#### Date of Award

7-1976

#### Degree Type

Thesis

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Physics

#### Supervisor

Dr. A.A. Harms

#### Abstract

This study is concerned with the analysis of the neutron transport equation with anisotropic phenomena. The mathematical method chosen for this analysis is the partial-range orthogonal function for the representation of the angular dependence of the pertinent angular neutronic functions. These functions are the neutron angular flux and the external sources of neutrons as well as the scattering functions. Of the several existing models of neutron transport analysis, the time-independent one-group neutron transport equations for plane and spherical geometries have been selected.

The partial-range Legendre polynomials, which represent an orthogonal set of functions over an arbitrary range of the angular variable, have been used for the analysis in this research. The reproduction of the partial-range polynomials using the Gram-Schmidt orthogonalization theorem as well as the linear transformation of variables is mathematically examined; moreover, the properties of orthogonality, the recurrence relations, and the full-range integrations have been established. The distinguishing feature of this mathematical formalism is that is combines the features of both methods of the discrete ordinate methods and the spherical harmonics approximations for the neutron transport analysis. The features of the discrete ordinate methods are incorporated by the arbitrary segmentation of the angular variable into N number of intervals. Then, the features of the spherical harmonics methods are included by expanding the neutronic functions in terms of the partial-range Legendre polynomials over these intervals. Hence, this formalism is very appropriate for the transport problems involving highly varying angular fluxes and strong anisotropic scattering.

For plane geometry, two different partial-range formalisms have been systematically developed. In both formalisms, the neutron angular flux and the external sources of neutrons are expanded in terms of partial-range Legendre polynomials. In the first formalism, which is designated by the NPL approximation, the scattering function has been represented in terms of full-range Legendre polynomials; in the second formalism, which is designated by the NPL-MPK approximation, the scattering function is reconstructed using the partial-range Legendre polynomials. The two formalisms allow for discontinuities in the angular flux as well as the external sources of neutrons at arbitrary point of the angular variable. However, it is only the second formalism of the partial-range scattering function at arbitrary points of the scattering angular variables. This permits the representation of the scattering function to a high accuracy with few terms.

An indication of the computational usefulness of these formalisms was obtained by calculating some neutronic parameters using low-order approximations. The NPL approximations have been used to calculate the eigenvalues associated with the homogeneous neutron transport equation which gives the diffusion length. Moreover, the end point, the linear extrapolation length and the ratio of the asymptotic flux to the total flux associated with the vacuum boundary of the Milne's problem in plane geometry have been examined. It has been found that this formalism of the partial-range analysis is better than the conventional methods of analysis especially for highly absorbing media and strong anisotropic scattering processes. In a certain sense, the low-order 2Po approximation doubles the range of c for the same accuracy compared to the usual double-Po approximation; the constant c is the average number of secondary neutrons per collision.

Three different low-order approximations of the NPL-MPK analysis have been examined and compared with each other as well as with the alternative methods of the same complexity. A highly anisotropic scattering function, for which the exact eigenvalue is known, has been used for the comparison. It has been found that the DPL-2PK approximation is very adequate for the analysis of problems with highly anisotropic scattering. Moreover, the DPo-DPo approximation is used to examine the critical thickness of a bare slab reactor. For practical compositions of the critical reactors, the critical thickness changes considerably with the degree of anisotropy. The effect of anisotropic scattering, therefore, must be considered in the analysis and design of nuclear reactors especially when strong anisotropic scattering is involved.

A general formalism, which allows for discontinuities in the neutron angular flux and its angular derivatives at position dependent points of the angular variables, has been established for spherical geometry. This formalism is exact and free of any functional assumptions and exactly represents the actual behaviour of the discontinuities in the angular flux, and hence it satisfies the boundary conditions exactly. The low-order 2Po approximation has been used to study the spherical Milne's problem. The results show discontinuities in the angular flux at position dependent angular points which are expected from the physical processes of the problem. The important feature of this formalism is that the 2Po approximation, which is of the same computational complexity as the diffusion theory, gives the total flux with much higher accuracy especially close to the surface of the sphere.

Finally, the partial-range formalism has been used to study some reactor physics problems of current practical interest. This is concerned with the reconstruction of elastic scattering cross section as well as the group-to-group transfer cross sections. For a numerical illustration, the elastic scattering cross sections of 14.0 MeV neutrons of U238 and Bi209, which are highly anisotropic, have been reconstructed. The results show improvement over the usual full-range representation of the scattering function. Further, the group-to-group cross sections of hydrogen, oxygen and water from (3.3287-3.0119) MeV to (2.7253-2.4600) MeV have been reconstructed using low-order representation. The results are in good agreement with the exact values and more accurate compared to the full-range Legendre polynomials approximation; the reason for this is that the group-to-group cross section is well-behaved only over certain ranges of the scattering angular variables. Moreover, this representation provides an additional degree of freedom: it is not necessary to employ the same order of approximation over the various allowable directional ranges of the scattering angular variable. The order can be varied according to the extent of anisotropy of the scattering cross section over each range. This suggests that the partial-range approximation of the scattering functions and the group-to group cross sections represents a potentially useful representation redundant in the neutron transport analysis with highly anisotropic scattering; a case in hand is the fast breeder reactor, the calculation of the blanket of the proposed fusion reactors, and detailed neutronic calculation in interface regions of thermal reactors.

#### Recommended Citation

Attia, El-Tantawy Abdel Azim, "Anisotropic Neutron Transport Analysis" (1976). *Open Access Dissertations and Theses.* Paper 907.

http://digitalcommons.mcmaster.ca/opendissertations/907