collective motions in nuclei.

The effect of spin-dependent forces on collective motion in nuclei

The vibrational state in even-mass, single closed shell nuclei are studied using a pairing force plus the part of the tensor force which can give rise to a 2+ collective state in the approximation of a dilute quasi-particle system (Sawada approximation). From the empirical energies of vibrational states the strength of this spin-dependent force is estimated, from which the B(E2) as well as the energy levels of the odd-mass nuclei are calculated. The results of this calculation are compared to the calculations which have been done with a pairing plus a quadrupole force and to experimental results. The conclusion is drawn that the spin-quadrupole force cannot be the dominant component of the residual interaction leading to the vibrational states, but that a choice of the strenght parameter of about 20% of the ordinary quadrupole force is not ruled out, and might improve the systematic calculations.

Collective rotations in nuclei

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.

Connection between the Nuclear Shell Model and the Unified Model

The main point of this paper is to discuss the connection between the unified model and a modified form of the nuclear shell model. In the latter, the effective interactions between nucleons, i.e., those not included in the central average potential, are assumed to be factorable. It can be argued that inside nuclear matter the Pauli principle greatly suppresses the off-diagonal effects of the interactions, i.e. it prohibits most inelastic collisions; so that the nucleons move nearly freely within the nucleus, at least as far as low-energy phenomena are concerned. Short-range correlations due to the interactions are not suppressed, but these are expected to manifest themselves at rather high energies.According to this view, the effective interactions between nucleons occur mainly at the nuclear surface and give rise to surface oscillations. While quadrupole oscillations predominate, higher modes also arise in a natural way. Under certain conditions, the problem of particles subject to mutual interactions may be solved by introduction of additional collective variables, which is the method of the unified model. Physically, the nuclear motion separates, at least approximately, into intrinsic and collective motions. The resulting wave functions are very similar to the ones used by Bohr and Mottelson, except that they are integrated over the collective variables. Also, the energy spectra are of the form obtained by Bohr and Mottelson, but the collective excitation spectrum is cut off, i.e., only a finite number of states occur.This method is applied to a simplified two-dimensional nuclear model. The particles are assumed to move in an isotropic harmonic oscillator potential, and to be in addition subject to (a) one-body spin-orbit forces, and (b) mutual quadrupole-quadrupole interactions. In the absence of spin-orbit coupling, the spectrum separates into a series of rotational bands, while in the absence of mutual interactions we have an independent-particle spectrum. The intermediate coupling problem is also treated in the hope that it may provide some insight into the competition between independent-particle and collective motions in nuclei. In the present example, the transition between the two limiting schemes occurs quite suddenly, in agreement with the experimental evidence.Another case of interest is the situation at the beginning of the nuclear $1p$ shell. In the limit of pure $\mathrm{LS}$ coupling, the states of maximum spatial symmetry form a rotational band, though only very few members can appear, and collective effects are not pronounced, because of the small number of particles involved. The effect of spin-orbit coupling can also be described in the language of the unified model.

Collective Motions in Nuclei by the Method of Generator Coordinates

For an $A$-particle system, a trial wave function is constructed of the form $\ensuremath{\Psi}({\mathrm{x}}_{1}, \ensuremath{\cdots}, {\mathrm{x}}_{A})=\ensuremath{\int}\ensuremath{\phi}({\mathrm{x}}_{1}, \ensuremath{\cdots}, {\mathrm{x}}_{A}; \ensuremath{\alpha})f(\ensuremath{\alpha})d\ensuremath{\alpha}.$ The preliminary nucleonic wave function, $\ensuremath{\phi}$, solves the problem in a "construction potential." This potential depends upon a "deformation parameter" or "generator coordinate," $\ensuremath{\alpha}$. The collective wave function, $f(\ensuremath{\alpha})$, or "generator function," is folded into $\ensuremath{\phi}$ to produce a system wave function that depends only upon the coordinates, ${\mathrm{x}}_{i}$, of the particles. In the integration, the deformation parameters dissolve away. They do not appear in the final state function; they only generate it. No collective coordinates ever come into use nor do such coordinates ever have to be defined. In typical cases when the generator function contains one or more nodes, it generates nodes in the system wave function $\ensuremath{\Psi}$ of the kind that describe collective kinetic energy. The energy of the system is extremized with respect to choice of the generator function, $f(\ensuremath{\alpha})$. No Hamiltonian ever appears except the $A$-particle Hamiltonian. All nucleons are treated on the same basis whether in or above closed shells. The appropriate variational calculation leads to an integral equation or "generator wave equation" for $f(\ensuremath{\alpha})$. This equation is solved in two limiting cases: the quadratic approximation, and the $\ensuremath{\delta}$-function approximation. An analysis is made of the Peierls-Yoccoz procedure to calculate the effectivemass parameter in cases where the forces acting in the system are invariant with respect to translation or rotation. There is no external machinery to drive the construction potential. The effective inertia constant does not appear likely to agree in general with that calculated for the essentially different problem of particles in such a machine-driven potential, though the latter value is presumably more nearly correct for physical applications. The trial wave function in the method of generator coordinates is designed for simplicity, not for precision. It is applied in the following paper to the dilatational and shape oscillations of ${\mathrm{O}}^{16}$.