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Masters Theses Student Theses and Dissertations

1969

Solution of the two-dimensional multigroup neutron diffusion Solution of the two-dimensional multigroup neutron diffusion

equation by a synthesis method equation by a synthesis method

William Ray Heldenbrand

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SOLUTION OF THE TWO-DIMENSIONAL MULTIGROUP NEUTRON DIFFUSION EQUATION BY A SYNTHESIS METHOD

BY tlt-fO WILLIAM RAY HELDENBRAND /946

I

A

THESIS

submitted to the faculty of

UNIVERSITY OF MISSOURI-ROLLA

in partial fulfillment of the requirements for the

Degree of

MASTER OF SCIENCE IN NUCLEAR ENGINEERING

Rolla, Missouri

1969

Approved by / / ;()/ J ' ~ /} .£!. ~ <}%au4¢':._ (advisor) ~1/.D,g/~

/)df_~6 (} I

ABSTRACT

A method, called the highe~ mode synthesis method, for

the solution of the two-dimensional neutron diffusion equa­

tion is developed. In this method, the two-dimensional

ii

eigenfunction is expanded in terms of one-dimensional funda-

mental and higher eigenfunctions. A substitute, weight, and

integration procedure is applied and the two-dimensional . equation is reduced to a one-dimensional equation in terms

of expansion coefficients. The expansion coefficients are

combined with the trial functions in order to obtain the two-

dimensional eigenfunction. This procedure results in a

significant reduction of computation time as compared with

standard iteration methods.

iii

ACKNOWLEDGMENTS

The author wishes to thank Dr. Doyle R. Edwards for

suggesting this project and assisting in its completion. The

cooperation of the computer staff at the U.M.R. computer

science facility was an integral part of the project and is

gratefully acknowledged.

The author would also like to thank his wife, Pat, for

her assistance in preparing the rough draft and in drafting

the figures and for her continual encouragement.

The typing of the final draft which was done by

Mrs. Judy Guffey is very much appreciated.

The author wishes to acknowledge the financial support

by the Atomic Energy Commission during the course of this

study.

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . LIST OF TABLES . . . . . . . . . . . . . . .

I . INTRODUCTION . . . . . . . . . .

iv

Page

ii

iii

vi

vii

1

II. LITERATURE SURVEY . . . . . . . . . . . . . . 4

A. Iteration Methods . • . . . . 4

B. The Stabilized March Technique . . . . 5

C. Comparison of SMT and Iteration Methods . . . 6

D. Synthesis Methods . . . . . . . . • . 7

III. DISCUSSION . . . . . . . . . . . . . . . . . . 13

A. The Multigroup Approximation . . . . . 13

B. Formulation of the Higher Mode Synthesis

Method . . . . . . . . . . . . . . . . 15

C. Results . . . . . . . . . . . . . . . . . 1. Problem One

2 •

3 .

Problem Two

Problem Three

. . .

. . . . .

. . . . . . . . . . • • . . . . . . . . • • . . . . . . . .

IV. CONCLUSIONS . . . . . . . . . . . . . . V. RECOMMENDATIONS

BIBLIOGRAPHY . . . . .

APPENDICES

. . . . . . . . . . . . . . .

. . . . . . . . . . .

21

22

30

36

42

43

45

A. The MUD-SYN Code . . . • . • . . . • . . . . . 47

v

Page

B. Listing and Sample Input and Output for the

MUD-SYN Code • . . . • . . . . . • • . . . 53

1. Fortran Listing of the MUD-SYN Code . . . 54

2. Sample Input for the MUD-SYN Code 99 •

3. Sample Output for the MUD-SYN Code . . . . 100

VITA . . . • . . . . . . . . . • . . . . . . • • • . . • 113

vi

LIST OF FIGURES

Figures Page

1. Problem One Configuration . . . . . . . . . 23

2. Flux Shape for X = o.o em. . . . . . • . . 27

3. Flux Shape for X = 31.5 em. • . • . . . . . 28

4. Flux Shape for y = 0.9 em. . . • . . . . . 29

5. Problem Two Configuration . . . . . • . . . 31

6 . Flux Shape for X = o.o ern. • . • . . . . . . . 34

7. Flux Shape for y = 0.0 ern. . • . . . . 35

8. Problem Three Configuration . . . . . . . . . . 37

9. Flux Shape foro z = 21.4 ern. . . . . . • . . . . 39

10. Flux Shape for z = 35.7 ern. • . . . • . . . • • 40

Tables

I.

II.

III.

IV.

v.

LIST OF TABLES

Relative Size of Expansion Coefficients

Problem One Data . . . . . • . . . • •

Relative Size of Expansion Coefficients

Problem Two Data . . . . . . . • . . • • .

. . • •

Problem Three Data . . . . . . . . . . . .

vii

Page

25

26

32

33

38

1

I. INTRODUCTION

The purpose of this thesis 1s to demonstrate a method of

solving the equation

+ vi>. f(E,~) r Lf(E' ,£)~(E' ,~)dE' .

0

= J~Ls(E'~E,~)~(E',~)dE' 0

(1)

This is the energy dependent, steady state diffusion equation

of reactor physics. The symbols used in this equation are

defined as follows:

D(E,~) = diffusion coefficient

~(E,~) = neutron flux

Lrem(E,u) = removal cross section

L (E'~E,u) = scattering kernel s -

= fission yield

= fission cross section

= energy

u = spatial variables

= number of neutrons/fission

= effective multiplication constant

Equation (1) has been used extensively to ascertain

basic parameters of nuclear reactors when they are operating

* in the steady state. Many methods of solving this equation

*niscussed in the Literature Survey

2

have been proposed and used. Almost without exception, the

method of solution is so complex that digital computers must

be used to do the calculations. Because of the high cost of

computer time, a method of solution must be used that is no

more accurate than is needed for a particular application.

For example, in the final stages of designing a reactor, a

method must be used which results in a high degree of accu­

racy. On the other hand, for survey studies a high degree of

accuracy is not a necessity, and a method should be used

which requires a minimum amount of computer time.

Many methods of solving the diffusion equation have been

used in the past. These include the iteration method, a

method which uses the assumption that the diffusion equation

can be treated as an initial value problem, and various

synthesis techniques. These are discussed in the Literature

Survey.

The method of solution which is presented in this thesis

is an approximate one requiring fewer calculations and, there­

fore, less computer time than comparable methods. This

method is referred to as the Higher Mode Synthesis (HMS)

method. HMS is based on the assumption that the spatial

dependence of the eigenfunction of Eq. (1) can be expanded in

terms of one-dimensional eigenfunctions. This method was

first proposed by Edwards [1].

In the Discussion the multigroup approximation and a

derivation of HMS are presented. A description of the code,

• 3

MUD-SYN, used to verify HMS is also given. Next, results

from MUD-SYN are compared to a more conventional production

code. This thesis is concluded with a summary of the results

and recommendations for further study.

II. LITERATURE SURVEY

Most methods of solving Eq. (1) use the multigroup

approximation and usually divide the space interval into dis-

crete segments [~]. Equation (1) is normally solved by

iteration methods. However, other methods have been devised,

such as the stabilized march technique [!] and various syn­

thesis techniques. These methods are described below.

A. Iteration Methods

For simplification, Eq. (1) may be written in operator

form as:

(2)

where L and M are linear operators, 1/l is the eigenvalue,

and x represents all the independent variables [4]. The

general iterative solution of Eq. (2) proceeds [~] as follows.

Equat.ion ( 2) is solved for ~(~) to give l +(!_) = L -lM+(!_). The

iterative method requires an initial guess for both ~(~)

and for l, denoted by +(O)(!_) and leo>, respectively. A new

• • ( 1 ) ( ) . b t . d b th 1 t . approx~mat~on, + ~ , ~s o a~ne y e re a ~on

After +(l)(!_) is known, a new approximation may be obtained

from the equation

5

This iteration may be continued until the difference between

K-1 K ~ (~) and ~ (~) is smaller than some preset value. This is

what is normally called the inner iteration. After the ~(~)

have converged, a new A may be calculated using the relation

( 1/ >. ( 1 )) = f~(x)L~(x)d!$, f~(~)M~(~)d~ •

This is the outer iteration. The new value of A is then used

in the inner iteration. This procedure continues until both

~(~) and A have satisfied the convergence criteria. Other

iterative schemes have been used, but the general procedure

varies only slightly from that presented above. Many compu-

ter codes have been written which use the iteration method

[6-9] •

B. The Stabilized March Technique

Another method for solving the multigroup diffusion

e_quation is the Stabilized March Technique (SMT) [~J. The

basis of SMT is that the diffusion equation can be treated as

an initial value problem. This difference equation is

inherently unstable. The instability cannot be eliminated,

but it can be controlled by performing a conditioning trans-

formation at various intervals during the march. The eigen-

value is determined by requiring that the flux goes to zero

at the outer boundary or at the extrapolated boundary. The

SMT requires an initial guess of the eigenvalue only. If

the flux is ·not zero at the outer boundary, another guess for

'

the eigenvalue is made and the march is repeated. This pro­

cedure is continued until an eigenvalue is found which makes

the flux satisfy the boundary conditions.

6

A one-dimensional code, MUD-MO, which utilizes SMT, has

been written. This code is used to solve the one-dimensional

form of Eq. (1) in order to calculate the trial functions,

adjoint functions and expansion coefficients used in HMS.

C. Comparison of SMT and Iteration Methods

The iteration methods have a disadvantage in that they

are very sensitive to the form of the scattering matrix.

When up-scattering is present, the convergence of the itera­

tion methods is slowed conside~ably. For problems involving

down-scattering only, the iteration methods are faster than

SMT. However, SMT is independent of the type of scattering

matrix; and for problems involving both up-scattering and

down-scattering, SMT is faster.

SMT has an advantage in that it can be used to calculate

higher eigenfunctions and eigenvalues. The higher eigenfunc­

tions and eigenvalues can be calculated in the same amount of

time as the fundamental. Only one iteration method is avail­

able to calculate higher eigenfunctions. This method is the

. Wielandt fractional iteration [~]. It requires a matrix

inversion at each point along one of the spatial axes; there­

fore, it can run into trouble near internal zeroes of the

higher eigenfunctions [11]. Both SMT and the iteration

methods can be used to calculate the adjoint function.

7

D. Synthesis Methods

In many situations a one-dimensional method does not

provide an adequate representation, and the problem must be

treated two-dimensionally. The normal approach to this pro­

blem is to use an iteration method for a two-dimensional

mesh; however, calculations on such a mesh require an inor­

dinate amount of computer time. Because of this fact, a

considerable amount of effort has been devoted to devising

methods of constructing approximations to two-dimensional

flux shapes using one-dimensional calculations. These

methods are referred to as synthesis methods. Most synthesis

methods rely on the assumption that the neutron flux is, or

almost is, spatially separable. For most reactors this

assumption can be made. However, if the reactor is small or

highly heterogeneous, this is a bad assumption; then other

methods must be used.

In the case of a cylindrical reactor, the flux is not

truly separable if the materials are not uniform axially.

However, the reactor may be divided into a small·nurnber of

axial zones and the assumption may be made that the flux is

separable within each zone. A method based on this assump­

tion has been used extensively (12].

Two variations of the procedure may be illustrated by

considering a reactor which is divided into two axial zones

by a control rod bank partially inserted into the core in the

axial direction. The first technique is to write:

j = group index

z < z < L r

where r is the radial direction, z is the axial direction,

8

{3)

zr is the point where the control rods begin, L is the height

of the reactor and H1 j(r) and H2j(r) are solutions of the

one-dimensional equation in zone (1) and zone (2), respec­

tively.* This approximation is substituted into the two­

dimensional equation and a one-dimensional equation in ej(z)

obtained. The major failing of this method is that the

synthesized flux normally has a large discontinuity at the

zone interface.

The second technique eliminates the discontinuity by

using an expansion of the form

K t

k=l (4)

where the aa (z) are the expansion coefficients and the Ha(r)

are the trial functions which satisfy the boundary conditions

in the r direction. The trial functions must be chosen .

intuitively such that this expansion would be expected to be

a good approximation.

*zone (1) is.the region without control rods, and zone (2) is the region _with control rods.

An expans~on of the form in Eq. (4) is much preferred

over the form in Eq. (3) because it will not result in a

discontinuity at the zone interface as will the form in Eq.

( 3) •

There is no set rule as to how to determine the expan­

sion coefficients, aa(z). The procedure to determine the

expansion coefficients depends on the choice of trial func-

9

tions. Several methods have been proposed and a few of these

are described below.

In order to illustrate these methods, a right cylindri-

cal reactor will be studied. The following notation will be

* used.

~ ~ ~ -V'•DV'~(r,z) + A~(r,z) = 1/>. M~(r,z) ( 5)

where

l<r,z) =col [~ 1 (r,z), ... , ~j(r,z)] ,

D = diag [D1 ,D2 , . . . '

M = [v fit~] '

~j(r,z) = neutron flux of the j-th group, . nJ = diffusion coefficient of the j-th group,

*Throughout this thesis the convention will be used that sym­bols representing matrices will be underscored and vectors will be represented with arrows above the symbols.

10

= fission yield,

= fission cross section,

Iij = scattering cross section from group i to group j.

The boundary conditions to be used with Eq. {5) are

-+ ~(r,z) = 0 at the outer boundary

-+ and ap(r,z)

ar

-+ = a~(r,z) = az 0 at the center.

The problem is to synthesize the flux of the form

-+ + ~(r,z) = H{r)a(z) • (6)

If K is- the number of trial functions, a( z) has as its ele­

ments [a]k(z)/k = 1,2, ••• K and each [a]k(z) has as its

elements [a].(z)/j = 1,2, .• ,J. H(r) is a row matrix having J

as its elements [H]k(r)/k = 1,2, ••• K, and each [[Jk(r) is a

matrix having as its elements [H] .. (r-) 6 ..• l.J l.J

-+ In order to determine the expansion coefficients, a(z),

Eq. (6) is substituted into Eq. (5), the governing equation.

*T *T Equation (5) is then multiplied by H (r), where H (r) is

the transpose of the matrix obtained when the adjoint func­

tion is used instead of its corresponding trial function.

The result is integrated over r. If this is done, the follow-

ing equation will result:

_ D • d 2 a ( z) + [A' + ( D' B 2 ) ] Z: ( z) = 1 I>.. M':; ( z) dz2 r

( 7)

ll

and A', (D'Br2 ), and M' are similarly defined.

In order to obtain Eq. (7) it is assumed that the flux obeys

h H lmh l · Th n1 a a · 1 b t e e 0 tz equat~on. e -r ar r ir term ~s rep aced y

DB 2 • Equation (7) can also be derived from a variational - -r

principle [12, 13].

In the development of Eq. (7), H*(r) was used as the

weighting function. The choice of the weighting function is

not necessarily limited to the adjoint function. If the

exact solution is. given by Eq. (6), Eq. (7) can be obtained no

matter what weighting functions are used. An interesting

method is to use H(r) as the weighting function. This method

has the advantage that the necessity of calculating the

adjoint function is eliminated. Hereafter, this method will

be referred to as "Galerkin's Method."

Methods similar to the ones described above have proved

themselves to be very useful in reactor analysis. Synthesis

methods have been used to construct three-dimensional power

shapes [14-16]. Very good results have been obtained for

this application. The flux synthesis technique has also been

used in burnup studies (17].

Another interesting synthesis technique is the Natural

Mode Approximation, (NMA). This technique is similar to the

one developed in this thesis except that the Natural Mode

Approximation deals with reactor kinetics problems. The NMA

12

is based on a modal expansion technique where the time-

dependent and space-dependent variables are approximated by a

series of products of time-dependent expansion coefficients

and space-dependent expansion modes. The space-dependent

modes are eigenvectors of the steady-state diffusion equa-

tion. The details of this method are not given here; however,

the reader can find a more complete discussion in a paper by

Foulke and Gyftopoulos [18]. NMA is similar to the other

techniques described in this section. One difference is that

instead of assuming that the spatial variables are separable,

it is assumed that the time variable can be separated from

the spatial variables. The major difference between NMA and

other synthesis approximations lies in the choice of trial

functions. The normal methods use as trial functions the

solutions for di££erent regions of the reactor. However, NMA

uses as trial functions the fundamental and higher eigenfunc­

. tions of the equation which approximates the equation to be ~

solved. NMA has been applied successfully to the calculation

and interpretation of reactor kinetics experiments.

13

,

III. DISCUSSION

A. The Multigroup Approximation

The form of Eq. (1) which will be discussed is the two-

dimensional, multi~group, multi-region diffusion equation.

The coordinate ·systems used are the cartesian and cylindrical

systems (x-y and r-z). The multigroup approximation of Eq.

(1) may be made by multiplying by dE and integrating from E. l.

to E. 1 . If this is done Eq. (1) becomes, for the i-th ].-

group,

where

•

- ~ JL [Di(r,z)raa~i(r,z)] ra ar ar

a az

[Di(r,z)ati<r,z)] az

i i Ng • • • t tJ' 1 (r z)AJ(r z) + trem (r,z)~ (r,z) = s , .,. ,

j=l j:t i

JE. 1 l.- 0

= ~(E,r,z)dE

Ei.

IE. 1 . l.-

Ei trem(Elr,z)p(E,r,z)dE l.

~ (r,z)

IE. 11E. 1 . l.- J-

E . E. E (E'+E,r,z)~(E',r,z)dE'dE l. J s

( 8)

14

and

ti, fi, and Di are defined in a similar manner.

The exponent, a, has the value 0 for cartesian coordi-

nates and 1 for cylindrical coordinates.

When the entire energy spectrum is considered, a set of

coupled differential equations is the result. The number of

these equations is equal to the number of energy groups being

considered. This set of equations may be represented in

matrix form by defining

;cr,z) = Col[. 1 (r,z), • 2 (r,z), ... , •Ng(r,z}]

where Ng is the number of groups [19]. The resulting equa­

tion is a matrix differential equation of the form

where

+ + Irem(r,z).(r,z)

+ ~ [D(r,z) ap(r,z)] az· az

+ + = r (r,z).(r,z) + viA F(r,z).(r,z} -s

(9}

i D(r,z} is a diagonal matrix with [D(r,z)] ii = D (r,z),

Irem(r,z) is a diagonal matrix with [Erem(r,z)] ii = Iirem(r,z),

~(r,z} is the scatter matrix with [Is(r,z}] ji = l:j,i(r z) s , ,

and

F(r,z) is the fission matrix with [F{r,z)]

ti<r,z) ·

i = f (r,z)

B. Formulation of the Higher Mode Synthesis Method

Equation (9), the multigroup diffusion equation, is

repeated here for easy reference. For simplicity, the form

of Eq. (9) in cartesian coordinates, a=O, is considered,

+ + - ~ [D(x y)ap(x,y)J- JL [_D(x,y) ap(x,y)]

ax - ' ax ay ay

15

(10) + + + + ~rem(x,y)~(x,y) = ~(x,y)~(x,y) +vi>. F(x,y)~(x,y) •

The validity of HMS depends upon the amount of error

introduced when the following approximation ·is made

+ ~(x,y) = ( 11)

where the ek(x) are the fundamental (k=l) and higher eigen­

functions (k=2,3,4, •.• ,K) which satisfy the one-dimensional

form of Eq. (10). The expansio~ Eq. (11). is substituted into

Eq. (10). Each side of Eq. {10) is pre-multiplied by l*T(x), . n

h •• , ) ( . • where T denotes t e transpose and the en x the adJ01nt

flux) are the fundamental and higher eigenfunctions obtained

by using the adjoint operator in the one-dimensional case of

Eq. (10). The result is then integrated over x, where x goes

from center to the outer boundary. These operations on

16

Eq. (10) will now be done. For clarity, the equation is con-

sidered term by term. D, ~rem, ~s and F are considered

independent of spatial coordinates within each region.

FIRST TERM

+ -

I< J + 1: k=l

+*T a2 + -+ e (x) D ----ek(x)dx a(y) n ax2

X

SECOND TERM

+ -

X

where [D'ln,k = J e:Tcx> QBkCx)dx

X

THIRD TERM

+ !_rem(x,y)~(x,y)

+ -

K -+ I:

k=l f e~T(x) Lrem6k(x)dx~(y) ~ L'rem~(y) X

where [t'remln,k = f X

FOURTH TERM

K -+ I:

k=l f -+a *T ( x) -+ ( ) ( ) ' + ( ) n ~sek x dxa y -+ ~ a y

X

where [£ 8 ']n,k = Je~T(x) ~8k(x)dx X

FIFTH TERM

+ vi>.. F(x,y)<P(x,y) +vi>.. F

-+ vi>..

X

where [F']n,k = f 6~T(x) F6k{x)dx·

X

These terms are substituted into Eq. (10) to give

17

- D' d 2 ~(y) + [E'rem­dy2

K E

k=l

18

X

-+ -+ E ' a ( y } + v />.. F ' a ( y} • -s (12}

d 2ak<x> The term can be approximated by finite differences

dx 2

[20]. In general, the one-dimensional Laplacian operator can

be written as

v2 e a 2 e + a ae = - ,

ar2 r ar

a2 e. ej+l - 2 e . + e. 1 but J = J J-

ar2 (~ r} 2

ae. ej+l - e. 1 and _J_ = J- ,

ar 2flr

therefore = ej+l - 2ej + ej-l + ~ <ej+l - ej_1 > •

(~r) 2 r 2~r

If this is done, the expansion coefficients can be determined

by solving Eq. (12).

At this point both the expansion coefficients and the

expansion functions are known; and by using Eq. (11), the

two-dimensional flux, l<x,y), can be calculated.

This solution represents the solution of a perturbed

system. For example, when the multigroup diffusion equation

19

is solved in one dimension for a cylinder in the radial

direction, the cylinder is considered infinite in the axial

direction. Therefore, leakage in the axial direction is

ignored. If the cylinder is not infinite, the axial leakage

represents a perturbation to the one-dimensional solution.

This situation is analogous to the solution of systems

which are time dependent. In this case, a common expansion

for a slab reactor is

~(x,t) = t n,odd

~ (t)cosB x , n n

where Bn = ~and a = thickness of the reactor [21]. a It has

been found that the higher modes are present immediately after

a perturbation, but they die out in time. This illustrates

the major difference between the time-dependent and the time­

independent expansion. The higher modes in the time-

independent case do not die out in time, because they

represent a perturbation due to tionstant leakage in the axial

direction. The leakage is due to the system not being

infinite in the axial direction and, obviously, the dimen-

sions of a given system are time independent. Thus, it can

be said that the system possesses ''steady-state transients."

If, in Eq. (11) an infinite number of terms is used,

Eq. (12) would be exact within the framework of diffusion

theory. However, in actual practice only a finite number of

terms can be used, and a truncation error is present. The

magnitude of the truncation error depends upon the conver-

gence of the series, and the convergence of the series in

Eq. (11) depends upon the degree of separability of the

spatial dependence of the eigenfunction of Eq. (10). Most

20

reactors are very complex in the horizontal plane, and a high

degree of inseparability would be expected. However, in the

vertical plane the reactor configuration is usually fairly

simple and convergence of the series in Eq. (11) would be

expected to be fairly rapid. If this criteria is used, the

trial functions (the eigenfunctions of the one-dimensional

equation) should be those which describe the flux in the

direction of highest complexity; and the synthesis should be

in the direction of least complexity. Another aspect which

should be considered is the size of the perturbation of the

one-dimensional equation, i.e., the amount o£ axial leakage.

The trial functions, ek(x), should approximate the two­

dimensional flux, ;(x,y). Therefore, the trial functions

should be calculated for the direction of largest leakage,

and the synthesis should be done in the direction of least

leakage. These are two criteria which should be used to

decide in which direction the synthesis approximation should

be made. These may be conflicting criteria, but normally

the direction of greatest complexity is also the direction of

largest leakage in a given reactor; and, when this is true,

no conflict will result.

The advantage of HMS is the reduction in the number of

computations which must be performed, thus the computation

time should be considerably less than when the more conven-

tional methods are used. Another advantage of HMS is that

21

the trial functions (solutions of the one-dimensional equa­

tion) remain unchanged even if changes of parameters are made

in the direction of synthesis. The trial functions may be

stored and used at a later time if it is desired to make

parameter changes in the synthesis direction. This would •

eliminate the need to calculate new trial functions. It

would be necessary to calculate only the expansion coeffi-

cients and do the summation of Eq. (11), which would reduce

the required computer time to a small fraction of the time

that would otherwise be needed.

C. Results

A computer code, MUD-SYN, has been developed to test the

Higher Mode Synthesis method. The code, MUD-MO, used to

determine the trial functions, adjoint functions, and expan-

sion coefficients, is based on the Stabilized March Technique.

As mentioned previously, this code has the ability to calcu-

late higher eigenfunctions; therefore, it is very adaptable

to this application. MUD-MO is incorporated into MUD-SYN as

a subroutine.-

The capability of MUD-SYN was tested by studying three

different types of reactors. These reactors were selected

because of their various geometrical and material properties.

22

I

The main points of interest are the effective multipli-

cation constant, the flux shape, and the computation time.

Another important factor is the number of terms needed in the

series in Eq. (11) in order to obtain accurate results. The

results from MUD-SYN are compared with the results from a

production code, EXTERMINATOR-2. EXTERMINATOR-2 is a two-

dimensional code which uses an iteration method. Problem one

is relatively simple and has an analytical solution. The

results using HMS are compared directly to this analytical

solution. In problems two and three, the flux shape and

effective multiplication constant from EXTERMINATOR-2 are

considered to be "exact."

·1. PROBLEM ONE

The first problem studied is one selected from the

Benchmark Problem Book [22]. This book is a compilation of

problems of varying complexity for which analytical or very

accurate approximate solutions exist. One of the main objec­

tives of the book is to assist in evaluation of computer

programs.

The problem selected, a homogeneous, two-dimensional

slab, is one for which an an-alytical solution exists. The

reactor configuration and boundary conditions are shown in

Figure 1. Seven energy groups are used, four fast and three

thermal; and full up-scattering is treated in the thermal

23

13.5cm.

({}=Q b7.5em.

I I I -l--~------------

1 I 4><1 -1--~-------------

1

X

PROBLEM ONE CONFlGURAT ION

FIGURE 1.

24

groups. Because of the criteria discussed previously, the

synthesis was done in the x - direction. The mesh size used

is 16 x 31 (16 points in the y - direction and 31 points in

the x- direction).

Because of the simplicity of this problem, it was

expected that convergence of the series in Eq. (11) would be

fairly rapid and truncation error would be small. This

proved to be true, and it is demonstrated by the relative

size of the expansion coefficients. The first three expan­

sion coefficients for various points in the x - direction are

shown in Table I. For each point the third expansion coeffi­

cient is approximately 0.0001 times the first expansion

coefficient. This suggests that good results could be

obtained by truncating the series in Eq. (11) after only one

or two terms. This is done and the results are summarized in

Table II. Excellent agreement with the analytical solution

is obtained by using only two terms in the series in Eq. (11).

The calculation time as compared with that of EXTERMINATOR-2

is very low.

The relative flux shape as calculated by MUD-SYN is in

good agreement with the flux shape as calculated by EXTERMI­

NATOR-2. This is shown by Figures 2, 3, and 4. Figure 2

shows the group 1 flux shape in the y direction for x = O.Ocm. Figure 3 shows the group 1 flux shape in the y -

direction for x = 3l.Scm. Figure 4 shows the group 1 flux

shape in the x - direction for y = 0.9cm.

'

25

TABLE I

RELATIVE SIZE OF EXPANSION COEFFICIENTS

Distance From Center Expansion Coefficients of Reactor

(em.) First Second Third

X 10-3 X 10-4

0.0 0.99999 0.19891 -0.95055

4.5 0.99452 0.18984 -0.94532

9.0 0.97815 0.18102 -0.92991

13.5 0.95105 0 .17916 -0.90408

20.25 0.89098 0.16805 -0.84698

36.0 0.66900 0.12534 -0.63595

42.75 0.54445 0.10138 -0.51769

51.75 0.35809 0.06910 -0.34040

65.25 0.52142 0.02205 -0.04925

TABLE II

PROBLEM ONE DATA

Machine Time !5eff (min •)

Analytical Solution 0.7745

EXTERMINATOR-2

MUD-SYN

2 Terms

3 Terms

*% Difference

34.03 0.7731

5.50 0.7747

10.10 0.7747.

Keff(Analytical) - Keff : 100 X

Keff (Analytical)

26

% Difference*

0.18

0. 025

0.025

0

""" 0 • en

t­..... ..... I

~ ')­a: a: a: t­..... m a: a: -

I l/) f"""

• 0

I

X ::l ...J lL

o• lO

• 0

z £) a: ..... :::) lLJ l/) Z N

• lLJ 0 > ..... t­a: ...J lLJ a: 0

0

:::::s.......__ Fl GURE 2

FLUX SHAPE FOR X= 0.0 c.m. ~

" MUD-SYN

'"'" +-~~- EXTERMINATOR-2

. ~------------~~----------~--------~ 0 0.00 5.00 10.00 --DISTANCE FROM CENTER OF REACTOR (CENTIMETERS)

,..., "

e C)

\{') • ,....,

z II

G) X

I

r')

0

a:: ;::::)

w

0 ~

a:: lL

:::> <

.!) w

lL..

S: :c (j')

X

:::> _

J

lL

CS

!INn

l~l:J\:11!9\:Il:U

~ 0 ~

z -2:

0::: w

t-X

w

j

0 Ln .

C't')

-t

0 0 • 0 .... 0 0 •

0 0 • 0

oo·o

28

- (/') a:

UJ

1-

LLJ 2

: .... .... z LLJ u -a

: co .... u a

: ~

lL

co a

: LLJ .... z LLJ u 2

: D

a

: lL

LLJ u :z

a

: .... C

J') ...... 0

en ..... ..... !5 )-

~ t­..... ~ $

~ ...1 lL

~ 5 lLl z

!i! ...... t­a: ..J lLJ a:

0 0 • .... II I l::::t::l I

FIGURE 4

Ln ......

• 0

0 Ln •

0

l/) N •

0

0 0

FLUX SHAPE FOR Y=0.9cm.

- MUD-SYN

-IHf- EX TERM I NATOR-2

. ~----~--~~--~----~-----+-----+--~ 0 0.00 10.00 20.00 30.00 ijO.OO 50.00 60.00 DISTANCE FROM CENTER OF REACTOR (CENTIMETERS) I'V

U)

30

2. PROBLEM TWO

The second problem studied is a slab with a reflector.

The configuration is shown in Figure 5. The mesh size is

21 x 41, and the energy spectrum is divided into two groups,

one fast group and one thermal group.

This problem is not as separable as problem one and

convergence of the series in Eq. (11) is not as fast for this

problem as compared to problem one. This is illustrated by

the expansion coefficients given in Table III. The third

expansion coefficient is approximately one tenth the size of

the first expansion coefficient.

Synthesis was done in the x direction. In order to

determine the weighted cross sections 1n the upper reflector,

the reactor is divided into two zones in the x direction as

shown by Figure 5. The zone (2) trial functions are approx­

imated by the zone (1) trial functions. Problem three is

treated in a similar manner.

The computation time and value of Keff for each run are

given in Table IV. The value of Keff obtained by using HMS

is in good agreement with the value calculated by EXTERMINA­

TOR-2 while the computation time of MUD-SYN is considerably

less.

The flux shape as compared with the flux shape calcu­

lated by EXTERMINATOR-2 is illustrated. by Figures 6 and 7.

Figure 6 shows the flux shape in the y direction for x = O.Ocm. Figure 7 shows the flux shape in the x direction for

y = O.Ocm.

X

31

=z~=NE~===========R=E=F=L=E=C=T=O=R====~~~~~---s-I

ZONE .1.

FUEL REGiON

R E F L <D= 0 85un. E c T 0 R

- - ~--------~~--------------~--~ o/(/? -0 olx - I t5 c.m. \

47.792 q em.

Mesh size ·21x41

PROBLEM TWO CONFIGURATION

FIGURE 5

32

TABLE III

RELATIVE SIZE OF EXPANSION COEFFICIENTS

Distance From Center Ex12ansion Coefficients of Reactor

(em.) First Second Third

X 10-1 X 10-1

0.00 0.99999 -1.0391 0.76671

7.08 0.99240 -1.0311 0.76114

14.17 0.96969 -1.0075 0.74368

23.60 0.91650 -0.9 52 2·0 0.70298

30.69 0.86024 -0..89375 0.65993

42.50 0.73785 -0.76659 0.56613

59.03 0.51556 -0.53568 0.39577

66.11 0.40605 -0.42199 0.31182

73.19 0.29036 -0.30081 0.22252

80.28 0.17025 -0.17556 0 .130 2 8

85.00 0.08849 -0.09026 0.06769

EXTERMINATOR-2

MUD-SYN

2 Terms

3 Terms

*% Difference

33

TABLE IV

PROBLEM TWO DATA

Machine Time (min.) ~ff % Difference*

19.00

6.42

7.77

0.9548

0 • 9 56 3

0. 9 745

0.15

2.1

Keff(EXTERMINATOR-2) - Keff(MUD-SYN)_ : 100 X

Keff(EXTERMINATOR-2)

Mesh size = 21 x 41 &

,_ 0 0

(f) t-

• ... I I l<:::s -FIGURE 6 ....

z ::J

)­a: a: a: t-.... m ~ ~

X ::J ...J IJ..

z D a: t-

Ul I'

• 0

0 IJ)

• 0

::> l&J IJ) Z N

• lLI 0 > ..... t­a:

FLUX SHAPE FOR X•O.Oem.

- MUD .. SYN

.._ EXTERMJNATOR-2

irl a: 0

0 . L~:oo-30:~~ 0 0.00 to. oo .20. oo s·o. oo DISTANCE FROM CENTER OF REACTOR

llO.OO 50.00 CCENT I METERSl w

+:

"

! ... !

I .... I

~

~ ....

i

8 ..: 1'''t 1 .... Fl GURE 7

FLUX SHAPE FOR Y= O.Ocm.

Je • 0

- MUD-SYN

i .._...... EXTERMlNATOR-2

• 0

~ • 0

8 a d.oo 1'o.oo 2b.oo s'o.oo aab.oo s'O.oo &o.oo 7o.oo b =- ~

DISTANCE FIDt CENTER aF AEACTGR lCENTit£TERSl w (1'1

3. PROBLEM THREE

The third problem is a cylinder with axial and radial

reflectors. The configuration is shown in Figure 8. The

energy spectrum is divided into two groups, one thermal

group and one fast group. The mesh size is 31 x 48 (31

36

points in the r direction and 48 points in the z direction).

Synthesis is done in the axial direction.

In this case, convergence of the series in Eq. (11) is

similar to that experienced with problem two. Two, three,

and four terms are used in the expansion. The results are

summarized in Table V. Because of available storage, the

mesh size for EXTERMINATOR-2 is 31 x 32. Even though the

mesh s~ze for EXTERMINATOR-2 is much smaller, the computation

time for MUD-SYN compares favorably. The value of Keff

calculated by MUD-SYN varies only slightly f-rom the Keff

calculated by EXTERMINATOR-2.

The relative flux shape calculated by MUD-SYN and

EXTERMINATOR-2 is illustrated by Figures 9 and 10. Figure 9

shows the flux shape in the r direction for z = 21.4cm. and

Figure 10 shows the flux shape in the r direction for z = 35.7cm •.

In problems two and three it is observed that as more

terms are used in the expansion the value of Keff calculated

by MUD-SYN is converging to a value higher than the value of

Keff calculated by EXTERMINATOR-2. This error is partially

due to the way the axial reflector was treated. As mentioned

in problem two, the trial functions in the axial reflecto~

z

r

~=0

REFLECTOR

FUEL REGlON

R E F

37

L ([)-:. 0 IOOelll.

E c T 0 R

f 5 C:.J'I'I. \

5.5 c:.M.

PROBLEM THREE CONFlGURATlON

FIGURE 8

EXTERMINATOR-2

MUD-SYN

2 Terms

3 Terms

4 Terms

*% Difference

38

TABLE V

PROBLEM THREE DATA

% Mesh Machine ~ff Difference* S1ze T1me

( m1n.)

31 X 32 18.51 0.8376

31 X 48

6.5 0.8392 0.19

7.5 0. 84 30 0.64

11.13 0.8527 1. 80

Keff(EXTERMINATOR) - Keff(MUD-SYN) : 100 X Keff (EXTERMINATOR)

39

<}I 8 •

• a

: 0

~ 0

CD

~

~

N

z z

~

-0~

" >-

2:: N

(f)

~

en o

w

I w

.....

0:: 0

t-0~

w

:::> X

0

2:: a.n...,.

cr lL..

w

~

:::>

I t {!)

w

1&.1 u

lL

a.. s-

<{

I •

<.J)

gl5

~

.... ~ _

J

lL.. 8~

diB

CW

') ffi ~ 8~

~~ La..

8~ .....

ocn

......... c

8 +-------~~-------.--------~--------~ . 0

oo·o

tSliNO

~Y~Yll~

Xn1~ NQY1n3N

3AI!~13Y

"" en .... ..... ~ )-

IE a: .... ..... m i

~ ..J La..

z D a: a z

0 0 • ... 11 I 5

U') r-•

0

0 U')

• 0

U') N • LLJ 0

> .... ~ rrl a: 0

0

L..a.._

Fl GURE I 0

FLUX SHAPE FOR Z =35. 7 em.

- MUD-SYN

"*-*- EX TERM I NAT OR-2

·~----+-----~----~----~-----+----~ 0 0.00 10.00 20.00 30.00 ~0.00 50.00 60.00 DISTANCE FROM CENTER OF REACT6R U:ENTIHETERSl +=

0

41

were approximated by the trial functions in the core.

Since the flux shape in the axial reflector is not the same

as the flux shape in the core this approximation results in

convergence to a Keff slightly higher than the true value.

The rest of the error is due to error in the integration and

error in calculating the trial functions and expansion

coefficients.

42

IV. CONCLUSIONS

The Higher Mode Synthesis method for the solution of the

multigroup diffusion equation is very useful when this equa­

tion is used to describe systems of intermediate complexity.

For most problems only two or three one-dimensional eigen­

functions are needed to provide an adequate representation.

Using HMS, the flux shape and effective multiplication con­

stant can be calculated with only a small error. This small

error is not significant if the results are to be used in the

first stages of reactor design, survey studies, or similar

applications. HMS is from two to five times faster than

EXTERMINATOR-2, depending on the complexity of the problem

and the form of the scattering kernel. This savings in

computation time is very ~mportant because of the high cost

of computer time.

43

V. RECOMMENDATIONS

Three-dimensional studies of reactor systems are much

preferred to one- or two-dimensional studies. However, the

problem of obtaining three-dimensional flux and power shapes

is, at best, very difficult. Three-dimensional computer

codes, using iteration methods, have been written; but their

requirement of large amounts of computer time restricts

their use to problems of utmost importance.

Using the synthesis method tested in this thesis, three­

dimensional problems could be studied. This could be done by

substituting the expansion

into the three-dimensional diffusion equation and using the

adjoint weight and integration procedure to obtain a relation

for the ak(x).

The two-dimensional trial functions and adjoint function

could be obtained by using MUD-SYN or they could also be

obtained by using a two-dimensional code based on SMT. At

present, the latter code is not available; however, the pro-

blem is being studied.

In problem two and problem three, difficulty was exper­

ienced in obtaining proper trial functions for the reflector

in the synthesis direction. The proper method is to use

solutions of the one-dimensional diffusion equation for the

•

44

reflector. In order to do this the core eigenfunctions must

be considered as a source. This would involve the solution

of an inhomogeneous equation. A code, similar to MUD-MO,

which has the ability to solve inhomogeneous problems has

been developed [23]. This code could be incorporated into

MUD-SYN and an axial reflector could be treated more rigor­

ously.

45

BIBLIOGRAPHY

1. EDWARDS, D. R. (1963) A numerical method for eigenvalue problems. Sc. D. thesis, Massachusetts Institute of Technology, p. 112.

2. CLARK, MELVILLE, JR., and KENT F. HANSEN (1964) Numeri­cal methods of reactor analysis. Academic Press, New York, p. 168.

3. EDWARDS, D. R. and KENT F. HANSEN (1966) The stabilized march technique applied to the diffusion equa­tion. Nuclear Science and Engineering 25, p. 58-65.

4. EDWARDS, D. R. op. cit. p. 1.

5. EDWARDS, D. R. op. cit. p. 3.

6. WACHSPRESS, E. L. (1967) CURE: a generalized two-space dimension coding of the IBM 704. KAPL-1724, 132 p.

7. CADWELL, W. R., J. P. DORSEY, H. B. HENDERSON, J. M. LISKA, J. P. MANDELL, and M. C. SUGGS (1960) PDQ-3: a program for the solution of the neutron diffusion equation in two-dimensions on the IBM 704. WAPD-TM-179, 56 p.

8. BRINKLEY, F. W. and C. B. MILLS (1959) A one-dimensional intermediate reactor computing program. LA-2161, 45 p.

9. FOWLER, T. B., M. L. TOBIAS and D. R. VONDY (1967) EXTERMINATOR-2: A Fortran IV code for solving multigroup neutron diffusion equations in two dimensions. ORNL-4078, 77 p.

10. WACHSPRESS, E. L. (1960) A numerical technique for solv­ing group diffusion -equations. Nuclear Science and Engineering 8, p. 164-170.

11. EDWARDS, D. R. op. cit. p. 4.

12. KAPLAN, S. (1961) Some new methods of flux synthesis. Nuclear Science and Engineering 13, p. 22.

13. SELENGUT,D. S. (1958) Variational analysis of multi­dimensional systems. HW-59126, p. 89-124.

46

14. HORST, R. B. (1958) Synthesis methods in R-Z geometry. Bettis Technical Review, WAPD-BT-8, p. 26-38.

15. BALL, T. W. (1958) Synthesis of lifetime power distri­butions in R-Z geometry. Bettis Technical Review, WAPD-BT-8, p. 39-45.

16. LORENTZ, W. N. (1958) Synthesis of three-dimensional power shapes-application of the flux-weighted synthesis technique. Bettis Technical Review, WAPD-BT-8, p. 46-52.

17. CANNON, G. A., E. A. COLBETH and F. M. OLSEN, SYBURN--A synthesized two-dimensional Pl or DSN burnup code. Transactions American Nuclear Society 4, No. 2, p. 253-254.

18. FOULKE, L. R. and E. P. GYFTOPOULOS (1967) Application of the natural modal approximation to space­time reactor problems. Nuclear Science and Engineering 30, p. 419-435.

19. EDWARDS, D. R. op. cit. P· 74.

20. EDWARDS, D. R. op. cit. P· 122.

21. LAMARSH, J. R. ( 19 66) Introduction to nuclear reactor theory. Addison-Wesley, Reading, Massachusetts, P· 429.

22. BLAINE, R., R. FROEHLICH, H. GREENSPAN, K. LATHROP, W. TURNER, D. VONDY, and E. WACHSPRESS (1968) Argonne code center: Benchmark problem book. ANL-7416, 73 p.

23. LUCEY, J. W. and KENT F. HANSEN (1968) Numerical solu­tions of boundary value problems. Nuclear Science and Engineering 33, p. 327-335.

47

APPENDIX A

The MUD-SYN Code

The MUD-SYN code solves the multigroup diffusion equa­

tion for a slab and for a cylinder in the r-z plane. It was

written to prove that the Higher Mode Synthesis method is

valid. Thus, it is a proof of principle code and not a pro­

duction code, even though it was compared to a production

code. The size of the problem is restricted to 10 energy

groups and 50 mesh points in each direction. This limitation

is not due to theoretical aspects, but is due to available

storage capacity.

The MUD-SYN code is divided into subroutines. Subrou­

tine MUDMOD calculates the trial functions, adjoint functions,

and expansion coefficients. Several other subroutines are

included in MUDMOD, but their functions are not significant

to this application. Subroutine VOLINT does the integration

by Simpson's rule. Subroutine TWOSYN calculates the two­

dimensional flux by combining the expansion coefficients and

trial functions in a manner dictated by Eq. (11). It also

prints the two-dimensional flux.

The input required for MUD-SYN is described below. The

first card is a title card with any characters permissible

between columns 2 and 72. All the fixed point data is

punched using a 20I3 format. Some of the variables are

relevant only to the MUD-MO code. In these cases, recommen­

ded values will be given. The fixed point data is:

NREG

NFCT

NOG

NPTS

KCOND

I FOUND

IFUNCT

INDEXM

NUMAX

KPOW

MORE

KAPP

KBCI

KBCO

This is the number of regions in the direction

in which the trial functions are to be ca1cu-

1ated, ~ 3.

48

This is a dummy variable equal to the number of

groups.

This is the number of groups, s 10.

This is the number of mesh points in the direc­

tion in which the trial functions are to be

calculated, .s so. The recommended value is s. If IFOUND equals 1, search for K-eff; 2, K-eff

is read in.

The recommended value . 2. l.S

The recommended value is 120.

The recommended value . 1. l.S

The recommended value . 1. l.S

The recommended value • 1. l.S

The recommended value • o. l.S

This is the inner boundary conditions. If KBCI

is 1, the flux goes to zero at the zeroth space

point. If KBCI is 2, the derivative of the flux

goes to zero at the zeroth space point.

This is the outer boundary condition. If KBCO

is 1, the flux goes to zero at the outer space

point. If KBCO is 2, the flux at the outer

. space point·is governed by the relation,

•

KREG

KCHO

KHUNT

49

y~npts-1 + d~'npts-1 = 0 where d is the extra-

polation distance andy is 1 or 0, depending on

how the coordinate system is established.

This is the region in which the criticality

search takes place.

The recommended value is 2.

If KHUNT is 1, no search for K-eff will be done.

If KHUNT is 2, a search for K-eff will be done.

IA This variable has the value 0 for a slab and 1

Kl

for a

If Kl

cylinder. -1

is 1, Keff' radius, EPA, EPB, and DEL are

read in. If Kl is 2, these variables are not

read in.

K2 If K2 is 2, the expansion coefficients and trial

functions are punched on cards. If K2 is 3,

these variables are printed.

K3 The recommended value is 0.

KG The recommended value is l.

KGA This denotes the group in which the trial func-

tions are normalized.

KNA This is the space point where the trial func-

MODE

KNORM

tiona are normalized.

This is the number of terms used in the expansion

in Eq. (11).

If KNORM is 1, the flux in the fuel region is

divided by the reciprocal of the fission cross

NPTZ

NREGZ

KNR

50

section. If KNORM is 2, the flux is not normal­

ized as stated above.

This is the number of points in the synthesis

direction.

This is the number of regions in the synthesis

direction, ~ 2.

This is the mesh point in the direction perpen­

dicular to the synthesis direction where the

flux is to be normalized.

KNZ This is the mesh point in the synthesis direc-

tion where the flux is to be normalized.

KNG This is the group for which the flux is normal­

ized.

MPOW

KGEL

If MPOW is 2, the power shape will be calculated.

If MPOW is 1, the power shape will not be calcu­

lated.

If KGEL is 1, the adjoint functions are used as

the weighting functions. If KGEL is 2, the

trial functions are used as the weighting func­

tions and the adjoint functions are not calcu­

lated.

INDZ(I) This is the mesh point at the outer boundary of

each region in the direction of synthesis.

There are NREGZ values and INDZ(NREGZ) = NPTZ

+ 1-KBCO. INDZ(I) is given on a separate card

from the rest of the fixed point data.

51

· IND(I) This is the mesh point at the outer boundary of

each region in the direction perpendicular to

the direction of synthesis. There are NREG

values and IND(NREG) = NPTS + 1 - KBCO. IND(I)

is also given on a separate card.

The floating point data is read on a lSX 4El5.8 format.

The first 15 spaces are reserved for card identification.

The subscripted data is read as (((S(I,J,K),I=l,NI),J=l,NJ),

k=l,NK). Each subscripted variable is begun on a new card.

RRO and RZO are on the same card and EPA, APB, and DEL are

on the same card.

RRO This is the space point corresponding to the

first mesh point in the direction perpendicular

to the direction of synthesis.

RZO This is the space point corresponding to the

first mesh point in the direction of synthesis.

RRO and RZO are normally input as 0.0.

The following data is not read if Kl=2.

RAD(I) This is the thickness of each region in the

direction perpendicular to the direction of

synthesis. There are NREG values.

CEFT(I) This is the reciprocal of the guess of Keff in

each region. For a reflector region the value

is usually input as 1.0. There are NREG values.

GNU(!) This is the number of neutrons per fission in

each region. There are NREG values.

EPA

EPB

DEL

This is the criteria for convergence of Keff•

It is recommended that EPB = EPA.

This is the increment of Keff before the true

value is located.

52

At this point there are NOG + 2 cards which are used for

problem identification. Any characters are permissible in

columns 2 through 72. The cross section data is the next

input.

SSIG(I,J,K) This is the scatter cross section. There are

RSIG(I,J)

FSIG(I,J)

F(I,J)

BUCK(I,J)

D(I,J)

GAM( I)

EXTP(I)

NOG X NOG X NREG values.

This is the removal cross section. There are

NOG X NREG values.

This is the fission cross section. There are

NOG X NREG values.

This is the fission yield. There are NOG X NREG

values.

The recommended value is 0.0. There are NOG X

NREG values.

This is the diffusion c;:qcffi:~.~·;~. There are

NOG X NREG values.

This is the y for the outer boundary condition.

There are NOG values.

This is the extrapolation distance for the outer

boundary condition. There are NO~ values. If

KBCO is 1, GAM(I) and EX!'rt~!l:fiY be left blank.

53

APPENDIX B

Listing and Sample Input and Output for the MUD-SYN Code

In the following pages a Fortran listing of the MUD-SYN

code and sample input and output are given. The sample input

and output are for problem three with three terms used in the

expansion.

54

c C 1. FORTRAN LISTING OF THE MUD-SYN CODE c c c C MUO-SYN A TWO-DIMENSIONAL, MULTIGROUP DIFFUSION THEORY CODE C USING THE HIGHER MODE SYNTHESIS METHOD c

c

DIMENSION PHI(l0,50) 1 RHOZ(3),FISIG(l0,3,3) 1 ZZC100) 1

3PHIC(20),8KWOC20t20JtfRWD(20 1 20),CURR(20 9 3),POWCl90),N(l0) 1 5DOG(2)

DIMENSION RAOR(3) COMMON IN0(3J,RA0(3J,CEfT(3J,GNU(3),SSIGC20t20t3),RSIGA(l0,10,3),

2FSIG(20,3J,ZC20,20J,WA(20J,WBC20J,GAMA(lO,lOJ,EXTPA(l0,1Q), 3PC20,20,3),T(20,20J,S(60t20JtTEMPC20,20J,TEMPA(20 1 20J,RH0(3), 5PSIClO,lO,lOJ,YC20),V(20),KSOLV(20J,IHUNT,KQUIT,KPRINTtNUMBER,KCON 60tKSToR,ISOLVtiTtH,KH,IST0P,IREG,KINDM,NPTS,NREG,KREG,KIND, 7MARSTP,LOLIM,MNOEX,KBCO,K8CI,IFOUNOtlfUNCTtlNDEX,INDEXM,NUMAX, 8KPOW,MORE,NFCT,NOG,KAPP,KCHo,KHUNT,IAtKl,EPA,EPB,OEL,A,NA,NB,x, 9XA,XB,KA,K8,KC,KO,KNtXCtALPHAtBETA,DT,DT0,ISOLVM,KROSS,KENO,K2

COMMON K3 9 KG,KGA,KNA 9 BKWD,FRWD,NSCR 1 NMOOE,ACEFT,EPC,KNORMt 2PHIAC10t 50t6JtfLXClO, 50,6),CURRN(l0,3,6J,RAOZC3J,RRO,RZO,NPTZ, 31NOZ(3J,NREGZ,KNG 1 KNR,KNZ,IHOP,SSIGA(l0tlOt3)tRSIG(l0,3J,F(l0,3J, 4FSIGAC10 9 10 1 3),0A(l0,10 9 3t,D(l0,3J,GAMClO),BUCKC10,3J,EXTPClOJ, 5ZAC10 1 10J,BCEFT(2,8J 9 NREGRtNPTRtiNDRC3),RH0RC3J,MPOW,KGEL

EQUIVALENCE CPSIC1J,PHI(l)),(PS1(50l),PHIC(l)),(TEMPA(l),POW(l)J, 5(TEMPAC19l),POWC)t(TEMPAC192),POWAVG),ITEMPAC193),POWCAV), 6(TEMPAC194l,XTJ 1 CTEMPAC195),XTA),(PSI(52l)tCURRCl))

IHOP•l DEFINE FILE 5C51,500,U,IFFILEJ

C CALCULATE TRIAL FUNCTIONS AND WEIGHTING FUNCTIONS c

CALL MUOMOO c C CALCULATE PSEUDO CROSS SECTIONS c

10 CONTINUE CALL CALCRS ItA• lA 00 69 l•l,NMODE DO 69 J•ltNMOOE

69 ZA(I,JI•O.O DO 103 l•ltNMODE WACIJ•l.O

103 ZA(I,IJ•l.O NA•NMODE NB•NMODE NC•NMODE KAPP•-1 IHOP•2 CEFTil J•BCEiif(;l, 1) NREGfl1*HRH ~ • :., • · ~TR-NPTS

c

00 3 1=1 ,NREGR RAOR (I )=RAO( IJ RHO R ( I J =RHO ( I J

3 INOR(IJ=INO(l) INOEX=l NPTS=NPT l NREG=NREGZ 00 2 I=l,NREGZ RAO( I )=RAOZC I J

2 INO (I)= I NOZ C I) IA=O 00 4 1=1 ,NA DO 4 J=l,NA

4 ZCI,J)=O.O DO 5 1=1 ,NA

5 Z(I,IJ=l.O DEL=O.OOt

55

C CALCULATE EXPANSION COEFFICIENTS c

c

CAll MUOMOO I A= II A

C CALCULATE 2-0 EIGENFUNCTION c

c

CALL TWOSYN RETURN END SUBROUTINE MUOMOO

C MUO-MO A ONE-DIMENSIONAL, MULTIGROUP DIFFUSION THEORY CODE C USING THE MARCH-OUT METHOD c

INTEGER SKIP DIMENSION PHIC10,50J,RHOZC3J,FISIG(l0t3t3J,ZZClOOJ,

3PHIC(20J,BKW0(20 1 20J,FRW0(20,20J,CURR(20,3),POWC190J,N(l0), 5DOG(2)

COMMON IN0(31,RADC3J,CEFT(3J,GNUC3J,SSIG(20t20t3J,RSIGAC10,10,3J, 2FSIGI20,3J,ZI20,20J,WAC20),W8(20),GAMAC10tlO),EXTPAClOtlOJ, 3P(20,Z0 1 3) 1 T(20t20JtSC60,20J,TEMPC20,20),TEMPAC20,20J,RHOC3J, 5PSIClO,lO,lOJ,YC20),V(20),KSOLVI20),1HUNTtKOUIT,KPRINT,NUMBER,KCON 6D,KSTOR,ISOLV,IT,H,KH,ISTOP,IREG,KINOM,NPTS,NREG,KREG,KINO, 7MARSTP,LOLIMtMNOEX,KBCO,KBCI,IFOUNO,IFUNCT,INOEX,INOEXH,NUMAX, 8KPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNT,IAtK1tEPA,EPB,OEL,A,NA,NB,X, 9XA,XB,KA,KB,KC,KD,KN,XCtALPHA,BETA,DT,DTO,ISOLVM,KROSS,KENO,K2

COMMON K3,KGtKGAtKNA,BKWO,FRWO,NSCR,NMOOE,ACEFT,EPC,KNORM, 2PHIA110, 50 1 6J,FLX(l0, 50,6),CURRNC10t3t6l,RAOZ(3),RRO,RZO,NPTZ, 3INOZC3J,NREGZ,KNG,KNR,KNZ 1 IHOP,SSIGAC10,10,3),RSIGC10,3J,FC10t3)t 4FSIGAC10t10t3J;OAC10tl0t3I,OC10,3J,GAMClOJ,BUCKCl0,3),EXTPClOJ, 5ZAClO,lOJ,BCEFTC2,8J,NREGR 9 NPTR,INORC3JtRHORl3J,MPOW,KGEL

EQUJ VALEMCE'.' (PSI C 1), pHI ( U J, C P St C 501 J, PH ICC 1 I J, l TE MPAC 1 It POW C 1 J I, 5(T!MPAC191J•POWCJ 9 (TEMPAI192J,POWAVGJ,CTEMPAC193J,POWCAV), 61tEMPA1194,.MIIflliMPAC195JtXTAJtiPSIC52lt,CURRilJ) .. sea a · 1'..-, ··,... Ill • ,., 1. t ;'(h tb

JUMP•l

c

SK IP=2 MMM=l GO TO (5,10J,IHOP

5 SKIP=1

C CALCULATE TRIAL FUNCTIONS c

c

10 CALL MUOONECSKIP) IFCKQUITJ 20,20,60

20 CALL MUOTWO 28 SK IP=2

IF(NUMBER-NUMAXJ 30,601 60 30 IFOUNO=l

GO TO (40,50),KCHO 40 RAOCKREGJ=RAO(KREGJ+OEL

GO TO 10 50 CEFTCKREGJ=CEFTCKREGJ+DEL

GO TO 10 60 GO TO (70,5),MORE 10 DO 1 3 I= 1 , 4

GO TO (13,169),JUMP 13 N(l)=O

KK=NMODE+1 DO 73 J=KK,4

73 N(J)=l JJJ=NPTS-2

169 KONTE=l DO 12 1=2,JJJ IF (PHI(l,I+l)*PHI(1,IJ.LT.O.OJ KONTE=KONTE+l IF (KAPPJ 71,12,12

12 CONTINUE DO 31 I=1,NOG

31 FLX(I,1,KONTEJ=PHIC(I) NN=NPTS+1 DO 4 J=2,NN DO 4 1=1 ,NOG

4 FLXCI,J,KONTEJ=PHI(I,J-1) 00 99 I= 1,NOG 00 99 J=l,NREG

99 CURRNCI,J,KONTEJ=CURR(I,J) GO TO C51,52),KGEL

51 CONTINUE

C TRANSPOSE MULTIGROUP OPERATOR c

DO 41 I= l,NREG DO 41 Jel,NOG EXTPACJ,I)=FSIGCJ,I)

41 GAMA(J,IJ•FCJ,It 00 42 I= ltNREG DO 42 J•l,NOG F(J,fJ•EXTPACd,I)

42 fS IGi:J'fl J•GAMA( J:tifJ · DO ~· K*ltNREG

56

c

DO 6 I=l,NOG DO 6 J=l,NOG

6 TEMP(I,JJ=SSIG(ItJtK) DO 7 l=l,NOG 00 1 J=ltNOG

7 SSIG(I,J,K)=TEMP(J,IJ

57

C CALCULATE ADJOINT FUNCTION c

CALL MUOONECSKIPJ CALL MUDTWO 00 32 I=l,NOG

32 PHIACI,l,KONTEJ=PHICCIJ DO 8 J=2,NN DO 8 1=1,NOG

8 PHIACI,J,KONTEJ=PHICI,J-1J DO 2 2 K= 1 , NR E G DO 21 l=l,NOG DO 21 J=1,NOG

21 TEMPCI,JJ=SSIGCI,J,KJ DO 22 I=l,NOG DO 22 J=l,NOG

22 SSIGCI,J,KJ=TEMPCJ,IJ DO 43 I= l,NREG DO 43 J=1,NOG EXTPA(J,IJ=FSIGCJ,IJ

43 GAMACJ,IJ=f(J,IJ 00 44 1=1,NREG DO 44 J=l,NOG f(J,I)aEXTPACJ,IJ

44 FSIG(J,IJ=GAMA(J,IJ 52 CONTINUE

BCEFTCltKONTEJ=CEfTClJ NCKONTEJ=1 IF CNC1J.EQ.1.ANO.NC2J.EQ.O.AND.N(3J.EQ.O.ANO.NC4J.EQ.OJ GO TO 1 If (N(1J.EQ.O.ANO.NC2J.EQ.1.AND.N(3J.EQ.Q.ANO.N(4J.EQ.OJ GO TO 2 IF CNC1J.EQ.O.ANO.NC2J.EQ.O.ANO.N(3J.EQ.1.AND.NC4J.EQ.OJ GO TO 2 IF CNC1J.EQ.O.ANOeNC2J.EQ.O.ANO.N(3J.EQ.O.ANO.NC4).EQ.1J GO TO 2 IF CNC1J.EQ.1.ANO.NC2J.EQ.1.AND.N(3J.EQ.O.AND.NC4J.EQ.OJ GO TO 1 IF CNC1J.EQ.1.AND.NC2J.EQ.O.ANO.N(3J.EQ.1.ANO.NC4J.EQ.OJ GO TO 2 IF CNC1J.EQ.1.ANO.NC2J.EQ.O.ANO.N(3J.EQ.O.ANO.NC4J.EQ.1J GO TO 1 IF CNCl).EQ.O.ANO.N(2J.EQ.1.ANO.NC3J.EQ.1.AND.NC4J.EQ.OJ GO TO 2 If CNC1J.EQ.O.ANO.NC2J.EQ.l.ANO.NC3J.EQ.O.ANO.N(4J.EQ.l) GO TO 2 IF CNC1J.EQ•O•AND.NC2J.EQ.O.ANO.N(3J.EQ.1.AND.NC4J.EQ.1J GO TO 2 IF CNC1J.EQ.O.ANO.N(2J.EQ.1.ANO.NC3J•EQ.l.AND.NC4J.EQ.lJ GO TO 2 IF CNC1J.EQ.1.ANO.NC2J.EQ.O.ANO.N(3J.EQ.1.ANO.NC4J.EQ.lJ GO TO 1 IF CNC1J.EQ.l.ANO.NC2J.EQ.1.ANO.N(3J.EQ.1.AND.NC4J.EQ.OJ GO TO 1 IF CNC1J.EQ.1.AND.N(2).EQ.l.ANO•NC3J.EQ.O.AND.NI4J.EQ.1J GO TO 1 IF (NI1J.EQ.1.AND.NC2J.EQ.l.ANO.NC3J.EQ.1.AND.NC4J.EQ.lJ MMM=2

1 ACEFT=ACEFT*2.5 IF (KONTi.EG.3J ACEFT=ACEFT*0.59 IF fKONTE.EQ.2J DEL=0.1 1~ U~ONtE.EQ.3) DEt.•l.O

;.:; '5 &Q N'[:8 f liJJ l¥ 2 ACEFT•ACEFT/2•0

c

GO TO 172 3 ACEFT=ACEFT/1.5

172 CEFT(lJ=ACEFT JUMP=2. If (KONTE.GT.NMODEJ CEFT(l)=CEFT(1)/3.0 IFOUND=1 GO TO (10,71J,MMM

71 RETURN END SUBROUTINE MUOONECSKIPJ INTEGER SKIP DIMENSION PHIC10,50),RHOZC3J,FISIG(l0,3,3J,ZZ(100J,

3PHICC20J,BKWOC20,20J,FRWD(20t20J,CURR(20,3),POW(l90),N(l0), 5DOGC2)

58

COMMON INDC3J,RADC3J,CEFT(3J,GNU(3J,SSIGC20,20,3J 9 RSIGAtl0,10,3), 2FSIGC20,3J,ZC20,20J,WA(20JtW8(20J,GAMA(lO,lOJ,EXTPA(lO,lOJ, 3P(20,20,3J,TC20,20J,S(60,20J,TEMPC20t20J,TEMPAC20t20J,RH0(3J, 5PSIClO,lO,lOJ,Y(20J,V(20l,KSOLVC20J,IHUNT,KQUIT,KPRINT,NUMBER,KCON 6D,KSTOR,ISOLV,IT,H,KHtiSTOP,IREG,KINOM,NPTS,NREG,KREG,KIND, 7MARSTP,LOLIM,MNDEX,KBCO,KBCI,IFOUND,IFUNCTtiNDEXtiNDEXM,NuMAX, 8KPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNT,IA,Kl,EPA,EPB,OEL,A,NA,NB,X, 9XA,XB,KA,K8,KC,KD,KNtXC,ALPHA,BETA,DT,DTO,ISOLVM,KROSS,KEND,K2

COMMON K3 1 KG,KGA,KNA 9 8KWO,FRWO,NSCR,NMOOEtACEFT,EPC,KN0RM, 2PHIAC10, 50 9 6J,FLX(l0, 50,6),CURRN(l0,3,6),RAOZ(3J,RRO,RZO,NPTZt 31NOZ(3J,NREGZ,KNG,KNRtKNZ,IHOP,SSIGAClO,l0,3J,RSIGCl0,3J,FCl0,3), 4FSIGA(l0,10,3J,OAll0,10,3),DC10t3J,GAMClOJ,BUCKC10,3J,EXTPClOJ, 5ZAClO,lOJ,BCEFTC2,8J,NREGR,NPTR,INOR(3J,RHORC3J,MPOW,KGEl

EQUIVALENCE (PSI(lJ,PHIClJ),(P$1(501J,PHICCl)J,CTEMPAClJ,pQWClJJ, 5(TEMPA(l91) 9 POWCJ,CTEMPA(l92J,POWAVGJ,(TEMPA(l93),P0WCAV), 6CTEMPAC194) 9 XTJ,(TEMPA(l95J,XTAJ,CPSIC521J,CURR(l)J

GO TO (10tl5),SKIP 10 CALL DATA

C SET INITIAL PARAMETERS c

INDEX= 1 NUMBER:: 1

11 ACEFT=CEFTCl) 15 IHUNT=-1 26 KQUIT=O

KPRINT=l KCOND=KSTOR

20 ISOLV=O REWIND NSCR IT=-1 H=l. KH=l I STOP= 1 IREG=l KINDM=NPTS

30 If C NR EG•l I , 40 t 40, 50 40 K I NOaffPt S .. _, ·: ...

GO TO 60v ; ? x 1 'i> t , i. ~ · , .:'~·~ ~so K ffl~ttt8tlli.Ot: ltiM.t ~::.o ~ 4 , ,, ~ ·,, ··.!. 0 ::;n TO .\ l!Hh ~.!:Clr l~Uf~ l .

c

60 CALL SET 70 IF(KQUIT) 80,80,430 80 CALL START 90 IF(KQUITt 100 9 100,430

100 CALL C0NOIT(MARSTP-1,MARSTP) 11 0 I F ( K H ) 120 , 12 0 ,17 0

C BACKWARD MARCH c

c

120 IF(KQUIT) 130,130,150 130 IF(JSTOP) 400,400,140 140 LOLIM=3

MARSTP=MNOEX-ISTOP+2 GO TO 260

C CONDIT FAILED RESET FREQUENCY OF CONDITIONING c

c

150 KCOND=KCON0-1 KQUIT=O

160 IF(KCON0-1) 165,165,20 165 CAll YEGAOS( 3,KSTOR,OJ

GO TO 430

C FORWARD MARCH c

c

170 IF(KQUITJ 200,200,180 180 MARSTP=MARSTP-1

KQUIT=O MNOEX=HNDEX-1

190 IFIMARSTP-2J 195t195,100 195 CAll YEGAOSC4,MNDEX,OJ

GO TO 430

C NUMBER OF STEPS TO NEXT CONDITIONING c

200 IFIITJ 220,210,210 210 LOliM=3

MARSTP=3+K8CO IT=l GO TO 260

220 LOLIM=3 MARSTP=2+KCONO

230 IFIMNOEX+MARSTP+KBCO-NPTS) 260,250,240 240 MARSTP=NPTS-MNOEX-KBCO 250 IT=O 260 CALL MARCH(lJ 27 0 IF ( I T J 100, 10 0, 2 80 280 GO TO C290,310),1FOUND 290 CALL EVAl(HARSTP-KBCO+lJ

CALL ZERO CALL OUTPUT C 1 J I FOUND• I FOUND

300 GO TO' I 305,118l, IFOUNO 305 IF( INOEX-lNDEXNt 20,430.,It30 310 GO TO (380t3201t1Ft1NCT

59

c c c

c

320 330 340

350 360 370 380 390 395

396

397

400 430 440

START BACKWARD MARCH

IT=O IF(NREG-1) 340,340,350 KINO= 1 GO TO 360 KIND:::IND(NREG-1) CALL RESTRT IFIKQUIT) 100 9 100,150 IF(INDEX-INDEXM) 390,430,430 IFCNUMBER-NUMAXJ 395 9 430,430 IFOUND=l GO TO (396,397J,KCHO RAO(KREGJ=RADIKREGJ+DEL GO TO 15 CEFT(KREGJ=CEFTfKREGJ+DEL GO TO 15 RETURN GO TO (440,10),MORE CALL EXIT END SUBROUTINE MUOTWO

CALCULATE EIGENFUNCTION

DIMENSION PHifl0,50J,RHOZ(3),FISIG(10,3,3J,ZZ(100), 3PHICC20JtBKWD(20t20J,FRW0(20 9 20J,CURR{20,3J,POWI190),N(lO), 5DOG(2J

60

COMMON IND(3J,RADf3J,CEFT(3J,GNUt3J,SSIG(20,20,3J,RSIGA(l0,10t31t 2FSIG(20,3JtZC20t20J,WA(20J,WB(20J,GAMA(10,10J,EXTPAClO,lO), 3P(20,20,3),T(20,20J,S(60,20J,TEMP(20t20),TEMPAC20,20J,RHQ{3), 5PS1(10,10 9 lOJ,YC20J,V(201,KSOLVC20),IHUNT,KQUIT,KPRINT,NUMBERtKCON 6D,KSTOR,ISOLVtiT,H,KH,ISTOP,IREG,KINDM,NPTS,NREG,KREG,KINO, 7MARSTP 1 LOLIM,MNDEX,KBCO,K8Citlf0UND,IFUNCT,INOEX,INDEXM,NUMAX, 8KPOW,MORE,NFCT 9 NOG 1 KAPP,KCHO,KHUNT,IA,Kl,EPA,EPB,OEL,A,NA,NB,X, 9XA,XB,KA,KBtKC,KO,KN,XC,ALPHA,BETA,OT,OTO,ISOLVM,KROSS,KENO,K2

COMMON K3tKG,KGA 9 KNA,BKWO,FRWO,NSCRtNMOOE,ACEFT,EPC,KNORM, 2PHIAC10, 50,6J,FLX(10t 50,6J,CURRNt10,3,61,RAOZC3J,RRO,RZOtNPTZ, 3INDZC3J,NREGZ,KNG,KNR,KNZ 9 IHOP,SSIGAC10,10,3),RSIGC10,3J,F(l0,3), 4FSIGAC10,10 9 3),0AC10tl0,3JtOC10t3J,GAMf10J,BUCKC10,3J,EXTPC10), 5l A ( 10 , 10 ) , BC EFT ( 2, 8 ) , NR E GR , N P T R, I NOR (3 J , RHO R ( 3 ) , MPOW, KGE L

EQUIVALENCE CPSIClJ,PHI(lJJ,(PSIC501J,PHICCl)),(TEMPA(l),POW(lJ), 5(TEMPA(191J,POWCJ,CTEMPAC192JtPOWAVGJ,(TEMPA(l93J,POWCAVJ, 6(TEMPA(l94J,XTJ,CTEMPA(l95J,XTAJ,(PSIC521J,CURR(l)J

CALL FUNCT CALL OUTPUT(2) GO TO (20 9 10J,KPOW

10 CALL POWER CAll OUTPUT(3J

20 RETURN END . ·· ·. · ~- t , i

SU8ROUTIM&j DATA't

61

C READS All INPUT c

c

DIMENSION PHIC10,50),RHOZI3J,FISIG(l0t3,3J,ZZ(l00), 3PHICC20),BKWD(20,20),FRW0(20,20),CURR(20,3),PQW(l90),N(l0Jt 500G(2J

COMMON INOC3J,RA0(3),CEFT(3J,GNU(3J,SSIGI20t20t3J,RSIGAC10,10,3), 2FSIGC20t3)tZl20,20J,WA(20J,WB(20J,GAMAClO,lO),EXTPA(lO,lOJ, 3P(20,2Q,3),T(20,20J,S(60,20J,TEMPC20t20)tTEMPAC20,20),RH0(3), 5PSI(l0 9 10,10J,Y(20J,V(20J,KSOLVC20),IHUNT,KQUITtKPRINT,NUMBER,KCON 60,KSTOR,ISOLV,tT,H,KHtiSTOPtiREG,KINOM,NPTS,NREG,KREG,KINO, 7MARSTP,LOLIM,MNOEX,KBCO,KBCI,IFOUNO,IFUNCT,INOEX,INOEXMtNUMAX, 8KPOW,MOREtNFCTtNOG,KAPP,KCHO,KHUNT 9 IA,Kl,EPA,EPB,OELtAtNA,NB,X, 9XA,XB,KA 9 K8 9 KC,KO,KN,XC,ALPHA,BETAtDTtOTOtiSOLVM,KR0SS,KENO,K2

COMMON K3,KG,KGA,KNA,BKWD,FRWO,NSCR 9 NMOOE,ACEFTtEPC,KNORM, 2PHIAC10, s0,61tflXClOt 50t6)tCURRN(l0,3,6J,RAolC3),RRO,RZO,NPTZ, 3INOZ(3J,NREGZ,KNG,KNR,KNZtiHOP,SSIGA(l0tl0,3J,RSIG{l0t31tF(l0,3), 4FSIGAC10tl0,3J,OAC10,10,3J,O(l0,3J,GAM(lOJ,BUCKCl0,3J,EXTP(l0), 5ZAC10 9 10J,BCEFT(2 9 8) 9 NREGRtNPTRtiNOR(3J,RH0R(3J,MPOW,KGEL

EQUIVALENCE (PSIC1),PHIClJJ,(PSI(50l),PHICCl)),(TEMPAClJ,POW(lJJ, 5CTEMPA(19l),P0WCJ,(TEMPA(192J,POWAVGJ,(TEMPA(l93),POWCAV), 6(TEMPA(l94J,XTJ,(TEMPA(195JtXTAJ,CPSif521J,CURR(l))

READ C1,270) CPOW(I),I=1,18J W R I T E ( 3 , 30 0 J ( POW ( I J , I= 1 , 18 ) READ (1 9 280) NREG 9 NFCT,NOGtNPTS,KC0NO,IFOUNO,IFUNCT,INOEXM,NUMAX,

2KPOW,MORE,KAPP 9 KBCI,KBCO,KREG,KCHO,KHUNT,IA,KltK2,K3,KG,KGA,KNA, 3NMOOE,KNORM,NPTZ 9 NREGZ,KNR,KNZ,KNG,MPOW,KGEL

READ (1 9 280) (INDZ(IJ,I=ltNREGZJ KSTOR=KCOND READ (1 9 280) CINO(I),I=l,NREGJ READ (1,290) RRO,RZO READ (1,290) (RAOZ(IJ,I=l,NREGZJ

10 GO TO (20,30J,K1 20 READ (1,290) (RAOCIJ,I=1,NREG)

READ (1,290) CCEFTCIJ,I=1,NREG) READ (1,290) (GNU(I),I=1,NREGJ READ (1,290) EPAtEPB,DEL,EPC IF ( EPC) 25, 25, 30

25 EPC=O.l*EPA 30 A=IA 40 IFCKAPPJ 50,60,70 50 NA=NFCT

NB=NFCT GO TO 80

60 NA•NOG NB=NOG GO TO 80

70 NA•NOG NB=NFCT

C COMMENT CARDS c

80 READ ( 1, 270 J: f f Y ( I J , I •1, 18) WRITE (3,310) (Y(IJ,I•ltl8J ll!Ailuf1i214Jt t~:u·tt~tl•l,l8J WRITE C3tJiOt lYCittl•l•l8t

c

WRITE (3 9 330) 90 DO 100 J=l,NOG

READ (1,270) (Y(IJ,I=1,18J 100 WRITE (3,320) (V(JJ,I=lt18) 110 IF(KAPP) 120,180,180

C VARIATION I c

c

120 READ {1,290) (((SSIGA(I,J,KJ,I=l,NAJ,J=1,NAJ,K=1,NREGJ READ (1,290) (((RSIGA(ItJtKJ,I=l,NAJ,J=l,NAJ,K=l,NREGJ READ (1,290) (((FSIGACI,J,KJ,I=l,NAJ,J=l,NAJ,K=ltNREGJ READ flt290J ((FSIG(l,JJ,I=l,NOG),J=l,NREGJ READ (1,290) ((fDA(I,J,K),J=ltNA),J=ltNA),K=l,NREGJ READ (1 9 290) ((ZA(I,JJ,I=1,NOGJ,J=1,NAJ READ (1,290) fWA(JJ,I=l,NOGJ READ (1,290) (fGAMA(I,JJ,I=1,NAJ,J=l,NAJ READ (1,290) ((EXTPACI,JJ,I=1,NAJ,J=1,NAJ

130 DO 160 I=l,NA 140 DO 150 J=l,NA 150 ZCI,JJ=O. 160 Z(I,IJ=l. 170 RETURN

C NORMAL MARCH-OUT AND VARIATION II c

c

180 READ (1,290) f(CSSIG(I,J,KJ,I=l,NAJ,J=ltNAJ,K=1,NREGJ READ (1,290) ((RSIG(I,JJ,I=l,NAJ,J=1,NREGJ READ (1,290) ((FSIG(ItJJti=l,NAJ,J=l,NREGJ READ (1,290) ((f(I,JJ,I=l,NAJ,J=l,NREGJ READ (1,290) ((8UCKfi,JJ,I=l,NAJ,J=l,NREGJ READ (1,290) ((O(I,JJtl=l,NAJ,J=ltNREGJ READ (1,290) CGAMCIJ,I=1,NAJ READ (1,290} (EXTPCIJ,I=l,NAJ

190 IFCKAPPJ 200,200t250 200 DO 230 I=l,NA 210 00 220 J=l,NA 220 ZII,JJ=O. 230 l(I,IJ=l. 240 RETURN 250 READ (1,2901 (WA(IJ,I=l,NAJ

READ (1,290) CWBCIJ,I=l,NAJ READ (1 9 290) ((l(I,JJ,I=l,NAJ,J=l,NBJ

260 RETURN 270 FORMAT ClX 18A4J 280 FORMAT (20131 290 FORMAT Cl5X 4El5.8J 300 FORMAT ClHl 18A4J 310 FORMAT ClHO 18A4J 320 FORMAT ClH 18A4J 330 FORMAT llH J

END SUBROUTINE SET

C CALCULATES P MATRIX ~\~t!-r~ ~~ .r~:.: f-1{ J;.:t~· ( ( .. ~- ~: .. _r·._.r:r_ ~-~~~<;;· -:--~:.:.-~~~-;-:- -~ ;:~-~- r; ~-;: ~ t

62

c

c

DIMENSION PHI(10,50),RHOZ(3),FISIG(l0,3,3J,ZZll00), 3PHIC(20),8KW0(20,20)tfRW0(20,20),CURR(20,3),P0W(l90J,N(l0), 5DOG(2)

63

COMMON IN0(3)tRA0(3),CEFT(3J,GNU(3),SSIG(20,20,3J,RSIGA(10,10,3), 2fSIG(2Q,3),ZC20,20),WA(20),W8(20J,GAMA(l0t10)tEXTPA(l0,10), 3P(20,20,3J,T(20,20J,S(60,20J,TEMP(20,20J,TEMPAI20,20),RH0(3), 5PS1(10,10,10),Y(20),V(20),KSOLV(20),1HUNT,KQUIT,KPRINT,NUMBER,KCON 6D,KSTOR,ISOLV,IT,H,KH,ISTOP,IREG,KINOM,NPTS,NREG,KREGtKINOt 7MARSTP,LOLIMtMNOEX,KBC0,KBCI,IFOUNO,IfUNCT,INOEX,INOEXM,NUMAX, 8KPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNT,IAtKlrEPAtEPB,OEL,A,NA,NB,X, 9XA,X8,KA,KB,KC,KD,KN,XC,ALPHA,BETA,OT,DTO,ISOLVM,KROSS,KENO,K2

COMMON K3,KG,KGA,KNA,BKWOtFRWOtNSCR,NMOOE,ACEFT,EPC,KNORM, 2PHIA(lO, 50,6),fLXC10, 50,6J,CURRNC10,3,6),RAOZ(3JtRROtRZO•NPTZ, 3INOZC3)tNREGZtKNG,KNR,KNZ,IHOP,SSIGA(l0,10,31,RSIG(l0,3),F(l0,3), 4fSIGACl0,10,3),0A(10,10,3),0f10,3JtGAMfl0)tBUCK(10,3J,EXTPilOJ, 5ZAC10,10) 9 8CEFTf2,8J,NREGR,NPTR,INOR(3J,RHOR(3J,MPOW,KGEL

EQUIVALENCE (PSI(l),PHI(1) Jr(PS1(501J,PHIC(l)),CTEHPA(lJ,POW(lJ), 5(TEMPA(191J,POWC),(TEMPA(192),POWAVG),(TEMPA(l93),POWCAV)t 6(TEHPA(194)tXT),(TEMPA(l95),XTAJ,(PSI(52l),CURR(l)J

10 IF(INDEX-lJ 30,30,20 20 GO TO (30,100J,KCHO 30 RHO(l)=RAO(l)/INO(l) 40 IFCNREG-lJ 70,70,50 50 DO 60 1=2,NREG 60 RHO( I J=RAOC I)/( INO( I J-INO( 1-1) J 70 DO 90 I=l,NREG 75 lf(RHOCIJ) 80,80,90 80 CALL YEGAOS(1,I,21

GO TO 230 90 CONTINUE

100 IFCKAPPJ 110,240,240

C VARIATION I c

c c

110 DO 220 K=1,NREG

120 130

140

150 160 170

180 190 200 210 220 230

X=RHOCKJ*RHOCK) G=GNUCKJ*CEFTCKJ 00 140 I=l,NA DO 140 J=l,NA TEMPA(I,JJ=RSIGACJ,J,KJ-SSIGACI,J,KJ-G*FSIGA(I,J,K) S(I,JJ=OA(I,J,KJ CALL DETINV(1,2) IF(KQUITJ 160,160,230 DO 200 I=l,NA DO 200 J=l,NA P(I,J 9 KJ=O. DO 190 l=lr NA PCI,J 9 K.t=Pfi 9 J,KJ+T(I,LI*TEMPA(L,JI PCI,J,KJ=X*P(l,J,KJ DO 220 .l•lt.NA P(J,I,KJ=2.+PCI,I,KI RETURN ·

NORMAl MARCH-OUT AND VARIATION II

c

c c c

c

240

250 260 270 280 290 300 310 320 330 340 350

16 17

DO 340 K=l,NREG G=GNU(KJ*CEFT(KJ X=RH0(K)*RHQ(K) DO 270 1=1 9 NA DO 270 J=l 9 NA P(I,J,K)=-SSIG(ItJtK)-G*F(I,KJ*FSIG(J,KJ DO 290 l=l,NA P(ltltKJ=RSIG(ItK)+D(I,KJ*BUCK(I,KJ+P(J,I,KJ Do 320 l=l,NA DO 320 J=l 9 NA P(I,J,KJ=(X*P(I,J,K)J/D(l,KJ DO 340 I=l,NA PCI,I,KJ=2.+P(I,I,KJ RETURN FORMAT (2013) FORMAT (15X 4El5.8J END SUBROUTINE START

BEGIN MARCH

DIMENSION PHIC10,50J,RHOZ(3J,FISIGC10,3,3J,ZZClOOJ, 3PHIC(20JtBKW0(20,20J,FRW0(20,20J,CURRC20,3),POWtl90),N(lOJ, 500Gl2J

64

COMMON IN0(3J,RA0(3J,CEFT(3J,GNU(3J,SSIGC20,20,3J,RSIGAC10,10,3), 2FSIG(20t3J 9 Z(20,20J,WAC20J,WBC20),GAMAC10,10),EXTPA(l0,10), 3PC20,20,3),T(20t20),S(60t20JtTEMP(20,20),TEMPAC20,20J,RH0(3J, 5PSI(lO,lO,lOJ,Y(20J,V{20J,KSOLV(20J,IHUNT,KQUIT,KPRINTtNUMBER,KCON 6D,KSTOR,ISOLV,IT,H,KH,ISTOP,IREG,KINOM,NPTS,NREG,KREG 9 KINO, 7MARSTP,LOLIM,MNOEX,KBCO,KBCitiFOUNO,IFUNCT,INDEX,INDEXM,NUMAX, 8KPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNT,IA,Kl,EPA,EPB,OEL,A,NA,NB,X, 9XAtXB,KA,KB,KC,KD,KN,XC,ALPHA,BETA,OT,DTO,ISOLVM,KROSS,KENO,K2

COMMON K3 9 KG,KGA,KNAtBKWOtFRWO,NSCR,NMODE,ACEFT,EPC,KNORM, 2PHIAC10, 50 9 6) 9 FLX(l0, 50,6J,CURRNC10t3t6),RAOZ(3)tRRO,RZO,NPTZ, 3INOZ(3J~NREGZ,KNG,KNR,KNZ,IHOP,SSIGAClO,lQ,3),RSIGC10,3),FC10,3), 4FSIGA(l0 9 10 9 3J,OAC10tlOt3J,DC10,3J,GAM(lOJ,BUCKI10,3),EXTP(lOJ, 5ZA(lO,lOI,BCEFT(2 9 8),NREGR,NPTR,INDRC3J,RHOR(3)tMPOW,KGEL

EQUIVALENCE (PSJ(lJ,PHI(l)J,(PSl(50l),pHfC(lJJ,CTEMPA(lJ,POW(lJ), 5CTEMPAI191JtPOWCJt(TEMPAC192),P0WAVG),ITEMPA(l93J,POWCAV), 61TEMPA(l94J,XTJ,CTEMPA(l95),XTAJ,CPSI(521JtCURRClJJ

MNDEX=l 10 IFCKAPPJ 20,20,50 20 DO 40 I=l,NA 30 DO 40 J=l,NA 40 PSICI.J,lJ=Z(I,JJ

GO TO 80 50 DO 70 I=l,NA 60 00 70 J=l,NB 70 PSICI,J,lJ=WA(IJ*Z(I,JJ 80 GO TO C90t190I.KBCI

C PHifOJ • 0-C

c c c

c c c

90 X=2*MNDEX ALPHA=X/(X+A)

100 00 130 l=l 9 NA 110 00 130 J=1,N8

PSiti,J 9 2J=O• 120 DO 130 L=ltNA 130 PSI(I,J,2)=PSICI,J,2J+ALPHA*Pft 9 L,1J*PSICltJtlJ

MNDEX=2

140 150

160 170

180

190 200 210 220

230 240 250 260

270 280

290 300

X=2*MNOEX ALPHA=X/CX+AJ BETA=tX-A)/(X+AJ DO 170 l=l,NA 00 170 J=l 9 NB PSI(I,J,3J=-8ETA*PSI(I,J 9 1) DO 170 L=l,NA PSICI,J,3)=PSICI,J,3J+ALPHA*Pfi,L,lJ*PSI(L,J 9 2J MARSTP=3 RETURN

OtPHICOJJ/OR = 0

DO 220 I=l 9 NA DO 210 J=l,NA S ( I, J J =PC I, J, 1 J SCI, I J=SC I, I J-2. X= 1./ ( 2. * ( 1. +A J J DO 260 1=1,NA DO 250 J=l,NA S(I,JJ=X*S(I,JJ S ( I, I J =S C I, I J +1. CALL DET INVC 1, 3J IF(KQUITJ 280,280,330 X=2*MNOEX ALPHA=X/(X+AJ BETA=CX-A)/(X+AJ DO 320 I=l,NA 00 320 J=l,NB PSI C I,J, 2J=O.

65

310 320

00 320 l=l,NA PSI(I,J,2J=PSICI,J,2J+CALPHA*PCitltli-8ETA*TCitlJ)*PSICL,J,l)

330 MARSTP=2 RETURN END SUBROUTINE OETINVCM,I1J

INVERTS S TO GIVE T OR EVALUATES DETERMINANTCSJ

DIMENSION PHIC10,50J,RHOZC3J,FISIGC10,3,3J,ZZ(l00), 3PHICC20J 9 8KWDl20,20J 9 FRWDf20,20J,CURR(20t3J,POW(l90J,NllOJ, 5DOGC2J

COMMON'IN0(3J,RAOC3J 9 CEFTC3J,GNUf3J,SSIGf20,20,3J,RSIGAfl0,10t3J, 2FSIGC20t3JjZf20t20J,WAC20J,W8C20J,GAMA(l0tlOJ,EXTPA(lO,lOJ, 3Pf 20,20t 3 J tTfc20.20 J, S( 60,20 J, TEMP( 20,20 J t TEMPA C 20,20 J, RHO( 3 J t

· 5PSI f!Ot lOt-lOf iYf>20t ,V( 201, KSOL V( 201, I HUNT ,KQUIT, KPR INT ,NUMBER, KCON 6D,KSTOR,ISOtV,ITtHt«HtiSTOP,IREG,KINDMtNPTS,NREG,KREG,KINOt

c

66

7MARSTP,LOLIM,MNOEX,KBCO,KBCI,IFOUND,IFUNCT,INDEX,INDEXM,NUMAX, 8KPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNT,IAtKl,EPA,EPB,OEL,A,NA,NB,X, 9XA,XB,KA,KB,KC,KD,KN,XC,ALPHA,BETA,DT,DTO,ISOLVM,KROSS,KEND,K2

COMMON K3,KG,KGA,KNAtBKWOtFRWO,NSCR,NM00E,ACEFT,EPC,KNORM, 2PHIA(l0, 50,6),FLX(l0, 50,6J,CURRN(l0,3,6),RAOZ(3),RROtRZO,NPTZ, 31NDZ(3)tNREGZtKNG,KNR,KNZ,IHOP,SSIGAClO,l0,3J,RSIG(l0,3),F(l0,3), 4FSIGA(lO,l0,3J,OAC10,10,3),0(10,3),GAM(l0)tBUCK(l0,3t,EXTP(lO), 5ZAC10,10),BCEFT(2,8J,NREGR,NPTR,INDR(3),RHOR(3),MPOW,KGEL

EQUIVALENCE CPSI(lJtPHI(l)Jt(PSIC50l),PHIC(lJ),(TEMPA(l),PQW(ll), 5(TEMPA(l91J,POWC),(TEMPA(192),POWAVG),(TEMPAC193JtPOWCAVJ, 6CTEMPACl94)tXTJ,(TEMPA(l95J,XTA),(PSIC521J,CURR(l)) si G= 1.E25 GO TO (2,2,1,2,2,1,1,2,2,1,1,2,2,2,2),11

1 NC=NA GO TO 10

2 NC=NB 10 GO TO C20,70),M

C UNIT MATRIX NEEDED FOR INVERSION c

c

20 DO 40 1=1,NC 30 00 40 J=1,NC 40 TEMPCI,JJ=O. 50 DO 60 I=l,NC 60 TEMPCI,IJ=l. 70 IFIABS(S(l,lJJ-EPRJ80,80,130 80 GO TO (90,110J,M 90 CAll YEGAOSC2tltllJ

100 RETURN 110 DT=O. 120 RETURN

C CROUT REDUCTION c

130 IFCNC-1J 160,160,140 140 DO 150 1=2,NC 150 SCl,IJ=SCl,I»ISCltlJ 160 GO TO C170,180),M 170 TEMP(l,lJ=TEMP(l,lJ/5(1,1J 180 IF(NC-1) 370,370,185 185 DO 360 K=2,NC

IMIN=K-1 190 DO 200 I=l,IMIN 200 SCK 9 KJ=SIK,KJ-SCK,I»*SCI,KJ 210 IFCABS(S(K,KJJ-EPBJ220,220,250 220 GO TO C230,240J,M 230 CALL YEGAOSC2,K,IlJ

GO TO 460 240 OT=O.

GO TO 540 250 IMAX•K+l .. 260 IFINC-IMAXJ 310.,270,270 270 00 ,)OQ .· I•IMAX~N£ 280 Ofl't~~ iJ•lt IHlN"<;,,~ :<.

c c c

c c c

c c c

290 300 310 320 330 340 350 360 370

380 390 400 420

430

440 450 460

470 480 500

510 520 530 540

S(I,KJ=S(I,KJ-S(I,J)*SCJ,K) SCK,IJ=SCK,IJ-SCK,J)*S(J,I) S(K,IJ=S(K,I)/S(K,KJ GO TO (320,360),H DO 350 I=l,K DO 340 J=l,IHIN TEMPCK,I)=TEMP(K,IJ-S(K,JJ*TEMP(J,I) TEHP(K,IJ=TEMPCK,I)/SlK,K) CONTINUE GO TO (380,470),M

INVERSE MATRIX

DO 390 J=l,NC TCNC,JJ=TEMPCNC,JJ IFCNC-1) 460,460,420 DO 450 K=2,NC IMIN=NC+l-K IMAX=IMIN+l DO 450 J=l,NC TCIMIN,JJ=TEMP(IMIN,JJ DO 450 I=IMAX,NC TCIMIN,J)=TCIMIN,JJ-SCIMIN,IJ*TCI,J) RETURN

DETERMINANT

DT=SCl,l)/10. IFCNC-lJ 540,540,500 DO 530 I=Z,NC DT=OT*SCI,IJ/10. IF(ABSCOTJ-BIGJ530,530,520 OT=SIGN(BIG,OTJ CONTINUE RETURN END SUBROUTINE CONOITCN,M)

CONDITIONING TRANSFORMATION

DIMENSION PHIC10,50J,RHOZC3J,FISIGC10,3,3J,ZZ(l00), 3PHICC20J,BKW0(20,20J,FRWOC20,20J,CURRC20,3),POW(l90), 500GC21

67

COMMON INDC3J,RA0(3),CEFTC3),GNU(3),SSIGC20,20,3J,RSIGAC10,10,3), 2FSIG(20,3J,ZC20,20J,WAC2QJ,W8(20J,GAMA(l0,10J,EXTPAClO,lOJ, 3P(20,20,3J,T(20,20J,S(60,20J,TEMP(2Q,20J,TEMPA(20,20J,RH0(3J, 5PSI(lQ,lO,lOJ,Y(20),V(20J,KSOLV(20),1HUNT,KQUIT,KPRINT,NUMBER,KCON 60,KSTOR,ISDLV,IT,H,KH,ISTOP,IREG,KINOM,NPTS,NREG,KREG,KINO, 7MARSTP,LOLIM 9 HNOEX,KBCO,KBCI,IFOUNO,IFUNCT,INOEX,INOEXM,NUMAX, 8KPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNT,IA,Kl,EPA,EPB,OEL,A,NA,NB,X, 9XA,X8,KAtK89KC,KOtKN,XC,ALPHA,BETA,OT,OTO,ISOLVM,KROSS,KENO,K2

COMMON K3,K&,KGA,KNA,BKWDtfRWDtNSCR,NMOOE,ACEFT,EPC,KNORM, 2PHIAC10t 5016l,FLX(10t 50,6J,CURRNC10,3,6J,RAOZ(3J,RRO,RZO,NPTZt 31NDli,Jt'tNREGJ t'KNGti(;NR ,KNZ, I HOP ,SSIGA ( 10, 10,3 J, RSI G ( 10,3 I ,F ( 10,3), 4FSIGACIOtlOt3t,OAC10tlOt3ltDC10t3),GAMC10J,BUCKC10,3),EXTPC10),

c c c

c c c

68

5ZA(l0,10),BCEFT(2,8),NREGR,NPTR,INDR(3),RHOR(3J,MPOW,KGEL E Q U I V AL EN CE ( P S I ( 1 J , PH I( U J , ( P S I ( 5 0 1 J , PH I C ( 1 J ) t ( T E M P A ( 1 ) , P 0 W ( 1 J ) ,

5(TEMPA(19l),P0WC),(TEMPA(192J,POWAVG),(TEMPAC193),POWCAV), 6(TEMPA(l94J,XTJ,(TEMPA(195),XTAJ,(PSI(52lJ,CURR(l))

10 IFCKAPP) 20,20,50 20 DO 40 1=1,NA 30 DO 40 J=1,NA

S(I,JJ=PSICI,J,MJ 40 TEMPA(I,JJ=PSI(J,J,N)

GO TO 90 50 DO 80 I=1,N8 60 DO 80 J=1,NB

S(I,JJ=O. TEMPA(I,JJ=O.

70 DO 80 K=l,NA S(I,JJ=S(I,J)+Z(K,IJ*WB(KJ*PSICK,J,MJ

80 TEMPA(I,J)=TEMPA(I,JJ+Z(K,I)*WBIK)*PSI(K,J,N) 90 CALL OETINV(1,5)

100 lffKQUIT) 110,110,210 110 lf(H) 270,120,120

120

130 140

150 160

170 180 190

200 210 220 230

240 250 260

270 280

290 300

310

FORWARD MARCH

ISOLV=ISOLV+1 KSOLVCISOLVJ=MNOEX+1 DO 160 1=1,NB DO 160 J=l,NB FRWO(I,JJ=O DO 160 K=1,NB FRWD(I,JJ=FRWOCI,JJ+TEMPACI,KJ*T(K,J) WRITE(NSCRJ FRWD lf(KAPPI 180,180,220 DO 200 l=l,NA DO 200 J=l,NA PSI(I,J 9 1J=FRWDCI,JJ PSI(I,J,2J=Zfi,JJ RETURN DO 250 I=l,NA DO 250 J=l,NB PSICI,J,li=O. PSICI,J 1 2J=WACIJ*ZII,JJ DO 250 K=l,NB PSI ( I, J, 1) =PSI( I, J, 1 J +WA ( I J *Z ( I, K J *FRWD ( K, J)

RETURN

BACKWARD MARCH

DO 300 l=ltNB DO 300 J=1,NB BKWDCI,JJ=O DO 300 K=l,NB BKWOCit~I•BKWO(I,JJ+TEMPAII,KJ*TCK,JJ WRITE .CNSCRJ BKWD tF ,,., .. , .Jzo. ~28.350

c

320 D~ 340 1=1,NA 330 DO 340 J=1,NA

PSI(I,J,1J=BKWD(I,J) 340 PSI(J,J,2J=Z(I,JJ

GO TO 390 350 DO 380 1=1,NA 360 00 380 J=1,NB

PSI(I,J,lJ=O. PSICI,J,2J=WA(IJ*ZCI,JJ

370 DO 380 K=l,NB 3AO PSI(J,J,lJ=PSI(I,JtlJ+WA(Il*ZCI,KI*BKWO(K,J) 390 ISOLV=ISOLV-1 400 IFfiSOLV) 410,410,420 410 ISTOP=O

GO TO 430 420 ISTOP=KSOLVCISOLV) 430 RETURN

END SUBROUTINE CROSS(KL,KR,K,KFOPJ

69

C MARCH ACROSS BOUNDARY c

DIMENSION PHI(10,50J,RHOZ(3J,FISIG(l0,3,3J,ZZ1100), 3PHIC(20J,BKW0(20,20J,FRWD(20,20J,CURRC20,3J,POW(190J,N(l0J, 5DOG(2)

COMMON IND(3J,RA0(3J,CEFT(3J,GNU(3J,SSIG(20,20,3J,RSIGA(10,10,3J, 2FSIG(20,3J,Z(20,20J,WA(20J,WBC20),GAMAC10,10J,EXTPA(10,10), 3P(20,20,3),T(20,20J,S(60,20J,TEMP(20,20J,TEMPA(20t20),RH0(3), 5PSI(10t10,10J,YC20J,V(20J,KSOLV(20J,IHUNT,KQUIT,KPRINT,NUMBER,KCON 6D,KSTOR,ISOLV,IT,H,KH,JSTOP,IREG,KINDM,NPTS,NREG,KREG,KINO, 7MARSTP,LOLIM,MNOEX,KBCO,KBCI,IFOUNO,IFUNCT,INDEX,INDEXM,NUMAX, 8KPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNT,IA,Kl,EPA,EPB,OEL,A,NA,NB,X, 9XA,X8 9 KA,KB,KC,KD,KN,XC,ALPHA,BETAtOT,OTO,ISOLVM,KROSS,KEND,K2

COMMON K3,KG,KGA,KNA,BKWO,FRWO,NSCR,NMOOE,ACEFT,EPC,KNORH, 2PHIA(l0t 50,6J,FLXC10, 50,6J,CURRNI10,3,6),RAOZ(3J,RRO,RZO,NPTZ, 31NOZC3J 9 NREGZ,KNG,KNR,KNZ,IHOP,SSIGA(l0,10,3J,RSIG(l0,3J,F(l0,3J, 4FSIGA(l0 9 10,3),0AI10,10,3),0(10,3J,GAH(10J,BUCKI10,3J,EXTP(l0), 5ZAI10tlOJ,8CEFT(2,8J,NREGR,NPTR,INOR(3),RHORl3J,MPOW,KGEL

EQUIVALENCE (PSI(lJ,PHI(lJJ,(PSI(501J,PHIC(lJJ,(TEMPAClJ,POWClJJ, 5(TEMPAC191J 9 POWCJ,CTEMPAC192),POWAVGJ,(TEMPA(l93J,POWCAVJ, 6(TEMPA(l94J,XTJ,(TEMPA(l95J,XTAJ,(PSIC521J,CURR(lJJ

KAaK+l KB=K-1 X=O. KTEMP. (Kl+KR) /2 DO 1 1=1 ,KTEMP

1 X=X+RAO( IJ XA=0.5*H*A*RHO(KRJ ALPHA• X/ (X+XA) BETA•(X-0.5*A*H*RHOCKLJJ/(X+XAJ

10 IFCKAPP) 20tl00tl00 20 00 40 l•ltNA 30 00 itOIJ•ltNA 40 S C I,~_,,..,.DM:I· t"dt.K•J ··

CALL OETINVClt6t

c c c

c c c

c c c

50 IFCKQUITJ 60,60 9 300 60 DO 90 I= l,NA 70 DO 90 J=l,NA

TEMP(I,JJ=O. 80 DO 90 L=l,NA 90 TEMP(l,J)=TEMP(I,JJ+TCI,LJ*OACL,J,KLJ

GO TO 150 100 DO 120 I=l,NA 110 DO 120 J=l,NA 120 TEMP(I,JJ=O. 130 DO 140 1=1,NA 140 TEMP(I,IJ=O(I,KLJ/D(I,KRJ 150 X=RHOCKRJ/RHO(Kl)

160 170

180 190

200 205 210

220 230

XA=BETA*X Q=ALPHA/2. QQ=(BETA-ALPHAJ*X QQQ= 1. -ALPHA DO 205 1=1, NA DO 200 J=l,NA S (I, J )=0. DO 190 L=l,NA SCI,JJ=SCI,JJ+X*TEMP(I,LJ*P(L,J,KLJ T(I,JJ=XA*TEMP(I 1 J) Sfi,JJ=Q*(P(J,J,KRJ+S(J,JJJ+QQ*TEMP(I,J) S ( I, Il =S ( I, I J +QQQ GO TO (220,270),KFOP

EXPANSION VECTORS

DO 250 I=l,NA DO 250 J=l,NB P S l( I, J, KA) =0.

70

240 250

DO 250 L=l,NA PSI(I,J,KAJ=PSI(I,J,KAJ+S(I,LJ*PSI(L,J,KJ-T(I,LJ*PSI(L,J,KBJ

270

275

280 290 300

GO TO 300

EIGENFUNCTION

KA=MNOEX+KH KB=MNDEX-KH DO 290 I=l,NA PHI(I,KAJ=O. DO 290 L=l 1 NA PHI(I,KAJ=PHI(1 9 KAJ+S(I,LJ*PHI(L,MNOEXJ-T(I,LJ*PHI(L,KBJ RETURN END SUBROUTINE MARCHCMJ

MARCH BETWEEN POINTS OF CONDITIONING

DIMENSION PHII10,50J,RHOZC3J,FISIGI10,3,3J,ZZ(l001, 3PHIC(20)

18KW0(20t20J,FRW0(20,20J,CURR(20,31,POWC190),N(lOJ,

500Gt Zlf "'~ < ~, - c •: '·

COMMON JN0(31,RADC3J,CEFT(3JtGNU(3),SSIGC20,20,3JtRSIGA(l0tl0,3J,

c

71

2FSIG(20,3),Z(20,20J,WA{20),W8(20J,GAMA(10,10),EXTPA(l0,10), 3P(20,20,3J,TC20,20),S(60,20),TEMP(20,20),TEMPA(20,20J,RHOC3J, 5PSIC10,10,10J,YC20J,V(20ltKSOLV(20J,IHUNT,KQUIT,KPRINT,NUMBERtKCON 60,KSTOR,ISOLV,IT,H,KH,ISTOP,IREG,KINOM,NPTS,NREG,KREG,KINO, 7MARSTPtlOLIMtMNOEX,KBCD,KBCI,IFOUND,IFUNCT,lNOEX,INOEXM,NUMAX, 8KPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNT,IAtKltEPA,EPB,DEL,A,NA,NB,X, 9XA,XB,KA,KB,KC,KD,KN,XC,ALPHA,BETA,OT,OTO,ISOLVM,KROSS,KENO,K2

COMMON K3,KG,KGA,KNA,BKWOtFRWO,NSCR,NM00E,ACEFT,EPC,KNORM, 2PHTACIO, 50,6),FLX(10, 50,6J,CURRN(10,3,6J,RAOZC3J,RROtRZO,NPTZ, 3INOZ(3),NREGZtKNG,KNR,KNZ,IHOP,SSIGACl0,10,3),RSIGC10,3),FC10,3), 4FSIGA(l0,10,3),0AC10,10,3),0(10t3J,GAMC10JtBUCK(10,3),EXTP(10), 5ZAC10,10),BCEFTC2,8),NREGR,NPTR,INORC3J,RHOR(3),MPOW,KGEL

EQUIVALENCE CPSI(l),PHI(l)),(PSIC501),PHIC(l)),(TEMPAC1J,POW(l)), 5(TEMPA(l9l),POWC),(TEMPA(192J,POWAVGJ,(TEMPA(193J,POWCAV), 6(TEMPAC194JtXTJ 9 (TEMPA(195J,XTAJ,(PSI(52l),CURR(l))

IREG=IREG 10 00 240 J=LOLIM,MARSTP

MNOEX=MNOEX+KH BETA=BET ( IREG)

C TEST FOR BOUNDARY CROSSING c

c

20 IFCMNOEX-KINOJ 150,30,150 30 IFCKINOM-KINDJ 50,40,50 40 IT=J-1

GO TO 250 50 CALL CROSS(IREG,IREG+KH,J-l,MJ 60 IF(KQUIT) 70,70,250 70 IREG=IREG+KH

KROSS=KR OSS+ 1 80 IF(KH) 90,90,120 90 IF ( IREG-1) 100,100,110

100 KINO=l GO TO 240

110 KINO=INDtiREG-1) GO TO 240

120 KINO=IND(IREGJ 130 IFCNREG-IREGt 140,140,240 140 KIND=KIND+KBC0-1

GO TO 24-0 150 GO TO Cl9lt160J,M

C EIGENFUNCTION c

c

160 KA=MNDEX +KH KB=MNDEX-KH

170 00 190 1=1,NA PHI(I,KA)=-BETA*PHI(I,KB)

180 00 190 K•l,NA 190 PHI(I,KAJ=PHICI,KA)+ALPHA*P(I,K,IREGJ*PHICK,MNOEXJ

GO TO 2•0

C EXP,ANSION VECtORS·· .. c

c

DO 192 K=l,NA 192 TEMP(l,KJ=ALPHA*P( I,K,IREGJ 200 DO 230 I=l,NA 210 DO 230 K=l,NB

PSI(I,K,JJ=-BETA*PSI(I,K,J-2) 220 DO 230 L=l,NA 230 PSI(I,K,JJ=PSI(I,K,JJ+TEMP(I,LJ*PSilL,K,J-1) 240 CONTINUE 250 RETURN

END FUNCTION BET ( M)

72

C ALPHA AND BETA .c

DIMENSION PHIC10,50J,RHOZ(3J,FISIG(l0,3,31,ZZ(l00J, 3PHIC(20),8KW0(20,20J,FRWDl20,20J,CURR(20,3J,POW(l90J,N(lOJ, 500G(2J

COMMON IN0(3J,RA0(3J,CEFT(3J,GNU(3J,SSIGl20,20,3J,RSIGA(l0,10,3), 2FSIG(20,3),Z(20,20J,WAC20),W8(20J,GAMA(l0tl0),EXTPA(l0,10), 3PC20,20,3J,T(20,20),S(60,20l,TEMPC20,20J,TEMPA(20,20),RH0(3J, 5PSIC10,10,10J,Y(20J,Vl20J,KSOLV(20J,IHUNT,KQUIT,KPRINT,NUMBER,KCON 6D,KSTOR,ISOLVtiT,H,KH,ISTOP,IREG,KINOM,NPTS,NREG,KREG,KINO, 7MARSTP,LOLIM,MNOEX,KBCO,KBCI,IFOUNO,IFUNCT,INDEX,INOEXM,NUMAX, 8KPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNT,IA,Kl,EPA,EPB,OEL,A,NA,NB,X, 9XA,XB,KA,KB,KC,KO,KN,XC,ALPHA,BETA,OT,OTO,ISOLVM,KROSS,KENO,K2

COMMON K3,KG,KGA,KNA,BKWO,FRWO,NSCR,NMOOE,ACEFT,EPC,KNORM, 2PHIA(l0t 50,6J,FLX(l0, 50,6J,CURRN(l0,3,6J,RADZC3J,RRO,RZO,NPTZ, 31NOZ(3J,NREGZ,KNG,KNR,KNZ,IHOP,SSIGA(l0tl0,3J,RSIG(l0,3t,F(l0,3J, 4FSIGAll0,10,3J,DAfl0,10,3),0(10,3J,GAMllOJ,BUCK(l0,3J,EXTP(l0J, 5ZA(l0,10),8CEFT(2,8J,NREGR,NPTR,INOR(3J,RHOR(3),MPOW,KGEL

EQUIVALENCE (PSillJ 9 PHillJJ,(PSI(501J,PHIC(lJJ,(TEMPA(l),POWllJJ, 5lTEMPA(l91J,POWC),(TEMPA(192J,POWAVGJ,(TEMPA(l93),POWCAVt, 6(TEMPA(l94J,XTJ,(TEMPA(l95J,XTAJ,(PSil52l),CURR(l)J

10 IF(IAJ 20,20,30 20 ALPHA=!.

BET=l. GO TO 80

30 GO TO (40,50,60J,M 40 X=MNDEX

GO TO 70 50 X=RAD(l)/RHOI21+MNOEX-INO(l)

GO TO 70 60 X=lRAOClJ+RA0(2)J/RH0(3J+MNOEX-INOf2J 70 XA=0.5*H*A

ALPHA=X/lX+XAJ BET=CX-XAJ/lX+XAJ

80 RETURN END SUBROUTINE EVAL(MJ

SET UP MATRIX FOR DETERMINANT EVALUATION

DIMENSION P+4lCIGj50J,RHOZ(3),FISIGfl0t3t31tZZll00), 3PHICf201i8lUtiU20•20JtFRW0'(20,201 ,CURRC20,3) ,POW( 190) ,NC 10J t

c c c

c c c

73

500G(2J COMMON INOC3J,RAOC3J,CEFTC3J,GNUC3J,SSIG(20,20,3),RSIGA(10,10,3J,

2FSIG(20,3),l(20t20J,WA(20JtW8(20J,GAMAC10,10),EXTPA(l0 9 10), 3P(20,20,3J,T(20,20),SC60,20J,TEMP(20,20),TEMPA(20 9 20),RH0(3), 5PSI(10,10tlO),Y(20),V(20),KSOLV(20J,IHUNT,KQUIT,KPRINT,NUMBER,KCON 60,KSTOR,ISOLV,IT,H,KH,ISTOP,IREG,KINDM,NPTStNREG,KREG,KINO, 7MARSTP,LOLIM,MNOEX,KBCO,KBCI,IFOUNO,IFUNCT,INDEX,INDEXM,NUMAX, 8KPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNT,IA,Kl,EPA,EPB,OEL,A,NA,NB,X, 9XA,XB,KA,KB,KC,KO,KN,XC,ALPHA,BETA,OT,OTO,ISOLVM,KROSS,KENOtK2

COMMON K3tKGtKGAtKNA,SKWO,FRWO,NSCR,NMODE,ACEFT,EPC,KNORM, 2PHIAC10, 50,6),FLXC10, 50,6),CURRN(l0,3,6J,RADZ(3J,RRO,RZO,NPTZ, 3INOZC3),NREGZ,KNG,KNR,KNZ 9 1HOP,SSIGAClO,l0,3J,RSIGC10,3J,FC10,3), 4FSIGAC10,10,3),0AC10,10t31tDC10t31,GAM(lOJ,BUCK(l0,3J,EXTPC10J, SZA(l0,10),BCEFT(2,8J,NREGR,NPTR,INDR(3),RHOR(3),MPOW,KGEL

EQUIVALENCE lPSIC11tPHIClJJ,(PSIC501J,PHICC1JJ,CTEHPA(lJ,powtlJJ, 5(TEMPA(19lJ,POWCJ,CTEHPAC192),POWAVGJ,CTEMPAC193JtPOWCAVJ, 6(TEMPA(194) 9 XTJ,CTEMPAC195J,XTAJ,CPSIC521J,CURR(1JJ

10 GO TO C20,100J,KBCO

20 30 40 50

60 10

80 90

100 110 120 130 140 150 160

170 180

190 200 210

220 230 240 250 260

PHI (H J = 0

IFCKAPPJ 30,30,60 DO 50 1=1,NA DO 50 J=l,NA SCI,JI=PSI(J,J,MJ GO TO 290 00 90 I=l,NB DO 90 J=l,NB SCI,JJ=O. 00 90 K=1,NA SCI,JJ=S(I,JJ+Z(K,I1*WB(K)*PSICK,J,MJ GO TO 290

GAM*PHICMJ + EXTP*DCPHICM)J/OE = 0

X=l./C2.*RHO(NREGJ) IF(KAPPJ 120,190,220 00 140 I=ltNA 00 140 J=l,NA TEHPCI,JI=X*(PSICitJ,M+lJ-PSICI,J,M-11) 00 180 I=ltNA DO 180 J=1,NA SCI,JJ=O. 00 180 K=1,NA SCI,JJ=SCI,JJ+GAMACI,KI*PSI(K,J,MJ+EXTPACI,KJ*TEMPCK,JJ GO TO 290 00 210 I=1,NA 00 210 J=1,NA SCI,JJ•GAMtii*PSiti,J,MJ+X*EXTPCIJ*CPSICI,J,M+1J-PSICI,J,M-lJJ GO TO 290 00 240 I•l,NA DO 240 J•l,NB TEMP (I, J )•GAM( I) *PSI (I, J, M J+ X*EXTP C I ) * (PSI C I' J' M+ 1) -PSI (It J' M-1) I DO~. 260',: l*ltNB 00 280 J•l,NB

c

S ( I, J J=O. 270 DO 280 K=l,NA 280 Sti,JJ=Sli,JJ+WB(KJ*ZCK,IJ*TEMP(K,J) 290 CALL DETINVC2,8J 300 RETURN

END SUBROUTINE ZERO

74

C SEARCH FOR DETERMINANT EQUAL ZERO C THREE POINT INTERPOLATION c

c

DIMENSION PHI(10,50),RHOZ(3J,FISIGC10,3,3J,ZZCIOOJ, 3PHIC(20t,BKW0(20,20J,FRW0(20,20),CURRC20,3),POW(l90),N(l0), 5DOG(2J

COMMON INDC3J,RA0(3J,CEFTC3J,GNUI3J,SSIG(20,20,3J,RSIGA(l0,10,3J, 2FSIG(20,3J,ZC20,20J,WAC20J,WBC20J,GAMA(l0,lOJ,EXTPAC10tl0), 3PC20,20,3J,T(20,20),S(60,20J,TEMP(20,20J,TEMPA(20,20),RH0(3J, 5PS1(10tlOtlOJ,Y(20J,V(20),KSOLV(20),1HUNT,KQUIT,KPRINT,NUMBER,KCON 60,KSTOR,ISOLV,IT,H,KH,ISTOP,IREG,KINOM,NPTS,NREG,KREG,KIND, 7MARSTP,LOLIM,MNOEX,KBCO,KBCI,IFOUND,IFUNCT,INOEX,INDEXM,NUMAX, 8KPOW,MORE 9 NFCT,NOG,KAPP,KCHO,KHUNT,IA,Kl,EPA,EPB,OEL,A,NA,NB,X, 9XA,XB,KA,KB,KC,KO,KN,XC,ALPHA,BETA,OT,OTO,ISOLVH,KROSS,KENO,K2

COMMON K3,KG,KGA,KNA,BKWO,FRWO,NSCR,NMOOE,ACEFT,EPC,KNORM, 2PHIA(l0, 50,6J,FLX(l0t 5Q,6),CURRN(l0,3,6I,RAOZ(3),RRO,RZO,NPTZ, 31NDZ(3J,NREGZ,KNG,KNR,KNZ,IHOP,SSIGAC10tl0t3J,RSIG(l0,3J,F(l0,3), 4FSIGAC10,10,3),0A(l0,10 1 3J,O(l0,3J,GAM(l0),8UCK(l0,3J,EXTP(l01, 5ZAC10,10),8CEFT(2,8J,NREGR,NPTR,INOR(3),RHORC3J,MPOW,KGEL

EQUIVALENCE CPSIC1J 9 PHI(l)),(PS1(50l),PHIC(l)),(TEMPA(lJ,POW(l)), 5(TEMPA(l91J,POWCJ,(TEMPA(192),POWAVGJ,CTEMPA(l93),POWCAV), 6(TEMPAC194) 9 XTJ,(TEMPA(l95),XTAJ,(PSI(52li,CURR(ltl

KB=l 10 GO TO (20,70J 1 KHUNT

C NO SEARCH c

c

20 GO TO C30,40J,KCHO 30 V(l)=RAO(KREGJ

RAOCKREGJ=RAOCKREGJ+OEL GO TO 50

40 VllJ=CEfT(KREGJ CEFT(KREGJ=CEFTIKREGJ+OEL

50 INOEX=INOEX+1 60 RETURN

C SEARCH c

70 IFCOTJ 90,80,100 80 GO TO C85,86J,KCHO 85 V(lJ=RADCKREGJ

GO TO 210 86 Vt 1 Ja.CEFT,UUlEGI

GO TO 210 90 K8:1J12' :·., \. i (J:'·~ ;·~; k"' i .; 1 • · ., ~ '.: i " :

100 JF..CJttUttJt~ tGa:t-l-N.Zf8 .. -t·•·.lfJ' .. d /' , ·

c c c

c

105 GO TO (110,120),KCHO 110 V(KB)=RADCKREGJ

GO TO 125 120 V(KBl=CEFT(KREGt 125 VtKB+3)=0T 130 IFCIHUNTl 140,150,250 140 DTO=DT

IHUNT=O GO TO 20

150 X=1.

160 170 180 190 200 210

220 230

240

250

X= SIGN( X, OTO) XA=l. XA=SIGNC XA, OT)

TEST FOR PROPER CHANGE IN SIGN

IFfX*XA) 170,180,140 IF(XA) 180,180,140 IHUNT=l VV=0.5*(V(1)+V(2)) IF(ABS((V(l)-V(2))/VVJ-EPA)210,210,220 IFOUND=2 NUMBER=NUMBER+1 GO TO (230,240),KCHO RAD(KREG)=VV GO TO 60 CEFT(KREGl=VV GO TO 60 V(3)=V(KBJ V(6)=VCKB+3J GO TO (260,270J,KCHO

260 VCKBJ=RAO(KREGJ GO TO 280

270 V(KBJ=CEFT(KREGJ 280 V(KB+3l=DT

H=V(4)*V(5)*V(6) VV=O DO 1020 1=4,6 PP=V(IJ DO 1010 J=4,6 IF(J-J) 1000,1010,1000

1000 PP=PP*CV(IJ-V(J)) 1010 CONTINUE

IFCEPA*ABS(PPJ-ABSCHJ) 190,190,1020 1020 VV=VV+H*V(I-3)/PP 1040 IFCV(2J-VVJ 190,190,290

290 IF(VV-VC1Jt 190,190,200 END SUBROUTINE RESTRT

C BEGIN BACKVARD MARCH c .,. OIMENSIONtPHIC10,50J,RHOZC3J,FISIGC10t3,3J,ZZClOOJ,

~ttlCt!lOt·~BfQfO( 20t20) t FRWDC 20t20J ,CURRC20,3 J ,POWC 190) ,N( 10), 5DOGf2J

75

c c c

c c c

10 20 30 40

50 60 70 80

90

100 110

120 130

76

COMMON IN0(3l,RA0(3J,CEFTC3J,GNUC3J,SSIG(20,20,3J,RSIGAI10,10,3J, 2FSIG(20,3),Z(20,20J,WA(20J,WB(20J,GAMA(10,10J,EXTPA(l0,10t, 3PC20t20,3),TC20t20J,S(60,20J,TEMP(20,20l,TEMPA(20,20),RHOC3), 5PSI(lO,lO,lOJ,YC20J,VC20),KSOLVC20J,IHUNT,KQUIT,KPRINT,NUMBER,KCON 60tKSTOR,ISOLV,IT,H,KH,ISTOP,IREG,KINOM,NPTS,NREG,KREG,KIND, 7MARSTP,LOLIM,MNOEX,KBCO,KBCitiFOUND,IFUNCT,INOEX,INDEXM,NUMAX, BKPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNT,IA,Kl,EPA,EPB,OEL,A,NA,NBtXt 9XA,XB,KA,KB,KC,KO,KN,XC,ALPHA,BETA,OT,DTO,ISOLVM,KROSS,KENO,K2

COMMON K3,KG,KGA 9 KNA,BKWO,FRWO,NSCR,NMOOE,ACEFTtEPC,KNORM, 2PHIAC10, 50,6l,FLX(l0, 50,6),CURRN(l0,3,6l,RADZ(3J,RR0,RZ0,NPTZ, 3INOZ(3),NREGZ,KNG,KNR,KNZ,IHOPtSSIGA(10,10,3),RSIGC10,3J,FC10,3J, 4FSIGAC10,10,3),0A(l0,10,3),0(10,3),GAMC10J,BUCKC10,3J,EXTP(l0Jt 5ZA(l0,10J,BCEFT(2,8),NREGR,NPTR,INOR(3J,RHOR(3),MPOW,KGEL

E Q U IV Al EN CE ( P S I ( 1 ), PH I ( 1 ) ), ( P S I ( 50 1 ) , PH I C ( 1 ) J t ( T E M PA ( 1 J , P 0 W ( 1 J ) t

5CTEMPA(l9l),POWCJ,CTEMPA(l92),POWAVGJ,CTEMPAC193),POWCAV), 6(TEMPAC194J,XTJ,CTEHPA(l95),XTAJ,(PS1(52l),CURR(1JJ

KH=-1 H=-1. MNOEX=INO(NREGJ IF(KAPP) 20,20,50 00 40 I= l,NA 00 40 J=l,NA PSICI,J,li=Z( I,J) GO TO 80 DO 70 I= l,NA DO 70 J=1,NB PSICI,J,1J=WA(IJ*ZCI,JJ GO TO (90,140J,KBCO

PH I (Hl = 0

MNDEX=MNDEX-1 BETA=BETCNREGJ 00 130 1=1,NA 00 130 J=1,NB PSICI,J,2J=O. 00 130 K=l,NA PSICt,J,2J=PSICI,J,2)+ALPHA*P(I,K,NREGJ*PSI(K,J,lJ GO TO 440

GAM*PHICMJ + EXTP*OCPHICMJJ/DE = 0

140 BETA=BET(NREGJ X=1./(2.*RHOCNREGJJ

150 IFCKAPPJ 160,210,210 160 DO 200 1=1,NA 170 DO 200 J=l,NA

SCI,JJ=O. 180 DO 190 K•1tNA 190 S(I,JJ•SfliJl+EXTPA(I,KJ*P(K,J,NREGJ 200 S (I, J J•ALPHA*X*S (I, J J +8ETA*GAM AC It J)

GO TO ~50·'': 210 ·oo ;240 ·l•·l~MA· i ~,; .~.; 220 'O'(f (23-0 ;J•J;,'ftA- I i f £ ~ ;·, ' ' ,, ..•

t- 'J u ! v !"~~ 1:: l'-< • i ~H~ I { -~ ·~ ,, t' . . ; i ! . ', l· '

c

230 SC I,JJ=X*ALPHA*EXTP(IJ*P(I,J,NREGJ 240 SCI,IJ=SCI,IJ+BETA*GAMCIJ 250 CALL DETINVCl,lOJ 260 IFCKQUITJ 265,265,470 265 X=Cla+BETA)*X 270 IF(KAPP) 280,330,330 280 00 320 I=l,NA 290 DO 320 J=l,NA

TEMPCI,JJ=O. 300 DO 310 K=l,NA 310 TEMP(I,JJ=TEMPCI,J)+T(I,KJ*EXTPA(K,J) 320 TEMPCI,JJ=X*TEMP(I,J)

GO TO 360 330 DO 350 I=l,NA 340 DO 350 J=l,NA 350 TEMPCI,JJ=X*EXTPCJJ*T(I,JJ 360 MNDEX=MNDEX-1

BETA=BETCNREGJ 370 DO 390 l=l,NA 380 00 390 J=l,NA 390 SCI,JJ=ALPHA*PCI,J,NREGJ-BETA*TEMP(I,JJ 400 00 430 I=l,NA 410 DO 430 J=l,NA

PSI (I ,J, 2)=0. 420 DO 430 K=l,NA 430 PSICI,J,2J=PSICI,J,2J+S(I,KJ*PSICK,J,lJ 440 ISOLVM=ISOLV

MARSTP=MNOEX+2-KSOLVCISOLVJ lOLIM=3 IREG=NREG

450 IFCMARSTP-LOLIHJ 470,460,460 460 CAll MARCH(l) 470 RETURN

END SUBROUTINE FUNCT

77

C SOLVES FOR EIGENFUNCTION c

DIMENSION PHIC10,50J,RHOZC3J,FISIGC10,3,3J,ZZC100J, 3PHICC20J,BKWDCZ0,20J,FRWOC20,20J,CURRC20,3J,POWC190J, 5DOGCZJ

COMMON INDC3),RADI3J,CEFTC3J,GNUC3J,SSIGC20,20,3J,RSIGAC10,10,3J, 2FSIGCZ0,3J,ZCZ0,20J,WAC20),WBC20J,GAMAC10,10J,EXTPAC10tlOJ, 3PCZO,Z0,3),TC20,20J,S(60,ZOJ,TEMPC20,20J,TEMPACZ0,20J,RHOC3J, 5PS I I lO, lOt 10 J, Y( 20 J, VI ZO J ,KSOL VI ZOJ, I HUNT ,KQUI T, KPRINT ,NUMBER, KCON 6D,KSTOR,ISOLV,IT,H,KH,ISTOP,IREG,KINOM,NPTS,NREG,KREG,KINO, 7HARSTP,LOLIM,MNOEX,KBCO,KBCI,IFOUNO,IFUNCT,INOEX,INOEXH,NUMAX, 8KPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNT,IA,Kl,EPA,EPB,DEL,A,NA,NB,X, 9XA,X8,KA,KB,KC,KO,KN,XC,AlPHA,BETA,OT,OTO,ISOLVM,KROSStKENO,K2

COMMON K3,KG,KGA 9 KNA,BKWD,FRWD,NSCR,NMOOE,ACEFT,EPC,KNORM, 2PHIAI10t 50,6J,FlXC10t 50,6),CURRN(l0,3,61,RAOZC3J,RRO,RZO,NPTZt 31NOZC3J,NREGZ,KNGtKNR,KNZ,IHOP,SSIGAC10tlOt31,RSIGC10,3J,FC10,3J, ltFSI~,AC lOt lOt 31 ,OAI lOtiO, 3J ,o C 1 O, 31 ,GAM I 101 t BUCK( 10,3 J t EXTPC 10 J t 5ZAC lOt 10 J tBCEFTt-2t8 I ,NREGR tNPTRt INORC 31 ,RHORC 3 J ,MPOW,KGEL

EQUIV Al.ENCE CPS It It • PHI ( 1 I It C PSI C 501 I, PH ICC 1 I J , C TEMP A C 1 J t POW C 1 J J t

c

5(TEMPA(l9l),POWC),(TEMPA(l92J,POWAVGJ,(TEMPA(l93),POWCAVJ, 6(TEMPAC194),XTJ 9 CTEMPAC195),XTAJ,(PSI(521),CURR(l))

IREG=NREG KENO=NPTS+l

10 DO 520 J=1,1SOLVM

C MARCH FORWARD TO lAST VALUE OF PHI c

c

K= I SOl VM+1-J H= 1. KH=1 N=KSOLVCKJ CAll HOSIEQll,K)

20 IF(IREG-1) 50,50,30 30 IFCN-INO(IREG-1)) 40,40,50 40 IREG=IREG-1

GO TO 20 50 MNDEX=N-1

MARSTP=KEND-N+l LOLIM=3

60 IF(MARSTP-LOLIMJ 110,70,70 10 KROSS=O

K IN D= IN D ( IRE G ) 80 IF(NREG-IREGJ 90,90,100 90 KIND=KIND+KBC0-1

100 CAll MARCHC2l 101 IF(KQUITJ 109,109,1000 109 IREG=IREG-KROSS 110 IF(J-11 300,300,120 120 IF(KENO-NJ 130,130,140 130 KN=N

GO TO 145 140 KN=KEN0-1 145 KENO=KN+l 150 00 180 l=l,NREG 160 IFCKEND-INO(IJ) 180,170,180 170 KA=I

GO TO 190 180 CONTINUE

KA=IREG+KROSS IFCKN-INO(KA)J 186,182,186

182 KA=KA+l 186 KB=KN+l

KC=KN+2 KO=KN KH=-1 H=-1. GO TO 200

190 KB=-KN KC=KN-1 KD•KN+l KH•l ... 1-.

78

C MATCH c

c c c

c

200 MNDEX=KB BETA=BET(KAJ

210 220

221

222 229

240

230 250

255

DO 220 L=1,NA Y(LJ=-BETA*PHI(L,KCJ DO 220 I= 1, NA Y ( L ) = Y ( L ) +A L PHA*P ( L, I , KA J *PH I ( I, K B) IF(KH) 222,221,221 CM=PHICKG,KDJ/Y(KG) GO TO 229 CM=YCKG)/PHICKG,KOJ LOL IM=N-1 X=((1+KHJ*CM+1-KH)/2 DO 250 1=1,NA Y(IJ=X*YCI) DO 250 L=LOLIM,KN PHICI,LJ=CM*PHICI,l) DO 255 I= 1, NA Y(I)=(Y(IJ-PHI(I,KDJJ*100./PHI(I,KO)

WRITE MATCH ERROR

WRITE (3,1010) KN,KG WRITE (3,1020) (Y(IJ,I=l,NA)

300 DO 310 I=l,NA 310 V(IJ=PHiti,NJ

C MARCH BACKWARD HALFWAY TO NEXT POINT OF CONDITIONING c

H=-1. KH=-1 CALL HOSIEQ(2,K)

320 IFCK-1) 330,330,340 330 KEN0=1

GO TO 350 340 KENO=N-1-CKSOLV(KJ-KSOLVCK-1))/2 350 LOLIM=3

MARSTP=l+N-KENO 360 IFIMARSTP-LOLIMJ 430,370,370 370 MNDEX=N

KROSS=O 380 IFtiREG-1) 390,390,400 390 KINO=l

GO TO 410 400 KINO=IND(IREG-1) 410 CALL MARCH(2) 420 IF(KQUITJ 430,430,1000 430 I REG= I REG

LOLIM=KEND CM=VtKGt/PHI(KG,NJ HARSTP=N-1

440 00\ .470 .,.,, .•. " .. 450 08~ 4.60·l-ult-l'&-.MARSTP 460 PHI ( t.L I•CM*PtiJ(:J,.l)

79

c

470 PHI(I,NJ=V(I) 520 CONTINUE

KH=l H=l.

80

C PHI(O) c

c c c

c c c

c c c

530 IF(A-1.2J 660,660,540

540 550 560 570 580 590 600 610

620 630

640 650

660

670

680 690 700

SPHERE

00 570 1=1,NA DO 560 J=l,NA S(I,JJ=P(I,J,lJ S( I, I J=S( 1, I J-2. DO 610 I=l,NA DO 600 J=1,NA S(I,JJ=S(I,JJ/6. S (I, I) =S ( I, I J + 1. CALL OETINV(l,ll) IFIKQUITJ 630,630,1000 DO 650 I=l,NA PHIC(IJ=O. 00 650 J=1,NA PHICI I J=PHIC( I )+T( I ,JJ*PHI (J, 1) GO TO 700

CYLINDER OR SLAB

ALPHA=2./(2.-AJ RETA=C2.+AJ/(2.-AJ DO 690 I=l,NA PHICCIJ=-BETA*PHI(I,2) DO 690 J=l 9 NA PHIC(IJ=PHICCIJ+ALPHA*P(I,J,lJ*PHI(J,l) IF(KAPPJ 710,900,900

VARIATION I CALCULATION OF PHI FROM EXPANSION COEFFICIENTS

710 GO TO C2790,720),KBCO 720 X•l./(2.*RHOCNREGJJ 730 DO 750 l=l,NA

Y(IJ=O. 740 DO 750 J=l,NA 750 Y(IJ=YCIJ+GAMA(I,JJ*PHI(J,NPTS-l)+EXTPA(I,J)*(PHI(J,NPTSJ-PHI(J,

2NPTS-2JJ*X 2790 00 2810 I=l,NREG

KA•INO(I) MNOEX=KA BETA•BETf I J X•BETA+l. 00 2800 J•l ,-NA

· VC:JJ•-X*PHl.t,t~•-A-lt ·oo· '-.. aoo ... ,. t &l&f ·• · -,~·. '• :"'CJ .. ~-..• -f~ ·~ t>.r<-:t !'"

81

2800 V(J)=V(J)+ALPHA*PCJ,L,IJ*PHICL KAJ X=0.5/RHO(J) ' DO 2810 J=l,NA CURR(J,IJ=O. DO 2810 l=l,NA

2810 CURR(J,IJ=CURR(J,IJ-OACJ,L,IJ*VlLJ*X GO TO 940

c C NORMALIZE TO PHI(KGA,KNAJ = 1. c

c c c

900 GO TO 12940,910J,KBCO 910 X=l./(2.*RHO(NREGJ) 920 DO 930 I=l,NA 930 Y(IJ=GAM(IJ*PHI(I,NPTS-lJ+X*EXTP(l)*(PHICI,NPTSJ-PHIII NPTS-2))

2940 DO 2960 I=l,NREG ' KA=INOIIJ MNOEX=KA BETA=BET(IJ X= BETA+!. DO 2950 J=l,NA V(JJ=-X*PHI(J,KA-1) DO 2950 L=l,NA

2950 V(JJ=V(JJ+ALPHA*P(J,L,IJ*PHI(l,KAJ X=0.5/RHO(IJ DO 2960 J=l,NA

2960 CURR(J,IJ=-X*O(J,IJ*VIJ) 940 IFIKNA) 942,942,946 942 CM=l./PHIC(KGAJ

GO TO 950 946 CM=l./PHI(KGA,KNAJ 950 DO 990 l=l,NA

Y ( I J =C M* Y ( I ) PHICIIJ=CM*PHIC(I)

960 00 970 J=l,NREG 970 CURR(I,JJ=CM*CURR(I,JJ 980 DO 990 J=1 9 NPTS 990 PHICI,JJ=CM*PHI(I,JJ

1000 RETURN 1010 FORMAT ClHO 14X 42HHERE IS THE PERCENT ERROR FOR MATCH AT X= 13,

215H FOR Y MATCH AT 13, lH.J 1020 FORMAT ( lHO 14X 4E 15.8 J

END SUBROUTINE HOSIEQIM,KJ

SOLVES IS-IJY = 0

DIMENSION PHICl0,50J,RHOZC3J,FISIGil0,3,3J,ZZ(lOOJ, 3PHIC(20J,SKWOC20t201tfRWOC20t20),CURRC20,3J,POWI190J,NI10J, 5DOGC2J

COMMON INOI3J,RAOI3J,CEFT(3J,GNUC3J,SSIGI20,20,3J,RSIGA(l0,10,3J, 2FSIG(20,3J,ZI20,20J,WAI20J,W8(201tGAMA(lO,lOJ,EXTPA(lO,lOJ, 3P( 20,2093Jt,TC'20,20J ,SI60,201 ,TEMP(20,20J ,TEMPAC20t20J,RH0(3J, 5PS ll·lO..l.O't'l01l,f.:'Voi 20), VI 20 J, KSOL V( 201, I HUNT ,KQUIT, KPRINT ,NUMBER, KCON 6D,KSTOR,:.tSOLV, IT ,H,KHt ISTOPt IREG,KINOM,NPTS ,NREG,KREG,KINO, 7MARSTPtLOLifltMNOEX 9 K8C09 K8CitiFOUND,IFUNCT,INOEX,JNDEXM,NUMAX,

c c c

c c c

c c c

82

8KPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNT,IA,Kl,EPA,EPB,OEL,A,NA,NB,X, 9XA,XB,KA,KB,KC,KO,KN,XC,ALPHA,BETA,OT,OTO,ISOLVM,KROSS,KENO,K2

COMMON K3,KG,KGAtKNA,BKWD,FRWD,NSCR,NMOOE,ACEFT,EPC,KNORM, 2PHJA(l0, 50,6),FLX(10, 50,6J,CURRNC10,3,6),RAOZC3J,RRO,RZO,NPTZ, 3INDZ(3),NREGZ,KNG,KNR,KNZ,IHOP,SSIGA(lO,l0,3J,RSIG(l0,3),F(l0,3J, 4FSIGA(l0,10,3),0A(10t10,3),0(10t3J,GAM(l0),8UCKl10,3J,EXTPC10), 5ZA(10,10l,BCEFT(2,8J,NREGR,NPTR,INDRC3),RHOR(3),MPOW,KGEL

EQUIVALENCE CPSIC1JtPHI(l)J,(PSI(50lJ,PHIC(l)J,CTEMPAClJ,POW(lJ), 5CTEMPA(19l),POWCJ,(TEMPAC192),POWAVGJ,ITEMPAC193JtPOWCAVJ, 6(TEMPA(194J,XT),(TEMPA(l95J,XTAJ,CPS1(521),CURRC1))

REWIND NSCR 00 2 1=1,K

8 10

2 READ CNSCRJ FRWD KK::K+l KKK=2*ISOLVM+l-K DO 8 I=KK,KKK READ CNSCRJ BKWO GO TO (20,90) ,M

20 30 40 50

60 10 75 80

90 100

110 120 125 130 140 150

FORWARD MARCH

IF(K-ISOLVHJ 40,30,30 WRITE (3,610) 00 80 I= l,NB 00 75 J=l,NB SCI,JJ=O. 00 70 L=l,NB S(I,JJ=SCI,JJ+FRWO(I,LJ*BKWD(L,J) T(J,J)=S(I,J) SCI,IJ=SCI,IJ-1. GO TO 140

BACKWARD MARCH

00 130 1=1,NB 00 125 J=l,NB S(I,JJ=O. DO 120 l=l,NB S(I,JJ=S(I,JI+BKWDCI,LJ*FRWD(l,JJ T( I,JJ=S( I,JJ SCI,IJ=SCI,IJ-1. IF(NB-1J 150,150,160 Y(1J=1. GO TO 400

WILKINSONaS METHOD

160 DO 270 1=2,NB KA=I-1

170 DO 270 J=l,KA 180 IFCSC·I,~l1 190,270,190 190 IF(A8SC5CJ,JJJ-ABS(SCI,JJ)J210,200,200 200 Xa5't 1-'YJ t/S( J' J J

1$0 'TU.J- 2M= ' ~.., '~

c

210 X=S(J,JJ/S(I,JJ 220 DO 230 L=1,NB

XA=SCJ,L) S(J,LJ=S(I,Ll

230 S(I,L)=XA 240 KB=J+l 250 DO 260 L=KB,NB 260 S(I,LJ=SCI,L)-X*S(J,LJ 270 CONTINUE

XA=S(NB,NBJ Y ( NB l = 1.

280 DO 350 1=2,NB KB=NB-1+ 1 X=O. KA=NB-1+2

290 DO 295 L=KA,NB 295 X=X+SCK8 9 L)*YCLJ 300 IFCABSCSCKB,KBJ)-l.E-101310,310,340 310 YCKBJ=1.

XA=O. 320 DO 330 J=KA,NB 330 Y(J)=O.

GO TO 350 340 Y(KBJ=(XA-XJ/S(KB,KB) 350 CONTINUE

IFCK3) 359,359,890

C ITERATION TO REDUCE ERROR c

890 00 1000 l=l,K3 DO 960 I=l,NB YCIJ=O. KA=I-1 KB=I+l IFIA8S(T(I,IJ-l.J-EPBI960,960,900

900 IF(KAJ 9309 930,910 910 00 920 J=l,KA 920 Y(IJ=Y(IJ+T(I,JJ*Y(~J 930 IFCKB-NBJ 940,940,955 940 DO 950 J=KB,NB 950 YCIJ=YCJ)+TCI,JJ*Y(JJ 955 YIIJ=YCIJ/(1.-T(I,IJJ 960 CONTINUE

X=ABSC Y( 1))

XA=l. XA=SIGNCXA,YCl)) DO 980 I=Z,NB IFIX-ABS(YCI))J970,980,980

970 X=ABSC Y( I J) XA•l. XA=SIGNCXAtY,(.JJ J

980 CONT INU!- • · X•XA/X' ·: DO 990 1•1,0

990 YC I J•X*Y( I)

83

c

1000 CONTINUE GO TO 400

C NORMALIZE TO MAXIMUM VALUE OF Y = 1. c

c

359 X=ABS( Y( l) J XA= 1. XA=SIGNCXA,Y(lJJ

360 DO 390 1=2,NB 370 IF(X-ABS(Y(I)JJ380,390,390 380 X=ABS(Y(IJ)

XA=l. XA=SIGN(XA,Y(I))

390 CONTINUE X=XA/X DO 395 I=l,NB

395 Y(IJ=X*Y(I)

C PHI(JJ AND PHI(J-1) c

400 GO TO C410,500J,M c C FORWARD MARCH c

c

410 KA=KSOLVCKJ-1 KB=KA+l

420 IFCKAPP) 430,430,460 430 00 450 I=l,NA

PHI(I,KAJ=Y(I) PHICI,KBJ=O.

440 DO 450 J=l,NA 450 PHICI,KBJ=PHICI,KBJ+BKWOCI,J)*YCJJ

GO TO 590 460 DO 490 l=l,NA

PHI(I,KAJ=O. PHI(I,KBJ=O.

470 DO 490 J=l,NB PHICI,KAJ=PHICI,KAJ+WACIJ*ZCI,JJ*YCJJ

480 00 490 L•l,NB 490 PHI(I,KBJ=PHICI,KBJ+WACIJ*ZCI,JJ*BKWO(J,LJ*YCLJ

GO TO 590

C BACKWARD MARCH c

500 KA=KSOLV(K) KB=KA-1

510 IFCKAPPJ 520,520,550 520 DO 540 l=l,NA

PHICI,KAJaY(I) PHifltKBJ•O.

530 DO 540· ~•l;NA 540 PHICitKBl•PHICitKBJ+FRWOCI,JJ*Y(JJ

GD T0·:590 550 00;~ 580ll•ltNA

84

c

PHI(I,KAJ=O. PHI(I,KBl=O.

560 DO 580 J=l,NB PHJ(I,KA)=PHI(I,KA)+WA(I)*Z(I,Jl*Y(J)

570 DO 580 L=l,NB 580 PHI(I,KBJ=PHI(I,KB)+WA(I)*ZCI,JI*FRWD(J,L)*Y(LJ 590 WRITE (3 9 620) KA

WRITE (3,630) (Y(IJ,I=l 9 NAJ 600 RETURN

85

610 FORMAT (lHl 19X 51HTHE EXPANSION COEFFICIENTS ARE GIVEN FOR THERA 2DIAL I 15X 21HMESH POINT INDICATED.)

620 FORMAT (lHO l4X 24HTHE RADIAL ~ESH POINT IS, I3J 630 FORMAT (lH l4X, 4El5.8)

END SUBROUTINE POWER

C CALCULATES POWER AS (FSIG,PHIJ c

DIMENSION PHI(l0,50J,RHOZ(3J,FISIG(l0,3,3),ZZfl00), 3PHIC(20J,BKW0(20,20t,FRW0(20,20),CURRf20,3J,POW(l90J,N(l0), 5DOG(2)

COMMON IND(3),RA0(3),CEFTC3J,GNU(3J,SSIG(20,20,3J,RSIGA(l0,10,3t, 2FSIG(20,3),Z(20,20J,WA(20),W8(20),GAMA(l0,10J,EXTPA(l0,10), 3PI20,20,3J,T(20,20J,S(60,20),TEMP(20,20J,TEMPAC20,20),RH0(3), 5PSI(l0,10 9 10J,Y(20J,V(20),KSOLV(20),1HUNT,KQUIT,KPRINT,NUMBER,KCON 6D,KSTOR,ISOLV,IT 9 H,KH,ISTOP,IREG,KINDM,NPTS,NREG,KREG,KIND, 7MARSTP,LOLIH,MNOEX,KBCO,KBCI,JFOUND,IFUNCT,INOEX,INOEXM,NUMAX, 8KPOW,MORE,NFCT,NQG,KAPP,KCHO,KHUNT,IA,Kl,EPA,EPB,OEL,A,NA,NB,X, 9XA,XB,KA,KB,KC,KO,KN,XC,ALPHA,BETA,OT,OTO,ISOLVM,KROSS,KENO,K2

COMMON K3,KG,KGA,KNA,BKWO,FRWO,NSCR,NMOOE,ACEFT,EPC,KNORM, 2PHIA(10, 50,6),FLXC10, 50,6J,CURRN(l0,3,6J,RAOZ(3),RRO,RZOtNPTZ, 3INOZ(3J,NREGZ,KNG,KNR,KNZ,IHOP,SSIGAC10tl0,3J,RSIGC10,3J,F(l0,3), 4FSIGA(10 9 10,3),0A(l0 9 10 9 3),0(10,3t,GAMC10),BUCK(l0,3),EXTP(l0), 5ZA(10,10) 9 BCEFT(2,8) 9 NREGR,NPTR,INOR(3),RHOR(3),MPOW,KGEL

EQUIVALENCE (PSI(l),PHI(l)J,(PSI(50l),PHIC(l)),(TEMPA(l),POW(l)), 5(TEMPA(l9l),POWCJ,(TEMPA(l92),POWAVGJ,(TEMPAC193),POWCAVJ, 6CTEMPAC194),XTJ,(TEMPAC195J,XTAJ,CPSI(52lt,CURR(l))

I REG= 1 PI =3.141592 6536

10 IF(IA-1) 20,30,40 20 XA=l.

GO TO 50 30 XA=2.*PI

GO TO 50 40 XA•4.*PI/3. 50 POWC:aO.

IA=IA+l 60 DO 70 I=l,NOG 70 POWC=POWC•PHICCIJ*FSIGfltlJ

XTA•XA*CCRH0(1J/2.J**(IA)J X T=XTA*POWC ' MARSTP•lND(:l 1-1 KA•INO l L) ~ ,· r. :· ~· ·, .

lOliiM-'l C ':0 · i '· , 1 .. ·

80 X•XA*CRHOIIREGf**(IAIJ

c

90 DO 120 I=LOLIM,MARSTP XB=X*l(l+0.5)**1A-(I-0.5)**1AJ POW(IJ=O.

100 00 110 J=1,NOG 110 POW(IJ=POW(IJ+PHI(J,l)*FSIG(J,IREGJ

XT=XT+XB*POW( I) 120 XTA=XTA+XB 130 XB=X*lKA**IA-CKA-0.5)**1A)

XTA=XTA+XB POW(KA)=O.

140 DO 150 I=l,NOG 150 POW(KAJ=POWfKA)+XB*PHI(I,KAJ*FSIG(I,IREGJ

IREG=IREG+l 160 IF(NREG-IREGJ 210 9 170,170 170 X=XA*(fRHOfiREGJJ**IIAJ)

XC=X*((KA+0.5l**IA-KA**IAJ XTA=XTA+XC

180 00 190 I=l 9 NOG 190 POW(KAJ=POW(KA)+XC*PHICI,KAJ*FSIG(I,IREGJ

XT=XT+POW(KA » POW(KAJ=POW(KAJ/(XB+XCJ LOLIM=INDCIREG-1)+1 KA=IND(IREGJ MARSTP=IND(IREGJ-1

200 IF(MARSTP-LOLIM) 130,90,90 210 XT=XT+POW(KAJ

POW(KAJ=POW(KAJ/XB POWAVG=XT/XTA POWCAV=POWC/POWAVG I A= I A-1

220 RETURN END SUBROUTINE OUTPUTIMJ

86

C WRITE RESULTS c

INTEGER Q DIMENSION PHI(l0,50J,RHOZC3J,FISIG(l0,3,3J,ZZflOOJ,

3PHICC20J,BKWD(20t20J,FRWD(20,20J,CURR(20,3J,POWC190J,NC10J, 500G(2J

COMMON IND(3J,RADC3J,CEFT(3J,GNU(3J,SSIG(20,20,3J,RSIGAC10,10,3J, 2FSIG(20,3J,Z(20t20J,WAC20J,W8(20J,GAMA(l0,10J,EXTPA(l0,10), 3P(20,20,3J,T(20,20J,S(60,20J,TEMP(20,20J,TEHPAI20,20J,RH0(3J, 5PSI(l0,10,10J,YC20J,V(20J,KSOLV(20J,IHUNT,KQUIT,KPRINT,NUMBER,KCON 6D,KSTOR,ISOLV,IT,H,KH,ISTOP,IREG,KINDM,NPTS,NREG,KREG,KINO, 7MARSTP,LOLIM,MNOEX,KBCO,KBCI,IFOUNO,IFUNCT,INOEX,INOEXM,NUMAX, 8KPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNT,IA,Kl,EPA,EPB,OEL,A,NA,NB,X, 9XA,XB,KA,KB,KC,KO,KN,XC,ALPHA,BETA,OT,OTO,ISOLVM,KROSS,KENO,K2

COMMON K3 9 KG,KGA,KNA,BKWO,FRWO,NSCR,NMOOE,ACEFT,EPC,KNORM, 2PHIAC10, 50,6J,FLX(l0, 50,6J,CURRN(l0,3,6),RADZC3J,RRO,RZO,NPTZ, 31NOZC3JtNREGZ,KNG,KNR,KNZ,IHOP,SSIGA(l0,10,3J,RSIG(l0,3J,F(l0t3), 4FSIGAC10,10,3J,OAC10tl0t3J,OC10,3J,GAMilOJ,BUCKI10,3J,EXTPClOJ, 5ZAC10,t0lt'BGIFTI29 8t,NREGR,NPTR,INORI3),RHOR(3J,MPOW,KGEL

EQUIVALENCE ( PSl(lt, PHI ( 1) ) , lP SI ( 501 J t PH ICC 1) J t ( TEMPAI 1 t ,POW C 1)),

c c c

c c c

c c

5(TEMPA(19l),POWC),(TEMPA(l92),POWAVG),(TEMPA(l93),POWCAV), 6CTEMPA(l94J,XT),(TEMPA(l95J,XTA),(PSIC521J,CURRC1J)

Q=K2 10 GO TO (20,260),KPRINT

GENERAL INFORMATION

20 KPRINT=2 30 IF (I A-1 J 40,50,60 40 WRITE (Q,570)

GO TO 70 50 WRITE (Q,580)

GO TO 70 60 WRITE (Q,590) 70 IFCKAPP) 80,90,100 80 WRITE ( Q' 600)

GO TO 110 90 WRITE (Q,610)

GO TO 110 100 WRITE (Q,620) 110 GO TO ( 120, 130) ,KBC I 120 WRITE (Q,630)

GO TO 140 130 WRITE (Q,640) 140 GO TO ( 150, 160) ,KBCO 150 WRITE (Q,650)

GO TO 170 160 WRITE (Q,660) 170 WRITE (Q,670) NREG,NA,NPTS 180 GO TO Cl90,260,260),M 190 WRITE (Q,680) KCONO,EPA

WRITE (Q,690) WRITE (Q,700) X=O.

200 00 210 I=l,NREG XA=l./CEFTCit X=X+RAO( I)

210 WRITE (Q,710) I,XA,INO(I),X 220 GO TO (230,240),KCHO 230 WRITE (Q,720) KREG

WRITE (Q,700) 235 WRITE (Q,740J VCKBJ,OT

GO TO 250 240 WRITE (Q,730) KREG

WRITE (Q,700) 245 XA=l./VCKBJ

WRITE (Q,740) XA,OT 250 RETURN 260 GO TO (270,290,490J,M

EIGENVALUE AND DETERMINANT

270 GO '-TO! 'C-'235t:l451 ,KCHO I ,, ' '1 ,,: ~(~~.~?"{(;# a,,. P'Ojj'~ i , ': ;,

EIGENFUNCTION

87

c

c

290 WRITE (Q,750) WRITE (Q,690) WRITE (Q,700) X=O.

300 DO 310 I=l,NREG X= X+RAO( I) XA=1./CEFT(IJ

310 WRITE (Q,710) I,XA,INO(I),X WRITE (Q,680) KCONO,EPA WRITE (Q,760J WRITE (Q,700) X=O.

320 DO 330 I=l,NA 330 WRITE (Q,770) X,I,PHIC(I)

IREG=l WRITE ( Q, 100)

340 00 405 J=l,NPTS 350 IFCJ-INO(IREGJ) 380,380,360 360 IFCNREG-IREGJ 380,380,370 370 I REG= IREG+1 380 X=X+RHO(IREGJ 390 DO 400 I=l,NA 400 WRITE (Q,770) X,I,PHI(I,J) 405 WRITE (Q,700)

WRITE (Q,780) 410 DO 435 1=1,NREG 420 DO 430 J=l,NA 430 WRITE (Q,790) I,J,CURR(J,I) 435 WRITE (Q,700) 440 GO TO C480,450J,KBCO 450 WRITE (Q,SOOJ

W R I T E C Q , 70 0 J 460 DO 470 I=l,NA 470 WRITE (Q 9 810) I,YCIJ 480 RETURN

C POWER c

490 WRITE (Q,820J WRITE (Q,830) XT WRITE (Q,840) POWAVG WRITE (Q,850) POWCAV WRITE ( Q, 860 ) X=O. WRITE (Q,700J WRITE (Q,870J X,POWC IREG= 1

500 DO 550 I•ltNPTS 510 IFCI-INDCIREGJJ 540,540,520 520 IFCIREG-NREGJ 530,5~0,540 530 IREG•IREG+l 540 X•X+f\ .. 0( IREG J I

550 MRil6N(tw8lOL x.POitl l f

88

c

89

560 RETURN 570 FORMAT (lHl l9X 36HTHE REACTOR BEING STUDIED IS A SLAB.) 580 FORMAT (1Hl 19X 40HTHE REACTOR BEING STUDIED IS A CYLINDER.) 590 FORMAT (1Hl l9X 38HTHE REACTOR BEING STUDIED IS A SPHERE.I 600 FORMAT (lHO 19X 54HTHE COEFFICIENTS OF AN INCOMPLETE SET Of FUNCTI

20NS ARE I 1H 14X 18HUSED IN THE MARCH.) 610 FORMAT (lHO l9X 49HA COMPLETE SET Of FUNCTIONS IS USED IN THE MARC

2H.J 620 FORMAT ClHO 19X 52HAN INCOMPLETE SET Of FUNCTIONS IS USED IN THEM

2ARCH.J 630 FORMAT (lHO 19X 53HTHE INNER BOUNDARY CONDITION IS THE FLUX EQUALS

2 ZERO.) 640 FORMAT CIHO 19X 51HTHE INNER BOUNDARY CONDITION IS THE GRADIENT OF

2 THE I lH 14X 17HFLUX EQUALS ZERO.) 650 FORMAT ( lHO 19X 47HTHE OUTER BOUNDARY CONDITION IS THE HOMOGENEOUS

2 I 1H 14X 14HDIRICHLET ONE.) 660 FORMAT (lHO 19X 47HTHE OUTER BOUNDARY CONDITION IS THE HOMOGENEOUS

2 I lH 14X lOHMIXED ONE.) 670 FORMAT (lHO 19X 28HTHE NUMBER OF REGIONS EQUALS 13, 22H, THE NUMBE

2R OF GROUPS I lH 14X 6HEQUALS 13t 39H, AND THE NUMBER OF SPACE PO 31NTS EQUALS 13, lH.J

680 FORMAT (lHO l9X 36HTHE FREQUENCY OF CONDITIONING EQUALS 12, l9H AN 2D EPSILON EQUALS I lH 14X El0.3, lH.J

690 FORMAT (lHO 14X 6HREGION 12X 5HK-EFF 12X 8HLAST PT. lOX 9HRAOIUS-C 2MJ

700 FORMAT 710 FORMAT 720 FORMAT 730 FORMAT 740 FORMAT 750 FORMAT 760 FORMAT 770 FORMAT 780 FORMAT 790 FORMAT 800 FORMAT 810 FORMAT 820 FORMAT

2 SCALE 830 FORMAT 840 FORMAT

C lH ) ( lH 17X ( 1HO 22X ( lHO 24X ( lH 21X ( lHl)

11, 13X F9.6, 13X 13, 12X F9.5) 7HRAOIUSC II, 4HJ-CM 20X llHDETERMINANTI 6HK-EFF( llt lH) 22X llHOETERMINANTJ El5.8, 16X El5.8)

(lHO 17X 9HRAOIUS-CM 13X 5HGROUP 12X l6HFLUX-NISQ CM/SECJ ( lH 17X F9.5, 15X 12, 14X El5.8) (lHO 17X 6HREGION 13X 5HGROUP llX 19HCURRENT-NISQ CMISECJ ( IH 2 0 X I 1 , 17 X I 2, 14 X E 15 • 8) ClHO 28X 5HGROUP 19X l8HBOUNDARY CONDITION) (lH 26X 12, 22X E15.8) (lHl l9X 53HPOWER IS IN FISSIONS/SECOND SUBJECT TO THE FLUX I lH 14X 32HFACTOR. 1 UNIT= 1 FISSIONISEC.J (lHO 19X 13HTOTAL POWER= 20X El5.8, 7H UNITS.t ClHO 19X 23HAVERAGE POWER DENSITY= lOX E15.8, 10H UNITSICC

2.) 850 FORMAT ClHO 19X 33HCENTER TO AVERAGE POWER DENSITY= El5.8t lH.) 860 FORMAT (lHO 23X 9HRAOIUS-CM l6X 22HPOWER DENSITY-UNITS/CCI 870 FORMAT (1H 23X F9.5, 19X E15.81

END SUBROUTINE YEGAOSCNl,N2,N3J

C WRITES ERROR MESSAGES c

0 I MENS lON ,pH I ( 10, 50 J , R HOZ ( 3 J , F IS I G ( 10, 3 , 3 J t Z Z C 100 I t 3PHIC (20J t"8KtUU20,20) ,FR1f0C 20t 20) ,CURR( 20t3 I ,POW( 190 J ,NC 10) t 500Gt·lif" :. ~ :d>, • :,;:;, ·; · ~- :. ·

C:QIIMOtf;_f.,_t:3JifYDt3lfi£6,.1'·f·3liGNU( 3J ,SSIGC 20,20,3) tRSIGAI 10 t 10,3), 2FSIGCZ0 1 3J,Zf20t20ltMAC20J ,WBI 20) ,GAMAC lO,lOitEXTPAClO,lOJ'

90

3P(20,20,3),T(20,20),S(60,20J,TEMPC20,20t,TEMPAC20,20J,RH0(3J, 5PSI(l0,10,10),Y(20),V(20J,KSOLV(20t,IHUNT,KQUIT,KPRINT,NUMBER,KCON 6D,KSTOR,ISOLV,IT,HtKH,ISTOP,IREG,KINDM,NPTS,NREG,KREG,KIND, 7MARSTP,LOLIM,MNDEX,KBCO,KBCI,IFOUND,IFUNCTtiNDEXtiNOEXM,NUMAX, 8KPOWtMORE,NFCT,NOG,KAPP,KCHO,KHUNT,IA,Kl,EPA,EPB,OEL,A,NA,NB,X, 9XA,XB,KA,KB,KC,KD,KN,XC,ALPHAtBETA,OT,OT0,ISOLVM,KROSS,KEND,K2

COMMON K3,KG,KGA,KNA,BKWO,FRWO,NSCR,NMOOE,ACEFT,EPC,KNORMt 2PHIA(l0t 50t6),FLXC10, 50,6l,CURRN(l0,3,6J,RAOZC3J,RRO,RZO,NPTZ, 31NOZ(3),NREGZ,KNG,KNR,KNZ,IHOP,SSIGA(l0tlOt31tRSIGC10,3J,F(l0,3J, 4FSIGAC10,10,3),0AC10,10,3t,DC10,3),GAM(l0),BUCK(l0,3),EXTP(l0), 5ZAC10,10),BCEFTC2,8J,NREGR,NPTRtiNOR(3),RH0R(3),MPOW,KGEL

E Q U I V AL E N CE ( P S I ( 1 ) , PH I ( U J , ( P S I ( 50 1 ) , PH I C C 1 ) ) , ( T E M P A ( 1 ) , P 0 W ( 1 ) ) , 5CTEMPA(l9lltPOWCJtCTEMPA(l92J,POWAVGJ,(TEMPACl93) 1 POWCAVJ, 6(TEMPAC194J,XTJ 1 CTEMPA(l95),XTA),(PSI(521J,CURR(l))

KQUIT=l 10 GO TO C20,30,40,50J,Nl 20 WRITE (3,70) N2

GO TO 60 30 WRITE (3,80) N2,N3

GO TO 60 40 WRITE (3,90) N2

GO TO 60 50 WRITE (3,100) N2 60 RETURN 70 FORMAT ClHO l9X 25HRHO IS NEGATIVE IN REGION 12, lH.J 80 FORMAT (lHO 19X 32HMATRIX INVERSION FAILED AT POINT 13, 22H. THE

2INVERTEO MATRIX I lH 14X 18HWAS FOR SUBROUTINE 13, lH.) 90 FORMAT ClHO l9X 27HKCONO WAS REDUCED TO 1 FROM 13, 22HBY REPEATED

2FAILURE OF I lH 14X 21HCONOIT ON MARCH BACK.) 100 FORMAT ClHO l9X 28HMARSTP WAS REDUCED TO 2 NEAR 13, 22HBY REPEATED

2 FAILURE OF I lH 14X 24HCONDIT ON MARCH FORWARD.) END SUBROUTINE CALCRS

C DATA PREPARATION FOR 2-D FLUX SYNTHESIS c

DIMENSION PHIC10,50J,RHOZC3J,FISIG(l0,3,3J,ZZC100), 3PHICC20) 1 8KWOC20 9 20),FRWOC20,20J,CURRC20,3),POWC190J,NClOJ, 4ft(20,20,3) 9 RC90J,RPC90),AA(20,20,3),8(10,3),G(l0,10t31,CClO,l00), 5DOGC2J

COMMON INDI31 1 RADC3J,CEFTC3J,GNU(3JrSSIGC20,20,3J,RSIGA(l0,10,3), 2FSIGC20 1 3) 1 ZC20,20J,WAC20J,WBC20J,GAMAClO,lO),EXTPAClO,lOJ, 3PC20 9 20 1 3) 1 TC20 9 20J,SI60 1 20J,TEMP(20,20J,TEMPAC20,20J,RH0(3), 5PSIC10 1 10 1 10J,YC20J 1 V(20J,KSOLVC20),IHUNT,KQUIT,KPRINT,NUMBER,KCON 60 9 KSTOR 9 1SOLV 9 IT,H,KH,ISTOP,IREG,KINDM,NPTS,NREG,KREG,KINO, 7HARSTP,LOLIM,MNDEX,KBCO,KBCI,IFOUND,IFUNCT,INDEX,INDEXM,NUMAX, 8KPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNT,IArKl,EPA,EPB,OEL,A,NA,NB,X, 9XA 1 XB,KA,KB,KC,KD 1 KN,XC,ALPHA,BETA,OT,OTO,ISOLVM,KROSS,KEND,K2

COMMON K3 1 KG,KGA,KNA,BKWD 1 FRWD,NSCR,NMOOE,ACEFT,EPC,KNORM, 2PHIA(l0t 50 9 6J,FLXC10t 50,6),CURRNC10,3,6J,RAOZ(3J,RRO,RZO,NPTZ, 31NDZC3J 1 NREGZ,KNG,KNR 9 KNZ,IHOP,SSIGAC10t10t3),RSIG(l0,3),F(l0,3), 4fSIGA(10t10 1 3),0AClO,l0 1 3J,O(l0,3J,GAMClOJ,BUCKCl0,3J,EXTPC10), 5ZAC10t10t,8CE~TC2,8J,NREGRtNPTRtiNORC3J,RHOR(3),MPOW,KGEL

COMMON .R~tAP.•fiW• DUFC lOt 101 EQUf¥M.ENCE C PSI C 11 t PHI C 11 J t CP Sl ( 501 J ,PH ICC 1 J), C TEMPAil J ,POW C 1)) t :J:tJ):;;X

c

5(TEMPA(191J,POWCJ,(TEMPAC192),POWAVGJ,(TEMPA(193J,POWCAVJ, 6CTEMPA(194),XTJ,(TEMPA(l95J,XTAJ,CPS1(521J,CURR(l))

NPT=NPTS ALPHA= lA GO TO (20,10),KGEL

10 NN=NPTS+1 DO 11 I= 1,NN DO 11 J= 1, NOG DO 11 K= 1 ,NHOOE

11 PHIA(J,I,KJ=FLX(J,I,KJ 20 CONTINUE

00 50 K= 1 ,NREG 00 50 I=l,NOG 00 40 J=l,NOG

40 Fl(I,J,KJ=FCI,KJ*FSIG(J,KJ 50 RSIG(l,KJ=RSIGCI,KJ+O(I,KJ*BUCK(I,KJ 60 IFCIA-1J 70,90,130

C SLAB c

c

70 BETA=l. 00 80 I= 1,NPT R(IJ=1.

80 RPCIJ=1. GO TO 160

C CYLINDER c

c

90 BETA=6.28318531 R(l)=O. RP(lJ=O.O LOW=2 KU=JNOClJ 00 120 l=l,NREG 00 100 J=LOW,KU RCJJ=RHOCIJ+RCJ-1}

100 RPCJJ=BETA*RCJJ IFINREG-IJ 120,120,110

110 LOW=KU+l KU==INOCI+lJ

120 CONTINUE GO TO 160

C SPHERE c

130 BETA==4.18879020 X=O. RClt=O. RPCli=O.O LOW=2 KU•INOClt DO ;160 I•'ltoNREG, 00 >140 : -l•LOW i KU; 0

""X'a,)(<ff\MQtll. t C l ~ l t ;{'{c ' :i'

RCJ)•X

91

c

140 RP(JJ=BETA*X*X IF(NREG-1) 160,160,150

150 LOW=KU+1 KU=INO( 1+1)

160 CONTINUE GO TO (190,220J,KNORM

190 CALL VOLINTCFI,O,FSIGA,l,lJ 00 210 K=l,NMOOE X=FSIGACK,K,l) XA=l. XA=SIGNCXA,XJ X=SQRT CABS( XJ) X=l./X XA=XA*X 00 210 I=l,NOG 00 200 J=l,NPT PHIA(I,J,KJ=XA*PHIA(I,J,KJ

200 FLX(I,JrKl=X*FLX(I,J,K) DO 210 J=l,NREG

210 CURRN(I,J,KJ=X*CURRNCI,J,K) 220 CONTINUE

CALL VOLINT(FJ,D,FSIGA,1,2) CALL VOLINT(FI,RSIG,RSIGA,2,3) CALL VOLINTCFI,O,DA,3,4J CALL VOLINTlSSIG,O,SSIGA,1,5J GO TO C300,240J,KBCO

240 00 250 K=l,NREG 00 250 I=l,NOG

250 RSIGCI,KJ=GAMCIJ CALL VOLINTIFI,RSIG,GAMA,3,1) DO 1 1=1, NMODE DO 1 J=l,NMODE

1 GAMA(I,J)=OUFCI,JJ 00 260 J=l,NREG 00 260 l=l,NOG

260 RSIGCI,JJ=EXTP(IJ CALL VOLINTIFI,RSIG,EXTPA,3,1) DO 2 I=l,NMOOE DO 2 J=l,NMODE

2 EXTPA(I,JJ=OUf(I,J) 300 RETURN

END SUBROUTINE VOLINTCAA,B,G,KHCO,KKKJ

92

C EVALUATE INTEGRAL OVER PHIA*AA*FLX c

DIMENSION PHIC10,50),RHOZ(3J,FISIGI10t3t31tZZClOOJ, 3PHICC20J,BKW0(20,20J,FRWDC20t20),CURR(20t31,POW(l90), 4FIC20 9 20,3J,RC90J,RPC90),AA(20t20t3J,B(l0,3l,G(l0tl0t31,CllO,l00), 5DOGC21

COMMON INDC3J,RAOC3J,CEFT(3J,GNUC3J,SSIG(20,20,3J,RSIGA(l0tl0t3J, 2FSIGCZOt31tZC20r20J,WAC20),W8(20),GAMAClO,lOJ,EXTPAClO,lOJ, 3Pt20.20t31tlf20 1 20J,S(60,20J,TEMPC20,20J,TEMPA(20t20J,RH0(3Jt

· 5ASI (Ill lOtiO li'ft18l,.YC20l ,KSOL VCZOJ, I HUNT ,KQUI T, KPRINT ,NUMBER, KCON ·:·: :·; ro 1 1 o. 1 l G'" ;·9o',"

93

6D,KSTOR,ISOLV,IT,H,KH,ISTOP,IREG,KINDM,NPTS,NREG,KREG,KIND, 7MARSTP,LOLIM,MNOEX,KBCO,KBCI,IFOUND,IFUNCT,INDEX,INDEXM,NUHAX, SKPOW,MORE,NFCT,NOG,KAPP,KCHO,KHUNTriAtKl,EPA,EPB,DEL,A,NA,NB,X, 9XA,XB,KA,KB,KC,KD,KN,XC,ALPHA,SETA,DT,DTO,ISOLVH,KROSS,KEND,K2

COMMON K3,KG,KGA,KNA,BKWDtFRWO,NSCR,NHODE,ACEFT,EPC,KNORM, 2PHIA(10, 50,6J,FLX(l0, 50,6J,CURRN(lQ,3,6t,RADZC31,RROtRZO,NPTZ, 31NOZC3J,NREGZ,KNG,KNR,KNZ,IHOP,SSIGA(l0,10,3),RSIG(10,3J,F(10,3J, 4FSIGAilO,l0,3J,DAC10,10,3J,D(10t3ttGAMflOJ,BUCKfl0,3J,EXTP(lO), 5ZA(10,10),8CEFT(2,8),NREGR,NPTR,INDR(3J,RHOR(3),MPOW,KGEL

COMMON R,RP,FUD,OUF(l0t10J E Q U 1 V ALE N CE ( P S I ( 1 ) , PH I ( U ) , ( P S I ( 50 1 J , PH I C C 1 J ) , { T E M P A ( 1 ) , POW ( 1 I ) ,

5(TEMPAC19l),POWCJ,(TEHPA(l92),POWAVGJ,CTEMPA(l93J,powCAVJ, 6(TEMPA(l94J,XTJ,(TEMPA(195J,XTAJ,(PSI(521J,CURR(l)J

DO 240 KK=1,NREGZ IF (KK.EQ.2J GO TO 11 GO TO 6

11 GO TO (6 9 22 9 33,33,22),KKK 22 DO 21 MM=1,NREG

DO 21 NN=l,NOG DO 21 JJ=l,NOG

21 AACNN,JJ,MMJ=AA(NN,JJ,2) GO TO 6

33 DO 32 MH=l,NREG DO 32 JJ=l,NOG

32 8(JJ,MMJ=B(JJ,2J 6 00 240 N=l,NHOOE

DO 240 M=l,NHOOE GO TO C10,60,60),KHCO

10 DO 50 1=1,NREG KU=INOCIJ IFCI-1J 20,20,30

20 LOW=l GO TO 40

30 LOW=INOCI-1) 40 JJ=LOW-1

DO 50 J=LOW,KU II=J-JJ Sf II, I J•O. DO 50 K=l,NOG 00 50 L=1,NOG

50 SCII,IJ=S(IItiJ+PHIA(K,J,NJ*AACK,L,IJ*FLX(L,J,HJ GO TO 190

60 DO 100 l=l,NREG KU=INDfl) IF(I-1) 70,70,80

70 LOW=l GO TO 90

80 LOW=INDCI-1J 90 JJ=LOW-1

DO 100 J=LOW,KU IIaJ-JJ SCIItii•O• OD·'.IOO K~lt NOG~.: · ' , · .

100 .. ~Sll$t ft•SC lit I-J+PHIAtK,J.Nl*8( IC,I l*Fl.XCK,J,MJ GO TO tlOtllO•l90J•KHCO

c

110 00 180 l=l,NREG X=l./IRHO(I)*RHOCitJ KU=INO(I)-1 I F ( I- 1 ) 120 , 12 0, 14 0

120 LOW=2 XA=2.*X*(l.+ALPHAJ 00 130 J=l,NOG

130 S(1,1J=S(l,ll-PHIA(J,l,N)*O(J,lJ*CFLX(J,2,MJ-FlX(J,l,MJI*XA GO TO 160

140 LOW=INO(l-11+1 00 150 J=l,NOG

94

150 S(l,IJ=SCl,IJ-PHIACJ,LOW-l,NJ*(O(J,l)*2•*X*(FlXCJ,LOW,MJ-FlX(J,LOW 2-l,MJ)+(2./RHOCIJ-AlPHA/RCLOw-lJJ*CURRNCJ,I-l,MJ)

160 JJ=LOW-2 I

XA=ALPHA/(2.*RHO(I)) DO 170 J=LOW,KU II=J-JJ DO 170 K=l,NOG

170 S(II,II=S(II,IJ-PHIA(K,J,N)*O(K,IJ*(X*(FlX(K,J+l,MJ+FlX(K,J-l,M)-22.*FlXCK,J,H)J+CXA/R(J))*(FlX(K,J+1,MJ-FlXIK,J-l,M))J

II=IND(I)-JJ DO 180 J=1,NOG

180 S{li,II=SCII,Il-PHIA(J,KU+l,N)*(X*2•*0(J,I)*(FLX(J,KU,MJ-FLX(J,KU+ 2l,M))-(2./RHOCIJ+ALPHA/R(KU+1))*CURRNIJ,I,M))

190 G(N,M,KK)=O.O DUF(N,MJ=O.O DO 240 1=1,NREG KU=IND(I)-1 IF(I-lJ 200,200,210

200 LOW=2 GO TO 220

210 LOW=INOCI-11+1 220 JJ=LOW-2

X=RP(lOW-ll*SCl,IJ 00 230 J=LOW,KU,2 II=J-JJ

230 X=X+4.*RP(J)*SCII,IJ+2.*RPCJ+ll*SCII+l,IJ II=KU+l-JJ X=X-RP(KU+li*SCII,II X=X*RHO (I) /3. OUFCN,M)=OUF(N,M)+X

240 G(N,M 9 KKJ=GCN,M,KKJ~X 250 RETURN

END SUBROUTINE TWOSYN

C COMBINE EXPANSION COEFFICIENTS AND TRIAL FUNCTIONS c

DIMENSION PHIC10,50J,RHOZC3),FISIG(l0t3t31,ZZC100), 3PHIC(20J 9 BK~Dl20,20),FRWOC20,20J,CURRC20,3J,POW(l90J, 4FI(20~20 9 31~R(lOOJ,RPClOOJ,AA(20,20,3),B(20,3),G(8,8,3J,C(8,100J, 5PHEC10,5-0J -"'~ '/

COMMON t<lft0(13-l;.tR;A0( 3) ,CEFT( 3 J ,GNUC 3 J, SSIG ( 20 t20 ,31 tRSIGA ClOt 10,3 J' 2F:S1G'f2a,-3.~'1.t20 9 20 I, WAf 20 J ,WB( 201 ,GAMA( 10,1 OJ, EXTPA( 10 t 10 J t

PD :J;) .J'"t.-Nf~r~

95

3P(20,20,3),T(20,20J,S(60,20J,TEMP(20,20) 9 TEMPA(20,20J,RH0(3J, 5PSI(l0,10,10J,Y(20J,V(20l,KSOLV(20),1HUNT,KQUIT,KPRINT,NUMBER,KCON 60,KSTOR,ISOLV,IT,H,KH,ISTOP,IREG,KINDM,NPTS,NREG,KREG,KINO, 7MARSTP,LOLIM,MNOEX,KBCO,KBCI,IFOUND,IFUNCT,INDEX,INDEXM,NUMAX, BKPOW,MORE,NFCT,NOG,KAPP,KCHOtKHUNT,IA,Kl,EPA,EPB,DEL,A,NA,NB,X, 9XA,XB,KA,KB,KC,KD,KN,XC,ALPHA,BETA,DT,DTO,ISOLVM,KROSSrKENOtK2

COMMON K3,KG,KGAtKNA,BKWD,FRWO,NSCR,NMODE,ACEFT,EPC,KNORM, 2PHIA(10, 50,6),FLX(l0, 50,6),CURRN(l0t3t61tRADZ(3),RRO,RZO,NPTZ, 3INDZ(3J,NREGZ,KNG,KNR 9 KNZ,IHOP,SSIGA(lO,l0,3),RSIG(l0,3),f(l0,3J, 4FSIGA(l0,10,3),DAC10,10t3),0(10t3),GAM(10J,BUCK(l0,3),EXTP(10J, 5ZA(10 9 1Q),BCEFT(2,8J,NREGR,NPTR,INDR(3),RHOR(3),MPOW,KGEL

E QU IV Al EN CE ( P S I ( 1 ) t PH Il U J , ( P S I ( 50 1 ) , PH I C ( 1) ) , ( T E M P A ( 1 ) , POW ( 1 ) ) , 5CTEMPA(l9l),POWC),CTEMPAC192),POWAVG),(TEMPA(193),POWCAV), 6(TEMPA(l94J,XT) 9 (TEMPA(l95J,XTAJ,(PSIC52l),CURR(l))

WRITE (3,11001 XXA=l.O/CEFT( 1) WRITE (3,1090) XXA NPHE=5 NPTZ=NPTZ+2-KBCO NPTR=NPTR+2-KBCO 00 1 1=1,NMOOE C( l,lJ=PHIC( I J 00 1 J=2,NPTZ

1 CCJ,JJ=PHI(I,J-1) RHOZC1)=RADZC1)/INOZ(l) If CNREGZ-1) 76,76,51

51 DO 61 1=2,NREGZ 61 RHOZCIJ=RADI(IJ/CINOZCII-INOZCI-1)) 76 DO 91 l=ltNREGZ

IF CRHOZCI)) 81,81,91 81 CAll YEGAOSCl,I,21 91 CONTINUE

00 72 K=l,NREGZ 00 72 J= 1 ,NREGR 00 72 1= 1 ,NOG

72 FISIGCI,J,KJ=FSIGCI,J) c C FLUX=SUMCA*TRIAl FUNCTIONS) c

c

DO 21 K=l,NPTZ 00 20 I= ltNOG 00 20 J=l,NPTR PHECI 9 JI=FLX(I,J,l)*C(l,K) 00 20 l=2,NMODE

20 PHECI,J)=PHE(I,JJ+FLXCI,J,LJ*C(L,KJ 21 WRITE (NPHE'K) PHE

C NORMALIZATION c

READ (NPHE'KNZ) PHE X=l.O/PHEIKNG,KNRJ DO ,'Jl K•lrNPTZ RE,AD'-tNPHE' K t PHE 09 ;\3Q•:-:·-11fl,tlt0G DO 30 J•l,NPTR

c

30 PHECI,J)=X*PHE(I,JJ 31 WRITE CNPHE'K) PHE

R(ll=RRO LOW=2 KU=INDR(l)+l DO 60 I= l,NREGR DO 40 J=LOW,KU

40 R(JJ=R(J-l)+RHOR(IJ IFCNREGR-IJ 60,60,50

50 LOW=KU+l KU=INDR(I+l)+l

60 CONTINUE ZZC1l=RZO LOW=2 KU=INDZ(1)+1 00 90 1=1,NREGZ 00 70 J=LOW,KU

10 ZZCJJ=ZZ(J-l)+RHOZ(IJ IFCNREGZ-IJ 90,90,80

80 LOW=KU+l KU=INDZCI+ll+l

90 CONTINUE IPR=l

95 00 180 I=ltNOG I PAGE=l KCK=O LOW=l KU=26

105 IF(NPTZ-KU) 100,100,110 100 KCK=l

KU=NPTZ 110 GO TO (111tll2t113J,IPR 111 PRINT 1040, I,IPAGE

GO TO 114 112 PRINT 1070, I,IPAGE

GO TO 114 113 PRINT 1080, IPAGE 114 KCQ=O

LOWR=1 KUR=6

115 IF(NPTR-KURJ 120,120,130 120 KCQ=1

KUR=NPTR 130 PRINT 1050, (R(KitK=LOWR,KURJ

00 140 K=LOW,KU READ (NPHE'Kl PHE

C PRINT FLUX c

140 WRITE (3,1060J ZZCKJ,(PHE(I,J),J=LOWR,KURJ IF (KCQl. 150t 150,160

150 lOWR•KUR+l, kUR•KUR+6 (l,AG'!•lPAGE+l

96

GO TO (151,152,153) 9 IPR 151 PRINT 1040, I,IPAGE

GO TO 115 152 PRINT 1070, I, IPAGE

GO TO 115 153 PRINT 1080, IPAGE

GO TO 115 160 IF(KCK) 170,170,180 170 LOW=KU+1

KU=KU+26 IPAGE=IPAGE+1 GO TO 105

180 CONTINUE NPTZ=NPT Z-2+KBCO NPTR=NPTR-2+KBCO IF (MPOW.EQ.l) GO TO 260 GO TO (181 9 240,260),1PR

181 LOW=l KU=INOZ(l) 00 230 I=l,NREGI LOWR=l KUR=INOR(1) 00 210 J=l,NREGR 00 191 K=LOW,KU READ (NPHE'K) PHE DO 190 L=LOWR,KUR DO 190 N=1,NOG

190 PHECN,L)=FISIG(N,J,I)*PHE(N,LJ 191 WRITE (NPHE 1 K) PHE

IF(NREGR-JJ 210,210,200 200 LOWR=KUR+l

KUR= INOR ( J+ 1)

210 CONTINUE IFCNREGI-1) 230,230,220

220 LOW=KU+1 KU=INOZCK+1)

230 CONTINUE IPR=2 GO TO 95

240 00 251 K=l,NPTZ READ (NPHE'K) PHE DO 250 1=2,NOG DO 250 J=1,NPTR

250 PHEC1 1 J)=PHEC1,J)+PHE(l,J) 251 WRITE CNPHE 1 K) PHE

READ (NPHE'KNIJ PHE X=1.0/PHEC1tKNR) 00 256 K=1 9 NPTZ READ CNPHE 1 K) PHE DO 255 I=1,NPTR

255 PHE(l,J)=X*PHECl,l) 256 WRITE CNPHE 1 KJ PHE

IPR•3 GO TO 95

260 RETURN

97

1000 FORMAT ( 72H 2 ,

1010 FORMAT (2013) 1020 FORMAT t5El5.8) 1030 FORMAT (58X El5.8) 1040 FORMAT (///54X 5HGROUP 12 9 5H FLUX 37X 4HPAGE 13/) 1050 FORMAT (/18X 6(2X 3HR = F9.5, 3X)/) 1060 FORMAT (2X 3HZ= F9.5, lX 6C2X El5.8)) 1070 FORMAT (50X 5HGROUP 12, 14H POWER DENSITY 32X 4HPAGE 13) 1025 FORMAT t6El3.8) 1080 FORMAT (50X 19HTOTAL POWER DENSITY 34X 4HPAGE 13) 1090 FORMAT (lHO 24X 'K-EFFECTIVE = 1 F9.6/) 1100 FORMAT C'l')

END

98

99

2. SAMPLE INPUT FOR THE MUO-SYN CODE

PROBLEM NUMBER THREE TOTALLY REFLECTED CYLINDER 2 2 2 30 5 1 2170 1 1 1 0 2 1 1 2 2 1 1 3 0 1 1 0 3 2 47 2 1 1 1 1 1

42 47 26 30

ORIGIN RAOZ 1.0 E 02 5.0 E 00 RAOR 5.5 E 01 5.0 E 00

1.136 1.0 1.0 1.0 1.0 E-05 1.0 E-05 0.01

THE REACTOR IS A CYLINDER THE NUMBER OF GROUPS EQUALS TWO GROUP ONE IS THE FAST GROUP GROUP TWO IS THE THERMAL GROUP

SSIG o. 0.46 E-02 o. o. o. 0.66 E-02 o. o.

RSIG 0.72 E-02 0.66 E-02 0.67 E-02 0.14 E-O;: FSIG 0.18 E-02 0.1 E-01 o. o. CHI 1. o. 1. o. BUCK o. o. o. o. 0 0.93 0.9 0.9 0.9 GAM o. o. o. o. EXTP o. o. o. o.

3. SAMPLE OUTFUT FOR THE MUD-SYN CODE

l = l :: 1 = l = l : 1 = l = 7_ :: 7 :: 7. = 1 = l = 1 = 1 = l ., 1 = l = l :: z = 1 : 1. = 1 = 7 :: i : 'l : l =

.J.C 7.3P(C') 4.7f.\OC 7.142FI6 ~.5?~A1

1l.QC476 14. ?fl"i7l }1,.f.66f-6 1 ~. C4 7 '>1 ?1.47.!il)f, n. flc<lc:; 1 7.6.lOC46 2A.57141 30.9~2~6 H.33J31 3'l.7142h 3~. c 'l<;? 1 4·).47617 4~.RG712 45.738(;7 4 7.61902 49.9CJ9<H '52.3~0Cl2 54o76}A7 "i7.142R1 S<J.5?377

~ o.o 1 2.1~09'.:' ~ 4.7619C 1 1.1478t. 1 9.51H1 l ll.QC476 z 14. z~r;n 1 = 16.66h66 7. = 1~.0476\ l : ?1.4?~56 l = 2l.~f1Qijl ~ -= ?.~.19040 7 = ?R.'Pl41 l = ji). qr, 7.36

K-EFFfCTIVF = 0.84?993

p = c.o C .9999Cl07t.E Cli o.oQa7?4f9f C'O 0.c;a~7540et: :c, Q.99f-4FOR?E v-. c.oc2,:~;nr c~ o. OR!l'::C: :)81E O•J c. ag l79513F. CC' C.9743C~96f 00 O.CJ655441t.E ·JC c. c 5 '55 7 61 7f c c c.a44?35llE CO (J.9~170249f OC· 0.91H4634£: CC O.OG29P450f % u. PA6A~?74~ ('(' C • 8n<P;448~F ·::lO O.Rr,H9S3?E: 'Ju C .PH 'i7413f 00 C'.81Cfl"'i032E OC 0. 7':l9110Hf I) I) o. 76f.J7l 1'31= 00 v. 742H628f ::JO C. 71 7 R6C 7Cf: 00 0. 61?.15375E OC G.M';5\'.41Ar: 00 C..63R 1ll5ZE 00

p :: 1?.6Cll31)

C.lll)7573AE 00 c. CJ1148606E 0(; C.91Z59462f OC• C.Ql(''H74!\E OC 0 .907Z3S92E CO C.907.75556E 00 ~ • ~97(1P7bCJF. 00 C. !'IJ024287E 00 ~.P8Z23654E CO o.~no7'ill£ co r.. 867.7591 7f 00 c.P.513oac;zr OJ C.A3A7396~F 00 0.825C6841E CO

~ = 2.11'53fl

:.~973'5?l?E OJ c.o97J7657F uO c.o:;Qf.,IC.q!)lt r:iO •J. 993°415,~ 0(, : • c;o v n c; 9 q r 0 ;) •j.qJ353P~7Sf: OJ :.s79t9452r c.o 0.<:717212JE OJ o. <;629~t,51f co :. 9'5299413?!: co 0.<l41731F4f OJ .J. 929?34 HE 00 0. 'H r; ') 14 5 ~ ~ 00 C.. 900'592VlE 00 ~.l3R450325E 00 C'. 'J6724114E OC C • 'J4l'JA404flF OJ G. fl29320Q7f 00 <.. '3~97Cl99f Ov C. 7!H03952E C'J c. 7<,4 3411 !'\F: 00 '.:.1406B61E 00 r. 715<l5R54F o.:. C.tt'LH1°7'JE J(; J. MlR0054F 0':'• c. 63.,410'HF: Cv

R = 14.ql'·76Q

c.s"'>747nr oo C.88647884E 00 o. 9~'>61 :n 1r oc• ~. 81' 3"i<Jt)l OE ::>0 o. 880~0 75 7E 00 0.'37606Jl9E 00 (1. 870560'i1Jf 00 0. %3Q181 'lE 00 (;. 135&14794~ 00 C.'t472'i'H5E 00 O.'J37244'51E CO v.82'->ll26bl: oo G.'tl3CJJ'iltb~ 00 0. 8006694?E JO

GRC'UP 1 fLU)(

I< = 4.23017

::l.CJ'3949CSIE OJ u.9P9ll51'tF OJ :).09R254S':if 00 v. 9'36CO<t'HE iJiJ o.qaz4~olvf ':Jj c.o77f1126U: CO C.91l472~0F. Cv J.q64t.6C:~lf O·:l J. 055 H2•J6E GO :>. q4S4783RE ·JO 0.9~4306141;: J•j 0.92190540E •'JO ').9C'829390f 00 O.B934fl93vE 00 J. S775?64of oJC J. 861.'4JC'iOE 00 J. ~4?144 HE 00 o. 322179"36(:: co O.ll023H16f Ov C.78083127E Ov ii.75831175E Uv -::>. 734 79~~7'= CJ :J.1ld1046f 0.) O.~A41!741AE :Jv :). 6'i8563HE C0 ·).63l3~944E 0J

(';~OU~' 1 flUX

R = lb.'l~307

O. A57WH45f. Ov 0. 357S6713t UO O.R567279CF .:'0 J.a5~77v2'•t oo 0.85l68A5CF Cv o.a474R340E :>o 0 • R4Z 16 lRbE 00 o. 835 H8J61:: 00 o.q21122onE oo 0.111961607.1: 00 O.IJ09 1J31111£ co c. 799182711: 00 0. 78731J344E CO ll. 7745491 9E 00

R = o.34o15

l. 9766 H36E 00 •J.97639H7E 00 0.97S44'}56F 0.:' r.HJZ2792f: OJ t .9697 336~£: 00 (..'.Q64Cl4323E 00 I-.95RAH311E: 00 C.9'jl56676E 00 C.H301!J81E Oi.J C .1 H22432E Ou c.n2191lOE c~ C.90995705E 00 C.'39652210E OC O.i'3190931E 00 G.B6io,l'i235f 00 C.34924q21E 00 ~.·B122'397f uO C .II 121 H4 8E 0 0 O.H19237CE 0·) 0.71C.70999E OC c .74114131 99[ J·) C~. n527051E oo C. Bll02l2E JO c.o7599630I:: oo 0.1)5C02644t: 0~ C.6?.32C·H5E CO

~ = 1Q.C3B44

C.~l761Z4UE CO ::, .il ~H5l85E Ol ('. 32653922£ 00 c,.'\2464921E lO C.3216H21F 00 O.R l76lo82E 00 0 .•H 24A'i40f 00 0 .8:J62'35q8E 00 0. 1990 3293E 00 0.79072934~ :Jv o.nl3B66lt: JO 0.77101654~ JJ C.75Q~l31lE 110 11.147251\JQE v~

R: fl.46154

J.<l'.>93924~E t)J O.Cl5911RHE IJC J.9581H514E Jv ..:.95bvul70E oo v.9~2'J6716E O;;

00 Q•.)

v • 9 41 a o 1 9 7r \.'o94L9C96H: 0.9 3472278E o.n63l769E ·~.9167\;787E J. 905B 7v9 7E O.B9384782E 0.8'K6~U64E C. A6o 2965 3E o.ao;c~l744E :; .a 3421242E O.Bl6'il175E J. 7977 H691:: C. 77790266E ;.; .1'J 70640'tE G. 7 35 2 29 L 3E u. 71242797E u .6B8b 1H(,JE J .664u2668E ;, .b ~A5159')E 0.612lb742E

Jv ')J ()I) 00 vv 0.) 0~ (I) 00 :){) 00 ""' .., ... oc oc 00 00 00 ,)Q 00

R = 21.1S3Bl

~.79b56589E uO o.79o30959E uo C..795'i2110f 00 u. 7Q31C677E 00 v.79084003E I)O IJ.78693628E 00 .J.78199786E ilO o.176CH02E OJ u.76904953E ~co J.76105583f oc J.752v63!'1£: Jv 0. HnA289€ •JC ~.73ll21.>o7f \JJ J.7192J925E 00

PAGE 1

R = 10.57692

0.93820125t: 0(.; 0.93192820E Cv J.93701434E 00 C.934A719l[ 00 C.93151665E OC O. 926915 8'3E 00 C.921C9573E OJ v.91406763E oo 0.90'ii\4779E 00 o. '39b4434C)f 00 o.aq;qsl14E or 0.87409383E 00 0. 861188 35E OC O.I34715140E 0(, 0.832Cl283E 00 0.8l571456E (10 o. 79846481E 00 G.78010180E 00 c.7607C946E co 0.74033076E 00 o.71891793E oo 0.6966799r;E 00 0.6H46430f 00 O. 649349 09E 00 0.6244Cl39£ 00 (.1.59'11')34361: (i(.l

PAGE 2

.R = 23.26918

C.76'iv2866E CIO o. 76~ 1786 7E 00 0.761t0l622E 00 O. 762Z7689E 00 o. 7'i952l78E 00 O. 75S71283E 00 0.751.>3045E 00 o. HS29982E OC o.ne59429E oo o. no ens ne oo j. 12227985£ oc u.H26CJ429E oo o.70217l04E co v. 6C)0726~6F OC

1-' 0 0

1 = 33.33331 0.81032288!: 00 o. 786357?.8F. 00 o.7&070321E oo 0.7HA9167E 00 0.70634985E 00 O. 678H524E 00 1 = l5. 71426 c·.7<}45076BE co O. 77100962E 00 0.74585593E 00 G.119'561J6E 00 C.692562C)4E 00 o. 66513413E: 00 z = 38.09521 (;.71764!181E 00 o. 75464898E 00 0. 73002 86 3E 00 0.70429760E 00 0.67786586E 00 0.65101892E CO l = 40.47617 0.75976634f co 0.13729'527E 00 0.71324104E 00 0.69810159E 00 0.66227758E 00 0.63604796t 00 7 = 4z.sr;nz C.740A7199E CO C.1l!\96~01E 00 0.69551021E 00 0.67099607E: 00 0.64581436E 00 0.6202~693£ 00 ., -= 45.7.3~07 C.72102994E 00 O. 6CICH042 ~E 00 0.67687607E 00 0.65301812[ 00 u.62F351048E OJ 0.60361797E 00 l = 47.619C2 C .70023316E CO O. 67952l98E 00 0.65135167E 00 0.634181261: 00 ~.61C37999E 00 0.58620512£ 00 7 = 49.9CI997 ('. 67 85156 7E 00 O. 6584460lE 00 0.63696235E 00 0.61450988E 00 c.~9144622E uo O. 56B020 64E 00 z = 52.38092 0.65590513E Ov o. 63650393E 00 c.61573613E: oa o.59403151E o" .J. 571 73634E 00 0.54909116!: co z = 54.76187 C .6324l94lE GO C.6131l36~E 00 0.59369016E OU 0.57276344E 00 0.5!>126703£ JO O. 52943307E OC 1 = 57.14282 0.6081?122E GO c. 5qJ13 325E 00 0.57087S1<;E 00 c.sso75467E oo 0.53008 354E 00 0.50908804£ 00 l = ~9.'i1377 <:-.58302492E 00 C. S6577814E 00 J.54731655E 00 0.52R02253E 00 0.5082036 3E 00 0.49807400E 00

GROUP 1 FLUX PAGE 3

R = 25.1fl4'i5 ~ = 27.49997 R = 29.615~0 R. = ~1.13067 R = 33.84604 R = 35.96141

1 = 0.0 C.13316914E C..O C. 70Q<Hl239f OJ J. 66e3J802E 00 C.63485038E OC 0.60021716E 00 o.56396770E oo l -= 2.3~095 C. 1329? 667E 00 G. 70074BA6F. 00 0.668Cg450E ·JO C.AV~63325E 00 u.6oool749E oo O. 56378168E 00 1 = 4.76190 G .7322C4e5E OJ L.700C5810E 00 0.6674?')81E OG o.~340l252E oo o.59942615E oo 0.56322634F 00 z = 7.14286 0 .13052704E 00 O. 6Q8454C?E 00 0.66589630E CO 0.632559541:: oc O.S9805268E 00 o.56193632E oo l = .:J.')73'31 C'.7278857AE CO 0. 6Q592714E OQ O.t>6348678E 00 G.!>3C27072E OC 0.5958A921E (;Q 0.55990422E 00 1 = 11.90476 0.72420~llE CC o. 692492371: 00 0.66071204E 00 0.627159831:: 00 J.59294784E JO 0. 55714023E 00 l = 14.28571 C. 71Q748SOF 00 0. 68'314743E 00 0.65606964F 00 0.6232 2468E 00 o.58922114E oc 0. 5 53643 94E 00 1. ;: 16.66666 C.1l425647E OC G.68299649t 00 o.651v6344E co C.6l846906E 00 C. 5 84 7308\lE CO 0.54941905£ Q(j 1 -= 1~.04761 C. 7C7e3·J01E 00 C.67675185E CO J.64520502F 00 C.61290389E 00 o.57946932E oo o. 5444 7550E: co "l = 21.42856 C.70C47015E 00 (,. 66971451E 00 0.638495501: 00 0.60653037E 00 v.51344195E oo o.53BR1466E oo 1 = 23.8(;951 C.69219404E CO 0.66180182E 00 0.63<.9515AE 00 0.59936404E OC o.56666839E oo o.53244Bl1E oc l = 26. 1904t> O.M300778E 00 o. 6'53Cl881E 00 0.62257802E 00 o.59140956E oo o.55914772E oo o. 52538151E 00 7 = ZIJ. 57141 0.67292~74f co o. 64337754F. 00 0.61338615E 00 0.58267778£ 00 ().550897.1 7E 00 o.5176243SE oo z -= 30.9'i:?36 C.66195488E ()•) G.632890z:?E 00 0.6G338759E 00 0.5Bl7984E 00 J.54191232E 00 0.50918674f 00 7 = ~3.33331 O.h501l746E OJ c.o2157220E oo o.59259707E oo 0.56292975£ OG O.'H222179E 00 O. 5000 82 08 E 00 l = 35.71426 o.&374:n69E oo C. 6094'194 JF 00 0.58102989[ 00 o.55194157E oo J.52183312E 00 0.49032068E 00 l. -:: 38.0°521 0.6239C02lf 00 C. 5965058?.E 00 0.568699()(;[ 00 G .540 l2795E 00 0.51075858£ 00 O. 47991496F 00 7 = 40.47617 0.60c:)'55280E 00 C.58278823E 00 0.55562091£: 00 0.52780455F 00 0 .49901211E 00 0.46887851E 00 z = 42.85712 C.594400<;Bf 00 C. SA830144E 00 :J.54190938f 00 0.51468444£: 00 o.48660S15t: oo 0.457222 B2E 00 z = 45.23807 C.'H8473'J1E 00 c. 55307353£ 00 0.5272913CiE 00 0.50089329£ 00 0 .41356957E JO 0.44497180E 00 z = 47.6}QQ2 C.5,17A570E 00 c. 5H11A26E 00 0.51207972E 00 0.49644316E 00 0.4599C71lE 00 0.43213516E· 00 1 -= 49. oqqcn (.5443'5831E 00 v. 52045584£: 00 0.496l9389E JO 0.47135276E 00 0 .44564056E 00 0.41872984E 00 z = 52.38092 <:.526717341:= 00 C.50Hll3:lE 00 0.47965717E 00 0.45564441E 00 0.43078917t: 00 a. 40477526E oo 1 = 54.76187 C. 50137~4CE 00 C. 48509979E 00 0.46248591E 00 0.43933213[ 00 0.41536659E Ov 0.39028364E: 00 z = 57.14282 O. 4A7!H~053f 00 o. 466457nE ou 0.444712'38£ 00 0.42244'lllE 00 0.39940476E 00 0.37528610E 00 z = 59.52377 0.46174125E 00 0 • 44 72G2SJE 00 0.42635'>06E 00 0.40501028E 00 o.38291134E oo 0. 35979468E 00

GROUP 1 FLUX PAGE 4

R = 38.07678 ~ = 40.19215 R = 42.30753 ~ = 44.42290 R = 46.53827 R : 48.65364

7 = o.c C.52'>M218E 00 C.4949l085E 00 0.4414'l617E 00 o. B505965E Ou o.3458vl59E oo O. 2CJ385715E 00 l = 2.~8095 C.5254C)Q64E OJ v.4847635f>l: oc 0.441?.9670t: 00 o. ~9493608E :JO 0.34569502E JO o. 2937682 9E 00 z = 4.7H9J L.'524<?7345E 00 C. 4842870.~E 00 U.44086337E 00 0. H454~ 11 E Ou .J • 34535635E 00 0.2934S087E 00 l = 7.147Ab 0.5?H715FIF t::O C.4AH7879F OJ 0.4~9B5504E OJ 0.3Q36467Bf 00 0. 34456724E Ov 0.29281062E 00 l ::: 9.5n?1 C..521P7854f CJ c. 4"14334'1!:: 00 C.43~26135E ':iO 0. 39222682t: ou J.34332514E JO 0.29175568E 00 z : 11.90476 0.5l930195E 00 v. 4H0'56421= 00 0.43610299E 00 0.39028966£ 00 0.34162939E 00 O. 29031450E 00

t-J 0 t-J

l = l4.2J3Cj71 l = 16.66666 Z = P~.0476l l = 21.4281)6 z "' 2 3. ~09'-i 1 z : 76.19046 z .., 28.57141 z = 30.9<;?:-6 1 = ll.33'H1 z = 35.71426 1 = 38.0<:1<;21 z = 40.47617 1 : 42.~5717 l -= 45.?.38C7 7 -= 47.6l«W2 l = 49.9QQC;I7 7.. ., 52.38092 z ; 54.761 ~7 7 = 57.14?82 1 = 50.52377

1 = OoJ z = 2.38095 l = 4.76100 z = 7.14286 z : 9.52181 z = 11.90476 l :: 14.2~571 7 = 16.66666 1 = l<l.C4761 'l "" 21.42856 1. = 23 .1:1rc:~s1 l = 26.19046 7. -= ?B.r;7141 ?. = 3C .95216 Z -= ·H.33331 7 :: 35.71426 7 = 38.0Q<;21 ., -= 40.47617 7 "' 42.85712 7 = 45.21807 z = 47.61902 Z : 49.CICl997 z ,.. s2.3~on l = 54.761Fl7 z "' 57.142~2 z = 159.152377

C. 51604277E 00 C.C:,l?lC475E 00 C. 5H49707E CO C.5C2221'52E 00 C. 406 2872 3E 00 C·. 4B970030E ()0 C.4R247Cl0E 00 C.474('.C544f: 00 0.466ll98lE 00 c..457Cll47E rc C.44732255E 00 C. 43103556E GO C.42611l24f OC C.41475?54( 00 0.4J27878~E uC C.39029318E OC 0.3772q63l[ C(l C .36377fi53E 00 C. 3497CI9C)3E CC C.3J5~608AF 00

P = 5L•.7690l

<.;.23959398E 00 C. 23952186F CO C .2l928713E vC C.23874l46E 00 c.23B!1178F oo 0.2367C65'>E 0:: c .23522002f co c. 2334?4~4E 00 0. 2 H '2 '52 7E ro C.22A9?3<J81:: ou C.2~621775E CO C. 22Hl5C4E CO C.2199l9(19E GO C.2163342FJF CO lo2l246856E CC C.20832U<;E C.v C.?03CI0075E 00 O.lQ<l?ll72E C,O v.19425R93E OG O.lq90'5491E 00 0.18360168E 00 C.1779072<?E OC 0 ol7197847E 00 0.16582018f 00 C .15944'lh4E 00 C.1'52M<n11:: 00

0.47604CJ48F 00 ('. 47?4l6'58E 00 C. 46!H66BF 00 o. 463300351: 00 O. 457R2584E 00 0.451749321'= 00 0.445079llE 00 C.4378?41'1E 00 c. 4?Q99697f: 00 C. 4216036?.F OC C.4126'i649E 00 e;.4C31b61lf 00 O. 393144llE GO c. 31:'261062f oc C.31l57339E 00 O. 36004 B4E OG 0. 34904R5H: C·l.l O. 33'55FJ70AE 00 o. 322692?81: 00 c,. 3C9372o6E oo

~ = 52.8A43A

O. l83'54768E 00 C. 18H9290E 00 C. 18 Hl361F. 00 O.l8289518E 00 o. 1922369lr 00 c.l8133652E oo 0.160l9760F 00 o. 11882246f 00 o. l7721397E 00 C.1H31391E 00 c. 11330134!: 00 J.1710C10ZE 00 0.16R47605E 00 o. 165729761; 00 c. 1627684«\E 00 0.15959150E 00 c.t'56204B8E cc o. 1516127'5f 00 (i. 14"8184''= 00 0. 144831 78E 00 G. 14065427E 00 O.l3629186E 00 o. l3114987E 00 O. 1270'4210E 00 C.12215179E 00 C.11711067f 00

0.43336535E GO 0.430C5812E 00 G.42618912E 00 u.42176038E oo 0.416 77660E 00 o. 411 ?4475E 00 J. 40517211 t ()J o.39856809E co 0. 391443HE 00 J.3~380277E CO o. 37'565792f 00 0.36701900E 'JO 0. 357894B4E OiJ 0.34830606E 00 0.33825856!: 00 C.3277663uE 00 o. 316A4327E 00 0.30549890E CU 0.2QH60491:: 00 o.28163~40E co

(;ROUP 1 FLU)(

R = S4.99Q7b

J.1263R43~t: GO C. lZb34677E CO O.l2622339E 00 ~.l259352bE 00 O.l2548ZOBE 00 C.l24Ab207E 00 CJ.l24C·7798f 00 O.l23l3C9AE 00 0.12202340E CO 0.12075651F 00 0.119329451: O·J 0. ll774546E 00 C .ll6C0685E 00 .:J.ll4ll!:>f\3E 00 O.ll2C7688E 00 O.l0988C)2lE 00 G .107551lt2'= 00 o.toa;oe3a"~E: co u.l0241135E oo 0.99716260E-01 o. 96M9680f-Ol 0.93845904E-Ol 0.907l8627E-\J1 0.874699q5E-Ol 0.~4109664E-01 •). 8063H468E-Ol

GRUUP 1 FlliX

o. 3878 3944~ 00 C.38487953E l.iO O.Hl41116E 0() 0.37745446E 00 0.372H4C7f 00 'J.36804330E 00 o.3,260903E oo G.35669822E 00 0.35032272E 00 C.34348470t JO C.31619553E 00 G.32846415E 00 C.32C29~37E 00 G.31171715E 00 C·. '\0272532E 00 o.2Q333550E oo J. 2fJ35599RE 00 C.2734C698E 00 0.2t.?902C3E 0•) 1.i. 2 5 20 5112 E 0 l.i

R = 56.24976

0.92487454E-Ol 0.9?4'iQC)77E-Ol C.92369616E-01 C.CJ2159854E-01 0. ~ 1!'1272 73f-O 1 O.ql37l444E-Ol C.C)G799o10f-Ol 0.1Gl06666[-J1 O.Rll2Q6222E-Ol 0.~8368952E-Ol C.~J732467CIE-01 O.B6165n07E-01 0.84893?.86E-C1 o.a 3509386E-01 C.R2011303E-Ol C.<JJ416441E-01 0.7q7Q9960E-OL c.7o89Q946E:-01 o.74988J08E-Ol O. 7lQ79211E-01 c.7v874155E-01 0. 6il676ll4E-O 1 0.663~7415E-OL 0.64010203f-Ol 0.615'H120f-01 o.59010949E-01

0.33948427E 00 0. Hb89144E 00 o.33386290E oo O.H039486E 00 G.32649046E 00 0.32215685E 00 C.31740016E Oil 0.31222618E 00 0.30664611f 00 o.30066073E oo 0 .29428047E 00 0. 2 87512 9 6E 0 0 0 • 2%3 6 '>0 5E 0 C J.27285391E JG 0.26498336E 00 o.25676441E oo 0 .24820775E 00 0. 2 39 3202JE 00 0. 2 30 12 5 37E 0 \; J.22062749E 00

R = 57.49976

0 .6040'H02E-O 1 0.60387A27E-Ol 0.60328834E-Ol 0.6019116ZE-01 0.59974533E-Ol 0.596782L9E-01 0 .593v3399E-Ol 0.58850806E-Ol C .5 8321480E-Ol o.~I715934E-Ol 0.570 33852E-Ol u. 56276791E-01 0. '5 544582 SE-0 1 o.54542016E-01 u. 5 3567465E-O 1 0.52521907E-Ol 0. '51407389E-O 1 o. 5u2 2518 7E-u1 0.48976470E-Ol 0.47664464E-01 vo46289623E-01 0.44853974E-01 0.433'.;CJ220E-O 1 \J. 4180656 7E-O 1 0 .402tJ0457E-Ol u.38541414E-Ol

0.28849149E 00 0.28628975E 00 0.2~31l453E 00 0.28076786E 00 0. 21744991E 00 O. 27316717E 00 0.2697249lE OC 0.26532817E 00 0.26058662E 00 lJ.25550026E 00 O. 25007844E 00 O.Z4432743E 00 0.238253COE CO O. 2 318 7C29E 00 0.22518206E 00 0.21819782[ 00 O. 21 092629E 00 o. 20331355E CO O.l9556004E 00 O.l8748903E 00

PAGE 5

R = 58.74976

o.29708762E-c.1 0. 29699929E- Cl o.29670913E-ot C.29603206E-01 o. 29496659E- Cl o. 29350'}25E-01 0.29lb65R3E-01 0.28943989E-01 0.28683655E-Ol o. 28385837E-01 o. 28050371E-01 0.27678039E-01 o.27269349E-CJ1 0. 26824843£-Cl 0. 26345536E-Ol 0.25831308E-01 0.25283165£-Cl 0.24701737E-01 0.24087'.;93£-01 O.Z3442321E-01 0.22766151E-01 o. 220600 70E- 01 0. 21324918E-01 O. 205612 89E-Ol 0.197713 7'.;E-01 0.1891)5428E-01

PAGE 6

1-' 0

"'

1 = o.o 1 = z. 38095 l = 4.76190 1. = 7.14286 l = 9.5231H l = ll.90476 l = llt.28571 l = 16.666f.6 l = 19.047H 1 : 21.4213!\6 z = 23.80951 l = 26.19046 l = 28.57141 l = 30.95236 1 = 33.33331 l = 35.71426 l = 3~.09'521 l = 40.47617 l = 42.85712 l = 45.23807 1 = 47.619C2 z = 49.99997 l = 52.38092 1 ,.. 54.761t'7 l = 57.14282 z = 59.'52377

1 = 61.90472 z = 64.28568 l = 66.66663 l = 69.01t758 l = 11.42853 z = 73.8(·948 z = 76.19043 z = 78.57138 1 = 80.95233 l = R3. 33328 l = 85.114?.3 z = 613.00~18 l = 90.47614 l = 92.85709 1 = <)5.23804 1 = 97.61899 z = ()9.99994 l =100.99q94 7 =101.99994 l =102.9<1994 1 =1C3.99994 l =104.99994

~ = 59.99976 -c .ll065.,26f-06 -C.l1062173E-06 -C.U05l336E-06 -c-.11C26106f-06 -C.109R638<)E-06 -G. 1Ci 9 3211 5f-06 -C.I0863459E-06 -C.107R0553E-06 -c .10683 584E-06 -C .IO'H2626f-06 -C. 1C447684E-06 -C.103CR997E-06 -G.l015h782E-06 -0.99912199£-07 -C. 9"126 74RE-07 -C.96211465f::-07 -C.94169764E-07 -C.920C4143E-07 -c.. a<n16 764E-o7 -o. 873132 54£-07 -C.847<~4692E-07 -0.82164718E-07 -().7942657(;E-07 -0.76582523E-07 -C •. 7364C251E-07 -0.70601061E-07

R = v.O

0.6098G242E 00 0.58C69402E 00 C.5'5C79067E 00 t.52019936E 00 t.48893221jf 00 t.45702571E 00 C.42450845E 00 C.39138430f 00 c. 35779810£ 00 C.32375228E 00 C.28928542E 00 G. 25442696E 00 C'.21914488E 00 C.\8367541£ CO C.147<)7'539E CO O.ll2088t.8E 00 0.76052189f-Ol 0.60921367E-Ol o. 45824539£-01 0.30761861\E-01 0.15374731£-01 -c. 5 &z7o; 74IE-tH

R = 2.11!>38

0. 60818619E 00 CI.5791'5497E CO 0. '54933101E 00 o. '51882070E 00 0.48763627E 00 0.45581406£ 00 o. 42338288f 00 0. 19034677E 00 O. 35684949E 00 o. 32289362E 00 o. 28'351771E 00 O. 25375152E 00 O. 21856350E 00 O. 18318766E 00 0.14758164E 00 o.nt78887E oo o. 15847924£-01 0.60760066£-01 0.457C2886E-01 o. "J0679606E-01 o. 1S333652E-Ol -o. 5619t723E-07

GROUP 1 FlUX

R = 4.23077

0.60338730E 00 O. 'H4~8490E 00 0.54499668£ 00 0. 514 72700E 00 :>.4837A795E 00 0.45221615£ 00 0. 42004055 E 00 0. ~8726586E 00 0.35403264E 00 0.32034391E 00 o.2862nt3E oo 0.251741j88E 00 0.21683717£ 00 0.1Al73903E 00 0.14641273£ 00 o.ll09oosar oo 0.75241685E-Ol 0. 6021H 016E-O 1 0.45341589f-01 O. 30435294E-01 0.152ll646E-Ol

-o. 559401 ozE-07

R = 6.34615

0.~9556377£ 00 0.56713456E 00 C.5H93067E 00 c .50805318£ 00 0.47751427E 00 0.44635063E O'l 0.41459179E 00 0.3g224328E 00 0.34944040E 00 0.31618720E 00 0.28252202E 00 0.24847615E 00 0.21402276E 00 0.17937756E 00 0.141t50711E 00 0.109451.72£ 00 0.74253619E-Ol 0.59499722E-Ol 0.44752281E-Ol O. l00l6766E-01 0.15012607E-Ol

-0.55521518E:-07

R = 8.46154

o.58501571E oo 0.'55708975E 00 0.5284C400E 00 0.49905521E 00 o.46905595E oo 0.43844259E 00 0.40724570E 00 0.37547177E 00 0. 34324920E 00 0.31058306E 00 :>.27751189E 00 0.24406815E 00 o.2102l826E oo 0.17619395E 00 O.l4193833E 00 0 .10750115E 00 0.1 Z922111E-Ol 0.5 8445718E-01 0.43<)57170E-01 0.29498924E-Ol 0.14743999£-01

-o. 549420B8E-07

PAGE 7

R = 10.57692

o.57208151F oo O. 54477245E 00 0. o; 16 7 22 14E 00 0.48802173E 00 0.45R684Zl£ 00 0.42874575E 00 o. 39823788E 00 o. 36716837E OC o. 33565742E 00 0.30371147E GO o. 271368 80E 00 0.23866338E (;0 o. 20 55 7564E 00 O.l1229062E 00 O.l3878906E CO O.lO!:il089ZE 00 0. 71290493E-01 0.5115Z230E-Ol o. 42981ZOitE-01 O. 28838601E-Ol 0.14414202E-Ol

-0.54204914E-C7

1-' 0 w

1 = 61.%472 z = 64.28";68 z = 66.66663 z "" 69. 047'58 l = 71.428'53 l = 73.8(948 l :: 76.19043 7 :: 7A.57138 7. .. AO. 9!5233 7 = fB. 333?8 1 = 8'5.7147.3 z = ~~.00'518 l = 90.47614 1 = q 2. f.!Ci 7C9 l : 9'5.2~804 z .. 97.bl899 l = ~9.99994 z "'100.99994 1 =1C1.o9Q94 1 =1C?.99994 l =103.09904 l =104.99Cl94

z = 61.9047? l = 64.2P'568 z = 66.1>6&63 l = 69.047t;6 l = 11.421J53 l = 73.8(948 z :: 7~.19043 l = 76.'57138 l = ~0.95233 l : P3.33328 z = 85.11423 1 = ~F\.00518 7. = 90.47614 z = 92.85709 Z = 95.231:lC4 l : 97.61699 z = Cl9.99994 7. =100.99994 1 =101.999<)4 7 -=102.QCI994 z =103.99994 7 =104.90994

p = 12.69230

C.. 55716300E Cv O. 53056'5'50E 00 c-.so3247nE OC; 0.47529'542F CO G.44672132E OC 0.417'56141£ OJ C. 3R784~44E 00 r.357'50t?7F oo C. 32690 102E 00 C.2957~590E 00 C. 2b42A'H~I; OC c.. 23242986f 00 C'.20C2G96RE 00 C•.l671A910E 00 0.1351 5753E CO G. 10235095E 00 C.f.9410145E-Ol 0.~565S78CE-Cl 0.41A54132E-Ol 0.2AC7~AC3E-01 C.140332C6E-Cl

-o. s 3314814E-o7

p = 25.38455

C.4469A63AE OC 0.42564613E OJ 0.403135A2E 00 Oo 3Pl308t;8E 00 0. 3'i837668E 00 C. 33497345E 00 C.311P255E CO 0.286867~0E 00 c .2622425CJf co Cl.23721000E 00 C • 2119P416f 00 O.l8b4233CE 00 c. l6059381E 00 o.l34573'>8E oo O.l0837954F 00 C.82040429E-01 0.555948Q5E-01 C.44576220E-Ol C.3l483125E-C1 (:. 221t06526E-01 0.1120011 Cjf-01

-0.449754b5E-07

R = 14.80769

o. 54067969f 00 O. 5l486'36RE 00 O. 48R36046f 00 0.46123421E 00 o. 43 350 375f 00 o. 40520430E 00 c. 371)16960f 00 C. 34 70C'l72E 00 0. 31 7 2 2 6 51 E 00 O. 213702<)62E 00 C.?5645618E 00 c. 22554356( co 0.19428104E 00 O. 162Fil61c;E 00 C. 13114643E 00 0. 90305'5llE -01 o. 61334712!:-01 G. 54006733E-Ol O. 40607Q61E-01 C. 27231533E-Ol o. 13611495E-01

-0. 522764F!OE-07

R = 27.49992

O. 42735839E 00 G. 40695512E CO O. 38600719E 00 O. ~6456406E 00 o. ~4263915E 00 0. 12026285E 00 O. 2Q746860E 00 0. 274?6982E 00 O. 25012557E 00 C. 2?M4A90E 00 O. Z0267266E 00 o. 11823368£' 00 0. 1'H53954E 00 o. l2866Z77E 00 o. 10361814E 00 o. 78414765E-Ol O. 53149439E-Ol O. 42t;93997E-01 O. 31986337E-01 0.21393005E-Ol O. 1069it150E-01

-o. 43l0123lf-o7

GROUP 1 FLUX

R = 16.92307

0.52303553E 00 0.49806631E 00 0.47242445E 00 0.44618285E 00 0.41935551E 00 C. 39197743E 00 G. 36408287f 00 o. ~3'>68323f 0:) 0.30687112E 00 0. 77765721 E 00 0.248C7864E Ju C.21817338E GO :J.lij793'565E 00 O.l5749454E CO u.l268c;448E oo 0.96047997f-Ol 0.651lbl07E-01 0.52235957E-Ol 0.39270025E-Ol 0.26326086E-Ol 0.1315911J9E-01

-0.51093910E-07

GROUP 1 FLUX

R = 29.61530

0.40743595E 00 0.38798368E CO o.3680124JE oo o. 34756869E 00 0.37666564E OJ 0.30533236E 00 0.28360045£: 00 0.26148331E 00 0.23903650E 00 0.21627271E 00 O.l 1H22318E 00 0.16992331E 00 O.l4638C42E 00 0.12266505E 00 o. 96787904E-01 J.74171:l557F-Ol 0.506718ft9E-01 0.4058161tOE-Ol 0.30466698~-01 0.20365383E-01 0.10180481E-Ol

-'). 410 86792E-07

R == 19.03844

o.50459599E oo 0.480'50660E 00 0.45576978E 00 0.43045288E 00 0.40456975E 00 C.HR15481E 00 C.35124796E 00 C.32384652E 00 G. 29604930E 00 0.26H6309F 00 C.239~2457E 00 c.2104124oe oo O.l!H30487E 00 0. 1 c; 19 3427E 00 o.l22371C8E oo 0. 92646 360E-Ol 0.62801003f-01 0.50382670E-01 0. 31870340E-01 o.zs377985E-ot O.l2M5534E-01

-0.4 1H70904E-07

R = 31.73067

C. 38703841E 00 0.36655984E 00 0. 34958833E 00 o. 33016801E 00 C.31031156E 00 o.Z9004633E oo o.26940238E oo c.Z4839222E oo C.22706932f 00 O. 2C'544523E 00 o.tB354994f oo O.l6141641E 00 o.l3905162E oo o.t1652583E oo 0.93844712E-Ol o.7l03B246E-Ol 0.48l38998E-01 0.38522266E-01 0.26912399E-01 0.19315679E-01 C.IJ6557215E-02

-0.38930256E-07

R :: 21. 15 381

0.48565519E 00 0.46246952E 00 0.4 3866235E 00 0.41429520E 00 0.38938218E 00 0.36395699E 00 0.3 3805460E 00 0.31168830E 00 0.2849H69E 00 0.25780368E OC C.23C.33404E Ou 0.202!:i6364E 00 O.l744q450E 00 G .146 2 2456E 00 O.ll776629E vO 0. 89155674E-O 1 0. 60427144E-0 1 0.48476167E-Ol 0.36430l73E-vl 0.24402313E-01 O.l2198v50E-Ol

-0.4 830960ZF-O 1

R = 33. A4604

0.36592609E 00 0.34845561E 00 0.33051860E 00 0.31215769E 00 0.29338461E 00 o.27422523E oo 0.25470752E 00 0.23484278E 00 0.21468321E JO o.19423920E oo o.11353874E oo o.15261281E oo O.l3146675E 00 0.110 17257E 00 0 o88729858E-Ol 0.6 7169070E-Ol O.lt5520153E-Ol 0.36393218E-Ol o.21306616E-Ol O.l8232968E-Ol o.91144219E-02

-o. 36610368E-o7

PAGE 8

R = 23.26918

0.46641755E GO 0.44414997E 00 O. 4212 86 70E 00 O. 3978 842 5E 00 O. 37395692E 00 o. 349'5 3749E: 00 o.324n6060E oo 0.29934001E 00 0.21364463E 00 0.24758762E 00 0.22120416E 00 O.l9453293E 00 O.l6757852E OC 0.14042729E 00 O.U309594E 00 O. 85 613966E-CL 0.58020696E-01 0.46531098E-01 0.34965198E-C1 O. 23409870E-01 O.ll702139E-01

- O. 46 711154E-07

PAGE 9

R : 35.96141

0.3438298U. 00 O. 3Z741439E 00 O. 3105600'5E 00 0.29330790E 00 0.27566898E 00 0.25766130E 00 0.23932809E 00 0.22066218E 00 O. 2017201SE 00 0.182511 39E 00 0.16306198E 00 0.14339983E 00 O.l23'52884E 00 Ool0352373E 00 0.833711t21E-01 0.63120782E-Ol o.~2780738E-Ol O.llt168899E-Ol O. 2563054\E-01 O.l7104786E-Ol 0.85503682E-02

-0. 34187Z73E-01

1-' 0 +='

1 == bl.9C472 z = tt4.28~6e l = 66.66663 l ,. 69. 04 7'58 z = 71.428'53 z ,. 73.Rt-94e l = 76.19043 l = 78.57138 l = 80.957.33 l = 81.33321) Z "' SS.71423 l ... 81J.09518 l = 90.47614 7 : 9?.R'H09 1 = 95.23804 l = 97.61~99 l = 99.99994 l =100.99994 l =101.99994 l -=10?.99994 l alC3.99994 l =104.99994

l = 61.90472 l .,. 64.2El'j68 l = 6h.666h3 l = h9.04758 l ... 11.42853 Z = 7::l.RC948 l = 76.19043 7 : 78.57138 l = llC.9'!>233 z ... 83.33328 l = 85.71423 Z = AS.C9518 1 = 90.47614 7 :; 92.85709 7. :: 95.238C4 7. = 97.,18C19 z : 1)11).(}9994 l =1C0.09994 z =101.99994 z =102.90994 1 =lr-3.99994 7 -=104.99994

" = 38.07678

0.32t4f'082E OJ C.3CI51803lf 00 C. 289469R4E CO C.27338946E CO G.256~895F 00 G.24Cl70t;4E 00 C. 22JC77C6E CO C.20567775E CC· 0.18S02243E CO c.t7011~A7F co C'.J5199143E CC C.. 1336b4'H E OC 0 • 11 5 1 41 2 7E 0 0 C. 964'l BC HE -(il <. .77721596f-01 C.'5~R47.957E-C1 ( .39~86121E-(11 c. 31821 t>65E:-~ 1 C. 23PMtl40E -01 C.1t;91RS94F-C1 C.79573C95E-02

-C.316C3140F-C7

R = 50. 76'l01

t .146088COE CO C·.Ball3'34f 00 C..l3J95C~OE ~C C.12462127E C0 0.11712885~ 00 O.lJ<?4~312E OC. O.lC.l692C?E 00 u.93757808E-Ol C. @5710943E-01 o. 775 sz ss 7E -c 1 0.69292903f-01 0. 60940 "32 5E -01 0.5249C462E-01 ( .44GOC506E-Ol c. ~5446 752f-r: 1 0.268479181'-0t C.l82124C)0E-Ol 0.14454059(-01 o.1oezt-927F-01 C.72C73042E-Ol 0.36C24251E-02

-C. Bh 429V~E-07

R = 40.19215

o. 2956461 9f 00 C. 2!H53151E OG 0.2t.7C3793E 00 C.25220382E 00 C. 237C3790E CO 0. ?21 56054E 00 o. 20579195[ co C. 1R913988E 00 c. l7345H5J: 00 c. 1569H90F. CC. C.l4C21641E CO C.l2H1Q4<;f OC C.1()622075F 00 C.'!90247nE-Ol c. 71 705341[-01 C.. <;429.?0C5f-01 C. 36AGh2lCJ!=-Ol c. 29B5~13E-01 C. 2l9q3350f-Ol C.. 14~6~465F -01 C. 7l2<HB4~F.-02

-l. 7M~2752E-07

q = 52.PFI43'3

O.lll91589E CC 0. 10657310E 00 C. 1U1G854Fif 00 v. :J5470428E-Ol c. e913C799E-01 o. q3CJ73149E-01 c. 77<J0'5059f-Ol 0.718266961;-01 c. ~566ll4nE-C.l O. Stl41'3n'H:-Ol 1). 530 ~50481;;-01 0. 466Bh254t:-Ol C. 40212516E-01 c. H709031JE-u1 c. ~715!,4911'-01 0.20569533E-Ol c. 1H54472E-Ol C. ll0699A3F-Ol 0. 82 H37F16E-02 :>. 'i51fi6674E-02 C. 27583700E-02

-o. 10l9P534J.:-o7

GROUP 1 FLUX

R = 42.30753

0.26914024E 00 0. 25629115E 00 0.24309647E 00 o.22959244E oo o. 215 78687E 00 0.20169801E 00 O.l87H330E 00 O.l7272949E 00 v.l5790319t: co 0.1428696')E 00 0.12764A47E GO C.ll22~~6H: JC' O.Q6n9'346JF-Ol O. 81047714F-Ol 0.652~1762E-01 0.4'l4329~2!:-0l O. 33516850!.=-C 1 o. ?6686ll3f -01 0.2JOC2559E-Ol O.l3H0635E-O 1 0.66634342E-0?

-0. 2h033984E-07

GROUP 1 FLUX

R = '>4.99976

J.71061594E-Ol J. 733P2H5f-O 1 0.696C4099E-vl C. 6') 737844E -0 l :J.bl785787E-Ol ·J.57152918E-01 o.c;3M3014E-i.Jl C.494575ROE-01 C. 452176 1HE-01 0.4090<.l506t:-Ol :).36c;52H4E-01 O.J2146Bl9E-Ol C.27689099E-01 0 • 2 3 2 1116 6 E -G 1 J.lR6<Jq4JOE-Ol O.l416396•JE-O 1 0.96091554!:-02 0.7621623nE-02 0.57083927E-C2 C.3799260BE-02 o. 18989568£-02

-0.71459q58':-08

p = 44.42290

G.24CA69CJ9E 00 0.22936970E 00 0.217560~3E 00 0.20'547521E 00 C.l93l2036E 00 o.ts051213E co C.l6766554E 00 0.1545 8596E 00 (. .141317431' oc 0.1Z796388E 01) C.ll424267E: 00 C .1J046977E 00 O.S6542249E-Ol 0.72538018E-Ol 0.584306lOE-Ol 0.442479361:-01 o. 300u5455E-Ol C.23867179E-Ol O.l78fi5927E-1H O.ll915646E-01 0.59%0426E-02 -0.2~0M427E-07

Q = ')6.24976

0.56393381E-Ol 0.53101226E-Ol 0. 5093 60213E-01 0.4810667BE-01 0.45214556E-Ol 0.42263255E-Ol C. 39255116E-Ol 0. 36192838E-Ol 0.330fi6587E-01 0.29937398E-Ol 0.26749171E-Ol C.l3524899E-01 0.202627591::-01 O.lS985826E-Ol o. 13684135E-Ol O.l0365125E-Ol 0.70319325E-02 (;. 5 5774!> 371::-02 o. 41773 766f-02 O. 27A02796E-02 O.l38964!HI:-02 -o. sz 29 39 56E-os

R = 46.53827

C.21083993E 00 o. 2vC.. 7744 3E o:; ~.~.t90437u8E oo 0.1 7985845f 01,) O.lb904449E 00 :; .l5800875E 01) O.l4676392E JJ 0.135 3142 3E 00 O.l2370C20E 00 0.11192453E 00 C.10000241E OC 0.8 7'l4fl892E-Jl 0. 75754225E-Jl 0.63498020E-01 0. 51150776E-O 1 0. 38737800E-01 u. 262 72397E-O 1 v.2087942lE-Ol O.l5644100E-Ol 0.10416862E-01 0.5 2077845E-02

-0.200Cl490E-07

R = 57.49976

C.36831919E-Ol 0.35013508E-01 0. 33267502E-Ol 0.31419583E-01 0.29530670E-01 0.27603105E-01 0 .2563BR15E-Ol 0. 2 3636375E-O 1 u.216096l2E-Cl O.l95t;i2801E-Jl u.17470501E-O 1 C.l5364651E-01 tJ.l3234075E-Ol O.l1093836E-Ol C...fi9174222E-02 0.67696944E-02 0 .45927167E-02 0. 36427758E-02 0. 27283435E-02 0 .1815B646E-02 J.Cl0761110E-03

- J. 3 415441 3E- 0 !l

PAGE 10

R = 48.65364

O.l79l7198E OC O.l7C61830E 00 0.16183H5E 00 O. 15 2R43 83E 00 0.14365435[ 00 o.13427669E oo 0.124 72093[ oc 0. 1149 904 7E 00 0.10512LC8E 00 O. 95114648E- 01 0.849B3826E-01 C.74H9397E-Cl o. 64316950E-Ol 0.539b3ll4E-Ol 0.43471396E-01 0.32924142E-Ol C. 22332158E- 01 O.l7734479f-Gl u.l32R5723E-01 0.88458916E-02 0.44214875F-02

-O.l6652333E-07

PAGE 11

R = 56.74976

O.l8ll4615E-01 O.l1249849E-01 O.l6361605E-CL 0.15452772E-Ol 0.14523767E-Ol O.l3575755E-Gl 0.12609672E-Ol O.ll625819E-01 0.10628033E-Ol 0.96l64532E-02 o. 85923336E-02 o. 7556635qf-C.2 0.65087751E-02 0.54561608E-02 O.lt3956004E- 02 o. 33294749E-02 0. 2258792'>E-02 O.l7915892E-C2 0.13418533E-02 0. 89307805E-C'3 o. 44638061 E- 03

-0.16797614£-08

I-I 0 c.n

GROUP 1 FLUX PAGE 12

R = 59.99<)76 1 = bl.90472 -(.. 6 74t:-9 :?.16f -07 z = 64.28568 -c. 642 4'329 7E-07 l = 66.66663 -C. t.C;c:»40 124E-C7 1 = 69.04758 -0. 5 75 ';503 8E-O 7 1 = 71.42853 -0. 54C94695E-07 l = 73.80948 -0.50563493E-07 z = 76.19043 -0.46965155~-07 z = 78.57138 -C.433C0950E-07 1 = 80.95233 -0.39584513E-C7 l = 'J3. ''H28 -('. 35816537£-07 l "" 85.71423 -o. 32C01793E-07 1 = 88.09'H8 -0.2R144C97E-07 l = 90.47614 -t. 24241746f-07 l = 92.8'5709 -c. 20321 053E-o7 l = 95.238C4 -0.163701i21E-07 l = 97.61899 -o .12399159E-o7 l = 99.99994 -O.R41C8400E-08 l =100.99994 -0.66M2678E-08 l =1C1.99994 -C..49936446E-OA z =102.99994 -C.33214567E-08 1 =103.99994 -O.l6601720E-08 l =104.99994 C.631021'59E-14

GROUP 2 flU)( PAGE 1

R = o.o R = 2.11538 R "' 4.21077 R = 6.3461'5 R = 8.46154 R = 10.57692

7 = o.o 0.')6396747F 00 o. 56290454E 00 0.55965763£ 1.)0 0.55428326E 00 0. 54690492E llJ o. 53 760213E OC z = 2.38095 0.56379837E 00 O. '56273556E CO 0.55948925E 00 0.55411583E 00 o.546HBB6E o\> o. 53743780E 00 1 = 4.76190 0.56324768f 00 0. 5621 f''S 76E 00 0.5'i894262E 00 0 .55351't32E 00 o.54620it27E oo o.51691202E oo l "' 7.14286 C.5619f>i'C7E 00 0. '5609C760E 00 0.'55766672E CO 0.'55231047E 00 o.54495710E oo o.5356B572E oo z = 9. '52381 C.559938 1HE 00 o. sse 8832 c;e oo 0.55565876£ 00 o.55032146E oo o.54299itOlE oo O. 53375554E 00 z = u.qc476 C.. sr; 71 H 73E 00 c.55612320E oo O.'i529l48CIE 00 0.5476039\E 00 0.54031265£: 00 o.53lll98BE oo 1 = 14.28571 G.55~67571E: 00 \i.55263l!l6E CO 0.549443'54f 00 o.51t4t6603E oo 0.53692079E 00 o.5277B584E oo l = 16.66666 c.54945099E oo o.r;~t8415tZE oo u. 54525113E 00 0.'54001397E 00 o.532B239ZE oo o.52375865E oo z s 19.04761 o.5445093'>f oo O. 51tH8284E 00 O. 51t0341l6E 00 0.53515697E 00 0.52f'031 59E 00 0.519047BOE 00 l = 21.42R'56 0.5388')4'54E 00 o.53783852E oo 0.53't7l532f 00 C,.52959883E 00 o.szzs4701E oc o.51365614E oo z ,. 23. 80951 o. 5321t8739E 00 o.r;3148347E oo O. t;2841693E 00 c.5233411BE oo o.st631Z6BE oo o. 50758684E CO z = 26.190't6 0.52542019f 00 o. 5241t2956E 00 0.5211t0373E 00 o.51639527E o., 0 .5095l940E 00 o. 50Ci85020E 00 l = 21J. 57141 o.5l1662TOE oo O. 'il6686~"E 00 0.51310555£: 00 0.50877106[ 00 o.sotCJq676E oc 0.4934551t7E 00 z • 10.9t;236 t.50922489E 00 o. 'ill8264 78E 00 ·J.'i053l229f co 0.'50047821E 00 O.lt938llt23E 00 O.lt8541218E 00 z = 33.33331 O. 50C 12174E 00 o. lt9918067E 00 0.49630052E 00 0.49153292£ 00 o.~tB498785E ao O.lt7613571£ 00 z .. 35.71426 0.49C36258E 00 0.41!9431!00E 00 0.4R661393E 00 O.lt8193949E 00 C.47552204E 00 0.46743089E 00 l ,. 38.C9'i21 0.47995734E 00 0. 479C'5236E 00 0.47628814E 00 0.4711l289E OC 0.46543157E 00 o.4'i751202E oo l ... lt0.47617 C.lt6892C41f CO o. 46803623F 00 0.46r;)31j6lf 00 o.460B651t4E oo O.lt5472854E 00 0.44699115f 00 l "' 4l.85712 C.457l6275f OC 0.45640063f 00 :l.45376712E 00 O.lt4940~17E 00 0 .ltlt34238 7E 00 8•431587887E 00 z = 4'S.23807 C.445Cl257E 00 C. 4441 7l40E 00 0.44161046E 00 0 .43736821E 00 0.43154418E 00 .lt21t201~9E 00 7 "' 47.619('12 0.4 ~217671 f 00 O.lt3l36185f OC 0.42887276£: 00 o.421t75277F og O.lt1909665E 00 o ... t1965 lE oo z = 49.99997 0.411'17270E 00 0. :A 79f'300E 00 0.41557109!: 00 O.ltll5788Jt: 0 O.lt0~091f9E 00 o.399l8761.E 88 l = ~2.380<12 Ce401t8175RE 00 O. 40t;lt28E 00 0.40l72261E 00 0.39786339f 00 0.39 565 3E 00 o.3eseeso6e Z .. 54.7l!lfH c. 3q(i32263E 00 o. 3,.CJ586'HE OC 0.38733858E 00 0. 3836l11t6E 00 o.H850910E oo o. 31206835E 00 l ': 57.147.82 C.37532580E 00 C. 31461 ROSE 00 0.372456Hf 00 C.36887819E 00 0.36396587E JO O. 3577721t7E 00 l = 1)9.t;2377 C. 3591'\362 RE 00 o. -,r;q 15171tE 00 0.35708'H1E 00 o.J5l651t50E ov o.31t8941t72E oc o.31t300613E co

1-' 0 m

GROUP 2 fLUX PAGE 2

Q = 1;?.692'\0 q = 14. All769 R = l6.92'\C7 R = 19.03844 ~ = llol5Hl R = 23.26'HB

7 : J.C C. 5265? 0~'5F 00 r:;.51~8C.25C)E CO 0.499584 7C)E 00 0.4R400396E 00 0.46H7703E 00 0.44920164E 00 z = 2.'380°5 C.5?635~BE 00 C.51~64126E 00 ').49947.86'3E CO vo4838'll55E OC 0.46702886E 00 0.44905829E 00 l '= 4.76190 C.52584362F OJ c. '51 H4014E 00 0.49AC)39HE uJ o. 4833 77v4E oo 0 .46651056E 00 0.44861734E 00 z ::: 7.14286 C.5?.464223F 00 v.'ill96748E 00 0.49779863E 00 0 .4AV7l79E 00 o.46550351E uo o. 44 759113E CO 1 = 9. '5?3 131 C.5?.275127E OC C.51Jl2158t: CO .'J.496C0321E 00 o.49053175E oo CI.46382338E 00 0.44597518E 00 z = 11.00476 C.520170C9f 00 0.50760?87F CO 0.49355429F CO 0.4Bl5919E 00 0.46153355E 00 o. 4437 735lf: 00 7 = 14.2F''57l C. 5169C41B!:: Q,J 0. 5C441~58J: 00 .J.4904'>64C.E 00 C .4 7515610E 00 C'.45863682E 00 0.44098830E 00 l = 16.6t.666 Oo '5l296073E 00 o. 500567791: co IJ.4867l395f 00 (.•.47153246E 00 0.45513719E 00 0.43762332E Gu z = 19.047h1 C.508l4674E 00 o. 4960650'\f 00 0.4q2335'37E CO C.46729070F 00 0.45104283E 00 0. 43368638E 00 z"' 21.4?856 c.r,o30fi58?E GO C.4909113·JE uO 0. 4773?413E 00 0.46243'>0lt 00 G.44635552E 00 0.429l7913E 00 l = 23.%9'51 C • 49 712 1 8 1 F C 'J c. 485110"'3f 00 0.47l68434F 00 C .45697105E 00 0.44l08164E 00 0.42410809E 00 1 = 26.1<>046 t.49C5?39C)E Cv C. 4 7 86 7?. ~ JE 00 0.46'5424l'~E 00 C.45090616E 0::> C .43522763f Ou O. 4184 7944E 00 l = 28.57141 C.48328167E 0:) C.4716v530E 00 C.45855254E 00 0.444248861: 00 0.42880177E 00 0.41230094E 00 l = 30.9'>236 C:.47S40426E 00 C.463Ql809E 00 C.45107806E 00 C.43700755F 00 0 • 4 21 8 1 2 2 4E 0 0 o.4055S034E oo Z = 33. 333'H C.4M·CI(·M1f Q(; O. 45562'il9E 00 0.443014~8!= 00 0.42919505E llO 0 .41427Ll8E 00 0.39832920E CO 1 .:: ~5. 71426 C .4577CI330E 00 J.446B2?8~'= 00 0.4343676CE ')·') 0.4208l791E 00 0.4061B527E 00 o.39055431E oo 7 = 3q.J9521 C.44~C7887E 00 C.43725246!: 00 0.4?.'H5010!:: :JO 0.4ll88788E 00 0.39756560E 00 O. 38226634E 00 7 = 40.47(,17 C .4H'174R4E OC C. 427l<H40~'= 00 0.4153737.7E 00 C.40241599E 00 0.38842309E 00 o.31347567E oo 1 = 42 o R5712 0.42689197E 00 G. 41657746E 00 0.40504736E CO o.~C)241230E oo J. 37876725E 00 o.36419141E oo l = 45.23RC7 C.41545480f CO o. 4u541649F oo J.39419514E 00 0. H 189852E 00 0. 3686189 7E 00 O. 35443360E 00 l = 47.t.l902 C. 4 0 34 7111 E 0 0 c. )Q.:H22"\0!: 00 0.1828?.454E 00 O. 37088233E 00 o.35798568E oo o. 34420943E 00 l = 49.99997 C.390C)5670f 00 C.18151008E 00 0.31095010E OJ C.35937622E 00 u. 34688145E 00 a. 3H53239E oo l = "i2.38092 C.l7792838E 00 <.. 36879641E 00 0.35858834E 00 0.34740191\E 00 0. 33532166E 00 o. 32241744E 00 z ... 54.76lf!7 0.36439f>75E 00 C. 35559195E OC o. 34574950!: co O.B49638uE 00 CI.32331610E 00 o. 31087387E 00 l = 57.14282 C.35(139538E Oil c. 341Q2872f 00 J. 3324642Rf JO 0.32209289E 00 0.31089264E 00 o. 29892838E 00 1 = 59.52377 C. 3359B86E 00 o. 32781643E 00 O. H874?28t: JO 0. 308798 HE 00 0.29806054E 00 0.28658992E 00

GRUUP 2 FLUX PAGE 3

R = 7 5. 38455 ~ = 27.49992 R = 29.61530 R = 31.73')67 R = 33.84604 R = 35.96141

z : o.o C.43C•15236E 00 O. 41007906F 00 J.3890B83E 00 G.36696720E 00 J.34394425E 00 0. 3199486 7E 00 l = 2.3800') C.43vC 1431E OC C.409947C4E 00 0.38888824£ ::lO 0. 3!>68 4865 E 00 0.343833l9E 00 0.319B4532E CO 1 = 4.76190 0. '•2Q5Cf2tl E OJ c. 4J95442 n oo ;).3885(15q4f 00 C.~b64A798E 00 0.34349501E 00 O. 31953096E 00 z = 7.14l~b (;.4286C'9C7E CO ~.4C~606<l5E llO 0.3876168lE 00 0.36564922E 'JO O. 34270889E OJ O. 318 799 56E llO l = q.52:iRl C.. 427C6126~ OC <,.40713108E OC :J. 38621 658€: c Q 0.36432827E 00 0.34147084E 00 o. 31 764799E: 00 l. = 11.904 76 C .42495292E 00 o. 4051212 7E 00 0.38431001E OJ 0.36Z52970E 00 0. H978498E 00 0.31607980E 00 l = 14.28571 0.42229591E 00 o. 40?57871F 0:> 0.381891HH: 00 0. 3602 544 7E 00 0. 3376'5756E 00 o. 31409603E 00 1 "' 16."t>666 Oo419(i63t>9[ 00 O. \9q5061J 1 E 00 0. 318'?9397E 00 0.35750544E 00 0.335J7603E 00 O. 3ll69qz 1 E 00 z ., 19.0471)1 0.4152935 7E: oc O. 39591265E 00 O. 31'557429E 00 0.354289COE O:J O. 33206129E 'JO o. 3088948lE 00 l .,. 21.42~'56 o.41C97730t: oo o. 391797541: 00 ·J. 371670541: 00 0.3S060638E 00 0.3286u982E 00 o. 30568421E 00 1 :: 23.lltq•;;t C.40612131E CO C.3ll716P.17E 00 v. 3672789<JE llO 0.34646374t: 00 o.3247znoe oo o.JOZ0722CJE og 1 = 26.1q046 C.40C73144E' O:i c. 382C2<H71.: 00 0. 36240453E 00 0.34186~54E CO v.3l041123E oo O.Z9806316E 0 .,_,.,. 2~.57141 ( .• 3 94 !114 A 5 E C 0 r. 37'>189HE 00 0.357053q8E CO O.H6811J04E OJ 0.31568640E 00 O.Z936621t3E 00 l .,. 3C .q52J6 C.3883797.BE: t;C L. H<J254101!: 00 :.'l512H78~ 00 0.33132780E 00 o. 3lv54062E OJ O. Z8887S52E 00 7 = 33.33331 c.3lll4~551E co O. 3':116342'\E OC 0.344953R9F 00 :i. H540381E OJ v.30498844E 00 o.Z837107ae oo z : 35.71426 C.373990l61:= 00 :;. 3'>fl'i36511: 00 0.33822CHE 00 C.H905228E 00 J.29903'H7E 00 0.27817303E 00 7 "' '\'t.(lQ'571 0.lb6C')376E 00 o. ~49q702<lE 00 C.B1C4318E 00 C. H228149E 00 u.292oR932E oo 0.27226973£ 00 l = 4:).47617 0.3t;763597E 00 c. 14~94530!= 00 0.32343042E 00 0.30510014E OJ 0.28595853E 00 o. 26600850E 00

1-' 0 -..J

l :: 42. ~5712 C. 34874552E CC C. 'H24M76E OC J.31~3Q03'iE 00 0.29751569E 00 0.27AA4990E OU 0.25939572E OC 1 : 45.238()7 Ci. :Bq4c 13H c.o c. ~7356L67f 00 J.3069399:>E: co G. B9'544L 7E 00 0.27L31846F 00 0.25244564[ c.o l = 47.61902 C. 32961076E 00 v. 31422794E OC u.29BOB55Lf 00 o.?.Bll9165F co u.26355004E o..> 0.24516326E 00 7 :: 49.99997 C.3193AM8E Gv O. 30448C HE Ov O. 2A863910E 00 0.27246910E OJ o.25537467E oo O. 23155 836E CO z = 52.38(\Q? c.30!.I74?fl2E oc v. 2943 338 ?.E 00 0.27921H7f 00 0. 263389(.5f 00 0 .24686420E 00 0.22964l62E 00 1 = '54.761A7 o. 797f.I;'IQ08E oJO G.2R'H96ClF 00 0.26921684E 00 0.253~5906E 00 0.2l8()2590E 00 0. 221419 8 7E 00 1 ., 57.14282 C.2A6?')0C'5f 00 C • .?72 890 75E CO :).25887179E 00 C.244200V>E CJ u. 228879HE OU o. 2129ll61E 00 1 = 59.52H7 O. 2744~48GE 00 ..;. 26162bBE 00 G.74R1Rb47E 00 o.B412049E: oo 0.21943194E Ou 0.20412326£ 00

G~OUP 2 FLUX PAGE 4

p = 38.07()79 ~ = 40.1'~?.15 R = 47.30753 R = 44.42290 R = 46.5387.7 R :: 48.65364

7 :: c.o C. 294Q9l72E 00 J. ~b91C812E 00 G.242351J10E 00 o.214B0566E oo J.18660l52E Ou 0.15790552E: 00 l = 2.3A:J95 C.294~9P15f OJ 0. 26902169E JO o.z4n7262F oo C..214137l8E 00 O.l96542BE OCI O. 15 785569E 00 1. = 4.1H9v C. ?9460835E 00 o. 26875134!: 00 0.2420H55E 00 0.2145263~E 00 0.18635911E CO 0.15770066£ GO 1 :: 7.142A'l l. 29393411E 00 C. 26814234E ::00 0.24l4BC77E 00 0. 2140 3557E 00 O.l8593287E 00 0.151340 llE uO 7 :: Q.52381 C.292R7243E' 00 C. 2611 739'>E 00 0.24060R75E: 00 0. 213262 86E 00 u.ts526l79E oo 0.15677226E 00 l :: 11.9('4 76 ('. 29142648f co O. 265854A4E 00 0.23942077[ vO 0 .212l0982E 00 0.18434697E 00 0.15599811E 00 l = 14.2!\";71 c. 2895Q73Qf 00 a. 2!>41%?.JF OJ 0.23791802E 00 o.210B7778E oo 0.18318975E 00 0. 15 50 18 81 F CO l = 16.66666 C.28138749E 00 o. ?6?.17020!= 00 0.23610252E OU 0. 20926851E 00 0.18179184E 00 O.l5383589E 00 z = 10.04761 C:.2848Ql<l6E 00 C.25QIH152f 00 0.23397833E 00 0. 2uH3584E 00 0.180 15635E 00 0.15245193E 00 l = 21.42P5b C.2'JlR4175E 00 O. 251lllllE 00 J.2l154670F 00 C .2()523059E 00 0.17828417E 00 0.15086776E 00 1 = 23."0951 C..27851152E CO O. 25401314E )0 0.22881061E 00 0.20280558E 00 O.l7617756E 00 o. 149085C4E 00 1 ::: 2(..1Q046 C.27413150BE 00 O. ~5070101E 00 0.2257738LE 00 C • .?JO ll389E 00 O.l1383927E 00 Ool41l0629E 00 l = 28.57141 C .27075756E 00 0. 246q99'5 6F 00 0.22244030E 00 o.19715917E oo 0.17l27258E 00 O.l4493430E 00 7 :: 30.9'5236 0.26634401E 00 C. ~4297327E 00 0.2188llt43E 00 O.l9394535E 00 O.l6848063E 00 0.14257l69E 00 1 = n.BHl 0.2615~226E 00 0.2'.\~62940E 00 0.21490252E 00 0.1q047821E 00 0.16546881E 00 0.14002311E CO z = 3 '5. 7147.6 C.2'5647640E 00 O.ZB97160E 00 0. 21C707fJ4E 00 o.1s&76031E oo O.l6223902E 00 0.13129000E 00 1. = 3a.oqs21 C.25103360E OC o. 2290062qE 00 0.206236361:: co C.18279701E 00 0ol5~79613E 00 o. 1 343 7659E 00 1 = 40.47617 C.24&j26066F. 00 o. zz:nJqne oo 0.201493'56E 00 O.l7R593?8E 00 v.15~14439E oo O.l3128638E 00 1 = 42.ft51l2 C'-. 23916 364E 00 O. 2l8171qlE 00 O.l96484HE 00 o.17ltl5357E: oo J.15126756E 00 0.12802261E 00 7 = 45.H807 C. Z 32 7'5 566E 00 o. 212H225E 00 0.19122022E CO 0.16948748E 00 C.14723414E 00 O.l2459260E vO 7. :: 47.61902 C. 22 6~ 413 2 E 0 0 o. 206201161; 00 0.18570411 E 00 0.164'59835E 00 0.14298689E 00 0.12099856E 00 l :: 4Q.C)q997 c. 21'~0296lf 00 C. l9981068E 00 O.l7994368E 00 0.15949267£ 00 O.l3855111E 00 0.117245 32E 00 1 :: 52.38'.)92 0.21113036f 00 o.t93151B9E oo 0. 1 13946913E 00 0.15417755E CC 0 .l339l438E 00 O.ll333811E OG z = 54.76187 C. 204l497~E OC o. 186236BE 00 O.l671lqOlE 00 0.1486'5732E: 00 0.129 l3901E 00 0.10928005[ 00 z = '57.142~2 r;,.1963C5?.2J:= 00 O.l7908025E 00 0.16l27449E 00 0.14ZQ4523E 00 Oo1Z417692E 00 0.10508114E 00 7. .. 1)9.'>?377 (. ol88l~238E 00 o. 111688501: 00 0.15461773E 00 O.l3104515E OJ o.Lt905152E ou 0.10074401E 00

G~OUP 2 FLUX PAGE 'i

R = 50.76901 !l = 52. ~A43~ !'. = 54. 99Q76 ~ = 56.24976 R :: 57.49976 R = 58.74976

l :: ').0 c.J2ACJ23l?E oo C. <;CJ89'H09E-01 O. 7lll25Hf-01 C.'53~23162E-Ol 0. 35998818E-O 1 Oo17956365E-01 1 = 2.3A095 C•.l2B88253E 00 0. 99868357E-Ol J.1l0903lt1E-01 o.53B06316E-Ol Oo35987560E-Ol 0.1 T950758E-01 l = 4.76190 C.l2R7t;6llE GO c. <;971IJ427E-01 0.1lC20603E-Ol 0.5H53551E-01 u.3595227JE-Ol O.l1933156E-Cl 1 = 7.142R.6 C•.HP4t-160f 0Q C. 9Q5422C1E-01 Oo 7CA58240E-01 0.5363J669f-Ol C.35870086E-01 0.178QZ160E-Cl 1 = 'l.573Pl C..l2799'H7E Cv C.9o1~H42E-Cl 0.70602b55E-Ol C • 5 3437 ZllE-0 1 o.357407ooe-:n O.l7827623E-C1 l = ll.QG476 ;).12H66C7F C:l c. 986'H252E-C:1 C. N253968E:-Ol 0.5 H 73307F.-01 v.3556419ZE-01 0.177H575E-01 7 : 14.2f\571 C .12656647E 00 C. QA073781E-01 0.69812894f-Ol 0.521\H4R8E:-01 u.3534C9ZOE-Ol 0.176282C8E-01 1 = 16.66666 u.17~bC·Ob4f OC. c. Q737526&jf -01 O.I:-Q28014RE-Ol o.~2436?55E-ot Oo350 71220E-O 1 O.l1493t.ME-Cl 1 ~ 19.04 761 O.l2447065f 00 C. 164497HE-~1 :J. 6~6&j6911E-C.l o. 'H964540E-01 Oo34755722E-Ol 0.1 H36313E-Ol 1 = zt.42R'>6 C.lP17H5f 00 G. ':l'54476v0f-Ol 0.67943573E-Ol 0.51424645E-Ol 0.34394622E-01 O.l71S6195E-Ol

._... 0 00

....

1 = 23.80951 z = 26.19046 l = 28.57141 7 = 30.95236 l = 33.33331 1 "' 35.11426 z = 38.09';21 1 = 40.47617 z = 42.85712 l = 4t;.Z38C7 z = 47.61902 7 = 49.99997 l "' 52.~8092 l ... 54.76187 l = 57.11t282 l .. 59.52377

l = c.o l : ?.38005 l = 4.76190 1 = 7.H2f'6 1 = q.S238l l = 11.901t76 z = 14. 21!511 l = 16.66666 1 .. 19.01t761 7 = 21.42856 1 ::: 23.80951 z = 26.19046 l ::: 28.571ltl l = 30.95236 z = :n. 33331 l = 35.71426 l = 38.09521 l = 40.47617 l = 42.85712 l = 45.23807

ll "' 47.6. 1902 : 4q.qqqq7 1 = 52.38092 l = 54.761Rl l = ~7.14282 l = '59.52377

z = 61.90472 z = 64.28r;68 1 = 66.66663

O.I2112186E 00 C.l2010628E OJ C .U833292E CO C.ll64C::400E OC O.ll432320E 00 C .112091 78E 00 C.10971314E 00 0 .l07190ClE 00 O.l01t52527E 00 0.101 72480E 00 0.987901rOTE-01 0.95726013E-01 C.92!\36032f-01 o. 89222841tf:-01 0.85794S08E-01 o.e2253tr56f:-Ol

R = 59.99976 O.I0036J37E-06 O.l0033148f-06 O.l0023297f-06 C .10l'00366E-06 C..99642648E-07 0.99150Mlf-07 C. 9852tH 19f-07 C. 97176308E-07 0.96896599E-07 0.95889132E-07 C.94756615E-07 C.93498954f-07 C.92118398E-07 &J.90616822E-07 C.ft8996956E-07 C.87259764f-07 Cl.851t080,3E-07 C. 8 34439l2E -07 0. 813691t 78E-07 c. .79189363f-07 O.l690':»053E-87 C. 14!!JlfJ,Uf• 7 Ct.72036130E-07 0.69456q74E-07 0.66788118E-07 0.64C31440E-07

R : 0.0

0.31t387410E 00 o.32745838E oo 0.31059855E 00

c. c;4H9701E-01 0. 93067884E-Ol O. q1~93759F-Ol o. 90199ll3E-01 0. 88586807f-Ol o. 86857617E-Ol o. a-;ot4403E-Ol 0. 830'i94JOE-Ol O. 80994546E-01 O. 18824520E-01 O. 765507821:-01 o. 7ltt76t ne-o1 O. 117043S8E-Ol O. 69136CH7E-Ol C. 66480'5 77E-01 C.63136618E-01

R = 2.11538

o. 34322560£ 00 o. 32684088E 00 o. 31001282E 00

0.67140758£-01 0.66249549E-01 0.6'5171378E-Ol o. 642071t91tf-01 O. 63059741E-Ql 0.6t828975f-Ol 0.60516894E-01 0.59125215E-Ol 0.57655334E-C1 o. 5611 0658E-01 0.5449209ltE-Ol 0.52801870E-01 0.510422HE-01 O. 492llt650E -01 0.47323696E-01 0.4~370460E-01

GROUP ?. FLUX

GROUP 2 FLUX

R = 4.23077

O. 34124494E 00 0.3249'545\E 00 0.30822312E 00

0.50Rl6976E-g1 0.5:l1421t86E- 1 O.ltt:Jit02l29E-01 0 .lt8596840E-O 1 0.47128203E-01 O.lt6n6598E-01 0.45803528E-01 C.lt4750202E-Ol 0.43631697£-01 0.42468574E-01 0 .ltl243527E-01 0.39964244E-01 0.38632426E-Ol 0.3724CJ178E-Ol 0. 3r;817973E-01 (i. 34339625E-01

R = 6.34615

C. 33796626E 00 0.321A3242E 00 C. 305262'51E 00

().33988169E-Ol 0. 33537067£-01 u. 3 JOlt l891tE-O 1 0.32503288E-Ol ~. 31922318E-Ol 0. 31299230E-Ol 0.30635025E-01 o.29930525E-Ol 0.29186435E-Ol o.28404489E-Ol o. 27585\HE-0 1 0.26729509E-Ol o.zsaJsnoe-ot 0.24913572E-Ol 0. 2 3956336E-0 1 0.22967566E-01

R = a. 46154

0.333't6522E 00 o. 31154601E 00 0.30ll9705E 00

0.169H461E-Ol O.l67281t46f-Ol O.l61tR1444E-Ol O.l6212787f-01 0 .15922997E-Ol O.l5612200E-Ol O.l5280891E-Ol O.llt9291t85E-Ol O.l4558326E-01 0.14168292£-01 0.13 759594E-Ol O.l3332803E-01 o.tzeeattsoe-ot O.l2427002E-Ol Oel1949528E-CJ1 O.ll456326E-Ol

PAGE 6

PAGE 1

R = 10.'57692

0.321790261::: 00 o. 31214201E 00 o.29607153E oo

1--' 0 <..0

z = 6Q.('4758 7. = 71.4?P53 z :: 73.A(048 7 :: 7h.}OC,4~

z = 78.57138 l = ~C.9523::\ l :: A3. 33~2~ l :: A5.71423 l. = I"'A.G9':i18 7_ = QQ.47614 1 = 9?.8570Q 1 :: 95.2'3804 l = Q7.6]~QQ 1 = Qc:).9Q9Q4 l. =100.<lQQC)4 l =10l.QC<Q<l4 z =102.QQQQ4 1 =103.99994 1 =lC4.ClC19CI4

l = 61.9(477 Z = 64. 28St>8 l = 66.M>663 1 = 69.047'>8 z = 1l.478'53 l = 73.~('948 Z = 76.19C43 l = 7A.57P8 l = RO.Cl5233 z = 83. 3:H28 1 = as. 71423 l = fl8.( 0 518 1 = q0.47614 Z = 92.8'57CQ 1 -= Q5.nA04 z = 97.61899 ., = 9Q.QQ994 7 =1C0.99994 1 =1Cl.'.19()Q4 l =10?.99~94 l =1':'l.Q~QQ4 l ::}.~4. 9°9Q4

7 : f>1.'lC:472 l :: M. 2~')6fl z = 66.f-6663

G. 2913463c;E OC 0.27571023f co c. 2'i771248f co 0.23CJ37374E 00 c.72C6Qoflc;E oo C. 201 75H7E 00 (;.}Q25"i341E cc C. H<? lC O%f uO C.14344954F GCi C.l?35M':i4E ::iQ u. 1035'J592E C:C' C'.83415'i6AE-Ol C.6316A.A57E-01 C.42A3"-344F-Cl O. 3432!>304E-Ol G.758C3Cl94£:-01 C.172'HM4f-J1 O. R646::lJ39E-07.

-C'.l2CI06393f:-C7

p = 1?.6023J

c. 32103091£: co :J. 3v~7C·513~ OJ c. 29()9661 7f oc C. 77 3S5962E OC C.2S739378E Ov C. 2'~(:5901 OE CO C.22346A90F OC C.206037Co4E 0(; C.1Rfi3'?2RlE 00 C • 17 C 42 2 ::- 2 E C· 0 C • p; 2 2 6 A 1 7f C· 0 C.l33912QAf O::l r.ur,3'>21SF oo C • 9 66 7C r, B 7E -0 1 r.17fl6'i6C1F-C1 C.5R.05P7P7E-Ol C..3CJ974'?CJ6[-01 C.H{'I3S239E-01 C.2407~205E'-Ol C.l6131J8~E-C1 C.806311e9E-C2

-C.'\\117~672f:-07

R = 2'>.~!\455

0.2t>?2'i8'.i'H 00 C. 24<t738G4E ·:iO C.236~€'161E OC

o. 29279310E 00 C.7.1519023E 00 C. 25722629E 00 c. 23892212E 00 O. 22028351E 00 c. ?01 H139E 00 C.lR22C9C1E 00 o. 16? 80204f 00 0. 14317H6F. CO C.·. 17.l3314'lF 00 \.o 103l6041E 00 C.l\32'57854'=-01 0. 6304734Qf-Ol t.. 42754043f-01 c. 34261783!::-::H c. 2'?7'54951E-01 O. 17264'H5E-Ol C.. f!62944R 7E-02

-C. 32947C21E-07

Q = 14.8071l9

0.31321319£: ')O O. 2'lfl31 713E CO c. 28295q~4F. 00 O. 267?4184E 00 C.25ll1356E 00 C. 234 77'5l6E 00 0.21RCi,777'= :>0 0.2G1057!-.1F 00 C. 18330052E 00 C. 1663C280!: 00 O. 1485RI:>75E 00 O.l3C6747':'E 00 C. 1175633 7E 00 C. 943329HE-01 C. 75GSl49qE-01 0.57'i3C664E-Ol (;. 390C4333f-Ol O. 312')6769E-01 C. ?34f\Ol74F-Ol o. 1<)734881£:-01 c. 7~6S13C<JE-02

-c.. 30'53C'i'45E-07

Q :: ?7.49947

J. 250( lll72F 00 o. 2lqC8?47f 00 t. ?7';R26?0E: (.Q

0.29110336E 00 o.21360l95E oo 0.25574166E 00 0.23754299E 00 0.21901202£ (;I) C.2C02149£1E JO 0.1811'>6'-NE 00 O.l6186l<JOE OJ 0.14235175E 00 O.l226l927E 00 0.10?76335E OJ ·J. R2716546E-Ol 0. 626P2331 E -01 f). 425V;71!JE-O 1 IJ. 340o269~E-J 1 C.756C.5187t.:-Cl J. 1 716 D 11 r -o 1 O.A<;78<J18QE:-02

-C. 32794560E-:H

GROUP 2 Fl.UX

R -= 16.9BC7

J.3046ClHE 00 o. 2900'5975E 00 o. 27512681E 00 o. zsoa442H :>o 0.24422G32t 00 ).2?.A275bCE 'JO O. 212C30Ht: 00 O.l9~49143E OU 0.178711951: OJ O.l6169804E CO 0.14447176~ 00 0. 1? 7C552 9E ')•) O.l0944623E 00 O.Ql720C4'iE-C1 0.13!175A44E-Ol O.SS934846E-01 O. 3 79204 71 E-01 0.30386388!:-01 o.na3zaaz~::-o1 O.l5292C37E-01 0.764B338E-02

-u. ?<H!H27uE-07

GROUP 2 FLUX

R = 2<l.6l53C

J.?l7174o'\E oo c.ussc;t'>4E' co 0.?.142?49~E JC

C.2A~30653E 00 o.2709H03E oo 0. 25~?A404f 00 C.23526013f Ou 0.216907'56E JO 0.1Qfi29094E CO C.l7941582E 00 0.16U30562E 00 0 • 1 40 9 A 2 R 7 E 0 0 O.l2144047E 00 O.lJ117'>05F 00 0. 81 C)79811E-01 f..'.62078193E-Ol G. 42C94'H9E-01 O. 3313397lf-Ol J. 25 35 7276E-0 l 0 .1~995978E-01 c.~4952749E-02 -c. H '>3965 6E-07

~ -= 19.031i44

0.29509860E 00 o.2B101057E oo 0.2665436<lE 00 J.2'3173783E 00 0.23660C94E 00 o.nusnze oo 0.20541465E: 00 O.l89H209E 00 O.l7313588E 00 O.l5665233E OC v .1399 !>285E 00 o.t2lu8949E oo C .lu60 3064E 00 0.8!H357ll4E-01 :. 715689661::-01 c-.5418'>321~-01 C. 36733624E-01 C.29432476E-01 c.?~113681E-ot 0.14806908E-01 0.74013881E-02

-o. ?8951anE-07

~ = 31.73067

0.?2H3271E 00 :J.2l305138E 00 0. 202C8365E: 00

(,.28446692E OJ v.26736397E 00 v.24991v1Af oo 0.2 3212624E 00 v.21401834E O\J u.l9564962E oo J.17102556E 00 J .l58l6'139E 00 :>.l3910371E OC 0.11'182232E OJ 0.100 4182 7f 00 J.R0886185E-Ol 0.61241:'8 76E-01 0 .415~lu93E-Ol 0. H282593E-O 1 0.250l6867E-Ol J .l6766123E-1Jl ~. 8 38047. 80!:-02

-v.321A5l8 3E-07

R = 21. 15381

o.za4a3o53E oo :.io27123827E 00 0.25727481E Ov 0.24298358E 00 0.22837287E 00 o.2l346188E oo O.l9827044E 00 ~.18280554E 00 C .lb7ll432E 00 0.15120363E 00 J. 13509405E 00 Ooll880108E 00 0.10234225E 00 v .8576571 9E-0 l v.6907826 7E-O 1 0.52299876E-Ol 0.35452895E-Ol 0.28402269E-Ol 0. 2l337077E-01 J .14283258E-01 0.71396865E-02

-0.280 26054E-O 7

R = 33.84604

Oo2iJ9b9599E 00 C.o19Q68468E 00 .:>.l8940502E 00

o. 2796258QE 00 O. 26281363E CO o. 2456566lE oc 0.22817510E 00 0.21037579E 00 o.t9231945E oo 0.114011 77E 00 0.155476C9f 00 o.l3673460E oo O.ll178212E 00 0.987078S5E-01 O. 795074 70E- 01 0. 60203541E- Cl Uo40820524E-01 o. 32713383E-Cl 0.24587605E-Ol O.l64763llE-01 0.82356185E-02

-o. 31730941E-o7

PAGE 8

p = 23.26918

o. 27 38 7464E 00 0.26079965E OC 0. 24731364E 00 c. 23363233E 00 0.21958363E 00 O. 20524621E 00 0.1906 3920E 00 O.l7576981E 00 0.16068232E 00 O.l4538360E 00 O.l29139348E 00 O.l1423326E 00 0.98402679E-Ol o. 82464099E-<n 0.66418350E-Ol 0.'30285067E-01 0. 34085855E- Ql 0.273Ql863E-Ol o. 20507719E-Ol 0.1372432QE-Ol 0.68603344E-02

- o. 27011 946E- 07

PAGE 9

R = 35.96141

o. 1 q t;v 66 s 1 f co C.l857'5370f 00 0.17619ll5E CO

1-' 1-' o

l = f.9.047'5!\ 7 = 71.42 ~53 7-= 71.RC94A l = 76.1°043 l = 7~.57138 ., = ao.qr-z~n

z = q.3.-:t~~26 7 = ~s. 71423 1 = 'HI. v9'5lfl 1 = qn.47614 1 = n. !\15 709 z = 915.23804 l : C7.~1Fl99 ., = <l9.90994 z = 1( 0. 90904 1 =1C'l.<l9904 l =1:1?.90994 l -=101.90994 z = 1 ()4. QQC)Q4

l 61.90472 7 64.?£156B l 6&.66o!>3 z 69.04758 z = 71.42853 l = 73.AC'048 l = 76.19043 l = 78.57138 l = A0.05233 l = 83.33328 7 = f\5.7142'3 1 = F18.C9518 l = 9:>.47614 Z = o2. A5 7C9 l o5.23fiC4 1 <H.6l809 7 99.99994 7_ 1G0.99"lo4 7 101. qooolt z 102.'>9'794 1 103.9C)Cl04 1 1J4.CJ9Cl94

z = 6 1 • Q(' 4 17 1 : 64.2FI'\68 z = 66.bt>66~

C. 223773C5E CO C.210269oJE 'JC Lol9t>':i4047F 00 c .1~25CJ293f cc C.16f'3l42AI:: 00 C.lC:.,3~o671F.: ~o G.t3o?l654'= 00 C.l24383l~E 00 Q.lJ93"'686E 1.0 c.o4228CA9f-Ol C:.7RC)657A3E-Cl C.t-31:0~242F-CH 0.4815C'841E-Cl C. 326 38256E-01 C. 26l36C 15'= -Cl O.lo6292BE'-01 C•.l "l12628F-Cl c.65645948F.:-v2

-c. 25911 977E-07

R = 1A.C767R

C.l79A'5213E ilO 0. 1 112f> 5 66 E 0 C Co 162 44 Af\2E 00 Col534?48CE 00 Co14419A83f 00 O.l347~3BE i:'C C. 12519C 79f 00 Goll5426C6F 00 C.10551Alf>E Ov r.o54712o3E-Ol C.~'529P777E-Cl C. 7SO 143~3E-Cl o.t-46184C..9t:-ot o .... 4l54fl65E-Cl 0.4361A012E-01 C. 3 30 23 7 4 1 E- 01 C. l23~5d39E-Ol C.l7~'l34'iCE-01 C.1341J;961F-Ol c.P9615732f-02 Co4479616<lE-07.

-C.l76A9434f-07

~ = 5c. 7oqot

0. 7~t>v416U-O 1 C.74A~l61?f-Ol 0.70<)97A94E-01

C.21328l63E 00 IJ. ~C045612E 00 O. lP736726f 00 u. l7403245E 00 c. 16045856F 00 0.146685l~f 00 O. l327185AE OC' Collf'I57718C: 00 Co 1042A065'= 00 0. 8982Q683E-Ol C. 7'i27995li;:-Ol C. 6C!>31502E-Ol o. 45902 781E-Ol 0. ~1113<)0~f-Ol C. 24()C7Q80E-Ol C. 1A704176E-Ol C.12509<)7QE-01 c. 625'.\316~!:-02

- c • 7.412 an r; r: -01

q = 40.1<}215

O.l6407096E 00 O.l%2379'.)1: JO 0. 141H 94 HE 00 0. l39q6?. 4t}E: 00 o. 131'54614f 00 C:. 172 9'56Q 3E JC O.ll4?CbO?F. 00 O. 1C5?.9A04F JO C. <;625Q594E-Ol C. FH0943(.,7E-01 1). 779145791:-01 O. 6343262CI':-01 o. '5894866'>E'-Ol 0. 49403'32'JE-Ol 0. 3<)191841E -01 0. lC1271574E-01 0. 20423416t.:-01 c. 163C79Q1~-01 C. 12235·11J4f-01 c. '1161\72 721=-i)2 C. 4CA '.\?832!= -02

-o. l609l!H lE-07

·~ = 52.88438

C.60909J14F-Ol c. 5!10011 7?E-01 c. <;'>Q150?qE-Ol

J.20232481E 00 0.19Cl5R.07E 00 O.l7774147E OC J.l650'H69E oa ').15221518E GJ 0.13914931!: 00 'J.12590003E 00 O.l1248505F 00 O.OAC)22849E-Ol O.fl5214316E-01 0.71412563F-Ol 0.57516579E:-G1 Oo43544449E-Ol 0 • 2 951 'H 92 E -0 1 'J.? 3619<lA3E-Ol O.l7734274E-Ol 0.11 ~57644f-01 0o5'n73094E-0?

-o. ?346536c;E-'J7

GROUP 2 ~='LUX

R = 42. 307'53

O.l4775711E 00 C.l407C344E 00 0.1 3345993E 00 0.126C·4f.l8E OJ 0.11P46679E 00 O.ll073172E 0(1 0.102A'5091E CIO 0.04R28427E-Jl O. P6683697E-01 0.78434885~-01 o. 700713Cil5E-J1 C.6l629064E-01 0. 5 3087631E-O l o. 44492451!:-01 O. 358364 HE-01 o • 211 3 H 1 1E -o 1 0.18394545E-Ol O.l4681112E:-Ol 'J.lllJ13042E-Ol 0. B5100 73f.-02 J. 3 614 51 12 F -c 2

-0.1444701\'iE-07

GROUP 2 FLUX

P = ~'t.'l9'l76

0.43357614E-C1 C. 'tl2F176~4E-Cl o. Hl62tluf-vl

Ool9085795E 00 C'.1793R.C89E 00 0.16766787E 00 0.15573490E 00 :).l't3588HE 00 0.1312621flf 00 O.lliH6434E 00 0.1061096?E 00 0 .q 331 '5959E:-Ol 0.8'J384433E-Ol OoS7365408E-01 0.54?57091E-Ol 0. 410 76928 E-0 1 0.27842801E-01 0.2:?272941E-01 0.16720291E-Ol O.lll7622QE-01 o.5586683!lE-oz

-c. 2212 szo3E-o7

~ = 44.42290

C.l3096492E OC 'J.12471235E 00 Ooll8.?9197E 00 0.11172086!' 00 C. • l 0 50 0 2 3 2 E 0 0 O.'lA146975E-Ol 0.911"1966E-01 0.840'510HE-01 0.76836526£-01 Oo6952J3311:-0l 0.62ll4064E:-01 o.54625295E:-01 Oo47054298E-(Jl 0.394365231:-01 0. H 76458C)E-01 0.240'i1104E-01 v.1o3050HE-01 C.1300fl125E-Ol 0.97563557E-02 0.'>5110475E-02 Cl.~2546343E-02

-0.127641 HE-07

R = <;6.24976

C.32fll6172f:-01 0.31249505E-01 G. 2964:J6 30E-C 1

v.l7881B67E 00 Oo16A12658E OC 0 o15714854E 00 Oo1459642JE 0\J O.l3457948E 00 J.l2302738E 00 O.l1Bl316E 00 O.Q94'52376E-01 0.8746l352E-Ol Oo75340986E-Ol J.6313<)498E-01 0.50853613E-Ol 0.3~500641E-01 J.26096940E-Ol 0.2u867430E-Ol O.l5662137E-Ol O.l046614JE-01 o.5231130bE-oz

-0.20712402E-07

p = 46.538l7

J • l 13 7 69 5 9F J 0 0.1v833'H2E I)J ::lo10276055E 00 0.970~2276E-Ol C.91216505E-01 0.85260808E-Ol Oo79192936E-01 Uo73015571E-01 J.66748261E-01 J.60393356E-Ol 0. '5 39 59079E-01 Oo47453h05E-01 U.40fi76381E-Ol U. 342 59 245E-O 1 C'. 2 75 94849E-O 1 0 .20894427E-O l Jol4l66l69E-01 C.ll296675E-01 (j .847225261'::-02 0.56526475E-Ol u.2825!'>H9E-Ol

-J.llC53299E-07

R = 57.49976

Oo2l94A624E-u1 u.2CC)CC786F.-01 .;, .19824110E-Ol

O.l6640371E 00 0.15639722[ 00 Ool4618504E 00 O.l3578099E: 00 Ool2519044E 00 Ooll444432E' 00 c. 10 354739E 00 Oo92514217E-01 o. 81359744E-Ol o. 70084751E-Ol 0. 58735244E-01 0.41306724E-C1 0.35815826E-Ol O. 24277706E- 01 Ool9403931E-Ol 0.14562022E-Ol 0. 972772 76E- 02 0.48626103E-02

-0.19231813E-07

PAGE 10

R = 48.65364

Oo962743L6E-Cl o.n&78143E-01 J.86958230E-OL 0.82127750E-Cl 0.77189445E-01 0.72149614E-01 o. 6101 4933E-vl o.&l787512E-vl 0.56484006E-Ol 0. 511063 75E-Ol 0. 4566l658E- Cl 0.40l56621E-Ol 0.34590617E-01 0. 289<)1401 f...;01 u. 233520 34E-Ol 0.17682236E-Ol O.ll988845E-Cl 0.95568672E-02 0.71667545E-02 0.47807880t-02 0.238971 90E-02

-0.93258556E-08

PAGE 11

R = 58.74976

(,.10948077f-Ol O.l0425407E-Ol 0.98886564E-02

t-' t-' t-'

1 .. t-9.04758 l .. 71.42'!53 Z = 73.POQ-48 7 • 76.19('43 l .. 7-..57138 l .. ~0.91)233 1 .. :n. B32B l .. IJ5.7142~ l • A8.09'il8 1 :: QQ.47614 l • CJ2.R570Q 1 .,. Q5.238C4 1 2 CJ7.611JCJC} l • Q9.99994 Z slOC.99CJ91t l a:10 1.99994 1 =107.9()994 1 .. 103.999Q4 7 "'104.99994

l = 61.90472 l a 61t.lf'!\68 l .. 66.66663 z ,. 69. 04758 7 .. 71.428'51 l = 73.8C'CJ48 l "' 76.l90ft3 1 .. 78.57138 l "' 80.95233 1 • 83.33328 l .. 8'5.71423 l .. 89.CQt;18 z .. 90.47614 1 .. 92.85709 l ,. 95.23801t z .. Q7.618CJ9 1 = 99.99CJ94 l s100.99991t Z =1Cl.999Q4 l •1C2.99CJ94 Z =1Cl.9QCJCJ4 l •1Cit.99994

C..67C54CHE-01 c.~3'l22196F-01 c.5AQC7583£-0l C. S41ll)2'5lE-Ol L .c;0441118E-01 C .461l1041E-01 C .lt172~485f-01 C. 37281182E-Cl c .32786559£-01 C. 28241 977E-Ol L.23670685E-01 r.190664QitE-Ol ( .1 44315(\ftf-01 t. 97fi92247E-02 (i. 78COCJ814£-02 c. 5&n5365E-02 <'• 39C 1 Ht;2E-02 r .195C2058E-02

-c.7r;941422E-oe

II a: 59.99976 C.61190633E-07 c .58269l08f-07 0. 5 52 6CJ449£ -07 0.52199230E-07 C.49C60425E-07 0.45857188£-07 c. 42')93 559£-07 O. 3CJ27l082E-07 (). 35900229E-07 C. 324821 98E-07 o. 2902152 8E-07 0.25'522'542£-07 l·. 21984CJ74~-07 c. 181t2618 ef-07 G.l4P42030E-07 C.ll238317E-07 c. 76191)725f-08 Ci. 6069942 BE-08 C • 45501t862E-08 C. 303361tHF-08 0.1'5163999£-0~

-0.59357054£-14

c. 51958CJ89E-01 (1. lt88 31t 711)6~- 01 0.4'56464?QF-Ol o. lt23Q7887E-Of Cl. 390CJ0'544E- 0 ~. 15735268f:-01 o. 32333H n-o1 o. 288(18624€ -01 O. 25405847E-01 0. 218842Hf-01 O.l8342182f-Ol O. 141745 76E-Ol O.ll1~7781E-01 O. l5Ab00CJ8E-02 "· 60436800f-02 C. 4'5315:36JE-02 C. 30220CJ54F-OZ 0.1510606'lE-02

-O.t;l.'71541iOE-OA

0. ,69~6!) q 7t: -(j 1 0 • 7.79CUt112E-0 1 c. H 7626~2 r -c 1 C.261lllR87f-Ol O. :,\24CJ3t.:'llE-O 1 O.HS9.U26E-Ol O. 30180M8f-Ol o. ?21Jit28CJ9£-01 0.21826Jl1E -01 C.21C.60966E-01 O.l Sit3 7ti95E -i)l :>.19753HlE-Ol o. 230 l615lf -o 1 0.174202CJ6E-Ol 0.2056422RE-Ol 0.1t;S64516E-01 o.uo8 so~t 8~ -o 1 O.l368810~f-01 0.15578158 -01 0.1179069 E-01 O.l30IJ686 7E -o 1 o.q8,Z4166E-oz O.l0517341tE-C1 o. 7CJ60 llt56f-02 J. 7964171i2E-02 o. b02791tltltE-02 0.~400H13E-02 o.lt0871tb67E-02 0.43014959£-02 O.l2'554576f-02 o. 3f21)0953E-02 Oo244:J7539E-OZ O.Z 5C6210E-07. c .1627'; 1931:-02 O.l07499HE-02 O.!ll151888E-Ol

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PAGE 12

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113

VITA

William Ray Heldenbrand was born on June. 17, 1946, near

Kidder, Missouri. He received his primary and secondary

education in the public schools of Winston, Missouri. He was

married to Patricia Jean Heldenbrand in 1966. He received a

Bachelor of Science Degree in Physics and Mathematics from

Central Missouri State College in May, 1968.

He entered the Graduate School of the University of

Missouri-Rolla in June, 1968, and has held an Atomic Energy

Commission Traineeship for the period August, 1968, to May,

1969.