CENTRALRESEARCHINSTITUTE FORPHYSICSBUDAPEST

3. G a d ó Z. Szatmáry

✓

SOPHIE and CECILY

Two Codes for Calculating Space Dependent Fast Neutron Spectra

S x u n ^ m n n S i c a d e m S c i e n c e / ,

CENTRAL RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

S O P H I E ^{a n d} C E C I L Y

TWO CODES FOR C A L C U L A T I N G SPACE DEPENDENT FAST NEUTRON SPECTRA

J. Gadó, Z. S z a t m á r y R e a c t o r R e s e a r c h D e p a r t m e n t

C e n t r a l R e s e a r c h I n s t i t u t e for Physics, B u d a p e s t Hungary

The codes SOPHIE and C E C I L Y are g e n e r a l i z e d v e r s i o n s of the a s y m p ­ totic slowing d o w n code G R A C E d e v e l o p e d at our Institute. Both codes are used for the c a l c u l a t i o n of the s l o wing d o w n n e u t r o n s p e c t r u m in o n e - d i m e n ­ sional /slab, cylind r i c a l , spherical/ g e ometries. The code SOP H I E is a c r i t i c a l i t y code, wh i l e C E C I L Y solves the s p a c e - d e p e n d e n t slowing down p r o b l e m w i t h i n the e l e m e n t a r y cell of the infin i t e fuel lattice. All three codes need the file c a l l e d R E A C T O R TAPE, c o n t a i n i n g the n u c lear data.

K I V O N A T

A S O P H I E és C E C I L Y k ó d o k a s z i n t é n i n t é z e t ü n k b e n k i f e j l e s z t e t t , a s z i m p t o t i k u s k ö z e l í t é s b e n m ű k ö d ő G R A C E kód á l t a l á n o s i t á s a i e g y d i m e n z i ó s g e o m e t r i á r a / sik, henger, gömb/. M i n d k é t kód l a s s u l á s i n e u t r o n s p e k t r u m s z á ­ m í t á s á r a szolgál. A S O P H I E kód k r i t i k u s s á g i s z á m í t á s o k r a alkalmas, mig a C E C I L Y kód a t é r f ü g g o la s s u l á s i p r o b l é m á t a v é g t e l e n f ü t o e l e m r á c s elemi c e l l á j á n belül o l d j a meg. M i n d h á r o m kód m ű k ö d é s é h e z a R E A C T O R T A P E nevű file h a s z n á l a t a s z ü kséges, a m e l y a n u k l e á r i s a d a t o k a t tartal m a z z a .

РЕЗЮМЕ

П р о г р а м м ы S O P H I E и C E C I L Y я в л я ю т с я о б о б щ е н и я м и а с и м п т о т и ч е с к о й п р о г р а м м ы з а м е д л е н и я GRACE, р а з в и т о й т о ж е в нашем И н с т и т у т е . Обе п р о г р а м м ы р а з р а б о т а н ы д л я р а с ч е т а н е й т р о н н о г о с п е к т р а з а м е д л е н и я в о д н о м е р н ы х / п л о с к о й , ц и л и н д р и ч е с к о й , с ф е р и ч е с к о й / г е о м е т р и я х . П р о г р а м м а S O P H I E с л у ж и т р а с ч е т а м к р и т и ч н о с т и , a п р о г р а м м а C E C I L Y р е ш а е т п р о б л е м у з а м е д л е н и я , з а в и с я щ у ю о т пр с т р а н с т в е н н о й к о о р д и н а т ы в э л е м е н т а р н о й я ч е й к е б е с к о н е ч н о й • р е ш е т к и Т В Э Л - о в . Для р а б о т ы о б е и х п р о г р а м м н у ж н а м а г н и т н а я л е н т а R E A C T O R ТАРЕ, с о д е р ж а щ а я не о б х о д и м ы е я д е р н ы е д а н н ы е .

sion a l - e q u a t i o n s a p p r o x i m a t i n g the n e u t r o n tr a n s p o r t e q u a t i o n in the fast e n e r g y range. The d i f f e r e n t i a l e q u a t i o n s are s u b s t i t u t e d by a finite d i f f e r e n c e scheme in the space variable. The m u l t i g r o u p picture is used w i t h a fixed 40 l e t h a r g y g r o u p s t r u c t u r e up to 10 M e V . The space interval c o r r e s p o n d i n g to the actual s y s t e m is d i v i d e d by m e s h points /max. 70 in S O P H I E and 25 in C E C I L Y / and into regi o n s /max. 15 and 6 resp./ c o n taining c e r t a i n m a t e r i a l c o m p o s i t i o n s .

The co d e s w o r k in three g e o m e t r i e s , name l y in the slab, c y l i n ­ d r i c a l and s p h e r i c a l ones. The two codes are d e v e l o p e d on the same basis as the m u l t i g r o u p z e r o - d i m e n s i o n a l code R J G 1 - G R A C E [1j . Quite similarly to it, the p r e s e n t v e r s i o n s of the codes do not take into account the t h e r m a l s p e c t r u m v a r i a t i o n in space immanently, but make use of the r e ­ su l t s of a p r e v i o u s T H E R M O S

[з]

The h o m o g e n i z e d c r o s s - s e c t i o n s for r e s o n a n c e m a t e r i a l s in their r e s o n a n c e r e g i o n s h o u l d be g i v e n in input r e q u i r i n g a p r e v i o u s run of the code R I F F R A F F [2] . A n o t h e r w a y of t r e a t i n g r e s o n a n c e s is av a i l a b l e in the codes, i.e. by u s i n g H e l l s t r a n d 's single pin r e s o n a n c e integrals and mutual s h i e l d i n g fact o r s [ l , 4] as input d a t a and l e t h a r g y gr o u p d i s t r i b u t i o n s of the r e s o n a n c e i n t e g r a l s /taken from a l i b r a r y / .

T h e p u r p o s e of d e v e l o p i n g these codes was twofold.

F i r s t l y they are s u i t a b l e to m a k e t h e o r e t i c a l investigations, e . g .

st u d y of some c u r r e n t l y used a p p r o x i m a t i o n s /fast advantage factors, m u t u a l s h i e l d i n g factors/;

s t u d y of the v a l i d i t y of the a s y m p t o t i c r e a c t o r theory e s ­ p e c i a l l y the e n e r g y - d e p e n d e n t b u c k l i n g a p p r o x i m a t i o n for the r e f l e c t o r s /used in G R A C E [l] / .

S e c o n d l y they m a y be used to c a l c u l a t e f e w —gr o u p d i f f u s i o n c o n ­ sta n t s in s y s t e m s of a h i g h n u m b e r of d i f f e r e n t m a t e r i a l regions.

It is w o r t h to mention, that the code S O P H I E may be used for c r it i ca l it y c a l c u l a t i o n s in cases w h e r e the a s y m p t o t i c a p p r o x im a ti o n is suff ic ie nt for taking into a cc o u n t the e f f e c t of axial leakage.

The codes SOPHIE and C E C I L Y were w r i t t e n in I C L - F O R T R A N . Typical r un ni ng times on the ICL-1905 c o m p u t e r are in the range of 5 to 10 minutes.

II. B A S I C E Q U A T I O N S A N D C A L C U L A T I O N A L M E T H O D

1. Ba si c e q u a t i o n s

The e q u a t i o n s solved by the co d e s S O P H I E and C E C I L Y can be w r i t ­ ten in the f o l l o w i n g form

1 9r ra

/lb/

- I Ф

N o t a t i o n s

u - l e t h a r g y

r - space v a r i a b l e

a - g e o m e t r y indicator, 0 for slab g e o m e t r y

1 for c y l i n d r i c a l g e o m e t r y 2 for s p h e r i c a l g e o m e t r y ф(г,и) - d i r e c t i o n i n t e g r a t e d flux:

1

2TT j" f ( r , u , y ) dy

r ' -!

J (r,u) - . " r a d i a l " curre n t : 1

S ( r ,u) - f i s s i o n source:

dy

oo

L ( r , u ) - i s o t r o p i c e l a s t i c s c a t t e r i n g rate l(r,u) - i n e l a s t i c s l o w i n g d o w n rate

L ^ ( r , u } - "radial" c o m p o n e n t of the a n i s o t r o p i c e l a s t i c s c a t t é r i n g rate L ^ ( r , u } - "axial" c o m p o n e n t of the a n i s o t r o p i c e l a s t i c s c a t t e r i n g rate.a x i a

The e x a c t m e a n i n g of L q , L^ L Z and I is given in ref. [l] . T / \

E ^r,u) — total c r o s s section.

The cod e s S O P H I E and C E C I L Y c a l c u l a t e the space depe n d e n c e of the n e u t r o n flux o n l y in one spatial variable, wh i c h is call e d the "radial"

var i a b l e . Sp a c e d e p e n d e n c e on the other spatial c o - o r d i n a t e s is taken into a c c o u n t by the "axial" q u a n t i t i e s : axial c u r r e n t J Z (r,u) and axial b u c k l i n g В .

T h e axial l e a kage is c a l c u l a t e d a s y m p t o t i c a l l y , quite similarly to the l e a k a g e c a l c u l a t i o n in G R A C E [ l ] . A c t ually, one of the main d i f ­ f e r e n c e s b e t w e e n the m u l t i r e g i o n codes SOP H I E and C E C I L Y is in the way of h a n d l i n g the axial leakage. The code S O P H I E is m e a n t to c a l culate the c r i t i c a l i t y and the fast s p e c t r u m of c o r e - r e f l e c t o r systems. The user jnay give d i f f e r e n t ax i a l b u c k l i n g s for d i f f e r e n t r e g i o n s and the set of

e q u a t i o n s /1/ is s o l v e d by t a k i n g the actual value of E (r,u) at every p o i n t r . The code C E C I L Y is m e a n t for cell c a l c u l a t i o n s and the same b u c k l i n g m u s t be u s e d for all r e g i o n s of the cell. P r a c t i c e has shown that in r e g i o n s w i t h small E T this m e t h o d fails, t h e refore in eq./lc/

E (r,u) is n o t the a c t u a l total c r o s s s e c t i o n at poi n t r , but a cell- a v e r a g e d value. C o n s e q u e n t l y , eq./ l c / m u s t be r e w r i t t e n in case of CECILY

as Г

J ET (r,u) ф (r ,u ) d V

В x / \ . С в 11 ,Z / ч T Z / \

- -J Ф (r , u ) + --- J (r,u) = L1(r,vi) 1 Ф ( r ,u ) dV

cell

T h e same a v e r a g i n g p r o c e d u r e is c a r r i e d out for all c r o s s - s e c Lions a p p e a r ­ ing in L Z .

T h e f o r m a l i s m d e s c r i b e d in the f o l l o w i n g holds for both codes w i t h m i n o r d i f f e r e n c e s . T h e r e f o r e the d e r i v a t i o n s fol l o w the line c o r e s ­ p o n d i n g to SOPHIE. W h e n n e c e s s a r y , the d i f f e r e n c e s b e t w e e n the two codes are indicated.

T h e e l a s t i c s l o w i n g d o w n source is a p p r o x i m a t e d in the same way as in G R A C E fl|. For e l e m e n t s h a v i n g mass numb e r s g r e ater than the a l u ­ m i n i u m /F- е lements/ the F e r m i - a g e a p p r o x i m a t i o n is used, while for a l u m i ­ n i u m and the l i g h t e r e l e m e n t s /f-elements/ the G r e a t l i n g - G o e r t z e l a p p r o x i ­ m a t i o n is used. The a n i s o t r o p i c s l o w i n g down source is taken into account

for h y d r o g e n e , d e u t e r i u m and b e r y l l i u m /h-elements/ a c c o u n t i n g for a lin e a r a n i s o t r o p y in the l a b o r a t o r y system.

2.• The m u l t i g r o u p e q u a t i o n s

The c o n t i n u o u s e n e r g y - d e p e n d e n c e of the n e u t r o n flux and c u r r e n t is s u b s t i t u t e d by a m u l t i g r o u p picture. The g r o u p str u c t u r e is identical w i t h that of GR A C E [l]. The j ^ gro u p s has the l e t h a r g y limits U j-i and u_. , and l e t h a r g y w i d t h Au_. = u_. - u_, ^ . The s u b s c r i p t j of

the thermal g r o u p is d e n o t e d by ^th * As t^ie last e n e r g y group,

= NTH, i.e. The g r o u p i n t e g r a l s of flux and c u r r e n t are the total numb e r of e n e r g y gro u p s / < 40/ u j

Ф . (r) = ^ ф ( r , u ) d u and

3 u j-i

u j

J j , Z (r) = J J r 'Z (r,u) du u j-l

w h i l e the o t h e r q u a n t i t i e s are a v e r a g e d for l e t h a r g y gro u p s in the same w a y as in G R A C E /see e q s . / 2 1 / - /33/, /35/ - /40/, /42/ of ref. [l] / . A f t e r simple t r a n s f o r m a t i o n s , e q s ./l/ c a n be w r i t t e n as follows!

d r a J r (r)

----^ --- + r a A . ( r ) ф.(г) = ra B . ( r )

/2а/

. d ф . ( r )

i + c-(r) о-(г) / 2b /

- § ф . ( г ) + C ^ (r ) J Z (r )= D ^ (r )

/2с/

w h e r e

Aj (r )

Bj(r)

C j ( Г )

Z h

h j

N. 5,.

, v f fj + 1 _{£ N.}

Lj F

r: AU .

f r fj ♦

V

AUj q F • T "

J-1 f r Au . q fj

r fj + T 1

£T - IN, M, . + £ 3 к k k 3 h

N h H k j Zhj + 2 Au .

3 n r 'Z

RA .

_____ 1_ T

Au. P h j-1 + T 1

V l (r)

B ‘

D 3Cj (r )

h B P h j-1

C * ( r )

- RA

The nomenclature is given in ref. [l].

The r-dependence of the cross-section, slowing down densities and the BIGG-type resonance absorption rate RA^ is not written here explicitly. Equations /2а-с/ are completed now with the slowing down e q u a t i o n s :

9 f j (r ) - -

q . j ( r ) - 1

A--- ф . (r) +

Л и . ~i '

r fj + 2

Ди .

_ £ J " Ди. q fj-i(r)

F(j + V

L N 5_. Ф •(r ) F j V"' E Ü T p "F “Fj

/За/

/ ЗЬ/

_ „ N. H, .

P '

=

--

hj u. J

Z. . 4- -5Г-

h j 2

и .

+ — hJ ' l 2 Pr 'Z C r )

Z . + h'i-1

h] 2

/ Зс/

The expressions of С , 2 (г) and p5.(r) are modified in CECILY:

J h 3

z „z, N cell

c ‘ = C j ( r ) =

N. H. . h hj

Ди zhi +

j

Ф j(r ) dV

j

N. H. .

h h 3 - ф . (r ) dV Ди . j v

„z , N cell Zhj + 2 p h j (r> ' --- r ---

z h

Jj(r) + --

Ди

1--- P Z

(r)

cell

Ф j(r) dV

h,j-l

Equations /2а/ - /2с/ and /За/ - /Зс/ are valid for every fast energy group /j = 1,2....N T H - 1 / with the initial conditions

q £ j (r) - q F j (r> - P hjr 'Z C O = 0 for j = 0 The equations of the thermal group are

d r “ (r) 2 th

dr + ra EA (r) + — ---

3 th 3CZ Cr)

3 th

Ф • ( r )

3 th

a

Z q fi - l (r) + q Fi - 1 (Г) "

f £3 th 1 1

£ В P Z , ( r )

h ^ t h “1

CÍ (r ) 3 th

/4а/

1 3

йф • (г)

Dth

dr C r. (r) Jj (r) =

'th th

E р Г . , ( r ) h h 3th-i

/ 4b /

- I ф. (r) + C Z (r) J* (r) = E p" - i ( r )

3 3th 'th 'th h3th_1

/4с/

where

C* (r) = Cr. (r) = Y T Cr ) - и- Í? (r )

th 3 th Jth Jth Jth

or in CECILY

C* = C* (r) 3 th 3 th

cell

^ (r ) - p. Y S. (r)

3 th 3 th 3 th

i Ф-: (Г) dV cell ]th

ф . (r ) dV 3 th

3. Finite differencing of the multigroup equations

The differential equations /2/, /3/, /4/ are defined on the space interval О < r < R , where R is the radius of the cylinder or sphere, or the thickness of a slab /half-thickness of a symmetric slab/.

The (P^-) boundary conditions for every group j are specified as a . / at the middle point of a symmetric slab, cylinder or sphere:

(О) = О /5а/

b . / at the left hand side of an asymmetric slab:

J j ( ° ) = ---j - Ф j ( О ) /5Ъ/

c./ at the right hand side of a slab, cylinder or sphere in the case of zero current /symmetry/ condition:

J^ (r ) = О /5с/

d ./ at the right hand side of a slab, cylinder or sphere in the case of a vacuum boundary:

J j (R) = 4>j(R) /5d/

Obviously, in CECILY always the conditions a./ and c./ apply.

In order to form a finite difference scheme from e q s . /2/ and /3/ the interval (0,R) is divided by N + 1 mesh points and the e q u a ­ tions are integrated for suitably chosen mesh intervals. The first mesh point r^ is at r = 0, while the last one rN + ^ is at r = R. The middle point of the interval (г гк+1 ^ denoted by г к +1/2‘ T ^e

and right boundaries lie at different mesh points depending on the bound-

ary conditions. The left boundary lies at r-^ in case a./ while at r 3y2 in case b ./. The right boundary lies at rN+1 in case c./, while at rN+l/2 in case d */* It is noted, the points r^ and/or rN+1 are fic­

titious points when boundary conditions b./ and/or d./ apply.

The mesh must be chosen in such a way, that the boundary of d if ­ ferent material compositions lies at the middle point of one of the mesh intervals /i.e. at some гк + 1 / 2 ^ ‘

I

Now integrating e q s . /2а/ and /2с/ between гк~1/2 and rK+l/2

/к

a 7r - ra J r + ra ГК-И ГК-1

ГК+1/2 J j,K+l/2 ГК-1/2 J j ,K - l /2 rK 2 A j,K(fc)4>j,K

■ ГК Г - - - - - В (t) /K-2,3,...N/ /6а/

1 ф . + Г К-И " ГК C j,K+!(*') + U j,K(&) a r+ C.

1 ^j,K+l “ 3 <i>j,K j » K + l /2

ГК+1 ~ ГК D -j,K+lU') * D i , K q ) /K = 1 ,2,...N/ / 6b /

В rK+l " rK-l

Ч к +

rK+l " rK-l „ Z Tz _ rK+l rK-l n z

2 C j , К (Л) J j,K 2 j,K(JO

/6с/

/К = 2 , 3 , . . .N/

where

A .j,K(A)

+ M j.t + f

В Д и . Au. p F F j , Ü opZ

r£j, Л + “21 3 J'KU)

_ Z _ r r

C j , K U ) j , K U ) zT j Д

. N h Hhj,*

£ N k 4 - - - ^

Zhj,£ + 2

B j , K U ) “ S j,k+ I j,K+qFj-l,K f

Au .

__ IAu. q fj-l,K

h B P >^ lTiS - RA . rfj,2+ 2

r zC j , K U )

j » K

Dj , K í O

Ди RA

E

h Д и . P hj-l,K+l/2 j/g— J r

Zh j , Л + 2

♦j-i.K 3- b K + l/2

D j , K U )

Ди RA .

E

h Ди. P h j - 1 ,К lziL

Zh j , Л + 2

4>j-l»K J j-i,K

Ди RA

D j,K+l(í.') E

h Ди. p hj-l,K+l/2 j ' K + l— j*

Zh j ,Л ' + 2

♦j-l.K+i j-1'K + i/2

Here Л and Л ' indicate the material compositions at the mesh points К and К + 1 respectively.

The expression of is modified in CECILY as

N+l K=1

I

c j = c j , K U )

T

Z j , K U ) - E N M . + E к k K 3 ' ^ h

N. H, . . h hj , Л

N+l

Ди 2М , Л * 2

r K+l rK-l

l <f>j K=1 J

rK+l " rK-l ,K

Now we introduce some quantities in order to get a well arranged set of equations

Al(j,K) =

-ra K+l/2

X r

, rK+l " rK C j , K +lU') ~ с 1,к(л)

3 2--- --- 2--- ----

/К = 1 , 2 , . . .N/

a ГК+1 rK-l

A0(J,K) = -A1(J,K) - A1(J,K-1) + r£ — V + 2 - - A j / K (£) /K = 2 , 3 , . . .N/

A V (J , К 'J - r “ - -K-! B j ( K ( 0 . ra+i/2 ° J fKtl(t*) ^ , . a ) +

J , K+l (Л') + C j,K(A)

Drj , K U O + D j , K - l U )

K - l / 2 r r

c j ,к (л') + c j , K-l( Л)

/К = 2,3,...N/

FI(J,K) = ф.^к

y y(j ,k ) = J ^ _ K

/К = 1,2,...N+1/ /К = 1 , 2 , . . .N/

/К = 1,2,...N+1/

Substituting /6Ь/ to /6а/ and using the notations in eq./7/ the following equation is obtained

Al(j,K>FI(j,K+l) + AO(j,K>Fl(j,K) + Al(j,K - l > F I ( j ,K-l) = AV(j,K)

/К = 2,3,...N; J = 1 , 2 , . . .NTH-1/ ^/8/ In the thermal group, one gets the same set of equations, using the nota­

tions

jj_u»K U ) 3<-h' ^

В

'th th' 3C

jt h ,K(°

£ B Рьч B jt h .K(l) = \ 4fjth-l.K + 4F j th-l.K

h Jth-1 'K C 3 th-K < 0

- c * T

= E

c jt h . K()0 c j t h . K ( D ^ t h'1 " Ujth £4 h .i

D jt h . K f l U ' ) = D jt h ,K(t) ■ I P hjth-l,K+ l/2

D it h 'K(° ■ h P h ith-1 'K

In CECILY, the expression of C jth ' K ^ is modified to

C Z = C Z

N+l l K=1

ET - ц .

^ t h ' *

3 th _3th'*J

j t h N+l

l ф ч к

K=1 3th'K

ф j

Jth,K

ГК+1 ” ГК-1 rK+l " ГК-1

Equations /3/ are integrated in the same way as e q s . / 2 / and /4/, i.e.

e q s ./3 a / and /3 b / and the axial component of e q . /Зс/ from rK _ i/2 to rK+l/2 ( K = 2 , 3 , ...N), and the radial component of e q . /Зс/ between rv and

rK+l = 1 '2 /***N )r giving

Ди .

‘f j,K

N f H f i , K ( Q

Au . ^{f i}

(

,

) +

f j,K(>e)

.Au.

rfj,K(€) + ~2

q fj-l,K

/9а/

/К = 1,2,...N+1/

V b K - é r $ ^{n f} EFj,K(t) F I (J 'K) /К = 1,2,...N+1/ / 9b/

r 1

P h j ,K + l/2 2

Nh Hh j , K ( Z ) Au .

"h j , К (í. ) + 2

Nh Hhj,K+l(&*) Au . 'hj,K+l(>') + 2

1

2

Ли. Ли,

h j , K+l (í.') Jhj,K(£) " 2

Л и . +

!hj,K(ü) + 2 Zh j , K+l (l ') + Ли

Y (J , К )

5h j-1,K + l /2 1 9 c

Ли . hj,K

Nh Hh j , К (Й.) Ли, Zh j , К (Ä.) + ~ T L

y y(j ,k) +

' h j , K ( M

Ли.

Zhj,K(£> + ~ T

P h j -1,K /9d/

/ К =

1

2

1

Y (j , к ) = (J /к ) ^ F i ( j ,к+i') - f i(j ,k)) + — + D j f K 0 B /9e/

YY(J,K) =

K + l /2

В F I

3C j,K(i) In CECILY, of course

2

(J,K) + —

UJLiil

C Z , /„ X

Г Г

c j I K+l (i-') + c j , K ( 0 /К = 1,2, . . .N/

/ К =

1

2

1

/ 9f /

hj ,K

N+l N, H. . .. ,

У __ !3__b j f K ( M ,

K

=1

hj,KU) 2______

rK+l r K-l

Ли . N+l

l

rK+l rK-l

y y(j ,k ) +

'hj,K(* )

K=1 j»K

Ли.

Zh j , К (S- ) + ~~2

z

P h j -1,К

/К = 1,2,...N+1/ / 9g /

These latter expressions of currents and slowing down densities are valid for j = 1, 2,... N T H - 1 , with the initial values

- _ r _ z _

q fo,K “ qFo,K “ Ph o ,K+l

" Ph o ,К ~ °

4. Boundary conditions

Two types of boundary conditions were specified in II. 4. at both ends of the interval (,0,r). A finite difference form of these conditions may be obtained in the following way.

a./ If r = 0 is the middle point of a symmetric slab, cylinder or sphere, then eq. /2а/ is integrated between ^ and r 2 / 2 ' Usin9 eq./5a/ and the notations

a

+1

A0(J,1) = A1(J,1) + A jj2(l)

a+1 D

AV(J,1) - B j , K D - r3/2 the following equation is obtained:

/ j=l,2 , . . .NTH /

C j,2(£)

Al ( J , 1) * FI ( J , 2) + A0(J,1) * FI ( J , 1) = AV ( J , 1) (J = 1, 2, ... NTH)

/ Ю а /

b./ If r 2 / 2 is the left boundary of an asymmetric slab, one may write

that

FI (J , 1) + FI (J , 2) =

2 - 2 Y C J , 1)

An equation formally identical to eq./lOa/ can be obtained with the notations

A O (J ,1) =

1 - 4

a r 3 / 2

A1(J,1)

1

4

1

a r 3 / 2

and

A V ( J ,

1

1

1+4 1^{U (j ,d} a Г 3 / 2

pj: ,

2

c jr , 2(&)

(J = 1 , 2 , . . .NTH)

c ./ if r = R is a symmetry boundary, then integrating eq./2a/ between r , and r using eq./5c/ and the definitions

N + l /2 N

a - гa

А О (J ,N + 1) = - A l (J ,N) + ^{N+l N+l/2}

a + 1 ‘j fN ( jj,) .a

AV(j,N+l) = " r.4tl

a a D

a + 1 ^B j , N ( Й,) + *N+1/2

(J = 1 , 2 , . . .NTH)

C j , N

( 0

one obtains that

A O (J ,N+l) * FI (J ,N+l) + A1(J,N)#FI(J,N) = AV(j,N+l)

(j = 1, 2, ... NTH)

d./ In case of a v a c u u m b o u n d a r y at r = rN + ^^2 one

can Proceecle

/ Ю Ь / w i t h

A O (J ,N + l ) =

1 - 4

a 1N+l / 2

A V (J ,N + l ) =

a N+l/2

A l (J ,N)

1 M a Í(j ,nT a

N+l/2

/ J =

1

2

D . j . N

( 0

C j,N<ll)

E q u a t i o n s /10а/ and / Ю Ь / t o g e t h e r w i t h e q s . /8/ give NTH by /N+1/

e q u a t i o n s for the NTH by /N+1/ u n k n o w n FI/K,J/ flux values. This set of e q u a t i o n s can be r e w r i t t e n as

a(j) $ (J) = V(J) (J = 1, 2, ... NTH) /11/

w h e r e the e l e m e n t s of the c o l u m n v e c t o r s and V(j) are the FI/J,K/ and A V /J ,К / q u a n t i t i e s (K = 1 , 2 , . . . N+l), w h i l e the /N+1/ by

/N+1/ m a t r i x A(j) has the form

А О (J ,1) Al (J ,1) 0 0

A O (J ,2) Al ( J ,2) 0

о

о A l (J , N-l) А О ( J , N ) Al ( J ,N)

О О A1(J,N) AO(j,N+l)

Eq. /11/ c o u p l e d w i t h eqs. /9/ can be solved starting from J = 1 and g o i n g on g r o u p by g r o u p to the thermal e n e r g y group.

5. S o l u t i o n of the e q u a t i o n s

The s o l u t i o n of eq. /11/ is q u i t e s t r a i g h t f o r w a r d by factorizing the m a t r i x A as

£ ( j ) = g(j) g(j)

Here

BO.(\J,l)

A l (J ,l) B O (J , 2)

!(j) =

о a i(j ,n) b o(j ,n+i)

w h e r e

B 0 ( J , 1)

b o(j ,k)

= A O (J ,1)

= a o (j ,k) A l (J ,K - l ) * A1 ( J , K - 1 $ B0(J,K-1)

/12а/

(К = 2,3, . . .N+1)

and

C ( J )

1 Cl(j,l)

1

О

C l ( j , N )

1

whe r e

c i(j ,iO = Al(j,K)

BO(J,K) /К = 1, 2, . . . N/ /

12

!(J) c (j ) ф (л ) = | (j ) Ф ( j) = v (j )

is i n v e r t e d in two steps as / d e noting the c o m p o n e n t s of Ф (,J ) by P S l (j,K)/

PSl(j,l) AV(J,1)

B 0 ( J , 1 )

PSI (j ,k ) A V ( j , K ) - A l ( j fK - l ) * P S l ( j fK-l) B O ( j fK)

/К = 2 , 3

/12с/

. . . N +

1

and

FI (J ,N+1) = P S l ( j , N + l )

F l ( j , N + l - L ) = P S I ( J , N + 1 - L ) - C l ( j , N + l - L ) * F I ( j ,N+2-l) /1 2 d /

I

/

= 1 , 2 , . . . N /

for e v e r y J /J = 1 , 2 , ... NTH/.

The n u m e r i c a l o p e r a t i o n s n e c e s s a r y to get the final s o l u t i o n gi v e n by eq. /12d / are c a r r i e d o u t in the f o l l o w i n g o r d e r / a s suming the source d i s t r i b u t i o n S . „ to be known/:

D I *

a./ F i r s t the q u a n t i t i e s A O / J , K / and A l / J , K / are c a l u c l a t e d ; they d e p e n d on the m i c r o s c o p i c c r o s s - s e c t i o n s , n u m b e r d e n s i t i e s and g e o m e t r y only.

b . / A V / 1 , K / is c a l c u l a t e d , then us i n g eqs./12/ the flux distri b u t i o n F I /J ,К/ is d e t e r m i n e d in the first m i crogroup.

c . / By the h e l p of eqs. / 9 / the s l o w i n g d o w n d e n s i t i e s and then the AV/J,K/

are c a l c u l a t e d for the next m i c r o g r o u p , and this p r o cedure is repeated un t i l the t h e r m a l g r o u p is r e a c h e d /i n c luding this, too/.

T h e o n l y p r o b l e m r e m a i n e d is that the source d i s t r i b u t i o n is not k n o w n as it can be c a l c u l a t e d o n l y from the FI/J,K/ flux distribution.

T h e r e f o r e a s o u r c e i t e r a t i o n m u s t be appl i e d w h i c h gives the flux and source d i s t r i b u t i o n s at the e n d of the p r o cedure. As far as C E C I L Y is concerned, a

f u r t h e r p r o b l e m of c a l c u l a t i n g the flux a v e r a g e d value of the total cross s e c t i o n s o c c u r s there. /See II. 7./.

6. S o u r c e i t e r a t i o n

In the c o d e G R A C E [l] the e n e r g y d i s t r i b u t i o n of the source was g i v e n by

S(u) = f(u)

i.e. by the f i s s i o n spectrum, b e c a u s e of the n o r m a l i z a t i o n

oo

j

w h e r e к .. is the e f f e c t i v e m u l t i p l i c a t i o n factor of the system. As in ef f

the p r e s e n t o n e - d i m e n s i o n a l space d e p e n d e n t codes the source is not c o n ­ st a n t in space b u t is g i v e n by

S(r.u) = - p i . ef f

(r,u') ф (r,u') du' /13/

an i t e r a t i o n p r o c e d u r e is nece s s a r y . The it e r a t i o n formula reads as

(i) = f(u)

( r , u ) k ( i " D ef f

v £ f ( r ,u ')ф (i-1) (r,u') du' ,

or in terms of the finite d i f f e r e n c e d m u l t i g r o u p quantities:

s (i\j,K)

NTH

b r - , 4 P I ^J ''K)

(i-1) /14а/

"eff

where

V syst e m and w i t h the initial value

/14b/

/ lo)

F l ( j , K J E 1 /J = 1, 2 , . . .N T H ; K = l , 2 , . . . N + 1 / /14с/

Thus the source d i s t r i b u t i o n is n o r m a l i z e d to u n i t y and the m a x i ­ m u m d e v i a t i o n of the source valu e s froTn those c o m p u t e d in the p r e v i o u s i t e r a ­ tion step gives the d e g r e e of c o n v e r g e n c e as

E s^-1^ ( j , k )

e . = m a x 1 - — ---77-r---

1 К

S ^ ' O j f K )

J

The source i t e r a t i o n is t e rminated, if the va l u e of is less than a small e s p e c i f i e d in the input.

It is w o r t h to remark, that the code c a l c u l a t e s the total d e s t r u c ­ tion rate

,(i) df Jr(i')

( r f ,u')

+ в

(r

w h e r e V is the v o l u m e of the system, w h i l e F is the o u t e r s u r f a c e of it and r^ is some p o i n t of this s u r f a c e / o b v i o u s l y if there is any v a c u u m b o u n d a r y of the s y s t e m / . The total d e s t r u c t i o n rates m u s t be equal to unity in e v e r y i t e r a t i o n step b e c a u s e of the n o r m a l i z a t i o n /14/, p r o v i d i n g a test

»

of the c a l c u l a t i o n .

7. C a l c u l a t i o n of the axial l e a k a g e in C E C I L Y

As it was m e n t i o n e d above, in the e q u a t i o n for the ax i a l c u r r e n t /see e q . / 2 c / / the c r o s s s e c t i o n s are a v e r a g e d o v e r the cell. S t r i c t l y s p e a k ­ ing, an inner i t e r a t i o n w o u l d be n e c e s s a r y in e a c h source i t e r a t i o n step to get the shape of the flux in e a c h group. It has b e e n found, however, that it is s u f f i c i e n t to c a l c u l a t e the cross s e c t i o n a v e r a g e s u s i n g the flux shapes o b t a i n e d in the p r e v i o u s sou r c e i t e r a t i o n step. The c o n v e r g e n c e rate of the source i t e r a t i o n is p r a c t i c a l l y not e f f e c t e d by that.

The u s e r m u s t s p e c i f y an axial b u c k l i n g value in the input which m a y be zero. As a first step the code solves the p r o b l e m for this value of the buckling.

At the user s o p t i o n a SEA R C H r o u tine can be switched on which finds the m a t e r i a l b u c k l i n g of the gi v e n fuel lattice, i.e. such a В v a l u e for w h i c h k eff = !• The user is warned, that the code is not p r o ­ t e c t e d f r o m c a s e s w h e r e к < 1.

oo

8. D e t e r m i n a t i o n of the c r i t i c a l sizes in SOPHIE

The code S O P H I E p e r m i t s to c a l c u l a t e the k e f f , the flux d i s t r i ­ b u t i o n and f e w - g r o u p c o n s t a n t s of a given c o r e - r e f l e c t o r system. If the u s e r is i n t e r e s t e d in the c r i t i c a l value of some parameters, a successive c h a n g e of the size of one or m o r e r e g ions is c a r ried out by the SEARCH r o u t i n e un t i l k 0 ff = 1 is reached. Of course, this c r i t i c a l i t y iteration can a l s o be c a r r i e d out by c h a n g i n g the b u c k l i n g values of one or more r e ­ g i o n s in SOPHIE.

9. Some r e m a r k s on t r e a t i n g r e s o n a n c e s

The user of b o t h co d e s S O P H I E and C E C I L Y has the option to make use of the two m e t h o d s for t r e a t i n g r e s o n a n c e s that were d e s c r i b e d in G R A C E [l] in detail.

T h e first m e t h o d c o n s i s t s in us i n g the h o m o g e n i z e d absorption, f i s s i o n and s c a t t e r i n g cr o s s s e c t i o n s of r e s o n a n c e e l e m e n t s in their r e ­

s o n a n c e regions, g i v e n by a R I F F R A F F [2] c a lculation. The outp u t of RIFFRAFF c o n t a i n s not o n l y the c e l l - h o m o g e n i z e d val u e s of these cross sections, but t h e i r v a l u e s in the fuel rod as well.

The so c a l l e d B I G G - t y p e r e s o n a n c e t r e a t m e n t [l, 4j m a k i n g use of H e l l s t r a n d 's r e s o n a n c e i n t e g r a l s and their e n e r g y gr o u p d i s t r i b u t i o n s are o b v i o u s l y u s a b l e o n l y in SOPHIE. In CECILY, further d e t a i l e d c o n s i d erations c o n c e r n i n g the ф - f u n c t i o n library, etc. are necessary. Here the quest i o n of m u t u a l s h i e l d i n g factors and fast a d v e n t a g e factor does not arise at all, b e c a u s e the code itself a c c o u n t s for the lattice structure of the system.

III. O R G A N I Z A T I O N OF T H E C O D E S

In this section a short a c c o u n t on the o r g a n i z a t i o n of the codes SOPHIE and C E C I L Y is b r i e f l y given. The flow d i a g r a m can be seen in A p ­ pen d i x 1.

Fi r s t of all it seems to be obvious, that this type of c a l c u l a ­ tion needs not only a w e l l - a r r a n g e d o v e r l a y st r u c t u r e of subroutines, but a suitable c o m m o n blo c k str u c t u r e b e c a u s e of the e n o r m o u s l y high number of the v a r i a b l e s to be stored. Practi c a l l y , 40 m a t r i c e s £ /see 11.4/

ou g h t to be stored each w i t h 141 e l e m e n t s /in the case of 70 m e s h points, w h i c h is a c t u a l l y p e r m i t t e d in SOPHIE/, al o n g w i t h the c o m p u t e d q u a n t i ­

ties /flux, current, etc./, e a c h w i t h 40 * 70 elements. This is impossible, u n l e s s m a g n e t i c tapes are u s e d for s t o ring the c o e f f i c i e n t s of the e q u a t i o n s to be solved and the c o m p u t e d q u a n t i t i e s . The storage is o r g a n i z e d in such a w a y that one or two reco r d s c o r r e s p o n d to e a c h m i c r o g r o u p .

The o v e r l a y s t r u c t u r e and the o r d e r of s u b r o u t i n e ca l l s satisfy the r e q u i r e m e n t s of e c o n o m i z i n g the r u n n i n g time.

The s u b r o u t i n e s m a y be d i v i d e d into three groups. The M A S T E R s e g ­ m e n t cal l s the i n p u t - o u t p u t s u b r o u t i n e s INPUT, T A P E M A K E R and ANITRA. The

INPUT s u b r o u t i n e reads in the b a s i c o p t i o n data, the g e o metry, and the e n e r g y g r o u p limits.

The T A P E M A K E R s u b r o u t i n e re a d s in the e l e m e n t - w i s e n u c l e a r d e n s i ­ ties, the r e s o n a n c e and t h e rmal data. A f t e r w a r d s it reads in the m i c r o ­ g r o u p c r o s s - s e c t i o n s for e a c h e l e m e n t o c c u r i n g in the actu a l p r o b l e m from the G R A C E L I B R A R Y T A P E and p r e p a r e s the m a g n e t i c tape MTl c o n t a i n i n g all m i x t u r e d e p e n d e n t q u a n t i t i e s n e e d e d for the s o l u t i o n of the equ a t i o n s , us i n g two r e c o r d s for e a c h m i c r o g r o u p .

For s a v i n g w i t h storage, T A P E M A K E R w r i t e s the H e l l s t r a n d - t y p e r e ­ sona n c e d a t á on the m a g n e t i c tape MT2, and a f t e r w a r d s it reads t h e m in from t h e r e .

T h i s s u b r o u t i n e sets a l s o the i n i tial source value. A f t e r e x e c u t i n g the s u b r o u t i n e g r o u p c o n t r o l l e d by S O L V E I G /i.e. the source i t e r a t i o n and the w h o l e c a l c u l a t i o n / s u b r o u t i n e A N I T R A p r i n t s out all the i n t e r e s t i n g

q u a n t i t i e s w h i c h can be o b t a i n e d from the s o l u t i o n c o n t a i n e d by the m a g n e t i c tape M T 2 . T h e A N I T R A c a l c u l a t i o n needs b o t h the tapes MTl and M T 2 . If the c r i t i c a l i t y s e a r c h o p t i o n is used, the c o n t r o l is g i v e n to s u b r o u t i n e SEARCH.

It c a r r i e s out the o u t e r i t e r a t i o n to ^ e ff = 1 /c h a n g i n g the sizes in the case of S O P H I E and c h a n g i n g the axial b u c k l i n g in the case of CECILY/. In e a c h o u t e r i t e r a t i o n step s u b r o u t i n e S O L V E I G is called, and upon f i n i s h i n g the o u t e r iteration, s u b r o u t i n e S E A R C H calls the o u t p u t s u b r o u t i n e ANITRA.

A n y n u m b e r of p r o b l e m s can be solved one after the other, the w h o l e p r o c e d u r e b e i n g r e p e a t e d s t a r t i n g w i t h subroutine INPUT.

L a s t but not least some w o r d s are n e c e s s a r y on the most important b l o c k of the codes, c o n t r o l l e d by s u b r o u t i n e SOLVEIG. For each m i c r o g r o u p this s u b r o u t i n e reads in the h o m o g e n i z e d m i x t u r e data from the magnetic

tape M T l p r e p a r e d by s u b r o u t i n e T A P EMAKER, then calls subroutine FLUX, which s o l v e s the space d e p e n d e n t e q u a t i o n s w i t h the actual source distribution.

F r o m the s o l u t i o n o b t a i n e d for the actual microgroup, subroutine S O L V E I G c a l c u l a t e s the c o n t r i b u t i o n of this m i c r o g r o u p to the source d i s ­ t r i b u t i o n of the n e x t source i t e r a t i o n step. T h e n it turns to the next m i c r o g r o u p d e s t r o y i n g the s o l u t i o n o b t a i n e d for the previ o u s one.

F i n a l l y a n e w source d i s t r i b u t i o n is set and /only in the case of C E C I L Y / the flux a v e r a g e d v a l u e s of the total cross sections are computed.

If the s o u r c e i t e r a t i o n c o n v e r g e d , the s u b r o u t i n e S O L V E I G makes an extra run, in o r d e r to p r o d u c e the m a g n e t i c tape M T 2 , c o n t a i n i n g the computed

flux, c u r rent, etc. d i s t r i b u t i o n s /one r e c o r d for e a c h m i c rogroup/. The tape MT2 is p r o d u c e d o n l y a f t e r the ou t e r i t e r a t i o n converged, if SOLVEIG is c a l ­

led by s u b r o u t i n e SEARCH.

IV. U S E R ' S M A N U A L / I N P U T - O U T P U T D E S C R I P T I O N

B o t h c o d e s are w r i t t e n for the ICL-1905 c o m p u t e r in ICL-FORTRAN.

B o t h S O P H I E and C E C I L Y r e q u i r e the w h o l e m e m o r y c a p a c i t y /32 К/ of the c o m p u t e r , as they c o n s i s t of four o v e r l a y units. The input data may be r e a d in f r o m p a p e r tape or cards, w h i l e the w h o l e outp u t appears on the line p r i n t e r . The co d e s m a k e use of a l i b r a r y tape and two scratch tapes for t e m p o r a r y d a t a storage.

T h e .input-output s y s tems of b o t h codes are very similar, so they are t r e a t e d together. T h e inp u t d a t a can be e a s i l y prepared, u s u a l l y they do n o t r e q u i r e any c a l c u l a t i o n by hand. However, in the p r e s e n t versions of the codes, a p r e v i o u s T H E R M O S

[з]

|2] run h a v e to p r e c e d e the S O P H I E or C E C I L Y runs giving the thermal and r e s o n a n c e d a t a of m i x t u r e s c o n t a i n i n g fissi o n a b l e elements.

The o u t p u t of the code is v e r y abundant, and some not too s i g n i ­ f i c a n t p a r t s of the o u t p u t can be o b t a i n e d by sw i t c h i n g on some switches /see later / .

1. Input data

1st card /10А8/ : title of the actual run

2nd card /2L4, E6.3, 13 , 912, 614/ : o p t i o n p a r a m e t e r s char. 1-4 /L4 / : T R U E - this is the last p r o b l e m

FA L S E - this is not the last p r o b l e m char. 5-8 / L 4 / : T R U E - c o m p l e t e o u t p u t of input data

F A L S E - short o u t p u t of input data

char. 9-14 /Е6.3/ : the c o n v e r g e n c e limit of the source i t e r a ­ tion / p r a c t i c a l l y 1.Е-5/

char. 15-17 /13/ : n u m b e r of m e s h poin t s

/in S O P H I E m a x i m u m 70, in C E C I L Y 25/

char. 18-19 /1 2/ 0: the code does not call the S E A R C H sub r o u t i n e 1: c r i t i c a l size d e t e r m i n a t i o n by SEA R C H

2: c r i t i c a l b u c k l i n g d e t e r m i n a t i o n by SEARCH char. 20-21 /1 2/ n u m b e r of r e g ions

/in S O P H I E m a x i m u m 15, in C E C I L Y 6/ char. 22-23 /1 2/ n u m b e r of m i x t u r e s

/in SOPHIE m a x i m u m 8, w h i l e in C E C I L Y 6, but r e ­ s o n a n c e i s o t o p e s m a y be p r e s e n t o n l y in the first 4 m i x t u r e s /

char. 24-25 /1 2/ 2: R I F F R A F F d a t a for e v e r y m i x t u r e c o n t a i n i n g r e s o n a n c e i s o t o p e s

1: B I G G - t y p e d a t a for e v e r y m i x t u r e c o n t a i n i n g r e s o n a n c e i s o t o p e s

0 : type of r e s o n a n c e d a t a is d e t e r m i n e d sep a r a t e l y for e a c h m i x t u r e c o n t a i n i n g r e s o n a n c e isotopes char. 26-27 /1 2/ 0: v a c u u m b o u n d a r y f r o m left

1: s y m m e t r y b o u n d a r y from left char. 28-29

<** /1 2/ 0: v a c u u m b o u n d a r y from rig h t 1: s y m m e t r y b o u n d a r y f r o m ri g h t char. 30-31 /1 2/ 0: slab g e o m e t r y

1: c y l i n d r i c a l g e o m e t r y 3: s p h e r i c a l g e o m e t r y

char. 32-33 /12/ m a x i m u m numb e r of source iteration steps char. 34-35 /12/ n u m b e r of m a c r o g r o u p s /max.6/

char. 36-59 /614/ up pe r b o u n d a r i e s of macrogroups.

3rd c ar d /13, 2E.3, 1513/: S E A R C H data

T h i s c ar d is to be o m i t t e d if no sea r c h c a l c u l a t i o n is requested.

c h a r . 1-3 /13/ m a x i m u m n u m b e r of outer i t e ration steps c h a r . 4-9 / Е 6 .3/ c o n v e r g e n c e limit of the outer iteration

/usually 1.Е-5/

c h a r . 10-15 / Е 6 .3/ an o v e r a l l c h a n g i n g factor /EV/

c h a r . 16-60 /15 I3/(S) 1 6- 3 9 / 8 13/ (c)

r e g i o n w i s e m o d i f i c a t i o n factor / M O D / N R / /

In o r d e r to u n d e r s t a n d the m e a n i n g of these last two quantities, we w r i t e the o u t e r i t e r a t i o n f or m u l a /e.g. for reg i o n w i s e b u c kl in g i t e r a ­ t io n /

b (n r)(n + 1 )= b (n r) 1 +

м е р

1 + M0D(NR) * VOL П ^

w h e r e V O L ^ ^ = EV and the later v a l u e s of V O L ^ n ^ are c a l c u l at e d from the k e ££ v a l u e and the p r e v i o u s V O L ^ n value. Here, NR specifies one of the regions.

4th c a r d / 8/ E6.3,l3// m e s h p o i n t data

char. 1-72 8 /E 6 .3,13/ The m e s h points are s p e c i f i e d by pairs of data.

The first n u m b e r or a pair gives in cm the d i s ­ tance of two m e s h poi n t s following, each other, w h i l e the second number gives the m es h point, w h e r e this d i s t a n c e changes. If one card is not sufficient, c o n t i n u e on the next cards.

5th c a r d /15/12, 13/ or 6/12, 13//: reg i o n s p e c i f i c a t i o n is given by a pair of data

char. 1-75 15/12, 13/ (S) The first numb e r of a pair gives the number , , ч of the m i x t u r e in the region, while the

char. 1-30 6/12, 13/ (C) . . ..

second one gives the first mesh point right to the reg i o n b o u n d a r y /which is at the m i d ­ po i n t of a m e s h interval/

6th card /8Е10.5/ : b u c k l i n g values

This is need e d o n l y if a slab or c y l i n d r i c a l g e o m e t r y c a l c u l a t i o n is carr i e d out.

char. 1-80 /8Е 10.5/ the valu e s of axial b u c k l i n g in the order of regions. If one card is not sufficient, c o n ­ tinue on the next one.

In the case of CECILY:

char. 1-10 /Е10.5/ the va l u e of axial b u c k l i n g

7th card /8Е10.5/ : b u c k l i n g i t e r a t i o n initial guess

This card is to pe p u n c h e d o n l y if a S E A R C H i t e r a t i o n is need e d for the d e t e r m i n a t i o n of the c r i t i c a l b u c k l i n g value. This card has the same format as the p r e v i o u s o n e .

T h e s e ca r d s form the first p a r t of input data, read in by s u b ­ r o u t i n e INPUT. A lot of small tests are b u i l t in the codes in order to p r o t e c t them from input errors. If an er r o r is d e t e c t e d the p r o b l e m is aborted. The second p a r t of in p u t d a t a are r e a d in by T A P E M A K E R w i t h a simi l a r care. A f t e r such an ab o r t i o n , the run of an e v e n t u a l next p r o b l e m r e m a i n s possible. The s u b r o u t i n e T A P E M A K E R reads in the m i x t u r e cards and the r e s o n a n c e and t h e r m a l d a t a cards.

M i x t u r e c a r d s : the f o l l o w i n g is r e p e a t e d for e a c h m i x t u r e 1st card /12, E6.3, L 4 / : m i x t u r e i d e n t i f i c a t i o n

char. 1-2 /1 2/ c h a r . 3-8 /E6.3/

char. 9-12 /L4/

m i x t u r e num b e r

fast a d v a n t a g e fac t o r in SOPHIE, w h e r e cel l - - h o m o g e n i z e d n u c l e a r d e n s i t i e s are to be given, In C E C I L Y it has no d i r e c t m e a ning, but it is used to c h a r a c t e r i z e m i x t u r e s . Namely, it has the val u e

+ 1.0 for m i x t u r e s in fuel;

any p o s i t i v e n u m b e r for m i x t u r e s in cladding;

any n e g a t i v e n u m b e r for m i x t u r e s in m o derator.

it is an i n d i c a t o r of r e s o n a n c e treatment. It is T R U E for B I G G - t y p e t r e a t m e n t and FA L S E for R I F F R A F F - t y p e treatment.

It has to be n o t i c e d that r e s o n a n c e isotopes m a y be p r e sent only in m i x t u r e s h a v i n g a m i x t u r e n u m b e r less or equal 4. For each element p r e s ­ e n t in this m i x t u r e p r e p a r e a 2nd m i x t u r e card.

2nd m i x t u r g — c a r d /А8, E8.4, 12/ : e l e m e n t d a t a . / T h e e l e ments are to be listed in the order of increasing mass n u m b e r s /

char. 1-8 /А8/ e l e m e n t identifier, c o r r e s p o n d i n g to the e l e m e n t i d e n t i f i e r list of the GRACE L I B R A R Y

char. 9 16 /Е8.4/ n u c l e a r d e n s i t y of this element

char. 17-18 /12/ it has the value

2: for fissile elements Is for fertile e l e ments

Os in SOPH I E for ev e r y ot h e r element, in C E C I L Y for e v e r y ot h e r e l e ment except

for H, D and Be.

In C E C I L Y a list is m a d e from all elements p r e s e n t in the syst e m ac c o r d i n g to the o r ­ der of the GR A C E LIBRARY. For H, D and Be,

the user m u s t give the num b e r of this e l e ­ m e n t in this list w i t h a n e g ative sign.

E.g. if the sys t e m contains H and Be, punch -1 for H and -2 for Be.

T h i s c a r d has to be r e p e a t e d as m a n y times as there are elements in the a c t u a l m i x t u r e . The m a x i m u m n u m b e r of e l e m e n t s in a m i x t u r e is restricted to 12, a m o n g the s e m a x i m u m 3 m a y be l i g h t e r than Be /including Be/, 6 m a y be l i g h t e r than Al, / i n c l u d i n g A l / and 9 m a y have a mass num b e r greater than t h a t of A l . T h e total n u m b e r of d i f f e r e n t e l e m e n t s in the syst e m is limited to 18, w h i l e no m o r e than 7 d i f f e r e n t r e s o n a n c e e l e m e n t s m a y be treated in a p r o b l e m . A f t e r the e l e m e n t data, a bl a n k card follows, ind i c a t i n g the end of e l e m e n t d a t a for the m i x ture. D e p e n d i n g on the res o n a n c e t r e atment cards of the type 3 1 h / or 3/В/ follow.

If there are no r e s o n a n c e e l e m e n t s in this mixture, these cards are o m i t t e d .

In the case of B I G G - t y p e r e s o n a n c e treatment, the data for r e s o n a n ­ ce c a l c u l a t i o n f o l l o w now.

3rd m i x t u r e card / А / /2/2Е8.4, 12// s B I G G - t y p e re s o n a n c e tr e a t m e n t data char. 1-36 2/2E.8.4, 12/ Single pin r e s o n a n c e integral for the total

e n e r g y range, m u t u a l s h i e l d i n g factor and index of \p - f u n c t i o n are to be gi v e n here for a b s o r p t i o n and for fission. In CECILY, the m u t u a l s h i e l d i n g factors m u s t be equal to u n i t y / s e e 11.9/

This card has to be r e p e a t e d for e a c h r e s o n a n c e isotope in the mixture, in the order of i n c r e a s i n g m a s s numbers.

In the case of R I F F R A F F - t y p e treatment, o n l y one card is to be read in here for this mixture:

3rd m i x t u r e card / В / /213/ : R I F F R A F F g r o u p limits

c h a r . 1-3 /13/ the first m i c r o g r o u p , w h e r e R I F F R A F F t r e a t ­ m e n t is to be used in the a c t u a l m i x t u r e c h a r . 4-6 /13/ the last m i c r o g r o u p , w h e r e this t r e a t m e n t

is applied.

This ends the m i x t u r e ca r d s for one m i x t u r e . A f t e r the last m i x ­ ture fol l o w the r e s o n a n c e cards for R I F F R A F F data, g o i n g a g a i n t h r o u g h all the m i x t u r e s . T h e s e ca r d s are o m i t t e d for m i x t u r e s not c o n t a i n i n g r e s o n a n c e i s o topes or if a B I G G - t y p e c a l c u l a t i o n is applied.

The s e cards h a v e an o r d e r w h i c h m a y a p p e a r arti f i c i a l b u t this o r d e r p e r m i t s to save c o n s i d e r a b l e storage. R e s o n a n c e d a t a are giv e n m i c r o - g r o upwise, in i n c r e a s i n g o r d e r of m i c r o g r o u p index. For a g i v e n m i c r o g r o u p r e s o n a n c e d a t a are g i v e n for all m i x t u r e s w i t h R I F F R A F F t r e a t m e n t in the o r d e r of i n c r e a s i n g m i x t u r e number. For a g i v e n m i x t u r e a r e s o n a n c e c a r d is p r e p a r e d fop e a c h r e s o n a n c e isot o p e in the o r d e r of i n c r e a s i n g m a s s number.

R e s o n a n c e card /ЗЕ12.6/: R I F F R A F F d a t a

char. 1-36 /ЗЕ12.6/ r e s o n a n c e a b s o r p t i o n , v times fission

and s c a t t e r i n g cr o s s s e c t i o n for the actu a l e l e m e n t for the a c t u a l m i x t u r e and m i c r o ­ group.

As to these ca r d s the f o l l o w i n g r e m a r k s have to be ob s e r v e d . On m i x t u r e car d s 3/В/ R I F F R A F F g r o u p limits w e r e s p e c i f i e d m i x t u r e w i s e .

T h e r e f o r e