General expressions for divergence relations and multipole expansions for arbitrary scalar functions

Journal Article · Sun Oct 01 04:00:00 EDT 1978 · J. Math. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1063/1.523555· OSTI ID:6985386

Davies, K T.R.; Becker, R L

In theories of the cohesion of a system whose matter density is constant (e.g., classical liquid drops) or nearly constant except in a thin surface region (e.g., nuclei or neutron stars), there occur computationally difficult single and double volume integrals of energy densities. Such integrals have been important in recent dynamical calculations of fission and heavy-ion reactions. Even if the matter density is diffuse, the integrals can be written as integrals over finite volumes by modeling the density as the convolution of a step function and a diffuseness function. If the integrands can be written as divergences of tensor fields, the integrals over finite volumes can be reduced to surface integrals by Gauss' theorem. We have found general expressions, which we call divergence relations, for a vector field whose divergence is a given scalar function and for a second-rank tensor field whose double divergence is the scalar. The equations derived are much easier to use and apply to a wider class of functions than formulas previously obtained in the literature. The generalization to nth order for application to many-body forces is included, and for all orders the dimensionality is arbitrary. The interaction energy of two nonoverlapping systems is often most simply evaluated by using the generalized Slater multipole expansion of the two-body interaction. A new expression is derived for the radial factor G/sub l/(r/sub 1/,r/sub 2/) appearing in the multipole expansion of an arbitrary scalar, two-body function. This G/sub l/ is expressed as an integral involving the product of the Fourier transform of the interaction and two Bessel functions. For some cases this integral can easily be evaluated by contour integration.

• Research Organization: Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830
• OSTI ID: 6985386
• Journal Information: J. Math. Phys. (N.Y.); (United States), Journal Name: J. Math. Phys. (N.Y.); (United States) Vol. 19:10; ISSN JMAPA
• Country of Publication: United States
• Language: English

Related Subjects

658000* -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
FISSION
FOURIER TRANSFORMATION
INTEGRAL TRANSFORMATIONS
INTEGRALS
MANY-BODY PROBLEM
MANY-DIMENSIONAL CALCULATIONS
MULTIPOLES
NUCLEAR REACTIONS
TENSORS
TRANSFORMATIONS
TWO-BODY PROBLEM
VECTOR FIELDS