SEC8LDModel - SECTION 8: Nuclear Models The Liquid Drop...

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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:
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2 The nucleus has a limited number of particles (<270) compared to chemical systems (~10 23 ). The net result is that there is a much larger fraction of nucleons on the surface relative to those in the bulk for nuclei compared to chemical systems.
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This note was uploaded on 07/02/2011 for the course CHEM-C 460 taught by Professor Staff during the Spring '10 term at Indiana.

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SEC8LDModel - SECTION 8: Nuclear Models The Liquid Drop...

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