Inverse methods and nuclear radii

Abstract

In considering spherically symmetric three-dimensional systems, inverse methods are applied to the nuclear bound-state problem. While retaining only the self-interactions of the (occupied)bound-state levels, an analytical solution is obtained for the potential. The simplest possible approximation to it corresponding to a single fictitious bound state is used to evaluate (root mean square) radii. Combining this formula with the well-known A ' dependence of the nuclear radii, a new formula is obtained containing the collective binding energy effect and the one of the saturation of nuclear forces. For absolute and relative radii (of isotopes of Sn, Xe, Nd, Dy, Yb, Os, Hg, Pb, and Pu), the results compare favorably with experiment. In spite of the crude approximations made, this approach yields the typical curvature of the plot of the experimental relative radii as a function of the mass number. The extreme simplicity of the formula recommends its use for global discussions or predictions. Yet, for a correct description of the finer details it is necessary to account explicitly for shell effects and deformations.

Correlated variations of nuclear radii and binding energies

“…In the process of writing this article, we became aware of a similar work by Hefter et al [4]. Starting apparently from a radically different method, these authors reach conclusions comparable to ours.…”

“…Finally, a relationship very similar to (4) (with y= 1/2) has been obtained in an elegant way by Hefter et al [4] by solving the inverse problem in the case of bound states. Note that from both the energy density method and the inverse problem it appears that (4) is only a first order approximation; the value of y is not dictated by strong theoretical arguments; it is better to consider it as an adjustable parameter.…”

Influence of the separation energy on the radius of neutron rich nuclei

“…On the other hand, inverse methods have been applied to the nuclear bound states by Hefter et al [7]. Some years ago, these authors yielded a compact expression for the variation of the mean square radius when going from a nucleus with A particles to an adjacent one with A + 1 particles.…”

“…This has been pointed out by several authors in connection with apparent large nuclear radii observed together with small neutron S, or two-neutron $2, separation energies [5,6]. On the other hand, it has been shown that the simple model of Hefter, de Llano and Mitropolsky [7], based on crude approximate solutions to the inverse method, does not account for such effects. Whereas this model describes the rough correlation observed between variations of nuclear radii and binding energies, it fails to predict a large increase as S, (or $2,) tends to zero.…”

Nonlinear approach in the ?-cluster model

“…This is the KdV equation. KdV equation, was applied many times in nuclear physics [12][13][14][15][16][17]. In the work [12] the connection was established between the solution of the KdVE and the nuclear potential in Schroedinger equation.…”