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Spatially dependent integral neutron transport theory for heterogeneous media using homogeneous Green's functions.

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Spatially dependent integral neutron transport theory for heterogeneous media using homogeneous Green's functions.
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Spatially dependent integral neutron transport theory for heterogeneous media using homogeneous Green's functions.
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Church, John Phillips,
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Approximation ( jstor )
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Geometry ( jstor )
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Neutrons ( jstor )
Plane geometry ( jstor )
Point sources ( jstor )
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SPATIALLY DEPENDENT INTEGRAL

NEUTRON TRANSPORT THEORY FOR

HETEROGENEOUS MEDIA

USING HOMOGENEOUS GREEN'S FUNCTIONS








By
JOHN PHILLIPS CHURCH


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
April, 1963































To my former teacher

Dr. Dudley E. South













ACKNOWLEDGMENTS


The author wishes to express his appreciation

to his chairman, Dr. G. R. Dalton, for his guidance and

assistance throughout the course of this investigation.

In addition, the author is indebted to the

staff of the Computing Center of the University of Florida

for their assistance, and to the Computing Center and the

Department of Nuclear Engineering for their combined

financial support of the computational work for this

dissertation.

The author has been ably assisted in the prepa-

ration of this dissertation by his wife, Nancy, who typed

the numerous drafts and the final manuscript; its final

form owes much to her concern. It is difficult to express

in words the author's appreciation of her faith in him.

She has made the path to his goal less difficult.














TABLE OF CONTENTS


ACKNOWLEDGMENTS . . . .

LIST OF ILLUSTRATIONS . . .

LIST OF TABLES . . . .
Y TI D~ T I


.N1 TR .W I A

CHAPTER I


CHAPTER II

I1.1



11.2


CHAPTER III

II1.1

111.2

111.3

CHAPTER IV

IV.1


IV.2

CHAPTER V

V.1

V.2


Page

S. iii

* vi

S. viii
1


. . . . # .

DEVELOPMENT OF THE HOMOGENEOUS
MEDIUM GREEN'S FUNCTION METHOD .

ANGULAR INTEGRATION . .

Expansion of Angular Dependence in
Spherical Harmonics and Zeroth
Harmonic Approximation ..

Advantages and Disadvantages of
Homogeneous Green's Function Method

SPEED DEPENDENCE . . .

General Speed Dependent Equation .

The Source Term .. .

One-Speed Equation . .

DIFFERENCE DENSITY METHOD .

Development of Difference Density
Equations . . . .

Equivalence of Equations for N and D

PLANE SLAB GEOMETRY . .

One-Dimensional Problem . .

Symmetry Considerations . .


* J.


S 4

S 15



S 15


S 26

28

S 28

S 30

S 34

S 36


*

.

.











CHAPTER VI


CHAPTER VII


VII. 1


VII.2

VII.3

CHAPTER VIII

VIII.1

VII.2

CHAPTER IX

IX.1



IX. 2


IX.3

IX.4


NOMENCLATURE

APPENDIX A

APPENDIX B

REFERENCES


COMPARISON OF N AND D METHODS
FOR SLAB GEOMETRY . . .

THE FIRST-FLIGHT GREEN'S FUNCTION
FOR A HOMOGENEOUS MEDIUM . .

Differential Equation and Boundary
Conditions . . .

Analytic Form of G(v;x x') ...

Monte Carlo Generation of G(v;xlx')

SOLUTION OF THE TRANSPORT EQUATION .

Spatial and Speed Dependence .

Iteration Techniques .. .

RESULTS AND CONCLUSIONS . .

Comparison of the Homogeneous Green's
Function Method with High Order P
and Sn Methods . . .

Monte Carlo Generation of the
Green's Function . . .

Iteration Convergence . .

Generation of the Green's Function
for the Multispeed Problem . .


. . . . *

DERIVATION OF Gf(v;xIx') .

SAMPLE PROBLEMS . .


4 0 0 0 0 . .


S 69

* 77

S 90

. 102

. 102

. 115

. 119


119


125

132


135

137

142

145

169


* 0



* 0














LIST OF ILLUSTRATIONS


Figure Title Page

1 The Spatial Dependence of the h Functions
of Equations (2), (7), and (8) . 13

2 Plane Geometry Transformation . 50

3 Symmetry of Neutron Flow'at the Boundaries
of Unit Cells in An Infinitely Repeating
Lattice . . . .. . 75

4 Slab Unit Cell with Mirror
Boundaries at x-+b . . .. 85

5 Mirror Symmetry of First-Flight Green's
Function About Plane at x-0 . . 97

6 Average Values of the Whole-Cell and
Half-Cell Green's Functions for
Problem 2, Appendix B . . 128

7 Average Value of the Whole-Cell and Half-
Cell Green's Functions for Problem 4,
Appendix B . . . . 129

8 Effect of Convergence Acceleration
Techniques Applied to Problem 2 of
Appendix B. Solution Iterated Until
Residuals Were Equal to 10-5 of the
Neutron Density at Each Spatial Point 133

9 Spatial Dependence of Scalar Neutron
Density for Gold-Graphite Cell of Problem 1. 148

10 Spatial Dependence of Scalar Neutron
Density for Fuel-Water Cell of Problem 2 .155

11 Advantage Factor for Fuel-Water Cell of
Problem 2 by Various Approximation Methods 156













12 Advantage Factor Calculated by HGI
Method vs. Number of Iterations for
Fuel-Water Cell of Problem 2 . . 158

13 Advantage Factors for Problem 3 by
Various Calculations . . . 161

14 Advantage Factor for Problem 4 by
Various Calculations . . .. 165

15 Spatial Dependence of Scalar Neutron
Density for Cases 1 and 2 of Problem 4 167

16 Spatial Dependence of Scalar Neutron
Density for Cases 3 and 4 of Problem 4 168


vii


Title


Page


Figure














LIST OF TABLES


Table
Number Title Page

1 Relative Effort for Methods of Solution
of End-Point and Midpoint Equations for
a Total of M Spatial Intervals . 109

2 Breakdown of Computation Time for
Problem 2, Appendix B .. .. 134

3 Parameters for Problem 1 . . 147

4 Parameters for Problem 2 . . 153

5 Parameters for Problem 3 .. . 159

6 Parameters for Problem 4 . . 163


viii














INTRODUCTION


Probably the most fundamental information needed

to describe static behavior of a nuclear reactor is the

speed and spatially dependent neutron collision rate,

that is, Z(r,v)N(r,v)v. Such information is needed in

order to predict long-term kinetic behavior of the reactor,

burn-up, power level, etc. Thus.one needs to know the

neutron density, N(r,v), over the entire speed spectrum

and space of the reactor.

Because of chemical binding and crystalline

effects, the scattering kernels are considerably different

for neutron speeds near thermal than for the higher speeds

implied when one speaks of epithermal and fast speed

ranges. In general, the absorption cross-sections of

reactor fuels also have a different functional dependence

on neutron speed at thermal speeds than at higher speeds.

Finally, because of the tremendous range of speeds in

reactor spectra, and because a critical reactor with no

independent sources is an eigenvalue problem if the entire

speed spectrum is considered, the problem of predicting

the neutron density is usually divided into several speed

ranges.











The present paper deals exclusively with the

prediction of the spatial and speed dependence of the

steady-state thermal neutron density in heterogeneous

media, although the basic method to be presented is

equally valid for any speed range.

There have, of course, been numerous approaches

to various aspects of the problem which can be very

loosely divided into two classes according to whether

the integro-differential or the integral form of the

transport equation was solved. The methods of solving

the integro-differential equation differ to a large extent

only in the choice of functions used to expand the angular

dependence of the terms in the equation. The expansion

functions include spherical harmonics (1-3), Tchebycheff

polynomials (4-7), and first order polynomials, that is,

trapezoids (8, 9).

The methods of solving the integral equation

have been a bit more varied, due perhaps to a more

intuitive derivation of the equation to be solved. As

Osborn (20) has pointed out, some approaches, particularly

as applied to thin absorbers (10, 11), have separated the

problem into two parts--first, a calculation of the

absorption rate in the absorbing media in terms of the

density at the surface of the absorber and second, a

density depression problem of evaluating the density











at the surface of the absorbing media in terms of the

density in an all-moderating media. Other integral

approaches (12-14), based on transport theory, depend

on the fact that the absorption rate depends only upon

the zeroth angular moment of the neutron density, the

solution for which is independent of higher angular

moments if scattering is isotropic.

The present paper describes the latter type

of approach. In brief, the method to be presented uses

a kernel for a homogeneous finite medium to formulate an

integral transport equation which predicts the speed and

spatially dependent neutron density in a heterogeneous,

finite unit cell.














CHAPTER I


DEVELOPMENT OF THE HOMOGENEOUS
MEDIUM GREEN'S FUNCTION METHOD

Assume that the neutron density in a general
three-dimensional medium satisfies the time independent

linear Boltzmann equation, and let N(r,vn)drdvd. denote

the steady-state angular neutron distribution, that is,

the number of neutrons in the volume element dr about r

having speeds in dv about v, and whose directions of

motion lie within the solid angle dn about n; N(r,vn)

will subsequently be called the angular density.

Define V(r,v) to be the probability per unit

time that a neutron of speed v will suffer a scattering

collision in dr about r with a nucleus which is in

motion with respect to the laboratory system. Further,

define y(r,v) to be the probability per unit time that

a neutron of speed v will be absorbed in dr about r by a

nucleus which is in motion with respect to the laboratory

system. Finally, define P(r,v-,v4*-nn) dv dgto be the

probability per unit time that a neutron of velocity v'r.'

will suffer a scattering collision in 4r about r with a

nucleus which is in motion with respect to the laboratory

system such that the neutron emerges from the collision











with speed in dv about v and direction of motion in dn

about n. Then let S(r,vr)drdvdn denote the source term,

that is, the number of neutrons per unit time being added

to the volume element dr about r, having speeds in dv

about v, and whose directions of motion lie within the

solid angle dn about n.

The transport equation which takes into

account the relative velocity between the neutrons

and collision nuclei which are in thermal motion can

then be written as


vn 7-N(r,vn + [V(r,v)+ A(r,v)] N(r,vr


S(r,vrn) + fdv*d' P(r,vN(v,-. N(r,v'rD. (1)

Equation (1) is an integro-differential equation; to

make the problem of its solution determinate it is

necessary to specify boundary conditions. The boundary

conditions follow from the physical interpretation of

N(r,vn). For example, at an interface between two media,

any packet of neutrons characterized by the vectors

r + Rn and vn will contain exactly the same number of

neutrons when it enters one medium as when it left the

other; in other words, N(r + Rn,vn) is a continuous











function of R for r + Rn at the interface. For a

complete discussion of this and other boundary conditions

applied in neutron transport theory, the reader should

consult Davison (2).

Consider the space of interest to be divided

into R regions, the medium of each region being assumed

homogeneous (the derivation of Eq. (1) assumes that the

medium is isotropic) so that the spatially dependent

probabilities V, ), and P in Eq. (1) are constant with

respect to r throughout each individual region.

Define a unit function, h, such that


h(p-s) 1 if p > s

= 0 ,if p s. (2)

For each interface between regions define an index k and

a position vector T such that -1k_ e JA. Then

h(rk-r) h(Ek-l-r) = 1, if Ek-r1 r Ek
(3)
= 0, if r e k-, or if r = rk

where r is on the line between rk and :k-l. One can

then use Eq. (3) to separate the r dependence of the

probabilities V, Y, and P in Eq. (1) in a special way.

In general, for any operator 0, one can write that









R
S(r,)- kl [h(Ek-r)-h(k-l-r)] (k,) (4)


Use Eq. (4) to rewrite Eq. (1) in the form

R
yvt* N(r,vn) + [ [h(rkE-r)-h(rk_,-r)]


S[v(k,v)+ (k,v)] N(r,vn)
R
8(r,vr) + 1 [h(rk-r)-h(!k-1-)]


dv'/ df' P(k,vI-v,n'-,O)N(r,v'n. ') (5)

Choose a particular region, say k', and remove
the term for the k' region from the summation on the LHS
of Eq. (5), meanwhile placing all other terms of that
summation on the RHS. Thus Eq. (5) becomes

vrL N(r,v*) + [h(Ek,-r)-h(rk'_-r1)

[V(k',v)+ /(k,v)] N(r,vn)

R
S(r,v) [(-r)-h( -r]
k+k'

[V(k,v)+ y(k,v)] N(r,vn)









R
+ 1 [h(Ek-r)-h(k_-1r)]
k-i

dv' dn P(k,vi-v,n'~-N(r,v'n') (6)

Now using the identity

h(p-s) I h(s-p) (7)

one can write

h(!k,-r) h(Ek, -r) [1 h(r-k,)] h(k,-_l-r)


-1 [h(r-r,.)+h(r,_-r)] (8)


Substituting Eq. (8) into Eq. (6) and putting the
[h(r-r,) + h(Ek'~,i-)] term on the RHS gives


va*VN(r,vm) + [V(k',v)+'(k',v)] N(r,vn)

SS(rvn) + [h(r-,)+h(r .,-r)

[V(k',v)+ '(k',v)] N(r,vn)


[ [h(rkr)-h( ~.-"r) [V(k,v)+ '(k,v)] N(r,vn)
k+k'









R
+ E [h(rki-r)-_h(!k-1L]
(9)
dv' dn' P(kv-'-,v,nt.) N(r,v'n')

Define a function gk' (r,vLr' ,vtn.') by requir-
ing that it satisfy the equation obtained by replacing
the RHS of Eq. (9) by the Dirac delta function product
6 (r-r') d (v-v') & (n-n'), subject to the same boundary
conditions as Eq. (9). Thus

TVgk T (rn r'Iv' 1)

+ [v(k',v)+ W/(k',v)] g' (r,vnlr' ,v'i')

g(r-r')&(v-v')s(n-n') (10)

The function gk (r,vnCr',v'.n') has a physical
significance; gk (r,vnr r',v'n') drdvdn is the number of
neutrons in the volume element dr about r with speeds
in dv about v and with directions of motion lying within
the solid angle dn about n due to a unit point source
at r' emitting one neutron per unit time in direction
n' with speed v' in a first-flight medium which has
the same properties as the medium in region k'.
It is important to understand fully the term
"first-flight medium" as used here. To be more explicit,
a first-flight medium is a medium in which all collisions










with nuclei result in removal of the neutron from the
population. Thus the removal probability for the Green's
function problem is equal to the total collision proba-
bility in medium k' of the real problem, and the neutrons
denoted by gkt(rvn l r*,v') are first-flight neutrons
in the sense that they have suffered no collisions
enroute from the source point to the field point. The
term first-flight neutrons will be used throughout the
following work.
The subscript k' will be dropped from

gk (r,vr'iE*,v't) in the following work, it being
understood that the function depends on the properties
of the particular region, k', chosen.
The function g(rvnjr',v'n') can be used as
a Green's function to convert Eq. (9) to the following
integral equation (see Chapter 7 of reference (16)).



N(r,vn) dr' dv' dn' g(r,vn r v'n) ,v'f)
rt v' n'

+ [h(r'-Ek)+hc(Ek.,-l-') [V(k',v')+ 2'(k',v')]N(r',v'n')


[h(!k-r' )-h(rkl-r')] [V(k' vt)+ Y'(k',v')] N(r' ,v')
k+k'










R
+ [h(Ek-r)-h(k-r1]


dv"d P(kvyrv',n"-)N(r'vlg) .


Since Eq. (10) describes the neutron density
in a medium in which all collisions result in removal
of the neutron from the population, the only way a neutron
which is emitted at r' may arrive at r is by direct
flights, that is, with no collisions enroute. It will
be assumed and later justified that events at the bounda-
ries do not change the speed of the neutron. Hence if a
neutron of speed v' is emitted at r' and arrives at r,
its speed must still be v'.
From this discussion it can be concluded that
the speed dependence of g(r,vfir',v'n') includes a delta
function in speed so that

g(r,vn r' ,v'') g(v;r,~ r' ,n.' )(v-v'). (12)

The product solution, Eq. (12), is a result of the fact
that since there can be no scattering in the medium,
then there is no interdependence of nr*n' and v-v'. Note
that g(v;r,- lr',') ) still has an implicit dependence on
v due to the v dependence of V(k',v) and ((k',v) in
Eq. (10).










Substituting Eq. (12) into Eq. (11) and carry-
ing out the integration over v' yields



N(rvn) dr d g(v;r,nl r,f' )S(r,vn')



+[h(r'-rk )+h(r, l-rt)] [V(k' v)+ Z(k',v) N(r',vn)



Sh(Ek-r)-h(-r'i )][V(k,v)+ (k,v)] N(r',vn')
k+k'

R
+ [h(rk-r')-h(k1-r')] (13)
k-1l --k -- --

dv" dn" P(k,v"--V ,n"-nr )N(r',v"n ) .

The integration over r' may now be considered.
The meaning of the functions [h(r'-rk,)+h(Ek1-r' ) and
[h(rk,-r')-h(rk_'-r') may be easily deduced from Eqs.
(2), (7) and (8) and is illustrated in Figure 1. The
function [h(r'-!k,)+h( k,l-r')] has unit value every-
where except in region k' where it has zero value, while
[h(k-r')-h(Ekl-r')] has zero value everywhere except
in region k where it has unit value. It can easily be
seen that the r' integration of the second term on the
RHS of Eq. (13) is an integration over all regions except



















1







0 -


1
h(r-'k +-r' )



0 ----


1

h(r'-r) + h(rk--r')
0 -

1

h(r~kt-re )-h(rk' _1-r')
0 ------




Figure 1 The Spatial Dependence of the h Functions
of Equations (2), (7), and (8)










the k'th one while the r' integration of the remaining
step-function terms is a sum over k of r' integration
over each individual region k. Thus one can write
Eq. (13) as



N(r,,vn) dr' dn' g(v;r,, r',n')S(r',vn')
rt n

R
+ ki [v(k',v)+ (k',v)] V(k,v)+ -k,v)]



dr' dn' g(v;r,nrr',nt)N(r',vn')
r'in k r'

R
+ Z dr' dnY g(v;r,n jr,nt)
k-1 ~ -
r'in k n' (14)


dv" d"P(k,v"- v,n'L.' )N(r',v"r") .

Note that the k-k' term in the first summation
is automatically zero, so that no special notation is
needed on the summation symbol.














CHAPTER II


ANGULAR INTEGRATION

11.1 Expansion of Angular Dependence in Spherical
Harmonics and Zeroth Harmonic Approximation
The angular integration of Eq. (14) may be

performed as follows. Expand all the angular dependent

terms of Eq. (14) in the complete orthonormal set of

spherical harmonics as defined in Chapter 7 of reference

(15). Thus

oo n
N(r,vn)- Nm(r v)y-(n) (15)
n-0O m--n

co n
S(r,vn) E Sm(r,v)Ym(n) (16)
n-O m--n n n

oo a
P(k,v'"Uv,n'!n) Pa(k,vv)yb()b*(). (17)
a-0 b--a a- a

The expansion for the Green's function is more

complicated. Because of possible events at the bounda-

ries enclosing the space of interest, g(v;r,nI r' ,')

depends explicitly on both n and nr' and not just on r*n'

as P(k,vn.v,n'n ) does. For the moment only the n.' de-

pendence of g(v;r,n r',nh) is represented by an expansion










in the complete orthonormal set of complex conjugate
spherical harmonics. Thus


g(v;r,n r'l,n') gs;rn r')Y (). (18)
s-0 t--s


Substitute Eqs. (15) through (18) into Eq. (14) to obtain


N(r,v ndr' dn' gt (v;rnJf)Yt* (A) : m(r',v)Ym(n!)
Ss,t nm
r' n'


+ R v(k',v)+ (kt,v)0 [V(kv)+ Y(k,v)



Sfdr' dn E gg(v;r, lr')Y t*(n) N(r (n'
S s,t s n,-
r'in k n'


+ rl drt dn' g (v;r,nr')Y (')
r'in k n1'
(19)


ab a a n -


Equation (19) may be simplified considerably
by interchanging the order of integration and summations
and by using the orthogonality relation of the spherical
harmonics, namely,











n)Yb n, a Smb (20)


where Sn,a is the Kronecker delta function.
After carrying out the angular integration,
Eq. (19) becomes


N(r,vn) dr' E ge(v;r,nlr')Sm(r',v)
n,m
rf


+ [V(k ,v)+ ](k ,v) [V(k,v)+ ?'(k,v)1



Sdr g gm(v;r,nr')Nm(r',v)
n,m
r'in k
(21)


+ R dr' dv" gn(v;rn l')Pn(k,vZ-v)o(r',v")
kl 1 f n,m
r'in k

Equation (21) is an expression for the total angular
neutron density and is exact; no approximation has been
made so far.
Now, the relative isotropy or anisotropy of
the n' dependence of g(v;r r',n') depends on the
boundary conditions applied to Eq. (10) and also on the











properties of region k'. Recalling the physical meaning
of g(v;r,nlr ,n.'), one observes' that n' denotes the

angular dependence of the unit point source in a medium

in which all collisions result in removal of the neutron

from the population. If the space of interest is of
infinite extent, then g(v;r,n r',n') has a double delta

function behavior in direction, namely S(-n*') -(-r')] ,

which merely describes the fact that since scatters

are not considered in the calculation of the Green's
function the direction of motion at r not only must be

the same as the direction of motion when emitted at r',

but also must be pointed along the vector connecting the

field and source points. In this case, the complete set

of coefficients g (v;r,,nir) would be required in Eq. (18)

to satisfactorily describe g(v;r,n r',n').

The problem is entirely different, however, if

one imposes perfectly reflecting boundaries around the

space of interest including the source as would be done
if one were calculating a unit cell in a repeating array.

Collisions or events at the boundaries will reflect

neutrons back into the space of interest with some

direction nL. If the neutron then arrives at r it will
have direction n, such that n + n' and n (r-r'),
although there llbe a definite relation between
although there will be a definite relation between











, n'_, and (r-r*). Further, if the medium characterized
by [V(k',v)+ ,(k',v)] has a mean free path that is long
relative to the distance between boundaries, then the
neutron may be reflected at more than one boundary before
arriving in the element of volume dr about r.
Now the integral of the angular density due to
a unit point source per solid angle, integrated over all
directions of emission n', is simply the angular density
due to an isotropic point source of one neutron per unit
solid angle or a total of 41T neutrons and is denoted by

g(v;r,nar'). Thus


g(v;r,nlr') g(v;r,nlr',n')dn'. (22)



From the discussion above regarding reflections at the
boundaries, it is obvious that in a finite space enclosed
by reflecting boundaries an isotropic point source will
produce a very nearly isotropic angular density

g(v;r, i r').
In order to take advantage of the relative
isotropy of g(v;r,n r'), one proceeds as follows. If
one asks only for information about the number density

N(r,v), the dependence of Eq. (21) may be integrated
out to obtain










N(r,v) dr' Z g(v;rlr )Sm rv)
nm
r'


k V (k ,v)+ k',v) [V(k,v)+ /(k,v)



dr' g(v;r r)lE (r',v)
n,m
r'in k
(23)
R m
+ dr' dv" n g(Vr'r)P (kv"-v)Nm(r',v")
k-1 n,mn n -
r'in k

where

N(r,v) -/N(r,v)dr (24)
n

Expanding the angular dependence of
g(v;rl r') in spherical harmonics gives


g(v;rn ') g(v;r r')Ym(n) (25)
n,m

The coefficients gn(v;rlr') in Eq. (23) are recognized
as being equal to the coefficients in the angular
expansion of g(v;r, nr'), a function which has been
discussed above and which has been concluded to be











nearly isotropic for a finite space enclosed by perfectly

reflecting boundaries and containing a medium having a

mean free path long relative to the distance between

boundaries.

Thus the coefficients gn(v;r r') form a rapidly

decreasing sequence for increasing n. Further, since

Eq. (1) includes a scattering term, the angular density

N(r,vn) should be rather isotropic, even for rather

strong absorbers. A unique advantage of the integral

equation method over the integro-differential equation

method thus becomes apparent; namely, the zeroth harmonic

approximation disregards only products of small terms,

that is, one assumes


-1 -1
1 1
0 0 0
g0 N0 g1 N *

g1 N1
1 1

Due to the orthogonality of the spherical harmonics,

there are no cross products to consider such as g0 No

or go N1 which might be of the same order of magnitude
0 0
as g0 NO'










The situation with regard to the scattering
term in Eq. (23) is even more favorable. One need not
assume completely isotropic scattering, but merely make
the less stringent assumption that the product of the
zero order coefficients of the Green's function, the
angular density, and the scattering kernel, is much
greater than the product of higher order coefficients,
that is,


g1 PI Ni1

g0 P NO 9 10 P NO

gl p N1
1 11


On the basis of the above discussion the
zeroth harmonic approximation is made in Eq. (23) and
only the n-0 terms are kept. Thus Eq. (23) becomes


N(r,v) dr' g0(v;r r')So(r',v)


r0

+ [v,(k.v)+/(kv) [V~f,v)+(kv

dr' g00V;r Ir)N (r',v)

r'in k










R O l o t
+ Z dr' dv" go(v;r r')P.(k,v.v)NO(r,v"). (26)
+-1 kl d -
r'in k

The zero order coefficients may be put in a
more recognizable form as follows. Substitute Eq. (15)
into Eq. (24) to obtain


N(r,v) Na(r,v) Ya(n)dn
n,m


Nm(r,v) Ym(c) -/f YO*()
n,m

-/41 N(r,v)


00*
where use has been made of the fact that Y* (n)
- YO n) 1/--1 Thus


N0(rv) N(r,v) (27)


Similarly, defining

S(r,v) -s (r,vn)dc (28)
rL










and using Eq. (16), one obtains


o 1
S(r,v) S(r,v) (29)
0 --


Define

g(v;rlr) g(v;r,lr')dn (30)

to be the number density of first-flight neutrons at r
having speed v due to a point source at r' emitting 4TT
neutrons isotropically with speed v in a homogeneous
finite medium. Substitution of the angular expansion
Eq. (25) into Eq. (30) then gives the result that


g0(v;r ') 1 g(v;r r') (31)


For the scattering kernel one has from Eq. (17)
that


P(k,v-.v, -)dn .- Z Pa(k,vi.v)Y b Y b d)
b 0



abb* /yb ,d
a,b










P(kv"-.v) ,/4f YO* ()

PO(k,v!',v) (32)

which is just the probability per unit time per unit
speed v that a neutron having speed v" will suffer a
scattering collision in region k and emerge from the
collision with speed v. The zero subscript will be
dropped and the function will subsequently be denoted
simply by P(k,v!3.v).
Finally, substituting Eqs. (27), (29), (31),
and (32) into Eq. (26) gives


N(r,v) dr' g(v;r r')S(r',v)
r'

R
+ 4 1 [(V(ktv)+ (k',v)] [V(k,v)+ Y(kv)]


Sf dr/ g(v;rlr)N(r',v)
r'in k

I R
k+ kl f dr' g(v;r r') dv"P(k,v'4v)N(r ,v").
r'in k (33)










11.2 Advantages and Disadvantages of Homogeneous
Green's Function Method
The preceding method of converting the integro-

differential transport equation to an integral equation

with a kernel defined by Eq. (10) has at least two

advantages over the more standard methods of solution.

First, as discussed above, the zeroth harmonic approxi-

mation involves discarding only products of small terms

as in Eqs. (22) and (23). For example, rather than neglect-

ing the higher moments of the neutron density, as must

be done when the integro-differential form of the transport

equation is used, one need only neglect the products of

the higher moments of the Green's function times the higher

moments of the neutron density.

A second advantage and a unique feature of the

Green's function method described in this work is that the

Green's function defined by Eq. (10) depends only on the

dimensions and geometry of the outer boundaries of the real

problem for which the neutron density is to be calculated,

the total cross section of the medium in the k'th region

of that problem, and the boundary conditions imposed on

Eq. (10). Once the Green's function has been determined,

it may be used to determine the neutron density in any

heterogeneous medium, the only restrictions being that

the outer boundaries of the neutron density problem must

coincide with those of the Green's function problem,











the boundary conditions must be the same for both g

and N, and the total cross section of the k'-th region

of the neutron density problem must be the same as

that for the homogeneous medium of the Green's function

problem.

A disadvantage of the method is implied above;

namely, if the outer boundaries and/or boundary conditions

of the neutron density problem do not agree with those for

which the Green's function has been determined, then a

new Green's function must be determined. Another dis-

advantage is that, as yet, the method has not been

extended to higher order approximations such as keeping

gm N P, g Pn N terms for n 0, although such an

extension to higher moments would, in general, follow

the work of Wilf (29), and would not alter the above

mentioned advantages.














CHAPTER III


SPEED DEPENDENCE

III.1 General Speed Dependent Equation

Equation (33) is a general speed-dependent

equation and is valid over the entire speed spectrum,

the limits of the speed integration being from v" 0

to v" -oo. As indicated in the introduction, the

present investigation will be confined to speeds in

the thermal region. In practice, there exist no true

sources for the thermal energy region other than slowing-

down sources, and hence S(r',v) 0 in Eq. (33). The

speed integration in Eq. (33) are then separated into

two parts, one for 0 < v i v* and one for v =>v*. The

speed v* represents the upper limit of the speed region

of interest, and is characterized by the following

assumptions (17):

1. For v = v*, the neutron density speed distri-
bution is proportional to 1/v2 (that is, the
neutron flux is proportional to 1/E).

2. The probability of a neutron being scattered
from a speed below v* to a speed above v* is
zero.

3. The motion of the collision atoms is ignored
with respect to that of the neutrons with'
speeds above v*.











4. The Maxwellian distribution for thermal
neutrons or collision atoms has negligible
magnitude for speeds above v*.
5. For neutron energies above v*, the energy
transfer is computed assuming that the col-
lision atoms are free from chemical bonds.
This implies that v* should correspond to a
neutron speed well above that required to
excite vibrations in the collision molecules,
so that the collision atoms would recoil
freely at the instant of impact.

6. For neutron energies below v*, the energy
transfer kernel P(k,vv-v) will depend on the
particular scattering model chosen. A
discussion of particular energy transfer
kernels and their scattering models is given
in references (17, 31).

The S(r',v) term in Eq. (33) is then modified
to include all neutrons scattered into a speed interval

dv about v from speeds above v*. Thus Eq. (33) may be

rewritten as


N(r,v) drI g(v;r r')S(r',v)

r'in k



+ 4 ( [V(kv)+ /'(k',v)] [V(kv)+ Y(kv)]


dr' g(v;r r)N(r',v)

r'in k











V*

+ k dr' g(v;r r') dv" P(k,v"!.v)N(r',v")

r'in k v"-0 (34)


where

co
S(r',v) dv" P(k,v!.v)N(r',v") (35)

v"-v*


III.2 The Source Term

Assumption (5) of III.1 implies that ordinary

slowing down theory may be used to derive the scattering

kernel for v v* (18, 19). Define

m-1
--- (36)
m+l


Then ov is the minimum speed that a neutron of initial

speed v can have after collision with a target nucleus

of free atom mass m; the energy transfer probability,

that is, the probability that a neutron of energy E'

will after scattering have energy in the interval dE

about E is (see (18))

T"(r,E..E)dE (dE for a2E' E E'


-0 ,for E < a E'

S0 for E E' (37)










or, changing variables,


7l(r,vi.v)dv 7T(rE'-,E)dE dv
rvdv

so that

7rf(r,v.v) 7r(r,EL-.E) dE


1 dE mv
SE' [la2(r)] dv l-o2(r)] (mv'
2v
2v
["I, vv for wav' v S v'


S0 for v Oav'

0 for v >v' (38)

Now, the differential scattering cross section,
Lg(r,vi-av) dv, that is, the probability that a neutron
of speed v' will be scattered at r into dv about v is
equal to the product of the total scattering cross
section at r for neutrons of speed v' times the proba-
bility that the speed of the neutron after collision
will be in dv about v. Thus

Zg(r,v4..v)dv s(r,v')7r(r,v'.-v)dv. (39)










Then the probability per unit time that a neutron of
speed v* will suffer a scattering collision at r such
that it emerges from the collision with speed in dv
about v is simply

P(r,v'.v)dv v'1%(r,v'Lv)dv (40)

so that for v' > v* one has from Eqs. (38), (39) and
(40) that


P(r,v-.v) 2vv's(r,v)
[1- (r ) v'2


2vZs(r,v')
for Ov' s v s v'
-[l-a(r)] v'

0, for v ctv'

(41)
0, for v => v

Assumption (1) of III.1 implies


A(r)
N(r,v') A for v' > v* (42)
vt2

where A(r) is an arbitrary constant with respect to
speed and is normalized to give a stationary distribution
for N(r,v'). Substituting Eqs. (41) and (42) into
Eq. (35) yields











S(r',v) dv" 2vls(k,v") A(r')
S1-a2(k) v" v"2
v"'v* [I I


2vs(k)A(r,) v/(k)
-n [ldv" (43)
[1-a2(k)] j v"3 d


where Zs has been assumed to be independent of v in the
speed range v* 5 v" v v/a(k).

Carrying out the integration in Eq. (43) gives


8(r',v) V A(r') v2 (k) if ca(k)v* v s v.
[1-o(k)] V*2 v2

0 if v < a(k)v* (44)

where the constant A(r) determines the source strength;
A(r) will be assumed to equal a spatially independent
constant A, that is, one assumes a spatially flat
slowing-down source distribution. As Honeck (22) has
pointed out, A(r) could, if necessary, be approximated
by the shape of the resonance group neutron density as
predicted by diffusion theory and the use of suitable
resonance group cross sections given in Weinberg and
Wigner (19).










II.3 One-Speed Equation

To obtain the one-speed equation, one must
assume separability of the spatial and speed dependence
of the neutron density. Then integrating Eq. (33)
over all speeds and using Maxwellian averaged cross
sections, one has


N(r) /dr'g(r r')S(r')

r'



+ 1- [v(k')+ $(k')] -[V(k)+ Y(k)) dr' g(r r')N(r')

r'in k


+ 1 V(-k) dr g(r r)N(r (45)
r'in k

where use has been made of the fact that


Co
P(k,vu.v)dv V(k,v") (46)
v-0

which is just the total probability per unit time that
a neutron in region k and having speed v" will suffer a
scattering collision.










In the one-speed case motion of the collision
atoms may be ignored, so that in conventional notation
Eq. (45) may be written as


N(r) 1 dr' g(r r')S(r')
r'



+ 1 (k')- (k)] dr' g(r|r')N(r')
r'in k

r'in k
kj g(rfrl')N(r ) (47)



Finally, the last two terms in Eq. (47) may be combined
to obtain


N(r) dr' g(r r')S(r')




+ k t(k')-a(k) dr' g(rr')N(r'). (48)

r'in k














CHAPTER IV


DIFFERENCE DENSITY METHOD


IV.1 Development of Difference Density Equations

An alternative to the preceding method of

describing the real neutron density, N, in a hetero-

geneous medium is to define a difference density, D,

as the deviation from a known density, N', for a similar

or reference space. Although it may appear at the outset

that such a method is analytically identical to that

developed above for the real density N, nevertheless,

a number of authors have found such a procedure to be

advantageous (13) and, in fact, necessary (12) in order

to obtain results for specific infinite geometry problems.

It will be of considerable interest, therefore,

to develop equations for the difference density, D,

parallel to those above for the real density N. A

direct comparison of the two methods will then be made

in order to determine their relative advantages when

applied to the specific problem to be considered in this

paper.

Thus the difference density D(r,vn) is defined

such that










D(r,vn) N(r,vri) N'(r,vn) (49)

or
N(r,vn) = D(r,vn) + N"(r,vn) (50)

Using primed symbols to denote the reference
space, one can write the transport equation for the
assumed known angular density as

v ~-IN'(r,v) + V'(r,v) + (rv) N'(r,Cvn)
.L (51)

S'(r,vn) + dv' dn P'(r,v-.v,n4 n)N' (r,v'n').

Substituting Eq. (50) into Eq. (1), subtracting Eq. (51)
from the resulting equation, and grouping all of the
N'(r,vn) terms on the RHS gives

vnVD(r,vn) + [V(r,v)+ (r,v)] D(rvi)

= S(r,vn) S'(r,vn)

v(r_,v)+ (r,v)-[,v) (v)+ /(r,v)] N'(r,vr)

+ dv dn' P(r,v--v,n'-.*-0D(r,v'n') (52)

+ dv' d.n P(r,vv,nI)-P' (r_,vv,_)] ,n N(rv,',).

As before, Eqs. (3), (4), and (8) are used to separate
the spatial dependence of the terms in Eq. (52). Thus





38



vn-VD(r,vn) + [v(k',v)+ (k',v)] D(r,vn)

- S(r,vn) S'(r,vn)

+ [h(r-rk,)+h(k'l-'r)] [V(k',v)+ K(k',v)] D(r,vn)


rh(k-r)-h(rkl-r)] [V(k,v)+ K(k,v)] D(r,vn)
k+k'


- k 1 h(Ek-r)(k-1-^r)] [V(k.v)+ Y(k,v)]

[V'(kv)+ 't(k,v)] N'(r,vr)


+ r h(rk-r)-h(kl,-r)] fdv dn' P(k,vi-*v ,nn)D(r,v'nW)
k=1 f


+ h(k-r)-h('k-1-r) dv' di' [P(k,vI+v n!*n)
(53)
-P' (k,vv,L).)] N' (r,v'n').

The boundary conditions imposed on Eqs. (51) and (53)
are the same as those for Eqs. (1) and (9). Thus the
Green's function which satisfies Eq. (10) may be used
to convert Eq. (53) into an integral equation as before.
Then, using Eq. (12), one may perform the integration
over v' to obtain








D(r,vn.) dr' d' g(v;r,g r' n') S(r' ,v')-S' (r' ,vn )

+ [h(r'- ,)+h(k, -rl') [v(k,v)+ ) k',v)] D(r',vn')

R
- [hr ktr')-h(h-1-r')1 [V(k,v)+ Y(k,v)]D(r',vn' )
k+k'

R
[h(Ek-r')-h(ik-lr')] [V(k,v)+ Y(kv)

yv' (k,v)+ 1 (k,v)] N' (r' ,v')

+ kl[h(Ek-r')-h(rek_-r') dv" dr' P(k,v" vn'n*')D(r',v'tn")


+ k E [h(Ek-r' )-h(Ek-i-r') dvt" d [P(kv' mt-.')
i (54)
P' (k,v-vn'".)] N' (r' vn") .

The r' integration for Eq. (54) is identical
to that for Eq. (13) and the result is

D(rv_) dr' dn' g(v;r,n r' ,n)[S(r',v f)-S'(r_',vn)]


+ V(k',v)+ (k',v)] [V(k,v)+ Yk,v)

dr' da g(v;rn ir'L l )D(r' ,v')
r'1in k 1'










R
+ dr' dn' g(v;r,nlr'l,.')
r'in k n

dv" a" P(k,v" vn'1,-c n')D(r',v"n')


- [V(k,v)+ Y(k,v)] [V'(k,v)+ (k,v)

S dr' dn' g(v;rnl r',n')N'(r',vn!')
rrin k '

R
+ dr, dn' g(v;r,nlr',n') dv"fdrl" P(kv4I.vI.nIn')
r'in k T.'
(55)
P' (k,v"-.v,n "-.')] N' (r' ,vI'") .

The angular integration presents no new
problems and may be performed as in Chapter II. Making
the zeroth harmonic approximation and asking only for
the scalar neutron difference density, one obtains


D(rv) d 1 dri g(v;rjr9)[ (r',v)-S'(r',v)]
rt


+ ([V(k',v)+ (k',v)]
47r k-i
[V(k,v)+ Y(k,v)]) dr' g(v;r r')D(r' ,v)
r'in k










R
k dr' g(v;rlr') dv" P(k,vU*v)D(r',v")
k-lr -
r'in k


R
k-i


+ 1

471


4"/


Sdr' g(v;rr')N'(r',v)
r'in k


dr' g(v; rr')
r'in k


(56)


*dv" P(k,vt v)-P' (k,v"-v) N' (r' ,v").

Writing Eq. (56) specifically for the thermal
spectrum yields


D(rv) dr' g(v;rr')[S(r'v)-S'(r',v)]
r'in k


R


[V(k',v)+ (k',v)]


[V(k,v)+ /(k,v)] dr' g(v;r r')D(r',v)
r'in k


S[vk,+ (k,v) )]


- [V'(k,v)+ Y'I(kv))


1
+1-


I R
+
kw,


v f











V*
1 R
4+ F i dr' g(v;r r') dv" P(k,v"!v)D(r',v")
r'in k v"'-0



-1 R i (k,v)+ (kv)]


[V'(k,v)+ '(kv dr' g(v;r r')N'(r,v)
r'in k

R
+ k~ dr' g(v;rlr')
r'in k

(57)
v*

dv" [P(k,v,-.u.v)-P' (k,v"--.v)] N' (r' ,v")

VV-O

where, in region k,


ro
S(r',v) S'(r',v) dv" P(k,v-!.v)D(r',v")
vl-v*


00 (58)
+ dv" [P(k,~,!v)-P'(k,v".v)] N'(r',v")
v-v*










and the scattering kernels for v > v* are determined
from ordinary slowing down theory as in 111.2.
Finally, the one speed version of Eq. (56) is


D(r) dr' g(r r')S(')- (
r'


1 R

+ -- k

+ Rl



1 R



+1 R
47( N-1


v[zt(k')- It(k)] dr' g(r r')D(r')
r'in k


v Zg(k) dr' g(r r')D(r')
r'in k


v [zt(k)- Z((k) drI g(r rf)N'(r')
r'in k


v[(k)- (k) dr' g(r r')N'(r')
r'in k


or if various terms are combined,


D(r) /dr' g(r r')[S(r')-S'(r')]
r'


(59)










+ v k')- Zak)] dr' g(r r)(r
r'in k


S1 v[Za(k)- (k)] dr' g(rr')N'(r') (60)
47r T l k f i
r'in k

IV.2 Equivalence of Equations for N and D
In order to establish the equivalence between
Eqs. (34) and (57) and between Eqs. (48) and (60),
the one-speed equations will be considered first.
Subtracting Eq. (60) from Eq. (48) and using
Eq. (50) gives


N'(r) 1 dr' g(r r')S'(r')
rt


+ k V[Zt(k*)- a(k)] dr' g(r )N(r')
r'in k


+ 1 v[Za(k)- Z(k)] f drv g(rlr')N'(rt)
k 47j
r'in k


_- fdr' g(r r')S'(r')
47r f
rt











+ RF v[ l(k)- .(k)] drt g(rr)N(r'). (61)

r'in k

Comparing Eqs. (61) and (48), one observes that the
requirement that Eqs. (60) and (48) be equivalent is
simply that It(k') -= (k'), that is, the total cross
section for the k'th region of the reference space be
identical to the total cross section of the k'th region
of the real space of interest. This simply says that
the Green's functions in Eqs. (48) and (60) must be
identical, that is, the Green's functions in Eqs, (48)
and (60) must both satisfy Eq. (10) and its boundary
conditions. But this is just the condition imposed in
the original derivation, and hence Zt(k') is equal to

Z (k'); equations (48) and (60) are therefore proved
to be equivalent formulations of the same problem.

Similarly, subtracting Eq. (57) from Eq. (34)
and using Eq. (50) gives



N'(r,v) = dr' g(v;rr')' (r ,)
*E k-lf ~ # ~
r'in k










1 R
47T kl


[V(k,v)+ a(k' ,v)]


Iv(k,v)+ Y(k,v)] dr' g(v;r r')Nt(r',v)
r'in k

V*
+ k dr' g(v;r r') dv" P(k,v"..v)N (r' ,v")
r'in k vIO


1
+


k [V(k,v)+ ~(kv)

[V'(k,v)+ f'(k,v)]


r'in g(v;rlr)N'(r',v)
r'in k


Sdr' g(v;rlr')
r'in k


* dv" [P(k,v-t..v)-P' (k,v"!.v) N' (r' ,v")
v" 0


1'


R
E-I


r


I


r'in k


dr' g(v;r r')8'(rt,v)











+ 1 [V(k'iv)+ K(k',v)]


['(k,v)+ (k.v)) dr' g(v;rjr')N'(r',v)
r'in k

v*
S R
+ 1 dr' g(v;r r') dv" P'(k,,v'la)N'(r'v,v")
r'in k V1=0

Comparing Eqs. (34) and (62) one again observes
that, just as in the one-speed equation, the requirement
that Eqs. (34) and (57) be equivalent is simply that
[v(k',v)+ (k' ,v)] be equal to [V'(k',v)+ ('(k',v)],
that is, the medium of the k'-th region of the reference
space must be the same medium that is in the k'-th
region of the real space of interest. As has been
mentioned above, this condition was imposed in the
original derivation so that the equivalence of Eqs. (34)
and (57) is established. A discussion of the relative
usefulness of the two formulations will be deferred
until the general equations have been specialized to


slab geometry.













CHAPTER V


PLANE SLAB GEOMETRY

V.1 One-Dimensional Problem
The equations of the previous chapters were
written for a general three-dimensional geometry, and
the Green's function g(v;r r') was interpreted as the

number of first-flight neutrons per unit element of
volume at r having speeds v due to an isotropic point
source at r' emitting 47r neutrons per unit time having
speeds v in a homogeneous medium. Before specializing
Eqs. (34) and (57) to an infinite slab geometry it will
be convenient to define a new Green's function G(v;r r'),
such that

G(v;r g(v;rr') (63)

Thus G(v;r r') is the number density of first-flight
neutrons at r having speed v due to a unit isotropic
point source at r' emitting neutrons of speed v in a
homogeneous medium. Rewriting Eq. (34) in terms of

G(v;rlr') gives












r'in k


+ 1 fV(k',v)+ /(k: ,v) -[V(kv)+ /(k,v)]


*/ dr' G(v;rir')N(r' ,v)
r'in k

v* (64)
+ 17 dr' G(v;r r') dv" P(k,vlvm)N(r',v")
k-.f -I- f
r'in k v"0O

and similarly for Eq. (57).
The transformation to plane geometry may be
performed as follows. Referring to Figure 2, the element
of volume dr' may be expressed as

dr' 2 7Ta da dx'

Define
R r r'

R r r'I (65)

From Figure 2, and Eq. (65) one obtains

R ( x-x'| 2 + a2)1/2 .x-_
In I















r fixed field point
r' variable source point
Y cos G


z /



//'r O
/ X



r x
/ L._


da


dx'


Figure 2 Plane Geometry Transformation










so that for constant Ix-x'l


dR = a da -Ix-x'
a_2 dr .
R 2

Hence
dr' 2 7 R dR dx'


_Hx-x'lX 2 d_
S2[Ix-x'I -dx' (66)


where the i integration is from -1l to y-0, and the R
integration is from R jx-x'I to R -oo .
Observing that in plane geometry the r de-
pendence of the functions N, N', D, S, and P is simply
a spatial dependence on x, one may write Eq. (64) for
plane geometry as


R
N(x,v) 1 dx' S(x',v) G(v;r r') 2 7r R dR
x'in k R- x-x'I


+ 1 [v(k v)+ '(k',v)] (k,v)+ (kv)]




Sdx' N(x',v) G(v;r r') 277R dR
x'in k R-|x-x'l










v* 00
+ 1 dxt dv" P(k,vl-.v)N(x',v") G(v;r r') 27TR dR
xin k v"-O R-Ix-x'i (67)

Define



G(v;xx') G(v;rr ') 2rrR dR (68)
R- x-x'l

The function G(v;x x') is the density of first-flight
neutrons at x having speeds v due to a unit isotropic
plane sourceat x' emitting one neutron per unit time
with speed v in a homogeneous medium. This function is
well known for an infinite medium (see Chapter 2 of
Case, et al. (21)) and will be discussed extensively
for a finite unit cell in Chapter VII.
Using Eq. (68) to rewrite Eq. (67) gives

R
N(x,v) 4 /dx' G(v;x ,x)S(x,,v)
x'in k


+ k1 [V(k',v)+ /(k',v)]-[V(kv)+ Y(k,v)]

dx, G(v;x x')N(x',v)
x'in k










V*

+ dx' G(v;x x')
x'in k v"0


Similarly, Eq. (57) for plane geometry becomes


D(xv) / dx' G(v;x x')B(x'.v)-S'(x'.v)]
x'in k


+ k1 [(k',v)+ (k',v)]-[V(kv)+ (k,v)]


/dx' G(v;x x')D(x',v)
x'in k


+ f dx' G(v;x x')
x'in k v"-0


R
-kx


3v" P(k,v-L.v)D(x',v")


[V(k,v)+ Y(k,v)] [V (k,v)+ YO (k,v)]


dx' G(v;x x')N'(x',v)
x'in k


(70)
v*

+ dx' G(v;x x') dv" P(k,v'-wv)-P'(k,vtlv) N' (x',v").
x'in k v-0


dv" P(k,v"L.v)N(x',v") (69)











V.2 Symmetry Considerations
For subsequent work the space of interest
will be assumed to be a plane geometry symmetrical
about the plane x-0. In the summations over k as
previously discussed, k-l denoted the left-most region
in space. The regions will now be counted differently,
letting k-l denote the region adjacent to, and to the
right of, the x-0 plane. Thus a summation over k from
k-l to k-R is replaced by a summation over k from
k -R/2 to k R/2. Let


R2 "- (71)

denote the number of regions in the half-space. For a
general function f(x) which is symmetric about x-0,
that is, f(x) f(-x), one can then write that

R
E f(k)J dx' G(v;x x')N(x',v)
k-l
x'in k

R2
R f(k) dx' G(v;xlx')N(x',v)
k-R f(k)
x'in k


+ ( f) dx' G(v;x x')N(x',v)
k--R2 k- f(k)
x'in k










0 b

= dx' + dx f(x')G(v;xlx')N(x',v) (72)
x'--b x'-O

where b and -b denote the right and left boundaries of
the space of interest. The notation used is defined

such that


cm c2 7C
+f ... + + k (73)
kl kn k-c1 k-en

and
(74)
C2 cm c2 cm
dx' + ... + dx' f(x') dx'f(x') + ...+ dx'f(x').
x'-c1 X' cn x'c 1 X'-cn


In the first integral in Eq. (72) let x'--y,

dx'=-dy. Observing that f(x)-f(-x) and N(x,v)-N(-x,v),
one can write

0 0

dx'f(x')G(v;x x')N(x',v) dy f(-y)G(v;x -y)N(-y,v)
x' -b y-b

b

/dy f(y)G(v;xl-y)N(y,v) (75)
yZ0










Then let y-x' so that Eq. (75) becomes

0 b
Sdx'f(x')G(v;xx')N(x',v) dx'f(x')G(v'x -x')N(x',v).
x-b -0 (76)

Substituting Eq. (76) into Eq. (72) yields

R
Z f(k) dx'G(v;x x')N(x',v)
fk-
x'in k

b
dx'f(x') [G(v;x x')+G(v;x -x')]N(x',V)
x'-0
(77)
f (k) dx' [G(v;x x')+G(v;x -x')]N(x',v)
k-1n
x'in k

As noted above, G(v;x x') is the scalar
density of first-flight neutrons at x due to an isotropic
unit plane source at x'. If one defines

H(v;x x') G(v;x x') + G(v;x -x') (78)

to be the half-space Green's function, that is, the
scalar density of first-flight neutrons at x due to
isotropic unit plane sources at both x' and -x', then









equation (69) may be rewritten for the half-space in
the form


N(x,v) dx' H(v;x x')S(x',v)
x'in k


+ kV (V(k',v)+ Y(k',v) -fV(k,v)+ 7(k,v)1


f dx' H(v;x x')N(x',v)
x in k

v* (79)
+ dx H(v;xlx') dv" P(k,v'.v)N(x',v")
x'in k v"-0

while Eq. (70) becomes


D(x,v) R 2dx' H(v;x x') S(x',v)-S'(x',v)]
x'in k

+ 2 ([V(k'.v)+ (k',v)]-[V(kv)+ (kv)]

dx' H(v}x x')D(x',v)
x'in k










v*
+ T dx' H(v;xlx') dv" P(k,v!".v)D(x',v")
x'in k v"-0


- ([V(k,v)+ /(k,v)]-[V'(k,v)+ Y'(k,v)]


dx' H(v;xIx')N'(x',v)
x'in k
(80)
v*
+ f dx' H(v;x x') dv" [P(k,v!.Lv)-P'(k,v!..v) N'(x'f,v")
x'in k v"10

The one-speed equations may also be written
for the one-dimensional half-space. Thus Eq. (48)
becomes

N(x) dx' H(x x,)S(x')
x >0


+ :2 v[Zt(k')- Ia(k)] fdx' H(x x')N(x') (81)
x'in k

while Eq. (60) becomes

D(x) .dx H(x xI) [(x')-BS(xv)
x') O





59



+ E2 v[t(k')- Za(k)] dx' H(xx')D(x')
x'in k


i2 v (k)-a( j) dx' H(xJx')N'(x') (82)
x'in k













CHAPTER VI


COMPARISON OF N AND D METHODS
FOR SIAB GEOMETRY

For the direct comparison of the formulation
in terms of real density,N, with the formulation in

terms of difference density, D, only the one-speed

equations will be considered, namely Eqs. (81) and (82).

Similar arguments would apply to the multispeed equations.

Suppose that all R regions of the reference
space contain the same medium and that the reference

space itself constitutes one spatial period of an

infinitely repeating lattice, that is, the reference

space is a unit cell. Then the neutron density N'(x)
in the reference space is that for an infinite homo-

geneous medium and may be assumed spatially flat.

Since the neutron density and medium are both
spatially uniform in the reference cell, one may assume
that the statement of neutron conservation holds not

only over the entire reference space, but also over any

arbitrarily small element of volume in that space. Thus

at any point


B'(x) S' v IN'. (83)










Substituting Eq. (83) into Eq. (82) and
cancelling terms where possible, one obtains

D(x) -dx H(xlx')S(x')
f',0
XI'-0


+ a vt(k')- (k)] dx' H(xjx')D(x')
x'in k


v a(k)N' dx' H(x x') (84)
x'in k

Then using Eq. (50), one may combine terms to write

D(x) dx, H(x x')S(x')
X'-O


+ k v 2(k') f dx' H(x x')D(x')




v Z.a(k) dx' H(x lx)N(x') (85)
k-1 f
x'in k

Adding and subtracting the quantity


v t(k') fdx' H(x x')N(x')
k-1in k
x'in k










to Eq. (85) gives


D(x) dx' H(x x)S(x')
x 'O


+ 1k2 v[Zt(kt)- Za(k) f dx' H(x x')N(x')
k-1
x'in k

R2
I v lt(k') dx H(x x')N'
k-in
x'in k


dxt H(x x')S(x')
x*=0


+ 2 v[t(k')- Za(k)] dx' H(x ')N(x')
x'in k

v I(k')N' dx' H(x x') (86)
x '0

But from Eq. (10)

v Zt(k')H(x x')dx' 1 (87)
x 50

so that using Eqs. (50) and (87), one can reduce Eq. (86)


directly to Eq. (81).










The direct reduction of Eq. (82) to Eq. (81)
having been established, the question remains whether
the use of Eq. (82) has any advantages over the use of
Eq. (81). It will be of particular interest to consider
whether or not the complexity of Eq. (82) can be reduced
by relating the real source S to the source 8' in the
reference space. A few such special cases are considered
below. In each case the assumption is made, as it was
above, that all R regions of the reference space contain
the same medium and that 8' and N' are spatially flat.
Case 1 S(x) 8' Over All Space
Equation (82) then becomes


D(x) 2 v I(k')- a(k)] dx' H(xJx')D(x')
x'in k


va(k)- i(k)]N f dx' H(xxt)




R- 2 tk'- a(k)] dxz H(x x')D(x')
x'in k


v Za(k)N' fdx H(x x') + vE(k'
x'in k











where Eqs. (83) and (87) have been used to obtain the
last term of Eq. (88).

Since S(x) = S' implies that S(x) is spatially
independent, Eq. (87) may be used to show that the first
term in Eq. (81) becomes


dx' H(x xt')(xt) = 8 dx' H(x x')
x'=O x '=O


S'
v St(k') (89)


Thus Eq. (88) can be reduced directly to Eq. (81) in
the same manner that Eq. (82) was reduced to Eq. (81),
so that there is no advantage of the difference density

formulation over that for the real density N. The
reason for this is that, although it is given that

S(x) S S', one must still calculate N' based on 8',
and the N' terms then act as source terms in the differ-
ence density equations.
Case 2 S(x') 0 For Region k-1, S(x') 8'
For Regions k 1

Equation (82) becomes

D(x) dx' H(xIx')S'(x')
x'in 1









+ 2
k-i


- 2
k-l


v[ZCk)- (k)]


dx' H(xIx')D(x')
x'in k


Sdx' H(x x')N'(x)
:'in k


To specialize further, let R2 2 and k' 2.


Eq. (90) becomes


D(x) S' dx'
x'in 1

+ v[Zt(2)-~a(1)


H(x x')


dx' H(x x')D(x')
x'in 1


+ v Z~(2) dx' H(x x')D(x')
x'in 2


v[a(1)- a(1)] N' dx' H(xl x').
x'in 1

Using Eq. (83) one may write Eq. (91) as


D(x) v[Zt(2), a(l)] idx' H(xI xtD(x')
x'in 1


(90)


Then


(91)









+ v zS(2) dx' H(x Ix)D(x')
x'in 2


a(l) N' dxv H(xlx') (92)
x'in 1

or, again substituting Eq. (83) into the last term of
Eq. (92), one finally obtains


D(x) v[Zt(2)- Za()] f dx' H(xx')D(x')
x'in 1


+ v ZS(2) dx' H(x x')D(x')
x'in 2

(1a(') 8'
7 a(2l) dx' H(x x') (93)
x'in 1

For this special case, Eq. (81) becomes

N(x) SI dx' H(xlx')
x'in 2


+ v[t(2)-Za(1)] /dx' H(xlx)N(xt)
x'in 1










+ v Zs(2) dx' H(xlx')N(x') (94)
x'in 2

The obvious difference between Eqs. (93) and
(94) is the source term. In a typical application of
calculating the scalar neutron density in a space
containing a gold foil in a graphite medium, the ratio

Za(1)/Za(2) would become


(1) Au 5.79
1 _7 1.94 x 104
a a
a(2) ^C .000299


It can be seen that in a finite medium containing a
highly absorbing region, even though it be of small
spatial extent, the neutron density will not necessarily
be a small deviation from a known density in a reference
space with no absorber. In fact, the deviation in
magnitude of the densities will be quite large, even
though the spatial shape of the neutron density N may
not be vastly different from the spatial shape of the
known density N'.
It seems obvious now that for problems involv-
ing finite geometries the analytic method of solving for
a difference density, D, offers no advantages over






68




the direct solution for the unknown density N; the

difference density method may, indeed, be limited in

usefulness to the solution of infinite geometry problems.












CHAPTER VII


THE FIRST-FLIGHT GREEN'S FUNCTION
FOR A HOMOGENEOUS MEDIUM

VII.1 Differential Equation and Boundary Conditions
The first-flight Green's function
6k' (r,v r',v'n') for a general three-dimensional homo-
geneous medium is defined such that it satisfies Eq. (10),
namely

va* Vgk' (r, vn r', v'n' )

+ [V(k',v)+ (k' ,v)] gk'(rvnr' ,v'n')

&(r-r )S(v-v )S(cn-n'); (95)

further, gk (r,vnr',v'_1') is subject to the same
boundary conditions that are applied to the angular
density N(r,vn).
Considering the RHS of Eq. (9) as a fictitious
source Q(r,vn), one may rewrite Eq. (9) as

vn.7VN(r,v) + [V(k',v)+ ((k',v)]N(r,v)) Q(r,vn) (96)










Thus N(r,vn) represents the first-flight angular density
in a medium as discussed in Chapter I, the removal pro-
bability being equal to the total probability of col-
lision in region k', with a source of neutrons Q(r,vn).
If the homogeneous medium k' is of infinite
extent, then the solution to the one-speed version of
Eq. (95) is known to be (see Chapter 2, of Case, et al.
(21))


t(k') -d
gi (rlr S 2 n-r!' )'(n f-r') (97)


where the superscript oo on g signifies that Eq. (97) is
valid only for an infinite medium or for a finite medium
enclosed by a non-re-entrant surface with vacuum boundary
conditions, and r-r' denotes the unit vector (r-r')/jr-r'
The extension to the speed dependent problem is obvious
from the discussion of Chapter I and Eq. (12). Thus


Zt(k',v) r-r'
-v-
S0, (r ', ,v'n') e
(98)

o(v-v')&(n-n')An ;Cc') .










Integrating Eq. (98) over v' and n! gives


t(k'v)lr-r'
gg?(v;r,n|r') e t-----(k ,v ); (99)
v r-r (99)

integrating over n then yields



k (v;r r') e -Ztk v) (-0oo
v r-r 2 (100)

where use has been made of Eq. (30).
The simple form of Eq.' (100) is due to the
fact that the medium k' is homogeneous so that the
collision probability is not a function of the flight
path of the neutron. As pointed out by Case, et al.
(21), the case of a spatially variable collision proba-
bility is a considerable complication because absolute
coordinates instead of only relative coordinates between
the field and source points enter the problem. It might
be well to point out again that a unique advantage of
the homogeneous Green's function method being described
is that it completely avoids the complication of a
spatially variable collision probability in the evalu-
ation of the Green's function.










From Eqs. (63) and (100) one can write the
expression for G(v;r r') for an infinite medium. Thus


(k';v _r-r'1
G- (v;rr') e(101)
47 v r-Ir' 2

From Eq. (68) one can then evaluate G(v;x x') for an
infinite medium. Thus, using the transformations of
V.1, one obtains



,3- ,,v Ir,-rIt(k'
Gco (v;x x') 27r R dR
47T v r1-r' 2
R-Ix-x'I



S e- t(k',v)R -1 dR (102)
R-I x-x'

The exponential integrals or En functions (21, 24) are
defined such that



En(x) e-xu u-n du (103a)
1












-i pn-2 e-X/P dp (103b)


00


Sxn-1 e-u u-n du (103c)

u-x

so that Eq. (102) may be written

Goo (v;x x') L El [Zt(kv)Xx-x] (104)

The Green's functions discussed above were,
in general, for an infinite medium. The problem
eventually to be considered, however, is that of a unit
cell; that is, the space of interest represents one
unit volume of an infinitely repeating set of unit volumes.
The application of the unit cell approach is
based on the fact that most heterogeneous nuclear reactors
have fuel and moderator arranged in a repeating lattice

structure. For calculational purposes a lattice is sub-

divided into a number of identical unit cells (18, 19).
The spatial and speed dependent neutron density, usually
just the scalar density, is calculated for the unit cell

and is subject to particular boundary conditions. Quanti-

ties such as the thermal utilization are then calculated











for the unit cell and are assumed to be valid throughout

the lattice. The reactor is then treated as a homo-

geneous reactor having the same thermal utilization as

the unit cell, and the scalar neutron density is calcu-

lated for the entire reactor.

In most reactors the lattice is very large

relative to the dimensions of the unit cell, so that

it is not unreasonable to treat the unit cell as if it

were in an infinite lattice. For a complete discussion

of this assumption, see Chapters 7 and 18 of Weinberg

and Wigner (19). If the assumption is then made that

the lattice is infinitely repeating, then at steady-

state the flow of neutrons in dn about direction ni

across a point on the boundary between adjacent unit

cells is equal to the flow in d_ about direction i

where L2 is the mirror image of nl as illustrated in

Figure 3. This is physically equivalent to the

assumption that the cell boundaries are perfectly

reflecting. Hence in the analysis that follows,

neutrons which actually arrive at r in the unit cell

by crossing the cell boundaries on first-flights can

be considered to have originated within the unit cell

at r' and suffered reflections at the cell boundaries

before arriving at r.





















S2 180- - -

T2 T 1



Figure 3 Symmetry of Neutron Flow at the
Boundaries of Unit Cells in An





problem in the following manner. Due to reflections at

the cell boundaries, first-flight neutrons from an
isotropic point source at r' can arrive at r by various

paths, so that the total density at r is the sum over
all possible paths i, of the densities of neutrons
arriving at r by path i and having traveled a total

path length Li.
Alternatively, the neutrons at r can be con-
sidered to have arrived by first-flights from i image
sources at r located a straight-line distance L from
soure i at located a straht-ine distance L from










the field point r. The total angular density at r is
then the sum over the i image sources of the contri-
butions to the density at r from each image source at
rt. (See Chapter 7 of Morse and Feshbach (16) for a
discussion of the method of images.)
Thus for a general homogeneous medium, one
has from Eqs. (63) and (99) that


t(k',v)L
G(v;r,n r') e (t( rv)i)
i 4 TTv2 -i (105)


where Li is the total path length traveled by a neutron
arriving at r along a first-flight path from an image
source at ri. From the alternate point of view, Li is
the path length traveled by a neutron along path i as
it suffered reflections at the cell boundaries before

arriving at r.
Integration of Eq. (105) over n yields


-" t(k',v)Li
G(v;r r') 4 7tvvJLi (106)
-O vLi2 (106)
4 7r vLi1

As one would expect, for an infinite medium or for a
finite medium bounded bya non-re-entrant surface with










vacuum boundary conditions Eq. (106) reduces to Eq. (101);
for a finite medium bounded by a re-entrant surface with

vacuum boundary conditions, Eq. (106) reduces to the

infinite medium Green's function with spatially variable

collision probability as given in Chapter 2 of Case, et al.

(21), namely



R (r sR/R, v) ds
G(v;r r') e 2(107)
4 7 vR2

where Zt(s,v) 0 outside the bounding surface.

VII.2 Analytic Form of G(v;xlx')

The physical effect of introducing reflecting
boundaries is to increase the density of neutrons at a

point due to reflections at the boundaries which, in

effect, send neutrons back into the finite unit cell.

The total neutron density at r may be separated into
two parts; namely, that due to first-flights from the
source at r' within the unit cell and that due to

neutrons that may be considered either as arriving

from i image sources located at r' outside the unit
cell or as having suffered one or more reflections at

the boundaries after emission at r'.










The first-flight angular density at r due to
a unit isotropic point source at r' in a homogeneous
medium is simply the infinite medium Green's function
obtained from Eqs. (63) and (99), namely,


0,,nv) rr-,e
oG (v;r,|r,) e- (a ^ ').
47T v r-r (108)

Let Gfc(v;r,n r') denote the angular density at
r of neutrons which do not arrive by first-flights
from the unit isotropic point source at r' but instead
arrive at r only after having suffered reflections at
the mirror boundaries of the cell, or alternatively,
arrive by first-flights from the i image sources
located at ri. Then let Li,n denote the total path
length traveled by a neutron along a path having a
direction that is characterized by the unit vector
ri-' pointing from the image source at r' to the
j -ji
field point r, the neutron arriving at the field point
r after having suffered n reflections at the mirror
boundaries of the cell. One can then write from Eq. (105)
that


Gfc(v;rnE I) z e- lt(k',v)Li,n _
n-l i 4 VLln 2 --
n (109)











The double summation over n and i is, of course, equiva-

lent to a single summation over all the image sources.

The total Green's function for the unit cell is then


G(v;r,n Pr') G (v;r,nlr') + Gfc(v'r,rl r') (110)

Thus it is seen that Gfc represents a finite medium

correction term to be added to GO.

In V.1 the plane geometry equations were

obtained by integrating the point source kernel over

all the sources on the yz-plane at x'. For a unit cell

with plane reflecting boundaries, a neutron which

arrives at r after having suffered n reflections, and

thus having traveled a path length Lin, can be con-

sidered to have originated at a fictitious or image

plane source located at a distance pLi,n from the field

plane. The cell boundary at which the neutron made its

last or n-th reflection lies between the field plane and

the image source plane.

The transformation from the point source

Green's function to the plane source Green's function,

that is Eq. (68), must be modified using the above

method of images to locate image plane sources at

distances pLi,n from the field plane. Referring to

Figure 2, G(v;x,plx') is the number of neutrons with










direction cosines between p and + du crossing a ring
element of area 27 Tper unit time, the ring element
being located in the plane at xt. If f is the azimuthal
angle indicated in Figure 2, then

dr du dP (111)

Hence, using the transformations of V.1,

Goo (v;x, Ix')dx dp dx' ,

2 0
x d d G I
fma0 f-2l1 (112)

Since the sources are assumed to be symmetri-
cal about the x-axis, G(v;r,flr') is a function only of
r,r', and u. Thus the integration over c may be
carried out in Eq. (112):

1
GC (v;x,p x') 47r2/ G(v;rl|r') X- 2 (113)

P O =0

For slab geometry G(v;r,nlr') has a delta function
behavior with respect to p' so that Eq. (108) becomes












Gc (v;rn.r') -


e- t(k',v) (x-x)/p| e r- -r .
4----x- ---------(114)
47yv(Ix-x'l /p)2 27r


The p'-integration may then be carried out in Eq. (113).
The result is


Gc(v;x,pIx') -


- (k',v) r-r

2v


or using Eq. (2),


G C(v;x,p x') -


-Z'(k'.V) I (X-X'f)^l
2v II

-Zt(k,v) (x-x')/P


h(x-x') for f > 0


(115)

h(x'-x) for < 0


2v I


where the step function has been introduced to account
for the fact that in an infinite first-flight medium,
the flow of neutrons in the positive P direction is
zero unless x is greater than x', while the flow of
neutrons in the negative p direction is zero unless x
is less than x'. For a more complete discussion see
Chapters 2 and 4 of Case, et al. (21).










In the same manner, one obtains the correspond-
ing expression for the finite medium correction term.
Thus

Zt(k',v) (x-xi)/p
GfC(v;x,pl x) e h(x-xi) for p 0
1 2v ||
(116)
g(k v) I(x-x) I
Ee -,- / h(xi-x) for p 0
i 2v i

where the single summation represents a sum over all i
image source planes located respectively at xi. The
first two image source planes are illustrated in
Figure 4.
Substituting Eqs. (115) and (116) into
Eq. (110), one can verify that each term in G(v;x,p x')
satisfies the one-dimensional form of the source-free
transport equation corresponding to Eq. (95), namely

oG(v;x,p x')
VU xy + v Zt(k',v) G(v;x,Ix') 0 (117)

where v Zt(k',v) is equivalent to [V(k',v)+ (k' ,v) .
Further, G(v;x,p x') satisfies the reciprocity theorem
(21), that is,

G(v;x,p x') G(v;x' x,-) (118)











Finally, each term of Eqs. (115) and (116) satisfies

the symmetry condition at the boundary x +b that the

flow of neutrons in dp about p is equal to the flow in

dp about -', that is,

G(v;x,u x') x-l b G(v;x,- ix') .x-b (119)
x-+b x-+b

Since each term of the Green's function G(v;x,ujx')

satisfies the original boundary conditions, it can be

concluded that the function itself satisfies the boundary

conditions. One can verify that, in agreement with

sections 4 and 5 of Case, et al. (21), a source plane
of strength qs at x' is equivalent to a discontinuity
of the normal component of the angular current, where

for an isotropic plane source, qs 1/2. The angular

current is simply the number of neutrons crossing
unit area perpendicular to the direction of flow per
unit time and unit solid angle. In particular, for
the infinite medium terms one can write

vpG(v;xuIx') -x-x'+- vuG(v;xGiux') x -x'- q5s() "


In an infinite medium the second term on the LHS

vanishes for p >0 while from Eq. (115) the first

term is equal to 1/2, that is, the source plane value.










In the solution for the scalar neutron density
in Eqs. (69) and (77) one needs the corresponding
scalar Green's function G(v;x x') obtained by integrat-
ing over all p the function G(v;x,p x'). Thus,

1

G(v;x x') G(v;x,p x')dp (120)
P--1

Recall that G(v;x x') is the first-flight
density of neutrons at x due to a unit isotropic plane
source at x' in a homogeneous medium. Then, referring
to Figure 4, one can see that for image sources to the
right of x, that is, x => x, the neutrons reaching x
must have a negative direction cosine u, while for
image sources to the left of x, that is, xi <= x, the
neutrons reaching x must have a positive direction
cosine. Thus the contribution of the image sources is

0 1

Gfc(vI;x x) = Gfc(v;x,plx.) dp + Gfe(v;x,,Ix')di (121)
0- 1 0

where the first and second integrals represent the
contributions of image sources placed respectively to
the right and left of x.



























c7G


x--2b-x'
Image Source
Plane at
x--b-(b+x')


Figure 4


x--b


x-O


01800-


-'



1,1




x-b x-2b-x'
Image Source
Plane at
x-b+(b-x*)


Slab Unit Cell with Mirror
Boundaries at x-+b










Letting Gtc denote one term of Eq. (116),
one may write the first integral in Eq. (121) as

0 0
f fe(v e- t(k'tv) I(x-X)/IU
Gi (v;x, |')du -d
iI2vl I2



e- Zt(k',v)(x-xi)/l d (122)
O-0

the last integral having been obtained from the preced-
ing one by a successive substitution of variables, first
letting y--u, and subsequently letting i-y.
In general then, one can sum over all image
sources and obtain the result that



G (v;x x') f e t(k',v) (x-xi) d (123)



Again, for a first-flight medium the range
of integration over ) for the infinite medium contri-
bution is from 0-O to P1. Thus one can write

1
G(v;xlx') f e- t(k',v) (x-x')/ d
-O0










1
+ -t(ck',v) (x-x') 4)
i 2v (124)
P-0

or from Eq. (103),


G(v;x x') El ( t(k'v)x-x'j]
(125)

+ E El[Zt(k',v) x-xil] )


To express G(v;xIx') as a function of the
unit cell dimensions it will be instructive to return
to the physical concept of reflection of neutrons in
order to locate the image sources with respect to the
cell boundaries. Using the notation for path length
previously discussed with regard to Eq. (109), one can
write the expression for GfC(v;x x') in Eq. (125) as


Gc(v;x x') E E El[7t(k',v)ILi.nl] (126)
n-l

where PLi,n is, as before, the distance separating the
field plane from an image source plane, the image
plane corresponding to a neutron having made n re-
flections at the mirror boundaries.











In Eq. (126), the summation over i extends

over all possible paths by which a neutron having its

direction cosine equal to u can arrive at the field

point after suffering n and only n reflections at the

boundaries. In plane geometry, it is easily seen that

for a first-flight medium there corresponds to any given

p only two such paths for any n. Hence the summation

in Eq. (126) runs from i-1 to i-2. Figure 4 illustrates

the two possible paths, L1,1 and L2,1 corresponding to

S= 0 and u < 0 respectively, by which a neutron can

arrive at x after suffering only one reflection. The

unit cell in Figure 4 has its midplane at x-O and the

reflecting boundaries at x b. The cell width,

that is, the distance between the two boundaries is 2b.

The expressions for pLi,n can be obtained by

simply adding the total path length a neutron travels

while making n reflections. The expressions for pi,n

depend on the orientation of the source plane and field

plane x relative to each other and to the x-0 plane.

It has been shown in V.2, however, that one need only

consider field planes for positive x, because the

solution of the half-cell is symmetrical with respect

to x.









The expressions for aLin are derived in
Appendix A where it is shown that Eq. (125) may be
written, for x > O, as

G(v;xx') 1. E (k',v)Ix-x']

+ 1 EI[Zt(k',v) 2nb-x'+(-1)nIxl
2v n-1
(127)
+ E[Zt(,v)12nb+x'-(-1)nx]) for x = 0.

Note that Eq. (127) can be shown to be term by term
equivalent to a more cumbersome series representation
of G(v;x x') derived by Aswad (25).
Finally, the half-cell Green's function
defined by Eq. (78) becomes

H(v;xlx') = E[l t(kO,v)|x-x'I] + E1[t(k',v)x+x'I

1 co
+ I E 1I[Zt(kv) 12"b'x-'l)n]
n=1

+ EL[Zt(k',v) 2nb.+x-(-1)nxi]

+ E,[Zt(k',v) 2nb+x'+(-l)nx]

+ El[Zt(k',v)12nb-x'-(-l)nxi] )(128)


for x =0 and x' O 0











VII.3 Monte Carlo Generation of G(v;xlx')

For complex finite geometries, the complete

analytic form of the first-flight Green's function may

be exceedingly difficult, if not impossible, to derive.

In such cases, it may be advantageous and even necessary

to represent the physical or mathematical system by a

sampling operation satisfying the same probability laws.

Such a process has come to be called a Monte Carlo

method (26).

Essentially, Monte Carlo methods are "paper

experiments." The experiment consists of performing

specific sampling operations many times, the sampling

operations satisfying the criteria mentioned above. As

in many other experimental procedures, the geometry

of the problem does not greatly affect the difficulty

of performing the Monte Carlo experiment. The con-

struction of a function, say neutron density, by

Monte Carlo is conceptually little more difficult for

a square or hexagonal unit cell than it is for a slab

or cylindrical unit cell. This is in striking contrast

to the analytic description of the same function;

there the solution for simple geometries does not, in

general, give any indication of how to proceed to more

complex geometries. Thus there is strong motivation










to generate the plane geometry Green's function G(v;x x')
by Monte Carlo. The transition to cylindrical or even
more complex unit cell geometries would not be difficult;
the general procedure, that is, the Monte Carlo sampling
operations would be essentially the same.
The Monte Carlo generation of G(v;xix'), that
is, the first flight Green's function for a homogeneous
finite medium in an infinite plane geometry with
perfectly reflecting boundaries has been discussed
extensively by Aswad (25) and will be outlined below.
In plane geometry, a cluster of neutrons is
considered to be emitted with unit statistical weight
at the source plane x'. Its direction cosine, P, is
chosen from a random distribution, the distribution
being such as to represent an isotropic source, that is,
a source which has equal probability for emission of
neutrons in an element of solid angle dn about any
direction n; thus in plane geometry the direction
cosine P occurs with equal probability for values
between -1 and +1. The random selection of P is
done by Monte Carlo sampling.
The projection of the path of the neutron
cluster on the x axis is divided into closed intervals

[Xm-l, xmJ [xm xm+1 ... and the statistical










weight of the neutron cluster is computed at points

Xm-l, xm, xm+l, *. The weight of the neutron cluster
at some point Xm, denoted by W(xmjx'), is subject only
to exponential attenuation along the path of travel, the
mean free path between removal collisions being equal
to the reciprocal of the total collision probability
Zt(k',v). Thus


W(xmIx') e- t ( ')/p (129)


If xm+l > xm x', and if W(xmlx') and W(xim+xl')
represent respectively the statistical weight of the
neutron cluster at x, and xm+1, then the quantity
[w(xmJx') W(xm1lx')] must represent the number of
removal collisions per unit time that occurred in the
closed interval [xm, xm+ .
The neutron cluster continues along its
original path until it arrives at a cell boundary
where it is considered to be reflected and sent back
across the cell with direction cosine p', where u*--u,
that is, Ip' Iu." The number of collisions that
occur in the interval [xm, Xm+1 as the neutron cluster
passes through that interval is again recorded. The
process is repeated as the neutron cluster suffers




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SPATIALLY DEPENDENT INTEGRAL
NEUTRON TRANSPORT THEORY FOR
HETEROGENEOUS MEDIA
USING HOMOGENEOUS GREEN’S FUNCTIONS
By
JOHN PHILLIPS CHURCH
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
April, 1963

To my former teacher
Dr. Dudley E. South

ACKNOWLEDGMENTS
The author wishes to express his appreciation
to his chairman, Dr. G. R. Dalton, for his guidance and
assistance throughout the course of this investigation.
In addition, the author is indebted to the
staff of the Computing Center of the University of Florida
for their assistance, and to the Computing Center and the
Department of Nuclear Engineering for their combined
financial support of the computational work for this
dissertation.
The author has been ably assisted in the prepa¬
ration of this dissertation by his wife, Nancy, who typed
the numerous drafts and the final manuscript; its final
form owes much to her concern. It is difficult to express
in words the author's appreciation of her faith in him.
She has made the path to his goal less difficult.

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . . . iii
LIST OF ILLUSTRATIONS vi
LIST OF TABLES viii
INTRODUCTION 1
CHAPTER I DEVELOPMENT OF THE HOMOGENEOUS
MEDIUM GREEN’S FUNCTION METHOD .... 4
CHAPTER II ANGULAR INTEGRATION 15
11.1 Expansion of Angular Dependence in
Spherical Harmonics and Zeroth
Harmonic Approximation 15
11.2 Advantages and Disadvantages of
Homogeneous Green’s Function Method . 26
CHAPTER III SPEED DEPENDENCE ..... 28
111.1 General Speed Dependent Equation ... 28
111.2 The Source Term 30
111.3 One-Speed Equation 34
CHAPTER IV DIFFERENCE DENSITY METHOD 36
IV.1 Development of Difference Density
Equations 36
IV.2 Equivalence of Equations for N and D . 44
CHAPTER V PLANE SUB GEOMETRY 48
V.l One-Dimensional Problem 48
V.2 Symmetry Considerations 54
iv

CHAPTER VI COMPARISON OF N AND D METHODS
FOR SLAB GEOMETRY 60
CHAPTER VII THE FIRST-FLIGHT GREEN’S FUNCTION
FOR A HOMOGENEOUS MEDIUM 69
VII.1 Differential Equation and Boundary
Conditions 69
VII.2 Analytic Form of G(v;x|x’) 77
VII.3 Monte Carlo Generation of G(v;xix’) . 90
CHAPTER VIII SOLUTION OF THE TRANSPORT EQUATION . . 102
VIII.1 Spatial and Speed Dependence ..... 102
VIII.2 Iteration Techniques 115
CHAPTER IX RESULTS AND CONCLUSIONS 119
IX.1 Comparison of the Homogeneous Green’s
Function Method with High Order P
and Sn Methods 119
IX.2 Monte Carlo Generation of the
Green's Function ........... 125
IX.3 Iteration Convergence 132
IX.4 Generation of the Green’s Function
for the Multispeed Problem 135
NOMENCLATURE 137
APPENDIX A DERIVATION OF Gfc(v;x |X») 142
APPENDIX B SAMPLE PROBLEMS 145
REFERENCES 169
v

LIST OF ILLUSTRATIONS
Figure Title Page
1 The Spatial Dependence of the h Functions
of Equations (2), (7), and (8) 13
2 Plane Geometry Transformation 50
3 Symmetry of Neutron Flow at the Boundaries
of Unit Cells in An Infinitely Repeating
Lattice 75
4 Slab Unit Cell with Mirror
Boundaries at x-+b .... 1 85
5 Mirror Symmetry of First-Flight Green*s
Function About Plane at x-0 97
6 Average Values of the Whole-Cell and
Half-Cell Green*s Functions for
Problem 2, Appendix B 128
7 Average Value of the Whole-Cell and Half-
Cell Green's Functions for Problem 4,
Appendix B 129
8 Effect of Convergence Acceleration
Techniques Applied to Problem 2 of
Appendix B. Solution Iterated Until
Residuals Were Equal to 10-5 of the
Neutron Density at Each Spatial Point . . . 133
9 Spatial Dependence of Scalar Neutron
Density for Gold-Graphite Cell of Problem 1. 148
10 Spatial Dependence of Scalar Neutron
Density for Fuel-Water Cell of Problem 2 . . 155
11 Advantage Factor for Fuel-Water Cell of
Problem 2 by Various Approximation Methods . 156
vi

Figure Title Page
12 Advantage Factor Calculated by HGI
Method vs. Number of Iterations for
Fuel-Water Cell of Problem 2 158
13 Advantage Factors for Problem 3 by
Various Calculations 161
14 Advantage Factor for Problem 4 by
Various Calculations 165
15 Spatial Dependence of Scalar Neutron
Density for Cases 1 and 2 of Problem 4 . . . 167
16 Spatial Dependence of Scalar Neutron
Density for Cases 3 and 4 of Problem 4 . . . 168
vii

LIST OF TABLES
Table
Number Title Page
1 Relative Effort for Methods of Solution
of End-Point and Midpoint Equations for
a Total of M Spatial Intervals 109
2 Breakdown of Computation Time for
Problem 2, Appendix B 134
3 Parameters for Problem 1 147
4 Parameters for Problem 2 153
5 Parameters for Problem 3 159
6 Parameters for Problem 4 163
viii

INTRODUCTION
Probably the most fundamental information needed
to describe static behavior of a nuclear reactor is the
speed and spatially dependent neutron collision rate,
that is, £(r,v)N(r,v)v. Such information is needed in
order to predict long-term kinetic behavior of the reactor,
burn-up, power level, etc. Thus,one needs to know the
neutron density, N(r,v), over the entire speed spectrum
and space of the reactor.
Because of chemical binding and crystalline
effects, the scattering kernels are considerably different
for neutron speeds near thermal than for the higher speeds
implied when one speaks of epithermal and fast speed
ranges. In general, the absorption cross-sections of
reactor fuels also have a different functional dependence
on neutron speed at thermal speeds than at higher speeds.
Finally, because of the tremendous range of speeds in
reactor spectra, and because a critical reactor with no
independent sources is an eigenvalue problem if the entire
speed spectrum is considered, the problem of predicting
the neutron density is usually divided into several speed
ranges.
1

2
The present paper deals exclusively with the
prediction of the spatial and speed dependence of the
steady-state thermal neutron density in heterogeneous
media, although the basic method to be presented is
equally valid for any speed range.
There have, of course, been numerous approaches
to various aspects of the problem which can be very
loosely divided into two classes according to whether
the integro-differential or the integral form of the
transport equation was solved. The methods of solving
the integro-differential equation differ to a large extent
only in the choice of functions used to expand the angular
dependence of the terms in the equation. The expansion
functions include spherical harmonics (1-3), Tchebycheff
polynomials (4-7), and first order polynomials, that is,
trapezoids (8, 9).
The methods of solving the integral equation
have been a bit more varied, due perhaps to a more
intuitive derivation of the equation to be solved. As
Osborn (20) has pointed out, some approaches, particularly
as applied to thin absorbers (10, 11), have separated the
problem into two parts—first, a calculation of the
absorption rate in the absorbing media in terms of the
density at the surface of the absorber and second, a
density depression problem of evaluating the density

3
at the surface of the absorbing media in terms of the
density in an all-moderating media. Other integral
approaches (12-14), based on transport theory, depend
on the fact that the absorption rate depends only upon
the zeroth angular moment of the neutron density, the
solution for which is independent of higher angular
moments if scattering is isotropic.
The present paper describes the latter type
of approach. In brief, the method to be presented uses
a kernel for a homogeneous finite medium to formulate an
integral transport equation which predicts the speed and
spatially dependent neutron density in a heterogeneous,
finite unit cell.

CHAPTER I
DEVELOPMENT OF THE HOMOGENEOUS
MEDIUM GREEN * S FUNCTION METHOD
Assume that the neutron density in a general
three-dimensional medium satisfies the time independent
linear Boltzmann equation, and let N(r,vn)drdvda denote
the steady-state angular neutron distribution, that is,
the number of neutrons in the volume element dr about r
having speeds in dv about v, and whose directions of
motion lie within the solid angle da about n; N(r,vn)
will subsequently be called the angular density.
Define V(r,v) to be the probability per unit
time that a neutron of speed v will suffer a scattering
collision in dr about r with a nucleus which is in
motion with respect to the laboratory system. Further,
define Y{r_tv) to be the probability per unit time that
a neutron of speed v will be absorbed in dr about r by a
nucleus which is in motion with respect to the laboratory
system. Finally, define p(r,vi->v,o¿*a) dv doto be the
probability per unit time that a neutron of velocity vVl*
will suffer a scattering collision in 3r about r with a
nucleus which is in motion with respect to the laboratory
system such that the neutron emerges from the collision
4

with speed in dv about v and direction of motion in dri
about n. Then let S(r,vrpdrdvdn denote the source term,
that is, the number of neutrons per unit time being added
to the volume element dr about r, having speeds in dv
about v, and whose directions of motion lie within the
solid angle dn about n.
The transport equation which takes into
account the relative velocity between the neutrons
and collision nuclei which are in thermal motion can
then be written as
vn *VN(r,vn) + V(r,v)+ ^(r,v)J N(r,vn)
= S(r,vn) + Idv’/dn’ PirjVi-^v^'-^rONCr,v*rO . (1)
Equation (1) is an integro-differential equation; to
make the problem of its solution determinate it is
necessary to specify boundary conditions. The boundary
conditions follow from the physical intepretation of
N(r,vn). For example, at an interface between two media,
any packet of neutrons characterized by the vectors
r + Rn. and vn will contain exactly the same number of
neutrons when it enters one medium as when it left the
other; in other words, N(r + Rn,vn) is a continuous

6
function of R for r + Rn at the interface. For a
complete discussion of this and other boundary conditions
applied in neutron transport theory, the reader should
consult Davison (2).
Consider the space of interest to be divided
into R regions, the medium of each region being assumed
homogeneous (the derivation of Eq. (1) assumes that the
medium is isotropic) so that the spatially dependent
probabilities V, Y , and P in Eq. (1) are constant with
respect to r throughout each individual region.
Define a unit function, h, such that
h(p-s) - 1
*= 0
if p =» s
if p s. (2)
For each interface between regions define an index k and
a position vector r^ such that
£t-i
N-
Then
h - h<£k-i-I> - 1» if Ik-1 - I * Ik
(3)
0, if r < Ii5_i> or if I ^ Ik
where r is on the line between rv and r,_ <,. One can
then use Eq. (3) to separate the r dependence of the
probabilities V, Y, and P in Eq. (1) in a special way.
In general, for any operator 6, one can write that

7
R
6 k~l
Use Eq. (4) to rewrite Eq. (1) in the form
vtl* N(r ,vn) + Jf] [h^Ek“£5”h(Ek-l“^]
• [v(k,v)+ / - S(r,vn) + |] [h(£k~£>~h(£k-i“£>]
dv'
da’ P (k, v-i—»v ,nWn)N(r,v *n')
(5)
Choose a particular region, say k*, and remove
the term for the kT region from the summation on the LBS
of Eq. (5), meanwhile placing all other terms of that
summation on the RHS. Thus Eq. (5) becomes
VO* N(r,vn) + [h(£k»-£)~h(rkt._i-£)]
• [v(k* ,v)+ /(k,v)] N(r,vn)
- s(E>v0) " ¿i [h(£k-£>-h(£k-rI>]
k+k*
• [v(k,v)+ /(k,v)] N(r,vrp

8
+
R
[h<£k-í)-h<£k-1-í>]
dv1/ dnf PíkjVi.VjnWijNírjV'n') . (6)
Now using the identity
h(p-s) - 1 - h(s-p)
(7)
one can write
Mr^t-r) - h<£kt„i*"£.) - [l - h<£-£k')] - h(£jj*_i“£)
-1 - [h+h] • <8>
Substituting Eq. (8) into Eq. (6) and putting the
[hCr-r^,) + h(£i£t_i”£)] term on the RHS gives
vn«VN(r,vn) + [v(k* fv)+^(k',v)j N(r,vn)
- s(£,vn) + [hCM^J+hír^j-r)]
• [v(k* #v)+ X'ik1 ,v)] N(r,vct)
“ 5 [h^£k’*£.^“,h^£k-l*’£^l [v(k,v)+ ^(k,v)J N(r,vfl)
k-1
k+k*

9
(9)
• /dv’ldn’ P(ktv-^-»v»q^»o) N(rfv’n*) .
Define a function £>k« (£#vn-|rf ^vtn.') by requir¬
ing that it satisfy the equation obtained by replacing
the RHS of Eq. (9) by the Dirac delta function product
conditions as Eq. (9). Thus
vrrVgfc» (£»V-G-|£' »v,rL*)
+ [v(k’,v)+ /(k’,v)j gk* (r,vn|£f ,v‘n*)
- (10)
The function gkt (£»v£>|£' »V,-A') bas a physical
significance; g^t(£»vn r*,v*n') drdvdA is the number of
neutrons in the volume element dr about r with speeds
in dv about v and with directions of motion lying within
the solid angle do. about A due to a unit point source
at r* emitting one neutron per unit time in direction
A* with speed v* in a first-flight medium which has
the same properties as the medium in region k*.
It is important to understand fully the term
’’first-flight medium” as used here. To be more explicit,
a first-flight medium is a medium in which all collisions

10
with nuclei result In removal of the neutron from the
population. Thus the removal probability for the Green's
function problem is equal to the total collision proba¬
bility in medium k' of the real problem, and the neutrons
denoted by g^t(£»VQ|£* »v,nf) are first-flight neutrons
in the sense that they have suffered no collisions
enroute from the source point to the field point. The
term first-flight neutrons will be used throughout the
following work.
The subscript k' will be dropped from
£kf »v'Q‘ ) *n the following work, if being
understood that the function depends on the properties
of the particular region, k*, chosen.
The function g(£»vn|r',v'n*) can be used as
a Green's function to convert Eq. (9) to the following
integral equation (see Chapter 7 of reference (16)).
r* v* n* v
+ jh(r*-rfct )-»h(rtc»_1-r')] [v(k' ,v')+ ¿^(k* ,v*)] N(r* ,v’nf)
N(r,vn) dv* / do* g(£»vn|r' ,v»n* ) f
- Jjjb(rjj-r*)-h(£k^i-r*)] [v(k,v')+ Y(k,v*)j N(r',v»a*)
k+k»

11
(11)
dv'Mdrp P(k,v'-í-í»v’ »Qn->n*)N(r’ ,v"ri")
Since Eq. (10) describes the neutron density
In a medium in which all collisions result in removal
of the neutron from the population, the only way a neutron
which is emitted at r* may arrive at r is by direct
flights, that is, with no collisions enroute. It will
be assumed and later justified that events at the bounda¬
ries do not change the speed of the neutron. Hence if a
neutron of speed v* is emitted at r# and arrives at r,
its speed must still be v',
From this discussion it can be concluded that
the speed dependence of g(r,vn.|r* ,v,n.f) includes a delta
function in speed so that
(12)
The product solution, Eq. (12), is a result of the fact
that since there can be no scattering in the medium,
then there is no interdependence of {}•£}' and v-vf. Note
that g(vjr,£\|r* ,n*) still has an Implicit dependence on
v due to the v dependence of V(k*,v) and iT(k* ,v) in
Eq. (10)

12
Substituting Eq. (12) into Eq. (11) and carry¬
ing out the integration over vf yields
N(r,vn) - / dr'1 / drV g(v;r,n r* ) r* nf v
♦ [hír'-rkO+hírjgt^-r*)] [v(k\v)+ ^(k* fv>] N(r* ,va)
R
- JUh<£k-l’>-h<£k_1-E,>][v+ y(k,v)j N(r* ,vn*)
k®l
k+kf
(13)
/ dvH / da” P(k, v^—v .n^n' )N(r *, v"rf )
The integration over r' may now be considered.
The meaning of the functions ^h(r*-rfet )+h(rh._^-r> )j and
[hir^t-r^ )-h(rj£t_^-r')] may be easily deduced from Eqs.
(2), (7) and (8) and is illustrated in Figure 1. The
function |h(r * -r^, )+h(rfct ^-r*)] has unit value every¬
where except in region k1 where it has zero value, while
[h(r^-rT)-h(rk_1-r*)] has zero value everywhere except
in region k where it has unit value. It can easily be
seen that the r* integration of the second term on the
RES of Eq. (13) is an integration over all regions except

13
hCr'-rjc»)
hCrjji-r*)
h<£k»-l-£,>
1
O
1
O
1
O
h(r*-r^.) + h(rk»_1-r’)
O - -
h (r^t -r* J-hCr^» _^-r' )
O -
£k'-l £k
r,,. r*
Figure 1 The Spatial Dependence of the h Functions
of Equations (2), (7), and (8)

14
the k*th one while the r* integration of the remaining
step-function terras is a sum over k of r * Integrations
over each individual region k. Thus one can write
Eq. (13) as
N(r,vn) “/dr' / da* g(v;r,n r* ,n’ )S(r* ,vn*)
r* nf
R
+ X,
k“l
[v(k»,v)+ T^k* ,v)J - [v(k,v)+ fi k,v)]
* Jdr* J dry g(v;r,n.|r* ,af )N(r* ,vry)
r*in k n!
+ £ /dFf / g(v;r,n.|r* )
k-1
r*in k nf
* / dv"/dryp^vi’-^n’^n* )N(r* ,v"n") .
Note that the k«k* term in the first summation
is automatically zero, so that no special notation is
needed on the summation symbol.
(14)

CHAPTER II
ANGULAR INTEGRATION
II.1 Expansion of Angular Dependence in Spherical
Harmonics and Zeroth Harmonic Approximation
The angular integration of Eq. (14) may be
performed as follows. Expand all the angular dependent
terms of Eq. (14) in the complete orthonormal set of
spherical harmonics as defined in Chapter 7 of reference
(15). Thus
oo n
N(r,vrO- £ £ N“(r,v)y“(rp (15)
n-0 m—n
oo n
S(r,vn)- £ £ s“(r,v)Y®(A) (16)
— n-0 m«-n
oo a . . *
P(k,v!4v,A^n) “EE Pa(k,v'-¡->v)Y_(A)Y° (n") . (17)
“ ~ a-0 b—a a _ a -
The expansion for the Green's function is more
complicated. Because of possible events at the bounda¬
ries enclosing the space of interest, g(v;r,njr',n.')
depends explicitly on both a and a' and not just on a»a'
as P(k,vu-*vtrP-*rú does. For the moment only the a' de¬
pendence of g(v;r,A¡r*,a') is represented by an expansion
15

16
in the complete orthonormal set of complex conjugate
spherical harmonics. Thus
g(v;rfn
r* ,n')
oo
e§0
s
E
t«-s
gg(v;r,n|r»)Yg (a*). (18)
Substitute Eqs. (15) through (18) into Eq. (14) to obtain
N(r,vrO
EsS(£f
n,ra
K
♦ E
k“l
[v(k',v)+ Y(k? ,v)j
• / dr» / do* Y 8s(vir,n|r,)Tg,>(n,) Z Í(£’» v J
J J s,t
r'in k n*
+ Y / d£* /dA* E 8g(v;r,n|rf )Yg Cq')
k~l J J 8tt
r*in k n*
(19)
* idv"/dn" £ Pa(k,vU-J-v)Yj(rL»)Y^CnM) £ Njj(r’ ,v")Ym(Q’1) .
/ / a»b . n
Equation (19) may be simplified considerably
by Interchanging the order of integrations and summations
and by using the orthogonality relation of the spherical
harmonics, namely,

17
*Sl¡fdG _ <>n,a 4»,1
(20)
where <ín,a is the Kronecker delta function.
After carrying out the angular integration,
Eq. (19) becomes
N<£»v£) • / d£* E ^s2í£.* *v)
J n,m ~
R
+ E
k-1
[v(k* ,v)+ '/(k* ,v)] - [v(k,v)+ ^k,v)]
' /d£f E s“(v;r,n|rf )N¡J(r» ,v)
J ~~ n,m
r*in k
(21)
+ E / dr' /dv,t E sñ k-1 / - / n, m n n “
rfin k
Equation (21) is an expression for the total angular
neutron density and is exact; no approximation has been
made so far.
Now, the relative isotropy or anisotropy of
the nf dependence of giv;*^!^* ,n') depends on the
boundary conditions applied to Eq. (10) and also on the

18
properties of region k'. Recalling the physical meaning
of g(v;r,a r*,n*), one observes'that n* denotes the
angular dependence of the unit point source in a medium
in which all collisions result in Removal of the neutron
from the population. If the space of interest is of
infinite extent, then g(v;r,n r* ,n*) has a double delta
function behavior in direction, namely 8(QrQ.' )¿jn-(r-r*)] »
which merely describes the fact that since scatters
are not considered in the calculation of the Green's
function the direction of motion at r not only must be
the same as the direction of motion when emitted at r*,
but also must be pointed along the vector connecting the
field and source points. In this case, the complete set
of coefficients g|(v;r,a r*) would be required in Eq. (18)
to satisfactorily describe g(v;r,njr* ,n.*).
The problem is entirely different, however, if
one imposes perfectly reflecting boundaries around the
space of interest including the source as would be done
if one were calculating a unit cell in a repeating array.
Collisions or events at the boundaries will reflect
neutrons back into the space of interest with some
direction n^. If the neutron then arrives at r it will
have direction such that + n* and + (£-£*) »
although there will be a definite relation between

19
fi-j, n*, and (r-r *). Further, if the medium characterized
by [v(k* ,v)+ '/(k* ,v)J has a mean free path that is long
relative to the distance between boundaries, then the
neutron may be reflected at more than one boundary before
arriving in the element of volume dr about r.
Now the Integral of the angular density due to
a unit point source per solid angle, integrated over all
directions of emission n', is simply the angular density
due to an Isotropic point source of one neutron per unit
solid angle or a total of 4TT neutrons and is denoted by
g(v;r,nr'). Thus
g(v;r,n r')
g(v;r,n r* ,n’ )da*.
(22)
From the discussion above regarding reflections at the
boundaries, it is obvious that in a finite space enclosed
by reflecting boundaries an isotropic point source will
produce a very nearly isotropic angular density
g(vir,n.|r*).
In order to take advantage of the relative
isotropy of g(v;r,njr*), one proceeds as follows. If
one asks only for information about the number density
N(r,v), the ri dependence of Eq. (21) may be integrated
out to obtain

20
N(r,v) « / dr* £ g“(v$rIr»)Sj(r*,v)
' nfm 1
+
£ [v(k*,v)+ '/(k*,v)] - [v(k,v)+ /(k,v)]
k**l \
dr’ E ®n(v5£ £,)Ií(íl,»v)
n,m
r*in k
(23)
E
+ E / dr’/dvM g“(v;r r* )P (kjV’-i-iv)!^!^ ,v”)
k«l / “ / n,m n
r’ in k
where
N(r,v) -J N(r,vn)dn. .
n
(24)
Expanding the angular dependence of
gCvir^nlr*) in spherical harmonics gives
) * E &nYn(£p *
1 n,m
The coefficients B^(v;rjr*) in Eq. (23) are recognized
as being equal to the coefficients in the angular
expansión of g(v;r,n rf)» a function which has been
(25)
discussed above and which has been concluded to be

21
nearly isotropic for a finite space enclosed by perfectly
reflecting boundaries and containing a medium having a
mean free path long relative to the distance between
boundaries.
Thus the coefficients g^(v;r r’) form a rapidly
decreasing sequence for increasing n. Further, since
Eq. (1) includes a scattering term, the angular density
N(r,vn) should be rather isotropic, even for rather
strong absorbers. A unique advantage of the integral
equation method over the integro-differential equation
method thus becomes apparent; namely, the zeroth harmonic
approximation disregards only products of small terms,
that is, one assumes
K° N°
B0 “o
t'í1 KJ1
< e? nJ
NÍ
\
Due to the orthogonality of the spherical harmonics,
0 n
there are no cross products to consider such as gQ
or gQ Nj which might be of the same order of magnitude
_ 0 „0
as ^o*

22
The situation with regard to the scattering
terra in Eq. (23) is even more favorable* One need not
assume completely isotropic scattering, but merely make
the less stringent assumption that the product of the
zero order coefficients of the Green's function, the
angular density, and the scattering kernel, is much
greater than the product of higher order coefficients,
that is,
g
0
0
/ 1 _i
6l ?! NXA
gO PlN®
On the basis of the above discussion the
zeroth harmonic approximation is made in Eq. (23) and
only the n«0 terms are kept. Thus Eq. (23) becomes
N(r,v)
0/ i
gQ(v;r|
r')S¡J(r»,v)
E
♦ £
k-1
[v(k',v)+/(k»,v)] - [v(k,v)+ /(k,v)]
• / dr’ go
r'in k

23
R
£
dr»/dv" gQ(v;r|r*)P0(k,v»i*v)N°(r»,v”)- (26)
The zero order coefficients may be put in a
more recognizable form as follows. Substitute Eq. (15)
into Eq. (24) to obtain
N(r,v) - £ N¡J(r,v) / Y“(£pdQ
n,m /
E n;<£»v) / Y^(n) Vür Y n,m J
*V4fr Ng(r,v)
0*
where use has been made of the fact that Yq (a)
â–  " 1/V47T . Thus
N“-’v) " vir N
(27)
Similarly, defining
S(r,v) -J S(r,vn)dn.
A
(28)

24
and using Eq. (16), one obtains
(29)
Define
g(v;r r')
/â– 
g(v;r,n. r* )dn
(30)
to be the number density of first-flight neutrons at r
having speed v due to a point source at r* emitting 4TT
neutrons isotropically with speed v in a homogeneous
finite medium. Substitution of the angular expansion
Eq. (25) into Eq. (30) then gives the result that
(31)
For the scattering kernel one has from Eq. (17)
that
P(k,v'i*v,n’i*n)dQ - £
a,b
“ E Pa(k,va»v)Y¡J*(n") / Y^(n)Y°*(n)-/47T dn.

25
- P0(k,vü^v) VÍT Y°*(a")
- P0(k,v!Vv)
which is just the probability per unit time per unit
speed v that a neutron having speed vM will suffer a
scattering collision in region k and emerge from the
collision with speed v. The zero subscript will be
dropped and the function will subsequently be denoted
simply by P(k,v’-4*v).
Finally, substituting Eqs. (27), (29), (31),
and (32) into Eq. (26) gives
N(r,v) - JL / dr* g(v;r|r* )S(r* ,v)
R /
47 f ([v(k*,v)+ /(k*,v)] - [v(k,v)+ Y(k,v)l
K** X \
• / d£* g(v;r|r*)N(r1,v)
r'in k
+ I dr* g(v;r |r* )/dv,,P(k,v,-i^v)N(r* ,vM).
(32)
r*in k
(33)

26
II.2 Advantages and Disadvantages of Homogeneous
Green*s Function Method
The preceding method of converting the integro-
differential transport equation to an integral equation
with a kernel defined by Eq. (10) has at least two
advantages over the more standard methods of solution.
First, as discussed above, the zeroth harmonic approxi¬
mation involves discarding only products of small terms
as in Eqs. (22) and (23). For example, rather than neglect¬
ing the higher moments of the neutron density, as must
be done when the integro-differential form of the transport
equation is used, one need only neglect the products of
the higher moments of the Green's function times the higher
moments of the neutron density.
A second advantage and a unique feature of the
Green's function method described in this work is that the
Green's function defined by Eq. (10) depends only on the
dimensions and geometry of the outer boundaries of the real
problem for which the neutron density is to be calculated,
the total cross section of the medium in the k'th region
of that problem, and the boundary conditions imposed on
Eq. (10). Once the Green’s function has been determined,
it may be used to determine the neutron density in any
heterogeneous medium, the only restrictions being that
the outer boundaries of the neutron density problem must
coincide with those of the Green’s function problem,

27
the boundary conditions must be the same for both g
and N, and the total cross section of the k’-th region
of the neutron density problem must be the same as
that for the homogeneous medium of the Green’s function
problem.
A disadvantage of the method is implied above;
namely, if the outer boundaries and/or boundary conditions
of the neutron density problem do not agree with those for
which the Green's function has been determined, then a
new Green’s function must be determined. Another dis¬
advantage is that, as yet, the method has not been
extended to higher order approximations such as keeping
&n Nn> £n pn Nn teriflS for n =» 0, although such an
extension to higher moments would, in general, follow
the work of Wilf (29), and would not alter the above
mentioned advantages.

CHAPTER III
SPEED DEPENDENCE
III.l General Speed Dependent Equation
Equation (33) is a general speed-dependent
equation and is valid over the entire speed spectrum,
the limits of the speed integration being from v" - 0
to vn “Co. As indicated in the introduction, the
present investigation will be confined to speeds in
the thermal region. In practice, there exist no true
sources for the thermal energy region other than slowing-
down sources, and hence S(r*,v) - 0 in Eq. (33). The
speed integrations in Eq. (33) are then separated into
two parts, one for 0 «= v ^ v* and one for v =» v*. The
speed v* represents the upper limit of the speed region
of interest, and is characterized by the following
assumptions (17):
1. For v =»v*, the neutron density speed distri¬
bution is proportional to 1/v2 (that is, the
neutron flux is proportional to 1/E).
2. The probability of a neutron being scattered
from a speed below v* to a speed above v* is
zero.
3. The motion of the collision atoms is Ignored
with respect to that of the neutrons with
speeds above v*.
28

29
4. The Maxwellian distribution for thermal
neutrons or collision atoms has negligible
magnitude for speeds above v*.
5. For neutron energies above v*, the energy
transfer is computed assuming that the col¬
lision atoms are free from chemical bonds.
This implies that v* should correspond to a
neutron speed well above that required to
excite vibrations in the collision molecules,
so that the collision atoms would recoil
freely at the Instant of impact.
6. For neutron energies below v*, the energy
transfer kernel P(k,viS».v) will depend on the
particular scattering model chosen. A
discussion of particular energy transfer
kernels and their scattering models is given
in references (17, 31).
The S(r*,v) term in Eq. (33) is then modified
to include all neutrons scattered into a speed interval
dv about v from speeds above v*. Thus Eq. (33) may be
rewritten as
dr' g(vjr r*)S(r*,v)
r'in k
r’in k

30
+
I dr*
rrin k
v*
g(v;r r’)
dv” P(k,vü
v”-0
.v)N(r' ,v”)
(34)
where
S(r’,v)
-oo
dv” P(k,v”,.v)N(r* ,v”)
(35)
III.2 The Source Term
Assumption (5) of III.I implies that ordinary
slowing down theory may be used to derive the scattering
kernel for v v* (18, 19). Define
of *»
m~l
m+1
(36)
Then ov is the minimum speed that a neutron of initial
speed v can have after collision with a target nucleus
of free atom mass m; the energy transfer probability,
that is, the probability that a neutron of energy E*
will after scattering have energy in the interval dE
about E is (see (18))
TT(r,E^E)dE - — n—=- for a^E* » E ^ E’
E» l-cr(r)
- 0 , for E «= <*2E»
0
9
for E =» E*
(37)

31
or, changing variables,
Tf (r ,vi»»v)dv » Tf(rfE*-»E)g~ dv
so that
7T(r,v-Uv) - If(r,EL»E)
1 dE mv
E* [l-a^r)] dv [l-c*2(r)]
2v
^l-o?(r)J v*2
for cor' S v v'
*=> 0 , for v «= ccv’
- 0 , for v >v*
Now, the differential scattering cross section,
rs(r,vls»v) dv, that is, the probability that a neutron
of speed v* will be scattered at r into dv about v is
equal to the product of the total scattering cross
section at r for neutrons of speed vf times the proba¬
bility that the speed of the neutron after collision
will be in dv about v. Thus
(38)
Is (39)

32
Then the probability per unit time that a neutron of
speed v* will suffer a scattering collision at r such
that it emerges from the collision with speed in dv
about v is simply
P(r,vi».v)dv - v*Zs(r,vi*-v)dv (40)
so that for v* =» v* one has from Eqs. (38), (39) and
(40) that
P(r,vl»v) - 2w£g(r,v»)
2vZs(r,v’)
jl-a^(r)J v*
, for ccv* -s v , for v <= ctv'
, for v = v’
(41)
Assumption (1) of IIX.l implies
.x A(r)
N(r,v’) - — ..tr.i —. , for v' v* (42)
v* 2
where A(r) is an arbitrary constant with respect to
speed and is normalized to give a stationary distribution
for N(r,v*). Substituting Eqs. (41) and (42) into
Eq. (35) yields

33
oo
2vIs(k)A(r»)
/
y/cc(k)
1
(43)
v'^v#
where LQ has been assumed to be independent of v in the
speed range v* s v” s v/a(k).
Carrying out the Integration in Eq. (43) gives
- 0 f if v c a(k)v*
(44)
where the constant A(r) determines the source strength;
A(r) will be assumed to equal a spatially independent
constant A, that is, one assumes a spatially flat
siowing-down source distribution. As Honeck (22) has
pointed out, A(r) could, if necessary, be approximated
by the shape of the resonance group neutron density as
predicted by diffusion theory and the use of suitable
resonance group cross sections given in Veinberg and
Wigner (19).

34
III.3 One-Speed Equation
To obtain the one-speed equation, one must
assume separability of the spatial and speed dependence
of the neutron density. Then integrating Eq. (33)
over all speeds and using Maxwellian averaged cross
sections, one has
dr* g(r r*)N(r*)
r*in k
r' in k
(45)
where use has been made of the fact that
/
oo
P(k,vÜ3»v)dv - V(k,v")
(46)
v-0
which is just the total probability per unit time that
a neutron in region k and having speed v" will suffer a
scattering collision.

35
In the one-speed case motion of the collision
atoms may be ignored, so that in conventional notation
Eq. (45) may be written as
N(r) -
*fr ^ g(£
r’)S(r')
+ ^7f E vDü k-1
r’in k
1 R f
+ jqf E v¿s 4 k-1 /
r’in k
(47)
Finally, the last two terms in Eq. (47) may be combined
to obtain
N(r)
47T / d£* g(£|£’)S(r')
+ ^ v[Zt )-Za] / dr* g(r|r*)N(r')
r’in k
(48)

CHAPTER IV
DIFFERENCE DENSITY METHOD
IV.1 Development of Difference Density Equations
An alternative to the preceding method of
describing the real neutron density, N, in a hetero¬
geneous medium is to define a difference density, D,
as the deviation from a known density, N*, for a similar
or reference space. Although it may appear at the outset
that such a method is analytically identical to that
developed above for the real density N, nevertheless,
a number of authors have found such a procedure to be
advantageous (13) and, in fact, necessary (12) in order
to obtain results for specific infinite geometry problems.
It will be of considerable interest, therefore,
to develop equations for the difference density, D,
parallel to those above for the real density N. A
direct comparison of the two methods will then be made
in order to determine their relative advantages when
applied to the specific problem to be considered in this
paper.
Thus the difference density D(r,vn) is defined
such that
36

37
D(r,vn) â– * N(r,vn) - Nf (r,vn)
(49)
or
N(r,va) = D(r,vn) + N”(r,vn)
(50)
Using primed symbols to denote the reference
space, one can write the transport equation for the
assumed known angular density as
vn* Vn* (r,vn)
- S *(r,vn) +
V‘(r,v) + V* (r,v) N* (r,vn)
dv' / dn* P'ir^-^v^njN' (rfv'n')
(51)
Substituting Eq. (50) into Eq. (1), subtracting Eq. (51)
from the resulting equation, and grouping all of the
N’(r,vn) terras on the RHS gives
vn* \7D(r,vn) + [v(r,v)+ /(r,v) D(r,vn)
« S(r,vn) - S*(r,vn)
- [v(r,v)+ X(r,v)j - V’(r,v)+ ^(r,v)
+ /dv’ / dn' Pir.vUvjnVODir^'n')
N*(r,vn)
(52)
+ dv* dn*
P(r,v^VjnU-rO-p* (r,v-*-*-v,rii->*n)j N* (r,v’n')
As before, Eqs. (3), (4), and (8) are used to separate
the spatial dependence of the terms in Eq. (52). Thus

38
v£\*yD(r,vn) + [v(k* ,v)+/(k',v)] D(r,vn)
- S(r,vn) - S' (r,vn)
+ [b<£l~£jk*>+hCjTfc• [v(k»,v)+ yCkf,v)J D(r,vn)
jj
- £ [h(£k"^~h^£k-l~r^] [v(k,v)+ iT(k,v)] D(r,vri)
k-1
k+k?
- ¿ [hCr^-rJ-hir^-r)]
R
+ ¿1 [h(-k"-)'h<^-l‘-)]
[v(k,v)+ /]
[v*(k,v)+ /*(k,v)]
N' (£»vn)
dv'/ da* PíkjV-Wv^n^n^OrpV'n*)
+ ^ [h(r^-r)-h(r^.i-r)] Jdv'J da* [ P (k,v-L>v^nUn)
(53)
- P' (k^vJ-^v.A^rp] N* (r,v'n').
The boundary conditions imposed on Eqs. (51) and (53)
are the same as those for Eqs. (1) and (9). Thus the
Green's function which satisfies Eq. (10) may be used
to convert Eq. (53) into an integral equation as before.
Then, using Eq. (12), one may perform the integration
over v* to obtain

39
D(r, vn) - J dr’J dn* g(v;r,n|r* ,n* )|S(rf ,vn* )-S* (r* ,va* )
+ [**(£♦-££, [v(k* ,v)+ /(k* ,v)] D(rffvn*)
R
- £ [h(£k-r' )-h(rjj.i-r*)] [ V(k,v)+ ^(k,v)J D(r' ,va')
k«l
k+kf
*" i?i [h^“-,)~h^k-i">-*] j[v+ ^]
- [v* (k,v)+ y* (k,v)] 1 N» (r» , vn’)
+ ^(£fc-l~£l* /dv"ídír Pík^v^Wj’ )D(£' jW)
+ J] [k^£k”£* )“h(£k.i~£* )] jdv" Í «^"[pik^iWir^-n*)
,(54)
- P’ (k,vüvy,n'-i-fi')] N* (£*, v"n" ) > .
The £* integration for Eq. (54) is identical
to that for Eq. (13) and the result is
D0r»v£p - e(v »£»{!£* »Q* )[s R /
+ £ [v(k\v)+ /(k«,v)] - [V(k,v)+ Xk,v)l
k-1 J
d£* J dn.* g(v}£,n|£* ^Q,)D(£* »vn*)
r? in k n*

40
+ ^ I dr’ / e(v;r,n. r* ,o*)
r * in k ÍV
dvM/ do” P(k,vü»v,n'W )D(r',v”n”)
R
&
[v(k,v)+ /(k,v)] - ¡V»(k,v)+ /f(k,v)j
• / dE* / »n' )N1 (r’ »vn*)
r*in kYi’
+ E I d£f j s(v!£>{! £* »G’ )fdv"i dQ" [P(k,v^»v,n.”»a*)
r*in k n'
(55)
- P* (k.v’ií.v.n’^A’)] N* (r* ,v"A") .
The angular integration presents no new
problems and may be performed as in Chapter II. Making
the zeroth harmonic approximation and asking only for
the scalar neutron difference density, one obtains
D(r,y) - / dr* gCvjrjr») [s(rf ,v)-S’(r* ,v)]
R
+ i-~ £ [v(k* fv)+ Y(k* ,v)l
47T J
\ i
- [v+ Y(k,v)]
dr* g(v;r|r*)D(r',v)
rfin k

41
~r V / dr1 g(v;r |r')/dv” P(k,vu»>v)D(r',v”)
47r k-1 / ~ /
r'in k
i R
— £
47T k-1
[v(k,v)+ >ík,v)j
- [v*(k,v)+ V' dr' g(v;r rf)N*(r*,v)
r'in k
R
+ ET . dZ' e r'in k
(56)
J dv” [p(k,vü»v)-P* (k,v'i^.v)] N' (r* ,v”).
Writing Eq. (56) specifically for the thermal
spectrum yields
D(r,v) - y /dr* g(v;r|r')[s(r»,v)-S'(r',v)]
r'in k
([V [v(k,v)+/(k,v)] / dr' g(v;r |r')D(r',v)
r'in k

42
+
v*
dv" P(k,v”
O
«.v)D(rf ,v”)
1
47r
[V(k,v)+
- [v* (k,v)+ ' (k,v)j / dr* g(v;r r’)N*(r’,v)
r'in k
♦ 3rr d^' B(v:£|í'>
r’in k
v*
(57)
dv” Jp(k,viU.v)-P* (k,víi*v)J N’ (r’ ,v”)
v”-0
where, in region k,
S(r’,v) - S*(r’,v)
co
dv” P(k,v’-i»v)D(r* ,v”)
V”-V*
-00
(58)
+j dv” [p(k,viVv)-P* (k#vii»v)j N’(r*,v”)
v-v*

43
and the scattering kernels for v =» v* are determined
from ordinary slowing down theory as in 111.2.
Finally, the one speed version of Eq. (56) is
D(r) m I dr» g(r|r* )[s(r* )-S» (r*)]
+ Irf X, vKt- ^t] / d£* *(£ r* )»(£•)
k-1
r'in k
+ hr £xv *.« J d£f g(£|r* JDCr»)
r*in k
“ |rf X, v f^t(k)~ ^t] / d£* g(r|r,)N,(r»)
k-1
r'in k
~jfr ^ v[ls(k)- Ig(k)] J" dr* gCrlr’jN* (£»)
(59)
rfin k
or if various terms are combined,
D<£> “ |rr / d£' sC£|£*>[s<£*>-s* <£*>]

44
+
X
Í7T
v[Zt(k’)-Ia(k)]
in
dr* g(r|rf)D(rf)
k
'w ¿ *<»- w
(60)
IV. 2 Equivalence of Equations for N and D
In order to establish the equivalence between
Eqs. (34) and (57) and between Eqs. (48) and (60),
the one-speed equations will be considered first.
Subtracting Eq. (60) from Eq. (48) and using
Eq. (50) gives
N'(r) - J-r / dr* g(r r')S'(r')
att
f »[Zt(k')- L(k)] / dr' e(rr*)N'(r
k-1 J ~
r'in k
3tT ^ vIXi"* I*(k>] j d£* s(£|£,>N,^£,>
r*in k
1
ATT
dr* E(£ £f)S,(r*)
r*

45
+ / d£* S(£|£* )N’ (r*).
r*in k
Comparing Eqs. (61) and (48), one observes that the
requirement that Eqs. (60) and (48) be equivalent is
simply that T^ik*) - £'(k'), that is, the total cross
section for the k’th region of the reference space be
identical to the total cross section of the kfth region
of the real space of interest. This simply says that
the Green’s functions in Eqs. (48) and (60) must be
identical, that is, the Green’s functions in Eqs.. (48)
and (60) must both satisfy Eq. (10) and its boundary
conditions. But this is just the condition imposed in
the original derivation, and hence £t(k*) is equal to
4 (k*); equations (48) and (60) are therefore proved
to be equivalent formulations of the same problem.
Similarly, subtracting Eq. (57) from Eq. (34)
and using Eq. (50) gives
1 R f
N’(r,v) “ ;|77=- £ / dr» g(v;r| r* )S’(r*, v)
r* in k
(61)

46
- [v(k,v)+ /(k,v)]
dr» g(v;r|r»)N»(r',v)
r*in k
v*
+ ]T I dr* g(vjr|r*)í dv" P(k,vü»v)N»(r*,v”)
r*in k v«0
4TT ¿
Ex j[v(k,v)+ /(k,v)]
- [v» (k,v)+ /» (ktv)] /dr» g(v;r |r» )N* (r» ,v)
r'in k
1
3tt
dr»
r’in k
g(v;r r»)S»
47
hr ¿ [v - [v* (k,v)+ y»(k,v)]j J dr' g(v;r|r* )N* (r* ,v)
r*in k
(62)
v*
^7f £ / d£.' 6(vir|r») / dv” P' (k,v!U.v)N* (r* ,v") .
r*in k v”-0
Comparing Eqs. (34) and (62) one again observes
that, just as in the one-speed equation, the requirement
that Eqs. (34) and (57) be equivalent is simply that
[v(k* ,v)+Y^k* ,v)J be equal to [v* (k' ,v)+ Y* (k* ,v)] ,
that is, the medium of the k*-th region of the reference
space must be the same medium that is in the k*-th
region of the real space of interest. As has been
mentioned above, this condition was imposed in the
original derivation so that the equivalence of Eqs. (34)
and (57) is established. A discussion of the relative
usefulness of the two formulations will be deferred
until the general equations have been specialized to
slab geometry.

CHAPTER V
PUNE SUB GEOMETRY
V.l One-Dimensional Problem
The equations of the previous chapters were
written for a general three-dimensional geometry, and
the Green's function g(v;r|r*) was interpreted as the
number of first-flight neutrons per unit element of
volume at r having speeds v due to an isotropic point
source at r* emitting 477' neutrons per unit time having
speeds v in a homogeneous medium. Before specializing
Eqs. (34) and (57) to an infinite slab geometry it will
be convenient to define a new Green's function G(vjr r'),
such that
G(vjr|r*> “ 4g Thus G(vjr r') is the number density of first-flight
neutrons at r having speed v due to a unit isotropic
point source at r* emitting neutrons of speed v in a
homogeneous medium. Rewriting Eq. (34) in terms of
G(v;r|r') gives
(63)
48

49
R í .
N(r,v) - / dr’ G(v;r|r*)S(r*,v)
r'in k
+
[v(k',v)+ /(k',v)]-[v(k,v)+ yOs,v)]
i
' J dr» G(vjr|r')N(rf,v)
r'in k
v* (64)
j dr* G(v;r|r')/ dv" P(k,vü»v)N(rftv”)
rfin k
v'*-0
and similarly for Eq. (57).
The transformation to plane geometry may be
performed as follows. Referring to Figure 2, the element
of volume dr' may be expressed as
dr* - 2 TT a da dx*
Define
R » r - r*
r
r*
From Figure 2,
R
and Eq. (65) one obtains
( |x~x*| 2 + a2)1/2
x-x'
(65)

r ■» fixed field point
r* ** variable source point
dx'
Figure 2 Plane Geometry Transformation

51
so that for constant lx-x'1
Hence
dR - a da
R
dr*
2 7T R dR dxf
(66)
where the integration is from to ^i-O, and the R
integration is from R » |x-x’| to R -oo .
Observing that in plane geometry the r de¬
pendence of the functions N, N*, D, S, and P is simply
a spatial dependence on x, one may write Eq. (64) for
plane geometry as
oo
N(xfv) - / dx' S(x»,v) / G(v;r r') 2 7ÍR dR
x*in k R-lx-x’i
R / /
£ [v(k\v)+ fl(k\v)l -fv(k,v)+ Y(k,v)l
S-l I
CO
• J dx’ N(x»,v)J G(v;r|r') 2 TT R dR
x'in k R-|x-x*|

52
dv" P(k,vü
x'in k v”-0
.v)N(x* ,v”)
00
G(v;rIr*) 27TR dR
R- |x-x'| (67)
Define
G(v;x x*)
oo
G(v;r|r*) 2TTr dR
R-lx-xM
(68)
The function G(v;x|x*) is the density of first-flight
neutrons at x having speeds v due to a unit isotropic
plane source at x' emitting one neutron per unit time
with speed v in a homogeneous medium. This function is
well known for an infinite medium (see Chapter 2 of
Case, e£ al. (21)) and will be discussed extensively
for a finite unit cell in Chapter VII.
Using Eq. (68) to rewrite Eq. (67) gives
N(x,v) * dx' G(v;x|x')S(x',v)
k-1 J
x'in k
+ X,
k“l
[v(k* ,v)+ /(k\v)]-[v(k,v)+ Y(k,v)]
dx* G(v;x x*)N(xf,v)
x'in k

53
v*
+ dx* G(v;x|x*) / dv” Pík^vü^ONte',v") . (69)
x'in k v”«0
Similarly, Eq. (57) for plane geometry becomes
D(x,v) ~ Z f dxt G(v;x|x») [s(x',v)-*S'(x* ,v)]
k-l J
x'in k
R
- X.
k-l
’
[v(k* ,v)+ Y(k* ,v)] ~[v(k,v)+ Y(k,v)J
dx* G(v;x|x*)D(x*,v)
x'in k
R
+ z
k-l
v*
dx* G(v;x x*) / dv” P(k,vü*v)D(x*,v”)
x'in k
yi'al)
R
¿i
[V(k,v)+ y(k,v)j -- JV* (k,v)+ ^(k,v;
j dx' G(v;x|x* )N* (x* ,v)
x'in k
v*
(70)
+ Z I dx' G(v;x x’) / dv” [p(k,v!Vv)-P' (k,v!i».v)J N* (x* ,v").
x'in k
v”—0

54
V.2 Symmetry Considerations
For subsequent work the space of interest
will be assumed to be a plane geometry symmetrical
about the plane x-0. In the summations over k as
previously discussed, k-1 denoted the left-most region
in space. The regions will now be counted differently,
letting k«l denote the region adjacent to, and to the
right of, the x~0 plane. Thus a summation over k from
k»l to k*»R is replaced by a summation over k from
k - -R/2 to k - R/2. Let
(71)
denote the number of regions in the half-space. For a
general function f(x) which is symmetric about x-0,
that is, f(x) - f(-x), one can then write that
x'in k
- T f(k) / dx* G(v;x x*)N(x*,v)
k"—R2 J
x’in k
/
E + É f k—R2 k«1 1
/
x’in k

55
dx1
x,=»~b
f(x»)G(v;x x’)N(x’,v)
(72)
where b and -b denote the right and left boundaries of
the space of interest. The notation used is defined
such that
£2
.L +
k“C^
«n '
+ ,¿*
k-°n(
02
k kWk
f V +
C_
+ JL fk
k»cn
(73)
and
(74)
m
dx’ + ... + / dx’
X’“Cj
f(x’) “ / dx’f(x') + ...+/ dx’f(x*) .
x'»c
n
ic’^c.
x’-c
n
In the first integral in Eq. (72) let x*«=-y,
dx’=-dy. Observing that f(x)~f(-x) and N(x,v)«=N(-x,v),
one can write
0 0
dx'f(x’)G(v;x|x’)N(x*,v) - - / dy f(-y)G(v;xj-y)N(-y,v)
"x'^-b y*=b
b
-/ dy f (y)G(v;x|-y)N(y,v)
y-o
(75)

56
Then let y-x* so that Eq. (75) becomes
_0 b
dx'f(x* )G(v;x|x* )N(x' ,v) *=* / dx'f(x' )G(v'x|-x* )N(x* ,v).
x'—b x'-O (76)
Substituting Eq. (76) Into Eq. (72) yields
f f(k) / dx'G(v;x|x')N(x*,v)
k-1 ' 1
x'in k
b
J~ dx'f(x’) [g(v;x|x')+G(v;x -x’)]n(x',v)
x'-0
J»2
(77)
px f / dx’ [g(vjx|x* )+G(v;x |-x')] N(xf ,v) .
x'in k
As noted above, G(v;x|x*) is the scalar
density of first-flight neutrons at x due to an Isotropic
unit plane source at x*. If one defines
H(vjx x')
G(v;x x*) + G(v;x -x')
(78)
to be the half-space Green's function, that is, the
scalar density of first-flight neutrons at x due to
isotropic unit plane sources at both x* and -x*, then

57
equation (69) may be rewritten for the half-space in
the form
N(x,v)
dx* H(v;x|x*)S(x*,v)
x' in k
52
+ ;
k-1
[v(k*,v)+ /(k* ,v)] -[v(k,v)+ 7(k,v)]
dx’ H(v;x|x*)N(x* ,v)
x’in k
*2
v*
+ ^ I dx* H(v;x|x’) / dv" P(k,v’-!*v)N(x* ,v")
x*in k v^-O
while Eq. (70) becomes
D(x,v) - /dx* H(v;x|x*)[s(x*,v)-S*(x*,v)j
k-1 I
xf in k
*2
+ y
k-1
[v(k’,v)+ /(k’,v)]-[v(k,v)+ ^(kfv)]
. / dx* H(v;x x*)D(x*,v)
(79)
x’in k

58
v*
&
+ ) / dx* H(v;x x») / dv" P(k,v!i*v)D(x' ,v")
RJ 1 J
x'in k v"-0
R2
[v(k,v)+ /] -[v'(k,v)+ V» (k,v)j
dx* H(v;x|x*)N'(x',v)
x'in k
v*
+ dx* H(vjx|ac* > / dv" £p(k,vü»v)-P' (k,v'l»v)j N* (x*,
x'in k v"«0
The one-speed equations may also be written
for the one-dimensional half-space. Thus Eq. (48)
becomes
N(x) « / dx* H(x|x*)S(x*)
x»>0
£2 v|jt(k')- La*)] / dx' H(x|x*)N(x')
k-1 J
x'in k
while Eq. (60) becomes
D(x) - /dx' H(x|x*) [s(x* )-S* (x* )J
x'=»0
(80)
v" ) .
(81)

59
+ v[It(k’)- Ia(k)J I dx* H(x|x’)D(x*)
x’in k
- *L v[la(k)- Ia] / dx’ H(X|X’)N, k~l J
x’in k
(82)

CHAPTER VI
COMPARISON OF N AND D METHODS
FOR SLAB GEOMETRY
For the direct comparison of the formulation
in terras of real density, N, with the formulation in
terras of difference density, D, only the one-speed
equations will be considered, namely Eqs. (81) and (82).
Similar arguments would apply to the multispeed equations.
Suppose that all R regions of the reference
space contain the same medium and that the reference
space itself constitutes one spatial period of an
infinitely repeating lattice, that is, the reference
space is a unit cell. Then the neutron density N*(x)
in the reference space is that for an infinite homo¬
geneous medium and may be assumed spatially flat.
Since the neutron density and medium are both
spatially uniform in the reference cell, one may assume
that the statement of neutron conservation holds not
only over the entire reference space, but also over any
arbitrarily small element of volume in that space. Thus
at any point
S*(x) - S* - v I^N’ . (83)
60

61
Substituting Eq. (83) into Eq. (82) and
cancelling terms where possible, one obtains
D(x) « / dx* H(x|x*)S(x*)
x'=-0
+ Y2 v[Zfc(k-)- !a(k)] / dx* H(x lx* )D(x*)
k-1 J
x*in k
- yf v I_(k)N* / dx' H(xlx*)
k-1 J
(84)
x*in k
Then using Eq. (50), one may combine terms to write
D(x) - / dx* H(x|x*)S(x*)
x*=-0
Ro r- f l
+ Y v ¿+(k*) / dx* H(x x*)D(x*)
k“l J
x*in k
R2 v f .
- £ v ¿a(k) / dx* H(x x*)N(x*)
k-1 J '
x*ln k
(85)
Adding and subtracting the quantity
112 v It(k») / dx’ H(x|x»)N(x')
k-1 J
x*ln k

62
to Eq. (85) gives
D(x) -J dx' H(x|x')S(x')
x’=-0
+ 5Z2 v[ZtOc*>- 5Ta] / dx’ H(x|x*)N(x‘)
k-1 J ‘
x'in k
R2 r- f ,
- Y v ¿t(k') / dx* H(x x*)N
k-1 T J 1
x'in k
dx* H(x|x*)S(x»)
x*=»0
£? v[Zt(k’)-Ia(k)] [ áx’
k-1 J
x’in k
H(x x*)N(x»)
- v
It(k»)N* /dx* H(x|x*) .
x*»*0
But from Eq. (10)
v It(k')H(x|x’)dx* - 1
x*=-0
(86)
(87)
so that using Eqs. (50) and (87)f one can reduce Eq. (86)
directly to Eq. (81).

63
The direct reduction of Eq. (82) to Eq. (81)
having been established, the question remains whether
the use of Eq. (82) has any advantages over the use of
Eq. (81). It will be of particular interest to consider
whether or not the complexity of Eq. (82) can be reduced
by relating the real source S to the source S' in the
reference space. A few such special cases are considered
below. In each case the assumption is made, as it was
above, that all R regions of the reference space contain
the same medium and that S' and N' are spatially flat.
Case 1 S(x) - Sf Over All Space
Equation (82) then becomes
dx' H(x x')D(x')
x’in k
x'in k
dx* H(x lx’ )D(x*)
x'in k
/
dx' H(x x') +
v£t(k»)
S*
x'in k
(88)

64
where Eqs. (83) and (87) have been used to obtain the
last tena of Eq. (88).
Since S(x) ** S' implies that S(x) is spatially
independent, Eq. (87) may be used to show that the first
term in Eq. (81) becomes
/ dx* H(x|x*)S(xf) - S' / dx' H(x|x*)
x*=»0 x*=»0
v £t (89)
Thus Eq. (88) can be reduced directly to Eq. (81) in
the same manner that Eq. (82) was reduced to Eq. (81),
so that there is no advantage of the difference density
formulation over that for the real density N. The
reason for this is that, although it is given that
S(x) • S «* S', one must still calculate N' based on S',
and the Nf terms then act as source terms in the differ¬
ence density equations.
Case 2 Six') «0 For Region k-1, S(x') - S'
For Regions k J 1l *
Equation (82) becomes
D(x) - - / dx' H(x x')S'(x’)
x'in 1

65
+ £ v[lt] / dx' H(x x* )D(x* )
k-1 J
x’in k
v[la(k)- £¡(k)] Jdx’ H(x|x»)N’(x’) .
x’in k
(90)
To specialize further, let R2 - 2 and k’ - 2. Then
Eq. (90) becomes
D(x) - - S’ / dx* H(x|x’)
x’in 1
+ v
[It<2>- ¿a (1)] J dx’ H(x|x’)D(x’)
x’in 1
+ v Za(2) / dx’ H(x|x’)D(x’)
x’in 2
- v[la(l)-I¡(l)] N’J dx’ H(x|x’).
x’in 1
Using Eq. (83) one may write Eq. (91) as
(91)
D(x) - v[lt(2)-Ta(l)] / dx’ H(x|X» )D(x’)
x’in 1

66
+ v I_<2) / dx’ H(x|x*)D(x’)
x’in 2
v Ia N* / dx* H(x x’)
(92)
x'in 1
or, again substituting Eq. (83) into the last term of
Eq. (92), one finally obtains
D(x) - v[lt(2)-Ia(l)] / dx» H(x|x' )D(x*)
x’in 1
+ v 1.(2)J dx’ H(x|x’)D(x’)
x’in 2
Ia(l) S'
C<2>
dx’ H(x x’)
x’in 1
For this special case, Eq. (81) becomes
N(x) - S’jdx* H(x|x’)
x’in 2
(93)
+ v[lt(2)- Ia(l)] J dx* H(x|x)N(x*)
x’in 1

(94)
+ v Is(2)
dx* H(x x»)N(x»)
x*in 2
The obvious difference between Eqs. (93) and
(94) is the source term. In a typical application of
calculating the scalar neutron density in a space
containing a gold foil in a graphite medium, the ratio
Ia(l)/ Za(2) would become
Au
_ 5.79 4
f ■*- “ — - 1.94 x 10* .
T (2) TC .000299
a a
It can be seen that in a finite medium containing a
highly absorbing region, even though it be of small
spatial extent, the neutron density will not necessarily
be a small deviation from a known density in a reference
space with no absorber. In fact, the deviation in
magnitude of the densities will be quite large, even
though the spatial shape of the neutron density N may
not be vastly different from the spatial shape of the
known density N*.
It seems obvious now that for problems involv¬
ing finite geometries the analytic method of solving for
a difference density, D, offers no advantages over

68
the direct solution for the unknown density N; the
difference density method may, indeed, be limited in
usefulness to the solution of infinite geometry problems.

CHAPTER VII
THE FIRST-FLIGHT GREEN*S FUNCTION
FOR A HOMOGENEOUS MEDIUM
VII.1 Differential Equation and Boundary Conditions
The first-flight Green*s function
gj^i (r,va|r* ,v*r\*) for a general three-dimensional homo¬
geneous medium is defined such that it satisfies Eq. (10),
namely
vry Vgk* (r,vn| r' ,v’n*)
+ [v(k* ,v)+ /(k* ,v)j gk» (£»vo. r* ,v*n*)
“ S(r-r* ) (95)
further, gk, is subject to the same
boundary conditions that are applied to the angular
density N(r,vn).
Considering the RHS of Eq. (9) as a fictitious
source Q(r,vn), one may rewrite Eq. (9) as
vn*V N(r,vfi) + [v(k* ,v)+ ^(k* ,v)] N(r,vrp - Q(£»vn) . (96)
69

70
Thus N(r,vn) represents the first-flight angular density
in a medium as discussed in Chapter I, the removal pro¬
bability being equal to the total probability of col¬
lision in region k*, with a source of neutrons Q(r,vn).
If the homogeneous medium k’ is of infinite
extent, then the solution to the one-speed version of
Eq. (95) is known to be (see Chapter 2, of Case, et al,
(21))
gk* <£*{} £* ) ■
v r-r*
S(n-n' )S(n - ^rf) (97)
where the superscript oo on g signifies that Eq. (97) is
valid only for an infinite medium or for a finite medium
enclosed by a non-re-entrant surface with vacuum boundary
conditions, and ft’ denotes the unit vector (r-r1)/ |r-r*|
The extension to the speed dependent problem is obvious
from the discussion of Chapter I and Eq. (12). Thus
£¡? <£f »vQ|r» ,v'n*)
- Itv>
r-r*
v r-r'
(98)
<£(v-v’ )¿‘(n-n* )£(n - "r-r *)

71
Integrating Eq. (98) over v' and n! gives
-L g£? e
r-r'
r-r
t| 2
S(a - r^r
integrating over n then yields
(99)
gg?(v;r r')
- It
r-r’
v r—r* 2
(100)
where use has been made of Eq. (30).
The simple form of Eq. (100) is due to the
fact that the medium k' is homogeneous so that the
collision probability is not a function of the flight
path of the neutron. As pointed out by Case, et al.
(21), the case of a spatially variable collision proba¬
bility is a considerable complication because absolute
coordinates instead of only relative coordinates between
the field and source points enter the problem. It might
be well to point out again that a unique advantage of
the homogeneous Green's function method being described
is that it completely avoids the complication of a
spatially variable collision probability in the evalu¬
ation of the Green's function.

72
From Eqs. (63) and (100) one can write the
expression for G(v;rlrf) for an infinite medium. Thus
.00
(vjrlr*)
- L(k\v)|r-r'|
e * I »
47T v| r-r’| 2
From Eq. (68) one can then evaluate G(v;x x’) for an
infinite medium. Thus, using the transformations of
V.l, one obtains
oo
G00 (v;x|x’)
r-r*
4 7r vlr-r'l 2
2 7T R dR
R-lX-X'l
R-lx-x’l
The exponential integrals or En functions (21, 24) are
defined such that
En(x)
.CO
e“xu u"n du
(101)
(102)
(103a)
1

73
1
í
O
^n-2 e-x//i á)1
(103b)
e"u u"n du
(103c)
so that Eq. (102) may be written
G°° (v;x|xf) - I
E1 [^t(k,»v>lx“xtl]
(104)
The Green’s functions discussed above were,
in general, for an infinite medium. The problem
eventually to be considered, however, is that of a unit
cell; that is, the space of interest represents one
unit volume of an infinitely repeating set of unit volumes.
The application of the unit cell approach is
based on the fact that most heterogeneous nuclear reactors
have fuel and moderator arranged in a repeating lattice
structure. For calculational purposes a lattice is sub¬
divided into a number of identical unit cells (18, 19).
The spatial and speed dependent neutron density, usually
Just the scalar density, is calculated for the unit cell
and is subject to particular boundary conditions. Quanti¬
ties such as the thermal utilization are then calculated

74
for the unit cell and are assumed to be valid throughout
the lattice. The reactor is then treated as a homo¬
geneous reactor having the same thermal utilization as
the unit cell, and the scalar neutron density is calcu¬
lated for the entire reactor.
In most reactors the lattice is very large
relative to the dimensions of the unit cell, so that
it is not unreasonable to treat the unit cell as if it
were in an infinite lattice. For a complete discussion
of this assumption, see Chapters 7 and 18 of Weinberg
and Wigner (19). If the assumption is then made that
the lattice is infinitely repeating, then at steady-
state the flow of neutrons in dn. about direction n^
across a point on the boundary between adjacent unit
cells is equal to the flow in dn. about direction
where is the mirror image of n^ as illustrated in
Figure 3. This is physically equivalent to the
assumption that the cell boundaries are perfectly
reflecting. Hence in the analysis that follows,
neutrons which actually arrive at r in the unit cell
by crossing the cell boundaries on first-flights can
be considered to have originated within the unit cell
at r* and suffered reflections at the cell boundaries
before arriving at r.

75
♦
Figure 3 Symmetry of Neutron Flow at the
Boundaries of Unit Cells in An
Infinitely Repeating Lattice
The infinite medium Green's function can then
be applied to the solution of the above finite geometry
problem in the following manner. Due to reflections at
the cell boundaries, first-flight neutrons from an
isotropic point source at r' can arrive at r by various
paths, so that the total density at r is the sum over
all possible paths i, of the densities of neutrons
arriving at r by path i and having traveled a total
path length L¿.
Alternatively, the neutrons at r can be con¬
sidered to have arrived by first-flights from i image
sources at r* located a straight-line distance Li from

76
the field point r. The total angular density at r is
then the sum over the i image sources of the contri¬
butions to the density at r from each image source at
r^. (See Chapter 7 of Morse and Feshbach (16) for a
discussion of the method of images.)
Thus for a general homogeneous medium, one
has from Eqs. (63) and (99) that
G(v;r,n|r’) - L 1 S(n - iQ\|)
1 4 TTvhl ~ ~ (105)
where 1»^ is the total path length traveled by a neutron
arriving at r along a first-flight path from an image
9
source at r^. From the alternate point of view, is
the path length traveled by a neutron along path i as
it suffered reflections at the cell boundaries before
arriving at r.
Integration of Eq. (105) over n yields
G(v;r rf)
„ - It(k’»v)Li
L ®
1 4 7T vLi2
(106)
As one would expect, for an infinite medium or for a
finite medium bounded by a non-re-entrant surface with

77
vacuum boundary conditions Eq. (106) reduces to Eq. (101);
for a finite medium bounded by a re-entrant surface with
vacuum boundary conditions, Eq. (106) reduces to the
infinite medium Green1s function with spatially variable
collision probability as given in Chapter 2 of Case, et al.
(21), namely
0
G(vjrlr’) - ®
(107)
where Zt(s,v) - 0 outside the bounding surface.
VII.2 Analytic Form of G(v;x|xt)
The physical effect of Introducing reflecting
boundaries is to increase the density of neutrons at a
point due to reflections at the boundaries which, in
effect, send neutrons back into the finite unit cell.
The total neutron density at r may be separated into
two parts; namely, that due to first-flights from the
source at r' within the unit cell and that due to
neutrons that may be considered either as arriving
from i image sources located at r^ outside the unit
cell or as having suffered one or more reflections at
the boundaries after emission at r*.

78
The first-flight angular density at r due to
a unit isotropic point source at r* in a homogeneous
medium is simply the infinite medium Green’s function
obtained from Eqs. (63) and (99), namely,
It(k* ,v)| r-r’|
— 477" vlr-r'l
G00 (v;r,n. r’) - ®
(108)
Let Gfc(v;r,n. r') denote the angular density at
r of neutrons which do not arrive by first-flights
from the unit isotropic point source at r’ but instead
arrive at r only after having suffered reflections at
the mirror boundaries of the cell, or alternatively,
arrive by first-flights from the i image sources
located at r^. Then let n denote the total path
length traveled by a neutron along a path having a
direction that is characterized by the unit vector
xQr’ pointing from the image source at r* to the
1 —i
field point r, tb© neutron arriving at the field point
r after having suffered n reflections at the mirror
boundaries of the cell. One can then write from Eq. (105)
that
(109)

79
The double summation over n and i is, of course, equiva¬
lent to a single summation over all the image sources.
The total Green's function for the unit cell is then
G(v;r,n|r') - G°° (v;r,n. r') + Gfc(v'r,n r')
(110)
Thus it is seen that G^c represents a finite medium
correction term to be added to G00.
In V.l the plane geometry equations were
obtained by integrating the point source kernel over
all the sources on the yz-plane at x'. For a unit cell
with plane reflecting boundaries, a neutron which
arrives at r after having suffered n reflections, and
thus having traveled a path length Li>n, can be con¬
sidered to have originated at a fictitious or image
plane source located at a distance from the field
plane. The cell boundary at which the neutron made its
last or n-th reflection lies between the field plane and
the image source plane.
The transformation from the point source
Green's function to the plane source Green's function,
that is Eq. (68), must be modified using the above
method of images to locate image plane sources at
distances from the field plane. Referring to
number of neutrons with

80
direction cosines between ji and p + dp crossing a ring
element of area 2 7T per unit time, the ring element
being located in the plane at xf. If

angle indicated in Figure 2, then
dn. - dp d
Hence, using the transformations of V.l,
G00 (v;x,p xf)dx dp dx' «
2 0
- dx dp/ df
(111)
G(v;r,/i
r')(-27T)
x-x'
f
r
dp'
dx*.
I>*
(112)
Since the sources are assumed to be symmetri¬
cal about the x-axis, G(v;r,nr') is a function only of
r,r’, and p. Thus the integration over carried out in Eq. (112):
G00 (v;x,p|x*)
ji*m0
r')
x-x*
ÍEL
1^*1
(113)
For slab geometry G(v;r,n.r') has a delta function
behavior with respect to p' so that Eq. (108) becomes

81
0“(v¡r^i|r’) - ® ~ •
477v( Ix-X'l /p)2 27T
The ji1 -integration may then be carried out in Eq. (113)
The result is
- It(k* ,v) Ir-r’l
G^CvjXjulx') - - ; ,
HH
or using Eq. (2),
G°°(v}x,ji x*)
- *v> I <*—** >//* |
HH
- ZtCkSv) |(x-x»)/n|
e 1 '
2v
h(x~x’) for ji =» 0
h(x‘-x) for
where the step function has been introduced to account
for the fact that in an infinite first-flight medium,
the flow of neutrons in the positive ji direction is
zero unless x is greater than x', while the flow of
neutrons in the negative ji direction is zero unless x
is less than x*. For a more complete discussion see
Chapters 2 and 4 of Case, et al. (21).
(H4)
(115)

82
In the same manner, one obtains the correspond¬
ing expression for the finite medium correction term.
Thus
Gfc(v;x,)i
x’)
E
i
- L(k,'v)
e x
(x-x^)/p|
r e- Zt05’.v)|(x-X¡)^|
i ¡7]^
h(x-x^) for jfi =» 0
(116)
h(x^-x) for «= 0
where the single summation represents a sum over all i
image source planes located respectively at x^. The
first two image source planes are illustrated in
Figure 4.
Substituting Eqs. (115) and (116) into
Eq. (110), one can verify that each term in G(v;x,pjxf)
satisfies the one-dimensional form of the source-free
transport equation corresponding to Eq. (95), namely
óG(v;x,u
v/> 3x r
x*)
+ v Zt(kf *v) G(v;x,jil x*) - 0
(117)
where v Zt(k*,v) is equivalent to [v(k’,v)+ X(kf ,v)] .
Further, G(v;x,ji|x’) satisfies the reciprocity theorem
(21), that is,
G(v;x,ji
x’) “ G(v;xf | .
(118)

83
Finally, each term of Eqs. (115) and (116) satisfies
the symmetry condition at the boundary x « +b that the
flow of neutrons in dp about p is equal to the flow in
dp about -p, that is,
G(v;x,p x')
x«=+b
G(v;x,-pj x*)
x»+b
(119)
Since each term of the Green’s function G(v;x,p x*)
satisfies the original boundary conditions, it can be
concluded that the function itself satisfies the boundary
conditions. One can verify that, in agreement with
sections 4 and 5 of Case, et al. (21), a source plane
of strength qs at x* is equivalent to a discontinuity
of the normal component of the angular current, where
for an isotropic plane source, qs - 1/2. The angular
current is simply the number of neutrons crossing
unit area perpendicular to the direction of flow per
unit time and unit solid angle. In particular, for
the infinite medium terms one can write
vpG(v;x,p|x*)
x«x’+e
- vpG(v;x,p x’)
x«x’-e
In an infinite medium the second term on the LHS
vanishes for p >0 while from Eq. (115) the first
term is equal to 1/2, that is, the source plane value.

64
In the solution for the scalar neutron density
in Eqs. (69) and (77) one needs the corresponding
scalar Green’s function G(v;x|x’) obtained by integrat¬
ing over all ji the function G(v;x,ji x’). Thus,
1
G(vjxjx') G(v;x,p|x’)dji . (120)
>—1
Recall that G(v;x|x’) is the first-flight
density of neutrons at x due to a unit isotropic plane
source at x' in a homogeneous medium. Then, referring
to Figure 4, one can see that for image sources to the
right of x, that is, x^ =» x, the neutrons reaching x
must have a negative direction cosine p, while for
image sources to the left of x, that is, x^ < x, the
neutrons reaching x must have a positive direction
cosine. Thus the contribution of the image sources is
Gfc(v,*x|x’)
0
1
G^c(vjx,jj |xf) dji +/ Gfc(v;x,ji|x’ )djw
•1 "p=0
(121)
where the first and second integrals represent the
contributions of image sources placed respectively to
the right and left of x.

85
Image Source
Plane at
x»-b-(b+x*)
Image Source
Plane at
x«b+(b-x*)
Figure 4 Slab Unit Cell with Mirror
Boundaries at x«+b

86
Letting G*c denote one term of Eq. (116),
one may write the first integral in Eq. (121) as
0
/ G^c(v;x,ji|x’ )dp
p°-l
£t |(x-x|)/^|
J1B-1
H^l
í e- Zt(k*,v)|(x-x*)/p| d£
Ln M ’
(122)
the last integral having been obtained from the preced¬
ing one by a successive substitution of variables, first
letting y—ji, and subsequently letting ji-y.
In general then, one can sum over all image
sources and obtain the result that
G^c(v;x|x*)
Sv
>0
j- iv) |(x-x¿)/u| d£
(123)
Again, for a first-flight medium the range
of integration over ji for the Infinite medium contri¬
bution is from ji*»0 to p»l. Thus one can write
(x-x»)/p|
G(v;x

87
1
/
e“ Zt(k,»v>| I'1!
(124)
or from Eq. (103)
G(v;x|x*)
1_
2v
Ei[ ^t (125)
+ L Ex [ It(k’ ,V) |x-x¿|]
To express G(v;x x*) as a function of the
unit cell dimensions it will be instructive to return
to the physical concept of reflection of neutrons in
order to locate the image sources with respect to the
cell boundaries. Using the notation for path length
previously discussed with regard to Eq. (109), one can
write the expression for G*c(v;xjxt) in Eq. (125) as
(126)
where }ü*±tn *s» as the distance separating the
field plane from an image source plane, the image
plane corresponding to a neutron having made n re¬
flections at the mirror boundaries.

88
In Eq. (126), the summation over 1 extends
over all possible paths by which a neutron having its
direction cosine equal to ± ji can arrive at the field
point after suffering n and only n reflections at the
boundaries. In plane geometry, it is easily seen that
for a first-flight medium there corresponds to any given
only two such paths for any n. Hence the summation
in Eq. (126) runs from i»l to i-2. Figure 4 illustrates
the two possible paths, and 1*2corresponding to
p > 0 and ji «= 0 respectively, by which a neutron can
arrive at x after suffering only one reflection. The
unit cell in Figure 4 has its midplane at x«0 and the
reflecting boundaries at x - ±b. The cell width,
that is, the distance between the two boundaries is 2b.
The expressions for can be obtained by
simply adding the total path length a neutron travels
while making n reflections. The expressions for
depend on the orientation of the source plane and field
plane x relative to each other and to the x-0 plane.
It has been shown in V.2, however, that one need only
consider field planes for positive x, because the
solution of the half-cell is symmetrical with respect
to x

89
The expressions for are derived in
Appendix A where it is shown that Eq. (125) may be
written, for x >0, as
G(v;x|x* ) - E^[Zt(k* ,v) jx-x’l]
Note that Eq. (127) can be shown to be term by term
equivalent to a more cumbersome series representation
of G(v;x|x*) derived by Aswad (25).
Finally, the half-cell Green’s function
defined by Eq. (78) becomes
H(v;x|x') - ^ E^[Zt(h* »v) |x-x*|j + EjJZj.Os' ,v) |x+x'|]
+ E i [ZtOs*»v> 2nb-x' + (-l)nx|]
n**l
+ Ej[Z^(k’,v) 2nb+x’-(-l)nxj j
+ E1[Zt(k’,v)|2nb+x’ + (-l)nx| ]
+ Ej_[Zt(k* ,v) |2nb-x'-(-l)nx|]
(128)
for x =» 0 and x’ =» 0

90
VII.3 Monte Carlo Generation of G(v;xlx*)
For complex finite geometries, the complete
analytic form of the first-flight Green’s function may
be exceedingly difficult, if not Impossible, to derive.
In such cases, it may be advantageous and even necessary
to represent the physical or mathematical system by a
sampling operation satisfying the same probability laws.
Such a process has come to be called a Monte Carlo
method (26).
Essentially, Monte Carlo methods are ’’paper
experiments.” The experiment consists of performing
specific sampling operations many times, the sampling
operations satisfying the criteria mentioned above. As
in many other experimental procedures, the geometry
of the problem does not greatly affect the difficulty
of performing the Monte Carlo experiment. The con¬
struction of a function, say neutron density, by
Monte Carlo is conceptually little more difficult for
a square or hexagonal unit cell than it is for a slab
or cylindrical unit cell. This is in striking contrast
to the analytic description of the same function;
there the solution for simple geometries does not, in
general, give any indication of how to proceed to more
complex geometries. Thus there is strong motivation

91
to generate the plane geometry Green*s function G(v;x x')
by Monte Carlo. The transition to cylindrical or even
more complex unit cell geometries would not be difficult;
the general procedure, that is, the Monte Carlo sampling
operations would be essentially the same.
The Monte Carlo generation of G(v;x xf), that
is, the first flight Green’s function for a homogeneous
finite medium in an Infinite plane geometry with
perfectly reflecting boundaries has been discussed
extensively by Aswad (25) and will be outlined below.
In plane geometry, a cluster of neutrons is
considered to be emitted with unit statistical weight
at the source plane x*. Its direction cosine, ji, is
chosen from a random distribution, the distribution
being such as to represent an isotropic source, that is,
a source which has equal probability for emission of
neutrons in an element of solid angle da about any
direction n; thus in plane geometry the direction
cosine ji occurs with equal probability for values
between -1 and +1. The random selection of jx is
done by Monte Carlo sampling.
The projection of the path of the neutron
cluster on the x axis is divided into closed intervals
[xm-l * xm\» [^m* xm+l]
, ... ,
and the statistical

92
weight of the neutron cluster is computed at points
^ín-l» xm» *111+1 > • The weight of the neutron cluster
at some point 3%, denoted by Wix^x*), is subject only
to exponential attenuation along the path of travel, the
mean free path between removal collisions being equal
to the reciprocal of the total collision probability
It WCXpjjx*) - e~ ^*t |(x¡m-x*)/p.|
If =* *m ^ » a^d if |x’) and WÍXm+ij )
represent respectively the statistical weight of the
neutron cluster at and x,^, then the quantity
[wCxjajx*) - WCXj^ijx*)] must represent the number of
removal collisions per unit time that occurred in the
closed interval ^xm, x^jJ .
The neutron cluster continues along its
original path until it arrives at a cell boundary
where it is considered to be reflected and sent back
across the cell with direction cosine y1, where
that is, |jufj - |ji| . The number of collisions that
occur in the interval jx,^ x^J as the neutron cluster
passes through that interval is again recorded. The
process is repeated as the neutron cluster suffers

93
successive reflections at the boundaries and, as a
consequence, makes successive passes through the
The statistical weight of the neutron cluster
never becomes zero, but only approaches that value as
a decreasing exponential as the path length becomes
infinite. It is, of course, very inefficient and, in
fact, impossible to follow the history of a neutron
cluster forever. It is also, fortunately, not necessary
to do so. Instead, such techniques as Russian Roulette
and statistical reweighting are used at the cell bounda¬
ries, the statistical weight of a neutron cluster at
the boundaries being compared to a randomly determined
survival probability in order to determine whether the
life history of the neutron cluster is to be terminated,
or whether the cluster is to be followed after having
been assigned a new statistical weight. This procedure
allows the statistically more significant histories to
be preferentially followed.
The statistical reweighting of the surviving
neutron clusters at the cell boundaries is such that
by following the surviving clusters one automatically
takes into account the weight of those clusters which
were terminated by Russian Roulette. The termination

94
of the history of a neutron cluster by use of the above
techniques eliminates an arbitrary cutoff of the path
length that would otherwise be required. A full
discussion of these techniques applied specifically
to the generation of the first-flight Green’s function
is given in reference (25).
After the termination of the life history of
one neutron cluster a new neutron cluster of unit weight
is started at x*, its new direction cosine having been
chosen from the random distribution of cosine values.
The life history of the new neutron cluster is followed
as before, recording the total number of collisions
occurring in the interval £2%, .
Let WjjCXjjJx’) denote the statistical weight
of a neutron cluster at when the cluster has made
its n-th pass across the plane at Xjj. Define
w(xm+l*xm|x,)
£
Wn-Wn(xIB+1|x»)
(130)
to represent the total number of collisions in the
interval [xmiXn+i] due to one cluster of neutrons
having made n passes through that interval. Let J
denote the total number of neutron clusters emitted
per unit time at the source plane x’, and let
wj denote W(xffl+1,xm |x*) for the j-th

95
cluster. Then the average number of collisions per
unit time per cluster in [x^Xj^jJ will be just the
total number of collisions per unit time in that
interval due to a unit isotropic plane source at x*
and is defined to be
-J
X»)
7 E'fj(xBM.l«Itm|3£’> •
J
(131)
But the total collision rate at x due to an isotropic
plane source at x* is simply v Zt(kf,v)G(v;x|x*). Hence
*ra+l
v Zt(k*,v)/ G(v;xjx*)dx
-j i
W <3W>xm|*')
(132)
3»+!
G(v;x x*)dx
-J
x”xm
(133)
Equation (133) represents the sum total of Information
obtained by the simulation of neutron histories and
may be used to obtain the value of the integral of
G(vjx|x*) over the interval [xm,xffi+1] , simply by
dividing the Monte Carlo results by vZt(k*,v).

96
It should be noted that the higher angular
moments of G(v;x,jj|x*) could be obtained by the above
technique by merely modifying the processing of the
neutron histories. Thus one would simply form the
product of the collision rate in an increment
about direction ji times the associated Legendre
functions P®(ji) and then sum the products over all
from p--l to r+l, thereby obtaining the G“(v;*|*'),
that is, the coefficients in the angular expansion
of G(v;x,jí¡x’ ).
To obtain a representation of the function
G(v;xjx‘), the interval [xnpX^ij must be chosen suf¬
ficiently small such that G(v;x|x’) may be assumed
constant over that interval. If this be the case,
then G(v;x|x') may be taken outside the integral in
Eq. (133). This, of course, is simply an intuitive
use of the mean value theorem of integral calculus.
Divide both sides of Eq. (133) by the spatial interval
width (Xjb+i - xm). Then the Green* s function is
G(v;x|x*)
-J ,
W
xra>3CBH-l v Zt(k* ,v)(xm+1-xm)
for .

97
The accuracy with which G(v;xjx*) is re¬
presented by the BBS of Eq. (134) without passing to
the limit depends on several factors. As smaller and
smaller spatial intervals are chosen, the average col¬
lision rate per neutron cluster in the interval [2%, *1^.1]
exhibits statistical deviations as the contributions
of succeeding clusters are added. Thus the total number
of clusters, J, must be increased to the point where the
-j
statistical deviations of W (x^j^Xjn x*), as a function
of J, are tolerable. Obviously, a balance must be
maintained between a desirably small spatial interval
and a number of neutron clusters, J, that can be
simulated in a reasonably short computation time.
It can be seen from Figure 5, that in a
Figure 5 Mirror Symmetry of First-Flight
Green*s Function About Plane
at x-0

98
space symmetric about the x-0 plane,
G(v;x|- x') - G(v; - x|x’) .
Hence the Monte Carlo generation of the half-cell
Green’s function, from Eqs. (78), (133) and (135)
yields
-xm+l
H(v;x xf)dx
xm
V ¡tt7k' .*>
-J .
* xm|x'>
-J ,
+ * <-xm.l"-xm x’>
or as in Eq. (134)
H(v;x|x’)
-j ,
w
lira ■— - ■—
*****+1 v It(k* ,v)(xm+1-xin)
+ lim
x—
i’Vx v It(k’,v)|xn+1-xm|
for x_«= x <=
SI
The last term in Eq. (137) represents the number of
collisions in the interval per unit time
per unit interval [-Xm+l»_xm]* After simulating a set
of histories of neutron clusters emitted from the
(135)
(136)
(137)
^+1 *

99
source plane at x’, one simply adds the collision rate
that in the interval
thus obtains the integral of the half¬
cell Green’s function over the interval xm «= x <
Geometricallyf this is equivalent to folding along the
midplane the surface which represents G(v;xjx’) on the
xx’-plane.
An investigation of the error bounds of the
Uonte Carlo results for H(v;xjx') compared to the
analytic results obtainable using Eq. (127) has been
carried out and indicates that values for H(v;x|x’>
having less than a five percent error are easily obtain
able with only a very modest amount of computing time.
In the preceding discussion the problem of
obtaining an explicit representation of the function
G(v;x|x’> was considered. It was concluded that in
order to obtain such a representation the interval
[xm,xm+il in Eq. (133) must be chosen sufficiently
small such that G(v;x|x’) may be assumed constant over
that interval. Thus the interval width depends on the
accuracy with which one wishes to represent G(v;x|x*).
This is a rather stringent requirement.
In an actual application, however, the
situation is considerably more favorable. In the

100
solution of Eq. (69) one has, in general, to evaluate
integrals of the type
xfm+l
Í f(x',v)G(v;x|x»)dx» (138)
-4»
m
where f(x*,v) is a fairly smooth function representing
either the neutron density or the integral over thermal
speeds of the product of the scattering kernel times
the neutron density. If the spatial interval [*¿.*Vl]
is chosen sufficiently small such that f(x'v) may be
assumed constant over that interval, then expression
(138) becomes
m+1
f(x*,v)G(v;xlx')dx’
ra+1
T(x*,v)/ G(v;x|xf)dx'
(139)
x
m
xm+l
f(x',v)/ G(v;x,|x)dx’ (140)
x *
xm
where Eq. (140) follows from Eq. (139) by the reciprocity
theorem (21), and f(x*,v) denotes the average value of
f(xf,v) over the [xa#x1¡l+i] interval.

101
The integral on the RHS of Eq. (140) is all
that need be generated by Monte Carlo, and, as indicated
in the discussion pertaining to Eq. (133), it is just
this Information which may be obtained directly from
the Monte Carlo results. Obviously, far fewer neutron
cluster histories will be required to represent the
Integral of G(v;x* x) over the [xm,xni+i] interval than
would be required to represent the function G(vjx'|x)
Itself.
Thus the maximum satisfactory interval width
[xm»xBH'l] is determined by the spatial dependence of
f(x*,v) and not by that of G(v;x’|x). Integrations of
Jt(x*,v)G(v;x|x*)dx* over a path containing the
singularity point of G(vjxJx’) at x-x’ are thus possible
with only a relatively few neutron histories being
required at each spatial point. No special consideration
need be given the point of singularity. Of course, the
same arguments apply to the half-cell Green’s function
H(vjxjx’) and its integral.
It is just the point discussed above, namely
that it is the integral of G(v;xjx*) or H(v;xjx*) over
an Interval that one needs and not the function itself,
that makes the Monte Carlo technique particularly useful
and economical as far as computation time is concerned.

102
CHAPTER VIH
SOLUTION OF THE TRANSPORT EQUATION
VIII.1 Spatial and Speed Dependence
In general, the equations to be solved, namely
Eqs. (79) and (81), are of the following types: for
the multispeed problem,
b
(141)
where
f(x,v) - S(x,v) + C^(x,v)N(x,v)
V*
1.
dv* P(x,vi-»v)N(x,vf)
(142)
while for the one speed problem
b
N(x)
Í
x«0
f(x*)H(x|x')dx*
(143)

103
where
f(x) - S(x) + C2(x)N(x)
(144)
In both Eqs. (141) and (143), b denotes the cell boundary.
In the above equations, S, C^, C2»and P are
spatially discontinuous functions, so that the integrals
in Eqs. (141) and (143) are evaluated as a summation of
integrals over spatial regions throughout which S, C]_,
C2»and P are continuous functions of x. Thus, as in
Chapter I, Eq. (141) can be written as
(145)
x’in k
where E2 denotes the number of regions in the half¬
space. Equation (143) can be written similarly.
Equation (145) can be solved numerically
by dividing the x space in each region k into
dependence first,
one may write Eq. (145) as
into J Intervals
(146)

104
If the intervals [x^x^J are chosen small enough such
that f(x',v,k) may be assumed constant throughout the
interval, then one can write Eq. (146) as
N(x,v) « Y? E^ f (m,m+l;v,k) I H(v;x* x)dx'
k°l m-1 J f
rrnl
(147)
km
where the reciprocity theorem (21) has been used to
interchange x and x' in the half-cell Green's function,
and f(m,m+l;v,k) denotes the constant value assumed
for f(x',v,k) throughout the ^x^x^jJ interval.
As discussed in VI1.3, it is just the integral
of the Green's function over an interval ^x^x^jJ that
is generated by the Monte Carlo technique. Let
x
m+l
H(v;m,m+1 x) -/ H(v;x'| x)dx'
x.
m
and rewrite Eq. (147) as
(148)
R2 Mjj.
N(x,v) “ E E f(m,m+l;v,k)H(v;m,m+l x)
k-1 m-1
(149)
The value of N(x,v) at the spatial coordinate xQ may
be obtained from Eq. (149) when written in the form
N(xn,v)
R2 Mk
,E, E, f (m,m+l;v,k)H(v;m,m+l| n) .
k«=l m-1 1
(150)

105
As pointed out by Kopal (27) the selection
of the xn*s is not completely arbitrary. If, as
Indicated above, N(xn,v) is to denote N(x,v) at the
coordinate xn, and one chooses f(m,m+l;v,k) to be the
average of the vhlues of f(x*,v,k) at the end points
of the [xjJjjX^jj interval, that is
f(m,m+l;v,k)
f(x^,v,k) + f(x^+1,v,k)
2
(151)
then Eg. (150) represents a system of n simultaneous
values of N J^that is, n values of N(xn,v) and
(1 + Y Mt) values of N(x',v) contained in f(x ,v) .
K m m J
In order that the system be determinate, the two sets
of spatial coordinates xn and ^ Bust be identical.
With this requirement satisfied, then N(xn,v) and
N(xn+i»v) denote the values of N(x,v) at the end points
of the[xn,*n+1] interval, and Eq. (150) is a determinate
system of (1 + V Mt.) simultaneous algebraic linear
k=¡l
R2
equations in (1 + Mjj) unknown values of N. The above
kml
method shall subsequently be called the end-point method.

106
An alternate method, and the one most commonly
used in the literature pertaining to numerical solutions
of the transport equation (23), is to consider the value of
N(x,v) at the midpoint of an interval [xm,xm+1j to be
representative of that interval. To be more specific,
let yn denote the midpoint of the [xn,xn+^j interval,
that is,
n
x
A:
n
n
x
n
+
xn + xn+l
(152)
Let f(ym,v,k) denote the value of f(x*,v,k) at the
midpoint of the
interval. Since that value
is to be representative of the entire interval, it is
obvious that f(ym,v,k) replaces f(m,m+l;v,k) in Eq. (150).
Recalling Eq. (148), one has that in the midpoint method,
the integral of the Green's function over the spatial
interval can be written
x
m+1
“/ H(v;x’| yn)dx' .
x
m
II(v;m,m+l
n)
(153)

107
The end points of the interval cannot be expressed in
terms of the midpoints exclusively so- that in the LHS
of Eq. (153) m denotes Xg. Keeping in mind that in
Eq. (153) the source for the Green’s function is located
at the midpoint of the [xn»xn+i] interval, in contrast
to Eq. (150) in which the source to the Green’s function
is located at an end point of an interval, one can
write Eq. (150) for the midpoint method as
N(yn,v) “EE f(ym»v,k)H(v;n,ttfl n) . (154)
k“l m-1
Equation (154) represents a determinate system of
&2 R2
E simultaneous algebraic linear equations in E Mk
k-1 k-1
unknowns if, and only if, the two sets of spatial
coordinates yn and y^ are identical. This method will
subsequently be called the midpoint method. Note that
unless equally spaced intervals are chosen, the two sets
of coordinates x^s and ym’s must be carried along in
the calculation in order to compute the integral of the
Green’s function H(v;m,BH-l n). This complication is,
of course, absent in the end-point method.
There are two major differences between the
end-point method and the midpoint method discussed

108
above. First, for any given total number M of spatial
simultaneous
intervals, where
equations for the end-point method involves one more
equation and one more unknown than does the system of
equations for the midpoint method. The amount of
effort expended in the solution of the sets of simul¬
taneous linear algebraic equations depends on the method
used in solving the equations. If the number of arith¬
metical operations such as addition, multiplication,
and division is chosen as the measure of computational
effort, then, as indicated in Chapter 11 of Richards
(28), the effort expended in solving the set of equations
by a matrix inversion method increases as the third
power of the number of unknowns, while the effort ex¬
pended per iteration in an iterative method of solution
Increases as the second power of the number of unknowns.
Table 1 gives a comparison of the relative effort required
for the two methods of solution applied to the end-point
and the midpoint methods. The Green's function must be
computed for a source position corresponding to each of
the unknown values of N(xn,v); the relative effort
expended in computing the Green's function for the end¬
point and midpoint methods is included in Table 1.

109
TABLE 1
Relative Effort for Methods of Solution
of End-Point and Midpoint Equations
for a Total of M Spatial Intervals
Matrix
Inversion
Iteration
(per cycle)
Green* s
Function
Calculation
Midpoint
Method
/v M3
~ M2
M
End-Point
Method
«/ (M+l)3
^ (M+l)2
a/ (M+l)
If matrix inversion is chosen as the method
of solution, then the midpoint method would require far
fewer computations than would the end-point method; if
the choice were to be based on number of computations
alone, then the midpoint method would be used rather
than the end-point method.
If an iterative technique is used, then the
choice is not so well defined. While the midpoint
method requires fewer computations per iteration
than does the end-point method, this difference is
not as extreme as the matrix inversion method; the
difference in effort varies as the square instead of
as the cube of the number of unknowns. More important
is the fact that the entries in Table 1 indicate only
the relative computations per iteration cycle and not
the total number of computations required by the two

110
methods. If the end-point method were to require fewer
iterations than the midpoint method to produce a
satisfactory solution because of providing a better
representation of the spatial dependence of the functions
throughout an interval, then the end-point method might
be the better choice. It should be noted that automatic
computing machines compute rapidly but may have limited
storage capacity. Richards (28) indicates that the
minimum storage needed for a matrix inversion solution
of the set of equations is of the order of M(M+1) while
that required for an iterative solution is of the order
of M + data. In such cases, an iteration method may be
more efficient than a matrix inversion method.
The second major difference between the end¬
point method and the midpoint method is the fact that
the end-point method can give explicit values of the
neutron density at cell boundaries and at interfaces
between regions while the midpoint method can only
give values at the midpoint of some interval. If the
midpoint of an interval coincides with an interface
between two regions, then the value of N(x,v) at that
midpoint will not be representative of the entire
interval because of the discontinuity in the gradient
of the scalar density at the interface. The space near

Ill
such interfaces must be divided into additional intervals,
thereby increasing the number of equations and unknowns.
Although the gradient of the scalar density
has a discontinuity at an interface, on either side of
the interface the density may be well represented by a
first order polynomial. Hence, if the space of interest
has many different regions so that there are many inter¬
faces to be considered, then the total number of spatial
intervals required for the end-point method may be far
fewer than that required for the midpoint method. The
above discussion does not apply, of course, at the cell
boundaries or midplane because there the gradient of
the scalar neutron density has no discontinuity, but
rather is zero on both sides of the midplane and each
cell boundary.
In this paper the set of simultaneous
equations obtained as an approximation to the Integral
equation, that is, Eq. (141), was solved by an iterative
technique; the end-point method of representing the
neutron density was chosen because the unit cell may
have many interfaces and because of the possibility
of reducing the number of iteration cycles required
to obtain a satisfactory solution. Thus Eqs. (150)
and (151) are the equations that were used to compute
the scalar density.

112
Consider now the speed dependence of N(x,v).
For a given spatial coordinate, the functions Ci, N,
and P are smooth functions of neutron speed; their
speed gradients have no discontinuities in the thermal
speed range considered in this paper. Thus there seems
to be no advantage in representing the speed dependence
of the scalar density by the end-point method; in fact,
there is a strong advantage in using the midpoint method
because of the reduced effort required in the solution
of the resulting set of simultaneous equations as
indicated in Table 1.
An equation analogous to Eq. (152) can be
obtained by letting uj denote the midpoint of the
[vj,vJ+l] interval» that is
(155)
. VJ + Vj+1
2
Let N(x,uj) denote the value of N(x,v) at the midpoint
of the [vj»vj+i] interval and assume that value to be
representative of the interval. Equation (142) can
then be written as

113
f(x,ut) - SÍXjUi) + Ci J
/.
/ duf P(x,u4uj)
U'“V."U.- J—1
(156)
Assume P(x,ui*ai^) to be representative of the interval
that
^j+1
/
P(x,Uj»U1) Avj
(157)
P(x,u-i^ui)duf .
It is characteristic of the midpoint method
that unless equally spaced intervals are chosen, the
limits of the integral in Eq. (157), that is, the end
points of the intervals, can not be expressed in terms
of the midpoints exclusively; it will be necessary to
carry along in the calculation the two sets of coordinates
u¿'s and Vj*s. With this in mind, one can write Eq. (156)
as
fiXjU^ - Six.u*) + C1(x,ui)N(x>ui)
(158)

114
To simplify notation the following definitions
are used
Njn - NCuj.x,,) (159)
pkjl - PCk.uj^Avj (160)
?lmk ‘ í(4':Vl¡ui>k> (161>
Slnn ‘ H <162>
With this notation Eq. (150) may be written
R2 — —
Ni„ - E E, flmk Himn <163)
k**l m-1
where
fimk “ simk + cik
and
Cik - [Vik' + 4k'] - [4k + 4k] •
For the one-speed case the i subscripts are dropped in
Eq. (163) and
fmk “ smk + ck
Nm + nhh-1
(166)
Nim + Ni,m+1
hHii
N
jm + Nj,®fl

115
where ck - v[lt(k') - laoo] . (167)
In the above equations, Sim denotes the source term
for the J^xm,xn+j_j spatial interval and the [v¿,v¿+1j
speed interval. The set of simultaneous linear
algebraic equations (163) can now be solved by an
iterative technique.
VUI. 2 Iteration Techniques
The initial guess for the iterative solution
of Eq. (163) is a spatially flat Maxwellian spectrum
normalized such that the total absorption rate in the
unit cell equals the total source rate. The normalization
requirement is that
v* -v*
dx’ / dv /(x,v)N(x,v) » /dx* / dv S(x,v)
¿11 x* v»0 all x* V“0
(168)
Denote the m-th iterate of N(x,v) by Nm(x,v). The
spatially flat first guess is then
N°(v)
4 no ,.S
Vrr n° y2 e
(169)
which is the Maxwell distribution expressed in dimension¬
less speed units. The value of N° in Eq. (169) is

116
determined by requiring that Eq. (168) be satisfied.
Thus, substituting Eq. (169) into (168), one has that
4
V/r
Nl
E2 J d3£t
k»l x’in k'v-O
dv S(x,v)
v*
E f dx' f dv y/(k,v)v^ e~v<
k-1 J J
x'in k v-0
(170)
Direct substitution of Eq. (170) into (169) gives the
first guess for N(x,v).
Assume that the calculation proceeds in the
direction of increasing n and i; then the iteration
proceeds as follows. The array of values for the first
guess or zeroth iterate having elements Njm is denoted
by N®, is obtained from Eqs. (169) and (170), and is
substituted into Eq. (164); the array of values then
computed by Eq. (163) is the first iterate of N
denoted by N1. This new array is substituted into
Eq. (164), and a second iterate is computed for the N
array by Eq. (163). The process is continued until the
solution converges, that is, until the residuals
defined by
n“ - N?-1
in in
are less than some finite
fraction of the neutron density at the mesh point i,n
for all i and n.

117
The rate of convergence can be materially
increased by requiring that each iterate, that is, each
array of values N® satisfy the normalization requirement
Eq. <168). If N® denotes the m-th Iterate and is
normalized in the above sense but is not a converged
solution, then N®** computed by Eq. (163) will not be
normalized. Define a normalization constant a®*3, such
that Eq. (168) is satisfied; thus
a"»1
R2 Mk J _
E E A*m E Sjnk Avj
k-lra-1 ' 1
R2 Mfe j
E E A*» E
k«l m-1 j«l
Yik
NJm + Nj fW,i
(171)
The array N®*1 can then be normalized by multiplying
each element of the array, N®+* , by a®4"*.
An additional technique that can be used to
accelerate convergence is that of over-relaxation (30);
the iterated values are adjusted according to the scheme
N®*1 « w N®*1 + (1-w) N®
(172)
where w is an over-relaxation factor which can be found,
in practice, by trial and error; that is, the rate of
convergence is determined for various values of w, and
the best value of w obtained in this manner will, for

118
similar problems at least, be satisfactory for subsequent
use (30). The combination of normalization and over¬
relaxation can be combined; the n®4’* iterate is then
computed according to the scheme
Navfl «, w aXJH-1 Nm+1 + (i^)^ (173)
where N®4"* on the RHS of Eq. (173) is computed from
Eq. (163). The iterative scheme defined by Eq. (173)
was used for all the calculations of this paper.

CHAPTER IX
RESULTS AND CONCLUSIONS
IX.1 Comparison of the Homogeneous Green*s Function
Method with High Order Pn and Sn Methods
The one-speed, slab geometry version of the
zeroth harmonic, homogeneous Green's function method
of integral transport theory, as developed in this paper
and subsequently called the HGI method, has been coded
for the IBM-709 computer for the specific case of two
regions in the half-cell. The code was then used to
solve several problems for which published results of
independent calculations by other methods were available
for direct comparison. The problems were chosen
primarily with a view towards comparing the EGI method
to the Pn, double-Pn, and discrete Sn methods frequently
used in integro-differential transport theory (2).
The specific results and conclusions of the individual
problems are presented in Appendix B. However, general
conclusions are discussed below.
The most important conclusion, based on the
results of the sample problems, is that the HGI method
gives results for finite unit cells comparable to
119

120
results of very high order Pn and similar methods. For
certain combinations of cell geometry and properties,
the HGI method yielded results comparable to those
obtained by Pj3 or higher order approximations. A
more general comparison would require either a detailed
study of the higher harmonic terms which were neglected
when the zeroth harmonic approximation was made in
Chapter II, or a very extensive set of unit cell
calculations by the HGI method for a wide variety of
cell properties and configurations.
In Problem 2 of Appendix B, the results for
a unit cell having two regions of equal volume of
absorber and moderator are discussed in detail. A
plot of the spatial dependence of the scalar neutron
density as calculated by the HGI method is compared
with a similar plot of results calculated by Gelbard,
et al. (34) using a discrete ordinate method "roughly
equivalent to a P^ approximation." In general, the
neutron density values calculated by the HGI method
agree within 3 per cent over the entire cell with the
values calculated by the Pj^ approximation.
Another parameter of interest is the ratio
of the scalar neutron density in the absorbing region
to that in the moderating region, that is, the advantage

121
factor for the unit cell. Values of the advantage
factor calculated by the HGI method agree within less
than one per cent of values calculated by Gelbard, et
al. (34) using the Pjj approximation mentioned above.
A similar calculation is discussed in detail
in Problem 3 of Appendix B for a thin unit cell having
regional thicknesses that are fractions of mean free
paths. There it is shown that the advantage factor
calculated by the HGI method agrees within 0.5 per cent
of the advantage factor, calculated by Meneghettl (9)
using a double-Pg approximation. For this problem the
double-P5 solution is demonstrated by Meneghettl to be
a converged solution, while for the same problem
Meneghetti’s results indicate that P13 and discrete
calculations do not give converged solutions.
The accuracy of the HGI method would seem to
confirm the assumption of Chapter II regarding the
isotropy of the angular first-flight density g(v;r,n.jr*)
due to a unit Isotropic point source in a finite cell
with reflecting boundaries, A unique feature of the
HGI method is that the assumption of isotropic angular
first-flight density g(v) depends only on the
geometry of the cell and the total cross section of
the medium for which the Green’s function is computed.

122
Furthermore, on the basis of the problems solved, the
accuracy of the HGI method seems to be less dependent
on the properties of the absorbing regions than are the
Pn, Sn, etc., methods. The extensive calculations of
both Meneghetti and Gelbard confirm that, in the latter
methods, the order of approximation required is: (1)
higher in cells having highly absorbing regions than
in cells having only weakly absorbing regions, (2)
higher in thin cells than in thick' cells, and (3)
higher in cells having absorbing layers which are thick
compared to moderating layers than in cells having
absorbing layers of thickness much less than the
thickness of the moderating layers.
Thus, on the basis of the results of Meneghetti,
if one considers unit cells having a fixed ratio of the
volume of the absorbing region to the volume of the
moderating region, then a decrease in the overall cell
size requires an increase in n for Pn and Sn calculation
to yield satisfactory results. However, the discussion
of Chapter II implies that a decrease in overall cell
size decreases the anisotropy of the angular first-
flight density gCv^r^r*) so that one might expect
that the HGI method would be quite suitable for thin
cells.

123
As mentioned above, the results of Meneghetti
imply that for cells which are larger overall, the Pn,
Sn methods become satisfactory for lower values of ”nM.
Since the larger the unit cell size, the poorer is the
assumption of isotropic g(v;r,n|r'), there should be a
cross-over point for cell dimensions beyond which the
Pn, Sn methods would be more satisfactory than the
HG1 method. Neither the existence nor the spatial
dependence of this cross-over point has been demonstrated
numerically as yet, and the subject would bear further
investigation.
The advantage of the BGI method over Pn, Sn
methods for parametric studies was mentioned in III.4
and has been verified by sample problems; namely, the
same Green*s function can be used repeatedly to solve
unit cell problems having different configurations and
different absorbing media, provided only that the overall
cell dimensions be the same for each problem and that the
total cross section of the k*-th region of the real
neutron density problem be the same as that for the
homogeneous medium of the Green*s function problem.
Once the Green's function has been computed, the solution
of the integral equation for the scalar neutron density
is a trivial problem, at least as far as machine

124
computation time required, and one can obtain the
solution, for example, to numerous cells which differ
both in the absorbing media and in the interface
position between the absorbing and moderating regions.
In contrast, the Pn, Sn methods depend explicitly on
the properties of all regions of the cell. The so¬
lution for one cell cannot be directly used in the
solution for another cell having absorbers of different
configuration and/or different absorbing and scattering
properties. In brief, one can save nothing from an
Sn solution for one cell that will facilitate the Sn
solution for a different cell.
In summary, the HGI method yields results
for finite cells comparable to high order Pn methods
for a wide range of cell dimensions and physical
parameters; the HGI method seems to be most accurate
for problems for which Pn approximations are least
accurate, that is, for very thin cells. Further, the
HGI method is particularly advantageous for parametric
studies of unit cells which have not only the same
overall cell size but also one region, the total
cross-section for which is the same for all cells.

125
IX.2 Monte Carlo Generation of the Green*s Function
The Monte Carlo generation of the Green’s
function was found to be useful and effective. Green's
functions were generated for several problems using
different numbers of neutron histories per spatial point.
In general, it was found that at least 2000 histories
per source position were required in order to adequately
represent an isotropic source and to reduce statistical
fluctuations of the Green's function to a satisfactory
level. This was determined not only by comparing the
statistical deviations of Green's functions which
differed only in the number of histories generated
per spatial point but also by comparing the calculated
values of the scalar neutron density which were obtained
using those Green's functions.
While the former criteria is the more
mathematically Justified, the latter criteria seems
to be more practical. The reason for this is that the
calculation of the neutron density tends to smooth out
the effects of the statistical fluctuations of the
Green's function. The value of the neutron density
at a point xn is obtained as the result of an integration
of the product of the Green's function for a source
at xn times the neutron density at all points 2%

126
including xn. Since the values of the neutron density
at all other points is obtained in a similar manner,
each N(xm) being calculated using a Green’s function
for a different source point xm, it would seem that
the value of the neutron density includes, in effect,
the statistical fluctuations of the Green’s functions
for all other source points. Since the statistical
fluctuations are just as likely to be positive as
negative, the statistical fluctuation of the neutron
density should be less than that for the Green’s
functions.
The number of histories generated by Monte
Carlo was considered to be sufficient when the contri¬
butions of additional histories did not appreciably
change the results of the neutron density calculations
in the real unit cell problem. The required sample
size of neutron histories agrees with the results of
Aswad (25) who computed standard deviations of the mean
from the expected values for various sample sizes.
Aswad also found an excellent agreement between the
values for the Green’s function calculated by Monte
Carlo with the values calculated using the analytic
expression, that is, Eq. (127).

127
Examples of whole-cell and "folded”, or half¬
cell, Green's functions generated by Monte Carlo are
given in Figures 6-7, for two different cell sizes.
The values plotted are the averages over the spatial
intervals of the Green's function.
Because the generation of the Green's function
is by far the most time consuming part of a unit cell
calculation, it is important to consider possible means
of reducing the sample size required to represent an
isotropic source. Define a constant Ci such that 1.
Then one can divide the neutron clusters into two classes,
according to whether 0 «= |ji|< Ci or Cj ■«= |ji| < 1. In
general, the neutron histories belonging to class 1,
that is, the histories having direction cosines the
absolute values of which are between 0 and C* can be
considered more important than those histories belong¬
ing to Class 2, that is, histories having direction
cosines the absolute values of which are between C¿ and
1. The reason for this is that the neutron histories
of Class 1 travel a much greater path length before
arriving at the cell boundary than do the histories of
Class 2. Thus the statistical weights of the Class 1
neutron histories will be much less at the boundaries
than the corresponding weights of the Class 2 histories;

hj Average Green’s Function for Spatial Intervals
H*
8SI

Average Green*a Functions for Spatial Intervals

130
therefore, the probability of surviving Russian Roulette
will be less for Class 1 histories than for Class 2
histories. This has the effect of causing Class 1
histories to contribute most of their statistical
weight to only one side of the source plane, while
Class 2 histories, because of their greater probability
for surviving Russian Roulette will contribute a signi¬
ficant portion of their statistical weight to both sides
of the source plane simply due to the fact that they
suffer a greater number of reflections at the boundaries.
Thus one would expect that the statistical
deviations in the Green’s function due to the contri¬
butions of Class 1 histories would be greater than that
due to Class 2 histories. A more sophisticated sampling
technique known as biased or importance sampling (35)
would permit preferential selection of the more important
Class 1 histories. Of course, the statistical weight
of the Class 1 histories would then be decreased to
compensate for their preferential selection, whereas
the Class 2 histories would be counted at full weight.
Such a process should materially reduce the computation
time required to generate a statistically satisfactory
Green’s function.
Another approach to the problem of reducing
the sample size and one which is of even greater interest

131
Is to evaluate the contribution of the known infinite
medium Green’s function analytically and to use the
Monte Carlo technique to generate only G*c(vjx,p|x*),
that is, the finite medium correction term which
accounts for the reflections at the boundaries. This
approach has the advantage of calculating analytically
that part of the Green’s function problem which causes
most of the statistical fluctuations in the Monte
Carlo solution, that is, the part of the Green’s
function that contains the source point singularity.
This approach would be applicable to any geometry since
the cell boundaries do not affect the infinite medium
part of the total Green’s function.
The Monte Carlo generation of the Green’s
function for slab geometry was used primarily to demon¬
strate the generality of the method, since from VII.2
one knows the analytic form of the slab geometry Green’s
function. The great value of the Monte Carlo technique
itself would become evident for complex geometries, as
discussed in VII.3, where the analytic form of the total
Green’s function may be exceedingly difficult or impossible
to derive although the infinite medium part of the Green’s
function is well known (21). In such cases the Monte
Carlo technique may be the only feasible way of obtain¬
ing the finite medium part of the Green’s function.

132
IX.3 Iteration Convergence
The normalization and over-relaxation
techniques discussed in VIII.2 were used to accelerate
the convergence of the iterative solution to Eq. (163)
for the one-speed case. The solution was iterated until
the difference between the calculated values and the
trial values was some fraction e of the neutron density
at each point. The number of iterations required for
the solution to Problem 2 of Appendix B to converge
such that e - 10-5 is illustrated in Figure 8 as a
function of over-relaxation factor for two cases,
namely, using the normalization technique and no
normalization. It can be seen that the normalization
technique is extremely useful. It was generally
observed that although the normalization technique
required more computation time per iteration, the tota?
computation time per problem was reduced by roughly
one-third.
The effectiveness of the. over-relaxation
technique was much less pronounced when used with the
normalization technique than when used alone. Also the
optimum over-relaxation factor was considerably closer
to the lower bound of 1.0 when the normalization
technique was used than when the over-relaxation

133
Figure 8 Effect of Convergence Acceleration
Techniques Applied to Problem 2 of
Appendix B. Solution Iterated Until
Residuals Were Equal to 10"5 Qf the
Neutron Density at Each Spatial Point

134
technique was used alone. On the basis of Figure 8,
an over-relaxation factor of W - 1.1 was used successfully
in all problems. However, a more complete study needs
to be made to determine the effects of cell configuration
on the optimum over-relaxation factor in order to es¬
tablish either a rule-of-thumb or a general method for
predicting the optimum over-relaxation factor for any
cell configuration. An indication of the division of
computation time for a typical problem is given in
Table 2. The problem discussed is Problem 2 of Appendix
B for which 2000 neutron histories were generated at
each of ten spatial points. It can be seen that the
generation of the Green's function consumes far more
TABLE 2
Breakdown of Computation Time for Problem 2, Appendix B
Operation
Time
in minutes (Approx.)
Generate Green's Function
20
Intermediate Tape Mounting
2.0
Solution of Problem
0.5
time than the solution of the problem itself. The
situation is actually far better than Table 2 Indicates,
however, because the Green's function tape can be used
for subsequent problems having the same moderator and

135
the same cell boundary dimensions. Thus the data for
Figure 8 were obtained using the same Green’s function
tape, ten problems being solved in approximately five
minutes. Slight modifications of the present program,
such as modifying intermediate tape usage, should
further accelerate solution time.
One would also expect that an algebraic
rearrangement of Eq. (163) consisting of factoring out
of the RHS the unknown value of the scalar neutron
density that one wishes to calculate and placing it
on the LHS would accelerate the convergence of an
iterative solution. Although it is expected that the
resulting equations would be exceedingly complex,
such an algebraic rearrangement is of future interest.
IX.4 Generation of the Green’s Function for the Multi¬
speed Problem
The extension of the present program to the
multispeed problem for J speed groups need not increase
the Green’s function generation time by a factor of J.
Rather than generate a new set of neutron histories at
each spatial point for each speed group, one can instead
obtain the neutron weights for all speed groups at each
spatial point by analyzing the histories for the one-
speed case. Thus if W(i;xm|xn) is the statistical

136
weight of a neutron cluster at ^ due to a source at
xn for speed group i, then from Eq. (129) the correspond
lng weight for speed group j is given by
W(j;Xm|xn) - WCijXnjIxjj) ^t<í)/^t (174)
After each pass across the unit cell, the
weights at the end points of each spatial interval are
subtracted to determine the number of collisions in
each spatial interval for each speed group. Russian
Roulette is then played at the cell boundary for the
speed group having the longest mean free path. This
procedure is then repeated for successive passes of
surviving neutrons and for successive histories as in
III.3 until a predetermined number of histories has
been generated. The average number of collisions per
unit time per neutron cluster in each spatial interval
is then computed for each speed group. Finally,
Eq. (133) is used to obtain the integral of the Green's
function over each spatial interval for each speed
group. The above procedure should permit generation
of the multispeed Green's function with a minimum
amount of computational effort.

NOMENCLATURE
D(r,v£p
D(rfv)
gk* <£»v£>
g(v;r,n r
g(v;r,n r
The angular difference density defined
as D(£>vn) - N(£tv£>) "* Nf(r,vn)
The scalar difference density defined
as D ',v’nf )drdvdn
The number of neutrons in the volume
element dr about r with speeds in dv
about v and with directions of motion
lying within the solid angle dn about n
due to a unit point source at rf emitting
one neutron per unit time in direction n’
with speed v* in a homogeneous finite
medium in which all collisions with
nuclei results in removal of the neutron
from the population and reflections occur
at the boundaries
* )d£<*G
The number of neutrons in dr about r
having speeds v and directions of motion
in dn. about n due to a unit point source
at r' emitting one neutron per unit time
per unit solid angle n with speed v, in
direction nf in a homogeneous finite
medium in which all collisions with
nuclei result in removal of the neutron
from the population and reflections occur
at the boundaries
)drdn
The number of neutrons in dr about £
with speed v having directions of motion
in dn about n due to a unit point source
at r’ emitting 4TT neutrons per unit time
of speed v isotropically in a homogeneous
finite medium in which all collisions
with huelei result in removal of the
neutron from the population and
reflections occur at the boundaries
137

138
g(v;r|r’)
The number density of first-flight
neutrons at r having speed v due to an
isotropic point source at r* emitting
47T neutrons per unit time with speed
v in a homogeneous finite medium in
which reflections occur at the boundaries
G(v;r |r')
The number density of first-flight neutrons
at r having speed v due to a unit isotropic
source at r' emitting one neutron per unit
time with speed V in a homogeneous finite
medium in which reflections occur at the
boundaries
G(v;x|x»)
The number density of first-flight neutrons
at x having speeds v due to a unit isotropic
plane source at x* emitting one neutron
per unit time with speed v in a homogeneous
finite medium in which reflections occur
at the boundaries
Li,n
The total path length traveled by a neutron
along a path having a direction charac¬
terized by the unit vector, (r^r1) pointing
from the image source at r' to the field
point e, the neutron arriving at the field
point r after having suffered n reflections
at the mirror boundaries of the unit cell
N(r,vn)drdvdn
The steady-state angular neutron distri¬
bution, that is, the number of neutrons
in the volume element dr about r, having
speeds in dv about v, and having directions
of motion lying within the solid angle dn
about _n
N* (£»vn)
The known angular density in a reference
space; N*(£ivrp corresponds to the unknown
angular density N(r,vn) in the space of
interest, that is, in the real unit cell
problem
N(r,v)drdv
The steady-state scalar neutron distri¬
bution, that is, the number of neutrons
in the volume element dr about r having
speeds in dv about v

139
P (r, v i->v, aWn) dvdn.
The probability per unit time that a
neutron of velocity v*iV will suffer
a scattering collision at r with a
.nucleus which is in motion with respect
to the laboratory system such that the
neutron emerges from the collision with
speed in dv about v and direction of
motion in dn about n
Po<£»vlL»v>dv
The probability per unit time that a
neutron having speed vn will suffer a
scattering collision at r with a nucleus
which is in motion with respect to the
laboratory system such that the neutron
emerges from the collision with speed
in dv about v; Po(rfvSL^v) is the zeroth
coefficient in the~~spherical harmonic
expansion of the angular dependence of
P(r, v^v,n'Un)
r
A vector Indicating a position in three-
dimensional space
A unit vector in three-dimensional space
with origin at the source point r’ and
pointing towards the field point r
S(r,vn)drdvdn
The angular source distribution, that is,
the number of neutrons per unit time
being added to the volume element dr
about r, having speeds in dv about v,
and directions of motion lying within
the solid angle dn. about n.
S(r,v)drdv
The scalar source distribution, that is,
the number of neutrons per unit time
being added to the volume element dr
about r and having speeds in dv about v
V(r,v)
The probability per unit time that a
neutron of speed v will suffer a scatter¬
ing collision at r with a nucleus which
may be in motion with respect to the
laboratory system

140
Ak»
7T(r ,v
) The probability per unit time that a
neutron of speed v will be absorbed at
r by a nucleus which is in motion with
respect to the laboratory system
The width of a unit cell in units of
total mean free paths of the medium of
the Green's function problem
JL*v)dv The probability per unit time that if
a neutron having speed v' suffers a
scattering collision at r, it will
emerge from the collision with speed
in dv about v

APPENDICES

APPENDIX A
DERIVATION OF Gfc(v;x|x')
The expressions for n and in
Eq. (126) for x > x* =» 0 are easily derived by
inspection of Figure 3. By the method of images,
|j*Ll,l| - 2b -X' - X
\f^2 l| “ 2b + x’ + x
|PLl,2|“|^L2,l|+ (2b-2x*)
|^L2,2|-|^Ll,l|+ (2b+2x')
| FL1,3I " \ph2,2I + (2b“2x' >
1/^2,3| “|^l1,2|+ (2b+2x*)
4b - x* + x
4b + x* - x
6b - x' - x
6b + x* + x
The pattern is now clear so that one
JjiLi nj * 2nb - x' + (-l)n x
|^L2,n| “ 2nb + *' “ (**l)n x
Similarly, for x* => x =* 0,
|jiL1>n » 2nb + x' - (-l)n x
IjiLaJ- 2nb - x* + (-l)n x
may write
) for x =» x*
/
\
>for x* =» x
/
0 (Al)
0 (A2)
142

143
Finally, for x =» 0 but x* < 0,
bLl,nl
l^2,nl
2nb +
2nb -
+ (-l)n x
- (-l)n x
>for x 0, x*
0 (A3)
/
Equations (Al) and (A3) are identical when proper account
is taken of the sign of x’. Hence for x =* 0,
\
l^l.n
l^L2,nl
2nb - xf + (-l)n x
2nb + x» - (-l)n x
fcfor x x1
From Eq. (126) and the above Eqs. (A2) and
(A4), the number density at x due to neutrons which
have suffered reflection at the boundaries can be
written as
(A4)
oo 2
Gfc(v;xjx') - I- £ £ E1[lt(k',v)|^i>n|]
- Sv E, Ki[lt|jaa>I1fl 4. H1[lt(k'.v)|i*8(I1|]
(A5)
Since each term of the summation includes E^ functions
of both Zt(k* ,v) and It(k’,v)|jiL2>n| » ^ seen
that G^ivjxjx*) for x =» x' is equal to Gfc(v;x|xf) for
x c xf. Thus no distinction need be made between the
two cases, and one can write for x =» 0 that

144
Gfc(v;x x«) - ^ - x* + <-l)n x|]
+■ Ej[l^(k*,v)j2nb + x’ - (-l)n x|]
, for x 0
Equation (A6) holds for both x* =* 0 and x* « 0.

APPENDIX B
SAMPLE PROBLEMS
The following problems are all for one-speed,
two-region, slab-geometry unit cells. Thus the solution
is obtained to the one-speed version of Eq. (163) for
R2“2. The properties of regions I and II will be de¬
noted by the subscripts I and II respectively. The
distance of the interface between regions I and II from
the midplane of the unit cell, that is, the half¬
thickness of region I is denoted by tj, while the
distance from the interface to the cell boundary is
denoted by tjjj the total half-cell width is thus
ti + tjj. All macroscopic cross sections are expressed
in cm-1 and all distances in cm.
For the following problems the a priori choice
was made to generate the first-flight Green’s function
for the moderator, that is, the medium in region II.
However, the results of several problems indicate that
different criteria should be examined. As discussed in
II.1, the first-flight angular density in a unit cell
due to an isotropic source will be more nearly isotropic
if the mean free path of the homogeneous medium of the
145

146
Green*s function problem is long relative to the
dimensions of the unit cell. Hence an important para¬
meter that should be examined for each problem is the
total optical thickness of the unit cell for the Green*s
function problem. This parameter is denoted by and
is the total width of the unit cell expressed in units
of total mean free paths of the medium of the Green*s^
function problem; thus
Afe» ** ^-t,k* * 2(^1 + ^ij)* (Bl)
For each of the sample problems Ab is tabulated, and
its relation to the results of each problem is discussed.

147
Problem 1 Thin Gold Absorber in Graphite
The first problem to be discussed is that of
a thin layer of gold absorber in a graphite unit cell.
In this problem, a spatially constant unit source
density of 1.0 neutron cm"3- sec**3- is assumed in the
graphite (region II) and zero source in the gold layer
(region I). Thus the source distribution represents
a slowing down source distribution, since there would
be negligible slowing down in the heavy absorber. The
physical parameters of the problem are given in Table 3.
TABLE 3
Parameters for Problem 1
Parameter
Region I
Region II
Ia cm*1
5.17
0.000235
Xs cm-1
0.549
0.385
Xj. cm"1
5.719
0.385235
t cm
0.01935
2.57595
-A-k
29.69
1.9996
S cm"1 sec"3-
0
1.0
The solution to this problem obtained by the
HGI method is compared in Figure 9 with solutions
obtained by an S4 calculation (32) performed at Oak
Eidge National Laboratory (33) and a P3 calculation

Figure 9 Spatial Dependence of Scalar Neutron Density
for Gold-Graphite Cell of Problem 1

149
using the special transmission boundary conditions of
section 10.3.8 of Davison (2). As indicated previously,
an important parameter for many problems may be the
optical thickness of a region, where optical thickness
is defined to be the geometrical thickness measured in
units of mean free paths. Davison has stated that for
an absorbing layer which is geometrically thin but
optically thick the degree of approximation required
for a satisfactory Pn calculation may be reduced by
applying optical or transmission boundary conditions.
The optical boundary conditions are
N(0+,p) » e""*1//1 N(0-,ja)
o
A
au.
N(0-,ji) - j/1) N(0+ ,p)
, ji «= 0
where h is the optical thickness of the absorbing layer
(region I of this problem) having negligible geometrical
thickness and being' situated at x *» 0, so that x > 0
and x c 0 are the scattering media on each side of the
absorbing layer. For full details the reader should
consult Davison.
It is seen in Figure 9 that the HGI method
predicts a greater perturbation of the scalar neutron
density by the absorbing layer than does either the S4

150
or P3 calculation. Neither the S4 nor the P3 calculation
should be expected to be too accurate near the interface,
because the angular density N(x,ji) is very anisotropic
near the interface between the absorbing and scattering
regions. Thus an approximation which depends on cutting
off the angular expansion of the angular density after
the first four moments may not yield a very accurate
description of the spatial dependence of the scalar
density near the interface, and one should check to see
if additional moments would improve the Pn and Sn results.
The HGI method, on the other hand, contains
a somewhat different cutoff of the angular expansion of
N(x,ji); in the HGI method one neglects only products of
higher moments of the angular density times the Green’s
function. Thus it is subject only to the assumptions of
II.1 regarding the isotropy of the corresponding angular
density of first-flight neutrons due to a unit isotropic
plane source in a homogeneous medium unit cell with re¬
flecting boundaries and the isotropy of the angular
neutron flux in the cell. Because the value of Ajj
is only about two, the assumption of nearly isotropic
first-flight neutrons should be fairly well justified.
This reasoning follows from the fact that a neutron
cluster making one complete pass across the unit cell

151
of the Green’s function problem would have a statistical
weight equal to 0.136. Most neutron clusters arriving
at a boundary for the first time will have a greater
statistical weight than the minimum value exp A-u
due to traveling less than the complete width of the
cell before arriving at the boundary. Therefore, many
neutrons will survive the Russian Roulette game at the
cell wall and will be reflected at the boundary, thus
making the first-flight angular density more nearly
isotropic. Thus the HGI method may give a more accurate
representation of the spatial dependence of the scalar
neutron density near the interface than either the 84
or P3 calculations.
Note that at distances greater than about
one-half of a mean free path from the interface the
84 and P3 solutions have the same shape as the HGI
solution as expected. Neutron conservation, however,
requires that the total absorptions in the cell must
equal the total sources so that the integral /vl aN(x)dx
over the unit cell must be equal to the integral
Js(x)dx over the unit cell, regardless of the method
used to calculate N. Since the scalar density calculated
by the HGI method is smaller than the S4 and P3 solutions

152
in the absorbing region, it must be correspondingly
larger in the scattering region in order to give the
same value
The close agreement of the S4 and P3 solutions
seems to indicate that the use of transmission boundary
conditions cannot be expected to improve greatly upon
the results of a P3 calculation using the more common
boundary conditions of continuity of angular moments
at the interface, this conclusion being based on the
rough equivalence of an S4 calculation to the latter
type of P3 calculation.
The results of this problem have not definitely
established the accuracy of the HGI method. Nevertheless,
there are doubts about the reliability of the P3 and S4
solutions near the interface as discussed above so that
the differences in the results obtained by the three
methods seem favorable to the HGI method. In order to
verify this conclusion, one should compare the HGI
method with higher order Pn and Sn calculations. This
is done in succeeding problems.

153
Problem 2 Equal Volume Regions of Water and
Strongly Absorbing Fuel
This problem is a fuel-water unit cell, regions
I and II being of equal volume. The problem has been
solved by Gelbard, et al. (34) by various spherical
harmonic approximations including Plf double-P0, P3,
and double-Pi, and by a discrete ordinate method "roughly
equivalent to a P-q approximation."
The explicit physical parameters of the problem
are given in Table 4. As in Problem 1, a slowing down
TABLE 4
Parameters for Problem 2
Parameter
Region I
Region II
Ia cm”*
1.0
0.02
ls cm*"1
0.3
3.12
¿L-t cm”*
1.3
3.14
t cm
0.127
0.127
Ak
0.660
1.595
S cm”* sec”*
0
1.0
source is represented by assuming a spatially constant
unit source density of 1.0 neutron cm*"*- sec”* in the water
(region II) and zero source in the fuel (region I).

154
The solution to this problem obtained by the
HGI method is compared in Figure 10 to the results
obtained by Gelbard using P^, P3, double-P^ (DP^),
and the discrete ordinate (BOR) calculations. It can
be seen that Pn- and Sn-type calculations appear to
approach the HGI results as n gets large. A similar
conclusion is reached if one considers the ratio of
the average neutron density in region I to the average
neutron density in region II, that is, the advantage
factor for the unit cell as calculated by Gelbard and
as calculated by the HGI method. These results are
indicated in Figure 11, the abscissa denoting merely
an increasing order of approximation.
As in Problem 1, the value of Ajj is relatively
small; the weight of a neutron cluster being equal to
0.203 after one complete pass through the cell for the
Green’s function problem. It is interesting to note
that if the fuel in region I had been chosen as the
medium for the Green’s function problem, then Aj would
be the optical width of the unit cell for the Green’s
function problem. The statistical weight of a neutron
cluster after one complete pass through the cell would
be equal to 0.516. Thus the probability of surviving
the Russian Roulette game at the cell boundaries would

Scalar Neutron Density, N(x), neutrons/cm
(Speed taken as 2200 m/sec)
Figure 10 Spatial Dependence of Scalar Neutron Density
for Fuel-Water Cell of Problem 2
Reflecting Boundary

Advantage Factor
156
Figure 11 Advantage Factor for Fuel-Water
Cell of Problem 2 by Various
Approximation Methods

157
be greater for a unit cell of optical width Aj than fox
one of optical width An* This means that for the
Green’s function problem, more reflections would occur
at the boundaries in a cell containing all fuel than
in one containing all moderator, and hence the first-
flight Green’s function would be more nearly isotropic
if the medium of region I were used for the Green’s
function problem than if the medium of region II were
used. This apparent contradiction is explained by the
fact that only the total mean free path is considered
in the Green’s function problem; the large scattering
cross section of the moderator makes the total mean
free path of the moderator less than that of the fuel.
In order to determine the convergence of the
HGI method as a function of the order of iteration,
advantage factors were calculated after each iteration
and the solution was iterated until the residuals were
less than 2 x 10“® of the neutron density at each spatial
point. The results of these calculations are illus¬
trated in Figure 12 and indicate that the HGI method
yields a rapidly converging solution which is of the
order of that obtained by an Independent calculation
’’roughly equivalent to a P11 approximation” (34).

158
Figure 12 Advantage Factor Calculated
by HGI Method vs. Number
of Iterations for Fuel-Water
Cell of Problem 2

159
Problem 3 Thin Cell Calculations
Meneghetti (9) has studied the convergence of
high-order Pn, discrete Sn, and double-Pn approximations
for a thin slab cell having regional thicknesses that
are fractions of mean free paths. Using the above
approximations, Meneghetti calculated the advantage
factor for a cell having the explicit physical parameters
given in Table 5 for the case of a spatially constant
TABLE 5
Parameters for Problem 3
Parameter
Region I
Region II
Za cm“*
0.12
0.028
Zs cm-*
0.08
0.042
Z^ cm*"*
0.20
0.07
t cm
0.32
2.24
Ak
1.024
0.358
S cm**1 sec-1
1.0
0
unit source in region I and zero source in region II.
This source distribution was used by Meneghetti in
order to investigate the convergence of the Pn, DSn,
and DPn solutions for speed groups in which the source
distribution was not that of a slowing down source
(as in Problems 1 and 2 of this Appendix) but instead
represented a fission source. Although the multispeed

160
equations derived in the preceding text are explicitly
valid only for the thermal speed range, the one-speed
equation is valid for any speed provided that appropri¬
ately averaged cross section data are used.
The results obtained by Meneghetti for various
orders of approximation are reproduced in Figure 13;
also shown is the single value of advantage factor
calculated using the HGI method. Since the HGI method
is a zeroth harmonic approximation, it is plotted at
n**0. Of the calculations by Meneghetti, only the DP5
solution appears reasonably converged. If one assumes
that the DP5 results represent the converged solution
for the cell, then the advantage factor calculated by
the HGI method is closer to the converged solution
than either the Pjg or the DS^g solution; in fact, the
HGI solution differs from the DPg solution by less than
0.5 per cent. While not definitely establishing the
relative order of approximation of the HGI method,
nevertheless, the data indicate that the HGI method
does yield solutions which agree with very high order
Pn, DSn, and DPn calculations.
For this problem the value of An is quite
small, and the statistical weight of a first-flight
neutron cluster after one complete pass through the

Advantage Factor
Figure 13 Advantage Factors for Problem 3 by Various Calculations
161

162
unit cell of the Green’s function problem would be
equal to 0.699. Such a high statistical weight means
that many histories would survive Russian Roulette at
the cell boundary so that the assumption of a nearly
isotropic first-flight Green’s function should be
particularly well justified for this problem. Accordingly,
the HGI method should give a very accurate result for
the advantage factor for this problem.

163
Problem 4 Comparison of Results for Thin Absorbing
Regions vs. Thick Absorbing Regions
Gelbard, et al. (34) have calculated advantage
factors for fuel-water unit cells which had the same
overall cell dimensions but which had absorbing regions
which differed both in thickness and in absorption
properties. In order to demonstrate the unique ad¬
vantages of the HGI method for parametric studies, the
same Green’s function was used to solve all four of the
cases discussed here. The physical parameters for the
four different cases to be considered here are given in
Table 6. In each case a slowing down source is represented
by assuming a spatially constant unit source in region II,
that is, the water, and zero source in region I. The
total cell width was 1.397 cm for each case.
TABLE 6
Parameters for Problem 4
Case
1
2
3
4
Region
Parameter^"'\^
I
II
I
II
I
II
I
II
Za cm-1
0.1
0.02
0.0
0.02
1.0
0.02
1.0
0.02
Zs cm”1
0.3
3.12
0.3
3.12
0.3
3.12
0.3
3.12
Xt cm-1
0.4
3.14
0.4
3.14
1.3
3.14
1.3
3.14
t cm
1.27
.127
.127
1.27
1.27
.127
.127
1.27
Ak
1.12
8.78
1.12
8.78
3.63
8.78
3.63
8.78
S cm"1 sec"1
0
1.0
0
1.0
0
1.0
0
1.0

164
The advantage factors calculated by Gelbard
using Pi, double-P0, P3, double-Pi, and a discrete
ordinate method comparable to a Pn approximation
(RDR) are compared in Figure 14 to the advantage
factors calculated by the HGI method.
As illustrated in Figure 14, the advantage
factors calculated by the HGI method do not, in general,
agree with the results that Gelbard obtained using the
RDR calculation. This is most noted for the two cases
of thick absorbing layers, that is, Cases 1 and 3. The
reason for the discrepancy seems to be the very high
value of An» The statistical weight of a first-
flight neutron cluster after one complete pass through
the cell of the Green*s function problem would be equal
to 0.00015; therefore, the probability of a neutron
cluster surviving Russian Roulette at the cell wall is
negligible. Thus the first-flight Green*s function for
this problem may be expected to be more anisotropic than
in the previous problems due to the fact that there are
fewer reflections at the boundaries.
The real neutron density would be expected
to be more anisotropic in the cells containing the
thick absorbers than in the cells with thin absorbers.
Therefore, since the zeroth harmonic approximation of

Advantage Factor
165
Case 2
Figure 14 Advantage Factor for Problem 4
by Various Calculations

166
the HGI method involves neglecting products of the higher
angular moments of both the first-flight Green's function
and the real neutron density, one would expect that the
deviation of the HGI results from the RDR results would
be greater for the cells having thick absorbing layers
than for the cells having thin absorbing layers. The
data of Figure 14 would seem to verify this prediction,
but more extensive calculations would be required to
establish this prediction for the general case.
The spatial dependence of the scalar neutron
density as calculated by the HGI method for each of
the four cases is illustrated in Figures 15 and 16.
All four cases were solved using the same Green's
function. The computation of the Green's function
required about one-half hour. However, once the Green's
function had been computed, the solution of each unit
cell problem required less than one-half minute of
computation time. This demonstrates the advantage of
the HGI for repetitious calculations. Once the Green's
function is available, additional unit cell calculations
can be generated quite cheaply, each problem requiring
only a few seconds of computation time.

Scalar Neutron Density, N(x), neutrons/cm
Figure 15 Spatial Dependence of Scalar Neutron Density
for Cases 1 and 2 of Problem 4
Reflecting Bounda;

Figure 16 Spatial Dependence of Scalar Neutron Density
for Cases 3 and 4 of Problem 4

REFERENCES
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AECL, 1958
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(3.962)
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AEC-tr-1691
11. T. H. R. Skyrme, MS-91 and MS-91A, 2nd Edition,
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Physics,” McGraw-Hill, New York (1953)
169

170
17. H. D. Brown and D. S. St. John, DP-33 (1954)
18. S. Glasstone and M. C. Edlund, ’’The Elements of
Nuclear Reactor Theory,” D. Van Nostrand,
Princeton (1960)
19. A. M. Weinberg and E. P. Wigner, ’’The Physical
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Chicago Press, Chicago (1958)
20. R. K. Osborn, Trans. Am. Nuclear Soc, 5, 31 (1962)
21. K. M. Case, F. de Hoffmann, and G. Placzek,
"Introduction to the Theory of Neutron
Diffusion,” Vol. I, U.S. Government Printing
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Their Application to Unit Cell Problems,”
M.Sc.E. Thesis, Unpublished, Univ. of
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BIOGRAPHY
John Phillips Church was born on July 14,
1934, in Columbus, Ohio. He graduated from Logan High
School, Logan, Ohio, in May, 1952. Mr. Church entered
the University of Cincinnati in September, 1952, and
graduated with the degree of Chemical Engineer in
June, 1957. During his undergraduate study, Mr. Church
gained cooperative work experience with the National
Cash Register Co., Inc., in Dayton, Ohio. After receiv¬
ing his degree, he accepted employment as Chemical
Engineer with this company from June, 1957, to August,
1958. He entered the University of Florida in September,
1958, and received the degree of Master of Science in
Engineering with a major in Nuclear Engineering in
January, 1960. From January, 1960, until the present
time he has pursued his work toward the degree Doctor
of Philosophy. -
Mr. Church is married to the former Nancy Carol
Snyder. He is a former AEG Fellow and is a member of
Tau Beta Pi and Phi Lambda Upsilon honor societies, the
American Nuclear Society, and the American Physical Society.

This dissertation was prepared under the
direction of the chairman of the candidate's supervisory
committee and has been approved by all members of that
committee. It was submitted to the Dean of the College
of Engineering and to the Graduate Council, and was
approved as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
April 20, 1963
Dean
bllegi
Dean, Graduate School
Supervisory Committee:
2.
Chairman

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