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This document was prepared under the sponsorship of the Commission of the European Atomic Energy Community (EURATOM)
Neither the EURATOM Commission, its contractors nor any person actin on their behalf
Io — Make any warranty or representation, express or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this document, or that the use of any information, apparatus, method, or process disclosed in this document may not infringe privately owned rights ; or
EUR 592.e
REACTOR PHYSICS PROGRAM AT EURATOM CCR - ISPRA by V. RAIEVSKI
European Atomic Energy Community — EURATOM
Joint Nuclear Research Center — Ispra Establishment (Italy) Reactor Physics Department
Paper presented at the IAEA Study Group Meeting on Research Reactor Utilization
Athens (Greece), September 9-13, 1963 Brussels, February 1964 — 31 pages
This report describes the main activities of the Ispra Reactor Physics Department of Euratom in the field of reactor physics.
It is divided into two parts :
The first one describes general reactor physics work including such topics as shielding, bum-up, fast reactor physics, and Monte-Carlo codes.
EUR 592.e
REACTOR PHYSICS PROGRAM AT EURATOM CCR - ISPRA by V. RAIEVSKI
European Atomic Energy Community ·— EURATOM
Joint Nuclear Research Center — Ispra Establishment (Italy) Reactor Physics Department
Paper presented at the IAEA Study Group Meeting on Research Reactor Utilization
Athens (Greece), September 9-13, 1963 Brussels, February 1964 — 31 pages
This report describes the main activities of the Ispra Reactor Physics Department of Euratom in the field of reactor physics.
It is divided into two parts :
The second part is connected with work required by the deve-lopment of the Orgel program and deals with Orgel recipe, thermal neutron spectrum, resonance escape probability and fast neutron multiplication factor.
EUR 592.e
EUROPEAN ATOMIC ENERGY COMMUNITY - EURATOM
REACTOR PHYSICS PROGRAM
AT EURATOM CCR-ISPRA
by
V. RAIEVSKI
1964
Joint Nuclear Research Center Ispra Establishment - Italy Reactor Physics Department
Paper presented at the IAEA Study Group Meeting on Research Reactor Utilization
C O N T E N T S
PART A GENERAL REACTOR PHYSICS
I. Shielding P h y s i c s
II. Monte Carlo Codes
III. B u r n - u p P h y s i c s
IV. F a s t R e a c t o r P h y s i c s
PART Β ORGEL REACTOR PHYSICS
I. Orgel Recipe
II. T h e r m a l Neutron Spectrum
III. Resonance E s c a p e Probability
REACTOR PHYSICS PROGRAM AT EURATOM CCR-ISPRA
INTRODUCTION
It is a v e r y interesting but not easy t a s k , to r e p o r t in the s h o r t time allowed for an o r a l p r e s e n t a t i o n , on the r e a c t o r physics work in p r o g r e s s in a c e n t e r . Each group is doing a good job, but a somewhat a r b i t r a r y selection has to be m a d e , in o r d e r to stay within the imposed time l i m i t s .
I hope that the topics p r e s e n t e d h e r e will sufficiently i l l u s t r a t e our activity without tiring the audience and without deceiving too large a n u m b e r of g r o u p s .
I have divided this p r e s e n t a t i o n into two p a r t s :
p a r t A deals with work not directly connected with the ORGEL p r o g r a m ; h o w e v e r , some of this work is or will be helpful for this p r o g r a m .
- 7
PART A - GENERAL REACTOR PHYSICS
I. SHIELDING PHYSICS
In the field of shielding nuclear energy p l a n t s , c e r t a i n problems of outstanding i m p o r t a n c e still r e q u i r e satisfactory solutions. Among t h e s e , p a r t i c u l a r i m p o r tance is a s s u m e d by that of determining r e m o v a l c r o s s sections for fission neutrons and that of calculating the propagation of neutrons and gamma r a y s in ducts. T h e o r e t i c a l and e x p e r i m e n t a l r e s e a r c h has been undertaken in this direction.
a) Removal c r o s s sections :
It is an useful i n t e g r a l p a r a m e t e r to d e s c r i b e the attenuation of neutrons through shielding. After some d i s t a n c e , an asymptotic distribution cha r a c t e r i z e d by an exponential behaviour is achiedVed , which can be p r o p e r l y d e s c r i b e d in t e r m s of a single c r o s s section, the r e m o v a l c r o s s section. F o r many m a t e r i a l s this p a r a m e t e r has been m e a s u r e d for a fission spectrum. As neutron s p e c t r u m s impinging on a shielding a r e often different from f i s
sion s p e c t r u m , n e e d i e r m e a s u r e m e n t s with monoenergetic s o u r c e s is obvious.
T h e s e m e a s u r e m e n t s have been p e r f o r m e d with water using the 5,5 Mev Van de Graaff of the Padova u n i v e r s i t y . A s o u r c e of neutrons with energy varying in the r a n g e 0-8 Mev and 15-20 Mev was obtained in the center cf a water tank 300 χ 300 χ 270 cm using L i , D2O and H3 t a r g e t s . Detectors
used w e r e indium, indium under c a d m i u m , sulphur in c a d m i u m , other with
m e a s u r e m e n t s Ni, F e , Al and boron and fission c h a m b e r s a r e in p r e p a ration.
R -1
The first r e s u l t s obtained for the water r e m o v a l c r o s s sections
Ε Σ
Mev cm
4 , 8 0,15
6,8 0,12
a r e in r e a s o n a b l e a g r e e m e n t with the previous one . (2)
A Monte Carlo prog r a m to deal with the problem has been pr e p a r e d , other m a t e r i a l s as iron and lead a r e now m e a s u r e d .
b) Propagation of neutrons through ducts :
The facility used for this experiment is the 1 MW SORIN swimming pool r e a c t o r (Saluggia, Italy). A cubical water tank (3 m edge) was put in front
fission
of the shielding facility, a s p e c t r u m is obtained by using an u r a n i u m c o n v e r t e r plate,the d i a m e t e r of which may vary between 5 and 45 c m . The n e u t r o n s
propagating through a tube extending from the u r a n i u m c o n v e r t e r p l a t e , w e r e m e a s u r e d with different t h e r m a l , r e s o n a n c e and t h r e s h o l d detectors and fission chamber s ( U , T h , N p ) . Two different d i a m e t e r tubes w e r e used in t h e s e e x p e r i m e n t s (5 and 15 c m . )
T h e o r e t i c a l r e s u l t s obtained with a Monte Carlo p r o g r a m w e r e fiound in good a g r e e m e n t with the e x p e r i m e n t a l ones.
II. MONTE CARLO CODE
(3) A g e n e r a l Monte-Carlo code for fast effect c a l c u l a t i o n s , named MOCA-II v '
was written by Rief for the IBM 7090. It is an extension and an improvement of an e a r l y work initiated at B N L . ' ' MOCA-II allows the solution of the g e n e r a l two or t h r e e dimensional energy dependent t r a n s p o r t equations under the a s
sumption of the following scattering k e r n e l s :
a) elastic s c a t t e r i n g , isotropic or anisotropic in the c m . s y s t e m .
b) inelastic s c a t t e r i n g either by the level excitation or by the s t a t i s t i c a l model.
c) fast fission for energy dependent V value
The p r i m a r y neutron s o u r c e can be spatially d i s t r i b u t e d , monoenergetic or as a fission s p e c t r u m , a n d a n i s o t r o p i c .
The p r e s e n t g e o m e t r y routines allow the calculation of s l a b , cylindrical and s p h e r i c a l l a t t i c e s , a g e o m e t r y of a r b i t r a r y rod bundles is also provided. The code allows the calculation of the following quantities :
1) The c l a s s i c a l fast fission factor.
2) The slowing down probability of neutrons below a c e r t a i n energy l i m i t .
3) The ratio of secondary to p r i m a r y n e u t r o n s .
4) Absorption p e r g e o m e t r i c a l region and energy group.
In addition to t h e s e r e s u l t s , the following quantities can be calculated :
5) The fast fission r a t i o ( o )
6) The neutron flux averaged for each g e o m e t r i c a l region as a function of energy.
7) The F e r m i age G and its x, y and ζ components.
8) The energy deposition due to elastic neutron s c a t t e r i n g .
(3) H. Rief - MOCA-II, a multipurpose Monte Carlo code for fast-effect calculations / T r a n s a c t i o n s of the A m e r i c a n Nuclear Society, Vol. 6 n ° l , J u n e , 1 9 6 3 .
10
A lot of applications of this g e n e r a l code have been made or a r e under way. F o r example, energy deposition in the different media of a heavy water m o d e -r a t e d , o-rganic cooled -r e a c t o -r has been calculated as a function of the deposited energy. This quantity is of i n t e r e s t for the fouling p r o c e s s ; of c o u r s e , m a t e r i a l damage r a t e cannot be directly i n f e r r e d from these calculations, but v e r y likely
some relation can be found between radiation damage r a t e and energy deposition.
These calculations w e r e c a r r i e d out for the energy range 10 Mev to . 5 Kev, which was subdivided into 26 energy groups with group a v e r a g e d c r o s s s e c t i o n s . Including a detailed flux calculation for each of t h e 12 g e o m e t r i c a l regions , reasonable v a r i a n c e can be achieved with 5.000 h i s t o r i e s , which need approxi-mately 7 minutes of IBM 7090 computer t i m e .
11
III. BURN-UP PHYSICS
T h e r e a r e two steps in b u r n - u p physics :
a) to d e t e r m i n e the isotopie composition of a sample as a function of the i r r a d i a t i o n in a given s p e c t r u m ;
b) the isotopie composition being known, to find the effect on the r e a c t i -vity.
This second step can be m o r e easily solved by m e a s u r i n g in a given r e a c t o r the reactivity effect of synthetic fuel elements made by homogeneous m i x t u -r e of depleted u -r a n i u m and Pu of diffe-rent i s o t o p i e composition.
As synthetic fuel e l e m e n t s a r e v e r y expensive m a t e r i a l s , one way to p e r -form the m e a s u r e m e n t s , is the substitution technique, (5) the r e s u l t s of which a r e generally i n t e r p r e t e d in t e r m s of two group t h e o r y , dealing with the substituted and r e f e r e n c e lattices as two homogeneous zones. Though m o r e economic than a complete c r i t i c a l loading, this method is still
expensive. Heterogeneous theory * ' i s a good way to interpret the r e s u l t of substitution e x p e r i m e n t s with a few e l e m e n t s . F r o m m e a s u r e m e n t s of c r i t i c a l level for different lattice pitches and using the k e r n e l h e t e r o g e -neous t h e o r y , the Y , ft p a r a m e t e r s of the substituted elements can be obtained in t e r m s of the corresponding p a r a m e t e r s of the r e f e r e n c e elemente T h e s e p a r a m e t e r s d e s c r i b e the absorption and production of neutrons when the fuel element is i m m e r s e d in the unperturbed flux at a lattice position. In p r i n c i p l e , m e a s u r e m e n t with a single fuel element can be meaningful in this t h e o r y . In such a c a s e , difficulty will a r i s e from the weak r e a c t i -vity effect due to the substitution of an element.
The g r e a t sensitivity of the oscillation technique will allow us to p e r f o r m (7)
the m e a s u r e m e n t in this c a s e .
(5) Y . G i r a r d and Al. - II. Geneva Conference 15/8/336
(6) G. B l ä s s e r - Symposium on Exponential and C r i t i c a l E x p e r i m e n t s , A m s t e r d a m ( 2 - 6 / 9 / 6 3 )
12
The c e n t r a l p a r t of a fuel e l e m e n t , b e i n g long enough to m i n i m i z e end e f f e c t s ,
i s s u b s t i t u t e d by a s a m p l e of the s a m e g e o m e t r y , but m a d e of a different m a
t e r i a l , s i m u l a t i n g the isotopie c o m p o s i t i o n after a given b u r n u p . A n o r m a l
fuel e l e m e n t is a t t a c h e d to t h i s c o m p o s i t e o n e , and t h e whole a s s e m b l y is
o s c i l l a t e d in the c e n t r a l t h i m b l e of a c r i t i c a l f a c i l i t y . The ECO r e a c t o r
will allow such m e a s u r e m e n t s thanks to a c e n t r a l t h i m b l e extending well
below the bottom face of the t a n k .
The r e a c t o r signal m e a s u r e d with a d e t e c t o r p l a c e d at a point r i s given for
a t h e r m a l r e a c t o r and non m o d e r a t i n g s a m p l e by the r e l a t i o n :
s = Gf (r*. ro,f) P(r*o) Ga(r*,£o, f ) A(?o)
w h e r e r o , a r e the s a m p l e position and the o s c i l l a t i o n f r e q u e n c y ,
w h e r e A ( r0) = v / ^ d E d c3 0^(E) V N ( r *0, E , ¿ o )
and P( r*o) = VffdE d 3 V(E) <Tf(E) Ι Τ Ν ( Γ £ , Ε , ¿? )
a r e the t o t a l a b s o r p t i o n and p r o d u c t i o n r a t e of n e u t r o n s in the s a m p l e in the
v N ( r o , E , <A> )flux. Ga and Gf a r e a b s o r p t i o n and p r o d u c t i o n r e s p o n s e function
which a r e c h a r a c t e r i s t i c of the r e a c t o r and d e t e c t o r u s e d . T h e s e functions
can be c a l c u l a t e d by a s t a n d a r d a p p r o x i m a t i o n m e t h o d as m u l t i g r o u p diffusion (7) t h e o r y , M o n t e C a r l o or Sjvj from the g e n e r a l e x p r e s s i o n given in. the r e p o r t ,
or can be b e t t e r deduced by c a l i b r a t i n g the r e a c t o r with b o r o n and U235
s a m p l e s .
O s c i l l a t i n g f i r s t a s t a n d a r d b o r o n s a m p l e , one obtains (the index " O " r e f e r r i n g
to the s t a n d a r d s a m p l e ) :
s° = Ga( r , r*o, />) A0{?0)
If a second s t a n d a r d containing U235 i s now o s c i l l a t e d , s i n c e the a b s o r p t i o n
c r o s s s e c t i o n of U235 i s an a l m o s t p u r e l / v b e h a v i o u r , the r a t i o :
Ζ = A 2 5 / A o
will be independent of the position in the r e a c t o r and can be obtained by a
13
The U235 signal can be written : 25
s = s° Ζ + Gf( r , r0) P2 5
For an arbitrary sample containing fissionable material, we can now write:
O " ι / ^ * o r r \ ' " S e f - SPf A - Λ " + (S Ρ - S . Z>
0 v f f ' P25
or :
·*■-*■ s A . , ,-*· -* ν Ρ
s f = a ( r , r0, f ) £ + b ( r , r0, f ) ·
P25
where a and b a r e now experimentally known functions from the standards. By performing the experiments for two detector positions one can vary the ratio b / a and measure separately the absorption and the production of
neutrons.
Question may now arise as to the neutron spectrum at the oscillation position. To answer this question, Diana * ' has investigated the influence of Pu on the spectrum in the fuel and in the moderator using the Thermos code (see ref. part B). As expected, this influence is very important within the fuel while calculation has shown that in the moderator the effect is negligible at a distance of about 2 cm from the fuel surface. From these r e s u l t s , the following conclusion can be drawn :
The moderator spectrum far from the fuel element is not sensitive to the details of thermal absorption. Therefore, the absorption integral, as measured by the oscillation technique, is already a sufficient characte ristic for the thermal spectrum in the :moderator far from the elements. This facilitates the interpretation of the behaviour of Pucontaining samples in a reactor containing only natural Ufuel.
Additional calculations have been made on the non uniform distribution of Pu in the fuel. Calculation has shown that the Ό fuel parameter is rela tively insensitive to the Pu distribution, the effect on y is bigger but as this parameter is less important for the reactivity, one can be confident that theoretical Pu distribution , small errors will be made, in predicting reactivity change in a leactor.from uniform Pu distribution measurements.
14
IV. FAST REACTOR PHYSICS
The I s p r a c e n t e r is not d i r e c t l y involved in the E u r a t o m fast r e a c t o r
p r o g r a m . T h i s work is p u r s u e d in different n a t i o n a l l a b o r a t o r i e s u n d e r
a s s o c i a t i o n c o n t r a c t s with E u r a t o m . H o w e v e r , a s s o m e knowledge in
fast r e a c t o r p h y s i c s m a y be of i n t e r e s t in the f u t u r e , a l i m i t e d t h e o r e t i c a l
p r o g r a m h a s r e c e n t l y s t a r t e d in I s p r a . Some efforts w e r e devoted to
Doppler effect and n e u t r o n s p e c t r u m in fast a s s e m b l i e s .
a)
D o p p l e r effect :
The influence of non uniform t e m p e r a t u r e d i s t r i b u t i o n within the fuel
w a s i n v e s t i g a t e d . T h i s p r o b l e m is of i n t e r e s t for p o w e r r e a c t o r s with
high power d e n s i t i e s and fueled with poor t h e r m a l c o n d u c t o r s a s for
i n s t a n c e UO?.
(9)
A M o n t e C a r l o a p p r o a c h of t h i s p r o b l e m w a s m a d e by M a t t h e s , who
i n v e s t i g a t e d the Doppler effect in the e n e r g y r e g i o n w h e r e r e s o n a n c e s
a r e w e l l s e p a r a t e d but not m e a s u r e d , so s t a t i s t i c a l d i s t r i b u t i o n of r e
s o n a n c e spacing and r e a c t i o n widths m u s t be u s e d . The n e c e s s a r y p a r a
m e t e r s for t h e s e d i s t r i b u t i o n s w e r e t a k e n from N i c h o l s o n ' s r e p o r t .
The multiplication f a c t o r is defined by :
K = - c p - Ζ Ni
w h e r e Ν is the n u m b e r of f i s s i o n n e u t r o n s s t a r t i n g in the r e a c t o r .
Ni the n u m b e r of f i s s i o n n e u t r o n s p r o d u c e d in the e n e r g y i n t e r v a l .A,.Ej,
the summation is extended over the whole r e a c t o r s p e c t r u m .
If the t e m p e r a t u r e i s c h a n g e d , a new value of the m u l t i p l i c a t i o n f a c t o r
K' is obtained :
K' =-1—
Σ
N'i
Ν "£
The change of r e a c t i v i t y due to t h i s change of t e m p e r a t u r e or the Doppler
effect, is t h e n given by :
Δ K = K ' K = -L·. ¿ Δ Ν
(9) W. M a t t h e s E u r a t o m I s p r a 380 (1963)
15
If i n t e r e s t s a r e p a i d only to t h o s e n e u t r o n s w h i c h a r e a b s o r b e d below a g i v e n e n e r g y EQ, t h e r e l a t i o n c a n b e w r i t t e n :
N N0 E i ^ E
o
M a t t h e s h a s p e r f o r m e d t h e s e c a l c u l a t i o n s f o r t h e e n e r g y r a n g e 1 K e v 1 0 K e v w h e r e r e s o n a n c e s a r e not e x p e r i m e n t a l l y r e s o l v e d but o v e r l a p p i n g c a n be n e g l e c t e d .
T h e c a l c u l a t i o n of t h e q u a n t i t y --¿j0· i s s t r a i g h t f o r w a r d a n d i s not g i v e n i n
t h e r e p o r t .
A b a s i c difficulty t o e v a l u a t e a d i f f e r e n t i a l effect i n t h e M o n t e C a r l o m e t h o d a r i s e s f r o m t h e i n h e r e n t s t a t i s t i c a l f l u c t u a t i o n s . T o a v o i d t h i s , t h e m e t h o d d e s c r i b e d by G o e r t z l a n d K a l o s * ' w a s u s e d . T h e s a m e s e t of r a n d o m p a t h i s s e l e c t e d t o c a l c u l a t e K a n d Κ' , t h e d i f f e r e n c e s of c o l l i s i o n p o i n t s d e n s i t y in t h e t w o c a s e s , d u e t o t h e d i f f e r e n c e i n c r o s s s e c t i o n s a r e t a k e n into a c c o u n t by p r o p e r l y m o d i f y i n g t h e w e i g h t of t h e n e u t r o n d u r i n g i t s flight.
(12) A p p l i c a t i o n of t h i s p r o g r a m w a s m a d e f o r t h e following s i t u a t i o n v ' :
D i a m e t e r of t h e fuel e l e m e n t s 0 , 3 1 7 c m
D i s t a n c e of t h e fuel e l e m e n t s 0 , 3 6 8 c m ( s q u a r e l a t t i c e ) C o m p o s i t i o n of t h e fuel e l e m e n t s 10% U 2 3 5 , 90% U 2 3 8
T e m p e r a t u r e s Τ c e n t e r (° C) 7 0 0 1 0 0 0 1 3 0 0 1 5 0 0 Τ s u r f a c e (°C) 500
If a m e a n t e m p e r a t u r e c a n be d e f i n e d , s u c h t h a t K ( TC, T8) = K ( Tm) w h e r e
T m c a n be c a l c u l a t e d out of Tc a n d Ts by a p r o c e d u r e w h i c h i s i n d e p e n d e n t
of Ts a n d TC 3 t h e following c o n d i t i o n m u s t be v e r i f i e d :
ì Κ . à Κ = c o n s t
ÙTC à Ts
C a l c u l a t i o n h a s s h o w n t h a t p r e s e n t l y t h i s i s not t h e c a s e . F o r t h e t y p e of r e a c t o r c o n s i d e r e d , one h a s t o e m p l o y t w o D o p p l e r c o e f f i c i e n t s , è K / d T8
and αΚ/òTc w h i c h c a n b e c a l c u l a t e d by t h e M o n t e C a r l o p r o g r a m d e s c r i b e d h e r e .
16
An a n a l y t i c a l a p p r o a c h of the p r o b l e m was a l s o m a d e by B l ä s s e r (13)
for a single r e s o n a n c e in the n a r r o w r e s o n a n c e a p p r o x i m a t i o n a n d a p a r a
bolic d i s t r i b u t i o n of the t e m p e r a t u r e in the fuel p l a t e s . A slab g e o m e t r y
c o n s i s t i n g of fuel e l e m e n t s and m o d e r a t o r was u s e d . As for the p r e v i o u s
Monte C a r l o p r o g r a m , t h i s a n a l y t i c a l p r o g r a m was w r i t t e n for t h e IBM7090. (12)
Application was m a d e to the following configuration. v
F u e l e l e m e n t t h i c k n e s s . 3 cm
M o d e r a t o r t h i c k n e s s 10 cm
R e s o n a n c e p a r a m e t e r s Er = 1000 ev
r = 1950 b
Γχ = 0 , 0 3 ev
C a l c u l a t i o n h a s shown that for the given r e s o n a n c e a m e a n t e m p e r a t u r e 2
defined by Tm = Ts + — ( Ts Tc) c a n be u s e d .
This r e s u l t is in s o m e c o n t r a d i c t i o n with KEANE c a l c u l a t i o n s * , which
w e r e m a d e in the Infinite A b s o r b e r M a s s a p p r o x i m a t i o n ; it s e e m s p o s s i b l e
that t h i s difference c o m e s from the smoothing of a b s o r p t i o n due to the
slowing down of n e u t r o n s within the fuel.
F u r t h e r c a l c u l a t i o n s will be m a d e with t h i s p r o g r a m m e to study the r a n g e
of a p p l i c a b i l i t y of the effective t e m p e r a t u r e c o n c e p t .
b) N e u t r o n S p e c t r u m in F a s t A s s e m b l i e s :
A t e n t a t i v e w a s m a d e to find a s i m p l e r r e p r e s e n t a t i o n of the s p e c t r u m in a
fast a s s e m b l y than the m u l t i g r o u p one. A s u p e r p o s i t i o n of a v i r g i n f i s s i o n
s p e c t r u m and an unknown s p e c t r u m was t r i e d .
ψ = A(c(X + ƒ>)
w h e r e Ç/> , X and f a r e n o r m a l i z e d s p e c t r u m tf> the neutron s p e c t r u m in the a s s e m b l y
X the fission s p e c t r u m f3 the unknown s p e c t r u m .
c\ is a proportionality constant, and A = l / l + o( as r e q u i r e d by the n o r m a l i z a t i o n .
(13) G. B l ä s s e r - Euratom Ispra 205 R (1962)
17
Furthermore, the constant o( can be determined by the condition that j (E) = o for high val ues of E. By instance E > E o
0 (E) = A o( X(E) for all E > E0.
This way, the unknown spectrum can be studied by subtracting the fission spectrum from the reactor spectrum
The reactor spectrum was calculated in the center of the assembly by a multigroup code (ZOOM 7090 version) using 16 groups and YOM cross
(15)
sections , and the properties of the unknown spectrum y° were studied following the PEARSON 'method. In this method, extensively used in
statistical problems, it is shown that distribution curves can be classified in eight types according to the values of their moments up to the fourth. This calculation was repeated for a lot of core assemblies including U metal, oxide and carbide with different enrichment and proportions of sodium and stainless steel. It turned out that all spectra could be fitted by type I curves of the Pearson system, the expression of which is :
f = const (l + -JL-)ml(l-_^-)m2
' v ai v a2
where χ is the lethargy, the origin of lethargies being taken at the point where the distribution reaches its maximum value.
Furthermore, a linearity was found between the parameter o( and the enrich ment down to 15% and the first and second moment of the distribution appears quite independent of core composition.
The results of these calculations seem to indicate the feasibility of two group calculations for fast reactors, using two overlapping groups. The cross
sections being properly averaged on these two groups. One can write :
° χ
χ +Σ ^
x= s
xZ
x^x
+ o p = s
f 18
w h e r e Οχ, O« a r e t r a n s p o r t o p e r a t o r s with c r o s s s e c t i o n s a v e r a g e d on
t h e s e d i s t r i b u t i o n s .
The t r a n s f e r c r o s s s e c t i o n Σ x _> <> from the f i s s i o n g r o u p can be t a k e n a s
the a v e r a g e on the f i s s i o n s p e c t r u m of the s c a t t e r i n g c r o s s s e c t i o n ( e l a s t i c
and i n e l a s t i c ) and ¿ p ^xi s s i m p l y the ^ ^ / p r o d u c t i o n c r o s s s e c t i o n a v e r a g
ed o v e r the Ρ s p e c t r u m . Another way to define the t r a n s f e r c r o s s s e c t i o n
Σ _^ is by the r e q u i r e m e n t that the r a t i o of the two fluxes far from
b o u n d a r i e s be equal t o . o(
The a v e r a g e d c r o s s s e c t i o n s o v e r the f s n e c t r u m c a n be obtained f r o m a
t a b l e a s a function of the c o r e composition. Or t h e ? s p e c t r u m c a n be f i r s t
computed for the given c o r e c o m p o s i t i o n u s i n g a m u l t i g r o u p t h e o r y in a one
d i m e n s i o n a l g e o m e t r y from which a v e r a g e c r o s s s e c t i o n s c a n be obtained.
This r e d u c t i o n of the n u m b e r of g r o u p s p e r m i t s the u s e of m o r e e l a b o r a t e d (17)
m e t h o d s such a s two d i m e n s i o n a l S ^ c a l c u l a t i o n s . M a r c h u k h a s shown
that one group t r a n s p o r t (P3) t h e o r y using a v e r a g e d c r o s s s e c t i o n s ( w h i c h a r e
obtained from 25 g r o u p P j c a l c u l a t i o n s ) give a l m o s t the s a m e r e s u l t s a s
d i r e c t 25 groups Ρ 3 c a l c u l a t i o n s .
The effect of t h e r e f l e c t o r w a s a l s o i n v e s t i g a t e d by using m u l t i g r o u p , one
d i m e n s i o n a l and two r e g i o n s c a l c u l a t i o n s . The s p a t i a l v a r i a t i o n of a v e r a g e
c r o s s s e c t i o n s is i m p o r t a n t only for the t h r e s h o l d f i s s i o n c r o s s s e c t i o n of
U238 n e a r the b o u n d a r y .
F r o m t h i s one can expect t h a t two group c a l c u l a t i o n s a c c o r d i n g to the h e r e
defined m o d e l c a n be m a d e for r e f l e c t e d a s s e m b l i e s .
Boundary conditions in t h i s c a s e a r e a d e l i c a t e point which can be solved
by using p r o p e r l y i n t e g r a l b o u n d a r y c o n d i t i o n s .
A n o t h e r i n t e r e s t of the p r o c e d u r e d e s c r i b e d , i s the i n t e r p r e t a t i o n of c r i t i c a l
e x p e r i m e n t s , by r e d u c i n g the n u m b e r of p a r a m e t e r s to be d e t e r m i n e d to 5
( o( , a.\ , a ¿ , m j , 1x12) i n t e g r a l q u a n t i t i e s can be e a s i l y and a c c u r a t e l y obtained
with a r e d u c e d set of d e t e c t o r s .
19
-PART Β - ORGEL REACTOR PHYSICS
Reactor design calculation from first principles, is still doubtful for a natural uranium fueled r e a c t o r , due to the poor reactivity of this system which must be calculated with a high accuracy, in o r d e r that criticality may be achieved during the whole life of the fuel element.
F o r this r e a s o n , calculations must rely on integral e x p e r i m e n t s which cover a range being wide enough for the d e s i g n e r ' s needs. This does not mean that refined t h e o r i e s a r e u n n e c e s s a r y . On the c o n t r a r y , needs for such t h e o r i e s a r e continuously i n c r e a s i n g , as they furnish a better insight in the physics of the s y s t e m . F u r t h e r m o r e , they may d e c r e a s e the number of high cost e x p e r i ments needed, thus allowing the designer to use safely the codes in a range where no or few experimental r e s u l t s exist, and enable one to obtain quantities of i n t e r e s t for the d e s i g n e r , which a r e not directly given by the e x p e r i m e n t s .
- 20
I. THE ORGEL R E C I P E :
Up to now, ORGEL c a l c u l a t i o n s u s e a r e c i p e c a l l e d CAROLINEI.
The ambition of a r e a c t o r p h y s i c i s t is of c o u r s e t o do good p h y s i c s . H o w e v e r ,
the r e a c t o r p h y s i c i s t h a s a l s o to p r o v i d e the r e a c t o r d e s i g n e r with m e a n s of
c a l c u l a t i o n . As r e a c t o r p h y s i c s is an applied s c i e n c e , m a i n l y justified by the
c o n s t r u c t i o n of r e a c t o r s , t h i s m e a n s that p r i o r i t y h a s to be given to t h i s second
t a s k . Calculations which do not t a k e into account all the p h e n o m e n a , which u s e
s o m e o v e r s i m p l i f i e d m o d e l s , a r e called by r e a c t o r p h y s i c i s t s "a r e c i p e " . T h i s
t e r m is by no m e a n s p e j o r a t i v e . A r e c i p e is good if it i s b a s e d on e x p e r i m e n t s
and it g e n e r a l l y gives b e t t e r r e s u l t s in the r a n g e c o v e r e d by the e x p e r i m e n t s
than m o r e refined t h e o r i e s b a s e d on f i r s t p r i n c i p l e s only. A r e c i p e is not a
s i m p l e c a l c u l a t i o n . Even the simplified m o d e l on which it i s b a s e d l e a d s to
r a t h e r lengthy c a l c u l a t i o n s which r e q u i r e the u s e of d i g i t a l c o m p u t e r s . F u r t h e r
m o r e , up to now, a l l r e a c t o r d e s i g n c a l c u l a t i o n s w e r e b a s e d on the u s e of m o r e
or l e s s s o p h i s t i c a t e d r e c i p e s .
C a r o l i n e I
C a r o l i n e I w a s developed f r o m the beginning for t h i s p u r p o s e . C a r o l i n e I / i s
b a s e d on the F r e n c h c o r r e l a t i o n for heavy w a t e r l a t t i c e s . In t h i s c o r r e l a t i o n ,
t h r e e p a r a m e t e r s : ï) , Α , Β a r e deduced f r o m buckling m e a s u r e m e n t s .
h i s t h e c l a s s i c a l n a t u r a l u r a n i u m t h e r m a l f i s s i o n f a c t o r , A and Β a r e the
c o n s t a n t s of the effective r e s o n a n c e i n t e g r a l . In o r d e r to m a k e allowance for
the c h a r a c t e r i s t i c s for which the studied l a t t i c e s differ f r o m t h o s e on which
the F r e n c h c o r r e l a t i o n w a s b a s e d , p a r t i c u l a r l y the p r e s e n c e of o r g a n i c liquid
i n s t e a d of heavy w a t a r in the fuel e l e m e n t s , a p p r o p r i a t e m o d i f i c a t i o n s to (2)
N a u d e t ' s c a l c u l a t i o n m e t h o d have been i n t r o d u c e d .
The p r e s e n c e of o r g a n i c modifies the slowing down p r o p e r t i e s of the l a t t i c e , thus
affecting the £ , p f a c t o r s and the slowing down a r e a . It modifies a l s o the
t h e r m a l n e u t r o n s p e c t r u m ; h o w e v e r , t h i s effect i s not d i r e c t l y dealt with by (3)
C a r o l i n e , but by a n o t h e r code called T h e r m i d o r * 'which p r o v i d e s a v e r a g e d (1) C a r o l i n e I , G . C a s i n i , W. de H a a n , E . Diana , C. F o g g i , A. K i n d , G . R o s s i
EUR134 e. (1962)
(2) R é s e a u x à eau l o u r d e R . N aud e t / Génie Atomique Β XV
- 21
t h e r m a l c r o s s sections as input data for Caroline.
F u r t h e r m o r e , as uranium carbide s e e m s a v e r y promising m a t e r i a l for ORGEL the resonance integral
r e a c t o r s , the c o r r e l a t e d values forV""were obtained from the c o r r e l a t e d values (4)
for U m e t a l and U oxide following the method given by Vernon v '
A set of buckling e x p e r i m e n t s was p e r f o r m e d with the F r e n c h c r i t i c a l facility Aquilon-II, in o r d e r to check the r e c i p e . During the e x p e r i m e n t s , t h r e e types
(5) of fuel e l e m e n t s w e r e used:
a) c l u s t e r of 1 9 m a g n e s i u m - c l a d u r a n i u m oxide rods - 12mm d i a m e t e r , i m m e r s e d in monoisopropyl diphenyl contained in a hexagonal a l u m i num tube; the distance between the boundaries of two adjacent rods was 1 m m .
b) the s a m e fuel e l e m e n t , but with adjacent rods in contact
c) c l u s t e r of 19 u r a n i u m oxide rods 16,2 m m in d i a m e t e r , with a distan ce of 1 m m between r o d s , a r r a n g e d as in the previous c a s e s .
The C a r o l i n e - I code is p r o g r a m m e d on the IBM-7090 in F o r t r a n language. It gives the following r e s u l t s : £ ,p,f, L ^ , LS ,k , Bm , and the t h e r m a l flux
in each region.
T i m e s of calculation a r e of the o r d e r of 1'.
When c o m p a r e d to the e x p e r i m e n t s , Caroline I gives buckling in a g r e e m e n t within + . 1 m~ .
T h e r m a l neutron s p e c t r u m and r e s o n a n c e escape probability a r e receiving m o r e and m o r e consideration since a few y e a r s . Improved solutions of the energy dependent Boltzman equation w e r e given, it is also a field where usually some quantum mechanics and nuclear physics a r e welcome and this explains t h e i r great a t t r a c t i o n for some runaway from these disciplines to r e a c t o r p h y s i c s .
(4) Effective resonance i n t e g r a l for uranium carbide R . V e r n o n -Nuclear Science and Engineering - 6(2), 163 (1959)
(5) C r i t i c a l e x p e r i m e n t s on natural uranium oxide, organic cooled,heavy water m o d e r a t e d lattices - G. Casini, C. Foggi, F . Toselli - Energia Nucleare , Vol. 9
22
-II. T H E R M A L NEUTRON SPECTRUM :
These effects a r e i m p o r t a n t , both for investigating neutron economy when Plutonium isotopes a r e p r e s e n t in the fuel e l e m e n t s and for dynamic through the t e m p e r a t u r e coefficient. In ORGEL lattices t h e s e effects a r e s t i l l
complicated by the p r e s e n c e of two m o d e r a t o r s at different t e m p e r a t u r e s .
A first approach of these effects was made in connection with r e a c t o r design calculation. As the p r o b l e m s dealt with a r e by no m e a n s c l a s s i c a l , this first approach can hardly be c o n s i d e r e d as a r e c i p e . It is b a s e d on the assumption that r e t h e r m a l i z a t i o n effect due to heating up of neutrons by the hot organic
coolant and hardening effect due to selective absorption of n e u t r o n s by fuel m a t e r i a l can be s e p a r a t e d .
The r e t h e r m a l i z a t i o n effect is dealt with by the two overlapping t h e r m a l (7)
group approximations proposed by Selengut . These two groups a r e Max-wellian distribution at the physical t e m p e r a t u r e s of the coolant and m o d e r a t o r . The fundamental p a r a m e t e r is the t r a n s f e r c r o s s sections from one group to the other. F o r the deuteron bound in the heavy water m o l e c u l e , the model
(8)
proposed by Brown and St. John can be r e g a r d e d as sufficiently a c c u r a t e . (9).
F o r the proton, the phenomenological model of Drozdov et al. is used.
The hardening into the fuel is dealt with by a s s u m i n g a maxwellian d i s t r i b u t i o n entering the fuel element. Inside the fuel, a multigroup diffusion equation is used without energy t r a n s f e r . Boundary conditions between m o d e r a t o r and fuel element a r e continuity of incoming c u r r e n t for each g r o u p , for the outgoing c u r r e n t only an i n t e g r a l continuity condition over the whole energy range is imposed.
(6) A simplified model for the determination of the t h e r m a l neutron s p e c t r u m in a fuel element - A. Kind, G. R o s s i - EUR 260 e.
(7) D. S. Selengut - Nuclear Science and Engineering, 9,94 (196 1) (8) Handbuch der Physik, XXXVIIl/2 page 465 - E. Arnaldi
23
-These perturbations of the main m o d e r a t o r s p e c t r u m due to the r e t h e r m a l i z a t i o n and hardening effects a r e summed up independently to find the fuel s p e c t r u m .
Thi s p r o c e d u r e is supported by the hot loop experiments made in Chalk River ; where a "neutron t e m p e r a t u r e " in the fuel was m e a s u r e d as a function of coolant t e m p e r a t u r e , the test event in these e x p e r i m e n t s is the fissbn reaction rate
.. D 239/ T T235 ratio Pu / U .
The T h e r m i d o r code based on this theory is already used for Orgel design calculations in connection with RLT2 ^ ', a code for burn up evaluation, and Caroline I.
F o r the m o r e refined t h e o r y , new approaches proposed by different authors a r e now investigated. Among t h e s e , the m o s t promising seems to be t h e
(12)
Horowitz and Tretiakoff v ' t h e o r y and the first flight collision probability (13)
used by Honeck in his T h e r m o s code. Which one is the most suitable for our purpose depends on s e v e r a l conditions. A decision will be made l a t e r on this y e a r , and work will be pursued on the selected one.
A g r e a t attention is paid to the energy t r a n s f e r of neutrons colliding with protons bound in organic m o l e c u l e s . A p r e l i m i n a r y analysis was made by
(14)
Ardente on the scattering by the proton bounded in the benzene m o l e c u l e , which i s taken as the fundamental dynamic unit. Different models were t r i e d and new ones proposed. The c r i t e r i u m for selection is the c o m p a r i s o n of the total s c a t t e r i n g c r o s s s e c t i o n as a function of neutron incident energy computed from the m o d e l , with the experimental v a l u e s .
(14 bis) The first and simplest model is the "gas m o d e l " introduced by Sachs and Teller. By m e a n s of one p a r a m e t e r , the effective m a s s , it takes into account the
t r a n s l a t i o n a l and rotational motions.
(10) C. B . B i g h a m , B . G . Chidley.R. B . T u r n e r - AECL 1350 (1961) (11) G. Bias s e r , G. Casini, J. Pillon - E U R / c / l S / 7 8 4 / 6 l f
(12) Effective c r o s s section for t h e r m a l r e a c t o r s - EANDC(E)14 (i960) (13) H . C . Honeck - T h e r m a l i z a t i o n t r a n s p o r t theory code for r e a c t o r lattice
calculations - BNL 5826
(14) V. Ardente - Colloque International de diffusion et de diffraction des neutrons - G r e n o b l e , 3-5 Sept. 1963
24
Comparison with experimental values , shows that it fails in the energy range (14 ter)
. 0 2 - . 3 ev. A m o r e elaborated model due to K r i e g e r and Nelkin , the "free molecular g a s " using two p a r a m e t e r s : the effective m a s s and a vibratio-nal constant taking into account the elastic vibratiovibratio-nal t r a n s i t i o n s from the ground s t a t e , gives - as expected - a better r e s u l t , but is still in d i s a g r e e m e n t with the experimental curve in the energy range above . 1 ev.
Improved models with an i n c r e a s e d number of p a r a m e t e r s would evidently give a better a g r e e m e n t . Ardente has investigated two of these models for the ben-zene m o l e c u l e . As the proposed p a r a m e t e r s a r e not c o r r e l a t e d directly with the total scattering c r o s s section, in which case a g r e e m e n t will be t r i v i a l , but with some known dynamical p r o p e r t i e s , as the vibrational frequency d i s t r i -bution deduced from optical measurements:, good confidence in the validity of the model can be gained and extension to m o r e complicated organic molecules can be m a d e .
The first of these models is the "two v i b r a t i o n a l m o d e l " . It t a k e s into account the t r a n s l a t i o n a l motion with the r e a l m a s s of the benzene molecule (78).
The rotational motion is r e p l a c e d by a t o r s i o n a l i s o t r o p i c oscillation with a single frequency W and m a s s Mr. F o r the vibrational s t a t e s , the frequency distribution of the benzene molecule shows concentration of the vibrational s t a t e s around . 12 ev and . 38 ev. T h u s , the simplified model r e t a i n s only two vibrational s t a t e s at t h e s e e n e r g i e s .
The c o m p a r i s o n with experiment shows a v e r y good a g r e e m e n t .
The second of t h e s e models is the "polycrystalline m o d e l " , which t a k e s into
account only the vibrational s t a t e s , in complete opposition to the "free gas model". A g r e e m e n t with experiment is also v e r y good. A further advantage of this
approach is the formal equivalence with a c r y s t a l l i n e model, for which extensi-vely developed mathe m a t i c a l methods a r e available.
25
These studies lead to the conclusion that the proposed models can be used for other polyphenyls. The refinement of the model would likely r e q u i r e a check with m o r e sensitive quantities than the total c r o s s section, as the differential scattering c r o s s section or the scattering law.
E x p e r i m e n t a l effort is not absent. E x p e r i m e n t s will be c a r r i e d out this year on the total c r o s s section of different organic liquids as a function of the t e m p e r a t u r e , using the t i m e of flight technique. M e a s u r e m e n t s of neutron s p e c t r u m in poisoned organic and at the interface of two diffe-rent t e m p e r a t u r e organics a r e also in p r e p a r a t i o n , in o r d e r to support A r d e n t e ' s calculations.
A m e a s u r e m e n t of the fine s t r u c t u r e of the t h e r m a l flux in simple geometry ORGEL type lattices , also r e l a t e d to neutron s p e c t r u m , has been p e r
-d o )
formed N in the F r e n c h c r i t i c a l facility Aquilon II, by activation of d y s p r o s i u m d e t e c t o r s . Two organic compounds, i. e. monoisopropyldiphenyl (Monsanto) and diphyl (Bayer) have been investigated. Good a c c u r a -cy was achieved in t h e s e e x p e r i m e n t s , + 1% on the disadvantage factors leading to about + . 1 % in the t h e r m a l utilization factor. T h e s e different factors w e r e computed by Sturm * ' using the T h e r m o s code and two dif-ferent models for the s c a t t e r i n g k e r n e l of the proton bound in the organic:
(17)
the Brown and St. John model and the Nelkin o n e \ G e n e r a l l y , the p r e -dicted disadvantage factors a r e slightly higher than the e x p e r i m e n t a l o n e s , leading to o v e r e s t i m a t e f by . 1 to . 3 % . As t h e s e different models a r e not satisfactory for the organic m o l e c u l e , calculation will be r e p e a t e d using the m o r e elaborated Ardente model.
These e x p e r i m e n t s , t o o , will be s y s t e m a t i c a l l y p e r f o r m e d with the ECO facility, with m o r e complicated g e o m e t r i e s including c l u s t e r s and p o s -sibly at different coolant t e m p e r a t u r e s .
(15) A. Boeuf, S . T a s s a n - EUR 206. e (1963)
26
III. RESONANCE ESCAPE PROBABILITY :
This i s a n o t h e r p r o b l e m which r e q u i r e s a good d e a l of e f f o r t s . In the
C a r o l i n e I c o d e , r e s o n a n c e escape p r o b a b i l i t y is given by the well known
r e l a t i o n : N f Vf Ieff
ρ = e Îf2"s Vm
Vs/i
w h e r e Ieff = A + Β γ S/M
A and Β t o g e t h e r with η a r e p a r a m e t e r s c o r r e l a t e d with buckling e x p e r i m e n t s ,
in t h i s c a s e , a c c u r a t e p r e d i c t i o n of buckling can be m a d e but wrong values of
the r e s o n a n c e e s c a p e p r o b a b i l i t y a r e obtained. T h i s s i t u a t i o n is r a t h e r
u n s a t i s f a c t o r y and b e t t e r f o r m u l a t i o n s a r e now u n d e r i n v e s t i g a t i o n .
Calculation of r e s o n a n c e escape p r o b a b i l i t y for c o m p l i c a t e d fuel g e o m e t r i e s
as c l u s t e r s for a p o w e r r e a c t o r , is a v e r y difficult t a s k . In f a c t , only
Monte C a r l o c a l c u l a t i o n s can d e a l with the p r o b l e m without o v e r s i m p l i f y i n g
a s s u m p t i o n s . But t h e s e c a l c u l a t i o n s , due to t h e i r i n h e r e n t l y s t a t i s t i c a l
e r r o r s n e e d long m a c h i n e t i m e , even with the b e s t a v a i l a b l e c o m p u t e r s , to
give the r e q u i r e d a c c u r a c y ; for t h i s r e a s o n it i s excluded that t h e s e c a l c u
l a t i o n s will be u s e d for r e a c t o r d e s i g n w o r k . M o r e l i k e l y , a c o m b i n a t i o n
of the M o n t e C a r l o and the a n a l y t i c a l m e t h o d will be u s e d . All t h e s e new
t h e o r i e s a r e b a s e d on the m u l t i g r o u p t e c h n i q u e . The f i r s t thing t o be done (18) is to c h e c k the validity of t h e a l r e a d y a v a i l a b l e c o d e s . G A M I , a code
developed by GA, allows the c a l c u l a t i o n of a v e r a g e d g r o u p c o n s t a n t s , it
u s e s the r e s o n a n c e p a r a m e t e r s in the r e s o l v e d r e g i o n and s t a t i s t i c a l d i s t r i
bution of t h e s e in the u n r e s o l v e d one. The b e s t thing to be done i s to c o m p a r e , (19)
the p r e d i c t i o n of G A M I with the H e l l s t r a n d e x p e r i m e n t a l r e s u l t s .
F o r t h i s p i r p o s e , Sturm^ h a s done the c o m p a r i s o n for c y l i n d r i c a l fuel
e l e m e n t s of U m e t a l , oxide and c a r b i d e s m a l l enough so that the a s s u m p t i o n
of a flat flux and l / E s p e c t r u m m a y be r e a s o n a b l y m a d e . T h i s c o m p a r i s o n
h a s shown that the r e s o n a n c e i n t e g r a l c a l c u l a t e d by GAMI is s y s t e m a t i c a l l y
lower by abovt 14% than the H e l l s t r a n d ' s v a l u e s . A f t e r a d e t a i l e d a n a l y s i s
it c a n be shown that t h e s e d i s c r e p a n c i e s can be a t t r i b u t e d to the n e g l e c t of
high a n g u l a r m o m e n t u m r e s o n a n c e s and a b s o r p t i o n of n e u t r o n s at e n e r g i e s
above 30 Kev.
(18) G . O . I o n o n and I. S. Dudek GA1850 / (1961)
- 27
The importance of these effects had already been d e m o n s t r a t e d by A.dler*" who fourri a value of 1,4 b for U metal and U oxide in a range v S/M > o, 2. A d l e r ' s calculations a r e coherent with H e l l s t r a n d ' s r e s u l t s except for UO_ where they a r e in e x c e s s of . 5 b over the extrapolated experimental r e s u l t s for V S/ M = 0 , 2 .
Vernon'^2'pointed out that about 0,8 b must be attributed for absorption at energies above 30 Kev, which a r e geometry independent.
(23)
A value of 1,4 b was also found by Nordheim v ; F o r UC, a difference of the same o r d e r was found by comparing GAMI calculations with the value d e -duced by Vernon from the H e l l s t r a n d ' s e x p e r i m e n t s , using the equivalence
(24)
t h e o r e m , which is a l s o coherent with M ü l l e r ' s x ' t h e o r e t i c a l calculations.
Another effect of i m p o r t a n c e for the l a r g e pitches (20-25 cm) encountered in power r e a c t o r s is the spatial dependence of the flux within the cell and deviations from l / E law due to absorption at higher r e s o n a n c e . These effects
(25)
were investigated by De Haanv , who derived an IBM-7090 multigroup code, named SKAL for r e s o n a n c e escape probability in heterogeneous l a t t i c e s .
(26)
NDA h a s a l r e a d y developed a code using the age theory to take t h e s e effects into account, but this method fails in c a s e of a cell containing m o r e than one m o d e r a t o r and for l a r g e fuel sections of the o r d e r of 25 c m¿,
The code developed by De Haan is a m u l t i g r o u p , multiregion code based on the solution of the t r a n s p o r t equation by the collision probability technique. Effective c r o s s sections by energy groups a r e calculated by GAM-I modified to take into account the higher angular momentum r e s o n a n c e s and absorption above 30 Kev, and the neutron distribution within the cell is calculated by SKÄL , which gives also the advantage-disadvantage factors and the resonance escape probability. T h e o r y is going along these lines and an experimental
supporr
p r o g r a m to\it is now in p r o g r e s s . The effective resonance integral of an (27)
ECO c l u s t e r was made by Tas san
(21) Adler et al. - PIGG 16 (1958) - 155
(22) A . R . V e r n o n - Nuclear Science by 6 (1959) 163
(23) Nordheim - L e c t u r e given in the Techn. Hochschule München, July 1963 (24) A . M ü l l e r - Nukleonik 3 (7) (1961) 303
2C
The ECO element consists of 19 Al clad n a t u r a l uranium m e t a l r o d s , 12 m m in d i a m e t e r , a r r a n g e d in an Al tube filled with o r g a n i c . M e a s u r e m e n t s have been performed by the activation technique, using the c e n t r a l thimble of ISPRA-I, a 5 MW D2O m o d e r a t e d and cooled r e a c t o r at the Ispra Center. Depicted uranium (300 ppm) d e t e c t o r s , 0, 1 m m thick and 12mm in d i a m e t e r were used and the induced Pu-239 activity of the detectors was counted
by a coincidence technique, actually one Y and one X r a y from de-excitation of P u - 2 3 9 . In o r d e r to avoid c o r r e c t i o n s for deviation from the / E law in the few cv r e g i o n s , as mentioned by the Swedes for the Rl r e a c t o r , the RI of the c l u s t e r was r e f e r r e d to the RI of a single U r o d , using the H e l l s t r a n d m e a s u -red value, a check was made by m e a s u r i n g the RI of the r e f e r e n c e rod by c o m p a r i s o n with the infinite dilution resonance integral of gold. The a g r e e ment obtained with the Hellstrand original r e s u l t s ( u n c o r r e c t e d for the d e -p a r t u r e of the flux from the l / E law); indicates that the above c o r r e c t i o n t e r m i s , as expected, very s i m i l a r for the Swedish Rl and the I s p r a I r e a c -t o r s .
Results obtained a r e given in the following table :
Organic F o r m u l a Spacing RI RI RI rod mm. m e a s u r e d calculated calcul ated
b a r n s Hellstrand H e l l s t r a n d Carl vïk rubber band P e s h a g e n b a r n s
barns Monoisopropyl
dyphcnil H ^ C i s 1 11,25 11,66 9,07
Dyphil
The last column gives the RI i n t e g r a l calculated by neglecting the slowing down of neutrons due to the organic.
Accuracy which may be achieved in these e x p e r i m e n t s is + 4 % .
This is not actually the case for these e x p e r i m e n t s , which a r e p r e l i m i n a r y and for these r e a s o n s , the experimental e r r o r s a r e not quoted.
The. m e a s u r e m e n t will be extended to a r a t h e r general investigation of the coolant and geometry effects, and successively to the determination of the effective r e s o n a n c e integral of UC cluster e l e m e n t s .
II
0 C12
29
-IV. FAST NEUTRON MULTIPLICATION FACTOR :
Another quantity of importance in r e a c t o r calculation, and which r e q u i r e s some efforts, is the fast neutron multiplication factor. This quantity was
(28)
investigated by L.Amyot , who performed a calculation on an Orgel c l u s t e r fuel e l e m e n t , using a DSN code, and eighteen g r o u p s of energy between 10 Mev and . 1 Mev. The average group c r o s s sections were
obtained with GAM. The DSN obtained s p e c t r u m exhibits a fission s p e c t r u m behaviour above t h r e s h o l d , and is sof/ter than the uncollided fission s p e c t r u m below t h r e s h o l d . As a m a t t e r of i n t e r e s t , the subthreshold s p e c t r u m r e s u l t ing from inelastic scattering in U-238 according to the evaporation model
- E
and the E e law suggested by Spinrad was compared to the uncollided fission s p e c t r u m and shows a striking r e s e m b l a n c e which justifies two
(29)
g r o u p s . a s sumption of Naudet . According to these calculations, the DSN eighteen groups w e r e collapsed into two groups (10 Mev - 1,4 Mev and 1,4 Mev - 0,1 Mev), the c r o s s sections averaged on these two groups and the £ factor calculated. Comparison with Naudet's and Spinrad's calculation show an a g r e e m e n t within 1 %o in i . Such a difference is c o m p a r a b l e with e r r o r s resulting in G""cg u n c e r t a i n t i e s , which is the important p a r a m e t e r .
Due to elastic collision with the coolant nucleus, the £. f a c t o r in Orgel fuel element is l e s s sensitive to \3Q inelastic c r o s s section u n c e r t a i n t i e s .
(31)
The second step in Amyot's calculation v 'was a general formulation
taking into account : the c l u s t e r g e o m e t r y , the non uniform distribution of fission s o u r c e s within the fuel, m o d e r a t o r b a c k s c a t t e r i n g , cell to cell i n t e r a c t i o n , ( Υ ,n) and (n,2 n) contributions.
(32)
The £ factor is defined according to Carlvik and Pershagenv definition,as
the number of neilrons slowing past a given energy anywhere in the lattice per p r i m a r y fission neutron. The boundary energy is taken at . 1 Mev.
(28) L.Amyot - EUR 305. e (1963) (29) R. Naudet - SPM n°401 (1958)
30
The £, factor is given by the following formula
c = f0 + (l-f0) y ^ y . i M e v g o ( r ' E ) C<r'E> d E cLr
where £Q is the fraction of fission neutrons born below . 1 Mev; g0(r,E) is
the probability for a neutron of energy E suffering a collision in r to be transferred below . 1 Mev ; C(r,E) is the collision density of neutrons of energy E at point r by primary fission neutrons.
Using a division in 3 regions :
fuel F diluent D moderator M
and in energy group i, this relation can be rewritten/with obvious notations
£ = f0+ ( l - f0) Σ ( ai o Ci F + bi o Ci o + ni o Ci m )
a,b,n are transfert cross sections independent of geometry.
The C's are obtained from a system of balance equations similar to the
. , , (33) Almgren's .
The balance equations depend on collision probabilities; in this report, the general expression of £ is given for a cluster geometry. The number of
(28)
groups is taken equal to two according to the study 'previously made.
31
CONCLUSION :
Caution m u s t be taken before introducing improved r e a c t o r physics t h e o r i e s in Orgel design calculations. One m u s t be s u r e that the i m p r o v e m e n t s thus introduced will not d e c r e a s e the a c c u r a c y of p r e d i c t i o n s . An extensive e x p e r i m e n t a l p r o g r a m is n e c e s s a r y to check and adjust the theory. A few of t h e s e e x p e r i m e n t s have a l r e a d y been mentioned in this r e p o r t .
(34)
The m a i n p r o g r a m will s t a r t e a r l y in 1964 with ECO , a c r i t i c a l facility now under completion i n I s p r a . Fuel elements made of 19 (0· 12) Al clad, u r a n i u m m e t a l , a s s e m b l e d in cluster, inside a p r e s s u r e tube filled with mono-isopropyldiphenyl and UC ( 0 2 5 , 2 and 30,9) a s s e m b l e d in c l u s t e r s of 7 and 4 rods would be investigated in 64. Buckling, k ^ , cell fine s t r u c t u r e p a r a m e t e r s will be m e a s u r e d using the ECO c r i t i c a l facility, the E X P O , an exponential facility, and the RBI r e a c t o r of Montecuccolino (Bologna, Italy), a PCTR type f a c i l i t y .
Some of t h e s e e x p e r i m e n t s will be repeated with h o t organic using loop type special fuel elements, and neutron s p e c t r u m m e a s u r e d both by activation and t i m e of flight techniques. The effect of b u r n - u p on reactivity will be m e a s u r e d by oscillating in the c e n t r a l thimble of the ECO r e a c t o r , samples of synthetic fuel elements made by m i x t u r e of depleted uranium and plutonium with two different isotopie compositions, 6% and 15% in Pu-240.
Such an e x p e r i m e n t a l and t h e o r e t i c a l p r o g r a m m u s t lead to a good u n d e r -standing of the physics of heavy water r e a c t o r s , especially of the organic cooled t y p e , and enables the r e a c t o r d e s i g n e r s to make a c c u r a t e predictions of the performance which can be expected for a power r e a c t o r .
