# TRANSPORT SOLUTIONS TO THE MONOENERGETIC CRITICAL PROBLEMS (thesis)

## Abstract

Transport solutions to the monoenergetic plane, spherical, and cylindrical critical problems with isotropic scattering are developed by the method of singular expansion modes. The results are given in the form of exact expressions for the neutron distributions and criticality conditions. These expressions depend on expansion coefficients that are shown to satisfy Fredholm integral equations of the second kind. Successive approximations on the coefficients lead to the asymptotic results of diffusion theory as well as to easily accessible transport corrections. The expansions for the neutron distributions are derived by two different, but equivalent, methods. In the first method a three-dimensional expansion for the angular density is developed from the elementary solutions of the transport equation and then specialized to the desired geometry. For problems with plane symmetry, the resulting expression are used directly to determine the expansion coefficients, whereas in spherical and cylindrical geometries the specialization of the three-dimensional solutions yields a variety of representations for the respective angular densities. The second method consists of replacing the angular distributions by suitable density transforms and then determining the transforms completely with the aid of the integral equations for the neutron densities. In plane geometry this method leads to the conversion of themore »

- Authors:

- Publication Date:

- Research Org.:
- Argonne National Lab., Ill.

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 4118021

- Report Number(s):
- ANL-6787

- NSA Number:
- NSA-18-006136

- DOE Contract Number:
- W-31-109-ENG-38

- Resource Type:
- Technical Report

- Resource Relation:
- Other Information: Orig. Receipt Date: 31-DEC-64

- Country of Publication:
- United States

- Language:
- English

- Subject:
- PHYSICS; ANGULAR DISTRIBUTION; CONFIGURATION; CRITICALITY; CYLINDERS; DENSITY; DIFFERENTIAL EQUATIONS; DIFFUSION; DISTRIBUTION; FREDHOLM EQUATION; MATHEMATICS; NEUTRONS; NUMERICALS; SCATTERING; SPHERES; TRANSPORT THEORY

### Citation Formats

```
Mitsis, G.J.
```*TRANSPORT SOLUTIONS TO THE MONOENERGETIC CRITICAL PROBLEMS (thesis)*. United States: N. p., 1963.
Web. doi:10.2172/4118021.

```
Mitsis, G.J.
```*TRANSPORT SOLUTIONS TO THE MONOENERGETIC CRITICAL PROBLEMS (thesis)*. United States. doi:10.2172/4118021.

```
Mitsis, G.J. Fri .
"TRANSPORT SOLUTIONS TO THE MONOENERGETIC CRITICAL PROBLEMS (thesis)". United States. doi:10.2172/4118021. https://www.osti.gov/servlets/purl/4118021.
```

```
@article{osti_4118021,
```

title = {TRANSPORT SOLUTIONS TO THE MONOENERGETIC CRITICAL PROBLEMS (thesis)},

author = {Mitsis, G.J.},

abstractNote = {Transport solutions to the monoenergetic plane, spherical, and cylindrical critical problems with isotropic scattering are developed by the method of singular expansion modes. The results are given in the form of exact expressions for the neutron distributions and criticality conditions. These expressions depend on expansion coefficients that are shown to satisfy Fredholm integral equations of the second kind. Successive approximations on the coefficients lead to the asymptotic results of diffusion theory as well as to easily accessible transport corrections. The expansions for the neutron distributions are derived by two different, but equivalent, methods. In the first method a three-dimensional expansion for the angular density is developed from the elementary solutions of the transport equation and then specialized to the desired geometry. For problems with plane symmetry, the resulting expression are used directly to determine the expansion coefficients, whereas in spherical and cylindrical geometries the specialization of the three-dimensional solutions yields a variety of representations for the respective angular densities. The second method consists of replacing the angular distributions by suitable density transforms and then determining the transforms completely with the aid of the integral equations for the neutron densities. In plane geometry this method leads to the conversion of the transport equation to a singular integral equation for the angular density. In spherical and cylindrical geometries the appropriate transforms are shown to satisfy separable equations whose solutions can be represented by unique expansions in terms of complete sets of singular expansion modes. The coefficients in the transform expansions are determined by resulting side conditions and the actual neutron distributions established by simple inversions. Although emphasis is on obtaining exact expressions for the neutron distributions and criticality conditions, the results are illustrated by approximate expressions and computations. (auth)},

doi = {10.2172/4118021},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1963},

month = {11}

}