The reflected slab and sphere criticality problem with anisotropic scattering in one-speed neutron transport theory

Progress in Nuclear Energy Vol. 31, No. 3, pp. 229-252, 1997

Pergamon

Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0149-1970/97 $32.00 + 0.00

0149-1970(95)00094-1

THE REFLECTED SLAB AND SPHERE CRITICALITY PROBLEM WITH ANISOTROPIC SCATTERING IN ONE-SPEED NEUTRON TRANSPORT THEORY

M. A. ATALAY

Institute for Nuclear Energy, Maslak,

Istanbul Technical University

Istanbul,

80626, Turkey

~ecei~dl6Mayl~5) Abstract

-- The

homogenous,

one-speed

transport

one-dimensional

equation

multiplying

is

medium

applied with

to

a

linearly

anisotropic scattering. Case's singular eigenfunction method is used to formulate the criticality conditions. In addition to available biortogonality relations in the literature,

some parallel relations are

derived to obtain the solution. In a previous study dealing with only isotropic

scattering,

it

has

been

shown

that

the

normal-mode

expansion of Case also serves to evaluate the complete spectrum of a

critical

system.

thicknesses

As

an

and eigenvalues

extension, of a slab

the

spectrum

of

critical

(also sphere because of its

relation to the antisymmetric solution of a slab) are determined by using

the

including

formulation established here. some numerical

data relevant

The results

of this study

to the method employed are

presented. C ~ y r i ~ t © l ~ 6 ~ v i e r S c ~ n ~ L ~

i. INTRODUCTION

One of the fundamental problems

in neutron transport theory is the

determination of criticality properties of a The criticality

of

homogenous

slabs

investigated by usinq different techniques 229

multiplying

and in

spheres

system.

have

one-speed

been

theory.

M.A. Atalay

230

The significant outcome of the relevant dimensions

or

equivalently

transport operator.

the

works

eigenvalues

To study this problem,

are the Case's singular

eigenfunction

are

the

critical

of

the

neutron

the

and

basic

approaches

Carlvik's

high-order

spatial expansion methods. Most works appeared in are primarily

interested in

isotropic

the

scattering.

literature

The

singular

eigenfunction method provides an analytic treatment based normal-mode

expansion

of

isotropic

scattering,

on

monoenergetic

transport equation.

One of the applications of this theory

study the critical

slab

and

sphere

problems.

the

The

is

to

fundamental

thicknesses for different eigenvalues has been reported originally by Mitsis al.

(1963) and also Case and Zweifel

(1974)

eigenvalues

evaluated

the

fundamental

(1967). Later, Kaper et thicknesses

and

also

in benchmark quality. Workers of the latter technique,

on the other hand, calculated

the

eigenvalue

spectrum.

1991). The effect of the reflexion on the eigenvalue studied by Garis and Sj~strand

(1994).

Lately,

(Garis,

spectrum

it has

been

is

shown

that the singular eigenfunction method also serves to evaluate the complete spectrum of critical thicknesses and eigenvalues for reflected slabs and spheres

with

isotropic

scattering

the

(Atalay,

1995).

The effect of anisotropic Using Carlvik's method,

scattering

is

treated

(1978)

The work performed here is an extension of

in the previous paper

(Atalay,

1995)

to

The

extension

anisotropic scattering

of

this

include

technique

to

the

effect

of

eigenfunction

In

general order

to the

bi-orthogonality relations have been constructed by McCormick

and

Ku~der

the

Perhaps,

bi-orthogonality relations,

because

of

half-space

Kohut

problems,

(1965).

scattering

have

reported

include

is performed by Mika (1961).

solve linearly anisotropic

and

that

linear anisotropic scattering by using the singular method.

rarely.

the eigenvalue spectrum calculations

been performed primarily by Dahl and Sj~strand (1993).

rather

some

deficit

in

the critical slab problem could not be

solved as in analogous manner to

isotropic

scattering

case.

In

~ e r e f l ~ d s l ~ d s p h f f e ~NRyNoblem this

paper,

we

obtained

some

new

231

relations

parallel

to

bi-orthogonality relations and derived the criticality conditions. With the help of

these

expressions,

thicknesses and eigenvalues can

be

the

spectrum

obtained.

of

critical

However,

we

restrict our study only to detect the real eigenvalues, existence and numerical values of complex

(1978) and

multiplying

of

medium,

the

range

bi-orthogonality relations

but,

eigenvalues

reported in both Dahl and SjOstrand

the

here the

have

Kohut

been

(1993).

validity

of

In the

is restricted in terms of the parameter

of the neutron secondaries per collision,

c. In other

words,

the

bi-orthogonality relations for linearly anisotropic scattering are valid in a range that the transport operator has one-pair discrete modes. On the other hand,

the

complex

eigenvalues

are

not

in

primary importance for the range, which our study covers, although we can detect considerable number of real eigenvalues.

2. THEORY

We consider a

homogenous

anisotropic scattering,

slab

of

thickness

2d.

For

linearly

the one-speed transport equation takes the

form I

H

~(x,u) + ~(x,u) = ~c f d~'w(x,~') (i + 3fl ~ @x

)

(i)

-i

where f

is

the

mean

cosine

of

the

scattering

angle

in

a

i

collision. material

If the medium is

surrounded

by

the

same

in both side, the symmetry in the problem to

will be preserved.

reflecting be

studied

Then the boundary conditions are given by

~(-d,~)

= R w(-d,-~)

for

~>0

C2)

W(d,-~)

= R w(d,~)

for

~>0

(3)

where R is the reflexion coefficient.

Then the symmetry

condition

M. A. Amlay

52 of the problem is

~,(x,g) According

to

Mika

V,(-x,-g)

=

(1965),

if

(4) the

eigenvalue

following condition in a multiplying medium

c-~ 1 +

then the

transport

1 3f

satisfies

the

(c>l)

(5) I

equation

given

by

Eq.(1)

will

have

only

one-pair of the discrete modes. Hence the normal-mode solution can be written as in isotropic scattering case in the form

I

V~(x,,u) = ao+~O+(,u)e-X/uO + ao_4~O_(,u)eX/'uO + f d'gAO")#l., (~)e-x'%' -1 where the discrete ~0±(~) and continuum #u(g)

eigenfunctions

(6)

for

linearly anisotropic scattering are given by

d(-+~0~)

c~ o

~o±(~)

-

2 c~

~ (~)

-

(7)

u ~ 0 d (~,~) p

-

-

+ A (u)6(~-u)

(8)

u--~

2

k(~) = d(u2)(l

- cu tanh-*u)

(9)

- 3f (l-c)Zu 2 I

d(ab) = 1 + 3f (l-c)ab

(i0)

i

Another similarity to the isotropic scattering

for

the

real c eigenvalues,

the corresponding discrete eigenvalues of

the

transport operator,

±~

the

are purely imaginary 0 roots of the dispersion relation in the form

A(u) = d(u2)[1

is

and

- cu tanh-*(1/~)] - 3f (1-c)2u 2 = 0 i

that

given

by

(11)

The reflectedslaband spherecriticalityproblem This equation

can be arranged

a relation with u

in quadratic

233

form for c to establish

as follows 0

{3f,u~

[u0tanh-*(llu0)-l]}c2

- {3f,u~

[~0tanh-*(ll~0)-l]

-I

+ u0tanh The symmetry

(i/u0))c + I = 0

(12)

condition given by Eq.(4) a

= a

0+

implies that (13)

0-

A(u) = AC-u)

(14)

Since we look for a solution to the homogenous normalize convenient

a

=I and then these last conditions 0+ form

equation, put Eq.(6)

we to

can more

~(x,~) = ~o+ I

+

Here, only unknown the boundary

- x/V

fod~A(~') [Ol~(/~)e

is the expansion coefficient

condition given by Eq.(3),

~0+(~)(Re-d/~0

+ #_u(~)e "~]

- e

0) + ~0_(~)

A(u).

If

(15)

we

use

we obtain

0 - e

0)

+ f s duA(u) [~ u(~)(Re -d/u - e d/u) + ~ _~(~) (Red/~ - e -d/~

I] =0

0

(161

In order to obtain this equation,

we have used the identities

(17) Eq.(16)

is

a

coefficient, linearly

singular

for

we need some half-range

anisotropic

bi-orthogonality Ku~6er

equation

(1965).

scattering,

A(~).

To

orthogonlity these

take

determine relations. the

form

this For of

relations and have been obtained by McCormick and

In our work, we need these expressions

234

M.A. Atalay

./

(~) ['~o+ (~')+Bc"o/2]a'(~') ("o -~) = -

~o+

x(~o)a(':)

(18)

0

; d/a4~o (~)[~o+(~)+gcuo/21g(~ )(uo_/J ) =

X(_uo)

o

(19)

d(uo~)

0 1

f dJu~u(~) [~O+(~)+Bcuo/2l¥(h~)(uo-'u)

= 0

(20)

o

2

, f d~# w(~)

c

[~0 (N)+Bcuo/2]¥(/~) (UO-'U) -

-

0

uOv 4

d(Uo)d(-uu)

(21)

X(-u) d(~o~' )

+

where

B =

3f l (1-c) (uO -~)

(22)

d (uOu')

a'(~)

- c~ 2

i

(1-c)(1-cfs.)(u

(23) 2

0 - h~Z)x(-,u)

= ~ (1)/(o)

(24)

1

¥

= f d~ ~"a'(,~)

(25)

0

, X(~)

= f

g(u)d(u 2) du

~ _ ~

(26)

o

We

next

multiply

Eq. (16)

by

[q~0+ (~)+Bc~0/2]F(~) (u0-~)

and

integrate over ~. With the expressions above, one obtains

uoX(~'o)d(Uo~) (Re-d/uo

- e d/~ o)

-

v 0X(_uo)d(

.~O~)(Red/~

o

-

e

O)

1

- f d~ v A ( v ) X ( - v ) d ( - ~ ) ( R e d/u - e -d/P ) = 0

(27)

0 This equation is an exact statement of the

criticality

condition

under arbitrary reflexion coefficient R. However, we still need an additional relation for A(~) that an iterative solution

for

this

Tbereflec~dslab~dspb~ecfidc~ity~oblem problem

can be found.

Eq.(16)

be c o n v e r t e d

Thus

far,

However, the

we have with

To apply

this

to a F r e d h o l m

proceeded

procedure

we

required

integral

complete

our

can

as

not

equation

formulation.

one

that

requires

equation.

in

the help of a v a i l a b l e

literature,

235

isotropic

scattering

bi-orthogonality

obtain

a

for A(u) Therefore,

relations

convenient

that

it would

we

case.

form

of

provide

suggest

the

d (]2op)

1

in the

us

to

following

relations 2

c PO]2

1

f dN~0+ (~u) [~]2(~)+cul2P]~(;~)

-

X(u o)

4

0

2 1

C

-

~

]2

o

X(-u o

[

] 1

d(-~o]2) +

]2o+ p

-

(28)

'1

/

-

(29)

J

L

0 It

(30)

6(~-]2') ]2 0 2 t

C

; d~¢_p, (p) [q~]2(U)+c]2/2"~]~,(/J)

d(-]2'u)

]2']2

-

1

X(-]2 ' ) [ ]2'+ ]2 + - -5]

4

(31)

0

These

relations

half-range

bi-ortogonality

make

use of the

and

Ku~er

identities

(1965).

[~u(~)+cu/2p]~(~) above

can be proven

and

in the same

relations given

Hence,

that derives

and thus one

by M c C o r m i c k we

integrate

fashion

now

needs

(1964)

also

the to

and M c C o r m i c k

multiply

Eq.(16)

over ~ and use the relations

by given

to obtain

~A(~)

-

~(c~12)

Z

a'(w)N(~) (Re -d/~

{

- e d/~)

[,

+ UoX(-P o) d(-UoS") %u uo

[ H_ + Uo + ~. .] (Rea/po u

+ ( du /~ - -+ u X(-p)d(-uU) ) 0

- e-d/uo)

5"d(-UoU) - +

]2

- e

~d (-u~)

(32)

M.A. Amlay

236

This gives the

required

Fredholm

integral

equation

for

A(u).

Hereafter, the usual approach is an iterative procedure

that

one

can continue with definite order approximation starting

from

the

zeroth order.

(See

Mitsis,

1961

or

Case

and

Zweifel,

However, we here skip the zeroth order and proceed the first order

approximation.

This

provides

us

1967).

directly the

with

required

optimum accuracy.

The

first

order

approximation

integral term in Eq.(32) and

necessiates

substitute

the

that

we

omit

resulting

the

equation

into Eg.(27). Using the definition Re d/~ - e -d/~

T(R,~) =

(33)

Re-d/~ _ ed/~ and also the expression of F(~) given by Eq.(23), we obtain X(~0)d(u0~) - X(-u0)d(-u0~)T(R,u0)

,

g, (c,u)

+ f du

(cu/2)xZ(-u)A(~)d(-uS)T(R,u)

5

o

z

d(uoD) u'dC-u ~) + u ~ + u ~ d ( u : ) - u: ] } _ X(_u0)d(_u05)T(R,u0) [

0

0

= 0

a(-~05)

(34)

where gi(c'w)

-

N(w)

(35)

N(w)

= ~,

(

X,=(u)

+

~

d(u =)

].)

A(~) = (l-c)(l-cf,)

(36) (37)

In Eq.(34), we take ,

Kj = f d~ u ~ 0

g

(c,w)

* 5

(cu/2)X z (-u)A(~)d (-~)T(R,u)

(38)

Themfl~tedslab~dsphe~'e~ficalityprob~m and

find

linearly

condition

anisotropic

scattering

237 slab

criticality

in the form

X(~o)d(~o~) T(R,v0)

=

X(-u0)d(-u0~)

( l + K 2 ) d ( u o ~ ) d ( - U o u)+d(- -~Ou)[K*~d(u~)+Ko (~0 ~ - uo)]z (39) (l+Kz)d(uau)d(-u0~)+d(u0~)[K,vd(~)+K0 We

note

that

the

appropriate

importance for half-range calculations. numerical

X-functions

treatment

in

Thus, we give more useful

calculation

(c

(-u0 ~ - u0)]z

instead of Eq.(26)

,

the

are

fundamental

relevant

expression

of

transport X(~)

for

in the form

}

2

X(~)=exp - ~ f dug, (c,u) [d z(;2) (I+ cu )+3fi(l-c)Zu2d(-p2)]in(~-~) l-u 2 8

(40) This expression

is required primarily

the integrand of Eq.(38) that (McCormick,

in numerical

in our procedure.

calculation

of

The Milne problem gives

1964)

x(-~ )d(-~ : ) o 0

= - e

-2z /v O 0

(41)

X(u0 )d(uo 5) We here introduced extrapolated for l i n e a r l y

(41),

anisotropic

the z Q is defined

z0-- - (u0/2)in

endpoint,

scattering.

z

0

of the Milne

Then c o m b i n i n g

problem

Eqs.(40)

and

by

0

d(~05) t + 4c I dk~g* 0

2

(~2)(I+

c,2

)+3f,(l-c)2k~2d(-~ 2) ]u O

I-~

Using Eq.(41) and the explicit expression of T(r,u Q ) on

0

(42)

0

the

left

M.A. Amlay

238

and somewhat

arranging

on

expressions

the

right

the

Eq~(39)

becomes

Re-~d-'0)llW0 I - e ' Re~Id-z0>llv01

~,d+z 0 ,tlwol

- e -~(d+z

o >/luol

K + Ki ~d(v:)-K ,~'d(~ ~) - ~ o o o o

~-

3f i (,-c)~

tz ~ d ( v : ) - L , ~ : J t l v o l i (43)

where

K = (l+Kz)d(v0~)d(-u0~)

(44)

~o : t~ol i

(4s)

and

It can be seen conjugate

easily

that

of the numerators

the

denominators

in both side of

anisotropic

_+,,

scattering

criticality

slab problem

the

Eq.(43).

fact, we take logarithm of both side of this obtain the final form of the

are

Using

this

and

then

equation,

condition

complex

of

linearly

in the form

,{ Rsin[(d-z o )/Iv o l]+sin[(d+z o)/Iv o I] } - iv- Rc°s [(d-z0)ll~0 1]-c°s [(d+z0)/Iv0 I] {KS-

3f (i-c)5

= tg-,{ __i0 ___, ..... (I+K)d(v 2

[K

5d(uZ)-K wz])Iv

i

±__/0__0_~0_]_o_:

}

(46)

~)d(-u ~) + E ud(vZ)-K uZd(~ z) 0

This form of the criticality

0

1

o

o

o

condition and hence the last step

our procedure are based on the fact that we are interested real eigenvalues

(c) which correspond to purely

eigenvalues

as stated before.

(v) 0

of

in only

imaginary discrete

3. THE SPHERE PROBLEM

The procedure described above and the criticality

condition

by Eq.(46)

and

provides us critical

slab thicknesses

given

eigenvalues

The~fl~tedsl~dspherecfificalityproblem for only even modes

depending

Eq.(4).

On the other hand,

related

to the slab

condition

on

the

the critical

problem

(Case and Zweifel,

239

symmetry

requirement

sphere problem

(odd

modes)

1967).

Hence,

with

an

instead of

of

is d i r e c t l y antisymmetry Eq.(4)

we

require

W(x,~) =-VJ( - x , - p ) Thus,

this

(47)

implies a

= -a O+

(48)

O-

A(u) = - A ( - p )

We then proceed exactly

(49)

in the same manner as above

and

use

the

definitions Re d/~ Re -d/p

, Lj

du

+ e -d/Ju

(50)

T (R,~) = 1

g (c,u) *

pJ

+ e d/p

(cp/2)XZ(-p)A(~)d(-uu)T

(51)

(R u)

0 and to obtain the c r i t i c a l i t y

Re

-

L(d-=o)/lUoi

Re~d-ZO>/i~O

i

~(d÷z

0

,/lu

0

(

-~(d+,

0

,/lu

0

J

+ e + e

condition

given by

i i (52)

where (53)

L = (l+Lz)d(u0~)d(-u0p)

Finally,

if we are

take logarithm

interested

in only real

of both side of this e q u a t i o n

eigenvalues, to find

we

will

240

M.A. Atalay sin[(d+z0)/lw01]-Rsin[(d-za)/lu01] cos[(d+z0)/lu01]+Rcos[(d-z0)/l~01]

(54)

This gives the criticality condition for a sphere corresponding to the odd modes of a slab.

4. NUMERICAL RESULTS

The first results of this study is given in first to zeroth order endpoint value, collision

z

moments

as

0

a

ratio

function

(c) have been tabulated.

of

Table ¥(H)

of

neutron

These results

for the rest of calculations and also quite

i.

and

Here,

the

extrapolated

secondaries are

per

fundamental

important

for

other

applications of the theory employed here. As far as we know, data has not been given in the literature

(except z

this

values in the 0

form of z c, for 0

isotropic

Plazcek and Hofmann,

scattering

are

presented

by

Case,

1961).

In Tables 2 through 5, the critical thicknesses for various values of c eigenvalue,

reflexion coefficient,

the

angle,

scattering

R

and avarage

cosine

of

are given. For these results, a i comparison is can be made considering only isotropic scattering.

Mitsis

f

(1963), Case and Zweifel

(1967), and Kaper

et

al.

(1974)

give only fundamental thicknesses.

Our results for the fundamental

mode agree very well with theirs.

On

the

report the first three critical thicknesses mfp.

Although

the

expressions

of

the

other

hand,

isotropic

here

if these are under criticality

obtained in this paper can not be reduced trivially the study considered only

we

scattering

to

80

conditions those

(Atalay,

of

1995),

there is good agreement between these two study results also.

Therefl~tedslab~dsph~e~c~i~oblem Table 1.

The ratio of the first function

~(u),

for various

per collision,

f

c

?

to zeroth order moments

u, and extropolated

values

241

endpoint,

of the mean number

(mfp) 0 of secondaries

z

~

7'

c.

c

1

0.0

of the

z

c

1.001

0.710274

0.709736

5.50

0.492957

0.132251

1.01

0.708733

0.703413

5.60

0.491525

0.129927

i.i0

0.694172

0.645971

5.70

0.490133

0.127684

1.20

0.679619

0.592392

5.80

0.488781

0.125517

1.30

0.666526

0.547144

5.90

0.487465

0.123423

1.40

0.654682

0.508410

6.00

0.486186

0.121399

1.50

0.643911

0.474869

6.10

0.484940

0.119439

1.60

0.634071

0.445536

6.20

0.483727

0.117543

1.70

0.625042

0.419659

6.30

0.482546

0.115706

1.80

0.616723

0.396659

6.40

0.481394

0.113926

1.90

0.609033

0.376078

6.50

0.480272

0.112200

2.00

0.601898

0.357551

6.60

0.479177

0.110526

2.10

0.595259

0.340784

6.70

0.478109

0.108901

2.20

0.589063

0.325535

6.80

0.477067

0.107324

2.30

0.583265

0.311607

6.90

0.476050

0.105792

2.40

0.577827

0.298833

7.00

0.475.056

0.104303

2.50

0.572715

0.287076

7.10

0.474086

0.102856

2.60

0.567899

0.276218

7.20

0.473138

0.101448

2.70

0.563352

0.266158

7.30

0.472211

0.100079

2.80

0.559052

0.256813

7.40

0.471304

0.098747

2.90

0.554978

0.248107

7.50

0.470418

0.097449

3.00

0.551112

0.239977

7.60

0.469551

0.096186

3.10

0.547437

0.232368

7.70

0.468702

0.094954

3.20

0.543940

0.225231

7.80

0.467871

0.093755

3.30

0.540607

0.218522

7.90

0.467058

0.092585

3.40

0.537426

0.212205

8.00

0.466262

0.091444

3.50

0.534386

0.206246

8.10

0.465482

0.090331

242

M. A. Atalay

T a b l e E.

Slab c r i t i c a l

R--O

R=

thicknesses for f =0.0. I

0.25

R=

0.50

R = 0.75

R =0.99

16.65904 52.79076

15.74156 51.87328

14.00221 50.13393

9.89338 46.02510

0.46986 36.60158 72.73331

i. I0

4.22674 15.26411 26.30148

3.51332 14.55069 25.58805

2.50594 13.54331 24.58068

1.24756 12.28493 23.32230

0.04582 11.08319 22.12056

1.20

2.57968 10.10860 17.63752

2.01041 9.53933 17.06825

1.32730 8.85622 16.38514

0.62088 8.14980 15.67872

0.02234 7.55126 15.08018

1.30

1 .87766 7 .82155 13 .76545

1.40621 7.35010 13.29400

0.89317 6.83706 12.78096

0.40758 6.35148 12.29537

0.01456 5.95846 11.90235

1.40

1 .47688 6 .46427 ii .45166

1.07602 6.06342 11.05081

0.66764 5.65503) 10.64243

0.30064 5.28803 10.27543

0.01070 4.99810 9.98549

1.50

.21523 .54516 .87510

5.19754! 9.52747

0.52993 4.85986 9.18980

0.23665 4.56659 8.89652

0.00841 4.33834 8.66828

1.60

.03039 .87331 .71623

0.724231 4.56715 8.41007!

0.43740 4.28032 8.12324

0.19423 4.03715 7.88007

0.00689 3.84981 7.69273

1.70

.89275 .35689 .82104

0.61975 4.08389 7.54803

0.37113 3.83528 7.29942

0.16413 3.62827 7.09241

0.00582 3.46996 6.93410

1.80

0.78630 3.94554 7.10478

0.54037 3.69961 6.85885

0.32146 3.48070 6.63994

0.14172 3.30096 6.46020

0.00502 3.16426 6.32350

i .90

.70157 .60902 .51646

0.47811 3.38555 6.29300

0.28292 3.19036 6.09781

0.12442 3.03187 5.93931

0.00440 2.91185 5.81930

63257 3. 32792 6. 02327

0.42805 3.12339 5.81874

0.25219 2.94754 5.64289

0.11069 2.80604 5.50138

0.00392 2.69926 5.39461

1.01

2.00)

l

O.

0.86760:

The reflected slab and sphere criticality problem

T a b l e 3.

Slab

critical

thicknesses

243

for f =0.I0. I

R = 0.25

R = 0.50

R = 0.75

R = 0.99

17 .48929 55 .59653

16.47157 54.57881

14.55300 52.66024

10.11262 48.21986

0.46978 3~.57702 76.68427

1.10

4 .40260 16 .09869 27 .79478

3.62049i 15.31658 27.01267

2.54580 14.24189 25.93798

1.25103 12.94712 24.64321

0.04574 1i.74183 22.43792

1.20

2 .67529 10 .69187 18 .70845

2.05681 10.07339 18.08997

1.33920 9.35578 17.37236

0.62064 8.63722 16.65380

0.02227 8.03885 16.05543

I .30

1 .94146 8 .29796 14 .65447

1.43265 7.78915 14.14566

0.89831 7.25482 13.61132

0.40688 6.76338 13.11989

C.01451 6.37101 12.72751

1.40

1 .52389 6 .87870 12 .23350

1.09347 6.44828 11.80308

0.67043 6.02523 11.38004

0.29999 5.65479 11.00959

0.01066 5.36547 16.72027

1.50

1 .25221 5 .91801 i0 .58380

0.88042 5.54621 10.21201

0.53184 5.19764 9.86344

0.23619 4.90199 9.56778

6.00838 4.67418 9.33997

1.60

1.06094 5.21574 9.37053

0.73450 4.88929 9.04409

0.43905 4.59384 8.74864

0.19399 4.34879 8.50358

0.00687 4.16167 8.31646

1.70

0.91901 4.67576 8.43250

0.62858 4.38532 8.14207

0.37281 4.12955 7.88630

0.16412 3.92087 7.67761

0.00581 3.76255 7.51930

1.8oi

O. 80961 4. 24538 7. 68115

0.54839 3.98416 7.41993

0.32330 3.75907 7.19483

0.14194 3.57770 7.01347

0.00502 3.44079 6.87656

O. 72281

0.48572 3.65592 6.82612

0.28499 3.45519 6.62538

0.12485 3.29505 6.46525

0.00441 3.17461 6.34481

0.43549 3.38154 6.32759

0.25454 3.20059 6.14664

0.11132 3.05737 6.00342

0.00393 2.94998 5.89604

=

1.01

1.90

0

3. 89301 7. 06321 O. 65236

2.00

3. 59841 6. 54446

244

M. A. AUday T a b l e 4. Slab critical thicknesses for f =0.20. i

R=0

R=

0.25

R=

0.50

R=

0.75

R

=

0.99

18.46196 58.90880

17.31947 57.76630

15.18058 55.62742

10.349441 50.79628

6.46971 46.91655

i.I0

4 .60486 17 .09464 29 .58442

3.73955 16.22932 28.71910

2.58786! 15.07764 27.56741!

1.25442 13.74419 26.23397

6.04565 12.53542 25.02520

1.20

2 .78406 II .39974 20 .01543

2.10703 10.72272 19.33840

1.35124 0.62022 9.96693 ; 9.23590 18.58261 17.85159

0.02220 E.63788 17.25357

1.30

2 .01346 8 .88675 15 .76004!

1.46072 8.33401 15.20730

0.90325 7.76654 14.64983

0.40599 7.27927 14.15256

6.01445 6.88773 13.76102

1.40

1 .57676! 7 .40052{ 13 .22428

1.11182 6.93558 12.75934

0.67300 6.49676 12.32052

0.29918 6.12294 11.94671

0.01062 5.83438 11.65814

1.50!

1.29391 6.39636 11.49880

0.89396 5.99640 11.09885

0.53366 5.63610 10.73854

0.23564 5.33809 10.44053

0.00835 5.11079 10.21323

Z .60!

1. 09579 5. 66332 i0. 23084

0.74564 5.31316 9.88069

0.44081 5.00834 9.57587

0.19377 4.76130 9.32883

0.00686 4.57438 9.14191

1.70

O. 94959 5. 10026 9. 25092

0.63862 4.78929 8.93996

0.37488 4.52555 8.67621

0.16427 4.31493 8.46560

0.00581 4.15648 8.30714

1.80

0.83758 4.65182 8.46606

0.55812 4.37236 8.18660

0.32587 4.14011 7.95435

0.14245 3.95669 7.77093

0.00503 3.81927 7.63351

1.90

0.74930 4.28486 7.82042

0.49563 4.03119 7.56675

0.28818 3.82374 7.35930

0.12575 3.66131 7.19687

.00444 .54000 .07556

2.00

0.67815 3.97817 7.27819

0.44589 3.74591 7.0459,3

0.25841 3.55843 6.85845

0.11260 3.41262 6.71264

.00398 .30400 .60402

1.01

The reflected slab and sphere criticality problem

Table 5.

c

Slab critical

thicknesses

245

for f =0.30. !

R = 0.25

R=

0.50

R = 0.75

P =0.99

19.62374 62.90185

18.32167 61.59978

15.90535 59.18346

10.60644 53.88455

6.46963 4~.74774

i.i0

4.84124 18.31306 31.78489

3.87306 17.34489 30.81672

2.63239 16.10422 29.57604

1.25773 14.72956 28.20138

L.04555 i~.51738 2£.98921

1.20

2.90947 12.28589 21.66232

2.16165 11.53807 20.91450

1.36337 10.73979 20.11621

0.61959 9.99601 19.37244

0.02211 9.39854 I~.77496

1.30

2.09571 9.64304 17.19037

1.490561 9.037891 16.58522i

0.90789 8.45522 16.00255

0.40487 7.95220 15.49953

t.01438 7.56171 15.10904

1.40

1.63694 8.08942 14.54191

1.13109 7.58358 14.03606

0.67526 7.12775 13.58023

0.29818 6.75067 13.20315

0.01056 6.46305 12.91553

1.50

1.34169 7.04618 12.75067

0.90838 6.61286 12.31735

0.53537 6.23986 11.94434

0.23501 5.93950 11.64399

6.00831 5.71280 11.41729

1.60

1.13657 6.28961 11.44266

0.75808 5.91112 11.06417

0.44286 5.59591 10.78495

0.19364 5.34669 10.499.73

C.00684 5.15989 10.31293

i .70

0.98679 5.71266 10.43854

0.65081 5.37668 10.10256

0.37780 5.10367 9.82955

0.16474 4.89062 9.61650

0.00582 4.73169 9.45757

1.80

0.87360 5.25689 9.64019

0.57121 4.95451 9.33780

0.33005 4.71334 9.09664

0.14364 4.52693 8.91023

0.00507 4.38837 8.77166

1.90

0.78595 4.88738 8.98882

0.51050i 4.61194 8.71337

0.29394 4.39538 8.49681

0.12774 4.22917 8.33061

0.00451 4.10594 8.30738

2.00

0.71692 4.58189 8.44686

0.46329 4.32826 8.19323

0.26606 4.13103 7.99600

0.11550 3.98046 7.84543

0.00407 3.86904 7.73401

1.01

R=0

M.A. Atalay

246 To evaluate

the

criticality

eigenvalues,

first,

the

discrete

eigenvalue of the transport operator ~ criticality conditions.

is determined by using the 0 corresponding c eigenvalue is

Then the

obtained to be the smallest root of the quadratic expression given by Eq.(12).

For even modes,

these eigenvalues are given in

6 through 9 for different slab thichnesses,

reflection

Tables

ratio

and

avarage cosine of the scattering angle to represent anisotropy. the other hand,

On

for odd modes the calculations must be carried out

by using rather the formulations given in section 4 and,

in

this

paper only the results of f =0.i case are given in Table I0. We i can compare these results with those of Garis (1991), Garis and Sj~strand Dahl

(1994) and Atalay

and

Sj~strand

(1979)

anisotropic scattering. closely with thickness

(1995) for

other

and

isotropic

Kohut

scattering

(1993)

for

linearly

Then it can be seen that our result

study

results,

especially,

is greater than 1 mfp. However,

in

when

terms

of

and

agree

the

slab

accuracy,

the potential of this method is more than the results presented in this paper.

Because

approximations.

As

we in

isotropic scattering, better convergence

here the

presented

work

Then

we

Kaper

the

et

first

al.

order

(1974)

for

one may consider to iterate further until

is

obtained,

linearly anisotropic scattering complex.

of

only

expect

although for

some

this

the

formulation

purpose

improvement

in

for

becomes the

a

more

accuracy

especially for the small slab thicknesses.

Finally, we remark some points about the calculations. tables, we have not presented any

for

R=I.

Because,

criticalitiy conditions become independent of

slab

thickness

this method, Second,

the

conditions,

when

the

integral

reflection expressions

except X-function

Gauss quadrature and extrapolated

integration. endpoint

subdivision of the integral

value

First,

is

perfect

involved

integrals, are

in

(Atalay, the

evaluated

The numerical values

z

of

in the in

1995).

criticality by

using

X-function

are evaluated by the succesive 0 interval near the right endpoint due

to improper behavior of the integrand at this point.

The reflected slab and sphere criticality problem

Table 6.

Eigenvalues

2d

Eigenvalue no.

0.20

1

20.0

0.0

R = 0.25

R=

3.798

2.954

1.277104

slab

(even mode~j

0.50

for f =0.0. I

R = 0.75

}~ = 0.99

2.235

1.602

1.024

1.203660

1.132324

1.064193

1.002487

2.873321

2.754703

2.651489

2.563145

2.490907

4.784651

4.676340

4.585603

4.509950

4.449126

6.748132

6.646529

6.561840

6.491478

6.435054

8.728176

8.630808

8.549558

8.482066

8.428020

1.007136

1.006578

1.005670

1.003962

2

1.061954

1.057880

1.051987

1.043422

1.032556

3

1.161612

1.153553

1.143461

1.131791

1.120480

4

1.292969

1.282199

1.270198

1.258050

1.247412

5

1.445171

1.432946

1.4204051

1.408624

1.398756

6

1.610866

1.597981

1.585493

1.574266

1.565066

7

1.785435

1.772321

1.760109

1.749421

1.740766

8

1.966002

1.952886

1.941012

1.930800

1.922585

9

2.150754

2.137747

2.126212

2.116407

2.108550

i0

2.338512

2.325667

2.314450

2.304990

2.297427

11

2.528490

2.515828

2.504899

2.495734

2.488417

12

2.720147

2.707673

2.697002

2.688090

2.680980

13

2.913100

2.900812

2.890372

2.881679

14

3.107074

3.094964

3.084730

3.076227

15

3.301864

3.289922

3.279874

3.271538

16

3.497317

3.485533

3.475652

3.467464

17

3.693314

3.681678

3.671947

3.663891

18

3.889764

3.878266

3.868672

3.860735

19

4.086597

4.075225

4.065755

4.057924

20

4.283750

4.272500

4.263144

4.255410

2.0

R=

(c) of multiplying

247

M.A. Auday

248

Table 7. Eigenvalues

(c) of multiplying

slab

(even mod.-s) for f =0.i0 I

2d

Eigenvalue

R=0.0

R = 0.25

R = 0.50

R=

0.75

R = 0.99

1

3.762

2.955

2.239

1.602

1.024

1

1.288864

1.209353

1.134469

1.064643

1.002487

2

3.103674

2.977924

2.868689

2.774869

2.697681

1

1.007816

1.007148

1.006082

1.004151

2

1.068076

1.063260

1.056538

1.047267

1.036159

3

1.178463

1.169124

1.157985

1.145755

1.134328

4

1.325259

1.313039

1.300110

1.287579

1.276851

5

1.496720

1.483110

1.469818

1.457746

1.447769

6

1.684606

1.670485

1.657385

1.645896

1.636544

7

1.883564

1.869375

1.856638

1.845679

1.836815

8

2.090173

2.076120

2.063766

2.053249

2.044761

9

2.302271

2.288424

2.276407

2.266233

2.258027

i0

2.518238

2.504646

2.492951

2.483078

2.475108

11

2.737230

2.723877

2.712443

2.702"800

2.695009

12

2.958498

2.945369

2.934154

2.924697

2.917049

13

3.181544

3.168617

3.157585

3.148278

14

3.406003

3.393256

3.382376

3.373191

15

3.631602

3.619014

3.608260

3.599172

16

3.858133

3.845683

3.835032

3.826026

17

4.085436

4.073107

4.062539

4.053598

4.290660

4.281771

no.

0.20

2.0

20.0

18

The reflected slab and sphere criticality problem Table 8.

Eigenvalues

2d

Eigenvalue no.

0.20

1

2.0

1

20.0

T a b l e g.

(c) of m u l t i p l y i n g

R = 0.0

R = 0.25

-

slab

(even modes)

for f = 0 . 2 0 I

R = 0.50

R = 0.75

R = 0.99

2.240

1.603

1.024 1.002488

1.301849

1.215398

1.136682

1.065097

1.008641

1.007826

1.006558

1.004357

1.075680

1.069889

1.062152

1.052086

1.040719

1.200267

1.189268

1.176880

1.164016

1.152421

1.369070

1.354975

1.340919

1.327882

1.316949

1.570086

1.554670

1.540389

1.527819

1.794405

1.778630

1.764600

1.752527

2.035884

2.020174

2.006474

1.994767

2.290353

2.274852

2.261414

2.249924

2.554948

2.539659

2.526373

2.514962

Eigenvalues

(c) of m u l t i p l y i n g

slab

2d

Eigenvalue no.

0.20

1

--

--

--

2.0

1

1.316289

1.221826

1.009662

20.0

249

R=

0.0

R=

0.25

R=

(even modes)

0.50

R=

0.75

for f =0.30. I

R = 0.99

1.603

1.024

1.138965

1.065557

1.002488

1.008645

1.007111

1.004583

1.085458

1.078341

1.069331

1.058359

1.046703

1.230305

1.217027

1.203067

1.189410

1.177524

1.435167

1.418356

1.402684

1.387770

1.377303

1.693396

1.674996

1.658831

1.644945

1.633599

2.003646

1.984334

1.967594

1.953197

250

M.A. Amlay

Table 10. Eigenvalues

(c) of multiplying slab (odd modes)

for f =0.I0 i

2d

Eigenvalue no.

0.20

i

2.0

20.0

R=

0.0

.

R = 0.25

.

.

R=

0.50

.

R = 0.75

R=

0.99

.

i

2.078849

1.929864

1.808212

1.711851

1.638876

2

4.207739

4.046303

3.933215

3.860274

3.819606

1

1.030872

1.028245

1.024297

1.018230

1.009516

2

1.117863

1.110058

1.100211

1.088991

1.078460

3

1.248125

1.236119

1.223189

1.210971

1.201104

4

1.408500

1.394042

1.380163

1.368364

1.359466

5

1.589000

1.573310

1.559411

1.548334

1.540317

6

1.782950

1.766700

1.7530981

1.742720

7

1.986072

1.969595

1.956364

1.946587

8

2.195621

2.179076

2.166199

2.156923

9

2.409802

2.393267

2.380701

2.371832

10

2.627411

2.610904

2.598596

2.590067

Ii

2.847614

2.831148

2.819050

2.810799

12

3.069824

3.053390

3.041467

3.033448

13

3.293616

3.227211

3.265424

3.257594

14

3.518678

3.502282

3.490604

3.482935

15

3.744761

3.728377

3.716778

3.709241

16

3.971697

3.955310

3.943766

3.936342

17

4.199335

4.182938

4.171433

4.164094

5. CONCLUSIONS

In the literature,

the

singular

eigenfunction

method

had

been

already extended to solve linearly anisotropic scattering problems using the

half-range

bi-orthogonality

application of this theory has been problems.

As a summary,

the work

relations.

limited

performed

to here

However,

only has

the

half-space aimed

two

T h e ~ f l ~ d ~ ~ds~ecri~c~ypr0bMm purposes.

First,

the

missing

treatment

of

251

the

critical

problem by the singular eigenfunction method in the literature performed

for linearly anisotropic

scattering.

The

use

method for the problem required the derivation of some half-range relations. bi-orthogonality

Constructing

relations,

some

parallel

of

slab is this

additional

relations

to

we applied this theory to the study of

finite multiplying medium criticality problems.

Second,

the criticality

conditions

obtained

detect the criticality spectra of a

here

multiplying

are

used

medium.

We

to had

already showed in a previous paper that the criticality conditions derived here could be

used

to

determine

not

only

fundamental

thicknesses and eigenvalues of a critical system but also complete spectrum of these quantities. detected eigenvalues Eq.(5),

is

In this study, however,

limited

given

by

because of the form of the solution and validity range

of

the bi-ortogonality relations. considering anisotropy

with

The

a

the range of

restriction

results

indicated that the

of

this

singular

paper

by

eigenfunction

method is also quite effective tool for this problem

in

addition

to Carlvik's method.

REFERENCES Atalay M.A.

(1995)

The

Boundary Conditions NucL.

En.

Critical

Slab

Problem

for

Reflecting

in One-Speed Neutron Transport Theory. Ann.

Accepted for publication.

Case K.M., de Hofmann F., Placzek G. TAeor~ o/ Neutron D6//~s~on.

(1953)

Introduction

to

tAe

U.S. Government Printing Office,

Washington D.C.

Case K.M. and Zweifel P.F.

(1967) t~neur

Transport Theorw. Addison

Wesley, Reading. Mass.

Dahl E.B. and Sj~strand N.G.

(1979) Nuc£.

Sc~.

En s. 6Q,

114.

~2

M.A. Atalay Garis N.S.

(1991) NutS. Sc~. EnM. 107, 343.

Garis N.S. and Sj6strand N.G.

(1994) Ann.

Kaper H.G., Lindemann A.J., and Leaf G.K.

Nuc~,

En, El, 67.

(1974) NutS. Sc£, En S.

54, 94.

Kohut P. (1993) NucL. S¢6.

McCormick N.J.

(1964)

En 8, 115, 320.

One-Speed

Neutron

Transport

Problems

Plane Geometry. Ph.D. Thesis, The University of Michigan.

McCormick N.J. and Ku~6er I. (1965) 3. Ma6A. PAys. 8, 1939.

Mika J.R. (1961) NucL. Sc~.

Mitsis G.J.

(1963) Nuc~,

En 8. 11, 415.

Sc~. En 8. 17, 55.

in