AEUDC 621
Different" Activation Techniques for the Study
of Epithermctl Spectra, A p p l i e d to
Heavy W a t e r Lattices of
Varying Fuel-To-Moderator Ratio
E* K. Sokolowski
AKTIEBOLAGET
ATOMENERGI
STOCKHOLM, SWEDEN 1966
AE-230
D I F F E R E N T A C T I V A T I O N T E C H N I Q U E S F O R T H E STUDY O F
E P I T H E R M A L S P E C T R A , A P P L I E D TO HEAVY WATER L A T T I C E S O F VARYING F U E L - T O - M O D E R A T O R R A T I O
by
Evelyn K Sokolowski
ABSTRACT
S p e c t r a l indices at the cell boundary have been studied as funct i o n s of l a t t i c e p i t c h in t h e r e f e r e n c e c o r e of t h e S w e d i s h RO r e a c t o r .
] 15
E p i t h e r m a l i n d i c e s w e r e d e t e r m i n e d by a c t i v a t i o n of In
, employing
t h r e e d i f f e r e n t t e c h n i q u e s : t h e t w o - f o i l , the c a d m i u m r a t i o a n d t h e s a n d w i c h foil m e t h o d s . T h e l a t t e r of t h e s e h a s t h e a d v a n t a g e of b e i n g i n d e p e n d e n t of a s s u m p t i o n s a b o u t foil c r o s s s e c t i o n s o r s p e c t r a l f u n c t i o n s ,
a n d it g i v e s a s p e c t r u m i n d e x t h a t l e n d s i t s e l f r e a d i l y t o c o m p a r i s o n s
with t h e o r e t i c a l m u l t i g r o u p c a l c u l a t i o n s . A l t e r n a t i v e l y the r e s u l t s can
be e x p r e s s e d in t e r m s of t h e W e s t c o t t p a r a m e t e r s r a n d T
when t h i s i s
n
j u s t i f i e d by the s p e c t r a l c o n d i t i o n s . T h e a g r e e m e n t b e t w e e n the t h r e e
m e t h o d s i n v e s t i g a t e d i s g e n e r a l l y good. Good a g r e e m e n t i s a l s o found
w i t h m u l t i g r o u p c o l l i s i o n p r o b a b i l i t y c a l c u l a t i o n s of d i f f e r e n t
of c o m p l e x i t y .
P r i n t e d a n d d i s t r i b u t e d in J u n e 1966
degrees
LIST OF CONTENTS
Page
1.
Introduction
2.
Methods of obtaining an epithermal s p e c t r u m
index
Analysis of sandwich foil data in t e r m s of
Westcott s p e c t r a l indices
2. 1
3
3
9
3.
Detector m a t e r i a l s
10
4.
4.1
4. 2
E x p e r i m e n t a l conditions
Calibration s p e c t r a
Geometry in R0
14
14
16
4. 3
Activity counting
16
5.
Analysis of data
5. 1
T r e a t m e n t of p r i m a r y data
5. 2
Determination of e p i t h e r m a l index
5. 2. 1
Two-foil method
5.2.2
C d - r a t i o method
5. 2. 3
Sandwich foil method
17
17
17
17
18
19
5. 3
19
6.
Determination of neutron t e m p e r a t u r e
Results
6. 1
6. 2
f9
Comparison of the different activation
techniques
19
Comparison with theory
22
Acknowledgements
26
References
27
F i g u r e Captions
29
- 3 -
1.
INTRODUCTION
The i n c r e a s i n g complexity of r e a c t o r lattices and the refinement
of calculating techniques have led to i n c r e a s i n g i n t e r e s t in r e a c t o r neutron s p e c t r a . Some time ago a s y s t e m a t i c investigation of s p e c t r a l indices in the reference core of the Swedish z e r o power r e a c t o r RO was
initiated. The core consists of a s q u a r e lattice of m e t a l l i c n a t u r a l u r a nium rods in heavy w a t e r . The lattice pitch can easily be v a r i e d . The
simplicity of the configuration m a k e s a t h e o r e t i c a l t r e a t m e n t v e r y
s t r a i g h t - f o r w a r d and facilitates an evaluation of different t h e o r e t i c a l
and e x p e r i m e n t a l techniques.
The neutron flux in the RO is considered to be too low to p e r m i t
the extraction of a b e a m for differential m e a s u r e m e n t s of the neutron
s p e c t r u m . T h e r e f o r e , the technique of integral reaction r a t e m e a s u r e ments is being employed. The p r e s e n t paper deals p r i m a r i l y with the
11 5
m e a s u r e m e n t of e p i t h e r m a l s p e c t r u m indices by activation of In
,
using t h r e e different techniques: the two-foil, the cadmium ratio and
the sandwich foil m e t h o d s . The l a t t e r of these leads to a s p e c t r u m index that does not depend on specific assumptions about foil c r o s s s e c tions or s p e c t r a l functions and that lends itself readily to c o m p a r i s o n s
with theoretical multigroup calculations. - Complementary m e a s u r e m e n t s
of a t h e r m a l s p e c t r u m index (neutron t e m p e r a t u r e ) have been made by
<-•
+•
' oft JLu
T
activation
1 7 6
2.
METHODS OF OBTAINING AN EPITHERMAL
SPECTRUM INDEX
The following relation holds between the reaction r a t e p e r a b s o r b e r atom, R, the neutron density per velocity i n t e r v a l , n(v), and
the absorption c r o s s section a(v):
00
R B J
n(v)v a(v) dv
(1)
o
If the c r o s s section v a r i e s as l / v , the i n c r e a s e in collision
probability, due to an i n c r e a s e in neutron velocity, is exactly compensated by the d e c r e a s e in c r o s s section, and hence the reaction rate
will be proportional to the total neutron density, independent of the
- 4-
spectral distribution. If, on the other hand, the cross section deviates
from a l/v function, the neutron density carries different weight in different velocity intervals, and the reaction rate, for a given total neutron
density, will depend on the neutron energy distribution.
If theoretical means exist for calculating the function n(v), and if
a(v) is known in detail, the normalized reaction rate itself suffices as a
spectral index,to be compared to the corresponding calculated quantity.
Often, however, it is desirable to extract information about n(v) from the
measured reaction rates without any previous knowledge of the spectrum.
One then usually makes certain assumptions about the function n(v), leaving open a limited number of parameters to be determined from the measured quantities. This is done in the Westcott formalism [l, 2], in which it
is assumed that the neutron flux follows a Maxwell distribution in the thermal energy region and varies as l/E in the epithermal region. A certain
empirical joining function is assumed in the intermediate region. The
parameters left to be determined are the neutron temperature T and
n
an epithermal index r, which is nearly equal to the epithermal fraction
of the neutron density. Westcott writes eq. (1) as
R n
* tot V 2Z00 C T 2 2 00 [ g< T n) + r S ( T n > :
&
In (2) g and s are tabulated for different nuclei as functions of the neutron temperature. Furthermore, a choice of s-functions is given, depending on the assumed spectral joining function.
Experimentally the reaction rate in a spectrum-sensitive absorber is often normalized with respect to the total neutron density by
dividing it by the reaction rate in a l/v-absorber (for which s»0 and gsl).
This is the basis of the so-called "two-foil method": foils are activated
in pairs of two, one containing the spectrum sensitive and one the l/v absorber. In the measured reaction rate ratio there enter several proportionality constants, characteristic of the nuclides involved, i. e. detector
efficiencies and cross section normalizations. To eliminate these, the
reaction rate ratio is also determined in a known reference spectrum.
In the "cadmium ratio method", two spectrum-sensitive foils
of the same kind are irradiated, one bare and the other under cadmium.
The cadmium cover effectively cuts off the part of the integral (l) between the energy limits 0 and E
, , corresponding to thermal absorp-
- 5 -
tion. If. the Westcott f o r m a l i s m can be a s s u m e d to be valid, the ratio
between the b a r e and the covered foil activities can be written
q 2 2 0 o ( g + rs)
R
Cd*
(3;
ITT
TTT
o
where T is the t e m p e r a t u r e corresponding to the velocity 2200 m / s ,
and I is the total r e s o n a n c e i n t e g r a l J ^ t f ( E ) dE, including the l / v
absorption,
Cd
(Expression (3) has to be used with some modification if the
e p i t h e r m a l flux s p e c t r u m does not go as 1/E around the cadmium cutoff e n e r g y . ) In cadmium ratio m e a s u r e m e n t s , the s a m e detector efficiencies and c r o s s section n o r m a l i z a t i o n s a p p e a r in the n u m e r a t o r
and the denominator. Thus calibration m e a s u r e m e n t s in a r e f e r e n c e
s p e c t r u m a r e superfluous.
If the c r o s s section function of the s p e c t r u m indicator is
c h a r a c t e r i z e d by a single s h a r p , dominating r e s o n a n c e , the r e l a tive strength of the neutron s p e c t r u m around the r e s o n a n c e energy
can be obtained by m e a n s of the "sandwich foil method". Since, in
the two methods d i s c u s s e d previously, a detailed knowledge of the
c r o s s section functions is r e q u i r e d (eqs. (2) and (3)), the d e t e c t o r
m a t e r i a l m u s t be so dilute as not to give r i s e to d i s t o r t i o n s of the
effective c r o s s section functions due to self-shielding. In the s a n d wich foil method, the foil thickness (or a b s o r b e r concentration) is
chosen such that the self-shielding is considerable at the main r e s onance but s m a l l over all other e n e r g i e s . F o i l s a r e i r r a d i a t e d both
singly and in sandwiches of t h r e e . The difference in activity of the
single foil and the central foil in the sandwich (indices " s " and "t"
respectively) is then due to shielding at resonance energy, and, for
a given foil t h i c k n e s s , is proportional to the relative strength of the
neutron density s p e c t r u m around that energy.
In o r d e r
a s s u m e that the
assigning to the
when multiplied
to retain the f o r m a l i s m introduced above, we shall
self-shielding effect in a foil can be d e s c r i b e d by
foil an effective differential c r o s s section such that,
by the u n p e r t u r b e d neutron flux and i n t e g r a t e d a c c o r d -
- 6 -
ing to ( l ) , it gives the c o r r e c t a v e r a g e atomic reaction r a t e . The difference between the m e a s u r e d reaction r a t e in the " s " - f o i l and that in
the "t"-foil is then given by
A R s k J n(v)- v [ (a (v) - a (v)] dv
(4)
The constant k includes detector efficiencies and c r o s s section n o r m a l i z a t i o n s . The a* s a r e the effective differential c r o s s sections for the
" s " - and " t " - foils respectively! and differ .only in the region of selfshielding, i. e. under the resonance peak. If self-shielding occurs over
a s m a l l interval only, then n(v) can be a s s u m e d to have the s a m e energy
dependence over this i n t e r v a l for all e x p e r i m e n t a l situations, i. e. we
can write n(v) * n F(v) where F(v) is a fixed function. We note that this
assumption is m o r e g e n e r a l than that made in the Westcott f o r m a l i s m ,
i. e. that the s p e c t r u m always has the s a m e shape over the total e p i t h e r m a l region. The p r e s e n t assumption is r a t h e r comparable to a m u l t i group t r e a t m e n t , the m e a s u r e m e n t s giving the relative strength of one
group, located under the main resonance peak. F o r indium, the group
extends roughly between the l i m i t s 1. 2 and 1. 7 eV. In the case of u r a nium-fuelled t h e r m a l r e a c t o r lattices the assumption of constant flux
240
shape over this interval is generally good; for l a t t i c e s containing Pu
it m a y be inadequate. We now write eq. (4) as
AR * k n
J F ( V ) - V 6 C T dV
(5)
k as well as the i n t e g r a l a r e c h a r a c t e r i s t i c of the foils and can be e l i m inated by calibration in a known s p e c t r u m . The strength of the neutron
density in the r e s o n a n c e energy group is suitably n o r m a l i z e d to the total
neutron density by m e a s u r i n g AR relative to the reaction r a t e in a l / v
d e t e c t o r . We then get
V R~7 J
Vn~~y
v
V
„ -J2L.
c^ A R
(6)
v Geo
^
'
c
, n
.
tot
c
where the index " c " denotes calibration quantities. In the following we
- 7 -
shall} for brevity, designate the ratios n / n . , by q. The quantity q/q
lends itself directly to a c o m p a r i s o n with t h e o r e t i c a l l y calculated
spectra.
The sandwich foil method, although e s t a b l i s h e d in s e v e r a l
v a r i a n t s for some t i m e (e. g. [ 3 ] ) , has not so far found wide application
in the study of r e a c t o r lattice s p e c t r a , one of the probable r e a s o n s b e ing the u n c e r t a i n t i e s due to the flux anisotropy effects. T h e o r e t i c a l
studies [ 4 ] indicate, h o w e v e r , that t h e s e effects m a y not be p r o h i b i tive. The method has usually been applied to d e t e c t o r m a t e r i a l s with
high energy r e s o n a n c e s . When used in t h e r m a l r e a c t o r l a t t i c e s , the
r e s o n a n c e capture in t h e s e m a t e r i a l s is m a s k e d by the t h e r m a l a b s o r p tion, and, to achieve r e a s o n a b l e a c c u r a c i e s , the foils have to be i r r a diated under cadmium, which leads to p e r t u r b a t i o n s of the flux. Howe v e r , indium, as well as gold, have s t r o n g , low-lying r e s o n a n c e s , which
r e n d e r s the use of cadmium superfluous.
The m e r i t s of the t h r e e methods under consideration a r e s u m m a r i z e d in Table 1:
Table I
Comparison of t h r e e methods for d e t e r m i n i n g
e p i t h e r m a l indices
Two-foil
method
Cd-ratio
method
Sandwich
foil method
Knowledge of
a(v) n e c e s s a r y
yes
yes
no
Calibr. in known
spectrum
necessary
yes
no
yes
Assumptions
about n(v)
necessary
yes
yes
no
Involves
l/v s t a n d a r d
yes
no
yes
Perturbation
of flux
small
large
small
good
poor
Spatial
resolution
good/poor
- 8 -
The decisive disadvantages of the C d - r a t i o method a r e the p e r turbation to the flux and, as a consequence, the poor spatial resolution
(the foils m u s t be placed at g r e a t distances from each other). These
disadvantages a r e n o n - e x i s t e n t for the two-foil method. F o r the sandwich foil method, the flux p e r t u r b a t i o n , which would be solely due to
the strong absorption at the r e s o n a n c e , is s m a l l , provided the r e s o nance is s h a r p , since any neutron, after once c r o s s i n g the detector,
would have to be s c a t t e r e d in the surrounding m e d i u m in o r d e r to be
r e t u r n e d to the d e t e c t o r , but would then be s c a t t e r e d out of the r e s o nance i n t e r v a l in the p r o c e s s . In the sandwich foil method, as it is d e s c r i b e d above, the spatial resolution is poor, since the single foil has
to be placed well away from the t r i p l e foil sandwich to avoid i n t e r f e r ence. If the m e a s u r e m e n t s a r e p e r f o r m e d in a flux gradient, such that
it is difficult to extrapolate the total neutron density over an a p p r e c i able distance, the method can be modified, so that the difference b e tween the outer foils and the inner foil of a sandwich is used [ 3 ] . This
difference will, however, be s m a l l e r than that obtained in the p r e s e n t
way. Both the two-foil and the sandwich foil method r e q u i r e calibration
in a known s p e c t r u m . F o r the l a t t e r method the calibration s p e c t r u m
m u s t have an a p p r e c i a b l e e p i t h e r m a l component, and hence the m o s t
readily attainable calibration facility - a t h e r m a l column - is of no
avail. The main advantage of the sandwich foil method over the other
two methods is that an unambiguous i n t e r p r e t a t i o n of the m e a s u r e m e n t s can be made without any detailed assumptions about the neutron s p e c t r u m or the c r o s s section function. (This, of c o u r s e , holds
only provided no s y s t e m a t i c e r r o r s a r i s e in the f o r m e r method, e. g.
due to anisotropy effects.) Thus, for instance, the degree of selfshielding in the foils does not have to be a c c u r a t e l y known. Also, the
subtraction of the t h e r m a l p a r t of the reactions is done e x p e r i m e n tally and not, as in the two-foil method, on the b a s i s of a s s u m e d t h e r m a l flux distributions, which have to be d e t e r m i n e d s e p a r a t e l y (e. g. in
s e p a r a t e m e a s u r e m e n t s of neutron t e m p e r a t u r e ) . T h e o r e t i c a l m u l t i group s p e c t r a can be checked against the e x p e r i m e n t a l s p e c t r u m index
by simple hand computations.
- 9 -
2. 1
Analysis of sandwich foil data in terms of
the Westcott spectral indices
Although the Westcott model is not required for the interpretation of the spectral index obtained by the sandwich foil method, it is of
interest to derive a relation between that index and the Westcott parameters in order to be able to compare the different methods. In [ 1 ]
Westcott writes the atomic reaction rate in a foil as
^
R « tfo [<1 - f) g a 2 2 0 0 + f ^
f
o
'
A
J £a(v) dv]
o
(7)
Here 0
is the conventional flux n^ • v o o r i r ,;
r
o
tot ZZUU
f is the epithermal fraction of the total neutron density;
T is the absolute neutron temperature
and T » 293
c
n
o
K.
The epithermal region is defined to lie between the energy limits p,kT and infinity. This definition introduces a dependence on neutron temperature into the epithermal neutron fraction. The epithermal
neutron density is taken to include those neutrons» in excess of the
Maxwellian distribution, which lie in the epithermal region. Westcott
chooses as his epithermal spectrum index no^the quantity f, but a
quantity proportional to \i,zi, namely r at f Mr'—. The function A describes the deviation of the epithermal density from a l/v
function
in the region joining the thermal and the epithermal parts of the spectrum. A approaches unity as the energy becomes much greater than
HkT. The last term in eq. (7) is the epithermal absorption, which can
be rewritten as
R
. * 20
J ~
r
epi
o 7 TTT
r
r[
o
f
^-a(v)dv + f I o (vv ) dv]
«J
v v '
•i v
'
no s.s.
s.s.
v(8)
'
The integral has been split up into a part, in which self-shielding is small and one in which it is appreciable. It may be noted that
the cross section appearing in the first integral is the infinite dilution
cross section, whereas the one in the second integral is reduced due
to self-shielding.
Forming the difference between " s " - and "t"-foil activities,
- 10 -
all the t e r m s in the reaction r a t e s cancel except the t e r m s c o r r e s p o n d ing to the l a s t one in eq. (8). Combining (8) and (6) we obtain for the
m e a s u r e d quantity
(9)
where T
is the neutron t e m p e r a t u r e of the calibration s p e c t r u m .
c
nc
3.
DETECTOR MATERIALS
The p r o p e r t i e s of the different detector m a t e r i a l s used a r e summ a r i z e d in Table 2.
115
In
(natural abundance 95. 77 %) has a strong and dominating
49
238
resonance at 1.457 eV. This energy is well below the r e s o n a n c e s of U
,
and hence In will not be subject to. strongly localized shielding n e a r fuel
r o d s . On the other handj the r e s o n a n c e lies well above the t h e r m a l neutron distribution for m o s t l a t t i c e s .
115
In
, upon neutron capturei predominantly gives the i s o m e r
In
(T-L » 54. 12m). To a l e s s e r extent In
in the ground state
s
(T"I_ =5 14 s) is formed, but this is of no importance in activation m e a s E
u r e m e n t s due to its short half-life.
Ref. [ 5 ] r e p o r t s an extensive investigation of the t h e r m a l activa1 15
tion c r o s s sections and resonance i n t e g r a l s of In
, undertaken to r e solve previously existing d i s c r e p a n c i e s . F o r the 54 minute activity the
data given a r e consistent,within e x p e r i m e n t a l e r r o r , with the s - v a l u e s
for In, tabulated by Westcott [ 2 ] .
F o r the two-foil and Cd- ratio methods the detector foils m u s t be
so thin that self-shielding is negligible. In the aluminium-indium foils
used, the resonance self-shielding was e s t i m a t e d to be 1 % [ 6 , 7 ] . In the
evaluation of the r e s u l t s it has been neglected, since it is m a s k e d by the
24
uncertainty in the resonance data. - The 15 hour Na
activity, resulting
from (n, a) - r e a c t i o n s in Al, was found to be negligible. The Al-In foils
were i n t e r c a l i b r a t e d on a rotating disc with an a c c u r a c y of 0. 3 %.
The r e q u i r e m e n t s of the foils for the sandwich method a r e a p p r e c i a b l e self-shielding at the m a i n resonance but negligible s e l f - s h i e l d -
- 11 -
ing at all other e n e r g i e s , notably in the t h e r m a l region. With the aid
of [ 6 ] it was found that the l e a d - i n d i u m foils,' specified in Table 2,
would m e e t these r e q u i r e m e n t s . The alloy has a melting point above
250 C and readily lends itself to the manufacture of the d e s i r e d foils.
Lead alloys have s o m e t i m e s been avoided in d e t e c t o r foils, since it
is feared that they may be deformed, p a r t i c u l a r l y when loaded between
fuel p e l l e t s . However, no deformation effects have been observed with
the p r e s e n t foils.
The foils were grouped into combinations of four - one single
foil and t h r e e in a sandwich - and these combinations w e r e always
retained. The single foils ("s") and the foils to be placed at the center
of the sandwiches ("t") were i n t e r c a l i b r a t e d b a r e on a rotating d i s c .
F u r t h e r , the "t"-foils were inter calibrated inside t h e i r r e s p e c t i v e
sandwiches on a rotating disc in a h a r d r e a c t o r s p e c t r u m . This l a t t e r
calibration gave good a g r e e m e n t with the inter calibration of the b a r e
foils, showing that second o r d e r effects due to differences in the cover
foils w e r e negligible.
Table 2
Detector
nuclide
T
115*
In
a ( 2 2 0 0 ) , l(inf. d i l u t i o n ) ,
b
b'
160±2 [ 5 ]
63
4 . 5 [ 8]
Foil composition
Foil
I(foil)
Foil
t h i c k n e s s (mm) d i a m e t e r (mm)
b
0. 2 w / o In, 9 9 . 8 w / o Al
0. 10
12. 0
2. 5 w / o In, 9 7 , 5 w / o P b
0. 10
12. 0
-
E l e c t r o l y t i c a l l y p u r e Cu
4.98
( E „ , * 0. 625 eV)
0. 10
12. 0
4.3
88 w / o M n , 1 2 w / o Ni
0. 127
12. 0
7. 2 w / o L u , 9 2 . 8 w / o Al
0. 15
12. 0
2500±85
( E - , » 1.3 eV)
cd[5]
Cu
P r o p e r t i e s of d e t e c t o r m a t e r i a l s
2500±85 [ 5 ]
[7]
cd[7]
Mn
T
Liu
K
55
13.2 [ 8 ]
176
2100
14.1
( E „ , » 0 . 5 eV)
C d
[8]
[13] W e s t c o t t ' s s values used [ 2j
T h e i n d i u m c r o s s s e c t i o n s g i v e n a r e t h o s e f o r t h e f o r m a t i o n of t h e 54 m i n u t e a c t i v i t y .
SK
I* J a(E)*§
'Cd
XKK
E „ , t h e s a m e a s for t h e i n f i n i t e d i l u t i o n i n t e g r a l .
Cd
1 1 . 4 C9]
-
The t h e r m a l flux p e r t u r b a t i o n at the "t"-foils due to the cover
foils e n t e r s as a c o r r e c t i o n into the final r e s u l t s and m u s t therefore be
d e t e r m i n e d with some a c c u r a c y . The flux p e r t u r b a t i o n is due to two different effects (see e. g. [ 10]): the effect of flux d e p r e s s i o n around the
a b s o r b e r , which depends on the p r o p e r t i e s of the surrounding m e d i u m ,
and the effect of shielding of the internal l a y e r s of the a b s o r b e r by the
outer l a y e r s , which is a p r o p e r t y of the a b s o r b e r only. Using the formulae of [ 1 0] it can be shown that for the foils under consideration the
t h e r m a l flux d e p r e s s i o n effect is negligible. The t h e r m a l shielding effect, e s t i m a t e d to be around 1 %, was d e t e r m i n e d experimentally by
exposing the " s " - f o i l s and the "t"-foils (covered) on a rotating disc in
a t h e r m a l column. The c o r r e c t i o n factor n e c e s s a r y to c o r r e c t the " t " foil activity for t h e r m a l shielding when r x 0 was found to be 1.013±002.
In section 6. 1 it will be shown that the t h e r m a l to total reaction r a t e
ratio is given roughly (not differentiating between " s " - and "t"-foils)
by l / 0 + "Ä r J "fr- )> where A and B a r e constants, c h a r a c t e r i s t i c of
the p a r t i c u l a r foils used. Hence the factor to be applied to the "t''-foil
activity to c o r r e c t for t h e r m a l shielding is
f^.
I + 0.013
'
„
nc
This c o r r e c t i o n has to be made i t e r a t i v e l y . The ratio B/A was d e t e r mined experimentally to be 1 2. 9 for the foils used.
In activation m e a s u r e m e n t s Cu
and Mn
a r e commonly used
as l/v s t a n d a r d s . However, the l / v dependence of the c r o s s sections of
both these isotopes is only a p p r o x i m a t e : both nuclides have n o n - n e g l i gible resonance i n t e g r a l s , for which c o r r e c t i o n s have to be applied.
In a s p e c t r u m , c h a r a c t e r i z e d by the e p i t h e r m a l index r, the reaction
r a t e in a n e a r l y - l / v a b s o r b e r m u s t be multiplied by the factor f, / to
c o r r e c t for r e s o n a n c e absorption. We have [ 2 ]
- 14 -
If s e l f - s h i e l d i n g o c c u r s in t h e r e s o n a n c e s , t h e i n f i n i t e d i l u t i o n r e s o n a n c e
i n t e g r a l I m u s t be r e d u c e d a c c o r d i n g l y . In F i g u r e 1 t h e r e s o n a n c e c o r r e c t i o n f a c t o r s f. / f o r t h e foils u s e d a r e p l o t t e d v e r s u s r for
different
v a l u e s of t h e n e u t r o n t e m p e r a t u r e . T h e s e l f - s h i e l d i n g d a t a of [ 7 ] a n d
[ 9*J w e r e u s e d .
1 7h
Due to i t s l o w - l y i n g r e s o n a n c e (0. 142 e V ) , L u
i s one of the
m o s t t e m p e r a t u r e s e n s i t i v e s p e c t r u m i n d i c a t o r s . It o c c u r s with a n a t 1 77
u r a l a b u n d a n c e of 2. 60 % a n d , upon n e u t r o n c a p t u r e , g i v e s L u
, which
1 77
d e c a y s t o s t a b l e Hf
with a 6. 8 day h a l f - l i f e . T h e r e s o n a n c e i n t e g r a l
of L u
77A
a b o v e t h e t h e r m a l r e g i o n i s s m a l l . T h e o t h e r c o n s t i t u e n t of
n a t u r a l L u is L u
. N e u t r o n c a p t u r e in L u
leads to Lu
with a h a l f - l i f e of 3 . 7 h o u r s , d e c a y s t o s t a b l e Hf
, which,
1 7 £>
. In t h e p r e s e n t
m e a s u r e m e n t s , t h e L u a c t i v i t y w a s a l l o w e d to d e c a y f o r a b o u t 30 h o u r s
175
b e f o r e b e i n g r e g i s t e r e d . In t h i s way t h e Lu
activity was eliminated.
4.
E X P E R I M E N T A L CONDITIONS
4. 1
Calibration spectra
S i n c e the m a i n p o t e n t i a l s o u r c e of s y s t e m a t i c e r r o r in t h e t w o foil m e t h o d l i e s in t h e a s s u m e d c r o s s s e c t i o n f u n c t i o n s , it is a d v i s a b l e
t o m a k e t h e c a l i b r a t i o n s in a s p e c t r u m t h a t m a y b e e x p e c t e d n o t t o differ too m u c h f r o m the s p e c t r a u n d e r investigation. F o r the sandwich
foil m e t h o d , a c a l i b r a t i o n s p e c t r u m with an a p p r e c i a b l e e p i t h e r m a l c o m ponent is n e c e s s a r y . The c a l i b r a t i o n s for the indium m e a s u r e m e n t s
w e r e t h e r e f o r e c a r r i e d out in t h e m i d d l e of the c e n t r a l c h a n n e l of t h e
R l [ 1 1 ] , w h e r e t h e s p e c t r u m h a s b e e n t h o r o u g h l y i n v e s t i g a t e d by
m e a n s of a f a s t c h o p p e r [ 12, 1 3 ] , foil a c t i v a t i o n s and o s c i l l a t o r m e a s u r e m e n t s [141 T h e c e n t r a l c h a n n e l c o n s i s t s of an a i r f i l l e d a l u m i n i u m
t u b e , e x t e n d i n g v e r t i c a l l y t h r o u g h the r e a c t o r c o r e . T h e c h a n n e l a x i s
i s l o c a t e d in a v a c a n t fuel r o d p o s i t i o n in t h e c l o s e d h e x a g o n a l l a t t i c e .
T h e a i r duct i s d i m e n s i o n e d in s u c h a way t h a t t h e s p e c t r u m i n s i d e t h e
d u c t s h o u l d be e q u i v a l e n t to the c e l l b o u n d a r y s p e c t r u m of a n o r m a l
c e l l . D e t a i l s of the l a t t i c e a r e g i v e n in F i g u r e 2 a .
It h a d b e e n found t h a t t h e t h e r m a l n e u t r o n s p e c t r u m of t h e
c h o p p e r b e a m could be w e l l r e p r e s e n t e d by a M a x w e l l i a n d i s t r i b u t i o n , a n d t h a t the n e u t r o n t e m p e r a t u r e in t h e e x t r a c t e d b e a m w a s
- 15 -
the same» within e x p e r i m e n t a l e r r o r , as in the empty channel. The neutron t e m p e r a t u r e in the channel was r e p o r t e d to be (22. 5±3. 5) C higher
than the m o d e r a t o r t e m p e r a t u r e . The e p i t h e r m a l s p e c t r u m , as m e a s ured with the chopper, could be r e p r e s e n t e d by a 1/E-function between
1 and 7 eV, above which energy some influence from the u r a n i u m r e s onances was o b s e r v e d . Special attention had been paid to the region
joining the Maxwellian and the i / E p a r t s of the s p e c t r u m . The e m p i r i cal s p e c t r u m joining function for the Rl central channel, r e p o r t e d in
[ 1 2 j , and confirmed for other l a t t i c e s , was used by Westcott [ 2 ] in
constructing his tables of c r o s s section p a r a m e t e r s s 4 « It is t h e r e fore a p p r o p r i a t e to use t h e s e p a r a m e t e r s in analyzing the p r e s e n t
measurements.
The e p i t h e r m a l fraction of the s p e c t r u m has been r e p o r t e d to
be 0. 0420±0. 0015 [ 12], c o r r e s p o n d i n g to a Westcott e p i t h e r m a l index
r of 0. 0400±0. 0015. The m o d e r a t o r t e m p e r a t u r e during the calibrations
was 28 c .
Since no calibration is r e q u i r e d for the C d - r a t i o method, a
m e a s u r e m e n t of the C d - r a t i o of indium in the r e f e r e n c e s p e c t r u m
gave an independent determination of the e p i t h e r m a l index. Using the
neutron t e m p e r a t u r e of e a r l i e r m e a s u r e m e n t s , we obtained
r
c
as 0.0425 ± 0. 0020
The e r r o r does not include the uncertainty in the c r o s s s e c t i o n s . Howe v e r , the value a g r e e s , all the s a m e , with the chopper m e a s u r e d value
within the quoted e r r o r s , giving i n c r e a s e d confidence in the r e s o n a n c e
integral data used. Since a c o m p a r i s o n between different methods was
d e s i r e d , the value of r , obtained from the C d - r a t i o m e a s u r e m e n t , was
c
used in the calibration for the two-foil method, and a l s o for the s a n d wich foil method when i n t e r p r e t e d in t e r m s of the Westcott model. F o r
d i r e c t c o m p a r i s o n s of the index q / q with t h e o r e t i c a l s p e c t r a , the l a t t e r
were n o r m a l i z e d against the c h o p p e r m e a s u r e d calibration s p e c t r u m (see
section 6. 2).
The calibration for the sandwich foil method in an a i r duct
r a i s e s the p r o b l e m of anisotropy effects. In o r d e r to investigate t h e s e ,
I n - P b foils w e r e repeatedly exposed both p a r a l l e l and p e r p e n d i c u l a r to
- 16 -
the channel axis. The values of AR/K, / . obtained in this way, a g r e e d
within e x p e r i m e n t a l e r r o r , and no s y s t e m a t i c differences w e r e found.
F o r the determinations of neutron t e m p e r a t u r e by m e a n s of
L u / M n r a t i o s , the Rl graphite t h e r m a l column was used as a c a l i b r a tion s i t e . The neutron t e m p e r a t u r e was taken to equal the ambient
t e m p e r a t u r e ) 25 C, and the e p i t h e r m a l fraction was a s s u m e d to be
zero.
4. 2
Geometry in RO
The RO r e f e r e n c e fuel a s s e m b l i e s consist of single rods of n a t u r a l m e t a l l i c u r a n i u m , clad in aluminium (see F i g u r e l ) . These rods
w e r e a r r a n g e d in s q u a r e l a t t i c e s , with pitches 110, 140, 170 and 190
m m . Details of the lattices a r e shown in F i g u r e 2b.
In t h e o r e t i c a l lattice calculations the cell is usually c y l i n d r i cized and the "cell boundary" is d e s c r i b e d by a single radial coordin a t e . In the e x p e r i m e n t s , the sites chosen for the m e a s u r e m e n t s were
the two e x t r e m e points on the s q u a r e cell boundary, i. e. the m i d points of the lines connecting n e a r e s t and n e x t - n e a r e s t neighbours
in the l a t t i c e . The differences in the s p e c t r a of these two points were
expected to be s m a l l , and a v e r a g e s were thought to c o r r e s p o n d well
to the t h e o r e t i c a l cell boundary.
All i r r a d i a t i o n s w e r e made close to the center of the r e a c t o r
core to m i n i m i z e m a c r o s c o p i c flux g r a d i e n t s .
The foils were suspended in the m o d e r a t o r on v e r t i c a l " l a d d e r s "
of 0. 1 m m aluminium, the distance between " s t e p " being 40 m m . In
the C d - r a t i o and sandwich foil m e a s u r e m e n t s , the b a r e and covered
foils were always s e p a r a t e d by six s t e p s , i. e. 240 m m . Enough Mn
or Cu foils were attached to each ladder to give an a c c u r a t e mapping
of the axial flux distribution and provide n o r m a l i z a t i o n .
4. 3
Activity counting
The foils were counted in a single channel a n a l y z e r , provided
with an automatic sample changer. Integral y counting was chosen, the
d i s c r i m i n a t o r t h r e s h o l d being set at r a t h e r flat p a r t s of the Y - s p e c t r a .
1 3"
The counting s y s t e m was calibrated p r i o r to each run. F u r t h e r , a Cs
- 17
s t a n d a r d was included in each loading of the s a m p l e changer. F o r each
nuclide a plot was m a d e of the change in integral count r a t e against the
corresponding change for Cs 1 37 as the photomultiplyer voltage was a l t e r e d . These plots w e r e then u s e d to c o r r e c t the count r a t e s for the
s m a l l drifts in amplification that did occur. It was e s t i m a t e d that in
this way the e r r o r s due to instability of the counting equipment was of
the o r d e r of a few tenths of a p e r c e n t .
The d e a d - t i m e of the a n a l y z e r was d e t e r m i n e d to be 1. 2 JJ,S for
In. D e a d - t i m e c o r r e c t i o n s w e r e ususally s m a l l .
5.
ANALYSIS OF DATA
.5. 1
T r e a t m e n t of p r i m a r y data
The p r i m a r y count r a t e s w e r e t r e a t e d with an IBM-7044 p r o gramme» which computes saturation a c t i v i t i e s , c o r r e c t i n g for b a c k ground, d e a d - t i m e l o s s e s , differences in foil weighting factors and
decay. C o r r e c t i o n s for counter drift w e r e applied s e p a r a t e l y a c c o r d ing to 4. 3
5. 2
5. 2. I
Determination of e p i t h e r m a l index
Two-foil method
The relat ion
U
v
[ g (T
n')
iA/
VRn / J
'V
+
r
) + r
x
Cg (T
n'
s
s
4 (T ) ]
(7 0)
x
4 (T
n') ]c
C
was u s e d to construct a T - v s - r plot for each m e a s u r e d reaction r a t e
n
r a t i o . The Westcott p a r a m e t e r s g and s , for indium w e r e used. Mn,
c o r r e c t e d for r e s o n a n c e absorption according to section 3, was u s e d
as a l / v s t a n d a r d .
r was d e t e r m i n e d uniquely by the i n t e r s e c t i o n with the T - v s - r
plots obtained from the L u / M n r a t i o s . (See section 5. 3.)
- 18 -
5. 2. 2
Cd-ratio method
Equation (3) has been employed, using the Westcott s, and g
factors in the numerator, and a modified resonance integral, based
on the value of [5] (see section 3) in the denominator. If the Westcott
s . factor is inconsistent with this resonance integral, the inconsistencyappears reduced in the r value.
The thickness of the Cd covers employed was 0. 75 mm. Ref.
[l]
gives the effective Cd cut-off energy as a function of Cd thickness for a
l/v absorber. For In the main resonance lies fairly close to the cut-off
andwill to some extent be shielded by the cadmium. This effectively
raises the cut-off energy with respect to that for a l/v-absorber.
Ref.
[ 15] gives the correction to the resonance integral of indium, due to
the increased effective out-off energy, as a function of Cd thickness.
The correction factor for d„ , » 0. 75 mm is 0. 939. The cut-off energy
in 0. 75 mm Cd for a l/v absorber is 0. 62 eV. Integrating numerically
over the cross section function for In
a 14.
-£EL * 0.276 [5] we obtain
a
epi
1. 3 eV
p
a
>J
0. 62 eV
[16] and using the ratio
r
r
5A4 Jd E
,
-T=r s o2r i0r >2 b
a
E
Adding this to the resonance integral from E
s 1,3 eV ([5], and table
2) and applying the correction factor for increased effective cut-off energy
we obtain
I 5 4 (d
d
* 0. 75 mm) * 2540 ± 100 b.
Strictly, the resonance integral in eq. (3) should be modified to
take into account the deviation of the flux from a l/E distribution above
the Cd cut-off. This deviation is described in [2] by the deviation of the
analytical function A from 1, A having been obtained by fits to experimentally determined reactor spectra. Numerical integration shows that
the resulting change in I is of the order of a few barns and hence much
less than the uncertainty in the resonance integral. It has therefore been
neglected. By means of eq. (3) T - v s - r plots for each measured Cd-ratio
were constructed, and r was determined uniquely by means of the intersecting plots for the Lu-Mn ratios.
- 19 -
5. 2. 3 Sandwich foil method
E q s . (6) and (9) w e r e used to obtain q/q and r. E i t h e r Mn or
Cu, with the resonance c o r r e c t i o n s d i s c u s s e d in section 3, was used
as a l / v standard. In evaluating AR, the "t"-foil activity was c o r r e c t e d
for t h e r m a l shielding by the cover foils according to section 3.
5. 3
Determination of neutron t e m p e r a t u r e
The two-foil method, using an e x p r e s s i o n analogous to eq. (10),
was employed, using the g and s . p a r a m e t e r s of Westcott. Mn r e a c tion r a t e s , with the a p p r o p r i a t e c o r r e c t i o n s , w e r e used for the neutron density n o r m a l i z a t i o n . T - v s - r plots w e r e constructed.
6.
6. ]
RESULTS
Comparison of the different activation techniques
Table 3 gives, for different lattice p i t c h e s , the Westcott r
v a l u e s , a r r i v e d at by the t h r e e different indium activation techniques,
and the neutron t e m p e r a t u r e , obtained from lutetium activations. The
activity r a t i o s , obtained with the two-foil method, a r e a l s o given. The
values a r e a v e r a g e s of those obtained for the two lattice positions investigated, which w e r e within e x p e r i m e n t a l e r r o r of each other, with
a slight trend towards a softer s p e c t r u m (lower r-values) at the c o r n e r
position. The quoted u n c e r t a i n t i e s include s t a t i s t i c a l e r r o r s and e r r o r s
pertaining to the v a r i o u s c o r r e c t i o n s to the m e a s u r e d quantities, both
for the t e s t - and the calibration s p e c t r a . They d o not, however, i n clude the u n c e r t a i n t i e s in the c r o s s section assumptions (notably in
the resonance i n t e g r a l or s . - f a c t o r for indium), nor do they include
the uncertainty in the calibration s p e c t r u m .
- 20 -
Table 3
Lattice
pitch
(mm)
Reaction rate ratios and Westcott s p e c t r a l indices (T
w
^u^/v
m
5 4
Wc
Two-foil
Two-foil
method
Cd-ratio
* 23
T
(°C)
n
Sandw. foil
190
170
140
0.755±.003
0.815±.004
0.941±.007
a0203±.0005 a0195±.0004 a0205±.0006 1.075±.015
a0258±.0007
a0255±.0012 1.085±.015
a0367±.0010 a0380±.0010 a0380±.0010
45±6
1 10
1.192±.006
a0575±.0007 a0626±.0010 0.0603±.0010 7.192±.015
75±6
48±6
57
(interpol.)
It should be noted that the calibration s p e c t r a a r e not the s a m e for
In and the Lu m e a s u r e m e n t s (see section 4. 1).
In ref. [ 17] a simple relation between neutron t e m p e r a t u r e and
m o d e r a t o r t e m p e r a t u r e is derived on the b a s i s of a homogenizing model.
F o r constant m o d e r a t o r t e m p e r a t u r e it reduces to
T
n
T ,
T"m " 1 +
N
a
a
N s" '
where a is a constant and N / N is the ratio between the number of aba' s
s o r b e r and s c a t t e r e r a t o m s . Accordingly, a plot of T / T
v e r s u s fuel° '
n' m
t o - m o d e r a t o r volume ratio should yield a s t r a i g h t line, passing through
1 at V../V ss 0. As seen from F i g u r e 3 this condition is fulfilled within
r
m
e x p e r i m e n t a l e r r o r . F o r the lattice pitch 140 m m no lutetium m e a s u r e m e n t s w e r e m a d e , and a neutron t e m p e r a t u r e was obtained by i n t e r p o l a tion in the figure.
Table 3 indicates that the e r r o r s in r, obtained with the sandwich foil method, a r e equal to or slightly l a r g e r than those obtained
with the other two m e t h o d s . This is due to the r a t h e r artificial i n t r o duction of a t e m p e r a t u r e dependent factor in the s p e c t r a l index of the
sandwich foil method. The relative e r r o r in the m o r e n a t u r a l index
- 21 -
q/q
( s e e below) i s g e n e r a l l y l e s s t h a n t h a t in t h e r - v a l u e s of a l l t h r e e
m e t h o d s . It s h o u l d be n o t e d t h a t w e h a v e n o t t a k e n i n t o a c c o u n t t h e s y s t e m a t i c e r r o r s t h a t m i g h t a r i s e in t h e t w o - f o i l a n d C d - r a t i o m e t h o d s
due t o t h e u n c e r t a i n t y in t h e i n d i u m r e s o n a n c e d a t a . F i n a l l y i t m u s t be
b o r n in m i n d t h a t t h e i n d e x r i s m e a n i n g f u l only u n d e r s p e c i f i c s p e c t r a l
conditions.
T o e l u c i d a t e t h e s a n d w i c h foil m e t h o d , l e t u s c o n s i d e r t h e r a t i o
R / A R , i . e. t h e r a t i o of t h e r e a c t i o n r a t e in a n " s " - f o i l to t h e d i f f e r e n c e
s'
in r e a c t i o n r a t e s of M s " - a n d " t " - f o i l s . F r o m e q s . (7) a n d (8) we h a v e
/4T
s
AR
'
n
o
f*T
I
n
o
°°
A
A
2200 o
7
r«2200
[ff v
H
,< >
o
/TTTT
(ny-r
(H)
-°tw*v
We n o w b e a r in m i n d t h a t g f o r i n d i u m i s c l o s e t o u n i t y a n d a l m o s t t e m p e r a t u r e i n d e p e n d e n t (the v a r i a t i o n in g a s T
t o 100
v a r i e s f r o m 20
C i s only 1. 75 % ) . T h e r e i s a s l i g h t t e m p e r a t u r e d e p e n d e n c e in
t h e b r a c k e t e d f a c t o r in t h e n u m e r a t o r . We e s t i m a t e t h e i m p o r t a n c e of
t h i s on t h e following g r o u n d s :
T h e i n t e g r a l i n t h e n u m e r a t o r i s of t h e o r d e r of m a g n i t u d e of t h e
t o t a l r e s o n a n c e i n t e g r a l f o r In, i . e. ~ 3000 b , a n d i n s e n s i t i v e t o
t h e n e u t r o n t e m p e r a t u r e . a ? p 0 0 * 1 60 b , h e n c e t h e f i r s t t e r m i n s i d e t h e b r a c k e t s i s of t h e o r d e r of 2 0 .
p. i s a p p r o x i m a t e l y 4 , a n d h e n c e t h e s e c o n d t e r m i n s i d e t h e
b r a c k e t s i s of t h e o r d e r of u n i t y a n d s m a l l c o m p a r e d t o t h e
_i
f i r s t t e r m . T h e t e m p e r a t u r e - d e p e n d e n t f a c t o r (T )
s
in t h e
s e c o n d t e r m v a r i e s only a b o u t 10 % o v e r t h e t e m p e r a t u r e r a n g e
of i n t e r e s t ( s e e F i g u r e 3), a n d h e n c e t h e w h o l e b r a c k e t e d f a c t o r
can, to a good a p p r o x i m a t i o n , be t r e a t e d as s p e c t r u m i n d e p e n d ent.
E q . (11) c a n n o w b e w r i t t e n a s
- 22 -
r
n
T
nc
w h e r e t h e c o n s t a n t s A a n d B a r e e s s e n t i a l l y s p e c t r u m i n d e p e n d e n t and
f u n c t i o n s of t h e foil p r o p e r t i e s only. ( R e m e m b e r i n g t h a t the f i r s t t e r m
of t h e n u m e r a t o r in (1 l) c o r r e s p o n d s t o the t h e r m a l a n d t h e s e c o n d t e r m
to the e p i t h e r m a l a b s o r p t i o n , t h e e x p r e s s i o n for the t h e r m a l - t o - t o t a l
r e a c t i o n r a t e r a t i o , u s e d in s e c t i o n 3 , can e a s i l y be d e r i v e d by c o m b i n i n g e q s . (1 1) and (1 2 ) . ) In F i g u r e 4 the r a t i o R /AR h a s b e e n p l o t 1
/' I'
S
i / nc
t e d a s a function of — J-ss— > t h e l a t t e r q u a n t i t y h a v i n g b e e n o b t a i n e d
rW i n
by C d - r a t i o o r t w o - f o i l t e c h n i q u e s . T h e e x p e r i m e n t a l p o i n t s i n c l u d e
the c e l l b o u n d a r y m e a s u r e m e n t s in RO, the m e a s u r e m e n t s in t h e a i r
d u c t of R 1 , a s w e l l a s m e a s u r e m e n t s i n s i d e t h e 1.2 % e n r i c h e d UO
r o d s of a 3 1 - r o d c l u s t e r [ 1 8 ] . All the e x p e r i m e n t a l p o i n t s l i e , w i t h i n
e x p e r i m e n t a l e r r o r , on t h e s a m e f i t t e d s t r a i g h t l i n e , s h o w i n g t h a t the
foils can in a l l c a s e s b e c h a r a c t e r i z e d by t h e s a m e c o n s t a n t s A and
B. This confirms that a n i s o t r o p y effects a r e negligible,
Obviously,
if A a n d B w e r e k n o w n , eq. (12) c o u l d be u s e d d i r e c t l y t o o b t a i n r / T ,
\
.
V n
w i t h o u t n o r m a l i z a t i o n t o a l / v a b s o r b e r or c a l i b r a t i o n in a k n o w n
s p e c t r u m . A and B a r e difficult to c a l c u l a t e with s u f f i c i e n t a c c u r a c y ,
b u t once a c u r v e of t h e t y p e of F i g u r e 4 h a s b e e n o b t a i n e d , it can b e
u s e d in f u t u r e m e a s u r e m e n t s with the s a m e t y p e s of f o i l s .
However,
a m e a s u r e m e n t a g a i n s t a l / v a b s o r b e r g i v e s a r a t i o p r o p o r t i o n a l to
r for a l l v a l u e s of r , w h e r e a s t h e r a t i o R /AR i s l e s s s e n s i t i v e to r
s
for high r - v a l u e s .
6. 2
C o m p a r i s o n with t h e o r y
A s shown in s e c t i o n 2, the q u a n t i t y q / q
is the ratio between
t h e n e u t r o n d e n s i t y a t i n d i u m r e s o n a n c e e n e r g y to t o t a l n e u t r o n d e n s i t y , n o r m a l i z e d to the s a m e r a t i o f o r a c a l i b r a t i o n s p e c t r u m .
This
s i m p l e i n t e r p r e t a t i o n m a k e s p o s s i b l e a d i r e c t c o m p a r i s o n with m u l t i group calculations. However, multigroup p r o g r a m m e s usually
refer
to a c e l l of an infinite l a t t i c e w i t h o u t l e a k a g e . In the e x p e r i m e n t a l c a s e ,
on t h e o t h e r h a n d , we h a v e a l a t t i c e with p o s i t i v e b u c k l i n g , a n d t h e
l e a k a g e p r o b a b i l i t y down to i n d i u m r e s o n a n c e e n e r g y will be l e s s t h a n
- Z3 -
the a v e r a g e for the total neutron population, which is concentrated
around t h e r m a l energy. Hence the e x p e r i m e n t a l quantity q should be
s y s t e m a t i c a l l y higher than the corresponding calculated quantity. To
c o r r e c t for this finite lattice effect, the e x p e r i m e n t a l r a t i o q/q
should
be multiplied by the r a t i o , P , between the non-leakage probabilities for
t h e r m a l neutrons and indium r e s o n a n c e n e u t r o n s , which can be written
approximately as
„ x 1 + 0.9 T B 2 ,
P
7 + M* B*
2
w h e r e B is the buckling, T the age from fission to t h e r m a l energy and
2
M the m i g r a t i o n a r e a for the lattice under investigation. This c o r r e c tion has been applied to the m e a s u r e d quantities, the lattice p a r a m e t e r s
having been obtained with the two-group diffusion theory code BURNUP-5
[19].
As pointed out in s e c t i o n Z, it is i m p o s s i b l e to a s s i g n s h a r p
limits to the energy i n t e r v a l over which r e s o n a n c e self-shielding o c c u r s , and this n e c e s s i t a t e s n o r m a l i z a t i o n with r e s p e c t to a s p e c t r u m ,
the shape of which around the indium resonance energy is s i m i l a r to
that of the s p e c t r u m under investigation. To obtain c o m p a r a b l e quantities from calculations, the t h e o r e t i c a l ratio q should a l s o be n o r m a l ized to a corresponding e x p e r i m e n t a l ratio for the calibration s p e c t r u m .
H e r e it i s , of c o u r s e , e s s e n t i a l that the calibration s p e c t r u m be t r e a t e d
in the s a m e way as the t h e o r e t i c a l s p e c t r u m with r e g a r d to group s t r u c t u r e , end points, etc. , and the differential s p e c t r u m from chopper m e a s u r e m e n t s , available for the calibration site [7 2 ] , was t h e r e f o r e used in
an a p p r o p r i a t e m a n n e r . In the chopper m e a s u r e m e n t s a neutron b e a m
was e x t r a c t e d from the Rl c e n t r a l channel by m e a n s of a graphite s c a t t e r e r , which was shown to cause a m i n i m u m of s p e c t r a l distortion. In
p a r t i c u l a r , the influence of the s c a t t e r e r on the q - r a t i o can be neglected.
The differential s p e c t r u m of the Rl c e n t r a l channel is given in Table 4.
The following relations hold between the channel n u m b e r i, the flight
time t and the energy E:
'- 24 -
t a (i - -i) • 200 u s ,
„ * 2340^ • 10 3 eV
Ar
E
.
r
The chopper m e a s u r e m e n t s w e r e p e r f o r m e d at a m o d e r a t o r t e m p e r a t u r e
of 37 C, while the foil calibrations w e r e m a d e at 28 C. However, this
d i s c r e p a n c y does not influence the q - v a l u e s .
Table 4
E• 0(E) f r o m a graphite s c a t t e r e r in the Rl central channel.
T m 37 °C [13]
m
i
00
05
10
0
1
2
3
4
149.4
146.4
147.7
142.4
145.4
147.6
150.8
160.3
164.7
174.6
183.4
151. 1
202. 2
15
20
222.7
253.9
310.9
393.3
492.0
616.5
746.9
913.8
1045. 1
1213.8
25
30
1323.7
1440.2
1562.3
1620. 6
1753.2
1796.5
1814.7
1815.5
1720.9
1778.2
1627. 0
1606. 1
35
40
1736.2
1737. 0
1677.0
1557.2
1494.5
1 440. 1 1412.2
45
1300.9
1222. 5
1166.5 ' 1149.4
50
1035.9
985.7
949.7
888.4
837.4
55
60
65
70
800. 7
770.5
725.9
688.3
638.9
600. 2
559.9
542.4
497.9
474.8
449.4
436.4
670.6
514.6
410. 1
408.5
394.7
361. 1
343.5
328.2
75
80
310. 1
301.8
300,7
281.6
281.2
270.2
254.5
243. 1
227.8
230.4
85'
90
203. 0
212.5
199.5
181.9
155. 1
160. 2
146.5
144. 0
174.3
139.4
95
135. 0
138. 6
134.2
124.4
125.4
1347.5
1094.3
T h e o r e t i c a l calculations of the cell boundary s p e c t r a of the R0
r e f e r e n c e lattices have been p e r f o r m e d with a 13-group collision p r o b a -
- 25 -
bility p r o g r a m m e C L E F [ 2 0 ] , as well as with a 30-group THERMOS
code [ 2 1 ] . Both of t h e s e p r o g r a m m e s a r e p r i m a r i l y designed for the
calculation of t h e r m a l s p e c t r a , but t h e i r ability to reproduce s p e c t r a
also in the low e p i t h e r m a l range is of i n t e r e s t e. g. in view of plutonium l a t t i c e s .
In C L E F the group s t r u c t u r e around the indium r e s o n a n c e is
r a t h e r c o a r s e : 4 . 0 - 1.5 and 1.5 - 0. 625 eV. It was found, however,
that the distribution of neutrons between these two groups was n e a r l y
the s a m e for all s p e c t r a studied (including the c h o p p e r - m e a s u r e d
calibration s p e c t r u m ) . Hence the neutron density in any one of these
groups, relative to the total neutron density, is proportional to q. An
a v e r a g e of the r a t i o s for the two g r o u p s , which n e v e r differed by m o r e
than 3 %, was taken. The t h e o r e t i c a l and e x p e r i m e n t a l s p e c t r u m i n dices a r e shown in Table 5 and in F i g u r e 5, where they a r e plotted
v e r s u s f u e l - t o - m o d e r a t o r volume r a t i o . This plot is l i n e a r over the
total range of i n t e r e s t .
Table 5 E x p e r i m e n t a l and t h e o r e t i c a l e p i t h e r m a l indices
for R0 cell boundary s p e c t r a
Lattice
pitch
(mm)
p-q/qc
q/qc
Exp.
Exp.
CLEF
THERMOS
190
0.480±.011 0.418±. 010 . 0 . 4 1 1
0.433
170
0. 600±.0 20 0.530±. 015
0. 528
0.550
140
0. 903±.018 0. 822±.016
0. 826
0. 840
110
1.475±. 017 1.397±.016
1.462
1.430
The t h e o r e t i c a l values may contain a s y s t e m a t i c e r r o r of about
2 % due to the n o r m a l i z a t i o n to the c h o p p e r - m e a s u r e d calibration s p e c t r u m . The a g r e e m e n t between the e x p e r i m e n t a l and the C L E F values is
excellent except for the m o s t undermode rated case» where the e p i t h e r m a l
level of C L E F is slightly too high.
- 26 -
The t r e n d indicates that this discrepancy will i n c r e a s e if the
s p e c t r u m is h a r d e n e d f u r t h e r . The THERMOS r e s u l t s a r e s y s t e m a t i cally too high by 3 %. The s a m e tendency has been observed in c o m p a r isons between THERMOS calculations and c h o p p e r - m e a s u r e d s p e c t r a
[22]. However, in those c a s e s the d i s c r e p a n c i e s were p r i m a r i l y a t t r i buted to difficulties in defining a c o r r e c t cell boundary. In the p r e s e n t
case p r e l i m i n a r y r e s u l t s indicate that the discrepancy might be r e solved by redistributing the energy groups s t r u c t u r e in the e p i t h e r m a l
region [ 2 1 ] .
ACKNOWLEDGEMENTS
I should like to e x p r e s s my gratitude to M e s s r s M Lindberg
and A Jonsson for p e r f o r m i n g the t h e o r e t i c a l work with THERMOS
and C L E F , r e s p e c t i v e l y . My thanks a r e also due to Dr. E Johansson
for communicating to m e the r e s u l t s of his unpublished chopper m e a s u r e m e n t s and to M r . Rolf Bladh for much help with the activation work.
Finally I should like to thank Dr. Rolf P e r s s o n for his e v e r - r e a d i n e s s
to discuss new p r o b l e m s .
- 27 -
REFERENCES
1.
WESTCOTT C H, et al.,
Effective C r o s s Sections and Cadmium Ratios for the Neutron
Spectra of T h e r m a l R e a c t o r s .
U.N. Int. conf. peaceful u s e s of atomic energy. Geneva 1958.
Vol. 16. p. 70.
2.
WESTCOTT C H,
Effective C r o s s Section Values for Well-Moderated T h e r m a l
Reactor S p e c t r a .
3 ed. c o r r . 1962. (EANDC (Can)-4.)
3.
EHRET G,
Die B e s t i m m u n g e p i t h e r m i s c h e r Neutronenspektren m i t
Resonanzsonden. (Sandwichmethode.)
A t o m p r a x i s 1_ (1961) 393.
4.
DALTON G R and OSBORN R K,
The Effect of F l u x Anisotropy on N e u t r o n - D e t e c t o r F o i l s .
Nuc. Sci. Eng. ^ 0 (1964) 4 8 1 .
5.
B E C K U R T S K H , et a l . .,
T h e r m a l Activation C r o s s Sections and Resonance I n t e g r a l s
of I n 1 1 5 .
Nuc. Sci. Eng. ]J_(}963) 329.
6.
ROE G M,
The Absorption of Neutrons in Doppler Broadened R e s o n a n c e s .
1954.
(KAPL-1241.)
7.
BAUMANN N P ,
Resonance I n t e g r a l s and Self-shielding for D e t e c t o r F o i l s .
1963.
(DP-817.)
8.
DAHLBERG R, et a l . ,
M e a s u r e m e n t s of Some Resonance Activation I n t e g r a l s .
R e a c t o r Sci. & Techn. 2 1 ( 1 9 6 1 ) 53.
9.
SOKOLOWSKI E K,
Not published.
10.
SOLA A,
Flux P e r t u r b a t i o n by Detector F o i l s .
Nucleonics JjS (i960) 3, 78.
1 1.
HELLSTRAND E , et al. ,
C h a r a c t e r i s t i c s of the Swedish Heavy Water R e a c t o r .
U.N. Int. conf. peaceful u s e s of atomic energy. Geneva 1955.
Vol. 5, p . 242.
- 28 -
1 2.
JOHANSSON E, et a l . ,
A F a s t Chopper and its Use in the M e a s u r e m e n t of Neutron
Spectra.
Ark. F y s i k _1_8 (I960) 513.
13.
JOHANSSON E ,
P r i v a t e Communication.
14.
SOKOLOWSKI E K, et al. ,
C r o s s Section M e a s u r e m e n t for Some E l e m e n t s Suited, as T h e r m a l
Spectrum I n d i c a t o r s .
Nukleonik_6 (1964) 245.
15.
MARTIN D H,
Correction F a c t o r s for C d - c o v e r e d F o i l M e a s u r e m e n t s .
Nucleonics U_ (1955) 3.
1 6.
HUGHES D J and SCHWARTZ R B,
Neutron C r o s s Sections.
2 ed. 1958. (BNL-325.)
17.
COVEYOU R R, et al. ,
Effect of M o d e r a t o r T e m p e r a t u r e upon Neutron Flux in an Infinite
Capturing Medium.
J. Nuc. E n e r g y 2, (1956) 153.
18.
SOKOLOWSKI E K and WIKDAHL C-E,
M e a s u r e m e n t s of Fine S t r u c t u r e and Spectral Indices in L a t t i c e s
of Marviken Type. 1965.
AB A t o m e n e r g i , Sweden,(Internal Report F F R - 4 7 ) .
19.
BJÖREUS K,
P r i v a t e Communication.
20.
JONSSON A and PEKAREK H,
Multigroup Collision Probability Theory in Cluster Geometry.
Comparison with E x p e r i m e n t s . 1965.
AB A t o m e n e r g i , Sweden, (Internal Report RFR-497).
Also p r e s e n t e d at the ANS Symposium on R e a c t o r P h y s i c s in the
Resonance and T h e r m a l Region, San Diego, F e b . 7-9, 1966.
21.
LINDBERG M,
P r i v a t e Communication.
22.
JOHANSSON E, et al. ,
Neutron S p e c t r a in Uranium Tubes.
Nuc. Sci. Eng. 22^(1965) 157.
- 29 -
FIGURE CAPTIONS
F i g u r e 1 Resonance c o r r e c t i o n factors for Mn and Cu foils as functions of e p i t h e r m a l index r for different values of neutron
temperature.
2
Lattice g e o m e t r y in Rl (calibration lattice) and RO
(test l a t t i c e ) .
3
Neutron t e m p e r a t u r e to m o d e r a t o r t e m p e r a t u r e as function of f u e l - t o - m o d e r a t o r volume r a t i o .
4
Sandwich, foil activity r a t i o R /AR as function of ( r j -=
)
nc
5 E x p e r i m e n t a l and t h e o r e t i c a l values of the infinite lattice
e p i t h e r m a l s p e c t r u m index P q/q
vs f u e l - t o - m o d e r a t o r
c
volume r a t i o .
Fig. 1
0. 1 m m MnNi foils
—'
——— 0. 1 m m Cu foils
0.99 W
'•^
* *
\
\ \
\
\ \
\
W
\
\
\
\ \
\ \
\
V
\
\
\\o°'
\
\
^
^60°C
o
I00°C
100 C
\
\
200°C
200°d
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Fig. 2
Aluminium tube
OD 1 54 m m
ID 145 m m
\
D z O (99- 55 %)
A i r duct
Measuring-site
/
/ F u e l rods (nat. m e t
uranium) OD 29.5 mm
Al cladding, thickn e s s 1. 1 m m
Hexagonal lattice pitch 145 m m
D 2 Q (99.71 %)
Central
measuring-site
Off-center
measuring-site
Fuel rod (nat. m e t .
uranium) OD 30.5 m m ,
Al cladding, OD 34. 5
m m , ID 3 1 . 5 m m
Square lattice pitch v a r i a b l e
Fig. 3
t
). 00
0.00
r
0.01
I *
0.02
i
0.03
190 170
b. 04
140
i
0.05
0. 06
0. 07
1 10 m m
V f /V
m
Fig. 4
Fig. 5
Pq/q,
Å
1.5
experiment
CLEF
-
THERMOS
1.4
7.3
1.2-1
1. 1
1. 0
0.9
0.8
0. 7
0. 6
0. 5
-|
0.4
0.3
-|
0. 2
0. 1
o.o
4
o. oo
i i -
0.01
I s
0.02
190
0. 03
170
0. 04
140
i
0. 05
<—
0. 06
0. 07
1 10 m m
V /V.
in
LIST OF PUBLISHED AE-REPORTS
1—157. (See the back cover e a r l i e r reports.)
158. A study o f the a n g u l a r distributions of neutrons from the Be' (p,n) B'
r e a c t i o n at l o w p r o t o n energies. By. B. A n t o l k o v i c ' , B. Holmqvist a n d
T. W i e d l i n g . 1964. 19 p. Sw. cr. 8:—.
159. A simple apparatus for fast ion exchange separations. By K. Samsahl.
1964. 15 p. Sw. cr. 8:—.
160. Measurements of the Fe M (n, p) Mn 5 4 reaction cross section in the neutron
energy r a n g e 2.3—3.8 M e V . By A . Lauber a n d S. M a l m s k o g . 1964. 13 p.
Sw. cr. 8:—.
161. Comparisons of measured a n d c a l c u l a t e d neutron fluxes in laminated
iron a n d heavy w a t e r . By. E. A a l t o . 1964. 15. p. Sw. cr. 8:—.
162. A needle-type p-i-n junction semiconductor detector f o r in-vivo measurement of beta tracer a c t i v i t y . By A . Lauber a n d B. Rosencrantz. 1964. 12 p.
Sw. cr. 8:—.
163. Flame speclro photometric determination of strontium in w a t e r and
b i o l o g i c a l m a t e r i a l . By G . Jönsson. 1964. 12 p. Sw. cr. 8:—.
164. The solution of a v e l o c i t y - d e p e n d e n t s l o w i n g - d o w n p r o b l e m using case's
e i g e n f u n c t i o n e x p a n s i o n . By A . Claesson. 1964. 16 p. Sw. cr. 8:—.
165. Measurements o f the effects of spacers on the burnout conditions for
f l o w of b o i l i n g w a t e r in a v e r t i c a l annulus a n d a v e r t i c a l 7-rod cluster.
By K. M . Becker a n d G . H e m b e r g . 1964. 15 p. Sw. cr. 8:—.
166. The transmission of thermal a n d fast neutrons in a i r f i l l e d annular ducts
t h r o u g h slabs of iron a n d heavy w a t e r . By J . Nilsson a n d R. S a n d l i n .
1964. 33 p. Sw. cr. 8:—.
167. The radio-thermoluminescense o f C a S O i : Sm a n d its use in dosimetry.
By B. B j ä r n g a r d . 1964. 31 p. Sw. cr. 8:—.
168. A fast r a d i o c h e m i c a l method for the determination o f some essential
trace elements in b i o l o g y a n d m e d i c i n e . By K. Samsahl. 1964. 12 p. Sw.
cr. 8 r - .
169. Concentration of 17 elements in subcellular fractions of beef heart tissue
determined by neutron a c t i v a t i o n analysis. By P. O . Wester. 1964. 29 p.
Sw. cr. 8 : — .
170. Formation of nitrogen-13, fluorine-17, a n d fluorine-18 in r e a c t o r - i r r a d i a t e d
H2O a n d D2O a n d a p p l i c a t i o n s to activation analysis a n d fast neutron
flux m o n i t o r i n g . By L. Hammar a n d S. Forsen. 1964. 25 p. Sw. cr. 8:—.
171. Measurements on b a c k g r o u n d a n d f a l l - o u t r a d i o a c t i v i t y in samples from
the Baltic b a y of T v ä r e n , 1957—1963. By P. O . A g n e d a l . 1965. 48 p. Sw.
cr. 8:—.
172. Recoil reactions i n neutron-activation analysis. By D. Brune. 1965. 24 p.
S w . cr. 8r—.
173. A parametric study of a conslant-Mach-number M H D generator with
nuclear i o n i z a t i o n . By J . Braun. 1965. 23 p . Sw. cr. 8:—.
174. Improvements in a p p l i e d g a m m a - r a y spectrometry w i t h germanium semiconductor d e t e c t o r . By D. Brune, J. Dubois a n d S. Hellström. 1965. 17 p.
Sw. cr. 8:—.
175. Analysis of linear M H D p o w e r g e n e r a t o r s . By E. A . W i t a l i s . 1965. 37 p.
Sw. cr. 8:—.
176. Effect of buoyancy on f o r c e d convection heat transfer in v e r t i c a l channels — a literature survey. By A . B h a t t a c h a r y y a . 1965. 27 p . Sw. cr. 8:—.
177. Burnout d a t a f o r f l o w of b o i l i n g w a t e r in vertical r o u n d ducts, annuli
a n d r o d clusters. By K. M . Becker, G . H e r n b o r g , M . Bode a n d O . Eriks o n . 1965. 109 p. S w . cr. 8:—.
178. A n a n a l y t i c a l a n d e x p e r i m e n t a l study of burnout conditions in vertical
r o u n d ducts. By K. M . Becker. 1965. 161 p. Sw. cr. 8:—.
179. H i n d e r e d El transitions in Eu>« a n d TO". By S. G . M a l m s k o g . 1965. 19 p.
Sw. cr. 8:—.
180. Photomulfiplier tubes for l o w level Cerenkov detectors. By O . Strindeh a g . 1965. 25 p. Sw. cr. 8:—.
181. Studies of the fission integrals of U235 a n d Pu239 w i t h cadmium and
b o r o n f i l t e r s . By E. H e l l s t r a n d . 1965. 32 p. Sw. cr. 8:—.
182. The h a n d l i n g of l i q u i d waste at the research station of Studsvik,Sweden.
By S. Lindhe a n d P. Linder. 1965. 18 p. Sw. cr. 8;—.
183. M e c h a n i c a l a n d instrumental experiences from the e r e c t i o n , commissioning a n d o p e r a t i o n of a small pilot plant for development w o r k on
aqueous reprocessing of nuclear fuels. By K. Jönsson. 1965. 21 p. Sw.
cr. 8:—.
184. Energy d e p e n d e n t r e m o v a l cross-sections in fast neutron shielding
t h e o r y . By H. G r ö n r o o s . 1965. 75 p. Sw. cr. 8:—.
185. A new method f o r p r e d i c t i n g the penetration a n d s l o w i n g - d o w n of
neutrons in reactor shields. By L. Hjärne a n d M . Leimdörfer. 1965. 21 p.
Sw. cr. 8:—.
186. A n electron microscope study o f the thermal neutron induced loss in
high t e m p e r a t u r e tensile ductility of N b s t a b i l i z e d austenitic steels.
By R. B. Roy. 1965. 15 p. Sw. cr. 8:—.
187. The non-destructive determination of burn-up means of the Pr-144 2.18
M e V gamma a c t i v i t y . By R. S. Forsyth a n d W . H. Blackadder. 1965.
22 p. Sw. cr. 8:—.
188. Trace elements in human m y o c a r d i a l i n f a r c t i o n d e t e r m i n e d by neutron
a c t i v a t i o n analysis. By P. O . W e s t e r . 1965. 34 p. Sw. cr. 8:—.
189. A n electromagnet for precession of the p o l a r i z a t i o n of fast-neutrons.
By O . A s p e l u n d , J. Björkman a n d G . Trumpy. 1965. 28 p. Sw. cr. 8:—.
190. O n the use of importance sampling in p a r t i c l e transport problems. By
B. Eriksson. 1965. 27 p. Sw. cr. 8:—.
191. Trace elements in the conductive tissue of beef heart determined by
neutron a c t i v a t i o n analysis. By P. O . W e s t e r . 1965. 19 p. Sw. cr. 8:—.
192. Radiolysis of aqueous benzene solutions in the presence of i n o r g a n i c
o x i d e s . By H. Christensen. 12 p. 1965. Sw. cr. 8:—.
193. Radioiy_sis of aqueous benzene solutions at higher temperatures. By
H. Christensen. 1965. 14 p. Sw. cr. 8:—.
194. Theoretical w o r k f o r the fast z e r o - p o w e r reactor FR-0. By H. H ä g g b l o m .
1965. 46 p. Sw. cr. 8:—.
195. Experimental studies on assemblies 1 a n d 2 of the fast reactor FRO.
Part 1 . By T. L. Andersson, E. H e l l s t r a n d , S-O. Londen a n d L. I. Tirén.
1965. 45 p. Sw. cr. 8:—.
196. M e a s u r e d a n d p r e d i c t e d variations in fast neutron spectrum when penet r a t i n g laminated Fe-D20. By E. A a l t o , R. Sandlin a n d R. Fräki. 1965.
20 p. Sw. cr. 8:—.
197. Measured a n d p r e d i c t e d variations in fast neutron spectrum in massive
shields of water and concrete. By E. A a l t o , R. Fräki a n d R. Sandlin. 1965
27 p. Sw. cr. 8:—.
198. M e a s u r e d and p r e d i c t e d neutron fluxes i n , a n d leakage t h r o u g h , a conf i g u r a t i o n of p e r f o r a t e d Fe plates in D2O. By E. A a l t o . 1965. 23 p. Sw.
cr. 8:—.
199. M i x e d convection heat transfer on the outside of a vertical cylinder.
By A . B h a t t a c h a r y y a . 1965. 42 p. Sw. cr. 8:—.
200. A n experimental study of natural circulation in a l o o p w i t h p a r a l l e l
f l o w test sections. By R. P. Mathisen a n d O . Eklind. 1965. 47 p. Sw.
cr. 8:—.
201. Heat transfer a n a l o g i e s . By A . B h a t t a c h a r y y a . 1965. 55 p. Sw. cr. 8:—-.
202. A study of the " 3 8 4 " KeV complex gamma emission from plutonium-239.
By R. S. Forsyth a n d N . Ronqvist. 1965. 14 p. Sw. cr. 8:—.
203. A scintillometer assembly for g e o l o g i c a l survey. By E. Dissing and O .
Landström. 1965. 16 p. Sw. cr. 8:—.
204. N e u t r o n - a c t i v a t i o n analysis of natural water a p p l i e d to h y d r o g e o l o g y .
By O . Landström a n d C. G . W e n n e r . 1965. 28 p. Sw. cr. 8:—.
205. Systematics of absolute gamma ray transition p r o b a b i l i t i e s in deformed
o d d - A n u c l e i . By S. G . M a l m s k o g . 1965. 60 p. Sw. cr. 8:—.
206. Radiation induced removal of stacking faults in quenched aluminium
By U. B e r g e n l i d . 1965. 11 p. Sw. cr. 8;—.
207. Experimental studies on assemblies 1 a n d 2 of the fast reactor FRO.
Part 2. By E. Hellstrand, T. L. Andersson, B. Brunfelter, J. Kockum, S-O.
Londen a n d L. I. T i r é n . 1965. 50 p. Sw. cr. 8:—.
208. Measurement of the neutron s l o w i n g - d o w n time distribution a t 1.46 eV
and its space dependence in w a t e r . By E. M ö l l e r . 1965. 29. p . S w . c r . 8 : — .
209. Incompressible steady f l o w w i t h tensor conductivity leaving a transverse
magnetic f i e l d . By E. A . W i t a l i s . 1965. 17 p. Sw. cr. 8:—.
210. Methods for the d e t e r m i n a t i o n of currents a n d fields in steady twodimensional M H D f l o w w i t h tensor conductivity. By E. A . W i t a l i s . 1965.
13 p. Sw. cr. 8:—.
211. Report on the personnel dosimetry at AB A t o m e n e r g i d u r i n g 1964. By
K. A . Edvardsson. 1966. 15 p. Sw. cr. 8:—.
212. Central reactivity measurements on assemblies 1 a n d 3 o f the fast
reactor FRO. By S - O . Londen. 1966. 58 p. Sw. cr. 8:—.
213. Low temperature i r r a d i a t i o n a p p l i e d to neutron activation analysis of
mercury in human w h o l e b l o o d . By D. Brune. 1966. 7 p. Sw. cr. 8:—.
214. Characteristics of linear M H D generators w i t h one or a f e w l o a d s . By
E. A . W i t a l i s . 1966. 16 p. Sw. cr. 8:—.
215. A n automated anion-exchange method for the selective sorption of five
groups of trace elements in n e u t r o n - i r r a d i a t e d b i o l o g i c a l m a t e r i a l .
By K. Samsahl. 1966. 14 p. Sw. cr. 8:—.
216. Measurement o f the time dependence of neutron s l o w i n g - d o w n a n d
t h e r m a l i z a t i o n in heavy w a t e r . By E. M ö l l e r . 1966. 34 p. Sw. cr. 8:—.
217. Electrodeposition of a c t i n i d e a n d l a n l h a n i d e elements. By N-E. B a r r i n g .
1966. 21 p. Sw. cr. 8:—.
218. Measurement of the electrical conductivity of He 3 plasma induced by
neutron i r r a d i a t i o n . By J. Braun a n d K. N y g a a r d . 1966. 37 p. Sw. cr. 8:—.
219. Phytoplankton from Lake M a g e l u n g e n , Central Sweden 1960—1963. By T.
W i l l é n . 1966. 44 p. Sw. cr. 8:—.
220. M e a s u r e d a n d p r e d i c t e d neutron flux distributions in a m a t e r i a l surr o u n d i n g av c y l i n d r i c a l duct. By J . Nilsson a n d R. S a n d l i n . 1966. 37 p.
Sw. cr. 8:—.
221. Swedish w o r k on brittle-fracture problems in nuclear reactor pressure
vessels. By M . G r o u n e s . 1966. 34 p. Sw. cr. 8:—.
222. Total cross-sections of U, UO2 a n d Th02 for thermal a n d subthermal
neutrons. By S. F. Beshai. 1966. 14 p. Sw. cr. 8:—.
223. N e u t r o n scattering in hydrogenous m o d e r a t o r s , studied b y the time
dependent reaction rate m e t h o d . By L. G . Larsson, E. M ö l l e r a n d S. N .
Purohit. 1966. 26 p. Sw. cr. 8:—.
224. Calcium a n d strontium in Swedish waters a n d fish, a n d accumulation of
strontium-90. Bv P-O. A g n e d a l . 1966. 34 p. Sw. cr. 8:—.
225. The r a d i o a c t i v e waste management at Studsvik. By R. Hedlund a n d A .
Lindskog. 1966. 14 p. Sw. cr. 8:—.
226. Theoretical time dependent thermal neutron spectra a n d r e a c t i o n rates
in H2O a n d D 2 0 . By S. N . Purohit. 1966. 62 p. Sw. cr. 8:—.
227. I n t e g r a l transport t h e o r y in one-dimensional geometries. By I. C a r l v i k .
1966. 65 p. Sw. cr. 8:—.
228. I n t e g r a l parameters of the g e n e r a l i z e d frequency spectra of m o d e r a tors. By S. N . Purohit. 1966. 27 p. Sw. cr. 8:—.
229. Reaction rate distributions a n d ratios in FRO assemblies 1, 2 a n d 3. By
T. L. Andersson. 1966. 50 p. Sw. cr. 8;—.
230. Different activation techniques for the study of epithermal s p e c t r a , a p p l i e d to heavy w a t e r lattices of v a r y i n g fuel-to-moderator r a t i o . By E. K.
S o k o l o w s k i . 1966. 34 p. Sw. cr. 8:—.
Förteckning över p u b l i c e r a d e AES-rapporter
1. Analys medelst gamma-spektrometri. A v D. Brune. 1961. 10 s. Kr 6:—.
2. Bestrålningsförändringar och neutronatmosfär i reaktortrycktankar —
n å g r a synpunkter. A v M . G r o u n e s . 1962. 33 s. Kr 6:—.
3. Studium av sträckgränsen i mjukt stål. A v G . Ö s t b e r g och R. A l l e r m o .
1963. 17 s. Kr 6=—.
4. Teknisk u p p h a n d l i n g inom r e a k t o r o m r å d e t . Av Erik Jonson. 1963. 64 s.
Kr 8;—.
5. Ågesta Kraftvärmeverk. Sammanställning av tekniska d a t a , beskrivningar
m. m. för r e a k t o r d e l e n . A v B. Lilhehöök. 1964. 336 s. Kr 15:—.
6. A t o m d a g e n 1965. Sammanställning av f ö r e d r a g och diskussioner. A v S.
Sandström. 1966. 321 s. Kr 15:—.
A d d i t i o n a l copies a v a i l a b l e at the l i b r a r y of AB A t o m e n e r g i , Studsvik,
N y k ö p i n g , S w e d e n . Transparent microcards of the reports a r e o b t a i n a b l e
through the International Documentation Center, Tumba, Sweden.
EOS-tryckerierna, Stockholm 1966