1
SECTION 8: Nuclear Models – The Liquid Drop
Theoretical models of the nucleus encounter two principal problems:
(1)
There is no exact mathematical expression that accounts for the nuclear force,
unlike the atomic case, for which the electromagnetic force is well-defined by
Coulomb’s Law, and
(2)
There is no mathematical solution to the many-body problem, a limitation shared
by both nuclear and atomic systems.
Fortunately, the power of the computer permits calculations that use sophisticated
approximations to minimize these problems and provide increasingly accurate models for
describing nuclear and atomic properties.
The starting point for theoretical models of the nucleus treats the problem from two
divergent perspectives: At the macroscopic extreme is the
Liquid Drop Model
, which
examines the global properties of nuclei, such as energetics, binding energies, sizes,
shapes and nucleon distributions. This model assumes that
all nucleons are alike
(other
than charge).
In contrast the
Shell Model
is designed to account for the quantal
properties of nuclei such as spins, quantum states, magnetic moments and magic
numbers. The basic assumption of the Shell Model is that
all nucleons are different
, i.e.
nucleons are fermions and must occupy different quantum states, as is the case for atoms.
The idealized goal of theoretical nuclear physics is to combine these two concepts into a
Unified Model that will describe both the macroscopic and microscopic aspects of
nuclear matter in a single comprehensive framework.
Justification for the Liquid Drop Model
The basic assumption of the Liquid Drop Model is that the nucleus is a charged, nonpolar
liquid drop held together by the nuclear force. In the simplest case, the chemical analogy
would be a droplet of composed of nonpolar
molecules such as CCl
4
or
isopentane held
together by Vander Waal’s attraction. For such systems, the following properties are
observed:
•
The attractive force is short-ranged; i.e. there is a relatively sharp boundary at the
surface, similar to our earlier discussions of a uniform density sphere or Woods-
Saxon nucleon distribution.
•
The force is saturated; i.e. all nucleons in the bulk of the liquid are bound equally,
independent of radius.
•
The nucleus is incompressible in its ground state, which accounts for the nearly
uniform density distribution (Fig. 7.3) and constant average binding energy (Fig.
2.1).
•
Surface tension is created by the loss in binding for nucleons on the nuclear
surface, an effect that leads to a spherical shape to minimize the surface energy.
There are, however, significant differences between a classical liquid drop and a nucleus
which must be accounted for in the model.
For example: