THE DIPOLE RESPONSE COEFFICIENT

IN NEUTRON DIFFUSION THEORY

by

M.D. LOVE and S.A. KUSHNERIUK

Chalk River Nuclear Laboratories

Chalk River, Ontario

May 1974

AECL-4826

THE DIPOLE RESPONSE COEFFICIENT FOR CYLINDRICALCHANNELS IN NEUTRON DIFFUSION THEORY

M . D . LOVE + AND S . A . KUSHNERIUK

Abstract

The influence of a cylindrically symmetric,neutronabsorbing and scattering channel on the asymptotic neutron flux inthe purely neutron scattering medium that surrounds the channel isderived for the case when, in the absence of the channel, the fluxdistribution is of a linear form. The consideration is one-velocity.The media are of infinite extent and the neutron sources areinfinitely far from the channel. Neutron scattering is isotropic.

The exact neutron flux distribution in the mediumis a solution of an integral equation. Its asymptotic part is ofthe same form as the solution of the diffusion differential equation.Thus the knowledge of the exact asymptotic solution effectively pro-vides a boundary condition for the diffusion solution. nuch condi-tions, expressed as "dipole response coefficients", are derived usingperturbation methods, variational procedures and continuity relations.Results are obtained first for homogeneous channels that absorbneutrons only. Configurations are then generalized to channels thatconsist of a purely absorbing inner region surrounded by an air-gapor a weakly neutron absorbing sheath or both. Finally channels areconsidered in which the inner medium scatters as well as absorbsneutrons.

ATOMIC ENERGY OF CANADA LIMITEDCHALK RIVER NUCLEAR LABORATORIES

CHALK RIVER/ ONTARIO, CANADAKOJ 1J0

May 1974

t Present address: Department of Nuclear Engineering,Queen Mary College, University of London, LondonEl 4NS, England.

AECL-4826

Le coefficient de response dipolaire pour les canauxcylindrigues dans la thgorie de diffusion des neutrons

par

M.D. Love* et S.A. Kushneriuk

*Adresse actuelle: Department of Nuclear EngineeringQueen Mary CollegeUniversity of LondonLondon, England, El 4NS

Resume

L1influence d'un canal de diffusion et d'absorp-tion des neutrons, cylindriquement symetrique, sur leflux neutronique asymptotique dans le milieu purementdiffuseur de neutrons qui entoure un tel canal estdeduite pour le cas ou en 1'absence de canal, la reparti-tion du flux se presente sous une forme lineaire. Laconsideration est celle d'une seule vitesse. Les milieuxsont d'une etendue infinie et les sources de neutronssont infiniment eloignees du canal. La diffusion neutro-nique est isotropique.

La distribution exacte du flux neutronique dansle milieu est donnee par la solution d'une equationintegrale. Sa partie asymptotique a la meme forme quela solution de 1'equation differentielle de diffusion.Ainsi, la connaissrnce de la solution asymptotique exactefournit effectivement une condition limite pour la solutiond e d* f f u s l o n' D e telles conditions exprimees comme etantdes "coefficients de reponse dipolaire" sont deduitesau raoyen de methodes de perturbation, de proceduresvariationnelles et de relations de continuite. Lesresultats sont obtenus tout d'abord pour des canauxhomogenes n'absorbant que les neutrons. Les configurationssont ensuite generaliseer pour des canaux comprenant uneregion interieure purement absorbante entouree par unentrefer ou une gaine absorbant peu les neutrons ou lesdeux. Finalement, des canaux dans lesquels le milieuinterieur diffuse aussi bien qu'11 absorbe les neutrons,sont pris en consideration.

L'Energie Atomique du Canada, LimiteeLaboratoires Nucleaires de Chalk River

Chalk River, OntarioKOJ 1J0

Mai 1974AECL-4826

1. INTRODUCTION

The integral equation for the neutron flux in

a capturing and isotropically scattering medium occupying a

volume V is in the one velocity approximation

j v —v

Here <f>(r) is the neutron flux, c(r) is the mean number of

secondaries produced per collision, H(v) is the macroscopic

total neutron mean free path at r, K(r_,r' ) is the probability

that a neutron born at r' will arrive at r, and S(r) is the

contribution to <J>(r_) due to the neutrons that have come directly

from sources or directly from outside the volume V, without

having been scattered in V.

In certain instances cj>(r) can be conveniently

expressed as (Davison 1957, p.94)

= cj>as(r)

in which the form of 4 (r) is identical with the form ofvas —

<j>diff(r), the solution of the diffusion differential equation

V2

while <{). (r) is of the order of e" , 5 being the optical

neutron path from r to the nearest discrete source or boundary,

- 2 -

whichever is tha smaller. In a typical thermal reactor,

which consists of cylindrical channels surrounded by a rela-

tively large volume of moderator for which c(r) is also close

to unity, a truly asymptotic flux region about the channels

exists. For this situation the transient flux is important

only within about one moderator mean free path from the sur-

face of a given channel. For many purposes in these reactors

it is of primary interest to know only <j> (r) in the moderator

rather than the complete $(v), and <}> (r) can be determined

from the solutions of the diffusion equation if the boundary

conditions to be satisfied by these solutions at the channel

surface are known. It is with the specification of the form

of the boundary conditions at the channel surface that we are

concerned in this paper.

The general solution to equation ( 1) in cylin-

drical polar co-ordinates for cylindrical channels, the pro-

perties of which are invariant under translation along and

rotation about the channel axis, can by correct choice of the

reference plane be written in a harmonic series cf the form

<}>(£) = ^ Q ^ ) + ^ ( r ) cose + 4>2(r) [cos 26 + b-,sin29]+• • • . (4)

(KCr) and (j^Cr) are respectively the monopole and dipole

fluxes. On a macroscopic scale the monopole flux has the

approximate form of a flat flux, while the dipole flux has

that of a linear flux. The higher order harmonics are generally

- 3 -

small except in closely spaced lattices. Taking into con-

sideration the harmonic expansion of the asymptotic flux 11

boundary condit' ~r"~: that we specify are the ratios

channel surface

(5)

These boundary conditions depend upon the dimensions and

neutron properties of the channel and moderator. The geometry

used above and below is specified later (see figure 1).

Expressions for the conditions to be applied in the determin-

ation of $ (r) have been developed by Davison, Kushneriuk and

McKay among, others (Davison 1951; Kushneriuk et al.. 1949, 1954, 1962,

1967). In this publication we present formulae for the dipolar

channel boundary condition. These formulae have been obtained

by considering an idealized problem, the open cell dipolar

problem. The underlying assumption in their practical appli-

cation is that the boundary conditions are not sensitive to the

differences between the idealized and practical problems.

We determine the form of X, for a cylindrical

channel by considering the effect of introducing the channel,

assumed to be of infinite height and invariant under transla-

tion along and rotation about its axis, into an infinite

volume of purely scattering moderator. In the absence of the

channel the neutron flux, maintained by sources at infinity,

is taken to be

- 4 -

<j>(r) = r cosO. (6)

In the presence of the channel the flux is everywhere propor-

tional to COG8 (see Appendix A). Specifically it may be repre-

:• • f • n L e ct a : '•

<|>(r) = (r + ^ + *trans ( r ) ) c O s 9' ( 7 )

where A,, the open cell dipole response coefficient, describes

the asymptotic response of the unperturbed flux to the intro-

duction of the channel. The first two terms on the right of eq.

(7) also are the solutions to the diffusion equation for this

problem, and accordingly we have (a is the channel radius)

(a2 + A,)X = a — = — . (5a)1 (a2 - A]_)

By solving the integral equation governing this idealized

problem exactly and in approximate ways, and by using physical

arguments, we have built up for a variety of channel config-

urations a general practical formula for A-, .

The dipole flux is of particular importance in the

calculation of streaming effects (Benoiirt 1968). Use of open cell

response coefficients is common to the work of many authors,

among whom we may note Auerbach et al. 1972; Berha 1971; Carter

1061; Stewart 1971; and Bonalumi 1973. There does not as

yet,however,exist in the literature an explicit set of practical,

f-vriejvjl formulae for A,. Carter and Jarv.is (]r:61) had Droviounly

- 5 -

considered the form of A, for void channels. They obtained,

by means of perturbation theory and perturbation theory

coupled with a variational principle,exact asymptotic values

for A-p and then interpolated between these using estimates

based on a P, spherical harmonics solution. They also gave a

simple analytic approximation to their results. Other authors

have developed computer codes for calculating A, (Maeder 19715

Alpiar 1969 i Berna 1971; Bridge and Cumpstey 1973).

In deriving our general formula we first con-

sider the simplest type of channels, namely homogeneous purely

absorbing channels. We derive by means of perturbation theory

exact asymptotic expansions for Ai, which are valid for the

limiting cases of channels of very large aiid very small radii.

In each case the order of successive terms and hence conver-

gence of the resulting series is examined. These results

enable us to check the accuracy of more general results obtained

by the application of a variational principle. We find that

we can re-interpret our general formulae in physical terms

using the continuity equation, and can a loo introduce simpli-

fying approximations. These approximations are based on the

inherent physical structure of our results and considerably

simplify our formulae with little overall loss in accuracy.

This enables us to extend the theory to include channel

inhomogeneity and the effects of the scattering of neutrons

from within the channel.

- 6 -

In this work we denote the channel radius by

a, the open cell dipole response coefficient for homogeneous

purely absorbing channels as A, (a,a/A), the moderator total

neutron mean free path as £, and the channel total mean free

path as If a. We denote the radial and azimuthal co-ordinates

about the channel axis by r and 9, and also make repeated use

of the geometry and coordinate labelling illustrated in Fig. 1,

Note also that when specifying in detail the integral equations

which are relevant to our analysis we automatically put

c(r) = 1.0 in the purely neutron scattering medium which sur-

rounds the channel. c(r) = 0 when the medium is purely

absorbing.

2. PERTURBATION EXPANSIONS FOR HOMOGENEOUS ABSORBING CHANNELSOF SMALL RADIUS

Here our concern is for the determination of

the asymptotic form of the open cell dipolar flux, as asso-

ciated with homogeneous absorbing channels of small radii.

By small it is meant that the channel radius is very much

smaller than the mean free path in the surrounding moderator.

The character of the method developed here is the same as that

of Kushneriuk, Davison and Seidel (1949), which method

was introduced for the determination of the extrapolation

length of small black cylinders.

2.1 THE PERTURBATION FORMALISM

As mentioned in the introduction and demonstrated

in Appendix A the general form of the solution (j>(r) for the

dipolar flux is <}>(r) = <})(r)cos8 where <j>(r) is a solution of

the integral equation (A9). (<j>(r) is identical with what is

denoted as g-,(r) in Appendix A.) In order to obtain <j>(r) in

a series of successive approximations of the form

(8)

in which <j>Q(r) is that part of <j>(r) that remains when we con-

sider the limiting case of a channel of vanishingly small

radius, we write the integral equation for <j>(r) in the form

- 8 -

/•« r r + r '4>(r)=coR0| r ' d r 1 I 4>(r') K ( p , r , r ' ) d p

JQ J\r-v'\

fI a ,

-cose I r ' d r ' j c|>(r ' )K(p , r , r ' ) dp

.r+r1

'dr1 jV(rz-a2)+

-cosG \ r ' d r ' { 4>(r') K-^p , r , r ' )dp (9)Ar'2-a2)

for r > a, and

-.00 y»rT '

<f>(r)=cose I r ' d r ' tj)(r ') K ( p , r , r ' ) d p

J o J | r . r . |

/•a r r + r 1

-cose I r ' d r ' t|>(r') K ( p , r , r ' ) d pJo ^ I r - r ' l

f°° r r+r '-cose 1 r ' d r ' • ( r 1 ) K0(p,r ,r ' )dp C9a)

J a • / | r - r ' |

for r < a. In the above (see also appendix A and figure 1)

- 9 -

K(p,r,r') = ~j J Ki-^p/X,) COSOQ,

= ^ J [Ki1(p/£) -Ki1((p + (a-l)pI)/£)]cor,(D,

K2(p,r,r') = |j J [Ki^p/A) - Ki-^ (p + (a-Dp2 )/& ) 3 cosu> (10)

and J is as defined in Appendix A, eq. (A9).

The subscripted terms of equation (8) should not be confused

with those of the harmonic expansion of eq. (4). The first term

of the above equations gives the unperturbed flux, and the

remaining terms describe the perturbation caused by the

channel. Consider now an ancillary function <j>(r;£), defined

for r > a by the equation

Too fr+r'<J)(r,£)=cosfl I r'dr' I cj, (r ' ;£ )K(p ,r,r' )dp

JO J|r-r'|

fa fr+r1

ScosB r'dr1 IJO J|r-r'

;E, )K(p ,r ,r' )dp

Too f6 r'dr1 IJa J

f-£,cos6 r'dr1 I <Ji(r' ;C)K1(p,r,r' )dp (11)

J J

- 10 -

and defined in a corresponding manner for r < a. <|>(r; O

can be expanded into an ascending series in powers of £,

namely

(12)

where setting £, = 1 we return to the series of eq. (8) with

the <\t (r) unambiguously defined. We proceed now for £ small

and a finite, considering later the question of convergence

of the series for the actual case of E, = 1 and a small.

On substituting (12) into the eq. (11) and collecting terms

of the same order in £, we obtain the perturbation equations

/"~ fr+r'<J>n(r)=cos6 I r'dr' $ (r' )K( p ,r ,r ' )dp (13)

Jo J|r-r'| U

and , i or n _> 1,

Too fr+r'

I r'dr1 IJO J|r-r'

> (r)=cos6 I r'dr1 I $ (r' )K(p ,r ,r' )dp + co:;9f (i)JO J|rr'| n n

where

ra rf n ( r ) = "I r'dr' IJ n J \

a rr+r'r'dr' I

0 •/|r-r'

^.r+r'

'dr' I *n x(r')K (p,r,r')dp (15)"//(r2-a2)+ /(r'2-a2)

- 11 -

for r > a, and

fa fr+r1

f ( r ) = - | r ' d r 1 / <f> n ( r ' )K(p , r , r ' )dpJO / | r - r ' |

r oo fr+r'I i . , y n - lJ r - r '

r ' d r 1 J ( | ( ( ) n _ 1 ( r ' )K 2 (p , r , r T )dp (15a)a

for r < a.

2.2 THE ASYMPTOTIC BEHAVIOUR OF (j>Cr)

By definition <j>n(r) corresponds to the sta-

tionary neutron distribution present in the absence of the

channel, which is given as (see introduction and eqs. (6) and (8))

(j)n(r) = r cose

so that tj)n(r)=r. This, of course, also satisfies eq. (13). In the presence

of the channel no mixing of harmonics occurs, i . e . the neutron flux is :;till

of only a cose azimuthal dependence, and we now sho*; t h a t the

asymptotic so lu t i ons to the equations ( I1-) take the f orm

= Cn(o,a/£) 52£e + O(e-r/Jl>. (16)

This immediately leads to

A1(a,a/jl) = I Cn(a,a/£)

n=l

- 12 -

The eqs. (14) can be solved directly by means of

Fourier transforms, though it is possible to construct solu-

tions, as we do, in terms of line sources. Each of the eqs.

(14) is interpreted as an equation for the determination of

the neutron flux due to a cylindrically symmetrical system of

line1sources in a uniform non-capturing medium filling the whole

of space. In this interpretation the source density is taken to

be f (r)/&; the neutron flux is then given by C'l'CrO-f (r) ) .

If L(r) is the flux distribution due to an isotropic line

source of unit strength per unit length situated along the Z

axis of our medium then the particular solution of eqs. (14)

can be expressed in terms of the source density f (r)/I and

the solution L(r), namely

I;1 / / 27T

/ r? fn ^

r ' ) d ' / cos9'L(p)d8' (18)

in which p = |r-r'| = (r +r' -2rr'cos(0-9'))2 and we have used

the fact that fR(r) = fn(r)coE6.

The solution L(r) for a line source is discussed

by Davison (1957, p.62). This solution is simplest

when the medium is also neutron capturing in which case it is

L(r) = til1 K 0 ( K r ) + Wi / g(y>K0(ry/£)dy (19)

in which < is the inverse diffusion length in the medium,

D ( K ) is the diffusion coefficient and

- ] 3 -

We therefore alter our medium by making it slightly neutron

capturing. In this case eq. (18) is changed by multiplying

the right hand side by c. Substituting for L(p) from eq.

(19) into the modified eq. (18), expanding K~(Kp) by the

Bessel function addition theorem (Watson 1952, p.3Gl),

integrating with respect to 91 and then passing back to the

limit of a purely scattering medium (i.e. c -• 1 whence K -+ 0,

D ( K ) ->• 3), we find,after combining with eq. (8),

<Kr) =00 / QO

n =

/ r'2f (r')dr') - + 0(e~r/£)Jo n r J

r + ( 3 I / r'2f (r')dr') - + 0(e~r/£) cos0. (70)n r J

A comparison with eq. (16) gives the form of Cn(cxa/£).

This also confirms the form of the open cell dipole flux as

assumed above.

2.3 EVALUATION OF C^a /JO

Because of the increasing complexity of the .integralr,

with the order of n, we content ourselves here, as is suffic-

ient for our purposes, with evaluating merely the lowest

order term, i.e. n = 1. Transforming co-ordinates from dr'

to dy we obtain

C, = - - ^ - y \ r dr dy1 2TTJT -^0 J 0

r7

I(r-p cosy)Ki-. (p/£)dp

i ( r - p c o s y ) K i ; L ( ( p + ( a - l ) p 2 ) / A ) d p

P2

2TTA J a

r°o /"I r^dr I

J a J

sin rdy (r-p c

(r-p

- c + c + c +~ 11 L12 L13

(21)

We row evaluate the integrals C-, ,

I) C11 r^dr = - " =25. 0

II) Keeping r fixed and transforming co-ordinates from y to GO

and then transforming from rdrdco to sdsdtj; ultimately gives

'12

in which x = aa/A and the P ( x ) ' s a re t h e f i r s t c o l l i s i o n

- 15 -

probabilities discussed in Appendix B. From a comparison with

formulae given in Appendix D we see that C12.may alterna-

tively be written as (eqs. (DID and (D18))

4-C12 = -j [PFq(x) + a

16£

III) Recalling the definitions of Ki^x) (see Appendix A)

we may establish that the integrals C13 and C1L|

are of the type discussed in Appendix C. In particular,

from eqs. (C6a) and (C7), with g(r) = g(r) = r,

O -. J I n 1 r, r, n 2

C13 + C 1 4 E - ^a

Also

C13

2.4 ORDER OF SUCCESSIVE APPROXIMATIONS AND CONVERGENCE OF SERIES

We present below only bare details of the

analysis for the simplest case, which is that of a black rod,

i.e. a = °°. The results are however more general in that the

integrals defining the C (a,a/&) are of the same form, and

for a << H the leading part of each arises in consequence of

the same terms. From such considerations we are led to

surmise that

C (a a/1) inL/nj.nfa/f.lr .(n..a/l)\. (?'n '

- 16 -

which establishes the order of contribution of the successive

approximations, and also the convergence of the series (12)

for t, = 1 and a small.

Expressing f,(r) as a definite, integral, we

can easily show that for a black rod

^(r) = a2cos6'0U/a) .

Combining this with (15) and (19) and using co-ordinate trans

formations akin to those used above leads to

Noting that, in general, for n > 2, C (a=°°,a/S.) depends upon

<j> i (r) in the same manner as C -,(a=oo,a/£) depends upon

<t> o(r), we have11 •" i~ —

for n >. 2

which further generalizing establishes the desired result

given in equation (22).

2.5 RESULTS

On substituting for the Cln(a,a/fc) from above in

eq.(17) and on taking the appropriate expansions of the

Fnm(a,a/£) integrals we obtain in the lowest order the general

perturbation expansion for A1. In particular we obtain

- 17 -

A,(a=l,a/JO = -a 2(| a2/12 +X O

and

= -a 2(| + S^i + . . . ) . (23)

Note that for the special case of a=0, cancellations occur and we

find that to determine A, to the order of a/A it is sufficient

to determine C-,(a=0,a/A) to the same order. This result is

in agreement with Carter 6 Jarvis (1961).

- 18 -

3. PERTURBATION EXPANSIONS FOR HOMOGENEOUS ABSORBINGCHANNELS OF LARGE RADII

We now consider channels for which the radius

is large compared to the neutron mean free path in the sur-

rounding moderator. The effects of channel curvature are

treated as a perturbation to the lowest order approximation,

which is the corresponding infinite half space, i.e. Milne

type problem. First the neutron flux is sought as a series

of successive corrections from which, when the different terms

of the same order of magnitude are collected together, it is

then possible to obtain an answer to the order of (a/£)~n.

In applying this procedure it is necessary to consider

separately the case of large void channels. We consider

first the case of non-void channels with aa >> I. 'The general

procedure is that due to Davison (1951) and we pre-

sent in what follows only an outline of the method and the

results as applied to our case.

3.1 THE PERTURBATION FORMALISM

The starting point of this method is the

integral equation for 4>(r) , which,for homogeneous

purely absorbing channels,is written in the form

Too

1>(r) = / r' <J)(r' )K(r,r')dr' r > a (24)

J a

- 19 -

where we have set

<j)(r) = cp(r)cos9

and

- ')//° ( t ) d t

(25)

9 9 9 9 " W ^ ' J dp I P Kn(t)dt,

(r'2+r2-p2)

in which J is as defined in equation (A9) and

p' = 0 for p < /(r2-a2) + /(r'2-a2)

= /(Ua2-l/(p2J2)) for P > /(r2-a2) + /(r'2-a2).

2 2Expanding in powers of (p -(r-r1) )/4rr' we eventually obtain

to the order of (e

P(x) = J ^ p(y)|El(z/i) - |i |^ - | E2(z/£)

2

128 [ri^7

[(F0(z/O-F0(z/£)) - (60<z/£)-G0(z/£))]

- 20 -

4(a+x)(a+y)

(F9(z/JD-F,(z/a))-(G,,(z/£)-G0(z/A)) )— * + •••> dy, (26)

(4(a+x)(a+y)r )

in which we have also used the transformations <Kr) = Cp(r)//r,

r = a+x, r' = a+y, where C is a constant of normalization,

and have set z = |x-y| ,

Fn(z/A) = Fn(z/£,(2a+x+y)/ii) , Fn(z/A) = T^z/SL , ( / ( 2ax+x2 ) + / ( 2ay+y2

Gn(z/£) = Gn(z/£,(2a+x+y)/Jl) , Gn(z/£) = Gn(z/4 , (/(2ax+x2 ) + / ( 2ay+y2

Fn(z,h) = if Cp2-z2)n^ dp f KQ(t)dt

and

Gn(z,h> = I j (p2-z2)n-1"2 dp J ^ KQ(t)dt.

The functions ER(z) are the "E" functions which are given as

/

CO

t'n e"zt dt,

and we also note that

Fn(z,z) = E,(z).

- 21 -

The coefficients appearing in equation (26)

-n.can be expanded in powers of (a ). Collecting the contribu-

tions from th<= different terms of the same order of magnitude

-2a/ £together we obtain to the order of (e ) the equation

• W /•"p(y)(E '21 Jo i1p(x) = yg- \ p(y){E1(z/£) - Z-£

[ f I E2(z/£)a

2 1^.(x+y)[| E2(z/Jl)+E3(z/a)] + • • •> dy,8a )

(27)

which equation we now solve by the perturbation method. Set

p(x) = po(x) + p2(x) + P3(x) + •••

and then substitute above to give the perturbation equations

( 2 8 )

P2(y)E1(z/)l)dy - j J P0(y)C|(f- E2(z/l)+E3(z/i))

u i3 U2a •'O

2_ §_ (GQ(z/£)-F0(z/£5)]dy, (29)

Ar

pQ(x) = ~ r pQ(y)E1(z/£)dy + % f p9(y)(Gn(z/£)-Fn(z/£))dyd 1% JQ 3 1 z ^Q ^

9 00

+ 1 * / p (y)(x+y) (f E9(z/£)+Eq(z/£))dy, (30)16a3 ^0 ° l Z

etc.

- 22 -

Here the order of contribution of the different terms is as

established in a general manner by Davison. For the present

purposes it suffices to note that the series converges for

all a > (0.7104... H , that the ordering of terms is strictly

true for a > 1 and that it breaks down for a = 0.

3.2 THE SOLUTIONS AND EXPANSIONS

We note that equation (28) is the equation

for the Milne problem and that its solution, arbitrarily

normalized so that PQ(°°) = l/&» is

po(x) = (qQ + x)/Ji + 0(e"x/il) = x/fi, + p Q(x),

with

qQ = 0.710446...1 and P Q ( M ) = qQ/£.

In general the solutions for p (x), n > 0,

are polynomials of degree (n+1) plus exponentially decreasing

functions. In order to preserve the correct order of magni-

tude of the P_.(x) we normalize the solutions so that (for

n > 0) they do not contain any term proportional to x. Thus

the Pn(x) may be written as

pn(x) = xn t l p<J> • x" p<2> • ... • x2p<"> • p n(x), (31)

in which Pn<x) is a bounded function (i.e. a constant plus

- 23 -

exponentially decreasing functions) and the p m''a arem

constants. On substituting for the p (x) from equation (31)

into the perturbation equation of the type of equation (30)

defining the PR(x)> an inhomogeneous integral equation for

the Pn(x) can be obtained. If we denote the free term of

this equation as f (x), then we have,by means of Davison's

lemma (Davison 1951),that

f°°Pn(») = 3 I fn(x) pQ(x)dxn pQ (32)

Combining the contributions of the various

p (x) , n >. 0, we have for p(x)

p(x) = x/SL + I pn(«) + I {xk + 1 vll{ + ... + x2p2(k)} + C(e"x), (33)

n k=2

i.e. a power series expansion for the solution. We also know

that Cp(x) = (a+x) 2<j>(x) has the representation (see eq. .

(7) and recall that r = a+x)

Cp(x) = (a+x)2 [(a+x) +

By expanding eq• (34) in a series and equating coefficients

with eq. (33) we finally obtain for A^, the dipole

response coefficient,

3 I p (°°) - 2a/&

A, = a2 - J l - ^ • (35)1 £ p (») + 2a/£

n

It can be seen that to derive A-, to any order of accuracy,

say (a/X.)"1", it is necessary to obtain £ p (°°) to the order

of (a/I)" 0"- 1'.

Note that in the procedure outlined above

for the determination of A,, there exist many auxiliary

equations relating A, to the constants p These

equations may be used to specify these constants though it

is perhaps preferable to obtain their values directly from

the solutions of the integral equations defining the various

p n(x).

3.3 EVALUATION OF £ p O ) AND THE RESULTING PERTURBATIONEXPANSIONS n n

The evaluation of p^C00) and p^C00) reduces to

the evaluation of certain definite integrals most of which

have been previously worked out by Davison (19 51) and

Kushneriuk and McKay (1954). Using their results it is

possible to readily determine J p (°°) in general to the ordern n

of (.a/I) and, for a=l and a=*>, to the order of (a/£)

Again we present only the barest details of the evaluation

of these terms.

- 25 -

Consider first p2(°°). On setting

p2(x) = x3p 3

( 1 ) + x 2p 2( 2 ) +

and then substituting into equation (29), we find, from

consistency requirements,

(1) 1 (2) 3%P3 = —y~ 5 P9 = — o —

8a £ 8a I

and for

P2(y)E1(z/i)dy + ij-J pQ (y) (G Q (z/£)-FQ (z/£) )dy

/16a 0

J E2(z/il)+E3(z/Ji))dy

y Eu(x/£) + -«• x E,(x/4)). (36)n

16a b H £

Using Davison's lemma, further integrating over the E

functions and then combining with the definite integrals

evaluated previously gives, for general a,

p (oo) = * [-0.20379 ... + -a a

for a = 1,

I2p (oo) = -0.20379... -s- to all orders1 a

- 26 -

and for a =

a[-0.20379... - 0.078125 |

- 0.09569 - + 0(^-a a1

(37)

Now consider fLC00). Set

P3(x) = x3 p 3

( 2 )• x 3p 3( 2 ) •

On s u b s t i t u t i n g in e q u a t i o n (30) we g e t

( 1 ) ( 2 ) ( 3 ) _

and

p , ( y ) E , ( z / l ) d y - ^ - I p o ( y ) ( F n ( z / A ) - <0 d I 2Ji J Q 2 U )dy

3 i ( P o ( y ) - ( q o+ y ) / ^ ) ( x + y ) ( f

1 6 a d ^ 0 U U £)dy

1 Dd

on I P o ( y ) E 1 ( z / £ ) d y + f Q ( x ) ( 3 8 )

- 27 -

In the same manner as before we finally obtain for the case

of a = 1

p3(°°) = -0.12742...£3/a3 to all orders

and for the case of a = °°

p3(») = -0.12684. . .A3/a3 + 0((£/a)An(a/JO). (39)

Substituting the above results into (35) leads

to the general perturbation expansion for both a >> I and

aa >> A. In particular we obtain for the general case

2A-.(a,a/£) = -a2[l - 1.4208922 - + 0.504734 K> +

a

3 4+ (0.22829 - —-—s ) —j + 0(—j7 An (a/A)) ] ,

a a a

and for the two special cases of a = 1 and a = °°

2 3An(a=l,a/A) = -a

2[l-l.4208922 ^ + 0.504734 Zj + 0.13453 —a a a

4 5+ 0.09558 ^_ + 0(=ij- An (a/A))]

a a

and

.2 n3A1(o=*,a/£) = -a

2[l-l.4208922 ^ + 0.504734 ^ + 0.22829 ^a a

0.15625 Lj- Jln(a/Jl) + 0a a

- 28 -

3.4 THE CASE OF LARGE VOID CHANNELS

The above perturbation equations and the

resulting expansions are valid for optically dense channels.

For large void channels we have developed an alternative

expansion procedure by means of which the associated pertur-

bation equations may be obtained. In the zeroth approxima-

tion this procedure leads to a Milne type of problem, the

kernel of which is modified by the contribution from the neu

trons which pass through the channel.

Bearing in mind the definition of p'

(see eq. (25)), the equation defining <|>(r) for the case

of a=0 (i.e. eq. (2"4)) is rewritten as

,2)Jdp/ Kn(t)dt

p/£ U

/*r+r'' ^ 2

T-<P )Jdp/ KQ(t)dtJ/(r2- * J°

,2. 2 2

/(r2-a2)+/(r'2-a2) *

As previously, two new variables r = a+x, r' = a+y and the

function C(j)(r) = p(r)//r are introduced as well as, for the

latter integrand, the transformation

- 29 -

s = p-p' a/(r -a sin ) + /(r' -a sin / ) - 2acos!|y

0 0 o o o ocos f+2ax+xi)+/(a cos i(j+2ay+y )- 2acost//

= (u+v-2a)cosi|/. (42)

Then eq. (HI) becomes

V(x2+2ax) + /(y2+2ay)

x-y | - Jp/iL, dp

- Jp

+x+y/

/(x +2ax) + /(y +2ay) /*•f9 ds I K (t)dt (43)

x+y -/s/A

with

f 1 = [(a+x)2-(a+y)2-p2][p2-(x-y)2]~is[(x+y+2a)2-p2]"1'5,

f2 = [a2(sec2^-l)-uv](sec2i|;-l)'l5[2uv-a(u+v)] 1

where ij/ is defined in terms of s in eq . (42) in which

equation u and v are also specified.

The first term in the integrand of eq. (4 3)

may be expanded in powers of a~n as in eq. (25); before the

second term may be expanded, it is necessary to

expand secij/ in powers of a . This expansion is

= t [ > -

- 30 -

(l-t2)(x2+y2)2

4a2(x+y)2

where t = s/(x+y). By expanding the integrands and neglecting

terms of the order of 0(e~ ) we obtain the equation

pCx) = |r J p(y)dy|E1p(y)dy|E1(|x-y|/U

+ ia U+yH

2

(x+y)

•Iwhere we have defined the H as the integrals

n - 2

and

n * ' " 2 a x + x 2 ^ " n 2

n - 2ds / K0(t)dt,

with

z = (x+y)/&.

- 31 -

The equation (4t»> may now be solved by the

perturbation method used earlier for the case 'a t 0. In

the lowest order approximation, i.e. taking a->-<», the equa-

tion, after some further integration, reduces to

1 f °°po(x) = JJ- J P0(y)[E1(|x-y|/£)+E1((x+y)/£)-2E2((x+y)/Jl)]dy

which is the equation obtained by Carter and Jarvis (1961).

The asymptotic solution of this Milne type equation was also

obtained by Carter in a variational approximation. The

result was

pQ(x) ^ (X+X Q)/£ with xQ = 0.21373...£ .

From this we immediately obtain

A, (a =0,a/A) = -a2[l - 0.4275 ... - + . . . ] . (45)l a

As in the case of optically dense channels

to carry the perturbation procedure further (i.e. to obtain

higher order approximations to A,(a=0,a/&)), it is necessary

to know the lowest order solution PQ(x) for small values of

x as well. This solution is not known and its determination

is a problem in its own right which we have not tackled.

- 32 -

4. VARIATIONAL APPROXIMATIONS FOR HOMOGENEOUS ABSORBINGCHANNELS OF ALL RADII

4.1 OUTLINE

Two approximate expressions are now derived

for A, (a,a/2.). These are obtained by the application of a

variational principle, in which use is made of a well-known

functional, F(q), some properties of which have been pre-

viouslv examined by Kushneriuk and McKay (1954). In the

simplest approximation A-, is found to be the solution of a

linear equation of the type

A1(a,a/Jl,)h22(o,a/4) + h^c^a/JO = 0, (46)

while a refinement of the procedure gives rise to an approxi-

mation in which A, is the root of a quadratic of the type

A12(a,a/S,)h3(a,a/Jl) + A^ct ,a/£)h2Ca,a/£) + h^Ca.a/Jl) = 0. (47)

These estimates for A, are based on the specific

properties of an inhomogeneous integral equation, which has

the form

qo(x) - J K(x,y)qo(y)w(y)dy - fQ(x) = 0 (48)

and for which the kernel is both symmetric and positive. It

is known that for q(x) = qQ(x) the functional F(q), where

F(q) = Iq(x)JqCx) -/ K(x,y)q(y)w(y)dy 'w(x)dx -2 L(x)fQ(x)w(x)dx,

(4 9 )

has an extremum with the stationary value

- /qQ(x) fg(x)w(x)dx,

which extremum is a minimum provided that

/ w(y)K(x,y) <. 1. (50)

In obtaining the above estimates we have taken the simplest

possible trial function for variation. In principle, A, can

be obtained to any desired degree of accuracy by the

employment of an iterative scheme. However the severe

increase in analytic complexity gives rise to only a marginal

improvement in accuracy (McKay 1960; Sahni 1964). Accordingly

our analysis is limited to that above.

In this,the h (a,a/£) are functions of the

parameters characterizing the channel. They are formally

related to each other and are initially given as multiple

integrals of cylindrically symmetric space, the integrands

of which involve the kernel of the integral equation. Of

these, h, can be evaluated explicitly in terms of elementary

functions, while, apart from the special case of a=l, it is

necessary to evaluate h2 and h3 by numerical techniques. In

what fellows and particularly in Appendix C we give some

.- 34 -

description of these integrals and also an outline of the

development of useful approximations to them.

From our numerical values for the h (a,a/Si)

and a comparison of equations (46) and C+7) with the

perturbation results obtained above, we are led to believe

that eq. (46) is accurate, in general, to better than 5%,

and that eq . (47) is accurate to better than, at worst,

2%, being considerably better for a large range of parameter

values. Both expressions go asymptotically into the correct

limits .

4.2 A VARIATIONAL ESTIMATE A. FOR A (a,a/£)

The integral equation for the dipolar flux for

r > a is as defined by equation (24) with K(r,r'), a symmetric

kernel, defined by equation (25). On setting

Al<J)(r) = r + — •!• q(r) r > a,

an inhomogeneous integral equation for q(r) may be obtained

from this equation, namely

q(r) = f(r) + / r'q(r')K(r,rI)dr', (51)Jaa

in which

f(r) = -r

j r ( r + ^)K(r,r)dr. (52)

\j r'K(r,r')dr' iObserving that the quantity / r'K(r,r')dr' is less than

the probability that a neutron scattered in the annulus

between r and r+dr is next scattered somewhere in the

region r > a, we see that the kernel of eq. (51)

satisfies the condition given by eq. (50). Accordingly

we may apply F(q) to the determination of q(r) , for which

we obtain, after rearranging and using the symmetry of K(r,r')

r00 r Ai r°° Ai lF(q) = - / r q(r)|-r - + I r• (r'+p^)K(r,r' )dr' dr

r00 r Aii r Ai rE J r r r "r r + J

i.e.

- J r q(r)f(r)dr = A;[2h3(a ,a/iL)+A1h2 (a ,a/£)+h;L(a ,a/£) (53)

where in the latter expression we have set

Jr°° r r 0 0 "I

r g(r) I r!g(r')K(r,r')dr' - g(r) dr, (5M)a L- a -J

with

g(r) = r and p-(r) = r in the integral defining h 1 ,

g(r) = 1/r and ?(r) = r in the integral defining h 2 1

g(r) = r and g(r) = 1/r in the integral defining h 2 2

p(r) = 1/r and p(r) = 1/r in the integral defining h;3,

and

The left side of equation (53) represents

the integral over space of two small quantities. Choosing

q(r) such that this vanishes and then solving the quadratic

gives

2h (a5a/JO±

1var -

From consideration of the asymptotic expansions of these

integrals for channels of both small and large radii, it is

found that the correct choice of sign for the radical is

the positive one. In particular the following estimates

for A, (a,a/A) are obtained from the asymptotic expansions

of the hn's

= -a2 i + ~ + OC2*- £n(a/JO)A, (a=0,a/£) = -a2 i + ~ + OC2*- £n(a/JO) a << SL,var ^ /iL Z J

A (o = l,a/S.) = -a2

ivar L8 £ 2 J

[l - /2 i + 1 ^ . + .

A (a=»,a/£) = a2 |^ i + 0(£n(a/JL)) a << I,var LdZ a J

- -a2

2 1 SL 1 9= -a 1 - /2 - + T ~ + ... a >> I. (56)

A comparison of these results with the exact perturbation

expansions obtained earlier (see eqs. (23), (400 and (u f,) )

shows these to be in excellent agreement. We have plotted

on Figs. 2-4 the values given by A, for the cases ofvar

ot = 0 , 1 and °°. We obtained these values by evaluating the

h integrals numerically, and we present later a practical

means for the general evaluation of eq. (55).

4.3 AN APPROXIMATE VARIATIONAL ESTIMATE A. fiDDDnYFOR A^c^a/JO l A P P R 0 X

This estimate is obtained by working with

the inverse dipole response coefficient. It should be

noted that formally a close analogy exists between this

determination of A-, and that of the linear extrapolation

length by Kushneriuk and McKay (1954).

Dividing throughout by A, the inhomogeneous

equation (51) is written as

q(r) = / r' q(r')K(r,r')dr' + f(r)3.

where

q(r) = (r

and

r°° if(r) = / K(r,r')dr' - £.

- 38 -

Applying F(q) to the solution of this inhomogeneous equation

~ -1with the trial solution q(r) = A, r, minimising with

respect to A,~ and then equating to zero, ultimately leads

to the estimate

1

1approx h 2 2 ( a ' a / U

This estimate, although obtained by transport theoretical

methods, assumes only the diffusion flux throughout the

moderator. Accordingly A, will be less accurate thanapprox

A, . It is,however, useful in that it is of a much simplervar

structure and, as can be seen from the following asymptotic

expansions and from Figs. 2-4, it contains the bulk of the

structure of A,var

Util izing the asymptotic expansions of the

var ia t ional integrals the estimate of equation (57) gives

o i -i - _ 2

approx L" i

= _ a 2 l l - ± . ^ . + . . . . I a » i9

A ( a = l , a / £ ) = - a 2 | £ ~ + . . . . | a « £ ,approx

= - a 2 11 - ^ - + . . . . | a » SL,

and

- 39- -

A (a=»,a/Jl) = a2 |j | + ()Un(a/Ji))| a << I,

approx L J

= -a2 II - T - + I a >> I. (58)

Comparison of these expansions with the exact perturbation

expansions (see equations (23), (M-O) and (M-5)) indicates

that (57) is a good estimate for channels of small radii

and a tolerable enough estimate for channels of large radii.

- 4ft -

5. A PRACTICAL FORMULA GIVING A. FOR HOMOGENEOUS PURELYABSORBING CHANNELS

The general formulae obtained above, which

give A-, for homogeneous purely absorbing channels, may, as

we now indicate, be encompassed in a more general formalism

by means of the continuity equation. By comparing these

formulae and also by introducing a certain type of approxima-

tion into the integrals appearing in them, we are able to

construct a simple practical formula for A,. As may be seen

in the following sections, this new formalism and these

approximations enable our theory to be extended to include

channel structures other than the homogeneous purely

absorbing type.

5.1 AN ESTIMATE FOR A^c^a/iO FROM THE CONTINUITY EQUATION

In the one velocity model the continuity

equation is written in the form (Case, et al. 1953, p.45)

= SCr) + (c(r) -

in which j_(r) is the neutron current.

Setting c=l and S=0 and then taking the radially

weighted integral over the volume of the moderator, we thus

obtain for the cper. cell dipolar problem

I 2I r V.j_(r)dr = 0. (59)cl

Now for this problem we may write (see eq. (7))

Al= <* + — + Al

where for purposes of later convenience we have taken A,

out of <|>. (r). Likewise we write"C DfcLrlS

( 6 0 )

where each component current arises from the corresponding

component flux. Substituting for j_(r) in eq. (59)

we obtain, on solving for A,5 the equation

°° 2r V.j- /

A1(a,a/A) = . (61)

J r2V.i2(r)dr + f

5.2 COMPARISON OF FORMULAE AND DEVELOPMENT OF A PRACTICALFORMULA

From a comparison of their integral represen-

tations we find that

£and

Noting the general definition of the hn integrals given in

eq. (54), we infer that

00

r2' [£These relationships are as expected and may be understood as

follows. They express the physical condition that the

divergence of each component flux across any element of

volume in the moderator is given by the difference of two

quantities. The first is the number of neutrons which

scatter into the volume element; this quantity is calcu-

lated from the component flux distribution. The second is

the number of neutrons given by the corresponding component

flux distribution which pass out of the volume element.

Thus, bearing in mind the above results

and also noting eq. (Cll) of appendix C, we have from the

continuity equation (see eq. (61))

h1(a,a/S,)A1(o,a/£) = - h 2 2 ( a j a / u + (transient terms)9 ( 6 2 )

dr.

- 43 -

from our variational estimate (see eq. (55))

h, (a,a/JO—± , (63)

and from our approximate variational estimate (see eq. (57))

h, (a,a/JOA, (a,a/£) * - — . (64)1 h22(a,a/JO

These expressions clearly bring out the relationships between

our different results. The expression obtained through the

continuity equation is exact, though the form of the transient

terms is not known. In our variational estimate the transient

effects have been approximated by a somewhat complicated

expression, which we have denoted as K(a,a/JO, while in our

approximate variational estimate they have been neglected.

We are now in a position to build up a simple

formula for A, . first we note from Appendix C that the h n

integrals may be written in the form

hn(a,a/£) = h n

h, is known explicitly in terms of elementary functions

- 44 -

In particular

h1(a,a/U = ACa/U + £ Anm<a/i)PnmCaa/Jl) , (65a)

(see equation (C7 )). h22 cannot be completely reduced;

however, on writing

h22(a,a/O =

we find that to better than 0.3% accuracy (see eqs. (Cll), (CIO) and (D6))

h22(a,a/£)

= BCa/4) + I Bnm(a/il)Pnm(aa/£), (65b)

where the coefficients B and B are tabulated coefficients

just depending on the channel radius. This type of analytic

approximation, which emerges in a natural manner from a

consideration of the structure of the collision part of the

h (a,a/£) integrals, is based on the physical fact that the

contribution to the integral from the neutrons that pass

through the channel is not too sensitive, providing that it

is correctly normalised, to the assumed angular distribution

of neutrons at the channel surface. Substituting into

equation (64) we obtain

- 45 -

A(a /£ ) + I A ( a / £ ) P ( a a / £ )A , C a , a / 4 > * - - i—2S 2* . ( 6 5 )

B(a/£> + I S n m C / ^ P C / £ )

The final step in the building up of our practical formula

is to try and incorporate some of the more precise features

of eq. (63) into eq. (65) in a simple manner. We

find from our numerical evaluation of the variational inte

grals that we can make the same type of approximation to

K(a,a/£). In particular we find on writing

K(a,a/£> =

that it is a very good approximation to set

K(a,a/Jl) K . ,(a/£) + Kc, (a/£)Pc, (oa/£) . (66)voxd bl ol

Combining eq. (66) with the denominator of eq. (65)

and setting (see also eqs. (65b), (Cll), (C9) and (D6))

a2B(a/)l) = B(a/£) -

and

a2B (a/A) = B (a/£) - &. _ .-, _ K, (a/£)P5,(aa/£) (66a)nm nm o,n,i,m ol ox

we obtain for the final form of our formula

- a

nm

In this expression

( 6 7 )

ACa/H) = - ~ (1 + - ) ,o a a.o a.

B(a/2.) = -3a (1.7175 + 3a/£)

+ BQ(a/JO

A,, (a/A) =16a

A1+2(a/Jl)3a

A 3 3 ( a M ) = 31 '

the coefficients Bn(a/£) and B (a/il) are tabulated (seeu nm

Table 1 and Fig. 5), and the Pnm(a a/I) are the collision

probabilities (see Appendix B and Table 2). The summation

in equation (67) is over the indices 31, 51, t+2 and 33.

The advantage of this formula is that the

tabulated coefficients depend on only the channel radius,

and are independent of channel content. The dependence on

channel content is contained in the P (a a/Jt) , and these

probabilities are known in terms of elementary functions.

This estimate for k^ will be accurate to at worst

3% and will be considerably better for a large range of the

parameter values . The maximum error is to be expected for

channels for which the radius is in the range of about 1 to 3

moderator mean free paths. Relaxing this accuracy, Bn(a/Jl)

may be dropped, P , P „ and P „ may be replaced by P (see Table 2)

and the coefficient B may be combined to yield an expression

requiring only 1 tabulated coefficient. The accuracy will

now be to within 8% at worst.

It should be noted that for void channels,

i.e. a=H,the P vanish, and on setting

B(a/£) = - 21'

3a (1.5 + 3a/J.)

our formula (equation 67) reduces to the Carter formula

(Carter 1961).

- 1*8 -

6. INHOMOGENEOUS ABSORBING CHANNELS OF ALL RADII

The practical formula, which we have built

up to give A, for homogeneous purely absorbing channels

(see equation (67)), may be generalized, as we now

indicate, so as to also give an estimate of A, for inhomo-

geneous absorbing channels. Restricting our attention to

channels for which the inhomogeneity is only along the

radius vector, we can derive in terms of the formalism

developed from the continuity equation the general expression

Ja

f- 2 ~ I"- 2 ~/ r v«j_2(r)dr + / r V«i

Ja J a

(r)dr

. (68)

Also, in complete analogy with the variational treatment of

homogeneous absorbing channels, two variational estimates

of the same structure as equations (55) and (57) can be

produced for this problem. The variational integrals, h ,

arising in these estimates, are defined in the same general

manner as the h of equation (54), with, however,the kernel

re-defined as

,2

(69)

2_ [~ ( r 2+ r ' 2 - p 2 ) /"(P-P'+T)/*

^ /./rr,2 2 W r .2 2, 2rr ( F JdP /, KQ(t)dt

- 49 -

In this and in what follows,the tilda indicates that the

identities refer to the case of an inhomogeneous channel,

and T/£ is the effective optical path length for neutrons

along the cross section of the channel. Now by examining

their integral formulation the corresponding identities

and relationships, to those found for the case of homogeneous

absorbing channels, can be seen to still hold between these

new expressions. In particular we have

hlA, = : , (70)

-h22 + K

where as before

,1/2K = 7

Also the h may still be split into void and collision parts.

The void parts are. identical to those obtained before, while

the collision parts differ. However we are able to make

the same type of approximation, in that by examining their structure

we find that the collision parts of the hn can be broken UD in

a natural way into a sum of the products of coefficients and

first collision probabilities. The coefficients are the same

as those obtained before (see Appendix B).

Thus, in short, eq. (67) still holds when

we consider inhomogeneous channels, with, however, the first

collision probabilities replaced by those associated with

_ 50 _

the type of channel under consideration. In Appendix B

explicit formulae are presented for these probabilities for

certain specific channel types.

- 51 -

7. CHANNELS WHICH BOTH ABSORB AND SCATTER NEUTRONS

The theory developed above for purely absorb-

ing channels is now extended to include effects due to

scattering of neutrons within the channel. The response

coefficient is derived using as a basis the continuity

equation and the particular channel configuration considered

is that of a homogeneous scattering and absorbing medium

surrounded by a weakly absorbing sheath or air gap.

In obtaining the response coefficient, a P,

neutron flux distribution in the scattering medium within

the channel is assumed, the normalization of the flux

being such as to maintain continuity of normal current at

the surface of the scattering medium. Thus for r < a

<f)(r) % B I (yr)cose (70)

where y = a /(3(l-c))/£, B is the normalizing constant and a

is the ratio of the mean free path in the moderator to

the total m.f.p. in the medium. For r > b the neutron

flux is still given by eq. (7) and for r > b we may write

for the current (see also eq. (60))

j_(r) = j^Cr) + A-J^Cr) + A^Cr ) + B ^ ( r ) (71)

in which the component :L(£) arises because of scattering

- 52 -

in the channel of the flux I1(yr)cos8. We now take

the radially weighted integral of the divergence of the

current across +he volume of the moderator. Bearing in

mind that V-^Cr) = 0 and setting B = B 1+ A

1B2 w e obtain,

after some rearrangement of terms,

A" (a,b/£) + I9(c,a,a,a,b)A:i(c,a,o,aJb) = — ^ (72)

1 + I ( c a a a b )

in which

A, (c,a,a,a,b) denotes the dipole response coefficient

for the case with scattering,

c is the mean number of secondaries per collision in

the inner medium of the channel,

A, (a,b/Jl) is the appropriate response coefficient for

the channel ignoring the effects of the scat-

tered neutrons (i.e. the capture m.f.p. is

effectively the total m.f.p.),

I1(c,a,a,a,b) = B1 hl+(rt,a,b,5)/n1(a

(73)

I2(c,a,a,a,b) =-B2 h1+(a,a,b,5)/n'1(a,b/Ji),

11 - C a, b / £ ) is as defined via eq. (54) for n=l for our particu-

lar channel configuration (see also eq. (65a), the

discussion in section 6, and Appendix B),

- 53 -

and h^(a,a,b,a) is related to j^Cr) through

a,b, a) .

As already stated, B is chosen sue 1 as to

maintain continuity of the normal current at the surface

of the scattering medium. The current within the channel

is taken to be that given by the diffusion theory. We

thus have

(a,b,a,6) + B 3

= -B 3a io(ya) -I-, (ya)

yacos8

= B J(a,6). (75)

Substituting for the B (i.e. B = B +A,B ) and neglecting

the transient current j (a,b,a,0), we have for B and BQ

B

(76)

so that A~1(c,a,ct,a,b) given by eq. (72) is now completely

specified. In the general expressions involving eqs. (72),

(73) and (75), j (a,b,a,9), j 9 (a,b,S,6), j n(a,6) and

- 54 -

h.(a,a,b,3) are new functions! J(a,0) is defined by eq,

(75). The new functions are written out explicitly

as integrals in appendix D where also possible approxi-

mations to some of them are indicated.

- 55 -

8. CONCLUSIONS

In this work procedures have been developed

that enable the influence of a cylindrical channel on a

tilted (i.e. linear) neutron flux distribution in an

infinite isotropically scattering, non absorbing (weakly

absorbing) moderator to be determined. In particular,

explicit formulae have been developed for the open cell

dipole response coefficient, and the associated extrapola-

tion length type boundary condition, for a variety of

channel configurations which may be applicable in the con-

sideration of fuel, control and void channels in a reactor.

It is found for the case of a void channel

that the Carter and Jarvis (1961) simple formula is

a very good approximation.

The results obtained in this work could

also be used for purposes of testing the accuracy of

various computer codes currently used to determine the

response coefficients in reactor calculations. Also

formulae for the determination of cell diffusion coeffic-

ients already exist (Carter 1961; Stewart 1971) in

which the aipole response coefficient is an assumed

known parameter. Our results provide that knowledge for

a wider class of channels.

- 56 -

ACKNOWLEDGMENTS

This work forms part of a Ph.D. thesis

submitted by M.D. Love to Queen's University, Kingston,

Ontario. The award by Queen's University of a McLaughlin

and a Reinhardt fellowship is gratefully acknowledged.

We wish to thank Dr. W.R. Conkie of Queen's University for

his support and encouragement of this project, and we

should also like to express our particular gratitude to

Dr. J.D. Stewart, formerly of the Chalk River Nuclear

Laboratories, who brought this problem to our attention.

- 57 -

REFERENCES

Alpiar R. (1969) Eidg. Institut fiir Reaktorforschung, Wiirenlingen,Report EIR-160

Auerbach T., Halg W. , and Menning J. (1972) Nucl. Sci. Engng. 49_, 509

Benoist P. (1968) Nucl. Sci. Engng. J34_, 285

Berna P. (1971) Centre D'études Nucléaires de Saclay, Report CEA-N-147 5

Bonalumi R. (1973) Trans. Am. Nucl. Soc. 16_, 285

Bridge M.J. and Cumpstey J.M. (1973) J. Nucl. Energy 2]_, 465

Case CM., De Hoffmann F., and Placzek G. (1953) Introduction to theTheory of Neutron Diffusion, Vol. 7, U.S. Government PrintingOffice, Washington 25, D.C.

Carter C. (1961) Reactor Sci. Technol. 15, 76

Carter C. and Jarvis R.J. (1961) Reactor Sci. Technol. ljj, 113

Davison B. (1951) Proc. Phys. Soc. A, 6>4_, 881

Davison B. (1957) Neutron Transport Theory. Oxford Univ. Press.

Davison B. (1959) Atomic Energy of Canada Limited Report AECL-867

Kushneriuk S.A. (1957) Atomic Energy of Canada Limited Report AECL-462

Kushneriuk S.A. and Jirlow K. (1962) Reactor Sci. Technol. Ij5, 464

Kushneriuk S.A. (1967) Atomic Energy of Canada Limited Report AECL-2709

Kushneriuk S.A., Davison B. and Seidel W. (1949) Proceedings of the2nd Canadian Mathematical Congress

Kushneriuk S.A. and McKay C. (1954) Atomic Energy of Canada LimitedReport AECL-137

Maeder C. (1969) Eidg. Institut fiir Reaktorforschung, Wiirenlingen,Report EIR-147

McKay C. (1960) Atomic Energy of Canada Limited Report AECL-1250

Sahni D. (1964) Atomic Energy of Canada Limited Report AECL-2083

Stewart J.D. (1971) Atomic Energy of Canada Limited Report AECL-4074

Watson G.N. (1952) Theory of Bessel Functions. Cambridge UniversityPress, 2nd edition.

- 58 -

APPENDIX A

SOME PROPERTIES OF THE SOLUTIONS OF THE INTEGRAL EQUATIONS

We wish to indicate some properties of the

solution of the integral equation given in eq. (1)

in the absence of sources. In particular we show that in a

solution based on a Fourier series expansion in the azimuthal

angle the various harmonic coefficients in the expansion

are uncoupled. We consider specifically the solutions for

the flux if>(r) in a purely neutron scattering, cylindrically

symmetrical medium surrounding a purely neutron capturing

medium as described in the introduction. Our conclusions

apply to solutions pertaining to more general inhomogeneous

but cylindrically symmetric medium configurations .

For the problem at hand, the integral equation

for the neutron flux distribution for r > a is (see Fig.

1 for the coordinate labelling)

" 1

ijp'.

r-sin"1 2:dy / Ki, (p/JO<j>(r')dp

' >dp

E 9TT // Q(r'r'>PH(r')dYdp (Al)

- 5 9 -

with

rcosY-/(a -r sin Y ) , P 2 = rcosY+/(a2-r 2sin 2y)p'=p o-px = r c o s Y - / ( a - r s i n Y ) , P 2 = r c o s Y + / ( a 2 - r 2 s i n 2 Y ) , p ' = p o - p n ,

2 2 2r' = r +p -2rpcosY »

1 = 9 + u , coso) = r - p c o s Y a n d

X00 /-tr/ 2

K (t)dt = / e"X s e c X

Consider a Fourier series solution

I T T J cosn0 + J h ( r ) s i n n 8 (A2)

n=0 n = l

w i t h

1 /*27T

r ? / <('(ri,8)cosn8d8,

1 /"27T

- — I 4>(r,8)sinn8d0 .w ^ 0

Substituting for (f>(r,6) from equation (Al) we find for g (r)

g n ( r ) = 2^j fj Q(r,r',p)dYdp i J *(r ' ,G ' )cosn6de . (A3)2TT

0

Now s i n c e 0 ' = 0+w we can change t h e i n t e g r a t i o n t o 0 ' , keep ing

Y , u , r , r 1 and p f i x e d . Using t h e f a c t t h a t <J>(r' ,2ir + E;) = <|> ( r ' , O

we have

- 60 -

( r) r * I I Q(r,r ' ,p)[g (r')cosnw+h (r')sinnto]dYdp (A4)

i . e . an integral equation for g (r) which i s apparentlyn

coupled to h (r). We examine this equation in more detail.

Using the property of cosw as defined in equation (Al), we

can show by induction that

cosnijj =.. in

= fn(r,r',Y) (A5)

and

sin nw = sinu)[cos(n-l)oj+coswcos(n-2 )u+cos u)cos(n-3)a)+• • •+cosn~ to]

sinto k ( r , r ' ,

(A6)

where the exact form of k (r,r',Y) may be inferred from eq.

(A5). The important property for us to notice is that

fn(r,r',-Y) 5 fn(r,r',Y); hence k^r.r'.-Y) = kn(r',r,Y) and,

therefore, sin(-Y)kn(r,rr ,-Y) = -sinvk (r,r* , Y). Taking into

account the details of the limits of integration and the

definition of Q(r,r',p) as implied by eq. (Al), the

symmetry properties of cosno), sin no), p,, p2, p1 and rf we

see that

- 61 -

II Q(r,r',p)h (r')sin nu dYdp = 0

while the equation for g (r) becomes

(A7)

g n ( r ) =1

n

/. . - l a r™ ?

dv / P lK i l

•'O JO X gn(r')cosnu)dn

I dy J K i , (p/Jl)g ( r ' ) c o s n a ) d p . (A8)

J s in" 1 ^ ^0r

Starting with an equation for h (r) similar to

that given by (A3) with' cosnB on the right hand side replaced by

sin n9, it is easy to show that h (r) in fact satisfies the

same integral equation as does g (r). Thus h (r) and g (r)

differ at most by a multiplicative constant. Thus we see

that in the expansion as given by equation (A2) there is no

coupling in the coefficients of the various harmonics and

\.hat the sin n6 coefficients are directly proportional to

the cosnB coefficients. The exact nature of the solution

given by equation (A2) will depend on the specification of the

solution at some boundary or region of the medium being

considered. Thus, in our infinite, purely scattering medium

case, if it is desired that the asymptotic solution have only

a cosn9 dependence (i.e. $ (r) is of the formas —

c|> (r) ^ (rn + —)cosn9 then the solution 4>(r) has only a3.S — XI

- 62 -

cosn6 dependence in all of the space (i.e. the complete

solution for <})(r) is <f)(r) = g (r)cosn6 where gR(r) is the

solution of the integral equation (A8) with the property

that gnCr) rn + -2-).r

For the particular case of the dipole flux,Althe asymptotic flux that is required is (r + —)cos9. The

complete solution in all of space in the dipole case is

then <J>(r_) = g-.(r)cos8 with g, (r) being the solution of the

integral equation defined in equation (A8 ) with n = 1.

We finally note that the integral equation (A8)

for the purely absorbing channel may be written in another

way which is often of use, namely,

g n ( r ) ~2 f» /7(r2-a2)+

Ja -Vr'l

2)+/(r'2-a2)J Ki-^p/Ocosntodp

r ' g ( r ' ) d r 1r/

v

with

2 2 J Ki^C (p+(a-l)p ' )/&)cosniodp

(A9)

cos. =

- 63 -

APPENDIX B

The expression (1-P ) may be physically

interpreted as the probability that a neutron, emitted by

an anisotropic neutron source of strength cosma cos11 B

located at the surface of a channel, will be transmitted

through the channel without undergoing a collision on the

first penetration of the channel. The P are termed thenm

first collision probabilities. For purely absorbing channels

they become the capture or impenetrability probabilities.

In general the P may be expressed as

definite integrals of the form

/ • • F T / 2 /V/2 n

•A) 7om. , . i n - L ,„

cos l dij; I cos 3d3

p ^ , *» . ( B l ,

Jo

in which x/£ is the effective optical path length of the

channel and the geometry is as shown in Fig. 6.

P FOR HOMOGENEOUS ABSORBING CHANNELSnm

Let the neutron mean free path in the channel

be Z/a and the channel radius be a. Call the first collision

probabilities the P m(aa/&). We have

F (aa/JL)P (na/l) - 1 n m (*2

f Tolnm

where

•TT/2

F (x) = -nm 77

m., . f7" n- l Q -2xcos^secBcos tydty I cos g e r

I t can be seen from (B2) t h a t

P (a=0 a/JO = 0 and P (a=°° a/JO = 1 .nm nm

Kushneriuk and McKay (1954) have studied the properties of

P31 and, more generally, Davison (1959) has studied

the integrals F . These authors have found that this° nm

type of integral may be expressed in terms of products of

the modified Bessel functions of the first and second kind.

In particular with aid of their results we have

P3, r 9 in(x)K (x) "1

1(x) = j x I1(x)K1(x)-l-2x+2x (IQ(x)K0(x) + I1(x)K1(x)+-= — = )

= | (fTx+^|x3+i|x5)I1(x)K1(x)+|(x

2+^x4)(K1(x)I0(x)-I1(x)K[)(xP 5 1(x) =

k «t / ^ « A

||x 3 - (|x2 + j |

and

- 65 -

P 3 3 ( X ) = " - ^

(B3)

P n m FOR HOMOGENEOUS ABSORBING CHANNELS SURROUNDED BY AN AIR GAP

Let the neutron mean free path in the absorbing

medium be A/a, the radius of the absorbing medium be a,

the outer radius of the air gap be b, and call the corres-

ponding first collision probabilities P (a,a,b). In terms

of Fig. 6

x = 2aacosnsec3 for 0 £. ty < sin" a/b,

= 0 for sin" a/b < ty <. ir/2

Thus from equation (Bl)

r -1 rI s i n a /b / TT/2 , , gI cos'Vip I cosn"i3(l-e~T/^)dB

p ( a a b) = Z° Z° —n m ' ' f-n/2 fit/2 .

I cos ipa.\|> I cos E dpJo Jo

which on transforming as a sin n = b sin \\i gives

P31(a,a,b) = |P 3 1 ( aa / JD ,

P51(a,a,b) -- |

- 56 -

P u 2 ( a , a , b ) * P1+2(aa/Jl)51(a5b)

in which

2 fa j n a 2 v + • - 1 a ] 8/2~ , b - a ^ 3 / 2 , - . 3_

andr 2 2 -i

PQ0(a,a,b) = | 4( 1 - 5->Fo-,(aa/Jl) + P,«(aa/£) .

3 3 b[_2 b d i fc -J

2 2 2The approximation made in PL^2(a,a,b) assumes that (b -a )/b

is small.

P FOR A THIN HOMOGENEOUS WEAKLY ABSORBING SHELLnm

Let the outer radius be b, the inner radius

be a, the neutron mean free path in the shell be Jl/'a, where

a << 1 and (b-a)a << I, and call the corresponding first

collision probabilities P (a,a,b). In terms of Fig. 6

— 1 aT = 2a(bcosip-acosn )secB for 0 <. \\> < sin r-

1 a= 2abcosiJ;secB for s in =• < \h £. TT/2b r

Substituting for T in equation (Bl) and expanding the exponent

gives to the order of a

2 2P31U,S,b) = ( -—-) + 0(5

2),

P51Ca.5.b) =

- 6 7 -

Pl+2(a,a,b) = !|£ [i _ C2(a,b)] + 0(a 2),l+2

in which

1 2b 2b b~ a 1 6 b

where K and E are the complete Elliptic Integrals of the

first and second kind, and

(B5)2 2 2

P33(a,S,b) =|« ( ^ _ ) ( 1 _ _) + Qca2).3 b

P FOR A HOMOGENEOUS, THIN, WEAKLY ABSORBING SHEATHm SURROUNDING A HOMOGENEOUS ABSORBING MEDIUM

Let the outer radius of the sheath be b, the

radius of the absorbing medium be a, the neutron mean free

path in the sheath be I/a, in the inner medium I/a, and

call the first collision probabilities P (a,a,a,b). Again

we take a(b-a) << I. In terms of Fig. 6

T = 2abcosiJ;sec3 for sin r- < ij; <_ IT/2

and

T = (2abcosi{>+2a(a-a)cosn)secB for 0 <. ty < s i n " ^ .

The integrals defining the P Ca,a,5,b) are complicated and

a simple explicit expression, even to the order cf Cl,

cannot be given. However we give an approximation which is

in the order of a. This is

- 68 -

Pnm(a,a,5,b) % Pnm(a,a,b) + TnmCa,a,b)Pnm(a,a,b)

-(l-Tnm(a,a5b))Pnm(a,a,b)Tnm(a,a,b) . (B6)

which approximation is derived from considering the physical

structure of the P (a.ct.S.b). The three terms of equationnm

(P6) are the absorption probability of the sheath in the

absence of the inner medium, the absorption probability of

the inner medium multiplied by the probability that neutrons

reach this medium, and finally a correction to the former

due to the fact that some neutrons are filtered out by the

inner medium. The underlying assumption in this is that the

distortion in the angular distribution of th-e impingent

neutrons due to the presence of the sheath does not appreciably

alter the fraction of neutrons absorbed in either the sheath

or inner medium. This should hold provided that a(b-a) << £.

Thus the evaluation of the P (a,a,a,b) reducesnm ' '

to that of the T (a,S,b). Expressed as definite integrals

(see Fig. 6)'sin" §• /"IT/2

b ^ /Tnm(a'5'b) = 4 JO

/Jn

cos llJdiii I cos11"

o n 9v : h e r e T = StCucosi^ - / ( a ' - b s i n \\)))sec$, f r o m w h i c h on

e x p a n d i n g t h e e x p o n e n t we o b t a i n t o t h e o r d e r o f ( a 2 ) ,

- 69 -

-2 2T,,(a,a,b) = 1 - |S- (£ (a,b) - ^jO + 0(a2),

b

~ 2T51(a,a,b) =

2T (a.o.b) = 1 ^ (1 _ 5_. _ |^ r ( a b ) ) + Q(S

2)*l 2Jl?(,b) 3b2 3 a 2

and

T33(a,a,b) = [*:••

2 2 "IV (l--^) + 0(a2). (B7)> 4b J

P FOR A HOMOGENEOUS ABSORBING MEDIUM SURROUNDED SUCCESSIVELYn m BY A THIN, HOMOGENEOUS, WEAKLY ABSORBING SHEATH AND AN

AIR GAP

Call the corresponding first collision proba-

bilities P (a,a,a,b,d), where d is the outer radius of the

air gap and the remainder of the parameters are as specified

for P (a,a,5,b). On performing the transformation

immediately preceeding equations (B4) we obtain in a

corresponding manner

- 70 -

P_ - . ( a , a , a , b , d ) = -r P ^ , ( a , a , a , b ) ,

( a , a , a , b , d ) = ^- P ^ , ( a , a , a , b ) ,

P 4 2 ( a , a , a , b , d ) % ^ ^ ( b , d ) P 4 2 ( a , a , a , b ) ,

a n d

P 3 3 ( a , a , a , b , d ) ) P 3 1 ( a , a , a , b ) + b1 - P 3 3 ( a , a J a , b ) ] . ( B 8 )

- 71 -

APPENDIX C

In this appendix some properties of the

hn(a,a/£) integrals defined by eq. (54) are discussed.

THE EXPLICIT EVALUATION OF THE h (a=l,a/£).

For the special case of ct=l, the kernel

K(r,r') defined by eq. (25) may be put in a particularly

simple form, by combining the integrals present

there and noting that 2Jdp = dw/p (see Appendix A and

Fig. 1),

1 f2Tr f00

K ( r , r ' ) = - i - - - I coswdu I Kn(pt/£)dt2TTJT ^0 "I u

I f°°= — I K, ( r t / £ ) I , ( r ' t / £ ) d t for r > r '

II J\ x x

f00

= i ^ J K^Cr't/JDI, ( r t / £ ) d t for r < r ' . (CD£ •'l

The latter relations were obtained by using the Bessel

function addition theorem (Watson 1952, p.361), Substituting

K(r,r') into eq. (54) we obtain, after some manipulation,

5 /-°°hn(a=l,a/£) = - f- I

3./%

- 72 -

I2(X)K2(X)

x"5 dX

r- h " r 1 ! ^ +2 i(IoKi-IiKo)+IoKo • 33.

3. I i. *» T 7 X i O ^ / T l ^ T V \ J _ T 1 / x l L

f y ( I K - I K - 2 ) + i % ( I 0 K 0 + I 1 K 1 ) ' ( C 2 )

JJC U J . X U o n u u X J - I

« I (A)K (X)h 0 1 ( a= l , a /« , ) = - ~ I ^ dX

LL ** ^a/A A

# riKo + I + J-P-+ IT2

(C3)/ M M I I I

9JT

a n d

r ° ° I n ( X ) K n ( A )h _ ( a = l , a / J l ) = - | I — n^ dX

" I (2 + IlV I0 Kl )

In the above we have abbreviated I (a/Jl) as I , K (a/Jl)

as K . The asymptotic expansions for the h 's may be

obtained by substituting the expansions of the Bessel func

tions directly into eqs. (C2), CC3) and (C4) or into the

associated integrals, if need be revising the limits of

integration. The expansions to lowest orders are

- 73 -

h1(a=l,a/Jl) = -a4[|- +

- -a

for a « I

for a >> I,

h21(a=l,a/A) = -af \~ + ^—^ (Y+An(a/£)-£n 2 - |) + *" for a << I

614a'for a >> £,,

2+1+

f o r a << I

64afor a >> I. (C5)

I n t h e s e y - 0 . 5 7 7 2 1 5 7 , . , i s E u l e r ' s c o n s t a n t .

INTEGRAL REPRESENTATION OF THE hn(a,a/£).

In general, from eq. (54), using the repre

sentation of the kernel given in Appendix A, we have

h (a,a/JOn = f rg(r)Ur Hdy ffo 0

(r-pcosY)Ki, (p/£)dp-g(r)

- Jr /

f rg(r)[^

sin"1 * 1)Ki.. Cp/Udp dr1 J

sin"1

dy f dr

(C6a)

- 74 -

With regrouping of terms and transformations this is

rewritten as

f / " " Pi /"•"• /*°° fffrM 1h ( a , a / A ) = < / r g ( r ) £*• / dY / a i - r r 1 C r - p c o s Y ) K i 1 ( p / A ) d p - g ( r ) d r

n ( y a L71* yo yo r x J_ A / 2 f°° r» r°°

+ a # d . / d u i F,Kin((w+u)/A)dw - I F_Ki, ((s + u)/A)ds

!

riT / 2 / • < » / • » )

Sv / dip I du I F1[Ki1((w+u+ap')/a)-Ki,((w+u)/A)]dw>

5 h . .(a/A) + h m- e - r t (a,a/A) (C6b)n,void n ,col l i s ion

in which h -,-,•• denotes that part of h (a,a/A) inn, collision r n '

which neutrons undergo a collision in passage through the

channel. To arrive at (C6) we first transformed as9 9 9

asinijj = rsinY for fixed r , then as u = p, = / ( r -a sin ijj)-acosi|j

keeping ty fixed and finally as p = u+s and p = u+w+p'

= u+w+2a cosij; keeping u and ty fixed. In (C6b)

v

S ^ J [a(a+ucosi|;)~(w+2a cosi|») (u+acosi(j) ] ,

= cosijj ^— ^i [a(a+ucosij;)-s(u+acosi|;)]

2 2 2 / - > 2 9 9 9and r1 = s +a -2ascosip = w +a +2awcos^, r = u +a +2aucos^.

Of the hR integrals, h^o^a/A), for which

case g(r) = g(r)=r, may be integrated out explicitly. We obtain

- 75 -

IF a £ P51(aa/£)

,202 3+ - g - p

3 3( a a / J L )

A^a/JO + I Anm(a/£) Pnm(cta/£) (C7)

in which the Pnjn probabilities are given in Appendix B.

When a=0 or when a=°° the P 's are 5 0 or unity so that

a3£h1(a=0,a/£) = - 2_±(l+i/a),

(C8)

h,(a=»sa/£) = - ( 1 " ^T>-

Expansions of h,(aja/il) in the general case may be made by

expanding the P 's.

The integrals h ., (a,a/£), h__(a,a/£) and

h,_(a,a/iO have not been integrated out completely. On

substituting g(r) = —, g(r) = r into (C6b) we have

2 1 » v o l dh Un) - - 24- rcoB2*d* rcosBdB f" U-2 1 v o l d ^ ^ ^ -'o u

= a2D](a/£), (C9)

- 76 -

h 2 1 , c o l l i s i o n ( a ' a / U

= -— I cosipdi I cos$d8 I0 a +u

e-usec3/5. d u

in which the summation £ is over the functions

= a2 , f51 = -Jiucosg,

and f33 = -2a2cos2iJJ. (C9a)

Combining the above i n t e g r a l s , and by analogy with the

representat ion of h-,(a,a/A), we f inal ly wri te h2-.(a,a/fi.) as

h 2 1 (a ,a / iO = h 2 1 j v o i

in which

/-TT/2 rir/2 /*«» f e - " ° « ^ # *B n m (a /£) = - | I cosi|Kii|» I cos3dB J -^HL^ d u , (C9b)

11 J0 J0 J0 u2 + a2+2aucost(.

Prn m(a,a/Jl) = 1

ir/2 rif/2 / - f e " U S e c e / A

/ 2 / 2 f

1 cos*d* cosBdS -° ° ° u

/ 2 / 2 f e

4—1 cos*d* cosBdS -HL^ e - u s e c 8 / J l duu +a^+2aucosi(>

- 77 -

with the f as given in eq. (C9a). The Pr 's are proba-

bilities. From a comparison of the numerical evaluation

of the Pr 's with the corresponding P m(aa/&) it is found

that to a very good approximation

h 2 1 , v o i d ( a / £ ) + I 5 nm ( a / £ ) P nm ( a a / £ > •

The h2 2(a,a/jl) integral is given by subst i -

tu t ing g(r) = r , g(r) = — in (C6). On collecting terms

we find

/-= - I

"'a.rKi 2 ( r /£)dr

a.

a I i^. (*/2 *QAa r°° a(a-scosT);)-(s-acosi|j))lcosB -ssec3/£,— I cosipdijj I cosgdg I —s K x •«• e ds^ -'0 / 0 •'O a +s -2ascos\|j0 •'O a +s -2ascos\|j

which on using the fact that

r cosw(a-tcosw) , _

^r for a > t2a

0 a -2atcosu)+t _ TT_ f o r a <• t

reduces to2

h22(o,a/JL) = - ~ + h21(a,a/JL), (Cll)

a most useful relationship. In view of(CIO) we also have

- 78 -

Bnm(a/*)Pnm(aa/lO. (C12)

It is found from a comparison of the numerical evaluation

of h 0 2 "that for parameter values in the range

(0 <. a £ "j 0 < a £ hi) the approximation given by eq. (C12)

is accurate to within 0.3%, the error being largest for a

small, a/ft large.

As expressed in eqs. (C9) and (C9b) it is

evident that hol . , and B are threefold integrals. These21,void nm &

integrals can be reduced to a onefold Laplace transform

type integral by means of the following transformations.

We consider in general an integral of the type

fn/2 fv/2 r°°= / cos(j;d^ I cosBdg / f (IJJ, B ,s )e" s s e c f * d s .

•TT/2 r-n/2(C13)

'0 "0 "'O

Transforming coordinates first as s = Rcosg for 3 fixed,

then as cotgsin^ = cotA for R and ty fixed and finally as

2 2cotip = tsecA//(R -t ), I may be rewritten as

r» R rI ~n- dR /./0 Rz "'O

•RI = / ~ - dR / t d t / f ( R , t , X ) d X (C1U)

w i t h s = / ( t +(R - t ) c o s A ) , c o s g = s / R , cosip = t / s . T h u s

f o r e x a m p l e , f o r h o , . . .^ ' 2 1 , v o i d '

- 79 -

,,6,3) = _ 2A_£_ COS^ 2S + A C O S ^ = - 2 A ^ ^ | [l_ + n A+t

E f(R,t,X).

Substituting into eq. (C1M-), using the fact that

TT/2dX TT ,

we have0 C2+B2cos2X 2C/(C2+B2)

/•oo - R /-R ? r 12 2 ' e ltd"1" 1 1

2 1 , vo id JQ R2 JQ ( A + 2 t ) LR /(A2 + 2At+R2)J

Integrating v;ith respect to t and then by parts ultimately

yields

¥ "n2A - 1 A

Jln(R+/(R2+A2) + R

18[R+/(R2+A2)]

/(R2+A2) + £n(A+/(R2+A2))

__

- 80 -

ASYMPTOTIC EXPANSIONS FOR THE hn(a,a/JO.

The expansions for h (a=l,a/£) have already

been given in eqs. (C5). Explicit values of h1(a=0,a/iL)

and h-, (a=°°, a/&) are given in eq. (C8). Here we list the

asymptotic expansions for h0, and h3 for c.= 0 and a=°°.

By series expansions of the type

T = 1 + -i-s- [bsin2e + b2sin36+ •••] for b2 < 12 sine

l-2bcos8+b

we obtain from formula (C6), after appropriate substitutions

1 ~for g(r) and g(r) (i.e. g(r) = —, g(r) = r for h 2 1 and

g(r) = g(r) = - for hg),

h21(a=0,a/£> = h2 oid(a/£) = -a< ±- - + *x 6a 10a

[ 9 o

3a 16a

"IJ

h3(a=0,a/£)

16a+ •••, a >> £. (C15)

The expansions for a << I are difficult to

make and only h21(a = °°,a/Jl) has been worked out extensively,

using the type of integral reductions discussed above.

For h21(a=°°,a/&)

f ( R , t , A ) = A TTTR

1 -

- 81 -

A(R+l)(A+t)

t2+(R2-t2)cos2A + A2+2At

so that by eq. (C14), after in tegrat ions ,

kl'

{ » • - | - + —• + ^- £n2A

|A

| (A 2 +R 2 ) 1 / 2 ^ - - l)in(A+/(R2+A2)HThe Laplace transform type integral in the latter expression

is related to Bessel and Struve functions and integrals

of these functions. The expansion for A = a/I small is

kl' )- An2A+

It may be noted that the first order term of this expansion

may be obtained very simply. Thus by writing f(ij/,3,s) as

O i-

, , , . v Ai" L (s + cosf ( i | i , 3 , s ) = = - U 9 2~11 L A +s +

s + cosB) (s+Acosip) 1

2As COST\I J

we have on substituting into (C13)

- 82 -

Al2

['•

a « f (S + COSB)(S+AJOS») -S secg .cosBdg I —* 5— ^^ e Hds^ A +s +2Ascosip

Integrating the latter integral by parts once and then

taking the limit as A •+ 0 we have

1|J

,2 .•— <± - £n2A-y).

Similarly we obtain, in the lowest order,

AV

h 9 , (a=0,a/O = * [1+JlnA +Y -£n2] ,

h 3 ( a , a / £ ) = l+«,nA+y-£n2 for a=0 and a=°°.

ds)

- 83 -

APPENDIX D

In this appendix the new integrals appearing

in the response coefficient formula for neutron absorbing

and scattering channels as given via eqs. (72), (73), (75)

and (76) are written out explicitly. The inner homogeneous

medium is of radius a and the purely neutron absorbing

sheath is of thickness b-a. In terms of the coordinate

labelling and the geometry illustrated in Fig. 6 and also the

representation used in appendices A and B, the integrals are

n ( a , b , a , 6 )

r-n/2 /-IT/2 r=° A -(p+T-.)secB/X.I cosndn / cosBdB I (1+—^) (a -pcos n )e dp0 J0 JSl r ' 2

-n/2 /•» An -(p + T2)secB/«-I dp

^TT r-n/2 /•» An11 I cosndn I cosgdB I (1+—=•) (a-pcosn)fl A / 2 •/0 • / s 1 r ' 2

1

(Dl)

in which

r ' = a +p -2apcosn, T 1 = <s1-

0 0 2T2 = S s , - s , , sQ = 2acosn, s 1 = acosn+Ab -a sin \\\

~ 84 -

C n C12 r2acosn I ( y r ' ) -apsecB/JUI cosndn j cos&dB J -p (a-pcosn)e K dp,

(D2)

and

rsin"1 | /-TT/2 rsl+S0 I^yr ' ) [„_ cab_ 1 b c o s l j j d^j cosBdBJ p — b -bscos^

+ Jl(bcosiJ;-s)cos3 e" tBC( ;p /*ds (D3)

with, in the l a t t e r ,

r' = b +s -2bscosi(i,

T = s1a+(s-s,)a,

2 2 2s, = bcosi|j-/( a -b sin ).

Note that j. depends only on a and a.4, n

EXPRESSIONS FOR HOMOGENEOUS CHANNEL WITHOUT A SHEATH

In the case b=a, j- may be evaluated explicitly

and is (we label j. in the case a=b as j. (a,6))i ,n i,n

- 85 -

(D4)

In an approximation to probabilities similar to that per-

taining to the specification for h21(a,a/Jl) in Appendix C

(see eq. (CIO)),we have

+ B33(a/£)P33(aa/£)] (D5)

in which

oO -» 9 G 9z + ^p)/a^ (D6)

(see eqs. (C9), (Cll) and (65b)), and B31(a/£), B 3(a/A), which

are = B31(a/A)/a and B33(a/£)/a (see Appendix C, eq. (C9b),

and also eq. (66a)), are tabulated functions. C^9(a/JO is

the integral

= - / cosgdB I ^ ^ ^-'O •'O u +a

(D7)+2au cosn

D (a/£) and C42(a/£) are tabulated in Table 3.

The current j^ (a,0) is

/

ir/2 AT/2 r2acosn I-^pr1)cosndnf cospdg I ——j

u J 0 •'0

(a-pcosn)e-apseceadp. (D8a)

- 86 -

This is written as

j 4 n(a,e) = j I2(jja)PES*(x)cos9 (D8)

in which P^'Cx) is like an escape probability (PEg (0) = 1,

P *(oo) E 0). (D8) arises by virtue of the fact that

Ar/2 Ar/2 /*2acosn I-Cyr1)I cosndr, / cosgdB I ^T—(a-pcosn)dp = 5~ I9(ya) (D9)JQ ^0 -'O r ^y

so that

-n/2 fn/2 /*2acosn

(D10)

If in eq. (D10) one takes the lowest order expansion of I,(yr)

and I.Cya) (i.e. approximates to the neutron flux distributic i

in the channel by a linear flux) then one has the approxima-

tion

PES ( X ) = k Cl " P33 ( x ) + | P31 ( X )

so that

(D12)

- 87 -

Here x denotes aa/&. More generally, by changing the inte

gration in eq. (D8a) from pdpdn to r'dr'doi and resorting

to BeSsel function addition theorems one can write for

j u (a,8) = coslH , n

c r00 (r XK (y)-i^j (y-x){ xK (y) + — 1 I(vx.y)x •'x (L y JKQ(y)+K2(y)

2 J(vx,y)> dy (D13)

with

2I(vx,y) = - ^ =- [xvl (xv)I (y)-ylo(y)l (xv)],

x v -y

J(vx,y) = g.-vy— \xvlo(xv)lo(y)-yl1(y)l1(xv)

CxvI1(xv)Io(y)-yI1(y)Io(xv)l|YT

In the absence of the sheath we denote

hj, (a,a,b,S) as hl^(a,a/£). From eq. (D3) with b = a (i.e. Sn=0),

comparing with eq. (D2), and by precedent we have and write

5,a n (a. B")

- 88 -

Eq. (D14) arises because

|\r/2 AT/2 f2acosi|j I-^yr')I cosipd^ I cos $dB I (acosijj-s) j ds = 0,Jo Jo Jo

Prr/2 AT/2 f2acos4/ I -^yr ' )- Y I cosi^d^ I cosBdB I s(acosij;-s) 1—

Jo Jo Jo

(D16)

E I3- I i2v 2

so that

A „ i n t i. • . . , ^ „ i 2acosi|j I

P^ *(y)=—=r^ r- I cosijjdil)| cos BdB I (acosi|j-s)—-—r/"-rT/2 A T / 2 . r1 cosijjd4)| cos^BdB /j Q j Q JQ

e - a s oC^P ,* d s < ( D 1 7 )

If in (D17) one takes the lowest order expansion for the

Bessel functions one has the approximation

TTx P 5 1 ( X ) ) ( D 1 8 )

so t h a t by eqs . (D12), (D14) and (D18)

h^Caja/U % J ? - I 2 (ya ) [P E S (x ) + a P £ S ( y ) ] . (D19)

In general, by resorting to Bessel function addition theorem

expansions, we have

- 89 -

2

t A

J (y-x)2 |[2xK1(y) + *- K2(y) ]I(Vx,y)

i 2My) )y (3KQ(y)+K2(y))J(xv,y) + — ^ jp(yJCxv,y)-xvJ(y,xv))>dy

(x v -y ) )(W 0)

where the notation J(y,xv) implies that the y and xv in

the J defined in eq. (D13) are interchanged. Thus a more

precise h^(a>a/SL) may be evaluated by combining the results

of eqs. (D13) and (D20) as required in eq. (D14).

EXPRESSIONS FOR A HOMOGENEOUS ROD SURROUNDED BY AN AIR GAP

As already indicated, J. (a,0) is unchanged4 ,n

from the j. (a,0) for the homogeneous channel. In the4- ,n

other expressions a is zero. In these expressions only

], , may be readily estimated. It is found that to a very

good approximation

j, (a,b,0) acoseI nnb 3 „ , vv Ifa

+ i (1 + Hg. e2(a,b)>P33<aa/n2(a,b)>P33

in which 52(a,b) is as defined in Appendix B, eq. (B5).

The j 0 (a,b,0) may be written as a sum of

certain coefficients multiplied by probabilities, thus,

(D21)

- 90 -

COS6 b1(a,b,Jl) + I Cnm(a,b,A)Pn*(a,b,a,Jl) (D22)

in which

0 b +u +2u^(b -a sin rp(D23)

1 r/2 p / 2 / - g e - U 8 e c B / £

C (a.b.A) = - cosndn I cosBciBI 9n m

0 .—= =- du (D24)/•ir/2 r S e - u s e c B / J l

0 0 0 b +u +2u/(b -a^sin^n)

and

"iT/21 p/2 r/2P (a ,b ,a ,JO = l—«— I cosndni cosBdB I

n m ^^nTn J n J n •'nO •'O •'O b V + 2 u / ( b 2 - a 2 s i n 2 n )

( D 2 5 )

The summation in eq. (D22) i s over the indices 3 1 , 42 and 33, and

the g are r e spec t ive ly

9 2 2g31 ~ ~a' g42 = ucosn, g33 = (acosn+ZCb -a sin n))cosn. (D26)

We have not examined either the coefficients or the probabili-

ties in detail. However it would appear that the probabilities

should be approximated well by the first collision probabili-

ties P (aa/S.) so that likely

- 91 -

Cnm(asb5il)Pnm(aa/£)j. ( D 2 7)

D1(a,b,£), C3l(a,b,Jl), C42(a,b,Si) and

reduce to D^a/A), B31(a/A), C42(a/£) and B33(a/A>

respectively, when b=a.

Finally for h4(a,a,b) we have

c fir/2 rrr/2 /*2acosn I,(prf) ,hh(a,a,b) = - ^ I cosridril cosBdfi I -=—, e~apsec$/ H4 * J0 Jo Jo ^

Cb2-(p + s1)/(b2-a

2sin2n)]dp+^ a2aI2(ua)PES*(y) . (D28)

We see that at least part of h^ is identical with that

derived for the homogeneous channel. In (D28),

2 2 2s-, = -acosn + A b -a sin n).

EXPRESSIONS FOR A HOMOGENEOUS ROD SURROUNDED BY A SHEATH

The sheath only absorbs neutrons. ;u (a,6)4- ,n

is unchanged. In the remaining integrals the effect of the

sheath (assumed to be thin) may be approximated by putting

in transmission probabilities in the manner discussed in

Appendix B. From the expansions for the T 's given in

that appendix we see that they are approximately equivalent

so that in the simplest approximation one might just use one

transmission probability, say T_,, and write

- 92 -

j i j n ( a , b s a , 6 ) % 3 i , n ( a ' b ' e ) T 3 1 ( a , S , b ) , i = 1 , 2,

(D29)

A better approximation is probably obtained by examining

the structure of the various terms involved in greater

detail so as to match the transmission probabilities more

closely to the likely sort of angular distribution of the

neutrons involved. Thus from the definition of jn (a,b,a56))l ,n

one might infer that (see also eq. (D2D)

j , ( a , b , a , 9 ) ^ acosSx ,n

33

31

- i - T h 0 ( a , o , b )

^ P l i 9(aa/S,)T l i 0 ( a , a ,

^ 5 2 ( a , b ) ) P 3 3 ( a a / J l ) T 3 3 ( a , a , b )3

(D30)

and so on.

- 93 -

TABLE 1 .

THE COEFFICIENTS Bn(a/«O AND B (a/iOnm

a/a)

0.1

0.3

0.5

0.7

1.0

1.5

2.0

2.5

3.0

3.5

4-0

4.5

5.0

5.5

B0

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

(a/JO

39062

12863

06268

03581

01822

00759

00380

00213

00130

00084

00057

00040

00029

,00021

B31(a/U

-0.26309

-0.20060

-0.16656

-0.14376

-0.12021

-0.09524

-0.07923

-0.06799

-0.05961

-0.05311

-0.04791

-0.04365

-0.04010

-0.03708

B51(a/£)

2.61465

0.38337

0.13463

0.06244

0.02514

0.00739

0.00240

0.00062

-0.00009

-0.00039

-0.00050

-0.00054

-0.00053

-0.00051

B42

1.

0.

0.

0.

0.

0.

0.

0.

0.

0,

0.

c.

0.

0.

(a/SL)

88667

54356

29031

18777

11569

06469

04194

02961

02211

01717

0137o

01127

00941

00798

B33(a/£)

0.33539

0.25596

0.21289

0 .18409

0.15432

0.12274

0.10243

0 .08812

0 .07743

0 .06912

0 .06245

0 .05698

0.05241

0.04852

TABLE 2

THE FIRST COLLISION PROBABILITIES Pnm(aa/S,)

Die

0

0

0

0

0

0

0

0

0

0

1

1

1

2

3

5

a/4

.01

.05

.1

.2

.3

.4

.5

.6

.7

.8

.0

.2

.5

.0

.0

.0

P31(aa/U

0.01974

0.09388

0.17700

0.31721

0.42989

0.52129

0.59595

0.65730

0.70796

0.74999

0.81429

0 .85961

0.90472

0 .94580

0.97719

0.99228

P51

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

(aa/£)

01760

08461

16125

29384

40327

49391

56921

63197

68442

72839

79651

84520

89426

93955

97454

99140

P42(aa/£)

0.01978

0.09462

0.17939

0.32387

0.44092

0.53620

0.61404

0.67787

0.73035

0.77363

0.83909

0.88430

0.92787

0.96511

0.98965

0.99799

P33(aa/A

0.02219

0.10518

0.19755

0.35152

0.47327

0.57038

0.64834

0.71122

0.76217

0.80359

0.86500

0.90629

0.94482

0.97603

0.99451

0.99937

- 95 -

TABLE 3

THE COEFFICIENT D^a/Jl) AND THE PRODUCT |

I

0 . 1

0 .3

0 .5

0.7

1.0

1.5

1.6

2 .0

2 .5

3.0

3.5

4 .0

4 . 5

5.0

5 .5

D 1 ( a / A )

•0 .88360

•0 .51175

•0 .37602

•0 .30077

- 0 . 2 3 3 2 1

-0 .17096

-0.13541

-0.11226

-0.09593

-0.08378

-0.07438

-0.06688

-0.06076

-0.05567

I C 4 2 ( a / U

0.02743

0.03837

0.04078

0.04084

0.03944

0.03617

0.03551

0.03298

0.03017

0.02775

0.02566

0.02385

0.02227

0.02089

0.01966

<r<sin-'f

r < Q

FIG. I. GEOMETRY AND CO-ORDINATE LABELLING FOR r > aAND FOR r<a, WHERE a IS THE CHANNEL RADIUS.

0

I PERTURBATION EXPANSION

E VARIATIONAL ESTIMATE

APPROXIMATE VARIATIONALESTIMATE

I Z PERTURBATION EXPANSION aȣ

X CARTER SIMPLE FORMULA

a= CHANNEL RADIUSf = MODERATOR M.F.P.

t/as CHANNEL M,F. P,

I 4 9a/j?

16 25

FIG.2. DIPOLE RESPONSE COEFFICIENT A,(a=O,a/f)

FOR VOID CHANNEL

ii

1 .Oi—

0.3 —

0.8

0.7

o.6

0.5

- 0.4<

CMI 0.3

0.2

0.1

0

m I PERTURBATIONEXPANSION o«Z

I VARIATIONAL ESTIMATEII APPROXIMATE

VARIATIONAL ESTIMATEM PERTURBATION

EXPANSION a » /

as CHANNEL RADIUSft £ MODERATOR M.R P.g/a = CHANNEL M.F.P.

4 9

a / /

16 25

FIG.3 DIPOLE RESPONSE COEFFICIENT A,(a = I,a/f)

a

8

a

1.0 r~

0.8

0.6

0.4

0.2

0

-0.2

- -0.4

CMIaI

-0.6

-0.8

-1.0

-1.2

m

I PERTURBATION EXPANSION Q«lU VARIATIONAL ESTIMATEHI APPROXIMATE VARIATIONAL

ESTIMATEET PERTURBATION EXPANSION a » £

a = CHANNEL RADIUSi = MODERATOR M.F.P.

je/a= CHANNEL M.F.P.

16 25

FIG.4 DIPOLE RESPONSE COEFFICIENT A,(a = ©.a/*)FOR BLACK ROD

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0

i

II1

1

- \ rw831

0

as CHANNEL RADIUS/s MODERATOR m.f.p.

•33

B42

FIG.5 THE COEFFICIENTS Bo (a / / )

AND Bnm (a /A)

FIG. 6 CHANNEL GEOMETRY