Case Method Research Articles

SO(2,1) and strong Coulomb coupling

It is shown explicitly, in the framework of the Klein-Gordon equation, that the algebraic method based upon unitary irreducible representations of the group SO(2,1) used to solve the problem of strong Coulomb coupling (e2Z>l+1/2) is equivalent to constructing solutions that are orthogonal with respect to some mixed scalar product, rather than the standard Klein-Gordon scalar product. This elucidates the difference between the spectra given by Case's method (1950) and by the algebraic method. By explicitly computing scattering states, it is further shown that algebraic solutions describe absorption of the particle as in the corresponding classical problem.

Transport solutions for multilayer slab-cell systems

Abstract Case's method is applied to one-speed neutron-transport problems in multiplayer slab cells. A set of singular integral equations for the eigen-function-expansion coefficients of the angular neutron flux is derived, and two methods of solution suitable for numerical calculation are presented. In the first method the solution of the Hilbert problem is obtained in the form of power series for a coupling parameter. In the second method the singular integral equations are reduced to a set of Fred Holm's integral equations. The numerical results are given.

Relation between the transfer matrix method and case's method

Abstract The relation between the Transfer Matrix for radiation transport and Case's method is elucidated. Completeness of the Case eigenfunctions is shown to be equivalent to the statement that the transfer matrix can be diagonalized. It is demonstrated that there is a one-to-one relation between the basic equations of the two methods. The implications of a symmetric scattering kernel are studied.

Solution of the single-velocity neutron transport equation in multilayer plane and spherical symmetric systems

AN algorithm is obtained for solving the single-velocity equation of neutron transport in stratified plane and spherical systems, using Case's method of eigenfunctions. Isotropic and anisotropic scattering are discussed.

On Continuous Eigenvalues in Neutron Thermalization

The continuous eigenvalue spectra of the linearized Boltzmann operator describing the energy distribution of neutrons in an infinite Einstein crystal are studied. This operator consists of two terms: a multiplication operator and an integral operator with δ-function-type singular kernel. The eigenvalue problem is transformed into the solution of an inhomogeneous integral equation by applying Case's method on the one-velocity transport equation. The existence of the solution of the integral equation is examined by the Neumann-series expansion. It is found that for sufficiently low temperature the range of numerical values of the multiplication operator forms the continuous eigenvalue spectra of the Boltzmann operator and the corresponding eigenfunctions are of δ-function type.

Generating Functions for the Exact Solution of the Transport Equation. I

Generating functions are used as analytic tools for the transformation of the integro-differential transport equations to partial differential equations. The transformations are constructed so that their inverses are known. Therefore, the problem of solving the integro-differential equations is reduced to the more familiar problem of solving partial differential equations. Part I of this series introduces the general method in detail for slab geometry. The general solution is given in analytic form, and the one-to-one correspondence with Case's method of elementary solutions is demonstrated. The results are extended to slab geometry without axial symmetry. Part II extends the method to the general time-dependent, anisotropic case for slab geometry. Parts III and IV treat the stationary and time-dependent problems in one and three space dimensions, introduce in addition the method of characteristics, and include numerical results.

On the critical problem for a two-region sphere

The one-speed critical problem is solved for a two-region sphere, where the total cross section is common, but different multiplication parameters in the two zones are permitted. A transform procedure is used to reduce the integral transport equation to a pseudo problem in slab geometry. This, in turn, is solved by Case's method of normal modes. The analysis leads to a Fredholm integral equation for the only unknown expansion coefficient. In addition, a critical condition, from which the allowable parameters of the problem are to be determined, is obtained.

Multigroup Treatment of Neutron Transport in Plane Geometry

AbstractThe eigensolutions of the multigroup neutron transport equation for the isotropic scattering medium in plane geometry are found by developing Case's method. The solution of the adjoint equa...

Radiative source-function predictions for finite and semi-infinite non-conservative atmospheres

The probabilistic method of Sobolev and Case's method of normal mode expansion are combined to predict source-function distributions for radiative transfer in non-conservative, planeparallel atmospheres. The solutions obtained for semi-infinite atmospheres are exact and can be expressed in terms of functions and parameters associated with the non-conservative Milne problem. The predictions for finite atmospheres are approximate and are constructed from the semi-infinite solutions. Tabular values of the requisite functions and parameters are provided to facilitate rapid numerical evaluation of the solutions. Although the finite solutions corresponds to the zeroth-order (optically thick) approximation by Case's method, an assessment of the accuracy indicates that the results are useful for optical thicknesses as small as one or even less. The close connection between the results obtained and the method of point-direction gain of Van de Hulst is discussed.

Radiative transport in an absorbing planar medium II. Predictions of radiative source functions

An accurate solution of the source function for radiative heat transfer through a nonisothermal absorbing and emitting grey gas between heated plates is given in terms of tabulated functions. The prediction is based on an approximate formula originally proposed by Yamamoto for a grey gas atmosphere and is extended to allow for the presence of a uniform independent internal source of energy release. It is shown that knowledge of the Hopf function and the moments of the Chandrasekhar-Ambartsumian X and Y functions is sufficient to calculate solutions with a maximum fractional error of about 0·01 per cent. An alternative approach, based on Case's method of solution, is developed in the appendix and serves to confirm the functional forms of the approximate formulas. The results of this paper in conjunction with an earlier paper suffice to calculate the most important physical quantities with remarkable accuracy and simplicity.

Radiative transfer in a free-electron atmosphere

The equations of radiative transfer for an electron-scattering atmosphere, as developed by Chandrasekhar, are solved using Case's method of singular eigensolutions. The eigenfunctions of the set of coupled transport equations have been found. There are two discrete eigenvectors and two linearly independent, degenerate, singular, continuum eigensolutions. These eigenvectors are shown to form a complete basis set for the expansion of arbitrary two-component vectors defined on the half-range. In addition, all of the necessary adjoint functions have been developed so that all expansion coefficients can be obtained by taking scalar products. As an example of the method, the Milne problem is solved and explicit results are obtained for the two components of the polarized radiation field at any optical depth in the stellar medium.

Two-group transport theory

Case's method of singular normal modes is used to construct solutions to the two-group neutron transport equations in one dimension. Full-range completeness and orthogonality theorems for these eigensolutions are proved and the necessary normalization integrals are presented. In addition, functions orthogonal to the degenerate eigensolutions are developed so that all expansion coefficients can be found by simply taking scalar products. As an example of the method, the exact solution for the infinite-medium Green's function is obtained.

Radiative Transfer. II

This paper presents an exact solution to the equations of radiative transfer for a generalization of the Uniform-Picket-Fence model discussed in a previous work. Here the absorption coefficient is allowed to take N different values over the frequency spectrum. Case's method is used to construct the normal mode solutions to the set of N coupled integral equations. Then half-range completeness and orthogonality theorems are proved that enable one to solve typical half-space problems. Explicitly, the asymptotic solution to the Milne problem is developed, including the extrapolated end point, while implicitly the complete solution is available.

Application of case's method to plane parallel radiative transfer

Application of case's method to plane parallel radiative transfer

First-Order Solutions of the One-Speed Transport Equation in Plane Multiregion Geometries

Case's method is applied to multiregion problems in plane geometry. Half-range and ``two-media'' orthogonality relations are used to obtain a system of coupled integral equations for the appropriate expansion coefficients. The system is explicitly solved in a first-order approximation, which is valid for wide regions. Numerical analysis and comparison with transport computer codes indicates that such solutions are essentially exact for region thicknesses of five mean free paths or more.

An exact solution of equations of radiative transfer for Local Thermodynamic Equilibrium in the non-gray case. Picket fence approximation

Case's method of singular normal modes is used to obtain an exact solution to the equations of radiative transfer in the “uniform” or “random” picket fence model assuming Local Thermodynamic Equilibrium. Completeness and orthogonality theorems are proved, and explicit results are obtained for the extrapolated endpoint, the temperature distribution, and law of darkening for the Milne problem. In addition, the method for solving other half-space problems is sketched. The case when the determinant of the “transfer matrix” does not vanish (corresponding to radiative transfer without Complete Local Thermodynamic Equilibrium or the neutron transport problem) is also discussed. Full range completeness and orthogonality theorems are presented for this case; however, no convenient form for the half-range case has been found.

Transport Solution of the One-Speed Slab Albedo Problem

Case's method of normal mode expansion is applied to the slab albedo problem (the problem of finding the angular density in a bare slab with monodirectional neutrons incident upon one face). The problem is reduced to determining expansion coefficients which are shown to depend upon the solution of two nonhomogeneous Fredholm integral equations. Using Neumann iteration to solve for the coefficients, explicit solutions are obtained for the angular density, scalar density, and net current in zero and first order approximations. (auth)

Orthogonality of case's eigenfunctions in one-speed transport theory

The eigenfunctions of the one-speed transport equation, as introduced by Case, are shown to have more general orthogonality properties than previously known. In particular, for isotropic scattering, partial range and “two-media” orthogonality relations are derived. An extension to linearly anisotropic scattering is indicated. These results facilitate the application of Case's method to one-speed transport problems in plane geometry.

An application of singular integral equation theory to a linearized problem in couette flow

This paper treats a linearized problem in Couette flow which has been derived from a simple version of the Krook statistical model for intermolecular collisions. The problem is reduced to the solving of Fredholm equations which are shown to converge rapidly for all values of the Knudsen number. The methods are based on singular integral equations and were first developed in neutron transport theory. Cercignani, in applying Case's method to the slip flow problem, has also studied singular equations dual to those of this paper.

One-Speed Neutron Transport in Two Adjacent Half-Spaces

Using Case's method for solving the one-speed transport equation with isotropic scattering, the Milne problem solution, the solution for a constant source in one half-space, and the Green's function solution are obtained for two adjacent half-spaces. These problems have been solved previously by other methods. Here the derivations are greatly simplified by using Case's method.