The objectives of this thesis include the development of an exact theory of neutron wave propagation in non-multiplying media as well as the application of this theory to analyze current experimental work.
An initial study is made of the eigenvalue spectrum of the velocity-dependent Boltzmann transport operator for plane wave propagation in both noncrystalline and polycrystalline moderators. The point spectrum is discussed in detail, and a theorem concerning the existence of discrete eigenvalues for high frequency, high absorption, and/or small transverse dimensions is demonstrated. The limiting cases of low and high frequency behavior are analyzed. A physical interpretation of the discrete and continuum eigenfunctions (plane wave modes) is given, and the point spectrum existence theorem is explained in the light of such interpretations.
Using this spectral representation, a technique for solving full-range boundary value problems for a general noncrystalline scattering kernel ~s presented. Orthogonality and completeness of the eigenfunctions are demonstrated, and the problem of a plane source at the origin of an infinite medium is solved. This solution is compared with that obtained by a Fourier transform technique. A procedure for solving half- range boundary value problems is presented for a one-term separable kernel model. For purposes of illustration, the problem of an oscillating source incident upon the boundary of a half-space is solved. The difficulty in extending the half-range theory to more general scattering models is discussed.
The second part of the thesis proceeds to demonstrate this theory in more detail by applying it to analyze recent neutron wave experiments in graphite and D2O parallelepipeds. To facilitate the interpretation of the general solution, the inelastic scattering kernel is approximated by a separable kernel, while the elastic scattering is modeled with a Dirac δ-function. The eigenvalue spectrum is analyzed in some detail, revealing several interesting conclusions concerning the experimental data and methods of data analysis. Several modifications in experimental design and analysis are suggested.
The agreement of the theory with experiment is sufficient to warrant its application to the analysis of more complicated experiments (multiple·-region, multiplying media, pulse propagation, etc.). Several suggestions for such extensions are indicated.
}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Lurie, Harold and Corngold, Noel Robert}, } @phdthesis{10.7907/DDW9-CY15, author = {Ludewig, Hans}, title = {Geometrical effects on the resonance absorption of neutrons}, school = {California Institute of Technology}, year = {1966}, doi = {10.7907/DDW9-CY15}, url = {https://resolver.caltech.edu/CaltechTHESIS:09242015-135757893}, abstract = {An investigation was conducted to estimate the error when the flat-flux approximation is used to compute the resonance integral for a single absorber element embedded in a neutron source.
The investigation was initiated by assuming a parabolic flux distribution in computing the flux-averaged escape probability which occurs in the collision density equation. Furthermore, also assumed were both wide resonance and narrow resonance expressions for the resonance integral. The fact that this simple model demonstrated a decrease in the resonance integral motivated the more detailed investigation of the thesis.
An integral equation describing the collision density as a function of energy, position and angle is constructed and is subsequently specialized to the case of energy and spatial dependence. This equation is further simplified by expanding the spatial dependence in a series of Legendre polynomials (since a one-dimensional case is considered). In this form, the effects of slowing-down and flux depression may be accounted for to any degree of accuracy desired. The resulting integral equation for the energy dependence is thus solved numerically, considering the slowing down model and the infinite mass model as separate cases.
From the solution obtained by the above method, the error ascribable to the flat-flux approximation is obtained. In addition to this, the error introduced in the resonance integral in assuming no slowing down in the absorber is deduced. Results by Chernick for bismuth rods, and by Corngold for uranium slabs, are compared to the latter case, and these agree to within the approximations made.
}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Lurie, Harold}, } @phdthesis{10.7907/00R6-9A13, author = {Erdmann, Robert C.}, title = {Time-dependent monoenergetic neutron transport in two adjacent semi-infinite media}, school = {California Institute of Technology}, year = {1966}, doi = {10.7907/00R6-9A13}, url = {https://resolver.caltech.edu/CaltechTHESIS:10022015-101429428}, abstract = {
An exact solution to the monoenergetic Boltzmann equation is obtained for the case of a plane isotropic burst of neutrons introduced at the interface separating two adjacent, dissimilar, semi-infinite media. The method of solution used is to remove the time dependence by a Laplace transformation, solve the transformed equation by the normal mode expansion method, and then invert to recover the time dependence.
The general result is expressed as a sum of definite, multiple integrals, one of which contains the uncollided wave of neutrons originating at the source plane. It is possible to obtain a simplified form for the solution at the interface, and certain numerical calculations are made there.
The interface flux in two adjacent moderators is calculated and plotted as a function of time for several moderator materials. For each case it is found that the flux decay curve has an asymptotic slope given accurately by diffusion theory. Furthermore, the interface current is observed to change directions when the scattering and absorption cross sections of the two moderator materials are related in a certain manner. More specifically, the reflection process in two adjacent moderators appears to depend initially on the scattering properties and for long times on the absorption properties of the media.
This analysis contains both the single infinite and semi-infinite medium problems as special cases. The results in these two special cases provide a check on the accuracy of the general solution since they agree with solutions of these problems obtained by separate analyses.
}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Lurie, Harold}, } @phdthesis{10.7907/K87B-FC98, author = {Lathrop, Kaye Don}, title = {Neutron thermalization in solids}, school = {California Institute of Technology}, year = {1962}, doi = {10.7907/K87B-FC98}, url = {https://resolver.caltech.edu/CaltechETD:etd-12122005-162942}, abstract = {To describe neutron thermalization in solid media, two simplified models are formulated to describe the motions of atoms bound in solids. One atomic model postulates that the atoms of solids are linear, classical, randomly oriented, harmonic oscillators characterized by a single energy; and the other model postulates the same basic oscillator but permits a distribution of oscillator energies. With atom speed distributions derived from these models, energy exchange cross sections are evaluated analytically assuming a free particle neutron-atom interaction. With these energy exchange cross sections, integral equations are formulated describing thermalization of neutrons in infinite homogeneous media containing a 1/v absorber. The integral equations of both atomic models are solved numerically for the neutron density speed distribution. Numerical results for the single energy atomic oscillator of unit mass are compared with experimental results for neutron thermalization in zirconium hydride. Results for the averaged energy atomic oscillator of unit mass are compared with the neutron density calculated from the Wigner-Wilkins monatomic gas model. This comparison is made for three values of absorption. Numerical results for averaged energy atomic oscillators of masses 1, 2, 9, and 12 are examined to determine the effect of atomic mass upon the neutron density distribution.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Lurie, Harold}, }