Citation

## Material Information

Title:
Analytic and numerical transport techniques in energy-dependent past neutron wave and pulse propagation
Creator:
Swander, James Elza, 1939- ( Dissertant )
Perez, R. B. ( Reviewer )
Mockel, A. J. ( Reviewer )
Ohanian, Mihran J. ( Thesis advisor )
Carroll, Edward E. ( Reviewer )
Selfridge, Ralph G. ( Reviewer )
Green, Alex E. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
1974
Language:
English
Physical Description:
x, 148 leaves. : illus. ; 28 cm.

## Subjects

Subjects / Keywords:
Continuous spectra ( jstor )
Eigenfunctions ( jstor )
Eigenvalues ( jstor )
Elastic scattering ( jstor )
Inelastic scattering ( jstor )
Neutrons ( jstor )
Spectral theory ( jstor )
Streaming ( jstor )
Wave propagation ( jstor )
Dissertations, Academic -- Nuclear Engineering Sciences -- UF ( lcsh )
Fast neutrons ( lcsh )
Neutron transport theory ( lcsh )
Nuclear Engineering Sciences thesis Ph. D ( lcsh )
Pulsed reactors ( lcsh )
Thermal neutrons ( lcsh )
City of Gainesville ( local )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )
Spatial Coverage:
United States--Tennessee--Oak Ridge

## Notes

Thesis:
Thesis -- University of Florida.
Bibliography:
Bibliography: leaves 143-147.
Also available on World Wide Web
General Note:
Typescript.
General Note:
Vita.

## Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
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14082761 ( OCLC )

Full Text

JAMES ELZA SWANDER

A DISSERTATION PRESENTED TO THE GRADUATE
COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA
1974

... ......

i "

]~

ACKNOWLEDGMENTS

The author would like to thank those who served at various times as

chairman of his committee, Drs. R. B. Perez, A. J. Mockel, and M. J.

Ohanian. In addition to the guidance given by the above individuals,

the author would like to acknowledge helpful discussions with Drs. J.

Doing and R. S. Booth.

Financial assistance was provided by a NASA Predoctoral

Traineeship in Space Sciences and Technology, a University of Florida

College of Engineering Fellowship, and Department of Nuclear Engineering

Support for the computations performed was provided by the

Northeast Regional Data Center.

The manuscript for this dissertation was prepared at the Oak Ridge

National Laboratory with the sponsorship of the U.S. Atomic Energy

Commission under contract with Union Carbide Corporation.

ACKNOWLEDGMENTS . . . . . . .

LIST OF FIGURES . . . . . . .

ABSTRACT . . . . . . . . .

CHAPTER

I INTRODUCTION . . . . . .

PAGE

ii

vi

viii

Purpose . . . . . . . . ... ... . .. .1

Early Neutron Wave Investigations . . . . . . 2

Fast Neutron Wave Investigations . . . . . . 4

Energy-Dependent Transport Formulation of
Wave and Pulse Propagation . . . . . . .. 4

Plane Symmetry and the Eigenvalue Equation . . . . 6

Interaction Operators for the Fast and Thermal
Neutron Regimes . . . . . . . . . . . 9

Spectrum and Eigenfunctions of the Thermal
Transport Operator . . . . . . . .... . 13

Completeness of the Thermal Eigenfunctions . . ... 28

Other Related Problems and Literature . . . . .. 32

II SPECTRUM AND EIGENFUNCTIONS OF THE FAST WAVE
TRANSPORT OPERATOR . . . . . . . .... . 35

Introduction . . . . . . . . ... .. . .35

Adjoint Eigenfunction Equations . . . . . ... .35

Biorthogonality of Eigenfunctions . . . . . . 37

Nonmultiplying Media: Spectrum of the
Slowing-Down Transport Operator . . . . . ... 38

CHAPTER PAGE

Forward and Adjoint Slowing-Down Eigenfunctions ..... 41

Discussion of the Slowing-Down Eigenfunctions . .. .. 46

Realistic Cross Sections: Nonmonotonic vZ 53

Fast Multiplying Media: Zero Scattering Cross
Section . . . . . . . . ... . . . . 56

Discrete Eigenfunctions for Fast Multiplying Media . .. 57

Continuum Eigenfunctions for Fast Multiplying Media . .. 59

Discussion of the Continuum Eigenfunctions . . ... 60

Degeneracy of the Continuum . . . . . . . .. 62

The Boltzmann Equation with Isotropic Interaction .... 62

III APPLICATION TO THE TRANSFER MATRIX METHOD . . . ... 65

Introduction . . . . . . . . . . . 65

Formal Operator Relationships . . . . . .... .66
~~ ~ -1
The Operators a6, X, and . 67

Spectrum and Eigenfunctions of a0 . . . . . ... 72

The Inverse Operator X 75

Diagonalization Operators . . . . . . . ... .76

Full-Range Orthogonality and Completeness . . . ... 78

Half-Range Orthogonality and Completeness . . . ... 79

Application to Fast Neutron Wave Propagation . . .. 81

IV APPLICATION TO DISPERSION LAW AND DISCRETE
EIGENFUNCTION CALCULATIONS . . . . . . . . 83

Introduction . . . . . . . . ... .. . .83

The Dispersion Function and Discrete Eigenfunctions . . 84

Algorithms for Evaluating the Discrete Spectrum
and Eigenfunctions . . . . . . . .... . 86

CHAPTER

Extension to Degenerate Kernels . . . . . . .

Isotropic Elastic and Inelastic Scattering . . . .

Illustrative Results: Dispersion Law and Eigenfunc
for Single Scattering Species . . . . . .

V SUMMARY AND CONCLUSIONS . . . . . . .

Summary . . . . . . . . . . .

Conclusions and Suggestions for Future Work . .

APPENDIX A

Introduction . . . . . . . . . .

General Formalism . . . . . . . . .

Algebra of the H-Matrix . . . . . . .

Form of the H-Matrix: T and R Operators . . .

Two-Region Transfer Matrix . . . . . .

Internal Sources . . . . . . . . .

Transfer Matrix for Homogeneous Slabs . . . .

The Operators a and . . . . . . . .

Diagonalization of the Transfer Matrix . . .

Transmission and Reflection Operators . . . .

Wave Transport Form of a and B . . . . .

APPENDIX B

Singularity of Inelastic Scattering Kernel Models

APPENDIX C

Macroscopically Elastic Scattering: The Elastic
Continuum . . . . . . . . . . .

BIBLIOGRAPHY . . . . . . . . . . . .

BIOGRAPHICAL SKETCH . . . . . . . . . .

PAGE

87

89

tions

100

100

102

107

107

111

112

116

117

119

124

127

129

130

133

LIST OF FIGURES

Figure

1.1. The continuum domain C in the spectral <-plane .

1.2. Structure of the continuum . . . . . .

1.3. Schematic dispersion law for a discrete eigenvalue

2.1. Orthogonality of forward and adjoint slowing-down
eigenfunctions . . . . . . . . .

2.2. Excitation of slowing-down eigenfunctions by a
monoenergetic source . . . . . . . .

2.3. Degeneracy of the continuum due to nonmonotonic vZ

4.1. Dispersion laws for constant cross-section, elastic
scattering model . . . . . . . . .

4.2. Zero frequency eigenfunction energy spectra . .

4.3. Eigenfunction energy spectra for moderate to high
frequencies . . . . . . . . .

4.4. Eigenfunction phases for moderate frequencies .

4.5. High frequency eigenfunction phase and amplitude
relationship . . . . . . . . . .

Page

S. 17

S. 18

S. 26

. 47

. 48

54

S. 92

S. 94

S. 95

S. 96

97

4.6. Eigenfunction energy spectrum for frequency approaching
the critical frequency . . . . . . . .

A.I. Entering and emerging fluxes for a single region . .

A.2. Entering and emerging fluxes for adjacent regions . .

A.3. Transmission . . . . . . . . . . .

A.4. Reflection . . . . . . . . . . . .

A.5. Transmission through adjacent regions . . . . .

A.6. Internal inhomogeneous sources . . . . . . .

98

109

109

113

113

118

118

Figure Page

A.7. Fluxes at an internal coordinate surface . . . ... 126

C.1. Schematic diagram of the "elastic continuum" for
macroscopically elastic scattering .. ... . 140

Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of Doctor of Philosophy

ANALYTIC AND NUMERICAL TRANSPORT TECHNIQUES IN ENERGY-DEPENDENT
FAST NEUTRON WAVE AND PULSE PROPAGATION

By

James El;a Swander

June, 1974

Chairman: Mihran J. Ohanian
Major Department: Nuclear Engineering Sciences

Neutron wave and pulse propagation analysis is a natural approach

to space-dependent kinetics in subprompt critical media. Prior to the

present work, analytic treatments of fast media have been few and

limited in scope, in contrast to thermal wave and pulse propagation,

which has been studied rather thoroughly and with quite sophisticated

techniques.

The principal difference between analysis of fast and thermal

systems is treating the slowing-down operator. A formal approach is

presented for arbitrary slowing-down operators; the spectrum, eigen-

functions, and adjoint eigenfunctions of the slab-geometry energy-

dependent wave transport operator are obtained, using the singular

eigenfunction technique. Both multiplying and nonmultiplying media are

treated. Fission is modeled by a one-term separable kernel, although

viii

the extension to a multiterm degenerate fission kernel, representing

several fissionable species, is apparent.

The fast neutron wave singular eigenfunction results are compared

with other energy-dependent transport work, particularly with previous

thermal eigenfunction analysis, and with work on static fast neutron

transport using an energy transform approach. Wave transport in fast

multiplying media and in thermal noncrystalline media (modeled by a

separable thermalization kernel) are rather similar in that due to

energy-regenerative interaction processes a discrete asymptotic separable

eigenmode exists for moderate values of wave frequency and absorption

cross sections. The dispersion laws obeyed by the eigenvalues associated

with these modes are qualitatively quite comparable. The fast nonmulti-

plying case has no direct thermal analogue other than the nonphysical

"absorption only" model. It is found that the presence of down-scattering

in this case gives rise to singular continuum eigenfunctions which are

not as simply interpreted as the straightforward streaming modes

obtained for zero scattering cross section. Nevertheless, these results

appear to be in qualitative agreement with other work on energy-dependent

fast neuton transport theory.

The formal analytic results are developed in several directions to

investigate their applicability to practical calculations. A major por-

tion of this work is devoted to obtaining the energy-dependent wave

transport representation of the transfer matrix method, which provides a

formalism for implementing calculations concerning wave and pulse propa-

gation through finite regions such as adjacent slabs of different compo-

sition. It is found that, for isotropic scattering, the basic operators

of the transfer matrix formalism can be constructed from eigenfujctions

of the wave transport operator. This result is general and is equally

applicable to fast and thermal analysis.

Finally, the dispersion law expression for a fast multiplying

medium is employed to develop an algorithm for computing the discrete

eigenfunctions and associated dispersion law for separable and degener-

ate fission kernels. A specific application of this method then is

made to the case of isotropic elastic and inelastic scattering from any

number of nuclides and levels, with arbitrary lethargy dependence of

cross sections. Elastic scattering is modeled by a free gas kernel, and

inelastic scattering by a constant energy loss per interaction per

level. Unlike techniques requiring inversion of matrices, computation

time increases approximately linearly with increases in lethargy steps,

making quite detailed computations feasible. Illustrative computations

are carried out using constant cross sections and a single elastic

scattering species.

CHAPTER I

INTRODUCTION

Purpose

As the demand for energy increases and supplies of economically

recoverable fuels diminish, fast breeder reactors will supply an

increasing proportion of base-load generated power. To operate most

economically these reactors will tend to be as large as is technologi-

cally feasible. Accordingly, as the size of fast reactor cores

increases, it will become increasingly important to understand spatially

dependent kinetic effects in fast systems.

A particularly straightforward method of investigating

propagation of neutronic disturbances in fast reactor materials is to

place a pulsed or oscillating source of neutrons at the face of an

experimental assembly, and then to observe the propagation of the neu-

tron "signal" .through the assembly. In this way one can study spatially

dependent flux oscillations such as might be expected to result from

flow-induced vibrations of core components, void formation and collapse,

and other such phenomena.

Neutron wave and pulse propagation experiments have been performed

in many different thermal media, both multiplying and nonmultiplying.

The theoretical basis of analysis of thermal wave propagation in

nonmultiplying media has attained a considerable degree of sophistication,

and fairly accurate numerical prediction of some experimental results is

possible. This is in contrast to the situation in the fast neutron wave

regime; few experiments have been performed, and analytic investigations

have been hampered by difficulties which do not arise in treatment of

thermal systems. The purpose of this dissertation is to present a

particular framework of approach within which these difficulties may be

addressed, extending techniques which have been applied primarily to

thermal analysis.

Three general objectives will be pursued:

(i) to develop the spectral representation of the energy-

dependent fast wave Boltzmann operator as far as possible

in sufficiently general form so that its potential for use

with realistic cross section data can be evaluated;

(ii) to extend a formalism which treats neutron transport in

finite and discontinuous media so that the above results may

be applied to wave transport in experimentally realistic

geometries and through successive regions; and

(iii) to illustrate applications of the analysis by computing the

fundamental eigenfunction and dispersion law for wave

propagation in fast multiplying media, using a modelled

kernel in the Boltzmann operator.

Early Neutron Wave Investigations

In 1948 Weinberg and Schweinler published the first description in

the open literature of the generation and analysis of neutron waves [1].

Using one-speed diffusion theory they were able to show that a localized

oscillation in neutron absorption within a reactor would produce a

perturbation in the neutron population which would propagate in wave-

like fashion. The first experiments with neutron waves were reported

in 1955 by Raievski and Horowitz [2], using a mechanically modulated

exterior source to generate waves in D20 and graphite. Uhrig [3] then

applied this technique to measurements in subcritical assemblies. Both

experimental and theoretical aspects of neutron wave propagation

subsequently received considerable attention and refinement, particularly

by Perez [4] and his associates at the University of Florida, although

experiments and most analytic efforts were restricted to thermal systems.

As investigation of the theoretical basis of neutron wave

experiments proceeded, it was realized that from an analytic standpoint

experiments involving spatially propagating pulses were equivalent to

neutron wave experiments, since any physically realizable pulse could be

time-Fourier analyzed to give its frequency components [5]. Also, it

became clear that neutron wave propagation was related to other linear

static and kinetic experimental techniques, in particular the classical

exponential experiment, which is the zero-frequency limit of the wave

experiment, and the pulse die-away experiment [5-8]. (The die-away

experiment monitors the time-rate of decay of a neutron population

which has been introduced into a finite assembly by a pulsed

external source. For technical reasons this type of experiment

was easier to perform than wave experiments, and enjoyed a more rapid

initial development [4].) As a result, these methods experienced

considerable parallel theoretical treatment [8-10]. Early work if this

area is reviewed extensively by Uhrig [7] and Perez and Uhrig [4].

Fast Neutron Wave Investigations

Neutron wave and pulse propagation has received proportionately

very little attention in the fast energy regime. The only experiments

described in the literature, performed by Napolitano et al. [11,12] and

Paiano et al. [13] at the University of Florida, have been in nonmulti-

plying media; no experiments in multiplying media have been reported.

The technical difficulty of such experiments probably has contributed

both to the lack of experimental data and the scarcity of methods to

predict and correlate results. Theoretical analysis also has been

retarded by the fact that even tractable energy-dependent analytic

models of fast media do not have convenient mathematical properties,

and consequently most of the elegant techniques which have been applied

to thermal neutron transport cannot be extended readily to this

problem [14]. Notable exceptions to the general absence of numerical

techniques and results are the multi-group, multiplying medium calcula-

tions of Travelli [15,16], and the calculations of Booth et al. [17],

using the multi-group discrete ordinates method of Dodds et al. [18] to

interpret Napolitano's experimental results.

Energy-Dependent Transport Formulation
of Wave and Pulse Propagation

Before discussing the various theoretical results which are

directly or indirectly applicable to fast neutron wave and pulse

5

propagation, it will be helpful to approach the general neutron wave

problem from the point of view of the energy-dependent transport method

which will be used in this dissertation. We begin with the classic

time-dependent Boltzmann equation for the neutron flux [19,20,21],

which we will write

1 at +(r ,EA,t) + Vp(r',E, ,t) + E (E)4(r,E, 5,t)

dQ dE' K(E',Q' E,Qn)(r,E',Q',t) = S(r,E,n,t) (1.1)
4n o

In standard notation, 0(r,E,Q,t) is the neutron directional flux, v is

the neutron speed, 5 is the unit vector in the direction of neutron

travel, E is the neutron energy, r is the spatial coordinate, Et(E) is

the total cross section, and S(r,E,j,t) is a flux-independent source

term. The interaction kernel K(E',t5' -E,) contains all neutron inter-

action processes which give rise to secondary neutrons with altered

energy and direction. All cross sections (and thus K) are taken to be

region-wise independent of r, and the medium described by the equation

is assumed to be isotropic in the sense that interaction properties do

not depend on the initial direction of neutron travel. Cross sections

also are assumed to be constant with respect to time and independent of

the flux; neutron wave and pulse propagation experiments meet these two

criteria quite well. Propagation of disturbances in sub-prompt critical

reactors also should be adequately described by this linear kinetic

model. (Nonlinear space-dependent kinetics are of interest prirftrily in

the context of excursion situations; such problems, while important, are

difficult to analyze, and thus far have been approached by use of

specialized and involved computational techniques [22,23,24].)

Any of a number of classic analytic approaches can be taken to the

solution of Eq. (1.1); here we will treat it as an eigenvalue problem.

This will enable us to extend our results to finite medium and multi-

region problems; the transfer matrix formalism, which will be discussed

in Chapter III, requires solution of a similar eigenvalue equation, and

we will be able to relate its solution to those of the wave Bolt:mann

equation for isotropic scattering. Furthermore, we can make use of

spectral analysis which already has been done on the thermal neutron

version of Eq. (1.1).

Plane Symmetry and the Eigenvalue Equation

The mathematical development of transport theory has reached its

greatest sophistication for the case of plane symmetry, and this is true

also for the particular subject of neutron wave and pulse propagation.

Since this geometry also is appropriate for the description of classi-

cal wave and pulse propagation experiments, we will turn our attention

to the specific case of plane neutron waves. The infinite medium plane

wave eigenfunctions which will be obtained may then be used in develop-

ing the corresponding transfer matrix formalism, which can be employed

to study the propagation of waves and pulses through finite slabs and

successive slabs of dissimilar materials.

Following customary arguments, we stipulate that all sources or

initial fluxes must be rotationally symmetric about the x axis and do

not depend on the transverse Cartesian coordinates y and z. Orienting

the x axis along the direction of wave propagation, we have for the

transport operator of Eq. (1.1)

A = p x
a

where x is the unit vector in the

the angle between the path of neu

restrictions of Eq. (1.2) the hom

written

j nE x

(1.2)

x direction, and p is the cosine of

tron travel and the x axis. With the

ogeneous Boltzmann equation may be

\$(x,E,U,t) + p -- (x,E,I ,t) + (E) q(x,E,W,t)

ldu' dE' K(E',p E,p)P(x,E',p',t) = 0. (1.3)
-1 c

We notice that time and space operators appear only in the first

two terms, respectively, while the integral operator acts on E and p.

Consequently, x and t variables may be separated. With appropriate

choices for the separation constants t may be expressed as a damped

plane wave

iwt -Kx
K (x,E,W,t) = F(E,u;<) e e

-ax i(wt-(x)
= F(E,p;K) e- e (1.4)

where w is the wave or Fourier-component frequency, and K is a complex

constant, < a + iC, the complex inverse relaxation length. Thus a is

the inverse relaxation length of the wave, while C is its wave number.

The frequency w will be regarded as a parameter of the equation, and we

will treat K as the eigenvalue to be determined. Introducing Eq. (1.4)

into Eq. (1.3)

(E ) 1 m-
i- U< + t(E) F(E,u;<) -J du dE K(E,u -+ E,u)F(E',' ;<) = 0,
-1 o
(1.5)

or defining

a(E,w) E E (E) + (E)
t v (E) (1.6)

Eq. (1.5) has the form

(a(E,w) UK)F(E,1;K) du' dE' K(E',u' E,.)F(E',V';<) = 0
-1 0 (1.7)

which will be referred to as the wave Boltzmann equation, or WBE,

throughout the rest of this work. This is the most general statement of

the Boltzmann equation in wave eigenvalue form with plane symmetry. As

an eigenvalue problem it should more properly be written

I du dE a( 6(E-E')6(-p) - K(Et', -V E,P) F(E',j';<)
-1 o

=

The pole at p = 0 causes no difficulties which we will need to

consider [25]; this value of u corresponds to a direction of neutron

travel perpendicular to the direction of wave propagation.

Interaction Operators for the Fast and
Thermal Neutron Regimes

The two general types of neutron interaction which are of

importance in wave and pulse propagation and which enter into the

kernel of Eq. (1.7) are scattering and fission. Adopting for a moment a

theoretician's perspective on reality, we may define a fast neutron

experiment as one in which the scattering kernel has a Volterra form in

energy. In a similar vein a thermal problem may be distinguished by the

presence of a Fredholm scattering kernel. These observations stem from

the fact that in the fast neutron regime one is concerned with neutron

energies from the eV range to about 10 MeV, the upper end of the fission

spectrum; hence only downscattering in energy is important. By contrast,

in the thermal regime neutrons are in or near thermal equilibrium with

their surroundings so that upscattering in energy occurs as well;

energies of interest range essentially over the thermal Maxwellian

spectrum.

Appropriate interaction kernel models reflect these properties.

We may write the thermal scattering contribution to the interaction

kernel as

dJ dfE' E (E',t -V E,-)"
-1 o

f1 f

-1 E

extending the notation of Eq.s (1.3) to (1.7).

In multiplying media the interaction kernel contains a contribution

due to fission in addition to scattering. Difficulties associated with

treating the slowing down of fission neutrons [26] have precluded trans-

port analysis of wave propagation in thermal multiplying media, although

other models such as age-diffusion have been employed [27]. No such

problem arises in the fast multiplying wave problem, since the energy

range of the fission spectrum is essentially the energy range of

interest.

For the fission contribution to the interaction operator we will

use the customary isotropic separable kernel

X(E)vZf(E').

When only one fissionable isotope is present this is a satisfactory

model. For two or more fissionable species, one can either construct an

equivalent separable kernel with averaged X and \)f or employ a degene-

rate kernel

2 (ji) (E (j)v ()
J X j (E) \)Z f (E').
j=l

Only the separable kernel will be treated in detail here but the

formalism of Chapter II can be extended in a straightforward way to

degenerate kernels for multiple fissioning species.

To avoid the appearance of unwelcome factors of 1/2 in connection

with the isotropic fission spectrum, we will make the following

notational distinction. Define the isotropic X(E) so that

JX(E)dE = 1. (1.9)
o

Define X so that

x E x(E,p) = 1/2 x(E);

(1.10)

then V.

f1 0
dP dE X = 1. (1.11)
-1 o

The fission interaction operator employing the separable kernel model

and Eq. (1.10) thus becomes

y fdu' dE' v- (E').
-1 o

which is the form which will be used throughout this work.

Using the above forms for contributions to the interaction kernel

the fast homogeneous WBE of Eq. (1.7) may be written

1 f
(a-u<)F(E,jp;) = dl< IdE' Z (E',p' E,I)F(E',u';<) (1.12)
-1 E

for nonmultiplying media, and

1
(a-p<)F(E,i; -1 E

+ J: du dE' d E(E')F(E',U';K) (1.13)
-1 o

for multiplying media.

It should be noted that we have taken into account only prompt

neutrons in Eq (1.13), and hence v here is the number of prompt neutrons

per fission. Delayed neutrons will contribute only at wave periods

greater than the shortest delayed neutron precursor lifetime, an effect

which has been investigated numerically by Travelli [16]. Also, it will

be assumed that the medium under consideration is subprompt critical.

The thermal nonmultiplying WBE, which will be discussed as a point

of departure for our work on Eq. (1.13), is

1 00
(a-pK)F(E,j;K) = du' dE Es(E',u -* E,u)F(E',u';K). (1.14)
-1 o

Spectrum and Eigenfunctions of the
Thermal Transport Operator

The fast neutron wave energy-dependent transport eigenvalue

problem can best be introduced by discussing work which has been done on

the analogous thermal problem, Eq. (1.14). This approach will be taken

because transport treatment of the fast problem is necessary to obtain

qualitatively correct spectral descriptions for passive media. Approxi-

mations such as diffusion theory can yield an estimate of the least

attenuated mode of propagation in fast multiplying media, where such a

fundamental mode exists, but can provide little other information rele-

vant to the properties one should expect of the exact transport treat-

ment.

The context of the work to be presented here is the "singular

eigenfunction method," which received its major impetus from a paper by

Case [28], and thus is frequently known as "Case's method." As ff

introduction to the literature on the singular eigenfunction method in

transport theory, including the wave problem, the review of McCormick

and Kuscer [29] is highly recommended, as it is both recent and

extensive.

Travelli [30] was the first investigator to arrive at an

essentially correct description of the spectrum of the energy-dependent

wave problem, based on a multigroup transport formulation. We turn now

to the energy-dependent analysis of the thermal wave eigenvalue problem,

corresponding to Eq. (1.14), performed independently by Kaper et al. [31]

and Duderstadt [32,33]; the former study employs an isotropic one-term

degenerate thermalization kernel, while Duderstadt discusses more gene-

ral types of scattering interaction models as well. Their results are

summarized in this section. Eq. (1.8) may be written in abbreviated

form as

AF E (A1 + A2)F = KF (1.15)

where the streaming operator Al is

A1 E d^ dEt + 6 (E'-E)6(p -).* (1.16)
-1 o

and the interaction operator A2 becomes

1 f 1
A = d' JdE'- Z (E',' E,p) (1.17)
2 f f 1" s
-1 o

using the scattering interaction kernel of Eq. (1.14).

The basic method for obtaining the spectrum and eigenfunctions of

this equation is a generalization of the work of Bednar: and Mika [34]

on the static Boltzmann operator, which in turn extended the classic

monoenergetic singular eigenfunction technique [25] to a continuous

energy representation. We begin by defining the domain C in the spectral

<-plane, which is the continuous spectrum of the streaming operator A :

C ---+E) -) = 0, P e [-1,1], E e [0,-) (1.18)

or in the notation of Eq. (1.14), those values of K for which a PK

vanishes. For any nonzero frequency w, a is complex, so that C will

occupy an area in the K-plane. It is instructive to consider both the

rectangular and polar forms of K e C; Eq. (1.18) implies that

v (E)
Re(<) =a -

Im() =v
gV(E)

., 1 f-2 u2} 1/2
r(K) I E 2 + w1

r(K) = n-1 +
(<) = tan- (1.19)
t

where r and 6 are the usual radial and azimuthal polar coordinate. In

general C will consist of two symmetric portions in the first and third

quadrants, due to p 6 (0,1] and W 6 (0,-1] respectively. This is represented

schematically in Figure 1.1. Since frequency w is a positive quantity,

C does not extend into the second and fourth quadrants. Figure 1.2

shows the first quadrant of the K-plane in more detail. The domain C

is bounded by the line P = 1, a = L ; from the rectangular form of
t
Eq. (1.19) it is apparent that the real part of this boundary line

assumes every value of E as E (and thus v) varies from 0 to m. The
t
polar form of Eq. (1.19) shows that as |[p varies from 1 to 0, values of

K corresponding to a fixed E generate a line of constant 6 which begins

at the boundary of C and extends to infinity. As the parameter W is

increased or decreased the domain C expands or contracts in the imaginary

K direction.

We note that if vE varies monotonically with E, each point of the
t
domain C will correspond to a unique E,W pair, E and ; for the ther-

mal analysis presented here, this is assumed to be the case. Two

important results then follow. First, Eq. (1.19) defines a one-to-one

mapping of E,u onto the spectral plane. Second, the spectrum of A1 is

not degenerate. The consequences of these results will be discussed

later.

The discrete and residual spectra [35] of A1 are empty [33]. The

singular continuum eigenfunctions of A satisfying the equation

(A -K)F(E,;K<) = 0

(1.20)

C Im(K)

a Re(K)

Figure 1.1

The continuum domain C in the spectral
K-plane.

Zt (E=O)

4p = const.

/

/

E-0 /

^7

Structure of the continuum.

Figure 1.2

are

F(E,u;K) a 6(E-E )6(u-u ) E= (a-WK) (1.21)

This gives a corresponding eigenmode, using Eq. (1.4) (for A = A1),

4 (x,E,P,t) = e a e i(wt"6(E-E )6(-pK ) (1.22)

which clearly represents neutrons of energy E streaming in the

direction p Since C is in one-to-one correspondence with all possible

E,p pairs, each point in the continuum corresponds to a unique neutron

speed and direction of travel. Referring to Eq. (1.19) we see that

modes with UK = 1 have relaxation length 1/Zt(E ) equal to the neutron

relaxation length; that is, modes corresponding to K on the boundary of

C represent neutrons streaming along the x axis. Other modes are more

attenuated, as the direction of neutron travel becomes more oblique to

the direction of wave propagation.

The spectrum of the streaming operator and its eigenfunctions are

qualitatively the same for both thermal and fast regimes, the only dif-

ferences being the values of E which are applicable, and the detailed

structure of t as a function of energy; Eqs. (1.19) (1.22) apply in

either case. It is the interaction operator A2, containing the descrip-

tion of the scattering and multiplication processes, which gives rise to

the qualitative differences between fast and thermal WBE eigenfunctions.

It seems likely for "reasonable" mathematical models of thermal

scattering that the spectrum of A = Al + A2 always contains the spectrum

of A1. This has been substantiated for A2 having the form of a parable

kernel with isotropic scattering [31,33]. This model, which was pro-

posed by Corngold et al. [36], has been used quite extensively in

analytic transport studies, since it represents fairly well the quali-

tative features of thermal scattering interactions [37]. Duderstadt [33]

has investigated more general operators A2, and while the spectral

analysis for less restricted models is somewhat more tentative, it does

appear to indicate that the spectrum of the streaming operator is in

general contained in the spectrum of the wave Boltzmann operator A. We

will see this in a more formal way from the technique used to construct

the continuum eigenfunctions.

To illustrate this method we obtain the eigenfunctions for the

thermal WBE with the separable isotropic thermalization kernel

A d dE Es
-1 o

fdp dE M(E) = 1. (1.23)
-1 o

(To be consistent with our later treatment of the fast WBE we will not

perform the usual symmetrization of this kernel, since for the fast case

A2 is not symmetrizable. The main result of interest which arises from

symmetry of A2 is that the eigenfunctions are mutually orthogonal, and

one avoids the adjoint problem; this and other considerations will not

be of direct concern here. Also note that M(E) is not the Maxwellian

distribution. To satisfy detailed balance M(E) = M'(E)s (E), where

M'(E) is proportional to the Maxwellian, subject to the above normaliza-

tion constraint.)

Using this form for A2, the WBE corresponding to Eq. (1.14)

becomes

(a-UK)F(E,i;K) = M(E) du' dE' Z (E')F(E',p';K). (1.24)
-1 o

Notice that this expression is exactly equivalent to the fast multiplying

WBE in the form of Eq. (1.13) when scattering is ignored in that equa-

tion. First we investigate the point spectrum. We see that K will be

an eigenvalue when the homogeneous equation (1.24) has a solution for

that value of K. Let us suppose that K 0 C so that (a u~) f 0; then

dividing by this factor,

F(E,; = M(E) d dE E (E')F(E',V ;<). (1.25)
-1 o

Defining the scalar product

f1
(P(E,P), iP(E,P)) E dp dE (E,i)f(E,p) (1.26)
-1 o

Eq. (1.25) may be written more compactly

F a- u (Z ,F) (1.27)
a-UK 5

Taking the scalar product of this equation with Z and eliminating the

scalar factor (Z ,F) we find that the condition for Eq. (1.26) to have a

solution is

1 = (1.28)
1 a-[S K]"

Defining the dispersion function

A(K, ) E [ (1.29)

Eq. (1.28) is simply the condition that this dispersion function vanish.

Eq. (1.28), which is referred to as the dispersion law, determines in

the present problem the regular eigenvalues < of the WBE as a parametric

function of frequency. Indeed, values of K which satisfy the dispersion

law for a given frequency w have been shown [31,33] to comprise the

point spectrum of A with A2 defined by Eq. (1.23); for these eigen-

values K. the corresponding eigenfunctions are given by Eq. (1.27):

F(E, ;K ) = A(K ) 1 ((1.30)
j j a-UK(

where X(K.) is an arbitrary constant. (Note that for small w and Z

this approaches a Maxwellian distribution in energy.)

When K e C, the term (a l<) is zero for a particular E = EK and

U = p As we have mentioned, C is contained in the continuous spectrum.

In the present case the continuous spectrum of A is identically the

domain C, and the continuum eigenfunctions are [31,33,34]

F(E,;K) = M(E) dp' dE' Z (E')F(E',P';<) X(K)6(a-pi)
a-p f j s
-1 o (1.31)

Kec

using the notation of Eq. (1.21); X(K) is an arbitrary constant. We see

that (a u<)-I has a pole at the "eigen-energy" EK and "eigen-angle"

p ; integrals over E,u involving this term will exist in the ordinary

sense, provided that its coefficients in the integrand are well-behaved

at the pole. Hence we may eliminate the scalar ( s,F) in Eq. (1.31) in

favor of the constant A(<) by taking the scalar product of the equation

with Zs(E) and solving for (Z ,F). We then find

F(E,u;< K)= () + 6(a uK)

(1.32)
K eC

so that X(<) is in fact a normalization constant; A is the dispersion

function defined in Eq. (1.29)

Eq. (1.31) may be obtained directly from Eq. (1.24) by a hetristic

argument [38]. Since for any variable x the function x6(x) is identi-

cally zero, apparently A(a pK)6(a u<) = 0 may be added to the right-

hand side of Eq. (1.24). Division by (a J<) gives Eq. (1.31) when

< e C. Evidently, then, the domain C always will be in the spectrum of

A, since it is contributed by the streaming operator, regardless of the

form of A2.

The continuum eigenfunction, Eq. (1.32), is composed of two

singular terms, one being the pure streaming mode of Eq. (1.211, and the

other having distributed E and u dependence, with the same formal struc-

ture as the discrete eigenfunction, Eq. (1.30), except that it has a

pole singularity since K e C. The scalar coefficient of the latter

term, s (E )/A, represents the relative excitation of the distributed

portion of the mode by the streaming portion (this can be seen more

clearly by comparing with the analogous fast continuum eigenfunction,

which will be developed in Chapter II). Hence the continuum eigenfunc-

tion may be interpreted as being due to direct streaming neutrons having

energy and direction E and p and an associated scattered distribution

which is excited by the streaming neutrons; the scattered distribution

is peaked at E and p due to the pole of the transport coefficient
-1
(a u~) but contains all other E,p values as well. Note, however,

that the entire mode has the phase velocity v : (cf. Eq. (1.4)) of
ph \$
the uncollided wave. This interpretation of the thermal continuum

eigenfunctions has not been given in previous treatments, as symmetriza-

tion of the kernel tends to obscure the physics involved.

Kaper et al. [31] have investigated the dispersion law A = 0 for

the separable kernel model. Their findings may be summarized as

follows. When w = 0, one has the classic exponential experiment [21];

there is exactly one pair of real eigenvalues <. = Kp provided that

the absorption of the medium is not too strong (of course the precise

condition will depend on the energy dependence of the cross sections).

Otherwise, the point spectrum is empty and will remain empty for all w.

As the parameter w is increased from zero, the pair of eigenvalues will

move symmetrically into the first and third quadrants of the complex

K-plane. Evidently for sufficiently large w there will be a limiting

frequency wu beyond which the discrete spectrum is empty; this value of

frequency appears to occur when K meets the boundary of the continuum

C. This situation is represented schematically, for the first quadrant,

in Figure 1.3. (We noted that in general the boundary of C is

frequency-dependent; here for simplicity it is shown for Et constant, in

which case the boundary remains a line perpendicular to the real axis.)

While for a time it was conjectured [32,33] that zeroes of the

dispersion function might exist within the continuum as "embedded eigen-

values" in a continuation of the dispersion law for w > c it now

appears [31,39] that this is not the case, although the dispersion

function apparently does vanish at points within the continuum [39];

referring to Eq. (1.32), this corresponds to points at which the delta-

function contribution vanishes. This subject will not be pursued here;

the interested reader is referred to Kaper et al. [31], Klinc and

Kuscer [39], and for an extensive discussion from a different point of

view, to the work of Doming and Thurber [40] and Dorning [41].

(L

I/
I/
/
/
/
/
/
/
/
/ Kp(w)
/
/
/
/
I
/
I
I

I /(J =0

Figure 1.3

Schematic dispersion law for a discrete
eigenvalue.

From the results of calculations based on the separable kernel or

comparable models [31,32] it is possible to draw some conclusions about

the physical interpretation of the frequency-dependent behavior of the

dispersion law. Near zero frequency the fundamental neutron wave mode

(if there is one) corresponding to K is less attenuated than the
p
streaming-associated continuum modes. As w increases, the wavelength of

the fundamental mode decreases and it becomes more attenuated. This

occurs because it becomes increasingly difficult for scattered neutrons

to remain in phase with the wave; from Eq. (1.24) we see that as K
p
approaches C the energy and angular distributions of the wave become

increasingly peaked for u = 1. Ultimately the fundamental mode

becomes nearly as attenuated as forward-streaming neutrons, and evi-

dently the distributed E,p term of the continuum eigenfunctions then

assumes the role formerly held by the fundamental mode as the frequency

increases beyond .
c
One additional remark should be made. For w = 0 the spectrum of

the static Boltzmann equation lies entirely on the real axis, and in

general it appears that it is the limit of the spectrum of the WBE as

w approaches zero. But obviously for K real the identification of E,p

pairs with points of C no longer can be made. Indeed it may be improper

to regard the static Boltzmann equation as the zero-frequency limit of

the WBE. No such problem arises in connection with the discrete spectrum,

in the sense that in the limit w = 0, Eq. (1.28) gives the correct

eigenvalues for the static case. This evidently is true of the disper-

sion law in general, and in that sense we speak loosely of the exponen-

tial experiment being the zero-frequency limit of the wave experiment.

In this work we will be concerned only with w / 0 except in the &

calculations of Chapter IV, which involve only the discrete eigenvalues

and eigenfunctions.

Completeness of the Thermal Eigenfunctions

In order to make use of a set of eigenfunctions such as those

obtained in the previous section, it is necessary to show that arbitrary

functions (suitably restricted) can be expanded using these functions

as a basis, and it is further necessary to evaluate the expansion

coefficients. First, then, one must prove that the set of eigenfunc-

tions is complete, or at least establish completeness within the context

of the problem one is to consider. Then either the eigenfunctions must

be shown to be orthogonal and normalized to unity scalar product, so

that orthogonality properties may be used to obtain expansion coeffi-

cients in the usual way, or some other procedure must be followed.

Normalization of continuum eigenfunctions is somewhat less than straight-

forward because, as may be seen from the form of these eigenfunctions in

Eq. (1.32), it involves products of delta functions of complex variables.

The alternative procedure is to find the continuum expansion coeffi-

cients G(K) of an arbitrary function Y(E,p) directly from the expression

for the expansion, which is a singular integral equation:

E) = f G(K) F(E,u;K)dK (1.33)
K

where TV is the portion of Y contributed by the continuum eigenfunctions

(the discrete mode contribution is found by the usual application of

orthogonality). F(E,J;K), which is now the kernel of the integral

operator, is known from Eq. (1.32) or a similar evaluation of the con-

tinuum eigenfunction based on another model.

In implementing either method the theory of generalized analytic

functions [42] has been the principle tool applied to date. Eq. (1.33)

also has been used to prove completeness of the WBE eigenfunctions,

since if it can be shown that an arbitrary Y(E,I) has a representation

in this form, the set of eigenfunctions F(E,W;K) must be complete.

This approach has been taken by Kaper et al. [31] and Duderstadt [32]

to show completeness for the eigenfunctions of the separable thermali-

zation model of the previous section; their treatments were based on

extension of the generalized analytic function technique as applied by

Cercignani [43] to problems in the kinetic theory of gases. The details

of this analysis are lengthy and will not be repeated here.

We will make reference to two types of completeness and

orthogonality. We note that values of K in the first quadrant correspond

to plane waves propagating in the positive x direction, and similarly

the third quadrant represents waves propagating in the negative x direc-

tion. In general, e.g., within a slab of finite thickness, a wave will

be made up of components traveling in both directions; to represent an

arbitrary wave (or pulse frequency component) Y(E,p,x,w) in WBE eigen-

functions, one must use all the eigenfunctions, corresponding to the

whole spectrum of the wave Boltzmann operator. Completeness of the

first type, in the sense that a unique representation of this sort can

be made, is termed full-range completeness. The corresponding fftl-

range orthogonality is simply orthogonality under the scalar product of

Eq. (1.26).

In applying eigenfunction techniques to boundary value problems,

one frequently wishes to represent an incoming source flux S(E,p,xo,w)

or to specify flux continuity for waves moving from one region into

another across an interface at a boundary point x In this case the

boundary condition will be specified for either p e [-1,0) or i e (0,1] and

will involve eigenfunctions for only one direction of wave propagation.

Completeness in this sense, termed half-range completeness, requires

that a function defined over p e (0,1] or p e [-1,0) can be represented

uniquely by WBE eigenfunctions corresponding to the eigenvalues in only

the first or third quadrant of the spectral plane, respectively. Half-

range orthogonality is orthogonality under integration the half range of

P .

Both full-range and half-range completeness requirements will be

seen to arise in Chapter III in connection with a formalized treatment

of the slab geometry boundary value problem. We should note that at

present half-range completeness can be proved only for quite restricted

kernel models, although full-range completeness can be shown for more

general kernels [29,33]. Our main interest in the completeness proper-

ties of the eigenfunctions of Eqs. (1.24) and (1.26) is that they are

indeed complete. We will use the same formal procedure to find the

eigenfunctions of the fast WBE, and will obtain qualitatively similar

results. Thus we may have considerable confidence, in lieu of proof,

that the fast eigenfunctions are complete as well.

There are two difficulties which will prevent us from extending

the generalized analytic function technique directly to the fast regime.

First, one must deal with the slowing-down operator. Second, the one-to-

one equivalence between values of E,p and points of C does not hold for

realistic fast cross sections (e.g., at resonances), and we will be

reluctant to consider more restrictive cross-section models (i.e.,

monotonic vZ ); this equivalence plays a central part in the generalized

analytic function method as it has been developed to date. Whether

these two problems are insurmountable is a matter for further investiga-

tion; however, it seems unlikely, in view of the results established in

the thermal case, that the fast eigenfunctions would not be complete

for "reasonable" cross-section models. (An example of an "unreasonable"

model is a strictly 1/v-dependent cross section

0
t
Z (E) -

or one which has this behavior over some energy range. When this occurs

the portion of C corresponding to this energy range collapses onto a

line. This case is discussed for thermal waves in polycrystalline

material by Duderstadt [33] and by Yamagishi [44]; it is necessary to

deal separately with the eigenfunctions on the line continuum which

results from this cross section.)

For an introduction to other literature on completeness of singular

eigenfunctions see the review of McCormick and Kuscer. It is interest-

ing in this connection to read the comments of Burniston et al. [45], and

the recent remarks of Zweifel [46] regarding the degree to whichlhe

rigorous mathematical basis for the singular eigenfunction transport

analysis has been established.

Other Related Problems and Literature

In the foregoing discussion we have seen that for the thermalization

model employed there the spatially dominant wave mode is due to the

regular eigenvalue < which is determined by the zeroes of the disper-

sion function A(K,w). Further, we see from Eq. (1.32) that the zeroes

or near-zeroes of A also will play a large part in determining the

character of continuum modes, since in regions where A is small the

scattering portion of the mode will dominate the streaming term. A

corresponding dispersion function appears to arise in general in the

treatment of regenerative media (i.e. those in which neutron interac-

tions can result in either a gain or loss in energy, and hence the

interaction kernel has a Fredholm form). Dorning and Thurber [40], for

example, find that in an alternative formulation of the wave problem and

in an initial value problem the nature of solutions is similarly

influenced by the behavior of a dispersion function. In addition,

dispersion laws are known to arise in nontransport approximations to

dynamic eigenvalue problems. For example, when the multi-group diffu-

sion approximation is used to obtain a matrix expression analogous to

Eq. (1.8), its determinant is the dispersion function, and the disper-

sion law is simply the requirement that the determinant vanish; the

solutions associated with values of < which satisfy the dispersion law

are then the desired eigenmodes. Indeed the multi-group diffusion

approach has been used rather extensively to compute dispersion laws for

moderators, and when sufficiently accurate scattering matrices are

employed, agreement of diffusion theory methods with experiment at low

frequencies can be quite good [47].

General discussions of wave and pulse propagation in the context of

its relationship to other dynamic problems, properties of the various

dispersion laws, and analytic methods which have been applied to these

problems will be found in Bell and Glasstone [21] and Hetrick [48]. An

excellent review of the literature in this area as of 1967 has been

given by Kucer [49], although it is of course somewhat dated. As an

alternative exact approach to transport problems, the Wiener-Hopf tech-

nique is finding increasing favor and must be viewed as a potential

method for analysis of the wave problem; Williams [50] recently has

published an expository review of the method. Also, the singular eigen-

function method review of McCormick and Kucer [29] should be mentioned

again in connection with the subject of transport treatments of various

static and dynamic problems.

Finally, with respect to the subject of Chapter III, note should be

taken of existing work treating neutron waves in geometry which is

finite or has discontinuities along the direction of wave propagation.

Interface effects first were investigated experimentally by Denning,

Booth and Perez [51]. This same problem was the subject of both numeri-

cal and analytic investigation by Baldonado and Erdmann [52,53]; their

work is of particular interest because one-speed and energy-dependent

diffusion and transport results are given. Mockel [54] has presented

both transfer matrix and invariant imbedding transport formulations for

wave transmission and reflection from a slab imbedded in an infinite

medium of different composition. Also to be noted is the treatment of

Larson and McCormick [55] of transport in a slab, in the static case,

using a degenerate scattering kernel. Recently much attention has been

given by Japanese and Indian groups to the problem of thermal neutron

wave propagation in assemblies of polycrystalline moderating materials

(e.g. graphite and beryllium) having finite transverse and longitudinal

dimensions; see for example Nishina and Akcasu [56], Kumar et al. [57],

and Yamagishi [44]. The latter is of particular interest because it

demonstrates, in a transport treatment, the presence of intermodal

interference.

CHAPTER II

SPECTRUM AND EIGENFUNCTIONS OF THE
FAST WAVE TRANSPORT OPERATOR

Introduction

In this chapter the singular eigenfunction formalism, presented in

Chapter I, will be extended to the fast WBE expressions, Eqs. (1.12) and

(1.13). Both the forward and adjoint eigenfunctions will be obtained

for general forms of the nonmultiplying, or "slowing-down," and multiply-

ing cases. The structure of these solutions will be discussed, and some

of the implications of using realistic cross-section and scattering ker-

nel models will be explored.

Adjoint eigenfunctions will be investigated for two reasons.

First, they will be necessary for the treatment of the transport formu-

lation of the transfer matrix in Chapter III. Second, as has been men-

tioned, analytic evaluation of expansion coefficients cannot be performed

using generalized analytic function techniques which have been applied

to thermal problems. For the same reason, we will not obtain normaliza-

tion constants analytically. However, biorthogonality of eigenfunction

sets will be shown in the classic way.

The appropriate scalar product under which to define adjoint

operators is given by Eq. (1.26). We consider the general wave eigenvalue

equation in the form of Eq. (1.8), which we may write as a

a1
F(E,vi;K) 1 ,
F(E,;K) du dE K(E',u E,)F(Ep,' ;<) = KF(E,U;r).
-1 o
-1 0 (2.1)

eigenvalue equation corresponding to the forward equation, Eq. (2.1), is

-1 o
a F=I(i;E,Ii) d-_o -r K(E,i -- Et')F"F=I(<;E",V") = 'CF-(ic;E,p)
-1 0 (2.2)

1
where we notice the factor is now within the integral. However, if

we define

t 1 -1
F (K;E,1j) E F (K;E,j) (2.3)

Eq. (2.2), becomes, upon substitution and rearranging,

(a K)Ft (K;E,u) di' dE' K(E,p -+ E',')F (K;E',') = 0 (2.4)
-1 o

which is the form which would have been obtained as the adjoint of the

homogeneous wave Boltzmann equation, Eq. (1.7). It will be more conven-

ient to deal with Eq. (2.4) since it differs from the forward WBE,

Eq. (1.7), only in the kernel of the interaction operator and hence we

will be able to apply the same techniques to the solution of both

It should be pointed out that K is used for the eigenvalue in

Eq. (2.2), with the implication that the spectra for forward and adjoint

equations are identical. That this is true for "well-behaved" operators

in the models we are considering will be apparent from the singular

eigenfunction formalism, although of course each case must be explained

individually. Nicolaenko [14] has exhibited an inelastic scattering

tions due to a singularity of the kernel at zero energy; he uses the

singular kernel in defining an energy transform for reduction of the

static transport slowing-down equation (for the model he considers) to

monoenergetic form. However it is shown in Appendix B that singularity

of inelastic scattering kernels is not an inherent attribute of fast

neutron transport problems. Thus for the forward and adjoint problems

the spectra and eigenfunctions can be regarded tentatively as being in

correspondence, subject to verification for specific interaction models.

Biorthogonality of Eigenfunctions

Biorthogonality of WBE eigenfunctions corresponding to different

eigenvalues can be shown by the usual argument. Writing Eq. (2.4) for

< and Eq. (1.7) for K, we take scalar products of the two equations

with F(<) and F (

(K <') (Ft(K'),F(K) = 0, (2.5)

noting that we are using a real-type scalar product, Eq. (1.26).. We

conclude that biorthogonality holds for F and F under a p-weighted

scalar product, while in view of Eq. (2.3) this is equivalent to

biorthogonality of F-1 and F with unit weighting.

Nonmultiplying Media: Spectrum of the
Slowing-Down Transport Operator

In fast nonmultiplying media the WBE is given by Eq. (1.12). The

corresponding adjoint WBE, Eq. (2.4), is found to be

(a Pj )F (K;E,p) = d jdE Z (E,p E',')F (<;E 1') (2.6)
-1 o

where the different energy limits for the adjoint Volterra scattering

operator are to be noted.

We have seen in Chapter I that for the absorption-only case

(a W<)F(E,p;K) = 0 (2.7)

the spectrum is the domain C in which (a uK) vanishes. The singular

eigenfunctions were

F(E,u;<) = F (K;E,U) = A(K)6(a P<) (2.8)

where the second identity occurs since Eq. (2.7), the streaming equation,

is self-adjoint. Thus in the limit of no scattering, the eigenfunctions

of the fast WBE tend to the delta-function form, Eq. (2.8).

We observe from Eqs. (1.7) and (2.6) that the domain C, due to the

streaming operator, also is contained in the continuous spectrum for the

slowing-down WBE; we now show that in fact it is identically the spectrum

since the scattering operator will cause no additional contribution to

the spectrum. To demonstrate this we show that all K E C are in the

resolvent set, which is the complement of the spectrum, and is defined

as those values of K for which (A K) has a bounded inverse. Therefore

we consider the existence of solutions to the equation

(A K)o = s(E,u). (2.9)

We examine first the case of isotropic scattering, for which the

scattering operator becomes

Sd' dE Z (E',i E,p)- du' dE' ZE(E E) (2.10)
-1 E -1 E

Using this operator we may write for Eq. (2.9) the equivalent equation

1 f 1
(a U1<)c(E,U1) jdVj dEVA E (E' E)O(E ,p) = S(E,)). (2.11)
-1 E

For values of K E C we may divide by (a K<) and integrate over p:

I 1(E,p)du E- t(E)

11
d I
= <
a- K
-1 E

1
S2 (E
2 "S

-* E).(E') +

S(E, )
du a
a-UK

Since K 0 C, both integrals over P exist and Eq. (2.10) is of the form

, (E) = f(E) dE'
E

1 f(E) dE'
E

1
2 s(E E)'(E') + g(E)

Zs(E'-E)*
s

S(E) = g(E).

(2.13)

(2.14)

Provided that the scattering kernel is bounded, the Neumann series

inverse

00
*(E) = o
n=o

f(E) dE'
E

s(E E).

g(E)

(2.15)

Thus (A K) has a bounded inverse, and we have

(2.12)

always exists [58].

the result that the complement of C is not in the spectrum of A. An

identical argument applies to the adjoint operator.

We can extend this result to anisotropic scattering by making a

PN expansion of the scattering kernel and 0; the procedure of Eq. (2.12)

then results in a set of coupled Volterra equations which must be

inverted. Thus for rather general scattering kernels, i.e., those which

can be developed in a finite bounded PN expansion, we have the result

that the spectrum of the wave Boltzmann operator consists only of the

continuum C.

Since the point spectrum for the slowing-down problem is empty,

there will be no regular eigenfunctions and corresponding space- and

E,p-separable eigenmodes. To obtain the singular eigenfunctions corre-

sponding to the continuous spectrum K< C, we may apply the technique of

Chapter I. Adding X(i)(a pi)6(a UK) to the right-hand side of

Eq. (1.12) and dividing by (a UK) we find

F= 1 fl f
F(E,p; -1 E

+ X(K)S6(a K<), K e C

(2.16)

or equivalently, ,

1 L du' dE' Z (E',w' E,p)- F(E,p;<) = A(<)6(a UK).
-1 E (2.17)

At this point it is necessary to proceed more formally. It has been
-1
observed in Chapter I that the integral of the factor (a PK) over

E,lJ exists, since it is a pole. We would like to extend the Neumann

series inverse, which we used in Eq. (2.15), to Eq. (2.17). Accordingly

we write

0 I n
F(E,p;<) = (d) du dE' Z (E ,I E,U)- 6(a uK)
n=o
-1 E

\(K) \$ 6(a WK)

SFSD (E,iJ;<)
SFK 6 C (2.18)

as the forward slowing-down eigenfunction. The formal "Case's Method"

derivation of Eq. (2.16) must be verified for specific scattering kernel

models by means of more careful arguments such as those used in substan-

tiating Eq. (1.32) [31,33,34]; it appears that this will succeed for

"well-behaved" scattering kernel models. For K in the continuous spec-

trum of A the inverse of the operator (A <) exists but is singular [35],

so it is with some justification that we write the second form of

Eq. (2.18), defining the formal inverse scattering operator \$. Further,

the Neumann series expansion has an interesting physical interpretation

in terms of familiar iterated collision integrals.

To see this we first recall that the zero scattering cross-section

eigenfunction of Eqs. (1.21) and (2.8) represent neutrons streaming with

eigen-energy E and direction u ; this delta-function distribution is
K K
also the n = 0 term of F(E,p;K). The second term is

F()(E,;<) a Z E(E ,p E,p) E < E
a-IJK S K K K

= 0 E > E (2.19)
K

which may be interpreted as the distribution resulting from one down-

scattering interaction, multiplied by the transport factor (a P<)1

which is peaked at E and K Similarly, higher terms in the expansion

may be interpreted as the result of n down-scattering interactions, so

that the entire eigenfunction may be regarded as the result of excita-

tion by neutron waves streaming with E and i along with an associated
K K
down-scattered contribution excited by the streaming portion. The

eigenfunction is nonzero only for E and below, since only down-

scattering can occur. (This deduction from Eq. (2.18) is valid whether

the Neumann series converges for E < E or not.) We see that the eigen-
K
function singularity consists of a delta-function contribution and a

pole contribution at E ; a similar structure occurs in the thermal

continuum eigenfunction, Eq. (1.32). Also we note that in the iterated

integrals each singularity is smoothed by integration, and that Ihe

unintegrated pole can be factored out from each term of the series, so

that we suspect that the Neumann series inverse will indeed converge for

rather general classes of scattering kernels.

The adjoint eigenfunctions may be obtained by an identical

t t Ii 1 E n
F (K;E,p) = A (K) 1 d dE' (E,u E',u')- 6(a PK)
n=o
-1 o

= A(K) \$ *6(a U<) < 6 C (2.20)

or

t t
F (K;E,l) E FSD(<;E,p) (2.21)

where A (<) is an arbitrary complex constant. The form of the adjoint

Volterra operator requires that F is identically zero for E < E ;

again the delta-function and scattering-associated term with pole

singularity at E = E occur. The properties of the forward and adjoint

eigenfunctions may be summarized by the rearranged expressions

F(E,p;<) = E(K)

1 E
-1 E

K< (E',U' E,- )
dE' s
a(E') WK

x E (E ,l E l', )]

E < E

E > E
E
K

= X(K) [(a -

p<) +

F t(;E,w)

+- I
a-PK
n=o

- 1 E
SduI dE
-1 F

E (E, -E E',p')
a(E') 1p

x E (E',ui E IJ]

E>E
-K

E K

00

n=o

= 0

(2.22)

= 0

(2.23)

5(a pK) +

Discussion of the Slowing-Down Eigenfunctions

An interpretation of the forward eigenfunctions in terms of

iterated collision integrals excited by monoenergetic unidirectional

(i.e. u = p ) streaming neutrons already has been given. We proceed by

considering their biorthogonality properties. In general, due to the

condition expressed by Eq. (2.5), forward and adjoint eigenfunction

pairs corresponding to different eigenvalues are orthogonal under a

i-weighed scalar product. For the same eigenvalue K, Eqs. (2.22) and

(2.23) clearly show that the scalar product will not vanish, due to the

coincident delta-functions. (This product of delta-functions of two

variables requires careful interpretation in terms of the theory of

generalized analytic functions or some other approach; for an introduc-

tion to the literature on this aspect of the singular eigenfunction

technique see McCormick and Kuscer [29].) The biorthogonality proper-

ties of the slowing-down eigenfunctions are illustrated schematically in

Figure 2.1 in terms of the energy variable. The eigenfunctions must be

orthogonal for overlapping energy-distributions as well as in the

trivial case when the distributions are nonoverlapping in energy.

It is interesting to consider the expansion of a monoenergetic

source in slowing-down eigenfunctions. This is schematically repre-

sented in Figure 2.2. We see from the first two sketches that such a

source will excite not only continuum modes having the eigen-energy E ,

but also will excite to some extent all modes with lower eigen-energies.

As is apparent in the third sketch, continuum modes with higher eigen-

energies will not be excited.

Ft(K)

I

F(K)

--- Ft'(K')

"-

I F (K)
I *

EK' EK

Figure 2.1

slowing-down eigenfunctions.

EK' EK E

F(,K)

F(K')

F (c)

F(K)

E, Eo E

I
I
I

Figure 2.2

Excitation of slowing-down eigenfunctions
by a monoenergetic source.

EK E E

F (K)

An analysis of the static slowing-down transport equation has been

performed by Maclnerney [59] for constant cross sections and elastic

scattering, in the lethargy variable. By performing a lethargy Laplace

transform he reduces the lethargy-dependent problem to one-speed trans-

port form. For the transformed problem (for slowing down in hydrogen)

both discrete and continuous spectra arise, as is usual in the one-speed

problem (see standard works such as Case and Zweifel [25]). However

due to inversion of the lethargy transform, the discrete modal contribu-

tion fails to give a space-separable solution for the isotropic space

and lethargy Green's function (i.e. a source 5(u)6(x)). This is in

accord with our result that a monoenergetic source excites a continuous

distribution of eigenfunctions. MacInerney tentatively attributes his

continuum eigenfunctions to streaming first-flight source neutrons;

confirmation of this, and further correlations between his work and the

present "exact" method must await more detailed investigation.

The existence of a discrete mode in the lethargy-transformed

problem raises an interesting point with respect to implementation of

the continuum singular eigenfunctions. A dispersion function, associ-

ated with both discrete and continuum modes, was seen to arise naturally

in the treatment of the thermal problem. We may associate the disper-

sion function with inversion of the Fredholm thermalization operator,

since in the slowing-down case only the Volterra operator is present,

and no such dispersion function appears. Physically we distinguish

between energy-regeneration which can occur through up-scatter in the

former instance and energy degradation in the latter. In the presence

of energy-regenerative mechanisms.we find.the potential for *

establishment of E,W-space-separable modes (for moderate frequencies and

absorptions) with attenuation length longer than the neutron mean free

path. For the slowing-down problem, with such mechanisms absent we have

t (E )
= Re < (2.24)

so that all modes are attenuated precisely as are the streaming-waves

with which we associate them.

However it is well known that the neutrons themselves (e.g. for

neutron pulses) are not attenuated in this manner, even though no

separable mode of propagation exists. Evidently, therefore, we are not

to regard a continuum mode as observable or capable of being excited

individually, since the neutrons which would constitute such a wave

certainly would not be attenuated according to the streaming mean free

path. This is further evidenced by the fact that a monoenergetic

unidirectional source excites modes having lower eigen-energies as well.

Apparently the identification of an individual mode with streaming and

associated scattered neutrons must be applied with some caution, although

it is clear that actual streaming source neutrons are represented by the

delta-function term of the appropriate eigenfunction. We must conclude

that the spatially persistent nonseparable neutron population (as

opposed to uncollided neutrons) excited by a delta-function source is

represented by constructively interfering continuum eigenfunctions,

where this constructive interference is due both to the distributed

part of the eigenfunction excited by the streaming, and to eigenfunctions

of lower eigen-energies. Evidently the discrete mode in MacInerney's

transformed problem corresponds to this constructively interfering modal

contribution.

It should be noted that the idea of interference of neutron waves

is not new, having been postulated as early as 1964 on the basis of

diffusion theory by Perez et al. [60] to explain phenomena observed in

wave experiments in subcritical assemblies. More recently, in the

transport treatment of polycrystalline materials by Yamagishi [44],

interference effects have been seen to arise from interaction of a

continuum contribution, due to neutrons with energies below the Bragg

cutoff, with the higher energy neutron population. In the present fast

nonmultiplying problem we have seen that modal interference is necessary

to describe neutron wave propagation in all but purely absorbing

materials.

In the same context it is interesting to consider elastic

scattering from very heavy nuclei. In this case the energy loss per

collision is sufficiently small that wave propagation in such a medium

is essentially monochromatic. Thus monoenergetic analyses may be per-

formed such as, for example, those of Ohanian et al. [61] and Paiano and

Paiano [62]. In this case, due to the energy-sustaining model of the

collision process, space-angle-separable monoenergetic eigenmodes occur

which are less attenuated than t We realize that in the actual energy-

dependent problem an energy loss does occur with each scattering interac-

tion, so that only continuum modes are present; nevertheless these

continuum modes must superimpose in such a way as to yield the Atmost-

separable wave behavior.

The macroscopically elastic scattering kernel model of the above

discussion may be written

se (E',p + E,p) = Ese(E')K(Pu u)5(E E). (2.25)

This kernel also is noteworthy because it is not bounded. Clearly our

discussion of bounded scattering kernels in establishing the resolvent

set, K 0 C, does not apply and we find spectral contributions do arise

for K C. The model of Eq. (2.25) is discussed in Appendix C, along

with several limiting procedures which may be used to attempt to derive

the strictly monoenergetic case as the limit of the almost-monoenergetic

case.

We conclude the discussion here by observing that another way of

viewing the problem of elastic scattering from heavy nuclei is to

consider a detector with an energy window AE wide enough to detect all

elastically scattered neutrons; one should then obtain experimental

results which are in accordance with monoenergetic theory. That is,

the detector response should show an asymptotic exponential signal decay

corresponding to the momoenergetic fundamental mode; this detector

response is the physical equivalent of solving for the zeroth moment of

the flux rather than the flux itself. In this instance we must agree

with Doming and Thurber [40] who remark in another context that in

attempting to correlate theory and experiment one can be mislead by

considering only the asymptotic behavior of flux solutions rather than

their moments.

Realistic Cross Sections: Nonmonotonic vE
t
In Chapter I analysis was restricted to total cross sections such

that vE is monotonic. This was done because the continuum values of K
t
and all possible E,u pairs are in one-to-one correspondence for monotonic

vZt, a requirement of the generalized analytic function treatment upon

which we rely for completeness results in the separable kernel case.

Here we explore briefly the consequences of relaxing the monotonicity

condition.

For this case degeneracy of the continuum results. From Eq. (1.19)

it is apparent that O(K) will assume the same value more than once when

vZ is not monotonic. This is illustrated in Figure 2.3, where it is
t
evident that for the same value of 0, but different energies, nondegene-

rate, singly degenerate, and doubly degenerate regions occur. Higher

degeneracies may result from more rapidly oscillating cross sections. We

exclude the case of constant vEt, which must be treated separately.

When the continuum is degenerate the coefficient (a PK) in the

forward and adjoint eigenvalue equations becomes zero for more than one

E,p pair at each degenerate < point. Thus in Eq. (2.16) and the corre-

sponding adjoint expression we may make the replacement

\(<)6(a wK) X m (K)6 (a ic)
m=l

S Xm(K)6(E E )K(M p ). (2.26)
m=l

Figure 2.3

Degeneracy of the continuum due to
nonmonotonic vE
t"

Thus from Eqs. (2.18) and (2.20) we have

NI
F(E,J;K) = I Xm(K) S-6m(a UK)
m=l

and

F (K;E,U) = X(K) \$ 6 (a pK)
m=l

(2.27)

(2.28)

for an M-degenerate K. Clearly since there are M arbitrary A's, NI

linearly independent eigenfunctions can be constructed. An obvious

choice is to set the A's equal to zero for all but one 6 ; we define the

M eigenfunctions

FSD,m(E,I;) = A (K) \$ 6 (a K)

F (K;E,) A (K) \$ 6 (a )
SD,m m m

(2.29)

(2.30)

which we notice are biorthogonal when the forward eigen-energy is less

than the adjoint eigen-energy but are not necessarily biorthogonal

otherwise. Also we see from Eq. (2.22) that the forward eigenfunctions

will have pole singularities at all eigen-energies En less than the

and

delta-functioneigen-energyE A similar structure occurs in til!

Fast Multiplying Media: Zero Scattering
Cross Section

The fast multiplying medium problem best may be approached by first

considering Eq. (1.13) with the scattering operator absent. Since the

fission interaction kernel is separable, Eq. (1.13) then becomes identi-

cal in form to the thermal WBE with separable kernel, Eq. (1.24), which

was discussed in detail in the first chapter. Identifying > with M(E)

and ifE(E') with Z (E'), we may write down immediately the results for

the nonscattering fast multiplying WBE from Eqs. (1.29), (1.30) and

(1.32). Thus we find that the discrete eigenvalues are given by the

dispersion law

A(K,w) 1 = 0 (2.31)

and the corresponding regular eigenfunctions are

F(E,;K.j) = X(j.) '.k
j a-pK.

K f C. (2.32)

The singular continuum eigenfunctions are

\7 v Lf a
F(E,p;K) = A(K) 7r + 6(a-p<)

(2.33)

K eC

Adjoint eigenfunctions, which we obtain for later comparison, are

Stt f(E)
F (K;E,p) = (K) -

K 0 C (2.34)

and

+ V f(E) ,(E)
F (c;E,p) = t(<) + 6(a-Kc)
a-Pa A

K e C (2.35)

where we have used the definition of x(E) from Eq. (1.10). We note that

the same dispersion function occurs in both forward and adjoint

expressions.

By analogy with the thermal problem we expect a symmetric pair of

eigenvalues for moderate frequencies and absorption. We further expect

that the set of eigenfunctions of Eqs. (2.32) and (2.33) will have full

and half-range completeness properties (although strictly speaking

these properties were demonstrated for a symmetrized kernel in the

thermal case; a similar symmetrization transformation could be per-

formed in the fast case).

Discrete Eigenfunctions for Fast
Multiplying Media

We now turn to solution of the fast WBE with down-scattering, as

represented by Eq. (1.13). For < e C, we may divide by (a ur) and

invert the identity minus the scattering operator (under the conditions

which were discussed previously) to obtain

F(E,U;K) = duv dE v (E')F(E ',p;<) (2.36)
a-FjK. = ja f
-1 o

using the inverse operator defined in Eq. (2.18). Taking the scalar

product of this equation with vZ we find that the condition for

solutions to exist is that

A(K,w) = 1 I, g I = 0 (2.37)

which defines the dispersion function and the dispersion law for the

discrete eigenvalues K.. The expression for the regular eigenfunctions

then is

F(E,U;'.) = ,(f .) \$ X
SJ a-Urc.

K P C. (2.38)

This expression may be compared with Eq. (2.32); making use of the

Neumann series interpretation of \$, we see that the presence of down-

scattering in the problem has resulted in an addition of all iterated

collision integrals of the nonscattering eigenfunction (cf. Eq. (2.22)).

Thus the discrete eigenfunction consists of the fission spectrum,

-1
weighted by the transport factor (a pK)1 (which is peaked at p = 1

but not singular, for K 0 C), and smeared down in energy by similarly

weighted scattering operators. We will discuss the regular eigenfunc-

tions and the dispersion law in more detail in Chapter IV.

The corresponding adjoint eigenfunctions similarly are found to be

t t t VZf(E)
F (Kc;E,i) = X (K.) (2.39)
J

It is readily verified that the same dispersion law is obtained here as

for the forward problem.

Continuum Eigenfunctions for Fast
Multiplying Media

By application to Eq. (1.13) of the arguments used in arriving at

Eq. (2.18), we find

F(E,p;K) = \$ -- du' dE' vE(E')F(E',u';K) +
a-UK Ir
-1 o

+ X(K) \$ 6(a-u6 ) K e C (2.40)

which in view of Eq. (2.18) may be written

F(E,p;K) = \$ a- (vz ,F) + X(K)FSD(E,u;) (2.41)
a-lK f S

Eliminating the scalar product term results in the expression *

F(E,;,<) = \(K) a f (--- SD + FSD

K e C (2.42)

for the forward continuum singular eigenfunction, where A is defined by

Eq. (2.37). The adjoint continuum eigenfunction is

t
t t t Xf ,FSD) +t
F (;E,p) = (K) + A F

< e C (2.43)

+ t
where in this case F \$ 6(a <).
SD -

Discussion of the Continuum Eigenfunctions

The eigenfunctions represented by Eq. (2.42) have an interesting

interpretation much in the same manner as that of Eq. (2.22), and with

similar reservations applicable. Making use of Eq. (2.37) for the dis-

persion function and expanding the inverse in a power series, Eq. (2.42)

may be written

F(E, la;<)
(E,<) = \$ 1 + ,\$ + jZ a +..**. .]fvF, +
A(K) a-U< I I f a- f SD

+ FSD(E,;<) 2

(2.44)

Now let us regard (vZfFSD) as the initial excitation of the fission

contribution to the mode. The resulting fission neutrons, after being

smeared in energy by scattering, have the energy and angular distribu-

tion S -- The form of the dispersion function expansion suggests
a- UK
that it be interpreted as a modal multiplication due to the sum over all

generations of fission neutrons. Thus we see that the continuum mode

again may be regarded as streaming-associated, since it consists of two

terms which we interpret as follows. The second term is FSD which has

been seen to be the down-scattered distribution associated with stream-

ing neutrons having the eigen-energy and eigen-angle. The first term of

Eq. (2.44) then may be interpreted as the fission-produced modal flux

distribution due to excitation in turn by the scattered term.

This attractive exegesis must be tempered, as in the slowing-down

case, by considering the scalar product of the adjoint eigenfunctions,

Eq. (2.43), with a monoenergetic source function. We observe first that

the source will excite modes with lower eigen-energy, due to the term

F In addition, modes having eigen-energies both above and below the
SD
t f
source energy will be excited due to the fission term \$ -
a-Uc
These results may be compared qualitatively with solutions for the

static fast multiplying medium transport problem obtained by Nicolaenko

and Zweifel [63] and Nicolaenko [14]. Energy-transform techniques were

used to treat fission and elastic scattering with constant cross sections

in the former study. Inelastic scattering, the model for which already

has been the subject of comment here, was added in the latter. Although

detailed comparison again is difficult due to the complex structure of

the continuum eigenfunctions, we find consistencies between their Green's

function results and the present.work. Specifically, in both studies,

Green's function solutions are found to contain both space-separable

contributions (which we ascribe to the discrete eigenmode) and nonsepa-

rable "slowing-down transients," which are solutions to the slowing-

down equation without the fission term, and which were found to be

necessary to achieve completeness for the eigenfunctions of the appropri-

ate Boltzmann equation. The correlation with our results is apparent.

Degeneracy of the Continuum

Should further complications seem desirable at this point,

consideration may be given to the effect of degeneracies in the continuum

upon the above treatment of continuum eigenfunctions. Since the

details are straightforward, we simply note that linearly independent

sets of eigenfunctions can be obtained; in particular a set correspond-

ing to those of Eqs. (2.29) and (2.30) may be derived by an identical

procedure. The eigenfunctions are given by Eqs. (2.42) and (2.43) with

the substitution of F and F for F and F
SD,m SD,m SD SD'

The Boltzmann Equation with Isotropic
Interaction

Finally, some general consequences of isotropy in the WBE operators

will be derived for use in Chapters III and IV. For the isotropic ker-

nel we write

K(E', E,p) = K(E' E)

(2.45)

so that the WBE for K 0 C may be written

F 1 l f
F(E,u;c) = du' dE' K(E' E)F(E',P';<)
a-i1
-1 o

= f K(E-
a-pK J
o

- E)F(E';K)dE'

with the definition

F(E;K)

E F(E,p;K)dj
-1

Integrating Eq. (2.46) over i we obtain

F(E;<) = f(E)

f K(E' E)F(E';)dE'

with f(E) defined as

f(E)

J 1 dvi
E a-iK
-1

(2.46)

(2.47)

(2.48)

(2.49)

Upon solution of Eq. (2.48) we then reconstruct the angular flux:'rom

F(E,P;

-1
= -1(E) F(E;) .
a-lK

For K 6 C the continuum eigenfunction equation becomes

F(E, P; ) = 1- -
a-K
-1

di', jcx

dE" K(E -* E)F(E',U';K) + A(<)6(a-u<).

Performing integration over I we have

F(E;K) = f(E) K(E' -+ E)F(E';K)dE' + X(K)6(E-E )

< e C.

(2.50)

(2.51)

(2.52)

CHAPTER III

APPLICATION TO THE TRANSFER MATRIX METHOD

Introduction

In this chapter the analytic results obtained for the WBE will be

applied to the transfer matrix formalism of Aronson and Yarmush [64] and

Aronson [65-70], making it available in a continuous-energy transport

representation. There are two aspects of this technique which make it

attractive as a potential method for numerical applications of transport

theory. First, it provides a convenient general framework for "problem

solving" in terms of certain basic operators (see Aronson [67] for a

number of examples). Second, it provides an explicit method for obtain-

ing transmission and reflection operators. As we will see, constructing

some of the required operator inverses will be equivalent to determining

the half-range orthogonality properties of the WBE eigenfunctions.

Although it is not possible in general to do this analytically, numeri-

cal inversion techniques certainly may be employed, so that the transfer

matrix formalism provides a straightforward approach to this difficult

aspect of finite medium problems.

The transfer matrix for slab geometry and its associated eigen-

value problem are derived in Appendix A. Essentially what one must do

is find the spectrum and eigenfunctions of a certain operator, 06, in

whatever representation the problem is formulated. From these

eigenfunctions all the relevant transfer matrix operators may be l

constructed, as well as transmission and reflection operators. Here we

will obtain the a, 6, and 06 operators for energy-dependent wave trans-

port with an arbitrary interaction kernel, and then show that for iso-

tropic scattering the eigenfunctions of o6 may be expressed in terms of

WBE eigenfunctions.*

Formal Operator Relationships

The operator relationships required to construct the transfer

matrix H for a slab of width T may be summarized as follows:

~ -TA -1
H =Se S (3.1)

where

B B

-

S = 3.2)

+
C C

*Definitions used here correspond to those used by Aronson in
Ref. [66] and earlier; some quantities differ by a factor of 2 in
Refs. [69] and [70].

e
-TA
e =

0

0

e

(3.3)

and F is diagonal. To obtain the operators B., C and F one must first

diagonalize an auxiliary operator 06:

X--1 6 X = 2
X =6x

(3.4)

where 2 is diagonal; then

-'-b-1
B+ = X X

C =X r x a.
4-= 1 l

(3.5)

(3.6)

The explicit wave

will be developed

transport representation of these formal relationships

in the following sections.

The Operators a6, X, and X-

The operators o and 6 are defined as the sum and difference,

respectively, of the operators a and 8, which were found in Appendix A

to be

and

68

m 1
a E dE' dI .(E'-E)6('-p) b KE(E',u E,U) (3.7)
o o

E dE du KBE(E',-P E,p) (3.8)
o o

where KBE is the kernel of the Boltzmann equation interaction operator

(operator A2 of Chapter I), including scattering and fission. In an

abbreviated notation Eqs. (3.7) and (3.8) become

4.
a K
a = - (3.9)

(3.10)

so that

S a 1
o = a- (K K ) (3.11)
p i

and

- = a 1 + --) -
6 =a- -(K + K)

(3.12)

69

Then their product is

2 -
6 = (K K) (a K K-)

2 (K + K )* (3.13)

in terms of an arbitrary interaction operator. This expression is
+
considerably more simple when K = K since the awkward middle term of

Eq. (3.13) vanishes. In particular this occurs when all interaction

processes are isotropic; we will assume this to be the case throughout

the rest of this chapter. We then obtain

S a 2a
6 K
2 2

0 1 2
= dE d", a 6(E'-E)6(U -W) 2a KB(E E) (3.14)
o o

which is the form we will consider here.
-1
We now wish to obtain the operator X and its inverse X which will

diagonalize 06 as in Eq. (3.4). This may be done by first finding the

spectrum and eigenfunctions of the operator o6, and then constructing
-1
X; X will be constructed in a similar way from eigenfunctions of the

adjoint operator. The validity of the diagonalization will of course

require that the sets of eigenfunctions are complete. We write She

eigenvalue equation for 06 as

06 X(E,p;y2) = 2s(E,u;2) (3.15)

Now 06 is an integral operator over E and j, and we will find that

in general it will have an area continuous spectrum as well as a possi-

ble discrete spectrum. To clarify the correspondence between the eigen-

function X(E,p;y ) and the operator X, let us consider for a moment the

simpler case which occurs when 06 is an ordinary N x N matrix (as it is,

in fact, for the multigroup diffusion representation). Then its spec-
2 2
trum consists of the N discrete eigenvalues y.; is the diagonal
2
array of the y., and X is the corresponding matrix made up of columns of
2 ~
eigenvectors, X.. E X(E ;y.). The matrix X is then a transformation

from the basis generated by the eigenvectors corresponding to the indi-
2 -1
vidual y., to the discrete-energy space; similarly, X is the inverse

transformation.

In the present transport case, the situation is entirely analogous,

but the summations over the discrete spectrum must be supplemented by an
2
integral over the continuum values of y and summation over the E. is

replaced by integrals over E and p. Thus P2 is the diagonal operator

consisting of both the discrete eigenvalues of 06 (if any) and the con-
2
tinuous spectrum, 2. The operator X is made up of "columns" of eigen-

functions X(E,p;2 ) with y2 as the "index"; it will involve both an

integral over the continuum and a possible sum over discrete contributions.

In other words, for the continuum, X(E,U;y ) is the kernel of an integral
2
operator over all continuum values of y2

Assuming that the set of eigenfunctions of o6 is complete, X may be

regarded as a transformation from the basis y2 to the basis E,u, while
-1
the operator X is the inverse transformation. Writing X formally as

X = dy2 X(E,;y2) (3.16)
2
Y

(the integral is understood to include the sum over the discrete
~-1
spectrum, if any) and X as

--1 = -1 2
X = dE du X- (' ;E,U) (3.17)
o 0

the left and right inverse relations become

~ T- X f1 ?2 f 2 2
X X = I = dE du X 'y ;E,p) dy X(E,U;Y )
o o

=f d2 6(y2 2) (3.18)
-2
Y

and *

XX1 = 1 = dy2 X(E','; y2) jdE du X ( 2;E,U)
2 o o

= dE Jdup 6(E-E )6(u-UP) (3.19)
o o

2 2 2
where 6(y -v ) is either a Dirac or Kroneker delta for y in the

continuous or discrete spectrum, respectively. The first of these

expressions is a biorthogonality relationship for the two functions

X(E,;y2 ) and X ( '2;E,p), while the second is a closure requirement over

U e (0,1] (although we will see later that this closure relation is

essentially a full-range condition).

Spectrum and Eigenfunctions of 06

Using the explicit expression for a6, Eq. (3.14), we may write the

eigenvalue equation, Eq. (3.15), as

2 d r2 a
2 IX = + ( d X

= 2 JdEA dp KBE(EA E)X(E,I';Y ) (3.20)
o o

Inspection of this equation shows immediately that the coefficients of

X on the left-hand side will give rise to a continuum which is identical

to the domain C which we defined in connection with the WBE; that is,

when y 6 C it is also in the continuous spectrum of 06. Further, we now

may apply the singular eigenfunction technique to obtain an expression

for the continuum X eigenfunctions. Noting that

-+ a--- + a----y (3.21)
-Y] 2a. a-ly

we find that for y 6 C

X = + (- dE" dj K (E' + E)X(E- ,i ; 2)
0 0
o o

+ :(y)6(a-py) (3.22)

where we explicitly consider only y's contained in the half of C which

is in the first quadrant of the spectral plane; including the other

half of the domain C gives redudnant results. For y 0 C we have simply

X = ( + 1, dE du E (E -* E)X(E',p ;y2) (3.23)
ta-wy a+py BE
o o

for the rest of the spectrum of 06. Now since the stipulation has been

made that KBE is isotropic, Eqs. (3.22) and (3.23) may be integrated

immediately over v to give

2r
X(E;y) = f(E) IdE' KB(E -* E)X(E ;y2) + X(y)6(E-E ), C
0 (3.24)

X(E;2) = f(E) dE. KBE(E E)X(E';y2), y C (3.25)
o

where f(E) is the same function defined in Eq. (2.49). But these

equations are the eigenvalue equations for the WBE with an isotropic

kernel. Thus we have the following results:

(a) The spectrum of the isotropic o6 operator is identical to the

spectrum of the related Boltzmann operator.

(b) The l-integrated eigenfunctions of the two operators are

identical.

That is, identifying y with <,

X(E;y2) = F(E;K) (3.26)

so

dE- d. KBE(E' E)X(E,U ;y) =
o o

-= dE KBE(E E)X(E;y2)

= dE KBE(E' E)F(E';<)
o

m I1
= dE dlj KBE(E E)F(E', ; o -1

Using this identity in Eqs. (3.22) and (3.23) and comparing with

Eqs. (2.46) and (2.52) for eigenfunctions of the WBE, we find that for

an isotropic KBE
B-E

X(E,u;

The Inverse Operator X1
-l
To obtain the inverse operator X- analytically the most obvious

approach is to construct it from the eigenfunctions of the operator

adjoint to 06; it then should have the required biorthogonality proper-

ties of Eq. (3.18). Referring to Eq. (3.19) we see that the appropriate

scalar product is

r r1
(, ,) dE dpi( (E,)' JJ(E,U) (3.29)
o o

a 2 X K2;E,) = d2 dE- dp a( ) KBE(E E)X-( 2;E',').

0 0 (3.30)

Defining the function

t 2 a 2- (
X (K ;E,p) =- (K ;E,P) (3.31)

and substituting in Eq. (3.30),

a2 2to2 2 1 t
K XK ;E,p) = 2 dE' du KBE(E E')X (K2;E',p')
0 o (3.32)

where KBE(E E') is the kernel of the adjoint WBE considered in

Chapter II. By duplicating the development of Eq. (3.20) we have

immediately

X (<2;E) = F (K;E) (3.33)
and

X (K2;E, ) = F (K;E,p) + Ft(-K;E,p) (3.34)

making use of the eigenfunctions of the adjoint WBE from Chapter II.

Then X- (K 2;E,p) is given by Eq. (3.31).

Diagonalization Operators

These results now may be used to obtain expressions for the

operators B+ and C+ which diagonalize the transfer matrix itself

(Eqs. (3.1) (3.3)). From Eqs. (3.5) and (3.6)

~- --'-1 -
B = X 6 X X

and

Since is diagonal, elements of may be written

Since F is diagonal, elements of ( may be written

5(E,;<) = I

mIdE fldV

aP 6(E'-E)6(uV-u)

0 0

KE(E E) X(E ',;K2)

UK 2
S- X(E,p;K )
a

making use of Eqs. (3.12) and (3.20). Then

B (E,;K) = a -a X(E,' ; 2)
a

For isotropic KBE we note that
BE

(a-u')F(E,u;<) = (a+p<)F(E,p;-<)

(3.35)

(3.36)

(3.37)

(3.38)

(3.39)

78

so that from Eq. (3.28)

B+(E,p;K) = 2F(E,U;<). (3.40)

"-1
Similarly is

m-1 1 m 1 a(E'
S dK -6(K'-K) dE dux1K2;E) dE d 'a (K- 6(E'-E)6(u'-u).
K f f fm
K 2 0 (3.41)

so that

-1 a -( 2
S(K;E,) =- X (K ;E,) (3.42)
pJK

and

C,(K;E,p) = (alJ)XN = + 2 Ft(' K;El) (3.43)
-aK K

making use of Eqs. (3.31) and (3.34), and Eq. (3.39) which also applies

to F

Full-Range Orthogonality and Completeness

At this point Bk and C. are still determined only to within a

normalization function of K. The normalization relationship may be

obtained by considering the operators S and S of Eq. (3.2). Substi-

tution of Eqs. (3.38) and (3.43) for B. and C into the left inverse
-1~
expressions S S = I yields the appropriate expression, which, with a

little manipulation, is also found (as for the monoenergetic case [68])

to be a whole-range biorthogonality relation for the WBE eigenfunctions

F and F Similarly, the right inverse equation (which in view of

Eq. (3.1) expresses explicitly the physical requirement that as the slab

width T is decreased to zero, the transfer matrix must reduce to the

identity) SS = I is found to be a whole-range closure expression for

F and F Thus we see that the validity of the diagonalization in

Eqs. (3.1) (3.3) depends on the full-range biorthogonality and com-

pleteness of the WBE eigenfunctions, properties which may be established

in the context of the WBE itself.

Half-Range Orthogonality and Completeness

The entire transfer matrix H now has been obtained without

explicit application of any half-range conditions. (It was not neces-

sary to solve Eqs. (3.18) and (3.19) for the inverse of X.) However,

we recall that the transfer matrix relates the angular flux over

u6[-l,l] at one slab surface to the corresponding flux at the other. It

is apparent that use of the full transfer matrix is completely equiva-

lent to applying full-range boundary conditions at one interface to an

infinite-medium eigenfunction expansion of the flux, which, of course,

completely determines the flux elsewhere within the slab, and in particu-

lar at the opposite surface. In situations where only incident fluxes

are known at interfaces, the transmission and reflection operators T and

R are more appropriate to the problem. It is in constructing these

operators that half-range conditions will appear.

The transfer matrix formalism provides expressions for T and R in

terms of the diagonalization operators Bk and C, (Eqs. (A.63) through

(A.66)), all of which are defined for E3 (0,1]. However, invertes of

the operators B, and C occur explicitly, and must be evaluated for the

interval (0,1] if one is to compute T and R. Thus, for B we have the

inverse relationships, using Eq. (3.40),

0 l1 -1
2 dE du B+ (K;E,v)F(E,p;rt) = 6(K'-<) (3.44)
o o

and

2 j dK F(E,U;i ) B (K;E',C ) = 6(E'-E)5(P'-p). (3.45)
2

Solving the singular integral equation, Eq. (3.44), for B_1 is

equivalent to obtaining the weight function W(E,p) and normalization

N(K) for half-range biorthogonality of F and Ft, since if they are

complete on the half-range we identify

2B 1(K;E,v) = N()W(E,p)F (K;E,p) (3.46)

so that Eq. (3.45) becomes a closure relation for the WBE eigenfunctions.

It was pointed out previously that for most realistic interaction

models analytic solutions of Eqs. (3.44) and (3.45) are not likely to be

readily forthcoming, and consequently the transmission and reflection

operators must be constructed without analytic expressions for the

required inverses. (In this respect the invariant imbedding method is

an alternative, as it provides a completely different formulation for

R and T. See Pfeiffer and Shapiro [71] for a review of the various

approaches to transmission and reflection, and Mockel [54] for an

example of application of both transfer matrix and invarient imbedding

methods to thermal neutron wave propagation.) Nevertheless, supposing

that some approximate numerical representation has been found for the

WBE eigenfunctions, one can always invert the diagonalization operators

numerically, which we have seen to be exactly equivalent to numerically

determining half-range normalization of the WBE eigenfunctions. After

the numerical inversion is performed, R and T also may be calculated

numerically.

Application to Fast Neutron Wave Propagation

We have seen that for isotropic interactions all the essential

operators required to apply the transfer matrix technique may be con-

structed from solutions of the WBE eigenfunction problem. Further,

requirements for half- and full-range closure and biorthogonality of

these eigenfunctions are implicit in the formalism. These results and

in particular Eqs. (3.28), (3.34), (3.40), and (3.43) extend Aronson's

static monoenergetic work [65-70], and are quite general with respect

to the energy-dependence of the kernel.

It is interesting to note that relationships between transfer

matrix and Boltzmann equation eigenfunctions, analogous to these equa-

tions, have been obtained for arbitrary anisotropic scattering and for

azimuthally dependent problems in the monoenergetic case [69]. While

there is no new physics per se in the transfer matrix approach beyond

that contained in the WBE, and thus one expects a direct interrelationship

between the two eigenvalue problems [68], it remains to be seen whether

the simple form of Eqs. (3.28) and (3.34) will hold for energy-dependent

formulations with anisotropic scattering. It should be stressed that

since no stipulations other than scattering isotropy have been made with

respect to the interaction kernel in this chapter, the results obtained

here apply equally to fast and thermal regions, with or without fission,

etc. Furthermore, the form of the term a(E,w), which we have taken to
1w
be Z + -, does not enter explicitly and therefore generalization is
t v
immediate to finite transverse dimensions through introduction of a

transverse buckling. Of course the static case w = 0 is included.

Thus the equivalence of the continuous energy transfer matrix and

WBE eigenfunction approaches for isotropic interactions is quite

apparent, so that we may regard the transfer matrix method as a possibly

convenient framework for application of the WBE analysis. In this sense

we now have in the results of this chapter a complete treatment of the

analytic basis of the isotropic scattering energy-dependent transfer

matrix. Application to fast neutron wave and pulse propagation thus is

a matter of finding suitable means of implementing the formal results

obtained in Chapter II, using the basic relationships developed here.

CHAPTER IV

APPLICATION TO DISPERSION LAW AND DISCRETE
EIGENFUNCTION CALCULATIONS

Introduction

Expressions for the spectrum and eigenfunctions of the fast

multiplying WBE were derived in Chapter II. Emphasis there was placed

on application of the singular eigenfunction technique to obtain formal

expressions for the continuum eigenfunctions. In this chapter we will

consider the mathematically more straightforward topic of the discrete

spectrum and regular eigenfunctions of the wave transport operator.

This subject now is quite thoroughly understood in principle for

the separable fission kernel model; Travelli [16] has presented an

essentially complete transport multigroup numerical treatment of the

fast wave slab-geometry eigenvalue problem which takes into account

scattering anisotropy, through a PN expansion, and delayed neutrons.

However, apart from Travelli's work and the present investigation [72],

this author is not aware of any other numerical fast neutron wave

results which have been reported. It is surprising indeed, in view of

the apparent timeliness of fast space-dependent kinetics studies, that

use is not being made of tools such as these to investigate neutron

disturbance propagation in detail.

In this chapter an interrelationship between the dispersion function

and regular eigenfunctions is made explicit, and some properties of the

83

dispersion law are noted. An application of these relationships is

made to the dispersion law for multiplying media with isotropic elastic

slowing-down and inelastic scattering. Numerical results are presented

for the case of one elastically scattering species.

The Dispersion Function and Discrete Eigenfunctions

The dispersion law concept in the context of neutron wave

propagation originally was introduced by Moore [73] as a relationship

between wave frequency and complex wave length; subsequently this con-

cept was rather broadly generalized [8-10], as discussed in Chapter I.

We have seen here, in an exact transport treatment of the fast neutron

wave problem, how the dispersion function occurs in the structure of

solutions in multiplying media, as indeed it does for all energy-

regenerative media, as a result of the presence of a Fredholm integral

operator. We now will proceed to formulate the discrete mode eigenvalue

problem for a separable fission kernel in a way which both appeals to

intuition and suggests a method for computing the dispersion law.

The dispersion functions for fast multiplying media with a

separable kernel were given by Eqs. (2.31) and (2.37) for the cases

when downscattering is absent and present respectively. These

expressions may be summarized by

A = 1 (vZf, G) (4.1)

where

G(E,p;<) = \$ *
a-w<

(4.2)

with \$ defined from Eqs. (2.17) and (2.18); when scattering is ignored

\$ = I. The operator \$ in Eq. (4.2) may be inverted and the equation

multiplied by (a-pK) to give

(a-pr)G(E,p;<) Jd l dE'Z (E',u'-E,u)G(E,;K) = y (4.3)
-1 E

Thus the function G(E,U;K) is the solution to a pure slowing-down WBE

with the normalized fission spectrum y as a source, corresponding
iwt -KX
(cf. Eq. (1.4)) to the space- and time-dependent source ye e

From Eq. (4.1) we see that K is an eigenvalue when G(E,p;K) satisfies

(vZf) G) = 1; in that case, comparing Eq. (4.2) to Eq. (2.38), we see

that G is an eigenfunction of the total WBE with the particular nor-

malization X(K) = 1. The term A-1 already has been interpreted in the

discussion of the form of the multiplying medium continuum eigen-

functions as a modal multiplication factor; here we may regard the

dispersion law as a modal prompt criticality condition which determines

the value of < for which a solution to the homogeneous eigenfunction

equation will exist.

An analogous interpretation of the dispersion law for the die-away

experiment in fast multiplying systems has been given by Moore [8] and

Doing [41], generalizing the work of Storrer and Stievenart [74] who

arrived at an expression for the fast pulsed neutron dispersion law by

considering successive generations of fission neutrons. The particularly

simple forms above and in the pulsed neutron case for the dispersion

function and eigenfunction in terms of vEf and X are a direct consequence

of the separable form of the fission kernel [8,41,74]. We will see how

these results generalize for a degenerate fission kernel.

Algorithms for Evaluating the Discrete Spectrum and Eigenfunctions

Two different numerical approaches to the solution of the eigenvalue

problem are suggested by the form of the WBE and Eqs. (4.1)-(4.3).

Travelli has employed both [16,30], having derived the techniques by

means of computational considerations. The first method is a direct

numerical solution of the PN multigroup representation of Eq. (1.13) for

< S C; other than the requirement for complex arithmetic this approach

is straightforward [30]. An alternative procedure is suggested by Eq.

(4.3). In form it is a familiar slowing-down equation, for which

solution techniques are well established. The function G(E,u;<) is

obtained readily for a particular value of < by solution of Eq. (4.3);

the dispersion function A may be evaluated by Eq. (4.1). Zeroes of A

for a particular w then may be found by application of a complex Newton-

Raphson procedure. This procedure has the advantage of not requiring

solution of a matrix eigenvalue problem, which can become prohibitively

lengthy for the large numbers of energy groups needed to achieve

accuracy in fast medium problems [16].

The Newton-Raphson procedure can be expedited by computing --

by the following scheme [73]. Differentiating Eq. (4.1) yields

aA DG
= f f ) (4.4)

and from Eq. (4.3),

1
(a-u<) --- d' dE' 1 (E ,p-*E,u) -- = uG (4.5)
-1 E

which is a slowing-down equation with pG as a source. Solution of Eqs.

(4.3) and (4.5) can be carried out in parallel.

Extension to Degenerate Kernels

Only a slight additional effort is required to formulate the

eigenvalue problem for a degenerate Fredholm kernel (e.g., multiple

fissioning species). Following our development for the separable kernel

model we obtain instead of Eq. (2.36)

1 0
NM (m) i (m)F
F(E,u;<) = f \$ X dP- dE- v/ F(E',u,<)
m=1 a-c
-1 o

= G (y F) (4.6)
m=l

using obvious notation for an M-term degenerate fission kernel.

Reducing this to the matrix equation

(yn), F) = y(n), G m)) (y() F) (4.7)
m=l

in the usual way, we find the condition for existence of a solution is

the dispersion law

Det ( I [(y(n) G )] = A = 0 (4.8)

where the quantity in brackets is the I x N matrix having elements

(/ m) (n) (mi)
(y(n), G(m) vZ f n (4.9)
f a-.I<

It is interesting to note that the G are solutions to M &

slowing-down problems like Eq. (4.3), each with the same scattering

operator but with the source energy distribution x/ characteristic
th
of the m species. Eq. (4.8) is more complicated than previous

expressions for the dispersion function, but evidently we may retain

our interpretation of A-1 as a modal multiplication factor; we notice

that all combinations of fission of the nth species due to down-

scattered neutrons from the mth species occur.

Values of K which satisfy Eq. (4.8) may be determined by

straightforward extension of the Newton-Raphson scheme discussed above.

The eigenfunction F(E,p;K) then may be reconstructed by means of Eq.
(i)
(4.6); the coefficients (y F) are the elements of the eigenvectors

of the matrix equation, Eq. (4.7), and the functions G(m) will have been

evaluated in satisfying the dispersion law. Thus we have achieved a

general extension of the separable kernel analysis to the discrete

spectrum and eigenfunctions of the WBE with slowing-down and a degenerate

fission (or thermalization) kernel.

As a postscript to the above discussion, we note that the entire

procedure is identical in the case of the adjoint eigenfunctions, with

the obvious transpose and interchange of X) and f (n), and with use

of \$t rather than \$. Further, it is easy to show that

(y(n), G()) (v(n), G(m) = Gt(n) (m)), (4.10)

where

t t 7f
G E (4.11)
a-W<

so that should one wish to construct forward and adjoint solutions

simultaneously it is necessary to solve Eq. (4.8) only once.

Isotropic Elastic and Inelastic Scattering

The slowing-down equations encountered in the previous sections are

solved readily by any of a number of methods available from the fast

reactor literature; see, for example, the review of Okrent et al. [75].

To illustrate the method described above we will develop the expressions

for a continuous slowing down [76,77] model, with the addition of a

simple inelastic scattering model as well.

For isotropic scattering, Eq. (4.3) may be written

G(E;K) f(E) 7 (E'E)G(E';K)dE' = f(E)/ (4.12)
E

using Eq. (2.49) for f(E); cf. Eqs. (2.48) and (2.52). Differentiating

with respect to K, we obtain

G(E;_ ) i a G(E;K) af
(E;)- f(E) E (E'-E) dE = (E) (4.13)

E

which is the isotropic equivalent of Eq. (4.5). (The P-dependent eigen-

function can be constructed from the P-integrated form by using

Eq. (2.50).)

For isotropic inelastic scattering from M species and inelastic

scattering fron N levels with a constant energy loss AE per interaction

we have, in standard notation, the interaction operator

E/a
f NE f
dE' Z (E-E) =
E m=I E

NI
+ 1 E
n=1
E

e m)(E)
dE _se +
dE' E' (1- C E ) +
m

d51Wn

Defining

H(E;K) = G(E;)
f(E)

(4.15)

and using Eq. (4.14), we find

E / m 1 (m)(E')
H(E;K) = f 2 se f(E')H(E')dE' +
m i1 mE'(
E

NI
+ N I s i(E+,E )f(E+AE )H(E+AE ) +
n 2 si n n n
n=l

(4.16)

and a similar equation for (4.13). Eq. (4.16) is solved most
E
conveniently by converting to the lethargy variable u = In -- and
F
differentiating with respect to u. The resulting equation

r (E s NE z(m) fu)H(u) 1(
8H(u;<) = 1 (u)e 1 E se (u) u
mu du 2 2 1 -A
m=1 m
u+lnn

NI
+1 I E(u) [ (n) (u )f(u )H(u )E(u )]
2 1 E(u ) du [si n n n n
n=1 n n

(4.17)

(4.14)

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ANALYTIC AND NUMERICAL TRANSPORT TECHNIQUES IN ENERGYDEPENDENT FAST NEUTRON WAVE AND PULSE PROPAGATION By JAMES ELZA SWANDER A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1974

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ACKNOWLEDGMENTS The author would like to thank those who served at various times as chairman of his committee, Drs . R. B. Perez, A. J. Mockel, and M. J. Ohanian. In addition to the guidance given by the above individuals, the author would like to acknowledge helpful discussions with Drs. J. Doming and R. S. Booth. Financial assistance was provided by a NASA, Predoctoral Traineeship in Space Sciences and Technology, a University of Florida College of Engineering Fellowship, and Department of Nuclear Engineering Sciences Graduate Assistantships. Support for the computations performed was provided by the Northeast Regional Data Center. The manuscript for this dissertation was prepared at the Oak Ridge National Laboratory with the sponsorship of the U.S. Atomic Energy Commission under contract with Union Carbide Corporation^ 11

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TABLE OF CONTENTS PAGE ACKNOWLEDGMENTS ii LIST OF FIGURES vi ABSTRACT viii CHAPTER I INTRODUCTION 1 Purpose 1 Early Neutron Wave Investigations 2 Fast Neutron Wave Investigations 4 Energy-Dependent Transport Formulation of Wave and Pulse Propagation 4 Plane Symmetry and the Eigenvalue Equation 6 Interaction Operators for the Fast and Thermal Neutron Regimes 9 Spectrum and Eigenf unctions of the Thermal Transport Operator 13 Completeness of the Thermal Eigenfunctions 28 Other Related Problems and Literature 32 II SPECTRUM AND EIGENFUNCTIONS OF THE FAST WAVE TRANSPORT OPERATOR 35 Introduction 35 Adjoint Eigenf unction Equations 35 Biorthogonality of Eigenfunctions 37 Nonmultiplying Media: Spectrum of the Slowing-Down Transport Operator 38 111

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CHAPTER PAGE Forward and Adjoint Slowing-Down Eigenfunctions 41 Discussion of the Slowing-Down Eigenfunctions 46 Realistic Cross Sections: Nonmonotonic vZ 53 Fast Multiplying Media: Zero Scattering Cross Section 56 Discrete Eigenfunctions for Fast Multiplying Media .... 57 Continuum Eigenfunctions for Fast Multiplying Media .... 59 Discussion of the Continuum Eigenfunctions 60 Degeneracy of the Continuum 62 The Boltzmann Equation with Isotropic Interaction 62 III APPLICATION TO THE TRANSFER MATRIX METHOD 65 Introduction 65 Formal Operator Relationships 66 The Operators o6, X, and X~ 67 Spectrum and Eigenfunctions of a6 72 The Inverse Operator X 75 Diagonal ization Operators ... 76 Full-Range Orthogonality and Completeness 78 Half-Range Orthogonality and Completeness . . 79 Application to Fast Neutron Wave Propagation 81 IV APPLICATION TO DISPERSION LAW AND DISCRETE EIGENFUNCTION CALCULATIONS ... 83 Introduction , . . 83 The Dispersion Function and Discrete Eigenfunctions .... 84 Algorithms for Evaluating the Discrete Spectrum and Eigenfunctions 86 IV

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CHAPTER PAGE Extension to Degenerate Kernels 87 Isotropic Elastic and Inelastic Scattering 89 Illustrative Results: Dispersion Law and Eigenfunctions for Single Scattering Species . 91 V SUMMARY AND CONCLUSIONS 100 Summary 100 Conclusions and Suggestions for Future Work 102 APPENDIX A Introduction ...... 107 General Formalism 107 Algebra of the H-Matrix Ill Form of the H-Matrix: T and R Operators 112 Two-Region Transfer Matrix 116 Internal Sources 117 Transfer Matrix for Homogeneous Slabs 119 The Operators a and 124 Diagonalization of the Transfer Matrix 127 Transmission and Reflection Operators 129 Wave Transport Form of a and 3 130 APPENDIX B Singularity of Inelastic Scattering Kernel Models 133 APPENDIX C Macroscopically Elastic Scattering: The Elastic Continuiim 137 BIBLIOGRAPHY 142 BIOGRAPHICAL SKETCH 148

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LIST OF FIGURES Figure Page 1.1. The continuum domain C in the spectral K-plane 17 1.2. Structure of the continuum 18 1.3. Schematic dispersion law for a discrete eigenvalue .... 26 2.1. Orthogonality of forward and adjoint slowing-down eigenfunctions 47 2.2. Excitation of slowing-down eigenfunctions by a monoenergetic source 48 2.3. Degeneracy of the continuum due to nonmonotonic vZ ... 54 4.1. Dispersion laws for constant cross-section, elastic scattering model 92 4.2. Zero frequency eigenfunction energy spectra 94 4.3. Eigenfunction energy spectra for moderate to high frequencies 95 4.4. Eigenfunction phases for moderate frequencies 96 4.5. High frequency eigenfunction phase and amplitude relationship 97 4.6. Eigenfunction energy spectrum for frequency approaching the critical frequency 98 A.l. Entering and emerging fluxes for a single region 109 A. 2. Entering and emerging fluxes for adjacent regions .... 109 A. 3. Transmission 113 A. 4. Reflection 113 A. 5. Transmission through adjacent regions 118 A. 6. Internal inhomogeneous sources 118 VI

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Figure Page A. 7. Fluxes at an internal coordinate surface 126 C.l. Schematic diagram of the "elastic continuum" for macroscopically elastic scattering 140 Vll

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Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ANALYTIC AND NUMERICAL TRANSPORT TECHNIQUES IN ENERGY-DEPENDENT FAST NEUTRON WAVE AND PULSE PROPAGATION By James Elza Swander June, 1974 Chairman: Mihran J. Ohanian Major Department: Nuclear Engineering Sciences Neutron wave and pulse propagation analysis is a natural approach to spacedependent kinetics in subprompt critical media. Prior to the present work, analytic treatments of fast media have been few and limited in scope, in contrast to thermal wave and pulse propagation, which has been studied rather thoroughly and with quite sophisticated techniques . The principal difference between analysis of fast and thermal systems is treating the slowing-down operator. A formal approach is presented for arbitrary slowing-down operators; the spectrum, eigenfunctions, and adjoint eigenfunctions of the slab-geometry energydependent wave transport operator are obtained, using the singular eigenfunction technique. Both multiplying and nonmul tip lying media are treated. Fission is modeled by a one-term separable kernel, although Vlll

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the extension to a multiterm degenerate fission kernel, representing several fissionable species, is apparent. The fast neutron wave singular eigenfunction results are compared with other energy-dependent transport work, particularly with previous thermal eigenfunction analysis, and with work on static fast neutron transport using an energy transform approach. Wave transport in fast multiplying media and in thermal noncrystalline media (modeled by a separable thermalization kernel) are rather similar in that due to energyregenerative interaction processes a discrete asymptotic separable eigenmode exists for moderate values of wave frequency and absorption cross sections. The dispersion laws obeyed by the eigenvalues associated with these modes are qualitatively quite comparable. The fast nonmultiplying case has no direct thermal analogue other than the nonphysical "absorption only" model. It is found that the presence of down-scattering in this case gives rise to singular continuum eigenfunctions which are not as simply interpreted as the straightforward streaming modes obtained for zero scattering cross section. Nevertheless, these results appear to be in qualitative agreement with other work on energy-dependent fast neuton transport theory. The formal analytic results are developed in several directions to investigate their applicability to practical calculations. A major portion of this work is devoted to obtaining the energy-dependent wave transport representation of the transfer matrix method, which provides a formalism for implementing calculations concerning wave and pulse propagation through finite regions such as adjacent slabs of different composition. It is found that, for isotropic scattering, the basic operators IX

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of the transfer matrix formalism can be constructed from eigenfunctions of the wave transport operator. This result is general and is equally applicable to fast and thermal analysis. Finally, the dispersion law expression for a fast multiplying medium is employed to develop an algorithm for computing the discrete eigenfunctions and associated dispersion law for separable and degenerate fission kernels. A specific application of this method then is made to the case of isotropic elastic and inelastic scattering from any number of nuclides and levels, with arbitrary lethargy dependence of cross sections. Elastic scattering is modeled by a free gas kernel, and inelastic scattering by a constant energy loss per interaction per level. Unlike techniques requiring inversion of matrices, computation time increases approximately linearly with increases in lethargy steps, making quite detailed computations feasible. Illustrative computations are carried out using constant cross sections and a single elastic scattering species.

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CHAPTER I INTRODUCTION Purpose As the demand for energy increases and supplies of economically recoverable fuels diminish, fast breeder reactors will supply an increasing proportion of baseload generated power. To operate most economically these reactors will tend to be as large as is technologically feasible. Accordingly, as the size of fast reactor cores increases, it will become increasingly important to understand spatially dependent kinetic effects in fast systems. A particularly straightforward method of investigating propagation of neutronic disturbances in fast reactor materials is to place a pulsed or oscillating source of neutrons at the face of an experimental assembly, and then to observe the propagation of the neutron "signal" through the assembly. In this way one can study spatially dependent flux oscillations such as might be expected to result from flowinduced vibrations of core components, void formation and collapse, and other such phenomena. Neutron wave and pulse propagation experiments have been performed in many different thermal media, both multiplying and nonraul tip lying. The theoretical basis of analysis of thermal wave propagation in

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nonmul tip lying media has attained a considerable degree of sophistication, and fairly accurate numerical prediction of some experimental results is possible. This is in contrast to the situation in the fast neutron wave regime; few experiments have been performed, and analytic investigations have been hampered by difficulties which do not arise in treatment of thermal systems. The purpose of this dissertation is to present a particular framework of approach within which these difficulties may be addressed, extending techniques which have been applied primarily to thermal analysis. Three general objectives will be pursued: (i) to develop the spectral representation of the energydependent fast wave Boltzmann operator as far as possible in sufficiently general form so that its potential for use with realistic cross section data can be evaluated; (ii) to extend a formalism which treats neutron transport in finite and discontinuous media so that the above results may be applied to wave transport in experimentally realistic geometries and through successive regions; and (iii) to illustrate applications of the analysis by computing the fundamental eigenfunction and dispersion law for wave propagation in fast multiplying media, using a modelled kernel in the Boltzmann operator. Early Neutron Wave Investigations In 1948 Weinberg and Schweinler published the first description in the open literature of the generation and analysis of neutron waves [1].

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Using one-speed diffusion theory they were able to show that a localized oscillation in neutron absorption within a reactor would produce a perturbation in the neutron population which would propagate in wavelike fashion. The first experiments with neutron waves were reported in 1955 by Raievski and Horowitz [2], using a mechanically modulated exterior source to generate waves in DO and graphite. Uhrig [3] then applied this technique to measurements in subcritical assemblies. Both experimental and theoretical aspects of neutron wave propagation subsequently received considerable attention and refinement, particularly by Perez [4] and his associates at the University of Florida, although experiments and most analytic efforts were restricted to thermal systems. As investigation of the theoretical basis of neutron wave experiments proceeded, it was realized that from an analytic standpoint experiments involving spatially propagating pulses were equivalent to neutron wave experiments, since any physically realizable pulse could be time-Fourier analyzed to give its frequency components [5] . Also, it became clear that neutron wave propagation was related to other linear static and kinetic experimental techniques, in particular the classical exponential experiment, which is the zero-frequency limit of the wave experiment, and the pulse die-away experiment [S-8] . (The die-away experiment monitors the time-rate of decay of a neutron population which has been introduced into a finite assembly by a pulsed external source. For technical reasons this type of experiment was easier to perform than wave experiments, and enjoyed a more rapid initial development [4].) As a result, these methods experienced

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considerable parallel theoretical treatment [8-10] . Early work in this area is reviewed extensively by Uhrig [7] and Perez and Uhrig [4] . Fast Neutron Wave Investigations Neutron wave and pulse propagation has received proportionately very little attention in the fast energy regime. The only experiments described in the literature, performed by Napolitano et al. [11,12] and Paiano et al. [13] at the University of Florida, have been in nonmultiplying media; no experiments in multiplying media have been reported. The technical difficulty of such experiments probably has contributed both to the lack of experimental data and the scarcity of methods to predict and correlate results. Theoretical analysis also has been retarded by the fact that even tractable energy-dependent analytic models of fast media do not have convenient mathematical properties, and consequently most of the elegant techniques which have been applied to thermal neutron transport cannot be extended readily to this problem [14]. Notable exceptions to the general absence of numerical techniques and results are the multi-group, multiplying medium calculations of Travelli [15,16], and the calculations of Booth et al. [17], using the multi-group discrete ordinates method of Dodds et al. [18] to interpret Napolitano' s experimental results. EnergyDependent Transport Formulation of Wave and Pulse Propagation Before discussing the various theoretical results which are directly or indirectly applicable to fast neutron wave and pulse

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propagation, it will be helpful to approach the general neutron wave problem from the point of view of the energy-dependent transport method which will be used in this dissertation. We begin with the classic time-dependent Boltzmann equation for the neutron flux [19,20,21], which we will write ^^ 4>(r,E,^,t) + ti ' V(j)(r,E,J^,t) + Z^CE)4)(r,E,J^,t) d^ 4lT dE" K(E^f^' ^ E,Q)
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model. (Nonlinear space-dependent kinetics are of interest primarily in the context of excursion situations; such problems, while important, are difficult to analyze, and thus far have been approached by use of specialized and involved computational techniques [22,23,24].) Any of a number of classic analytic approaches can be taken to the solution of Eq. (1.1); here we will treat it as an eigenvalue problem. This will enable us to extend our results to finite medium and multiregion problems; the transfer matrix formalism, which will be discussed in Chapter III, requires solution of a similar eigenvalue equation, and we will be able to relate its solution to those of the wave Boltzmann equation for isotropic scattering. Furthermore, we can make use of spectral analysis which already has been done on the thermal neutron version of Eq. (1.1). Plane Symmetry and the Eigenvalue Equation The mathematical development of transport theory has reached its greatest sophistication for the case of plane symmetry, and this is triie also for the particular subject of neutron wave and pulse propagation. Since this geometry also is appropriate for the description of classical wave and pulse propagation experiments, we will turn our attention to the specific case of plane neutron waves. The infinite medium plane wave eigenfunctions which will be obtained may then be used in developing the corresponding transfer matrix formalism, which can be employed to study the propagation of waves and pulses through finite slabs and successive slabs of dissimilar materials.

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Following customary arguments, we stipulate that all sources or initial fluxes must be rotational ly symmetric about the x axis and do not depend on the transverse Cartesian coordinates y and z. Orienting the X axis along the direction of wave propagation, we have for the transport operator of Eq. (1.1) ^ ' h = Mj^ ; ]s E n ' X (1.2) where x is the unit vector in the x direction, and y is the cosine of the angle between the path of neutron travel and the x axis. With the restrictions of Eq. (1.2) the homogeneous Boltzmann equation may be written :^-^ (})(x,E,y,t) + u ^ 4)(x,E,y,t) + Z^(E) (t)(x,E,y,t) 1 rÂ°Â° dy' dE' K(E',y' -^ E,y)(})(x,E',y',t) = 0. (1.3) ; -1 We notice that time and space operators appear only in the first two terms, respectively, while the integral operator acts on E and y. Consequently, x and t variables may be separated. With appropriate choices for the separation constants <)) may be expressed as a damped plane wave

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(t)^(x,E,y,t) = F(E,y;K) e^*^^ e"'^'' Â„,Â„ , . -ax i(wt-?x) = F(E,y;ic) e e ^ ^ (1.4) where u is the wave or Fouriercomponent frequency, and k is a complex constant, K = a + iC, the complex inverse relaxation length. Thus a is the inverse relaxation length of the wave, while C is its wave number. The frequency O) will be regarded as a parameter of the equation, and we will treat K as the eigenvalue to be determined. Introducing Eq. (1.4) into Eq. (1.3) Â— y< + Z^(E) F(E,y;K) dy^ dE' K(E^y-' ^ E,y)F(E^y^K) = 0, (1.5) or defining a(E,a.) EZ^(E) . ^^^^ (1.6) Eq. (1.5) has the form fl fÂ°Â° (a(E,a)) yK)F(E,y;K) dy dE K(E ,y -> E,y)F(E ,y ;k) = -1 o (1.7)

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which will be referred to as the wave Boltzmann equation, or WBE, throughout the rest of this work. This is the most general statement of the Boltzmann equation in wave eigenvalue form with plane symmetry. As an eigenvalue problem it should more properly be written f 1 rÂ°Â° dy^ dE' ^^^ 6(E-E')6(y-yO i K(E^y' -> E,y) F(E',y^K) = KF(E,y;K). (1.8) The pole at y = causes no difficulties which we will need to consider [25] ; this value of y corresponds to a direction of neutron travel perpendicular to the direction of wave propagation. Interaction Operators for the Fast and Thermal Neutron Regimes The two general types of neutron interaction which are of importance in wave and pulse propagation and which enter into the kernel of Eq. (1.7) are scattering and fission. Adopting for a moment a theoretician's perspective on reality, we may define a fast neutron experiment as one in which the scattering kernel has a Volterra form in energy. In a similar vein a thermal problem may be distinguished by the presence of a Fredholm scattering kernel. These observations stem from the fact that in the fast neutron regime one is concerned with neutron energies from the eV range to about 10 MeV, the upper end of the fission spectrum; hence only downscattering in energy is important. By contrast.

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10 in the thermal regime neutrons are in or near thermal equilibrium with their surroundings so that upscattering in energy occurs as well; energies of interest range essentially over the thermal Maxwellian spectrum. Appropriate interaction kernel models reflect these properties. We may write the thermal scattering contribution to the interaction kernel as 1 dy' dE' Z^(E^y" ^ E,y) -1 while the fast scattering operator has the form 1 fÂ°Â° dy' I dE' Z^(E^y' ^ E,y) 1 E extending the notation of Eq.s (1.3) to (1.7). In multiplying media the interaction kernel contains a contribution due to fission in addition to scattering. Difficulties associated with treating the slowing down of fission neutrons [26] have precluded transport analysis of wave propagation in thermal multiplying media, although other models such as age-diffusion have been employed [27] . No such problem arises in the fast multiplying wave problem, since the energy range of the fission spectrum is essentially the energy range of interest.

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11 For the fission contribution to the interaction operator we will use the customary isotropic separable kernel When only one fissionable isotope is present this is a satisfactory model. For two or more fissionable species, one can either construct an equivalent separable kernel with averaged x and vZor employ a degenerate kernel j = l Only the separable kernel will be treated in detail here but the formalism of Chapter II can be extended in a straightforward way to degenerate kernels for multiple fissioning species. To avoid the appearance of unwelcome factors of 1/2 in connection with the isotropic fission spectrum, we will make the following notational distinction. Define the isotropic x(E) so that X(E)dE = 1. (1.9) Define X so that X = x(E,y) = 1/2 xCE); (1.10)

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12 then .00 dy dE X = 1. (1.11) Â•1 The fission interaction operator employing the separable kernel model and Eq. (1.10) thus becomes 1 /'Â°Â° du' J 1 dE' vZ^(E')' which is the form which will be used throughout this work. Using the above forms for contributions to the interaction kernel the fast homogeneous WBE of Eq. (1.7) may be written Ca-viK)F(E,ja;K) = /Â•I dy' } i 1 E dE' E^(E',u' ^ E,y)F(E',y ;k) (1.12) for nonmultiplying media, and (a-yK)F(E,y;K) fl rÂ°Â° dy' dE' Z (E',y' E,y)F(E',y';K) 1 E Â•1 fÂ°Â° + X dy' dE' vZ_(E')F(E',y';K) (1.13) 1 o for multiplying media.

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13 It should be noted that we have taken into account only prompt neutrons in Eq (1.13), and hence v here is the number of prompt neutrons per fission. Delayed neutrons will contribute only at wave periods greater than the shortest delayed neutron precursor lifetime, an effect which has been investigated numerically by Travelli [16]. Also, it will be assumed that the medium under consideration is subprompt critical. The thermal nonmultiplying WBE, which will be discussed as a point of departure for our work on Eq. (1.13), is (a-u<)F(E,y;K) = -1 '1 fÂ°Â° du' dE' Z (E',m' ^ E,m)F(E',m';k). (1.14) Spectrum and Eigenfunctions of the Thermal Transport Operator The fast neutron wave energydependent transport eigenvalue problem can best be introduced by discussing work which has been done on the analogous thermal problem, Eq. (1.14). This approach will be taken because transport treatment of the fast problem is necessary to obtain qualitatively correct spectral descriptions for passive media. Approximations such as diffusion theory can yield an estimate of the least attenuated mode of propagation in fast multiplying media, where such a fundamental mode exists, but can provide little other information relevant to the properties one should expect of the exact transport treatment. The context of the work to be presented here is the "singular eigenf unction method," which received its major impetus from a paper by

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14 Case [28], and thus is frequently known as "Case's method." As an introduction to the literature on the singular eigenfunction method in transport theory, including the wave problem, the review of McCormick and Kuscer [29] is highly recommended, as it is both recent and extensive. Travelli [30] was the first investigator to arrive at an essentially correct description of the spectrum of the energy-dependent wave problem, based on a multigroup transport formulation. We turn now to the energydependent analysis of the thermal wave eigenvalue problem, corresponding to Eq. (1.14), performed independently by Kaper et al. [31] and Duderstadt [32,33]; the former study employs an isotropic one-term degenerate thermalization kernel, while Duderstadt discusses more general types of scattering interaction models as well. Their results are summarized in this section. Eq. (1.8) may be written in abbreviated form as AF E (A + A )F =
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15 and the interaction operator A becomes ^2 = fl j-oo dE' I (E^y' y s E,y) (1.17) -1 using the scattering interaction kernel of Eq. (1.14). The basic method for obtaining the spectrum and eigenfunctions of this equation is a generalization of the work of Bednarz and Mika [34] on the static Boltzmann operator, which in turn extended the classic monoenergetic singular eigenfunction technique [25] to a continuous energy representation. We begin by defining the domain C in the spectral K-plane, which is the continuous spectrum of the streaming operator A : ^^(E) 1 0) yv(E) K = 0, y e [-1,1], E e [o,Â«>) (1.18) or in the notation of Eq. (1.14), those values of < for which a yK vanishes. For any nonzero frequency co, a is complex, so that C will occupy an area in the K-plane. It is instructive to consider both the rectangular and polar forms of < 6 C; Eq. (1.18) implies that ^JE) Re(K)

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16 where r and 6 are the usual radial and azimuthal polar coordinates. In general C will consist of two symmetric portions in the first and third quadrants, due to u 6 (0,1] and y 6 (0,-1] respectively. This is represented schematically in Figure 1.1. Since frequency w is a positive quantity, C does not extend into the second and fourth quadrants. Figure 1.2 shows the first quadrant of the <-plane in more detail. The domain C is bounded by the line y = Â± 1, a = E ; from the rectangular form of Eq, (1.19) it is apparent that the real part of this boundary line assumes every value of Z as E (and thus v) varies from to Â°Â°. The polar form of Eq. (1.19) shows that as |y| varies from 1 to 0, values of K corresponding to a fixed E generate a line of constant 9 which begins at the boundary of C and extends to infinity. As the parameter o) is increased or decreased the domain C expands or contracts in the imaginary K direction. We note that if vZ varies monotonically with E, each point of the domain C will correspond to a unique E,y pair, E and y ; for the thermal analysis presented here, this is assumed to be the case. Two important results then follow. First, Eq. (1.19) defines a one-to-one mapping of E,y onto the spectral plane. Second, the spectrum of A is not degenerate. The consequences of these results will be discussed later. The discrete and residual spectra [35] of A are empty [33] . The singular continuum eigenfunctions of A , satisfying the equation (A^-K)F(E,y;K) = (1.20)

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17 esIm(K) a= Re(K) Figure 1.1 The continuum domain C in the spectral K-plane.

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18 Figure 1.2 Structure of the continuum.

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19 are FCE,y;K) 6(E-E^)6(u-u^) = 6(a-y<) . (1.21) This gives a corresponding eigenmode, using Eq. (1.4) (for A = A.), (x,E,]A,t) = e'^"" e^*^'^^'^''^6(E-E^)6(M-y ) (1.22) which clearly represents neutrons of energy E streaming in the direction y . Since C is in one-to-one correspondence with all possible E,y pairs, each point in the continuum corresponds to a unique neutron speed and direction of travel. Referring to Eq. (1.19) we see that modes with y = Â± 1 have relaxation length 1/E (E ) equal to the neutron relaxation length; that is, modes corresponding to k on the boundary of C represent neutrons streaming along the x axis. Other modes are more attenuated, as the direction of neutron travel becomes more oblique to the direction of wave propagation. The spectrum of the streaming operator and its eigenfunctions are qualitatively the same for both thermal and fast regimes, the only differences being the values of E which are applicable, and the detailed structure of E^ as a function of energy; Eqs . (1.19) (1.22) apply in either case. It is the interaction operator A , containing the description of the scattering and multiplication processes, which gives rise to the qualitative differences between fast and thermal WBE eigenfunctions. It seems likely for "reasonable" mathematical models of thermal scattering that the spectrum of A = A + A always contains the spectrum

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20 of A^. This has been substantiated for Ahaving the form of a separable kernel with isotropic scattering [31,33]. This model, which was proposed by Corngold et al. [36], has been used quite extensively in analytic transport studies, since it represents fairly well the qualitative features of thermal scattering interactions [37] . Duderstadt [33] has investigated more general operators A , and while the spectral analysis for less restricted models is somewhat more tentative, it does appear to indicate that the spectrum of the streaming operator is in general contained in the spectrum of the wave Boltzmann operator A. We will see this in a more formal way from the technique used to construct the continuum eigenf unctions. To illustrate this method we obtain the eigenfunctions for the thermal WBE with the separable isotropic thermal izat ion kernel -1 O dE' ^gCE'), .00 du J J 1 o dE M(E) = 1. (1.23) (To be consistent with our later treatment of the fast WBE we will not perform the usual symmetrization of this kernel, since for the fast case A is not symmetrizable. The main result of interest which arises from symmetry of A is that the eigenfunctions are mutually orthogonal, and one avoids the adjoint problem; this and other considerations will not be of direct concern here. Also note that M(E) is not the Maxwellian

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21 distribution. To satisfy detailed balance M(E) = M'(E)Z (E) , where M^(E) is proportional to the Maxwellian, subject to the above normalization constraint.) Using this form for A , the WBE corresponding to Eq. (1.14) becomes (a-yK)F(E,y;K) = M(E) dy' dE' Z (E')F(E',y';K). (1.24) Notice that this expression is exactly equivalent to the fast multiplying WBE in the form of Eq. (1.13) when scattering is ignored in that equation. First we investigate the point spectriim. We see that k will be an eigenvalue when the homogeneous equation (1.24) has a solution for that value of k. Let us suppose that k C so that (a y<) ^ 0; then dividing by this factor. F(E,y;K) M(E) a-yK dydE' Z (E')F(E',y'*;<) s -1 Defining the scalar product (1.25) ((|)(E,y), i)iE.]x)) = .00 dy dE (})(E,y)iJ;(E,y) (1.26)

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22 Eq. (1.25) may be written more compactly F = M a-yK ^ s (^.,F) (1.27) Taking the scalar product of this equation with E and eliminating the scalar factor (I ,F) we find that the condition for Eq. (1.26) to have a solution is Z -^ s' aUK (1.28) Defining the dispersion function A(K,w) s' a-iJK (1.29) Eq. (1.28) is simply the condition that this dispersion function vanish. Eq. (1.28), which is referred to as the dispersion law, determines in the present problem the regular eigenvalues k of the WBE as a parametric function of frequency. Indeed, values of k which satisfy the dispersion law for a given frequency o) have been shown [31,33] to comprise the point spectrum of A with A^ defined by Eq. (1.23); for these eigenvalues K. the corresponding eigenfunctions are given by Eq. (1.27): F(E,y;<.) = A(K.) ^^^ *Â• ' ' j'^ ^ ] a-yK (1.30)

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23 where A(k ) is an arbitrary constant. (Note that for small co and Z ' a this approaches a Maxwellian distribution in energy.) When K e C, the term (a y<) is zero for a particular E = E and K ^^ = \As we have mentioned, C is contained in the continuous spectrum. In the present case the continuous spectrum of A is identically the domain C, and the continuum eigenfunctions are [31,33,34] '^^'^'^^ =W^\^^^ \ ^E^ ^,^^')n^',V^;K) . AÂ«)6Ca-pK) -1 o (1.31) K e c using the notation of Eq. (1-21); A(k) is an arbitrary constant. We see that (a yK)" has a pole at the "eigen-energy" E^ and "eigen-angle" y^; integrals over E,y involving this term will exist in the ordinary sense, provided that its coefficients in the integrand are well-behaved at the pole. Hence we may eliminate the scalar (Z ,F) in Eq. (1.31) in favor of the constant A (k) by taking the scalar product of the equation with Z^(E) and solving for (Z ,F). We then find F(E,y;K) = A(k) M(E) ^s ^^<^ a-y< J Â— + (S(a yK) (1.32) K e c so that A(K) is in fact a normalization constant; A is the dispersion function defined in Eq. (1.29)

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24 Eq. (1.31) may be obtained directly from Eq. (1.24) by a heuristic argument [38]. Since for any variable x the function x6(x) is identically zero, apparently X(a yic)6(a y<) E may be added to the righthand side of Eq. (1.24). Division by (a ijk) gives Eq. (1.31) when K e C. Evidently, then, the domain C always will be in the spectrum of A, since it is contributed by the streaming operator, regardless of the form of A . The continuum eigenfunction, Eq. (1.32), is composed of two singular terms, one being the pure streaming mode of Eq. (1.21), and the other having distributed E and y dependence, with the same formal structure as the discrete eigenfunction, Eq. (1.30), except that it has a pole singularity since k 6 C. The scalar coefficient of the latter term, E (E )/A, represents the relative excitation of the distributed portion of the mode by the streaiming portion (this can be seen more clearly by comparing with the analogous fast continuum eigenfunction, which will be developed in Chapter II). Hence the continuum eigenfunction may be interpreted as being due to direct streaming neutrons having energy and direction E and y , and an associated scattered distribution which is excited by the streaming neutrons; the scattered distribution is peaked at E and y due to the pole of the transport coefficient (a yK) , but contains all other E,y values as well. Note, however, that the entire mode has the phase velocity v H ^ (cf. Eq. (1.4)) of the uncoil ided wave. This interpretation of the thermal continuim eigenf unctions has not been given in previous treatments, as symmetrization of the kernel tends to obscure the physics involved.

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25 Kaper et al . [31] have investigated the dispersion law A = for the separable kernel model. Their findings may be summarized as follows. When 03 = 0, one has the classic exponential experiment [21]; there is exactly one pair of real eigenvalues k. = Â± k , provided that J P the absorption of the medium is not too strong (of course the precise condition will depend on the energy dependence of the cross sections) . Otherwise, the point spectrum is empty and will remain empty for all co. As the parameter w is increased from zero, the pair of eigenvalues will move symmetrically into the first and third quadrants of the complex K-plane. Evidently for sufficiently large w there will be a limiting frequency u^ beyond which the discrete spectrum is empty; this value of frequency appears to occur when K meets the boundary of the continuum C. This situation is represented schematically, for the first quadrant, in Figure 1.3. (We noted that in general the boundary of C is frequencydependent; here for simplicity it is shown for Z constant, in which case the boundary remains a line perpendicular to the real axis.) While for a time it was conjectured [32,33] that zeroes of the dispersion function might exist within the continuum as "embedded eigenvalues" in a continuation of the dispersion law for w > oj , it now c appears [31,39] that this is not the case, although the dispersion function apparently does vanish at points within the continuum [39]; referring to Eq. (1.32), this corresponds to points at which the deltafunction contribution vanishes. This subject will not be pursued here; the interested reader is referred to Kaper et al. [31], Kline and Kuscer [39], and for an extensive discussion from a different point of view, to the work of Doming and Thurber [40] and Doming [41].

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26 le / / / / / I /'^=0 Figure 1.3 Schematic dispersion law for a discrete eigenvalue.

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27 From the results of calculations based on the separable kernel or comparable models [31,32] it is possible to draw some conclusions about the physical interpretation of the frequencydependent behavior of the dispersion law. Near zero frequency the fundamental neutron wave mode (if there is one) corresponding to Â± < is less attenuated than the streaming-associated continuum modes. As cj increases, the wavelength of the fundamental mode decreases and it becomes more attenuated. This occurs because it becomes increasingly difficult for scattered neutrons to remain in phase with the wave; from Eq. (1.24) we see that as < approaches C the energy and angular distributions of the wave become increasingly peaked for y = Â± 1. Ultimately the fundamental mode becomes nearly as attenuated as forwardstreaming neutrons, and evidently the distributed E,y term of the continuum eigenfunctions then assumes the role formerly held by the fundamental mode as the frequency increases beyond w . One additional remark should be made. For o) = the spectrum of the static Boltzmann equation lies entirely on the real axis, and in general it appears that it is the limit of the spectrum of the WBE as u) approaches zero. But obviously for k real the identification of E,u pairs with points of C no longer can be made. Indeed it may be improper to regard the static Boltzmann equation as the zerofrequency limit of the WBE, No such problem arises in connection with the discrete spectrum, in the sense that in the limit 00= 0, Eq. (1.28) gives the correct eigenvalues for the static case. This evidently is true of the dispersion law in general, and in that sense we speak loosely of the exponential experiment being the zerofrequency limit of the wave experiment.

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28 In this work we will be concerned only with co ^ except in the calculations of Chapter IV, which involve only the discrete eigenvalues and eigenfunctions. Completeness of the Thermal Eigenfunctions In order to make use of a set of eigenfunctions such as those obtained in the previous section, it is necessary to show that arbitrary functions (suitably restricted) can be expanded using these functions as a basis, and it is further necessary to evaluate the expansion coefficients. First, then, one must prove that the set of eigenfunctions is complete, or at least establish completeness within the context of the problem one is to consider. Then either the eigenfunctions must be shown to be orthogonal and normalized to unity scalar product, so that orthogonality properties may be used to obtain expansion coefficients in the usual way, or some other procedure must be followed. Normalization of continuum eigenfunctions is somewhat less than straightforward because, as may be seen from the form of these eigenfunctions in Eq. (1.32), it involves products of delta functions of complex variables. The alternative procedure is to find the continuum expansion coefficients G(k) of an arbitrary function 4'(E,u) directly from the expression for the expansion, which is a singular integral equation: r(E,y) = G(K) F(E,ii;K)dK (1.33) K

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29 where 'F' is the portion of ^ contributed by the continuum eigenfunctions (the discrete mode contribution is found by the usual application of orthogonality). F(E,y;<), which is now the kernel of the integral operator, is known from Eq. (1.32) or a similar evaluation of the continuum eigenfunction based on another model. In implementing either method the theory of generalized analytic functions [42] has been the principle tool applied to date. Eq. (1.33) also has been used to prove completeness of the WBE eigenfunctions, since if it can be shown that an arbitrary V(E,\i) has a representation in this form, the set of eigenfunctions F(E,y;K) must be complete. This approach has been taken by Kaper et al . [31] and Duderstadt [32] to show completeness for the eigenfunctions of the separable thermalization model of the previous section; their treatments were based on extension of the generalized analytic function technique as applied by Cercignani [43] to problems in the kinetic theory of gases. The details of this analysis are lengthy and will not be repeated here. We will make reference to two types of completeness and orthogonality. We note that values of k in the first quadrant correspond to plane waves propagating in the positive x direction, and similarly the third quadrant represents waves propagating in the negative x direction. In general, e.g., within a slab of finite thickness, a wave will be made up of components traveling in both directions; to represent an arbitrary wave (or pulse frequency component) '!'(Â£, y,x,u)) in WBE eigenfunctions, one must use all the eigenfunctions, corresponding to the whole spectrum of the wave Boltzmann operator. Completeness of the first type, in the sense that a unique representation of this sort can

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30 be made, is termed full-range completeness. The corresponding fullrange orthogonality is simply orthogonality under the scalar product of Eq. (1.26). In applying eigenfunction techniques to boundary value problems, one frequently wishes to represent an incoming source flux S(E,y,x ,03) or to specify flux continuity for waves moving from one region into another across an interface at a boundary point x . In this case the boundary condition will be specified for either ye [-1,0) ory G (0,1] and will involve eigenfunctions for only one direction of wave propagation. Completeness in this sense, termed half -range completeness, requires that a function defined over y 6 (0,1] or y 6 [-1,0) can be represented liniquely by WBE eigenfunctions corresponding to the eigenvalues in only the first or third quadrant of the spectral plane, respectively. Halfrange orthogonality is orthogonality under integration the half range of y. Both full-range and half-range completeness requirements will be seen to arise in Chapter III in connection with a formalized treatment of the slab geometry boundary value problem. We should note that at present half-range completeness can be proved only for quite restricted kernel models, although full-range completeness can be shown for more general kernels [29,33]. Our main interest in the completeness properties of the eigenfunctions of Eqs . (1.24) and (1.26) is that they are indeed complete. We will use the same formal procedure to find the eigenfunctions of the fast WBE, and will obtain qualitatively similar results. Thus we may have considerable confidence, in lieu of proof, that the fast eigenfunctions are complete as well.

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31 There are two difficulties which will prevent us from extending the generalized analytic function technique directly to the fast regime. First, one must deal with the slowing-down operator. Second, the one-toone equivalence between values of E,y and points of C does not hold for realistic fast cross sections (e.g., at resonances), and we will be reluctant to consider more restrictive cross-section models (i.e., monotonic vE ); this equivalence plays a central part in the generalized analytic function method as it has been developed to date. Whether these two problems are insurmountable is a matter for further investigation; however, it seems unlikely, in view of the results established in the thermal case, that the fast eigenf unctions would not be complete for "reasonable" crosssection models. (An example of an "unreasonable" model is a strictly 1/v-dependent cross section or one which has this behavior over some energy range. When this occurs the portion of C corresponding to this energy range collapses onto a line. This case is discussed for thermal waves in polycrystalline material by Duderstadt [33] and by Yamagishi [44]; it is necessary to deal separately with the eigenfunctions on the line continuum which results from this cross section.) For an introduction to other literature on completeness of singular eigenfunctions see the review of McCormick and Kuscer. It is interesting in this connection to read the comments of Burniston et al . [45], and

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32 the recent remarks of Zweifel [46] regarding the degree to which the rigorous mathematical basis for the singular eigenfunction transport analysis has been established. Other Related Problems and Literature In the foregoing discussion we have seen that for the thermalization model employed there the spatially dominant wave mode is due to the regular eigenvalue k , which is determined by the zeroes of the dispersion function A(K,aj). Further, we see from Eq. (1.32) that the zeroes or nearzeroes of A also will play a large part in determining the character of continuum modes, since in regions where A is small the scattering portion of the mode will dominate the streaming term. A corresponding dispersion function appears to arise in general in the treatment of regenerative media (i.e. those in which neutron interactions can result in either a gain or loss in energy, and hence the interaction kernel has a Fredholm form) . Doming and Thurber [40] , for example, find that in an alternative formulation of the wave problem and in an initial value problem the nature of solutions is similarly influenced by the behavior of a dispersion function. In addition, dispersion laws are known to arise in nontransport approximations to dynamic eigenvalue problems. For example, when the multi-group diffusion approximation is used to obtain a matrix expression analogous to Eq. (1.8), its determinant is the dispersion function, and the dispersion law is simply the requirement that the determinant vanish; the solutions associated with values of k which satisfy the dispersion law are then the desired eigenmodes. Indeed the multi-group diffusion

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33 approach has been used rather extensively to compute dispersion laws for moderators, and when sufficiently accurate scattering matrices are employed, agreement of diffusion theory methods with experiment at low frequencies can be quite good [47]. General discussions of wave and pulse propagation in the context of its relationship to other dynamic problems, properties of the various dispersion laws, and analytic methods which have been applied to these problems will be found in Bell and Glasstone [21] and Hetrick [48]. An excellent review of the literature in this area as of 1967 has been given by Kuscer [49], although it is of course somewhat dated. As an alternative exact approach to transport problems, the Wiener-Hopf technique is finding increasing favor and must be viewed as a potential method for analysis of the wave problem; Williams [50] recently has published an expository review of the method. Also, the singular eigenfunction method review of McCormick and Ku^Ser [29] should be mentioned again in connection with the subject of transport treatments of various static and dynamic problems. Finally, with respect to the subject of Chapter III, note should be taken of existing work treating neutron waves in geometry which is finite or has discontinuities along the direction of wave propagation. Interface effects first were investigated experimentally by Denning, Booth and Perez [51]. This same problem was the subject of both numerical and analytic investigation by Baldonado and Erdmann [52,53]; their work is of particular interest because onespeed and energy-dependent diffusion and transport results are given. Mockel [54] has presented both transfer matrix and invariant imbedding transport formulations for

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34 wave transmission and reflection from a slab imbedded in an infinite medium of different composition. Also to be noted is the treatment of Larson and McCormick [55] of transport in a slab, in the static case, using a degenerate scattering kernel. Recently much attention has been given by Japanese and Indian groups to the problem of thermal neutron wave propagation in assemblies of polycrystalline moderating materials (e.g. graphite and beryllium) having finite transverse and longitudinal dimensions; see for example Nishina and Akcasu [56], Kumar et al. [57], and Yamagishi [44] . The latter is of particular interest because it demonstrates, in a transport treatment, the presence of intermodal interference.

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CHAPTER II SPECTRUM AND EIGENFUNCTIONS OF THE FAST WAVE TRANSPORT OPERATOR Introduction In this chapter the singular eigenfunction formalism, presented in Chapter I, will be extended to the fast WBE expressions, Eqs. (1.12) and (1.13). Both the forward and adjoint eigenfunctions will be obtained for general forms of the nonmul tip lying, or "slowing-down," and multiplying cases. The structure of these solutions will be discussed, and some of the implications of using realistic cross-section and scattering kernel models will be explored. Adjoint eigenfunctions will be investigated for two reasons. First, they will be necessary for the treatment of the transport formulation of the transfer matrix in Chapter III. Second, as has been mentioned, analytic evaluation of expansion coefficients cannot be performed using generalized analytic function techniques which have been applied to thermal problems. For the same reason, we will not obtain normalization constants analytically. However, biorthogonality of eigenfunction sets will be shown in the classic way. Adjoint Eigenfunction Equations The appropriate scalar product under which to define adjoint operators is given by Eq. (1.26). We consider the general wave eigenvalue 35

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36 equation in the form of Eq. (1.8), which we may write as ^FCE,u;k) -1^ rl 1 o rÂ°Â° dE' K(E',u -> E,y)F(E^y ;k) = KF(E,y;K) (2.1) The adjoint eigenfunctions will be denoted by F" (K;E,y); the adjoint eigenvalue equation corresponding to the forward equation, Eq. (2.1), is ^F"^(K;E,y) dy' -1 dE" ^ K(E,y -^ E^y^)F"^(K;E^y") = KF"^(K;E,y) (2.2) where we notice the factor Â— is now within the integral. However, if we define f''" (K;E,y) E i F"^(K;E,y) (2.3) Eq. (2.2), becomes, upon substitution and rearranging. (a y<)F"'' (<;E,y) dy' dE' K(E,y E^y')F^(<;E^y^) = (2.4) 1 o which is the form which would have been obtained as the adjoint of the homogeneous wave Boltzmann equation, Eq. (1.7). It will be more convenient to deal with Eq. (2.4) since it differs from the forward WBE, Eq. (1,7), only in the kernel of the interaction operator and hence we

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37 will be able to apply the same techniques to the solution of both forward and adjoint equations. It should be pointed out that k is used for the eigenvalue in Eq. (2.2), with the implication that the spectra for forward and adjoint equations are identical. That this is true for "well-behaved" operators in the models we are considering will be apparent from the singular eigenfunction formalism, although of course each case must be explained individually. Nicolaenko [14] has exhibited an inelastic scattering operator for which the adjoint spectrum contains additional contributions due to a singularity of the kernel at zero energy; he uses the singular kernel in defining an energy transform for reduction of the static transport slowing-down equation (for the model he considers) to monoenergetic form. However it is shown in Appendix B that singularity of inelastic scattering kernels is not an inherent attribute of fast neutron transport problems. Thus for the forward and adjoint problems the spectra and eigenfunctions can be regarded tentatively as being in correspondence, subject to verification for specific interaction models. Biorthogonality of Eigenfunctions Biorthogonality of WBE eigenfunctions corresponding to different eigenvalues can be shown by the usual argument. Writing Eq. (2.4) for K and Eq. (1.7) for k, we take scalar products of the two equations f with F(K) and F (k') respectively, and subtract to find (< k') yF'''(K'),F(<) = 0, (2.5)

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38 noting that we are using a real-type scalar product, Eq. (1.26). We conclude that biorthogonality holds for F and F under a u-weighted scalar product, while in view of Eq. (2.3) this is equivalent to biorthogonality of F and F with unit weighting. Nonmultip lying Media: Spectrum of the Slowing-Down Transport Operator In fast nonmultip lying media the WBE is given by Eq. (1.12). The corresponding adjoint WBE, Eq. (2.4), is found to be (a u<)F'''(K;E,y) = fl dy' dE' E^(E,y ^ E^u")F^(K;E^y') (2.6) where the different energy limits for the adjoint Volterra scattering operator are to be noted. We have seen in Chapter I that for the absorption-only case (a y<)F(E,y;ic) = (2.7) the spectrum is the domain C in which (a \ik) vanishes. The singular eigenfunctions were F(E,y;K) = F''"(K;E,y) = X(K)6(a yK) (2.8) where the second identity occurs since Eq. (2.7), the streaming equation, is self-adjoint. Thus in the limit of no scattering, the eigenfunctions of the fast WBE tend to the deltafunction form, Eq. (2.8).

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39 We observe from Eqs. (1.7) and (2.6) that the domain C, due to the streaming operator, also is contained in the continuous spectrum for the slowing-down WBE; we now show that in fact it is identically the spectrum since the scattering operator will cause no additional contribution to the spectrum. To demonstrate this we show that all k (6 C are in the resolvent set, which is the complement of the spectriim, and is defined as those values of k for which (A k) has a bounded inverse. Therefore we consider the existence of solutions to the equation (A K)4> = s(E,y). (2.9) We examine first the case of isotropic scattering, for which the scattering operator becomes ^1 /<Â» n Â»00 J du' J dE' i:g(E',y' ^ E,ii)= J dM' J dE' ^ E^(E^ ^ E). (2.10) -IE -IE Using this operator we may write for Eq. (2.9) the equivalent equation (a yK)(j)(E,y) rl dy ' J dE^ I Z^(E' -V E)(})(E^y') = S(E,y). (2.11) For values of k g C we may divide by (a ]s<) and integrate over y:

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40 I (}>(E, y)dy E (|)(E) -1 _dy_ a-yK , E dE2 ^s^^' E)(|)(E') dU S(E,y) a-yK (2.12) Since k j2 C, both integrals over y exist and Eq. (2.10) is of the form (()(E) = f(E) dE' 2 ^s^^' ^ E)(J)(E') + g(E) (2.13) or 1 f(E) dE' ^ E (E'-E) 2 s ())(E) = g(E). (2.14) Provided that the scattering kernel is bounded, the Neumann series inverse *(E) = I n=o .00 f(E) dE' -;r E (E' ^ E)' ^ s n g(E) (2.15) always exists [58]. Thus (A k) has a bounded inverse, and we have

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41 the result that the complement of C is not in the spectrum of A. An identical argument applies to the adjoint operator. We can extend this result to anisotropic scattering by making a P expansion of the scattering kernel and cj); the procedure of Eq. (2.12) then results in a set of coupled Volterra equations which must be inverted. Thus for rather general scattering kernels, i.e., those which can be developed in a finite bounded P^^ expansion, we have the result that the spectrum of the wave Boltzmann operator consists only of the continuum C. Forward and Adjoint Slowing-Down Eigenfunctions Since the point spectrum for the slowing-down problem is empty, there will be no regular eigenfunctions and corresponding spaceand E,y-separable eigenmodes. To obtain the singular eigenfunctions corresponding to the continuous spectrum k 6 C, we may apply the technique of Chapter I. Adding A(K)(a yK)6(a yic) to the right-hand side of Eq. (1.12) and dividing by (a yK) we find 1 f^ ^Â°Â° -1 E dE' E CE'.y' ^ E,y)F(E,y;K) + + A(K)6(a yK), k 6 C (2.16)

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42 or equivalently. aUK .00 dy' dE' Z^(E%y' ^ E,u) F(E,y;K) = A(K)6(a mk) . (2.17) At this point it is necessary to proceed more formally. It has been observed in Chapter I that the integral of the factor (a yK) over E,y exists, since it is a pole. We would like to extend the Neumann series inverse, which we used in Eq. (2.15), to Eq. (2.17). Accordingly we write F(E,y;K) = A(k) I n=o a-y< (Â•1 'Â°Â° dy^ dE' Z (E',y' -^ E,y) s n 6 (a UK) 1 = X(K) \$ Â• 6(a yK) Fsp(E,y;K) K e c (2.18) as the forward slowing-down eigenfunction. The formal "Case's Method" derivation of Eq. (2.16) must be verified for specific scattering kernel models by means of more careful arguments such as those used in substantiating Eq. (1.32) [31,33,34]; it appears that this will succeed for "well-behaved" scattering kernel models. For k in the continuous spectrum of A the inverse of the operator (A k) exists but is singular [35], so it is with some justification that we write the second form of

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43 Eq. (2.18), defining the formal inverse scattering operator \$. Further, the Neumann series expansion has an interesting physical interpretation in terms of familiar iterated collision integrals. To see this we first recall that the zero scattering cross-section eigenfunction of Eqs . (1.21) and (2.8) represent neutrons streaming with eigen-energy E and direction y ; this delta-function distribution is also the n = term of F(E,y;K). The second term is Ff'^(E,y;<) .-l^E^(E^.y^^E,y) E < E^ = E > E (2.19) K which may be interpreted as the distribution resulting from one downscattering interaction, multiplied by the transport factor (a \ik) which is peaked at E and y . Similarly, higher terms in the expansion may be interpreted as the result of n down-scattering interactions, so that the entire eigenfunction may be regarded as the result of excitation by neutron waves streaming with E and y , along with an associated down-scattered contribution excited by the streaming portion. The eigenfunction is nonzero only for E and below, since only downscattering can occur. (This deduction from Eq. (2.18) is valid whether the Neumann series converges for E < E or not.) We see that the eigenfunction singularity consists of a delta-function contribution and a pole contribution at E ; a similar structure occurs in the thermal continuum eigenfunction, Eq. (1.32). Also we note that in the iterated

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44 integrals each singularity is smoothed by integration, and that the unintegrated pole can be factored out from each term of the series, so that we suspect that the Neumann series inverse will indeed converge for rather general classes of scattering kernels. The adjoint eigenfunctions may be obtained by an identical procedure; Eq. (2.6) leads to t f V F (k;E,u) = A'(k) I n=o a-yK du' ^n dE' E (E,y E%y^)6 (a UK) = ^"''(k) 3^* 6 (a UK) K e C (2.20) or F^(k;E,u) e fJ^(k;E,u) (2.21) where A (k) is an arbitrary complex constant. The form of the adjoint t Volterra operator requires that F is identically zero for E < E ; again the deltafunction and scattering-associated term with pole singularity at E = E occur. The properties of the forward and adjoint eigenfunctions may be summarized by the rearranged expressions

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F(E,u;k) = A(K) 6 (a y<) 1 " fl f K Z fE-,y' * E,p) 1" Â•1 E x23CE^,y,-E',y') E < E Â— K "*"(!<:) [fi ?*"(<;Â£, y) = A'Ck) I6(a yK) + 1 fE E fE.y ^ E^y') 1 1 r (^ (^ S^^'^ n y< 1 E K X E^CE'.y' -> E^,y^) E > E Â— K 45 = E > E (2.22) E < E (2.23)

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46 Discussion of the Slowing-Down Eigenfunctions An interpretation of the forward eigenfunctions in terms of iterated collision integrals excited by monoenergetic unidirectional (i.e. y = u ) streaming neutrons already has been given. We proceed by considering their biorthogonality properties. In general, due to the condition expressed by Eq. (2.5), forward and adjoint eigenfunction pairs corresponding to different eigenvalues are orthogonal under a y-weighed scalar product. For the same eigenvalue <, Eqs. (2.22) and (2.23) clearly show that the scalar product will not vanish, due to the coincident delta-functions. (This product of deltafunctions of two variables requires careful interpretation in terms of the theory of generalized analytic functions or some other approach; for an introduction to the literature on this aspect of the singular eigenfunction technique see McCormick and Kul'cer [29].) The biorthogonality properties of the slowing-down eigenfunctions are illustrated schematically in Figure 2.1 in terms of the energy variable. The eigenfunctions must be orthogonal for overlapping energy-distributions as well as in the trivial case when the distributions are nonoverlapping in energy. It is interesting to consider the expansion of a monoenergetic source in slowing-down eigenfunctions. This is schematically represented in Figure 2.2. We see from the first two sketches that such a source will excite not only continuum modes having the eigenenergy E , but also will excite to some extent all modes with lower eigen-energies. As is apparent in the third sketch, continuum modes with higher eigenenergies will not be excited.

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47 i N fUk') K E^' E K f-K i E/c' E^ Figure 2.1 Orthogonality of forward and adjoint slowing-down eigenfunctions.

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48 F^/c) F{k) 1 F'{k) Figure 2.2 Excitation of slowing-down eigenfunctions by a monoenergetic source.

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49 An analysis of the static slowing-down transport equation has been performed by Maclnerney [59] for constant cross sections and elastic scattering, in the lethargy variable. By performing a lethargy Laplace transform he reduces the lethargy-dependent problem to one-speed transport form. For the transformed problem (for slowing down in hydrogen) both discrete and continuous spectra arise, as is usual in the one-speed problem (see standard works such as Case and Zweifel [25]). However due to inversion of the lethargy transform, the discrete modal contribution fails to give a space-separable solution for the isotropic space and lethargy Green's function (i.e. a source 6(u)6(x)). This is in accord with our result that a monoenergetic source excites a continuous distribution of eigenfunctions. Maclnerney tentatively attributes his continuum eigenfunctions to streaming first-flight source neutrons; confirmation of this, and further correlations between his work and the present "exact" method must await more detailed investigation. The existence of a discrete mode in the lethargy-transformed problem raises an interesting point with respect to implementation of the continuum singular eigenfunctions. A dispersion function, associated with both discrete and continuum modes, was seen to arise naturally in the treatment of the thermal problem. We may associate the dispersion function with inversion of the Fredholm thermal ization operator, since in the slowing-down case only the Volterra operator is present, and no such dispersion function appears. Physically we distinguish between energyregeneration which can occur through upscatter in the former instance and energy degradation in the latter. In the presence

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50 of energyregenerative mechanisms we find. the potential for establishment of E,y-space-separable modes (for moderate frequencies and absorptions) with attenuation length longer than the neutron mean free path. For the slowingdown problem, with such mechanisms absent we have ? = Re K = ^ (2.24) K SO that all modes are attenuated precisely as are the streaming-waves with which we associate them. However it is well known that the neutrons themselves (e.g. for neutron pulses) are not attenuated in this manner, even though no separable mode of propagation exists. Evidently, therefore, we are not to regard a continuum mode as observable or capable of being excited individually, since the neutrons which would constitute such a wave certainly would not be attenuated according to the streaming mean free path. This is further evidenced by the fact that a monoenergetic unidirectional source excites modes having lower eigen-energies as well. Apparently the identification of an individual mode with streaming and associated scattered neutrons must be applied with some caution, although it is clear that actual streaming source neutrons are represented by the deltafunction term of the appropriate eigenfunction. We must conclude that the spatially persistent nonseparable neutron population (as opposed to uncollided neutrons) excited by a delta-function source is represented by constructively interfering continuum eigenf unctions, where this constructive interference is due both to the distributed

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51 part of the eigenfunction excited by the streaming, and to eigenfunctions of lower eigenenergies. Evidently the discrete mode in Maclnemey's transformed problem corresponds to this constructively interfering modal contribution. It should be noted that the idea of interference of neutron waves is not new, having been postulated as early as 1964 on the basis of diffusion theory by Perez et al . [60] to explain phenomena observed in wave experiments in subcritical assemblies. More recently, in the transport treatment of polycrystalline materials by Yamagishi [44], interference effects have been seen to arise from interaction of a continuiim contribution, due to neutrons with energies below the Bragg cutoff, with the higher energy neutron population. In the present fast nonmultiplying problem we have seen that modal interference is necessary to describe neutron wave propagation in all but purely absorbing materials. In the scime context it is interesting to consider elastic scattering from very heavy nuclei. In this case the energy loss per collision is sufficiently small that wave propagation in such a medium is essentially monochromatic. Thus monoenergetic analyses may be performed such as, for example, those of Ohanian et al . [61] and Paiano and Paiano [62] . In this case, due to the energy-sustaining model of the collision process, space-angle-separable monoenergetic eigenmodes occur which are less attenuated than Z . We realize that in the actual energydependent problem an energy loss does occur with each scattering interaction, so that only continuum modes are present; nevertheless these

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52 continuum modes must superimpose in such a way as to yield the almostseparable wave behavior. The macroscopically elastic scattering kernel model of the above discussion may be written E (E^y^ ^ E,u) = E (E')KCy' -> y)6(E E') . (2.25) This kernel also is noteworthy because it is not bounded. Clearly our discussion of bounded scattering kernels in establishing the resolvent set, K jS C, does not apply and we find spectral contributions do arise for < )S C. The model of Eq. (2.25) is discussed in Appendix C, along with several limiting procedures which may be used to attempt to derive the strictly monoenergetic case as the limit of the almost-monoenergetic case. We conclude the discussion here by observing that another way of viewing the problem of elastic scattering from heavy nuclei is to consider a detector with an energy window AE wide enough to detect all elastically scattered neutrons; one should then obtain experimental results which are in accordance with monoenergetic theory. That is, the detector response should show an asymptotic exponential signal decay corresponding to the momoenergetic fundamental mode; this detector response is the physical equivalent of solving for the zeroth moment of the flux rather than the flux itself. In this instance we must agree with Doming and Thurber [40] who remark in another context that in attempting to correlate theory and experiment one can be mislead by

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53 considering only the asymptotic behavior of flux solutions rather than their moments . Realistic Cross Sections: Nonmonotonic vE In Chapter I analysis was restricted to total cross sections such that vZ is monotonic. This was done because the continuum values of < and all possible E,vi pairs are in one-to-one correspondence for monotonic vZ , a requirement of the generalized analytic function treatment upon which we rely for completeness results in the separable kernel case. Here we explore briefly the consequences of relaxing the monotonicity condition. For this case degeneracy of the continuum results. From Eq. (1.19) it is apparent that 8(<) will assume the same value more than once when vZ is not monotonic. This is illustrated in Figure 2.3, where it is evident that for the same value of 6, but different energies, nondegenerate, singly degenerate, and doubly degenerate regions occur. Higher degeneracies may result from more rapidly oscillating cross sections. We exclude the case of constant vE , which must be treated separately. When the continuum is degenerate the coefficient (a yK) in the forward and adjoint eigenvalue equations becomes zero for more than one E,u pair at each degenerate < point. Thus in Eq. (2.16) and the corresponding adjoint expression we may make the replacement M X(K)6(a yK) -> y A (k)6 (a yK) ^ ^ '^ m m m=l M y X (K)6(E E )6(y y ). (2.26) '^, m "^ ^ Km-^ ^ Km m=l

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54 Figure 2.3 Degeneracy of the continuum due to nonmonotonic vl .

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55 Thus from Eqs. (2.18) and (2.20) we have M F(E,y;K) = I A^(K) \$-6^(a yK) (2.27) m=l and , M , , F"^(<;E,y) = I A^(hc) \$^ 6^(a \xk) (2.28) m=l for an M-degenerate k. Clearly since there are M arbitrary A's, M linearly independent eigenfunctions can be constructed. An obvious choice is to set the A's equal to zero for all but one 6 ; we define the M eigenfunctions ^SD.m^^'^'^^ = ^m^^^ ^ Â• ^m^^ " ^^^ (2-29) and fI^ ^(K;E,y) = xIm J5^ Â• 6 (a y<) (2.30) bL),m m m which we notice are biorthogonal when the forward eigen-energy is less than the adjoint eigen-energy but are not necessarily biorthogonal otherwise. Also we see from Eq. (2.22) that the forward eigenfunctions will have pole singularities at all eigen-energies E less than the isTl

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56 delta-function eigen-energy E Â• A similar structure occurs in the ^ "' Km adj oint eigenf unctions . Fast Multiplying Media: Zero Scattering Cross Section The fast multiplying medium problem best may be approached by first considering Eq. (1.13) with the scattering operator absent. Since the fission interaction kernel is separable, Eq. (1.13) then becomes identical in form to the thermal WBE with separable kernel, Eq. (1.24), which was discussed in detail in the first chapter. Identifying x with M(E) and vZ-(E') with Z (E^), we may write down immediately the results for the nonscattering fast multiplying WBE from Eqs. (1.29), (1.30) and (1.32). Thus we find that the discrete eigenvalues are given by the dispersion law A(K,oa) I vE., Â— ^ f a-yK = (2.31) and the corresponding regular eigenf unctions are F(E,y;K.) = X(K.) Â— ^^ J K ^ C, (2.32) The singular continuum eigenf unctions are F(E,y;K) = A(K) X f K a-yK X 6 (a-yK) K e C (2.33)

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57 Adjoint eigenfunctions, which we obtain for later comparison, are readily found to be F (K ;E,y) , vE (E) A (K.) Â— ^ ^ j' a-yK. K /g C (2.34) and F'^(K;E,y) = A^(k) vZ^(E) a-UK ix(E,) A + 6 (a-yK) K e c (2.35) where we have used the definition of x(E) from Eq. (1.10). We note that the same dispersion function occurs in both forward and adjoint expressions . By analogy with the thermal problem we expect a symmetric pair of eigenvalues for moderate frequencies and absorption. We further expect that the set of eigenfunctions of Eqs. (2.32) and (2.33) will have full and half-range completeness properties (although strictly speaking these properties were demonstrated for a symmetrized kernel in the thermal case; a similar symmetrization transformation could be performed in the fast case) . Discrete Eigenfunctions for Fast Multiplying Media We now turn to solution of the fast WBE with downscattering, as represented by Eq. (1.13). For k jg C, we may divide by (a \xk) and

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58 invert the identity minus the scattering operator (under the conditions which were discussed previously) to obtain F(E,U;k) = 3 X a-yK -00 dy^ dE' vZ^(E')FCE',y';K) (2.36) -1 using the inverse operator defined in Eq. (2.18). Taking the scalar product of this equation with vZ_, we find that the condition for solutions to exist is that A(k,(jo) = 1 vZ-, \$ Â• -^^ f a-yK (2.37) which defines the dispersion function and the dispersion law for the discrete eigenvalues k. . The expression for the regular eigenf unctions then is F(E,y;K.) = A(K.) 3 Â• Â—^^ 3 J a-yic. K ^ C, (2.38) This expression may be compared with Eq. (2.32); making use of the Neumann series interpretation of \$, we see that the presence of downscattering in the problem has resulted in an addition of all iterated collision integrals of the nonscattering eigenfunction (cf. Eq. (2.22)). Thus the discrete eigenfunction consists of the fission spectrum.

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59 weighted by the transport factor (a uk)~^ (which is peaked at y = Â± l but not singular, for K C) , and smeared down in energy by similarly weighted scattering operators. We will discuss the regular eigenfunctions and the dispersion law in more detail in Chapter IV. The corresponding adjoint eigenfunctions similarly are found to be t ft ^^ff^^ F CK.;E,y) = A^Ck.) \$^ . j^ . (2.39) It is readily verified that the same dispersion law is obtained here as for the forward problem. Continuum Eigenfunctions for Fast Multiplying Media By application to Eq. (1.13) of the arguments used in arriving at Eq. (2.18), we find F(E,y;K) = \$ a-yK j -1 dydE^ vZ^(E')F(E',y^;K) + + A(k) \$ Â• 6 (a-yK) < e C (2.40) which in view of Eq. (2.18) may be written F(E,y;K) = \$ . -X_ f^j^^^f) + A(K)Fgp(E,y;K) . (2.41)

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60 Eliminating the scalar product term results in the expression F(E,U;k) = A(K) X_ ^"^f'^SD^ a-VK + F SD K e c (2.42) for the forward continuum singular eigenfimction, where A is defined by Eq. (2.37). The adjoint continuum eigenfunction is f'''(k;E,u) = a'^(k) vE, a-yK A + F SD K e c (2.43) t _ t where in this case F = \$ Â• 6 (a yK) . Discussion of the Continuum Eigenfunctions The eigenfunctions represented by Eq. (2.42) have an interesting interpretation much in the same manner as that of Eq. (2.22), and with similar reservations applicable. Making use of Eq. (2.37) for the dispersion function and expanding the inverse in a power series, Eq. (2.42) may be written F(E,y;K) A(K) X a-yic 1 + f a-yK f a-yK ^^f'^SD F5p(E,y;<) (2.44)

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61 Now let us regard (vE ,F ) as the initial excitation of the fission contribution to the mode. The resulting fission neutrons, after being smeared in energy by scattering, have the energy and angular distribution \$ Â• Â— ^ . The form of the dispersion function expansion suggests a-yK that it be interpreted as a modal multiplication due to the sum over all generations of fission neutrons . Thus we see that the continuum mode again may be regarded as streaming-associated, since it consists of two terms which we interpret as follows. The second term is F which has been seen to be the downscattered distribution associated with streaming neutrons having the eigen-energy and eigen-angle. The first term of Eq. (2.44) then may be interpreted as the fission-produced modal flux distribution due to excitation in turn by the scattered term. This attractive exegesis must be tempered, as in the slowing-down case, by considering the scalar product of the adjoint eigenf unctions, Eq. (2.43), with a monoenergetic source function. We observe first that the source will excite modes with lower eigen-energy, due to the term F . In addition, modes having eigen-energies both above and below the ^^ \)T. t f source energy will be excited due to the fission term \$ Â• Â— -Â— . "Â•^ a-yK These results may be compared qualitatively with solutions for the static fast multiplying medium transport problem obtained by Nicolaenko and Zweifel [63] and Nicolaenko [14]. Energy-transform techniques were used to treat fission and elastic scattering with constant cross sections in the former study. Inelastic scattering, the model for which already has been the subject of comment here, was added in the latter. Although detailed comparison again is difficult due to the complex structure of the continuum eigenf unctions, we find consistencies between their Green's

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62 fimction results and the present work. Specifically, in both studies. Green's function solutions are found to contain both space-separable contributions (which we ascribe to the discrete eigenmode) and nonseparable "slowing-down transients," which are solutions to the slowingdown equation without the fission term, and which were found to be necessary to achieve completeness for the eigenfunctions of the appropriate Boltzmann equation. The correlation with our results is apparent. Degeneracy of the Continuum Should further complications seem desirable at this point, consideration may be given to the effect of degeneracies in the continuum upon the above treatment of continuum eigenfunctions. Since the details are straightforward, we simply note that linearly independent sets of eigenfunctions can be obtained; in particular a set corresponding to those of Eqs. (2.29) and (2.30) may be derived by an identical procedure. The eigenfunctions are given by Eqs. (2.42) and (2.43) with t t the substitution of F_^ and F for F and F . SD,m SD,m bu oU The Boltzmann Equation with Isotropic Interaction Finally, some general consequences of isotropy in the WEE operators will be derived for use in Chapters III and IV. For the isotropic kernel we write K(E',u' ^ E,y) = K(E' ^ E) (2.45)

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so that the WBE for k C may be written F(E,y;K) = a-yK J fÂ°Â° dy' dE' K(E' ^ E)F(E",y-';K:) 1 o 63 a-ytc J KCE' -V E)F(E';ic)dE' (2.46) with the definition F(E;k) 1 F(E,y;K)dy (2.47) Integrating Eq. (2.46) over y we obtain F(E;k) = f(E) Â•* N 1 1-. "^ K(E -y E)F(E ;K)dE (2.48) with f (E) defined as f(E) E dy a-yK -1 (2.49)

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64 Upon solution of Eq. (2.48) we then reconstruct the angular flux from F(E,U;k) = f_I|lF(E;K) (2.50) For K e C the continuum eigenfunction equation becomes fl fCO F(E,u;k) = a-y< J -1 du' dE' K(E' ^ E)F(E',y';K) + X(K)6(a-yK). (2.51) Performing integrations over y we have F(E;k) = f(E) K(E' ^ E)F(E';K)dE' + X(k)6(E-E ) 1^ K e c, (2.52)

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CHAPTER III APPLICATION TO THE TRANSFER MATRIX METHOD Introduction In this chapter the analytic results obtained for the WBE will be applied to the transfer matrix formalism of Aronson and Yarmush [64] and Aronson [65-70], making it available in a continuousenergy transport representation. There are two aspects of this technique which make it attractive as a potential method for numerical applications of transport theory. First, it provides a convenient general framework for "problem solving" in terms of certain basic operators (see Aronson [67] for a number of examples). Second, it provides an explicit method for obtaining transmission and reflection operators. As we will see, constructing some of the required operator inverses will be equivalent to determining the halfrange orthogonality properties of the WBE eigenfunctions. Although it is not possible in general to do this analytically, numerical inversion techniques certainly may be employed, so that the transfer matrix formalism provides a straightforward approach to this difficult aspect of finite medium problems. The transfer matrix for slab geometry and its associated eigenvalue problem are derived in Appendix A. Essentially what one must do is find the spectrum and eigenfunctions of a certain operator, aS, in whatever representation the problem is formulated. From these 65

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66 eigenf unctions all the relevant transfer matrix operators may be constructed, as well as transmission and reflection operators. Here we will obtain the a, 6, and 08 operators for energydependent wave transport with an arbitrary interaction kernel, and then show that for isotropic scattering the eigenfunctions of a6 may be expressed in terms of WBE eigenfunctions.* Formal Operator Relationships The operator relationships required to construct the transfer matrix H for a slab of width x may be summarized as follows: H = Se Â™ S ^ (3.1) where

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67 -tA e +Tr (3.3) and r is diagonal. To obtain the operators B^, C^ and ? one must first diagonalize an auxiliary operator o6: X'^ a6 X = r^ C3.4) n2 where T is diagonal; then B^ = X Â± 6 X r"-^ (3.5) and c^ = x"^ Â± r"^ x"^ a (3.6) The explicit wave transport representation of these formal relationships will be developed in the following sections. The Operators a6, X, and X" The operators a and 6 are defined as the sum and difference, respectively, of the operators a and 3, which were found in Appendix A to be

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Then their product is 69 05 =^-^ (k" K") (a K"" K") C3.13) in terms of an arbitrary interaction operator. This expression is considerably more simple when K = K , since the awkward middle term of Eq. (3.13) vanishes. In particular this occurs when all interaction processes are isotropic; we will assume this to be the case throughout the rest of this chapter. We then obtain ;;7 _ a 2a ~ dE' dp' ^ 6 (E'-E) 6 (y^-y) ^K^^CE^ E) ly y (3.14) which is the form we will consider here. We now wish to obtain the operator X and its inverse X which will diagonalize o6 as in Eq. (3.4). This may be done by first finding the spectrum and eigenfunctions of the operator a6 , and then constructing X; X will be constructed in a similar way from eigenfunctions of the adjoint operator. The validity of the diagonalization will of course

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70 require that the sets of eigenfunctions are complete. We write the eigenvalue equation for 06 as oS ' X(E,u;Y^) = Y^X(E,y;Y^). (3.15) Now 06 is an integral operator over E and y, and we will find that in general it will have an area continuous spectrum as well as a possible discrete spectrum. To clarify the correspondence between the eigen2 function X(E,y;Y ) and the operator X, let us consider for a moment the simpler case which occurs when 06 is an ordinary N x N matrix (as it is, in fact, for the raultigroup diffusion representation) . Then its spec2 ~2 trum consists of the N discrete eigenvalues y.; F is the diagonal 2 array of the y., and X is the corresponding matrix made up of columns of -2 eigenvectors, X.. X(E.;y.). The matrix X is then a transformation ij 1 J from the basis generated by the eigenvectors corresponding to the indi2 ~-l vidual Y-> to the discrete-energy space; similarly, X is the inverse trans format i on . In the present transport case, the situation is entirely analogous, but the summations over the discrete spectriom must be supplemented by an 2 integral over the continuum values of Y , and summation over the E. is ~2 replaced by integrals over E and y. Thus T is the diagonal operator consisting of both the discrete eigenvalues of a& (if any) and the con2 tinuous spectrum, y . The operator X is made up of "columns" of eigen2 2 functions X(E,y;Y ) with Y as the "index"; it will involve both an integral over the continuum and a possible sum over discrete contributions.

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71 In other words, for the continuum, X(E,ij;y ) is the kernel of an integral 2 operator over all continuum values of y . Assuming that the set of eigenfunctions of a& is complete, X may be 2 regarded as a transformation from the basis y to the basis E,u, while the operator X" is the inverse transformation. Writing X formally as dy X(E,y;Y ) Â• (3.16) (the integral is understood to include the sum over the discrete spectrum, if any) and X as x-1 fl dE du x"^(Y^;E,y) (3.17) the left and right inverse relations become X ^X = I = dE dy X"^(Y'^E,y) J dY^ X(E,y;Y^) 2 2 2 dy 6(y "Y^ ) (3.18)

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and 72 XX" "^ I = dY^ X(E',y';Y^) Y ,.00 dE dy x'-^(Y^E,u) fl dE du 6 (E-E') 6 (y-y") (3.19) 2 ^2 2 where 6 (y -y' ) is either a Dirac or Kroneker delta for y in the continuous or discrete spectrum, respectively. The first of these expressions is a biorthogonality relationship for the two functions 2 -12 X(E,y;Y ) and X (Y ;E,y), while the second is a closure requirement over y e (0,1] (although we will see later that this closure relation is essentially a full-range condition) . Spectrum and Eigenfunctions of o6 Using the explicit expression for a6, Eq. (3.14), we may write the eigenvalue equation, Eq. (3.15), as Y ^y ?.]

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73 to the domain C which we defined in connection with the WBE; that is, when Y 6 C it is also in the continuous spectrum of 06. Further, we now may apply the singular eigenfunction technique to obtain an expression for the continuum X eigenfunctions. Noting that f ^

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74 X(E;y ) = Â£(E) dE' K^pCE^ ^ E)X(E-;y^) * X(y)6(E-E^), y Â€ C BE' (3.24) X(E;y ) = f(E) dE' Kgg(E' ^ E)X(E';Y^), y^C (3.25) where Â£(E) is the same function defined in Eq. (2.49). But these equations are the eigenvalue equations for the WBE with an isotropic kernel. Thus we have the following results: (a) The spectrum of the isotropic a6 operator is identical to the spectrum of the related Boltzmann operator. (b) The yintegrated eigenfunctions of the two operators are identical. That is, identifying Y with k. so cÂ°Â° fl dE' o o X(E;y ) = F(E;k) dy' Kgg(E' ^ E)X(E',y';Y ) (3.26) dE' Kg^(E' E)X(E^Y ) fCO dE^ K3g(EE)F(E';k)

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75 ,00 dE ^ f'^^ Se (E' -y E)F(E^y^K) (3.27) Using this identity in Eqs. (3.22) and (3.23) and comparing with Eqs. (2.46) and (2.52) for eigenf unctions of the WBE, we find that for an isotropic KÂ„_ BE X(E,y;< ) = F(E,y;K) + F(E,y;-<) (3.28) The Inverse Operator X :;-! To obtain the inverse operator X" analytically the most obvious approach is to construct it from the eigenfunctions of the operator adjoint to o6; it then should have the required biorthogonality properties of Eq. (3.18). Referring to Eq. (3.19) we see that the appropriate scalar product is

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Defining the function 76 (3,31) and substituting in Eq. (3.30), a 2 v+r 2 Â„ . 2a X (k ;E,y) = -^ y dE' dy' Kgg(E -^ E')x'''(K^E',y') (3.32) where ^^^(E -Â» E') is the kernel of the adjoint WBE considered in Chapter II. By duplicating the development of Eq. (3.20) we have immediately and X'^(K^E) = F^(k;E) x"''(K^;E,y) = F'''(K;E,y) + F"'"(-K;E,y) (3.33) (3.34) making use of the eigenf unctions of the adjoint WBE from Chapter II. Then X"-^(K^;E,y) is given by Eq. (3.31). Diagonalization Operators These results now may be used to obtain expressions for the operators B^ and C^ which diagonalize the transfer matrix itself (Eqs. (3.1) (3.3)). From Eqs. (3.5) and (3.6)

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77 ?;-! B^=XÂ±6xr"EXÂ±^ (3.35) and C^ = X ^ Â± f"-^ X ^ a E x'-^ Â± ^ ^ (3.36) Since F is diagonal, elements of K may be written C(E,y;K) = ^ fl dE' dy[^Ci6(E--E)6(y^-y) o i hE^^' ^^ X(E',y;0 y< Y/-C 2Â— X(E,y;K ) (3.37) making use of Eqs . (3.12) and (3.20). Then B^(E,y;<) = ^X(E,y;K^) (3.38) For isotropic KÂ„Â„ we note that oc (a-y<)F(E,y;K) = (a+y<)F(E,y;-K) (3.39)

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so that from Eq. (3.28) 78 B+(E,y;K) = 2F(E,y;Â±<). (3.40) Similarly ^ is t^ = [ dK^-4 6(<^-K) dE o o dyx"-^(K"^;E,y) Â•00 dE^ J ; o o 1 dy'-^^p6(E'-E)6(y^-y). y (3.41) so that and r"^(K;E,y) = ;^x"^(<^E,y) C+(K;E,y) y , , -Â„t ^ 2y Â„t -^ (aÂ±yK)X = Â± Â— ^ F aK (Â±<;E,y) (3.42) (3.43) making use of Eqs . (3.31) and (3.34), and Eq. (3.39) which also applies to F . Full-Range Orthogonality and Completeness At this point B^ and C^ are still determined only to within a normalization function of k. The normalization relationship may be obtained by considering the operators S and S of Eq. (3.2). Substitution of Eqs. (3.38) and (3.43) for B^ and C^ into the left inverse expressions S S = I yields the appropriate expression, which, with a little manipulation, is also found (as for the monoenergetic case [68] )

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79 to be a whole-range biorthogonality relation for the WBE eigenf unctions F and F . Similarly, the right inverse equation (which in view of Eq. (3.1) expresses explicitly the physical requirement that as the slab width T is decreased to zero, the transfer matrix must reduce to the identity) SS" = I is found to be a whole-range closure expression for + F and F . Thus we see that the validity of the diagonalization in Eqs. (3.1) (3.3) depends on the full-range biorthogonality and completeness of the WBE eigenfunctions, properties which may be established in the context of the WBE itself. Half-Range Orthogonality and Completeness The entire transfer matrix H now has been obtained without explicit application of any half-range conditions. (It was not necessary to solve Eqs. (3.18) and (3.19) for the inverse of X.) However, we recall that the transfer matrix relates the angular flux over ye[-l,l] at one slab surface to the corresponding flux at the other. It is apparent that use of the full transfer matrix is completely equivalent to applying full-range boundary conditions at one interface to an infinite-medium eigenfunction expansion of the flux, which, of course, completely determines the flux elsewhere within the slab, and in particular at the opposite surface. In situations where only incident fluxes are known at interfaces, the transmission and reflection operators T and R are more appropriate to the problem. It is in constructing these operators that half-range conditions will appear. The transfer matrix formalism provides expressions for T and R in terms of the diagonalization operators B^ and C^ (Eqs. (A. 63) through

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80 (A. 66)), all of which are defined for y 6 (0,1]. However, inverses of the operators B^ and C^ occur explicitly, and must be evaluated for the interval (0,1] if one is to compute T and R. Thus, for B , we have the inverse relationships, using Eq. (3.40), 'Â°Â° f 1 dE dy B;;^(K;E,y)F(E,u;Â±K') = 6(k'-k) (3.44) o and 2 2 dK F(E,y;Â±K) B;^(<;E",y') = 6 (E'-E) 6 (y"-y) . (3.45) Solving the singular integral equation, Eq. (3.44), for B^ is equivalent to obtaining the weight function W(E,y) and normalization + N(k) for half-range biorthogonality of F and F , since if they are complete on the half-range we identify 2B;^(<;E,y) = N(K)W(E,y)F'''(Â±ic;E,y) (3.46) so that Eq. (3.45) becomes a closure relation for the WBE eigenf unctions. It was pointed out previously that for most realistic interaction models analytic solutions of Eqs . (3.44) and (3.45) are not likely to be readily forthcoming, and consequently the transmission and reflection operators must be constructed without analytic expressions for the required inverses. (In this respect the invariant imbedding method is an alternative, as it provides a completely different formulation for

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81 R and T. See Pfeiffer and Shapiro [71] for a review of the various approaches to transmission and reflection, and Mockel [54] for an example of application of both transfer matrix and invarient imbedding methods to thermal neutron wave propagation.) Nevertheless, supposing that some approximate numerical representation has been found for the WBE eigenfunctions, one can always invert the diagonalization operators numerically, which we have seen to be exactly equivalent to numerically determining half-range normalization of the WBE eigenfunctions. After the numerical inversion is performed, R and T also may be calculated numerically. Application to Fast Neutron Wave Propagation We have seen that for isotropic interactions all the essential operators required to apply the transfer matrix technique may be constructed from solutions of the WBE eigenfunction problem. Further, requirements for halfand full-range closure and biorthogonality of these eigenfunctions are implicit in the formalism. These results and in particular Eqs. (3.28), (3.34), (3.40), and (3.43) extend Aronson's static monoenergetic work [65-70], and are quite general with respect to the energy-dependence of the kernel. It is interesting to note that relationships between transfer matrix and Boltzmann equation eigenfunctions, analogous to these equations, have been obtained for arbitrary anisotropic scattering and for azimuthally dependent problems in the monoenergetic case [69]. While there is no new physics per se in the transfer matrix approach beyond that contained in the WBE, and thus one expects a direct interrelationship

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82 between the two eigenvalue problems [68], it remains to be seen whether the simple form of Eqs. (3.28) and (3.34) will hold for energydependent formulations with anisotropic scattering. It should be stressed that since no stipulations other than scattering isotropy have been made with respect to the interaction kernel in this chapter, the results obtained here apply equally to fast and thermal regions, with or without fission, etc. Furthermore, the form of the term a(E,(jo), which we have taken to be Z + Â— , does not enter explicitly and therefore generalization is immediate to finite transverse dimensions through introduction of a transverse buckling. Of course the static case w = is included. Thus the equivalence of the continuous energy transfer matrix and WBE eigenfunction approaches for isotropic interactions is quite apparent, so that we may regard the transfer matrix method as a possibly convenient framework for application of the WBE analysis. In this sense we now have in the results of this chapter a complete treatment of the analytic basis of the isotropic scattering energy-dependent transfer matrix. Application to fast neutron wave and pulse propagation thus is a matter of finding suitable means of implementing the formal results obtained in Chapter II, using the basic relationships developed here.

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CHAPTER IV APPLICATION TO DISPERSION LAW AND DISCRETE EIGENFUNCTION CALCULATIONS Introduction Expressions for the spectrum and eigenfunctions of the fast multiplying WBE were derived in Chapter II. Emphasis there was placed on application of the singular eigenfunction technique to obtain formal expressions for the continuum eigenfunctions. In this chapter we will consider the mathematically more straightforward topic of the discrete spectrum and regular eigenfunctions of the wave transport operator. This subject now is quite thoroughly understood in principle for the separable fission kernel model; Travelli [16] has presented an essentially complete transport multigroup numerical treatment of the fast wave slab-geometry eigenvalue problem which takes into account scattering anisotropy, through a P^ expansion, and delayed neutrons. However, apart from Travelli 's work and the present investigation [72], this author is not aware of any other numerical fast neutron wave results which have been reported. It is surprising indeed, in view of the apparent timeliness of fast space-dependent kinetics studies, that use is not being made of tools such as these to investigate neutron disturbance propagation in detail. In this chapter an interrelationship between the dispersion function and regular eigenfunctions is made explicit, and some properties of the 83

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84 dispersion law are noted. An application of these relationships is made to the dispersion law for multiplying media with isotropic elastic slowing-down and inelastic scattering. Numerical results are presented for the case of one elastically scattering species. The Dispersion Function and Discrete Eigenfunctions The dispersion law concept in the context of neutron wave propagation originally was introduced by Moore [73] as a relationship between wave frequency and complex wave length; subsequently this concept was rather broadly generalized [8-10], as discussed in Chapter I. We have seen here, in an exact transport treatment of the fast neutron wave problem, how the dispersion function occurs in the structure of solutions in multiplying media, as indeed it does for all energyregenerative media, as a result of the presence of a Fredholm integral operator. We now will proceed to formulate the discrete mode eigenvalue problem for a separable fission kernel in a way which both appeals to intuition and suggests a method for computing the dispersion law. The dispersion functions for fast multiplying media with a separable kernel were given by Eqs . (2.31) and (2.37) for the cases when downscattering is absent and present respectively. These expressions may be summarized by A = 1 (vZ^, G) (4.1) where G(E,p;k) = \$ Â• -^ Â— (4.2)

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85 with \$ defined from Eqs . (2.17) and (2.18); when scattering is ignored 3=1. The operator \$ in Eq. (4.2) may be inverted and the equation multiplied by (a-yK) to give 1 fÂ°Â° (a-UK)G(E,y;K) | dy' dE"Z^(E^y'->E,y)G(E,y;K) = x Â• (4-3) -1 E Thus the function G(E,y;ic) is the solution to a pure slowing-down WBE with the normalized fission spectrum x as a source, corresponding itot -
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86 Algorithms for Evaluating the Discrete Spectrtim and Eigenfunctions Two different numerical approaches to the solution of the eigenvalue problem are suggested by the form of the WBE and Eqs . (4.1)(4.3). Travelli has employed both [16,30], having derived the techniques by means of computational considerations. The first method is a direct numerical solution of the P^^ multigroup representation of Eq. (1.13) for K ^ C; other than the requirement for complex arithmetic this approach is straightforward [30] . An alternative procedure is suggested by Eq. (4.3). In form it is a familiar slowing-down equation, for which solution techniques are well established. The function G(E,v;<) is obtained readily for a particular value of < by solution of Eq. (4.3); the dispersion function A may be evaluated by Eq. (4.1). Zeroes of A for a particular oo then may be found by application of a complex NewtonRaphson procedure. This procedure has the advantage of not requiring solution of a matrix eigenvalue problem, which can become prohibitively lengthy for the large numbers of energy groups needed to achieve accuracy in fast medium problems [16] . 3A The Newton-Raphson procedure can be expedited by computing rÂ— a K by the following scheme [73]. Differentiating Eq. (4.1) yields |^=-(Â«vll> t^-^) and from Eq. (4.3) , (a-yK) -^ du'l dE' E^(E',M'-^E,y) |^ = yG (4.S) -1

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87 which is a slowing-down equation with yG as a source. Solution of Eqs . (4.3) and (4.5) can be carried out in parallel. Extension to Degenerate Kernels Only a slight additional effort is required to formulate the eigenvalue problem for a degenerate Fredholm kernel (e.g., multiple fissioning species). Following our development for the separable kernel model we obtain instead of Eq. (2.36) M (m) f f <Â• X F(E.y;<) = J: 3 . 1Â— dyM dE' vl/'"JF(E'.y';K) m=l ^"^"^ i i * M ^G(m)^y(m)^ F) m=l (4.6) using obvious notation for an M-term degenerate fission kernel, Reducing this to the matrix equation M (y^"\ F) = I (yC"). G^Â™)) (yf'"^ F) m=l (4.7) in the usual way, we find the condition for existence of a solution is the dispersion law Det I . t(y(n), G^Â™^)] = A = (4.8) where the quantity in brackets is the M x M matrix having elements (yCn)^ G^m)^ ^ (n) y^"^ f ' a-yK J (4.9)

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88 It is interesting to note that the G "^ are solutions to M slowing-down problems like Eq. (4.3), each with the same scattering operator but with the source energy distribution x characteristic of the m species. Eq. (4.8) is more complicated than previous expressions for the dispersion function, but evidently we may retain our interpretation of A as a modal multiplication factor; we notice that all combinations of fission of the n species due to downscattered neutrons from the m species occur. Values of k which satisfy Eq. (4.8) may be determined by straightforward extension of the NewtonRaphson scheme discussed above. The eigenf unction F(E,y;K) then may be reconstructed by means of Eq. (4.6); the coefficients (y , F) are the elements of the eigenvectors of the matrix equation, Eq. (4.7), and the functions G^ "^ will have been evaluated in satisfying the dispersion law. Thus we have achieved a general extension of the separable kernel analysis to the discrete spectrum and eigenfunctions of the WBE with slowing-down and a degenerate fission (or thermal ization) kernel. As a postscript to the above discussion, we note that the entire procedure is identical in the case of the adjoint eigenfunctions, with the obvious transpose and interchange of x and "^^n , and with use of \$ rather than \$. Further, it is easy to show that (yf'^), G^'")) E (vi:/"\ gW) = (G'^Cn)^ ^(m)^^ ^^^^^^ where t t "^^f G = \$^ ' -^(4.11) ^ a-yK

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89 so that should one wish to construct forward and adjoint solutions simultaneously it is necessary to solve Eq. (4.8) only once. Isotropic Elastic and Inelastic Scattering The slowing-down equations encountered in the previous sections are solved readily by any of a number of methods available from the fast reactor literature; see, for example, the review of Okrent et al, [75]. To illustrate the method described above we will develop the expressions for a continuous slowing down [76,77] model, with the addition of a simple inelastic scattering model as well. For isotropic scattering, Eq. (4.3) may be written G(E;k) f(E) E fOO 1 2lg(E'^E)G(E";K)dE' = f(E)x (4.12) using Eq. (2.49) for f(E); cf. Eqs . (2.48) and (2.52). Differentiating with respect to k, we obtain ^!iri-f(B)fi.^CB-.B)faB= =Iftfll (^-3) 9k E which is the isotropic equivalent of Eq. (4.5). (The y-dependent eigenfunction can be constructed from the y-integrated form by using Eq. (2.50).) For isotropic inelastic scattering from M species and inelastic scattering fron N levels with a constant energy loss AE per interaction we have, in standard notation, the interaction operator

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90 E/a lOO

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91 is readily integrated niimerically, beginning with zero lethargy located above the maximum fission spectrum energy, e.g., 10 MeV. In Eq. (4.17) we have defined H(u;k) = "',( Â— . (4.18) T^ Illustrative Results: Dispersion Law and Eigenfunctions for Single Scattering Species This method of solution was applied to Eq. (4.17) and a related 8G equation for -rÂ— , for a one-term elastic scattering kernel. These expressions were implemented as explicit difference equations. The Newton-Raphson method employing Eqs . (4.1) and (4.4) was used to find zeroes of the dispersion function. All cross sections were taken to be constant. Note that with constant total cross section the boundary of the continuum does not change with frequency. Two computed dispersion laws are shown in Figure 4.1, illustrating the effect of different values for multiplication. Values of k and co are expressed in units of Z . Computations were based on scattering from a nuclide of mass 230. A Maxwellian fission distribution having an average neutron energy of 1.98 MeV was used. Here, as in the work of Travelli [16], only one discrete mode per frequency was found, in accordance with the no-scattering case (i.e., by analogy with the thermal separable kernel results). As would be expected the less strongly multiplying eigenfunctions are in general more rapidly attenuated. Eigenfunctions corresponding to individual points on the dispersion curves may be reconstructed from the computed functions H(u) using

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92 Figure 4.1 Dispersion laws for constant cross-section, elastic scattering model.

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93 Eq. (4.18). With arbitrary normalization constant A the isotropic eigenfunction F(u;<) is F(u;k) = XG(u;k) = AH(u;k) f(u) e'" . (4.19) Zero frequency eigenfunctions, which are real-valued, are shown in Figure 4.2, corresponding to the zero frequency points in Figure 4.1. The eigenfunctions are normalized to coincide with the fission spectrum X(u) at lower energies; the effect of down-scattering is noticeable at higher energies. Amplitudes and phases for the complex-valued eigenfunctions are shown in Figures 4.3 and 4.4. Amplitudes have been plotted with different normalization to separate the curves. These and subsequent data refer to the vZ. = 0.09 dispersion law. For relatively moderate frequencies, =Â— < 1 x 10 , there is little departure from the zero^t frequency energy distribution for energies above about 0.2 MeV. With frequency increasing through the midrange of the dispersion law it appears that neutrons of lower energies experience increasing difficulty keeping pace with the wave. At y~ = 4 x 10 , oscillations in the phase 't occur at about 0.5 MeV accompanied by some noticeable structure in the amplitude. As frequency increases beyond this value severe phase lagging occurs for lower energy neutrons; Figure 4.5 shows this effect, with phase and amplitude values superimposed. It is interesting that the amplitude maximum occurs in a region of rapidly varying phase. As frequency increases toward the continuum, the eigenfunction becomes increasingly peaked, as shown in Figure 4.6. This bunching of the

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94 ro I I 3

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95 > 2 o cr u z UJ ^< 0) O e O 03 Â•4-> O (U ex, to X oo 0) c c o o c 3 c 0) Â•H UJ to 3 oo Â•H

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96 > 2 >e) Â•H O c (U D cr u Â•H a.

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97 a, Â•H o 3 s -13 c > 10 > o q: z UJ c o o c 3 c (1) Â•H 3 cr 0) ^1 tn 3 W) Â•H a.

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98 > >-* o UJ z Ul C Â• H ,c o o a o c O (J -H C Â•!-> 3 -H (H U C O i> bO (U Â•H X U5 +J vO 3 Â•H

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99 neutrons in energy occurs at an energy value corresponding to the maximum of (a-K) , so that we begin to see the onset of continuumlike behavior. The phase corresponding to Figure 4.6 is not plotted; the eigenfunction exhibits lagging of phase (which will result in rapid oscillations of phase relationships over small ranges of energy) even more severe than that shown in Figure 4.5, beginning at about 0.8 MeV and extending to lower energies. Peaking of the eigenfunction at the energy which maximizes (a-ic)" can be understood by noting the ydependent form of the eigenfunction, Eq. (4.2). Recalling that (a-yK)~ has a pole for y = 1 at the edge of the continuum, it is apparent that the discrete eigenfunction will become more nearly monoenergetic and forward-directed as cj approaches the critical frequency, due to the presence of this factor (note that it is also a coefficient of each of the collision integrals in \$) . Evidently, then, as o) approaches the critical frequency it will become increasingly difficult to excite a discrete mode without strongly exciting continuum modes as well, in view of the lack of phase coherence of the eigenfunction in its maximum amplitude region. Recalling previous remarks on the apparent interference of continuum eigenfunctions to produce almost-separable mode behavior, it is probable that the continuum contribution to the propagated wave will be important in this case as well. It would be interesting if a high-frequency wave were to assume asymptotically a sharp energy distribution as in Figure 4.6; this does not, however, seem likely. Further investigation of this question will require methods for determining superimposed contributions of both discrete and continuum modes.

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CHAPTER V SUMMARY AND CONCLUSIONS Summary This dissertation has explored the transport formulation of the fast neutron wave and pulse propagation problem in slab geometry. Both multiplying and nonmultiplying media were considered. The spectrum of the wave Boltzmann operator and formal expressions for discrete and continuum eigenfunctions were obtained. It was found that while the presence of the scattering operator prevents direct application to the fast case of the analytic techniques which have been used for the thermal regime, a consistent extension of the thermal eigenfunction results is found which takes scattering into consideration. In this extension an operator appears which arises from inversion of the identity minus the scattering operator; the inverse operator has a Neumann series expansion for eigenvalues in the discrete spectrum. This expansion represents familiar iterated collision intervals, each integral weighted by a transport factor (a-yK)~ . The same operator inverse appears in the continuum eigenfunction expressions; while the situation there regarding the existence of the Neumann series inverse is more tentative, apparently there is physical and some mathematical justification for interpreting the inverse operator in this way. All eigenfunction expressions reduce to forms consistent with more well-established thermal 100

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101 results in the limit of vanishing scattering cross section. Qualitative correspondence was noted with the few transport studies dealing with the static transport equation in fast multiplying and nonmultiplying media. A basis for application of the wave transport results to finite regions was established by generalizing the transfer matrix formalism to energy-dependent isotropic interaction models. The complete equivalence of the isotropic Boltzmann equation and transport transfer matrix eigenfunction problems was demonstrated, and specific simple relationships between the two sets of eigenfunctions were obtained. These results are more general than the scope of the present fast wave investigation; since the specific form of the interaction kernel does not enter into the expressions, they apply equally well to fast or thermal regimes. Finally, disperrion law and discrete eigenfunction expressions were considered in detail for separable and degenerate fission kernel models, an efficient algorithm for fundamental mode computation was presented, and specific expressions were developed for an isotropic elastic scattering and inelastic scattering model. Eigenfunctions were shown to consist of a superposition of solutions to wave transport slowingdown equations, each having the fission spectrum for one fissionable species as a source. Eigenvalues were found from the dispersion law, which was seen to be in the form of a modal criticality condition. Illustrative calculations were performed for a one-term isotropic elastic scattering, separable fission kernel model. The discrete eigenfunctions were found to have an energy distribution very similar to the fission spectrum at low frequencies, with small phase variation over the total energy range. At frequencies approaching the critical frequency, the

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102 eigenfunctions exhibited strong peaking in energy and in the forward direction, accompanied by rapid phase variation with energy in the vicinity of the peak. Conclusions and Suggestions for Future Work The theory of fast neutron wave and pulse propagation has received almost no attention, and the application of transport theory to the fast neutron regime has received little. While the difficulty of performing analysis in the fast regime is acknowledged, nevertheless this situation is both surprising and unfortunate. In this dissertation an attempt has been made to obtain specific results where possible, and to establish directions which future investigations should take. Theoretically, the situation with respect to the discrete spectrum and eigenfunctions appears to be well in hand, at least for moderate frequencies. This is not to say that interesting work does not remain; as noted before, very little numerical work exists on wave and pulse propagation in fast multiplying media, and evidently there is no experimental data available. The qualitative effects of material composition, crosssection characteristics and other parameters upon fundamental mode propagation have yet to be explored. In particular it should be ascertained whether traditional propagation experiments are sensitive to pareimeters which may be of interest in fast reactor kinetics and safety analysis. The implications of the peaked energy and direction exhibited by the discrete eigenfunctions for very high frequencies should be developed. It is possible that this behavior is simply an indication that discrete modes are not excited strongly, relative to the continuum modes, as frequencies approach the critical frequency. On

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103 the other hand if waves should be found to exhibit such behavior for deep penetration this would be an interesting result; this subject could be further explored theoretically, and studied experimentally by means of energy-sensitive detectors. To investigate high frequency waves in multiplying media, propagation in finite regions, and, of course, propagation in nonmultiplying media it will be necessary to take into account the continuum eigenfunctions in implementing an eigenfunction expansion approach (or the equivalent transfer matrix formalism) . Whether such an approach can succeed in practice is a question which must be addressed by further investigations. Either of two generic methods which suggest themselves may be employed in treating the continuum; one may attempt to extend analytic techniques, such as the generalized analytic function method, to the fast case, or one may search for some approximation to the continuum eigenfunctions which discretizes the continuum and which is amenable to niimerical implementation. The former suggestion involves significant extrapolation of present mathematical techniques, and may well prove too complicated in realistic applications, though valuable for phenomonological understanding through treatment of modeled interactions. The latter suggestion has, to the best knowledge of this author, no precedent. These comments are not intended as a criticism of the singular eigenfunction approach as applied in particular to the fast neutron regime, since the method is far from fully developed for thermal analysis. Indeed, it is not yet clear whether the singular eigenfunction method will prove useful as a framework for numerical application to practical problems [29,50] in the thermal regime although it is of

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104 course valuable both as a source of exact solutions against which to check other methods and as a means of understanding transport effects. It is apparent from the forms of the continuum eigenfunctions for both fast multiplying and nonmultiplying cases that it is necessary to understand the slowing-down continuum eigenfunctions in order to understand all continuum phenomena. These eigenfunctions are important not only as the appropriate ones for the important problem of propagation in passive media, but also since they evidently are associated with the modal excitation mechanism for the continuum for multiplying media. In this connection it should be remarked that the interpretation which was given for continuum modes can be viewed in rather general terms by recalling that the "Case's method" eigenfunction expression, e.g., Eq. (2.16), looks very much like an inhoraogeneous Boltzmann equation with a delta-function source j interpretation of the singular eigenfunctions in terms of modal excitation by such a streaming source therefore may be illusory. On the other hand, in the absorption-only case it is obvious that this interpretation is correct. Whether this is an accurate or helpful approach to understanding the continuum is a matter which can be resolved only by detailed investigation, just as the formal eigenfunction expressions themselves remain to be substantiated, either theoretically or by application and comparison with other methods. Since slowing-down eigenfunctions appear in all continuum expressions, a logical starting point for investigation of continuum phenomena would be to explore the slowing-down problem for a simple elastic scattering or comparable kernel. Results of such a study would be not only of theoretical interest but also helpful in interpreting pulse propagation experiments in heavy scattering materials [11-13].

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105 Mention should be made of the possibility of extending the present analysis to systems having finite transverse dimensions, and to other geometries. The essential analytic novelty presented here is the inversion of the scattering operator; applying this technique to other transport problems (again, see McCormick and Kuscer [29]) should be straightforward. Also it should be repeated that the methods presented here are applicable to thermalization problems in which separable or degenerate kernels are employed.

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APPENDICES

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APPENDIX A Introduction In Chapter III various transfer matrix operators in slab geometry are expressed in terms of WBE eigenfunctions, and the basic equivalence between the eigenfunction expansion and transfer matrix techniques is emphasized. Actually, the transfer matrix approach is a family of methods which may be applied to fairly arbitrary geometries and to physical quantities other than the neutron flux. In this appendix the transfer matrix will be derived from basic principles and its general properties will be discussed, essentially following Aronson and Yarmush [64] and Aronson [65] . It will be found necessary to introduce the geometry explicitly in order to progress beyond fundamental considerations. The most general expression for the basic wave-transport transfer matrix operators in slab geometry then will be obtained. General Formalism The transfer matrix approach to neutron propagation differs from the more common Boltzmann equation technique in that the latter is concerned with a pointwise description of the transport process within a medium, while the transfer matrix method considers the problem of transport through a medium in terms of incident and emergent fluxes at the boundaries. In this respect it greatly resembles the invariant imbedding technique, and in fact there is considerable common ground between the 107

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108 two [71]. Initially we consider a region oÂ£ space occupied by a homogeneous isotropic medium through which a neutron flux may propagate. (Here the flux will be the usual energy and direct iondependent "angular flux" in either time-dependent or Fouriertransformed representation. Obviously, more restricted models also may be treated.) All that is required at this point is that one be able to assign two "sides" to the region. These sides will be arbitrarily referred to as "left" and "right." No problem arises with the formalism in the treatment of voids and reentrant configurations, provided that all relevant surfaces of the region under consideration are assigned to either the left or right sides. In some cases, as, for example, for finite bodies, designation of the two "sides" may be entirely arbitrary, but for such bodies, once boundaries are assigned the transfer matrix is defined unambiguously. Since only slab geometry will be considered in detail here, this aspect of the method will not be pursued; it will be assumed that appropriate sides may be designated for all regions of interest [78] . The transfer matrix may be defined as the linear operator which relates entering and emerging fluxes at one side of a region to the corresponding fluxes at the other side. This situation is represented schematically in Figure A.l. Here, for example, (f) (L) = (j) (r ,E,\$^,t), where r is the coordinate of a point on the left surface of the region; motion in the positive sense is taken to be from left to right. Thus (fi (L) and
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109 H.(L) <^_(L) <^+(R) <^JR) Figure A.l Entering and emerging fluxes for a single region. <^+(L) i>JL) f^(R) <^_(R) Figure A. 2 Entering and emerging fluxes for adjacent regions.

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110 j^ 9 j^ Â• n > =^ Q 3 '^ ' n < => s emerging entering where n is the outward-directed unit vector normal to the surface at any point. The transfer matrix equation* for Figure A.l may be written " <}>^(R) "

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Ill region, it is necessary that the region be well defined by its bounding surfaces. These topological nuances will not be of interest here; it is sufficient to note that Eq. (A.l) certainly will be applicable to a singly-connected region totally bounded by its "right" and "left" sides. Algebra of the H-Matrix Consider two adjacent regions, 1 and 2, completely bounded by "left" and "middle," and "middle" and "right" sides, respectively. This situation is diagrammed in Figure A. 2. Applying Eq. (A.l) to regions 1 and 2 we may write (M) . (A. 3) Combining these two equations. (j)(R) = H2H^(()(L) . (A. 4) Evidently the transfer matrix for the combined regions 1 and 2, H , is given by the matrix operator equation H^2 = H2Â»l ^^-'^

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112 and similarly * H regions = H H , Â• * Â• HÂ„H, . (A. 6) n" nn-1 21 ^ ^ Obviously transfer matrices do not in general commute. Furthermore, it is apparent that for any region, in the limit of vanishing width the transfer matrix must tend to the identity operator: lim H = I. (A. 7) "width" -^ These characteristic properties must be satisfied by all transfer matrices [64] . Form of the H-Matrix: T and R Operators We will now express the transfer matrix in terms of linear operators which relate emerging fluxes to incident fluxes. This may be done without any further restrictions on the type of region considered. Figures A. 3 and A, 4 illustrate the situations relevant to the ideas of transmission and reflection, respectively. An incident flux (L) will give rise to an emergent flux ({) (R) which may be regarded as transmitted, and 4)_ (L) which may be regarded as reflected. In other words, any entrant flux at one side will produce an emergent flux distribution over the entire surface; the flux emerging from the same side is termed "reflected," and flux emerging through the other side is "transmitted." These distinctions apply even though the surface configuration may be such that the terms are not particularly descriptive of the physical

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113 <^+(L) <^+(R) Figure A. 3 Transmission. +(L) <^_(L) Figure A. 4 Reflection,

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114 processes. Transmission and reflection operators may be defined for (j) (L) incident, as in Figures A. 3 and A. 4, by the relations 4 (R) = T4 (L) (A. 8) (t)_(L) = R(()^(L) (A. 9) where T and R may be used to treat both attenuating and multiplying media. These operators are in fact Green's functions for the energy and direction-dependent flux, expressing outwarddirected fluxes at surface points of the region in terms of the "source" (\) (L) , which is completely arbitrary. That is, T = 7(^^,\,E^ -^ P^,n^^,E^^:t^-t) (A. 10) and R E R(^^,ti^,E^ ^ r'^,\$^'j^,E'j^;t'-t). (2.11) A corresponding pair of operators may be defined for an incident flux distribution at the right, (j) (R) : (J)_(L) = T*(i)_CR) (A. 12) (}>^(R) = R*(!)_(R) (A. 13) where the asterisks represent transmission and reflection in the sense of righttoleft. These operators in general will differ from their

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115 unstarred counterparts. Equations (A. 8), (A, 9), (A, 12) , and (A. 13) may be combined to give the emergent fluxes for arbitrary incident flux distributions over both faces : * (R) = T({) fL) + R*(t) (R) (()_CL) = R4)^(L) + T*(|._(R) (A. 14) (A. 15) But Eqs. (A. 12) and (A. 13) may be solved formally for the fluxes at the right face in terms of those at the left. (In the following it will be assumed that all required inverse exist; in applications, existence may impose restrictions on the problem. For example, subcriticality may be required.) ({)_(R) = -T*"^R(t)^(L) + T*"^(J)_(L) ^(R) = (T-R*T*"-^R)(|)^(L) + R*T*"-^()) (L) (A. 16) (A. 17) preserving the correct sequence in all operator manipulations. If the fluxes are now written in vector form and the operators combined into a matrix, we have
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116 which is precisely the transfer matrix equation, with the identification H = T R*T* ^R R*T*"-^ -T* R (A. 19) This is the most general form of the transfer matrix in terms of the four transmission and reflection operators [64] . Two-Region Transfer Matrix Equation (A. 5) now may be applied to the transfer matrices for two adjacent regions, H and H , as in Figure A. 2. The transfer matrix of the combined region, H Â„, is the matrix product of two matrices such as in Eq. (A. 19); the elements of the product matrix bear the same relationship to the combined region as the corresponding elements in Eq. (A. 19) bear to a single region. For example, we may compute the lower right-hand element 12 *_ 1 * *_ 1 *_ 1 *_i '-1 * *-l ^2^ri (A. 20) Inverting this expression. 12 ^2^ 1 * (A. 21)

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117 which gives the operator for transmission through both slabs from right to left, provided that the required inverse exists. Then, if * -1 (I R-jR-,) may be expanded in a Neumann series, we have 00 * * _ T = T y 12 'l ^ n=o ( h\ " T* . (A. 22) Physically, Eq. (A. 22) may be interpreted as in Figure A, 5. The total transmitted flux is the sum of contributions from flux which has suffered n pairs of internal reflections before exiting through the left face. This is a well-known result which has been obtained previously by "particle-counting" arguments analogous to the one illustrated here. Internal Sources Sources external to the region of interest are expressed in terms of the entering fluxes which they generate at the surfaces of the region, and therefore enter implicitly into the formalism through <^ (L) and <^_ (R) . Fission sources are linear functions of the flux and therefore may be included directly in the transmission and reflection operators. Thus the only type of source which would occur as an inhomogeneous contribution to the transfer matrix equation would be a flux-independent source imbedded in the region. Such internal sources may be taken into account by expressing the sources in terms of the emerging fluxes which they generate at the left and right faces, Q and Q , respectively, as

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118 Figure A. 5 Transmission through adjacent regions, Q_^ ^Q. Figure A. 6 Internal inhomogeneous sources.

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119 in Figure A. 6. Equations (A. 14) and (A. 15) may be modified to include the contributions of internal sources to the emerging fluxes: <}>^CR) = T(t)^(L) + R*(()_(R) + Q^ (f._(R) = R(j)^(L) + T*4)_(R) + Q_ (A. 23) (A. 24) These two equations may be combined to give equations in cj) (R) and (() (R) corresponding to Eqs. (A. 16) and (A. 17), but containing the source fluxes Q and Q_ ; the resulting equations similarly may be arranged into matrix form as (1)(R) = H
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120 X direction may be approximated arbitrarily well by layers of homogeneous sub-slabs.) It is assumed that all incident fluxes and all source distributions are (a) constant with respect to y and z (b) constant with respect to azimuthal angle about x. In this case the angular neutron flux becomes a function only of X, E, y, and t, where y is the usual cosine of angle with respect to the X axis. Values of y will be taken to range from to 1 for both (})^ and (}) . This designation is convenient as well as unambiguous, since the information regarding the "direction" of the fluxes, normally contained in the sign of y, is transferred to the subscripts + and -. For a homogeneous slab with these symmetries imposed T = T(y^,E^ ^ y-j^,E'j^;T;t'-t) = T*(yR,Ej^ -^ M\,E'^;r;t^-t) (A. 27) and R E R(yj^,E^ -y y'^,E'^;T;t'-t) = R*(\.Er "^ ^'r'^^r'^'^^"^^ (A. 28) where x is a slab width. Since 4) and (J) do not depend upon position on the surface, T, R, and H have spatial dependence only through slab width, as expressed in the above equations. Equality of the forward and backward operators is the result both of the symmetry of the physical situation and of the stipulation y 6 (0,1]. An immediate consequence of this is that

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121 H = H* (A. 29) and hence for two adjacent slabs, 1 and 2, of identical composition Â»12 = Â»12 = ("2"l^* = Â»l"2 = Â»1Â»2 t^-^O^ so that in this case "l2 = V2 = Vl = "21^^-''^ This equality also is obvious on physical grounds, since the composite slabs (1 + 2) and (2 + 1), and the total slab (12), all are indistinguishable with respect to neutron transport properties. The commutation property of Eq. (A. 31) is useful in determining the functional dependence of H upon slab width for homogeneous slabs, when width is regarded as a variable parameter. For slab width x + Ax, H(x + Ax) = H(x)H(Ax) (A. 32) independent of the manner in which Ax, which we now may regard as an arbitrary increment, is selected. The variation of H with respect to slab width then may be written HCx + Ax) H(x) = H(x)[H(Ax) I] (A. 33)

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122 so HCx + Ax) HCx) Ax " "'''-' HCAx) l" Ax (A. 34) We now follow the usual procedure of passing to the limit Ax -> 0. Define W = lim Ax > I HCAx) Ax (A. 35) That W exists is at least plausible, since the transfer matrix is required by Eq. (A.?) to have the property lim H(Ax) = I Ax -^ (A. 36) An explicit form for W will be obtained by use of Eq. (A. 35) In the limit Ax = 0, Eqs. (A. 34) and (A. 35) give 3H(x) 3x WH (A. 37) (note that W and H commute) which, with the initial condition H(0) = I yields the form H(x) = e xW (A. 38)

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123 for a homogeneous slab of width x. W is an operator, whereas slab width now appears only as a scalar multiplier. We now have two expressions for H. Equation (A. 19) becomes H = T RT"-^R RT"'^ t'^^r -1 (A. 39) taking into account the symmetry of the homogeneous slab geometry (Eqs. (A. 27) and (A. 28)). Equations (A. 38) and (A. 39) may be exploited to determine an explicit expression for W. Now the transmission and reflection operators must have a Taylor series development in terms of slab width x; in the limit x ->we should have to first order in x xa + Â• Â• ' (A. 40) R = xB + (A. 41) since T should tend to the identity operator and R should tend to the null operator as the slab width becomes vanishingly small. Once again, a and 3 are operators (which we must deduce from other considerations) , while x is a scalar. When these expression are inserted into Eq. (A. 39), the corresponding equation for H becomes, to first order in x. lim H = x ->xa x6 x6 I + xa (A. 42)

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124 Then from Eq. (A. 35) W a -6 3 -a (A. 43) It should be noted that W is dependent of slab width, since the operators a and 6 depend only on the properties of the slab medium. Thus through the introduction of slab geometry it is possible to reduce the transfer matrix from a form containing rather formidable Green's functions in all space and velocity variables to one in which the spatial coordinate enters only through slab thickness, and only as a scalar parameter. The Operators a and g The transfer matrix equation for a homogeneous slab extending from to X may be written compactly <^(x) = H(x)(})(0) (A. 44) Differentiating and making use of Eq. (A. 37), M^ = M^HO) = -WH(x)*(0) (A. 45)

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125 so that by use of Eq. (A. 44) ^^ = W4>(x) . (A. 46) This equation is especially interesting since it is a differential operator equation for the flux vector at the right surface only. While Eq. (A. 46) was obtained for a slab of width x, it applies equally well to the flux at a surface located at coordinate x, embedded within a slab of arbitrary width x, as in Figure A. 7. Thus Eq. (A. 46) is essentially a pair of coupled differential operator equations for the forward and backward-directed angular fluxes, (|) (x) and (^ (x) , at x within the slab. For example, the equation for the

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126 Figure A. 7 Fluxes at an internal coordinate surface.

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127 depends on rapid convergence of terms of the series (tW) . While this may be practical for some problems in which the full transfer matrix is required, especially for thin slabs, an alternative approach is to diagonalize W. Diagonalization of the Transfer Matrix Suppose a basis has been found upon which W is diagonal; that is, for some matrix S, S ^WS = A (A. 48) where A is the diagonal operator consisting of the spectrum of W. Then W = SAS"-^ (A. 49) and Â„ -TW -tSAS"'^ .. cr>^ H = e = e (A. 50) so that -tA -1 H = Se S . (A. 51) That is, if the spectrum and the eigenfunctions of W can be determined (the eigenfunctions constitute the matrix S) , the exponential behavior -tA of H with respect to x will enter only through e , which is itself diagonal. Rather than attempting to solve an eigenvalue equation for a 2 x 2 matrix of operators, it is more convenient to consider first an auxiliary eigenvalue problem. Define

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128 a = a + 6 6 E a e (A. 52) (A. 53) and their product operator, a6. Suppose that this operator may be diagonalized. r^ = x"-^ a6X , (A. 54) where T is the diagonal operator consisting of eigenvalues of o6 . Then it may be verified by substitution that, formally at least, the following equations result: A = -r (A. 55) -i (A. 56) ,-1 (A. 57)

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where 129 B^ = X Â± C (A. 58) and Explicitly, C^ = x"^ Â± C^ 5 E 6xr"-^ = a'hv (A. 59) (A. 60) W = SAS -1 1_ 4 and H = Se-"^S-^ = i 4 -r e-^^ e tF c c + (A. 61) C C + . (A. 62) These are the diagonalization operator equations which are discussed in detail in Chapter III. Transmission and Reflection Operators In most applications it will be necessary to compute transmission and reflection operators as well as (or in place of) the full transfer matrix H. In fact, the most utilitarian feature of the formalism, aside from its generality, may be that it provides relationships between transmission and reflection operators and eigenfunctions of the more familiar

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130 neutron transport equations. Expressions for these operators in terms of the diagonalization operators may be deduced by comparing the form of H as a function of T and R, Eq. (A. 39), with Eq. (A. 62). With a little manipulation one obtains T(T) = 4C:^ e""^^ E'^Ct) (A. 63) + + and defining and R(T) = E (t) E^^(t) CA.64) E+(T) = B^ + Be""^^ U e"^^ (A. 65) U = C C^^ = B^^ B (A. 66) which permit computation of T and R from operators related to the physical properties of the slab medium through o6 and the basic operators a and B . Wave Transport Form of a and B To obtain a and 3 in a wave transport representation, we return to Eq, (A. 47). Writing a and 3 as integral operators (recalling that T and R and thus, from Eq. (A. 40), a and 3 are defined over y 6 (0,1]) with kernels K and K^, Eq. (A. 47) becomes oc p

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131 a(()^(x,y,E,a)) "9r dE' dy' K^(E ,y' E,y;(jo)(j)^(x,y ,E ,03) rÂ°Â° fl dE' I dy" Kg(E ,y" ^ E,y;oa)(t)_(x,y",E ,0)) (A. 67) Now in similar notation the WBE is y -^ + a(E,a3) (})(x,y,E,a)) = dE' o -1 dy' Kgg(E',y'' -> E,y)(t)(x,y',E'',a)) (A. 68) including all interaction terms in the kernel KÂ„Â„. Rearranging, DC If = J (|) + ^ J dE' j dy' Kgg(E',y' ^ E,y)(l)(x,y',E',aj) (A. 69) When y is restricted to the interval (0,1], (y) . Furthermore, we have defined (y) E (j)(-y). If the integral in Eq. (A. 69) is divided into separate integrals over positive and negative y', and if the substitution y" = -y' is made in the negative integral. 3(j)^(x,y,E,(i)) Â— ({) + Â—

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132 where y, y^, y" 6 (0,1]. Comparing Eqs. (A. 67) and (A. 70), we find that a = J o dEJ dy' pL^ 6(E'-E)6(y"-y) \ Kgg(E^y' E, y) and (A. 71) 3 S dE' dy' yl^BE^^ ',-y' ^ E,y)l (A. 72) A similar treatment of the equation for ^ (x) gives identical results. The kernel K may, of course, contain any useful interaction model. Be This extends Aronson's work [64-70] to the most general wave transport representation of the one-dimensional slab transfer matrix.

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APPENDIX B Singularity of Inelastic Scattering Kernel Models In this appendix an inelastic scattering kernel model [79,80] will be discussed which gives rise to an additional spectral contribution in the adjoint problem. This observation is due to Nicolaenko [14], who made use of the kernel in treating fast static transport problems. We recall that in general the total isotropic scattering cross section is defined by 13(E) = E (E->-E')dE' . (B.l) s^ For downscattering (either elastic or inelastic) this becomes E^(E) = Z^(E-E')dE' . CB.2) We may write the kernel in question as K.^CE-^EO = g(E) h(E') E > E' =0 E < E" CB.3) For constant total inelastic scattering cross section, which Nikolaenko stipulates, we may write 133

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134 Z . (E) = const. SI = E SI g(E) H(E')dE(B.4) so that the constant cross-section condition requires that g(E) h(E')dE' = 1 or J" (E )dE = g(E) (B.5) If h(E) is bounded as E^-0, g(E) -^0 in this limit. Thus while the forward inelastic scattering operator dE' E . (E'-^E) SI Z^. h(E) I dE' gCE^) E (B.6) has a bounded kernel for E > 0, the adjoint inelastic scattering operator .E dE" Z . (E^E') Â• = Z . g(E) 51 SI "^ dE h(E') (B.7) clearly does not, as g(E) is singular. Thus our inversion of the identity plus scattering operator fails for the adjoint WBE employing this model. In fact operator (B.7) has eigenfunctions [14] g(E) with corresponding eigenvalues y foi" ^H ^ with ReX2.1; hence the spectra of forward and adjoint Boltzmann operators evidently are different, and the entire approach based on construction of biorthogonal eigenfunction sets is unsuccessful in this case.

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135 The question therefore becomes, is this unfortunate property a characteristic of inelastic scattering in general, or is it an artifact introduced by the specific model, Eq. (B.3)? Fortunately, the latter is true. From Eqs. (B.4) and (B.5) it is evident that the stipulation of constant total cross section at E^, together with the form of the kernel (B.3), forces g(E) to be singular. That is, it forces a divergent differential cross section, a result which is both inconvenient, as we have seen, and physically incorrect, since inelastic scattering does not occur at all for sufficiently low energies. (Nicolaenko avoids the difficulty of nonphysical results due to this divergence, as he does not apply the kernel below the inelastic scattering threshold; nevertheless the singularity does emerge, as we have seen, in the adjoint problem.) As we are dealing with fast systems we are in fact indifferent as to the precise form the inelastic scattering crosssection model at thermal energies and below, but we certainly are at liberty to allow the inelastic scattering cross section 2^^(E) to vanish as E^, for example, in some neighborhood of E = 0. This additional degree of freedom entirely resolves the present difficulty; retaining the kernel (B.3), we see that 2Â„. (E) gCE) SI (B.8) so that for E ^ all that is required is that E^^(E) vanish sufficiently rapidly. It is apparent that the singularity of the kernel adjoint to (B.3) is not a property which generalizes to more accurate models of inelastic

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136 scattering, and therefore the behavior of scattering kernels at the zero-energy limit should not necessarily hinder the development of transport theory for fast systems.

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APPENDIX C MACROSCOPICALLY ELASTIC SCATTERING: THE ELASTIC CONTINUUM It has been shown in Chapter II that for "well-behaved" slowing-down operators the point spectrum of the wave Boltzmann operator is empty and that K ^ C belongs to the resolvent set. Here we discuss two limiting processes for scattering models which express concretely some of the ideas presented in connection with almost-separable mode behavior, and which give rise to continuous spectra for k ? C. Duderstadt [32,33] discusses the effect of introducing the macros copically elastic scattering operator of Eq. (2,25) upon the spectrum of the wave Boltzmann operator. Such a scattering operator arises in the theory of thermal neutron Bragg scattering in polycrystalline materials. In Chapter II its use in modeling elastic scattering from heavy nuclides was discussed briefly. To pursue the ideas presented there, consider for simplicity an isotropic scattering kernel for the slowing-down WBE having macroscopically elastic and nonelastic terms Z^(E'->E) = SggCE') a(E',E.AE) + E^^^(E'^E) (C.l) where 137

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138 a(E',E,AE) = -^ , E' AE < E < E= otherwise (C.2) and E (E'->-E) is a bounded kernel. Repeating the resolvent set discussion for < ^ C, Eqs. (2 .9)(2 .15) , with Eq. (C.l) as the kernel, we find that the complement of C again is the resolvent set for any nonzero AE, since both kernels are bounded. However in the limit AE = Eq. (C.l) becomes the singular kernel of Eq. (2.25); once more repeating the resolvent set investigation we obtain for Eq. (2.13) KE) = f(E)iE^g(E) (j)(E) + f(E) j j^sne^^'^^^ *(EOdEg(E) (C.3) Defining AeCE) 1 jfCE) l3g(E) (C.4) Eq. (C.3) becomes (E')dE' + g(E) , K g C . (C.5) For values of k (entering through f(E)) and E such that A vanishes, Eq. (C.5) will not be invertible and k will be in the spectriom of the wave Boltzmann operator [32,33]. For a particular E, Eq. (C.4) is the dispersion law function for the point spectrum of the monoenergetic

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139 wave transport equation; as E varies, values of k which satisfy the dispersion law will sweep out a line, the "elastic continuum" [32,33], in the complex k plane exterior to the transport continuum C. For a monoenergetic source this additional spectral contribution evidently will give rise to modes which appear to be discrete modes having the source energy, in accordance with the discussion in Chapter II. This appearance of additional spectrum only in the limit AE = is not surprising from a mathematical point of view; however, physically it is evident that the role of the elastic spectral contribution must have been assumed for AE ;^ by superimposed continuum eigenfunctions for K e C. Thus we must be cautious in regarding continuum modes as more attenuated than discrete regular modes, since superimposed continuum modes clearly can exhibit highly persistent spatial behavior. Another simple limiting process can be employed to model the same phenomenon, making use of the elastic continuum. Define the kernel E (E'^E) = BE fE') 6(E'-E) + (1-6) E fE') K(E'^E) , s s s <_ 6 <_ 1 CC.6) so that B acts like a detector energy window, determining the fraction of neutrons which may be regarded as monoenergetic after one collision. The "elastic continuum" T corresponding to this model is shown schematically in Figure C.l. For a particular energy E the corresponding line in the transport continuum is denoted by < (E ) , while KqCEq) is the value of k which satisfies the elastic dispersion law Ag(E^) = 0. We note that for this model the apparent attenuation and wave length of

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140 Figure C.l Schematic diagram of the "elastic continuum" for macroscopically elastic scattering.

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141 the elastic mode depend on the value of 6; this may be seen by using the definition of f(E) Â£(E) = i ln(2^) (C.7) in the elastic dispersion function, Eq. (C.4) so that Ag = may be rearranged to give 1 _ 1 ^a+K, K = rr I ln( } 2 se ^a-K-^ = 1 8E^ ln(|il) . (C.8) For < not too near the continuum, so that variations in the logarithm e are slow, < varies almost linearly with 6, as shown in Figure C.l. Evidently, then, the observed modal propagation constants in an experiment would depend on the detector energy window.

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147 64. R. Aronson and D. L, Yarmush, J. Math. Phys ., 1_, 221 (1966). 65. R. Aronson, Nucl. Sci. Eng ., 27_, 271 (1967). 66. R. Aronson, "Some Applications of the Transfer Matrix Approach," Proc. Conf. Neutron Transport Theory, Virginia Polytechnic Institute, Blacksburg, ORO-3858-1 (1969). 67. R. Aronson, J. Math. Phys ., Ij^, 931 (1970). 68. R. Aronson, Transp. Theory and Stat. Phys ., 1_, 209 (1971). 69. R. Aronson, Nucl. Sci. Eng ., 5]^, 157 (1973). 70. G. Carroll and R. Aronson, Nucl. Sci. Eng ., 5J_, 166 (1973). 71. W. Pfeiffer and J. L. Shapiro, Nucl. Sci. Eng ., 38_, 253 (1969). 72. J. E. Swander and A. J. Meckel, Trans. Am. Nucl. Soc . , 14 , 865 (1971). 73. M. N. Moore, Nucl. Sci. Eng ., 2_1. 565 (1965). 74. F. Storrer and M. StiSvenart, "Contributions to the Theory of the Pulsed Neutron Technique Applied to Fast Multiplying Systems," Report EUR 593. e, European Atomic Energy Community (1964). 75. D. Okrent, W. B. Loewenstein, A. D. Rossin, A. B. Smith, B. A. Zolotar and J. M. Kallfelz, Nucl. Appl. Techno 1 ., 9, 454 (1970). 76. W. H. Stacey, Jr., "Advances in Neutron Continuous Slowing-Down Theory," Proc. National Topical Meeting on New Developments in Reactor Physics and Shielding, Kiamesha Lake, New York, CONF-720901 (1972). 77. F. E. Dunn and M. Becker, Nucl. Sci. Eng ., 47, 66 (1972). 78. M. Ribaric, Arch. Rati. Mech. Anal ., Â£, 381 (1961). 79. S. Yiftah, D. Okrent and P. Moldauer, Fast Reactor Cross-Sections , Pergamon Press, Inc., New York (1961). 80. M. Cadilhac and M. Pujol, J. Nucl. Energy, 21, 58 (1967).

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BIOGRAPHICAL SKETCH James Elza Swander was born in Youngstown, Ohio on December 3, 1939. He received the degree of A.B. from Earlham College, Richmond, Indiana, in June, 1961, with a major in chemistry. In September, 1961, he entered the University of California, Berkeley, for two years of graduate study in physics. He entered the Department of Nuclear Engineering Sciences of the University of Florida in September, 1963, and received the degree of Master of Science in Engineering in April, 1966. 148

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation _.f^r the degree of Doctor of Philosophy. lihran J. Ohanian Professor and Chairman of Nuclear Engineering Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissjext^tion for the degree of Doctor of Philosophy. Edward E. Carroll Professor of Nucl Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. R. B. Peri Research Staff Sdientist Oak Ridge National Laboratory I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. "^ Ralph G. Selfridge Professor of Computer Information Sciences \i

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I certify that I have read this study and that in luy opinion it copforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. j l : -W I 1>Alex E. Green Graduate Research Professor of Physics and Astronomy This dissertation was submitted to the Dean of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment, of the requirements for the degree of Doctor of Philosophy. June, 1974 Dean, College of^ Engineering Dean, Graduate School

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;'i i^, A ^^^ 4 2 4. 88Â«f