AE-223
UDC 539.125.5.164
539.125.5.523
CO
CN
CN
LU
Neutron Scattering in Hydrogenous
Moderators, Studied by Time
Dependent Reaction Rate Method
L G. Larsson, E. Möller and $„ N. Purohit
AKTIEBOLAGET ATOMENERGI
STOCKHOLM, SWEDEN 1966
AE-223
NEUTRON SCATTERING IN HYDROGENOUS MODERATORS, STUDIED
BY THE TIME DEPENDENT REACTION RATE METHOD
L G Larsson, E Möller and S N Purohit
SUMMARY
The moderation and absorption of a neutron burst in water, poisoned with the non-l/v absorbers cadmium and gadolinium, has been
followed on the time scale by multigroup calculations, using scattering
kernels for the proton gas and the Nelkin model. The time dependent
reaction rate curves for each absorber display clear differences for
the two models, and the separation between the curves does not depend
much on the absorber concentration. An experimental method for the
measurement of infinite medium reaction rate curves in a limited geometry has been investigated. This method makes the measurement
of the time dependent reaction rate generally useful for thermalization
studies in a small geometry of a liquid hydrogenous moderator, provided that the experiment is coupled to programs for the calculation of
scattering kernels and time dependent neutron spectra. Good agreement has been found between the reaction rate curve, measured with
cadmium in water, and a calculated curve, where the Haywood kernel
has been used.
Printed and distributed in March 1966.
LIST OF CONTENTS
Page
1.
Introduction
3
2.
Principles of the investigation
4
3.
Calculations of the time dependent reaction rate
5
4.
Experiments under infinite medium conditions
8
5.
Measurements
12
6.
Reaction rate curves for no absorption
13
7.
Conclusions
14
Acknowledgements
14
References
15
Appendix
17
Tables
19
Figures
1.
INTRODUCTION
The study of neutron scattering kernels is an important branch
o£ reactor physics, since the quality of detailed calculations of neutron
spectra in reactors depends on how well the neutron scattering is reproduced by the scattering matrix derived from a certain model. Many
kinds of integral experiments can serve as checks on such models. The
stationary spectrum in the pure moderator is sensitive to differences
in the scattering model. By introducing a poison in the medium, it is
possible to perturb the continuous "stirring" of the neutrons between
different energies and thus observe effects of scattering in selected regions. The comparison of measurements and calculations then shows
which model works best. Such investigations on stationary spectra have
been done at several laboratories. The pulsed neutron source measurements of the decay of thermalized neutrons in pure moderator geometries of different sizes are also used to judge thermalization models,
since the diffusion cooling coefficient obtained is an integral parameter of the scattering law. The experimental determination of this parameter is rather difficult and relatively large errors are connected
with the values obtained. Some principal difficulties with the method
seem not to have been resolved. Other ways to use the pulsed neutron
technique have also been proposed. Santandrea et al. [1 ], Verdaguer
[2, 3] and Friedman [4, 5] have introduced non-l/v absorbers in the
moderator and measured the decay constant. The decay constant will
depend not only on the absorption cross section and leakage, but also
on integral parameters of the differential scattering cross section.
Another method to get integral information about the scattering
kernel with pulsed source measurements has been used by Möller and
Sjöstrand [6"] and Möller [7 3 for light and heavy water. The moderation is followed during the slowing down, thermalization and absorption by the recording of the interaction of the time-dependent flux and
spectrum indicators, distributed in the moderator in small amounts.
Time analysis is performed on the gamma radiation emitted in neutron
capture in the indicator. If the indicator used deviates from the l/vlaw, the measurements imply a separation both in time and energy of
the flux, although the energy resolution is rather rough due to the
large energy region which contributes to the reaction rate. The results
- 4-
of the measurements have been analyzed by extracting an integral parameter, the thermalization time constant, to describe the approach
to equilibrium, and also by comparing with calculated reaction rate
curves. Such calculations have been performed by Ghatak and Krieger
[8], Purohit [9] and Poole [10]. This method is also sensitive to differences in the scattering kernels, used in the calculations. Since the
calculations are most easily made for the infinite medium case, the
experiment should be done under conditions, which correspond to this
case. Measurements at all points in the moderator are very time-consuming, however, and a method to perform the spatial integration directly would be preferable.
A modification of the time-dependent reaction rate method, which
should lead to a higher accuracy of the experimental results and allow
a more detailed interpretation of the curves obtained, has been suggested
by Möller [11 ]. Instead of using only small amounts of absorber as an
indicator, one could introduce large amounts in order to perturb the spectrum, and measure the time-dependent reaction rate as in the earlier
measurements. One will then be unable to find the thermalization time
constant in such an easy way as in the earlier measurements, but if r e action rate curves are calculated for the same absorption, the resulting
curves for different scattering models may be compared as earlier.
2.
PRINCIPLES OF THE INVESTIGATION
The aim of this work is to study whether the introduction of higher
concentrations , the influence of which on the neutron flux can not be disregarded, may lead to enhanced scattering model dependence in the timedependent reaction rate in the region for slowing down and thermalization,
and thus to find which absorber and which concentration should be used in
order to get the best separation between different models. In connection
with this study, the conditions under which reaction rate curves, representative for the infinite medium case, could be measured, will also
be explored.
The time dependent flux ^ ( E , t) in an infinite volume of a hydrogenous moderator, where an absorber with a macroscopic absorption
cross section E ^, corresponding to the concentration N, has been dissolved, is given by the Boltzmann equation
- 5 -
+ J§N(E',t)
S s ( E ' ~ » E ) d E ' +Q(E,t)
(1)
S (E) and H (Er—> E) are the total and inelastic scattering cross sections. Q(E,t) is the time dependent source of neutrons at the energy
E. 2
is the capture cross section of hydrogen. The reaction rate
RN(t) of the flux with the absorber added is
When Eq. (1) is solved under the initial conditionsof a short
burst of fast neutrons in the medium, the solution is the time dependent flux, describing how the neutrons are slowed down
and ther-
malized. After the time for complete thermalization, an equilibrium
spectrum will be established, which decays exponentially in amplitude.
During the whole process, the flux will be sensitive to differences in
the inelastic cross section in the integral term of Eq. (1). The reaction rate R^OO will thus depend on the model used for the scattering
process.
3. CALCULATIONS OF THE TIME DEPENDENT REACTION RATE
Eq. (1) was solved for the case of water by the multigroup method by the use of a 22 group computer program, NEFLUDI, previously
developed and reported by Purohit [9 3, in which absorption could be
included. An analytical expression for time-dependent slowing down
in hydrogen was used to feed neutrons down into the groups, which
covered the energy region below 1 eV. Different group structures
were used for the proton gas model and the Nelkin model for technical
reasons in the scattering matrix evaluation, and this difference was
retained. The group structures are shown in Table I. The integration
of the flux over the absorber cross section, according to Eq. (2),
was included in the program.
- 6-
The calculations were made for several concentrations of cadmium
and gadolinium in water. The scattering kernels were the same as used
in the earlier work of Purohit [9]. The absorption cross section of cadmium was taken from Ref.
[12] and reduced by 4 % according to Ref.
[13]. The gadolinium cross section was taken from Ref.
hydrogen absorption cross section from Ref.
[14] an the
[15"]. The cross section
values were taken in the center of each energy group.
The decay constants to be expected during the thermal decay, if
the flux were not perturbed by the non-l/v absorption, would be
X
unpert. = ^ H
where v
E
aH<Vo> + §A W
is 2. 200 • 10 cm s
and g
v
o»
and g.
^
are constants, corre-
sponding to Westcott' s g-factors [16] for hydrogen and the added absorber. These were recalculated for our rather rough group structure
and differed slightly from the values of Westcott. For cadmium,
a(v )
was 2450 barns and the g-factors 1. 32 (gas model) and 1. 33 (Nelkin
model). The hydrogen g-factors were 1. 00 within 0. 4 %.
The calculations were made for four concentrations, corresponding to neutron lifetimes of about 60, 22. 5, 10 and 5 \is.
The results of the cadmium calculations are given in Fig. 1.
Curves for the same concentration have been normalized at 30 jj,s, the
reason being that a comparison with an experiment must be made by
normalizing at some time when the thermalization has been completed.
A common feature of all the curves is, that the Nelkin model curve has
a higher maximum value in the time region 3 - 4 |JLS than the gas model
curve. This is caused by the slower transfer of the neutrons from the
higher thermal energies when chemical binding becomes important,
and the neutrons will therefore spend a longer time in the region of
high cadmium absorption around 0. 18 eV according to the Nelkin model.
(If the curves are normalized to unit source strength, the maximum
value is about the same, but the higher absorption for the bound model
case in the time region 3 - 1 5 \is reduces the flux so much that the r e action rate after thermalization is lower than that for the gas model. )
After complete thermalization, both curves have changed into an exponential decay due to absorption, with about the same decay constant for
the two models.
- 7 -
The gadolinium calculations resulted in curves (not reproduced) ,
which showed smaller differences for the two models, as found for the
case of no absorption [ 9 ] . During the thermalization period, the absorption is smaller in the bound model curves, since the gadolinium
cross section rises in the lower thermal region.
For both absorbers, the thermal decay takes place perfectly
exponentially, without change of the decay constant in the time region
35 - 160 |JLS , implying that the numerical accuracy is good.
The curves may be characterized by the time of the maximum
reaction, the ratio of the maximum value to the extrapolated value
of the asymptotic exponential at the time of the maximum, and the
amplitudes and decay constants found by fitting a sum of two exponentials to the calculated curves. Some of these parameters for cadmium are given in Table 2. It is clear, that the difference between
the results for the two models does not depend very much on the concentration. Our conclusion is thus, that the choice of absorber concentration is of no great importance for the purpose of obtaining distinct differences in the model calculations.
It is, in principle, possible to use our calculated curves to
study the dependence on concentration of the effective cross section
of the absorber for the equilibrium spectrum during the asymptotic
decay. This would be interesting, since an extensive numerical
study by Calame [17] gave results, contradictory to those of
Friedman [5], The decay constant should be given by the following
equation:
\ =\
o
+ Nv (a , , + N _ ! ! "
o x eff
öN
+ . . .)
'
(4)
'
K
where X is the decay constant for the pure moderator and
er .- = ga(v ) is the effective cross section of the non-l/v absorber
in a Maxwell flux. The spectrum is shifted to the energy region of
smaller absorption, and the effective cross section is reduced. The
concentration coefficient do rr/dN is thus a negative number for
any non-l/v absorber, and is a measure of the deviation of the a s ymptotic spectrum from a Maxwellian of moderator temperature.
- 8 -
The coefficient will depend on the scattering kernel, Calame calculated
da ,f/dN for water, using scattering kernels of the Radkowsky model
and the Nelkin model. The coefficients obtained were, in units of
10" 17 barn • cm 3 , -0. 286 and -0. 329 for cadmium, and -3. 70 and -4.18
for gadolinium. These results indicate a much smaller spectrum change
with increasing concentration than reported by Friedman and by Meadows
and Whalen [18]. We analyzed our decay constants, determined by least
squares exponential fittings to the thermal part of the calculated reaction
rate curves, in terms of Eq. (4) and determined the concentration coefficient. For cadmium, we obtained the values -0.16 and -0. 11 , in the
above mentioned unit, for the free proton gas and the Nelkin model,
respectively. We see, thus, a still smaller perturbance of the spectrum
than Calame. For gadolinium the coefficient was in the same unit, -26
and -13 respectively. These results deviate from those of Calame, the
probable explanation being that we do not have, in our calculations, the
precision required for the study of this aspect. The decay constants
obtained for slight poisoning did not agree with those predicted by Eq.
(G), as should be required for full consistency in the numerical material. The difference was 3 per cent. No attempt has been done to improve the calculations for the decay period. The use of the same group
structure for both models and proper weighting of the cross sections
within the groups, as attempted in the NEFLUDI
TDCS [9], anda
greater number of energy groups would lead to a higher accuracy.
4.
EXPERIMENTS UNDER INFINITE MEDIUM CONDITIONS
In order to use the reaction rate method for studies of neutron
scattering kernels, conditions corresponding to infinite medium conditions should be aimed at, since the correct treatment of leakage effects
complicate the problem. For heavy moderators, the effects of diffusion during the moderation make a spatial integration, i. e. measurements at several points, necessary. For liquid hydrogenous moderators, the slowing down and thermalization take place in a limited region around the source. Hence, it seems possible to make a measurement with a stationary detector, placed outside a moderator assembly
where the absorber has been homogenously distributed, and obtain the
spatial integration directly. Several conditions must be fulfilled in
order to get the correct results.
- 9
The neutrons must not leave the moderator during the time period of interest. For a chosen geometry, there is an upper limit for the
source energy in this respect. For a neutron source energy of 1 MeV
in water, 96 % of the neutrons will be found within a radius of 18 cm
when they are thermalized, as can be found from the age measurements
of Foster [191- If the neutron-producing reaction has a low yield under
this limitation, a higher concentration of the non-l/v absorber may be
chosen in order to get a reasonable counting rate in the detector, since
the preceeding discussion of the calculated reaction rate curves showed
no critical dependence on the concentration, and this will probably be
true also for other scattering kernels.
The gamma ray detector must also have the same efficiency
for each radius in the region, where the neutrons, assumed to be
spherically distributed around the source, are moderated. Exact
fulfilment of this requirement can only be reached for the case of
moderation in a liquid scintillator.
In practice, it is possible to obtain a very good approximation
to these conditions for a hydrogenous moderator, with the detector situated outside the medium.
Assume that a pulse of high energy neutrons is introduced in
the center of a moderator volume, where a spectrum indicator is
uniformly dissolved as to give a total macros copic absorption cross
section E TVJ(E). Gamma rays from neutron capture are recorded in
a detector, placed outside the assembly. Time analysis of the pulses
from the detector will give the time dependent reaction rate
R
where
N(t)
=
I J V
cLj(E,r,t)
E
' r > t )E a N ( E )
e
D(r'
9>
9)dVdE'
(5)
is the neutron flux density per unit energy, assumed to be spherically symmetric, at a distance
r from the center of the moderator volume with
the energy E at the time t,
and
e.p.(r, 9, cp)
is the probability that a gamma quantum from neutron capture at a point, defined by spherical coordinates (r, 8, cp), will be recorded in the detector,
placed at a distance D from the moderator center.
- 10 -
Separating the integral according to the different variables gives
dE
J ^N^'1"'^ ^
dr
r
I
e
D ( r ' 9' «P) d ( c o s 9 ) d <P (6)
8, cp
If we now can write
J e(r, 9, cp) d(cos 9)d<? = 4iTe(r) = constant
0, cp
(7)
e.g. if the integral is independent of r, we will have
= e J E TVJ(E) dE J &IVT(E, r , t ) 4rrr dr
(8)
Further assuming that we can neglect the leakage of neutrons
from the moderator volume during the time period of interest, we can
extend the integration in
r to infinity to get
§§NN (E,t)dE
(9)
where
i N (E,t) = J § N (E,r,t) 4xrr2dr
o
(10)
We will thus measure a time dependent reaction rate, proportional to the reaction rate in an infinite medium.
We have made calculations of the effeciency of a spherical gamma
ray detector, assumed to be totally absorbing, for various combinations
of the detector size, its distance from, the center of the moderator a s sembly, and the gamma ray attenuation coefficient. The details are given
in the appendix, and the results for the moderator radius 18 cm are shown
in Fig. Z. For all cases, the efficiency is seen to increase with the radius
in the moderator. The best conditions are found for a 5 inch detector,
placed close by the boundary of the moderator. Here the efficiency is almost constant in the range 0 - 7 cm.
- 11 -
Most neutrons from a source at the center will, after slowing
down, be found within the region 3 - 1 0 cm from the source. When
detecting gamma rays from the reaction of the spatially distributed
neutrons with the homogenously distributed absorber, the variation
of the efficiency with the radius is not so serious as might be expected
from Fig. 2, since the spatial distribution is a weight factor in the
total reaction, the neutrons at small and large radii being of minor
importance. Thus the effective variation of the efficiency will be r e duced to a much smaller value.
If, furthermore, the directional dependence of the detector
efficiency is taken into account, the radial efficiency variation may
be reduced still more.
Pönitz and Wattecamps [20] utilized the detection of gamma
rays from neutron capture in a water solution of manganese for flux
measurements. They made calculations of the efficiency, where a
cosine function represented the directional dependence of the
4 inch x 6 inch detector, and the spatial distribution was described
by a two-group diffusion theory expression. The detection efficiency was found to be constant within 1. 5 per cent for neutron source
energies between thermal and 1 MeV and a moderator radius of
23 cm. For the energy region 10 keV - 1 MeV, which is of most
interest to us, the corresponding figure is 0. 8 per cent.
In a pulsed source experiment, the independence of the detector efficiency on radius should be reflected in the thermal decay constant of the assembly. During a first period,, when no neutrons have
leaked out of the moderator geometry, the measured decay constant shall be given by the absorption term only. When leakage becomes important, the decay rate increases and finally an asymptotic decay constant is reached, which depends, as is well known,
also on the moderator geometry and the scattering kernel. Another
check on the desired properties of the detection is, that the measured curves shall, except for intensities, be insensitive to source
energy variations in a range, given by the moderator size.
- 12 -
5.
MEASUREMENTS
The results from the previous sections have been exploited in performing time dependent reaction rate measurements for light water with
cadmium. The equipment used has been described earlier [6], The geometry of the experiment is shown in Fig. 3. The water container was cylindrical with a diameter of 36 cm and a height of 30 cm. A lithium target
is situated in the center in a plexiglass tube with an inner diameter of
2 cm. The detector used was a plastic scintillator, 5 cm in diameter and
20 cm long. Its directional sensitivity was modified by means of a lead
shield. By rotating a RdTh source along selected radii, and integrating
the counts, the radial efficiency was determined. The detector geometry
was varied until acceptable conditions were obtained. The resulting radial dependence of the efficiency is included in Fig. 4. The gamma ray
discriminator was set at 2. 2 MeV, and since the gamma rays from RdTh
have an energy of 2. 62 MeV, this test simulates rather well the detection of cadmium capture gamma rays.
Measurements were first done for unpoisoned water with the discriminator set at about 1 MeV implying that mainly hydrogen capture
gamma rays are detected. The Van de Graaff accelerator voltage was
set at 3. 0 MV, corresponding to a mean energy from the thick target
7
7
Li (p, n)Be
reaction of about 0. 4 MeV. In the time period 0 - 75 p,s,
a pure exponential decay was obtained with a decay constant, found by
a least squares fit, of (0. 497 ± 0.006)» 10
s" , which is to be compared
with the infinite medium value of (0. 489 ± 0. 004) • 1 0 4 s" 1 . The agreement is seen to be good. Then measurements were done for water,
poisoned with boric acid to have an infinite medium lifetime of 26. 3 JJUS ,
corresponding to a decay constant of (3. 80 ± 0. 01) ' 1 0
s" . The de-
cay constant, measured with the same discriminator setting, was also
a perfect exponential with a decay constant of (3. 81 ± 0. 01) • 10
s" , also
in good agreement with theory, indicating the desired integrating properties of the detector.
Next, a measurement was made with the discriminator set at
2. 2 MeV, in order to get a background curve for the cadmium measurements. The later were done after having replaced about 1. 2 litres of
the boric acid solution by the same volume of cadmium sulphate solu-
- 13 -
tion with the same effective absorption for thermal neutrons. The con18
centration of cadmium in the moderator became 2. 21 • 10
atoms per
cm . (These conditions were chosen from estimates of the concentration needed to yield a suitable counting rate and to reduce the overlapping background from preceeding bursts. ) The result after the subtraction of the very small background, the reaction rate curve for cadmium,
was similar to the curves published earlier [6] and to the curve pair,
second from top in Fig. 1. After the passage of the peak and the transient thermalization period, the curve decayed exponentially, The decay
4 -1
constant was found to be (3. 79 ± 0. 05) • 10 s , in good agreement with
the value from the measurement with hydrogen capture gamma rays.
The neutron producing reaction used is not spherically symmetrical but forward directed. The measurements mentioned were done
with the detector in the forward direction, and at an identical position
at an angle of 90 degrees to the proton beam. The results were the
same, except for the intensities, which were about 10 per cent lower
in the later case.
From the results of these measurements we conclude, that the
conditions of the experiment are, within the limitations mentioned,
such as to yield a correct spatial integration.
6. REACTION RATE CURVES FOR NO ABSORPTION
The measurements were done with so-low cadmium concentration, that the decay of the neutron flux can be regarded as caused
only by l/v-absorption. The cadmium acts as indicator and not as a
poison. Consequently the curves can be corrected to the case of no
absorption by multiplication with an exp(lt) function, the decay constant being obtained from the exponential fit to the thermal period.
The resulting curve can then be compared with reaction rate curves,
calculated for different scattering kernels and no absorption. Fig. 5
shows the result of the experiment (the average from four measurements) with an estimated error of 2 % in the peak value, and of the
calculations performed by Purohit [9] for the proton gas and the
Nelkin model kernels and by Poole [10] for Haywood's kernel [21 ],
(The earlier published experimental curve [6], which was not
spatially integrated, has a lower peak height. ) We find a very good
- 14 -
agreement between the experiment and the curves for the Haywood kernel,
whereas the kernel for the Nelkin model does not lead to a good reproduction of the experiment. The unrealistic nature of the gas model is evident.
The Haywood kernel curve rises later than the other curves. This may be
due to a different method to describe the slowing down, used in the calculations.
7.
CONCLUSIONS
By the use of a suitable geometry of moderator and gamma ray
detector, time dependent reaction rate curves
representative for the
infinite medium case can be measured with a good accuracy in liquid
hydrogenous moderators. The ability to separate between different scattering kernels, used in the calculation of the corresponding reaction rate
curves, does not depend much on the concentration of the absorber in the
medium. Hence the choice of concentration may be based on experimental
requirements. These results make it possible to use the reaction rate
method more generally for thermalization studies in hydrogenous substances and for the study of temperature effects on the scattering kernel.
Obviously chemical problems may arise in organic moderators if cadmium, samarium or gadolinium must be dissolved. It must also be r e marked, that the results tell most about the goodness of the scattering
kernel in the upper thermal energy region. In order to get reliable comparisons with calculated curves, the latter must be calculated with quite
high accuracy. The comparison of our experimental results for water
with different model calculations shows the best agreement with calculations based on the Haywood kernel.
AC KN OW LE DGE ME NTS
The authors wish to express their thanks to Dr. M J Poole, Harwell,
for putting at their disposal results of one of his model calculations. The
interest in this study shown by Professor N G Sjöstrand is also gratefully
acknowledged.
15 -
REFERENCES
1.
SANTANDREA E, TOSELLI F and VIANO G,
Neutron temperature measurements with pulsed neutron sources.
Int. conf. on the peaceful uses of atomic energy, Vol. 16,
Geneva 1958, p. 265.
2.
VERDAGUER F,
Enfriamiento de los neutrones por captura y su aplicacion al
estudio de su termalizacion en agua ligera. 196 3.
(J. E.N. 127-DF/I 39. )
3.
VERDAGUER F e t a l . ,
Development of the pulsed neutron source technique at the Junta
de Energia Nuclar.
Int. conf. on the peaceful uses of atomic energy, Geneva 1964,
P/678.
4.
FRIEDMAN E,
A new method for measuring thermalization parameters.
Nucl. Sci. Eng. 1_4 (1 962) 420.
5.
FRIEDMAN E,
Studies of neutron thermalization in H?O by the pulsed source
and "non l / v " absorbers method.
Nucl. Sci. Eng. L9 (1964) 203.
6.
MÖLLER E and SJÖSTRAND N G,
Measurement of the slowing-down and thermalization time of
neutrons in water.
Arkiv Fysik 2_7 (1964) 501.
7.
MÖLLER E,
Measurement of the time dependence of neutron slowing down
and thermalization in heavy water.
Arkiv Fysik (in press).
8.
' GHATAK A K and KRIEGER T J,
Neutron slowing-down times and chemical binding in water.
Nucl. Sci. Eng. 21_ (1965) 304.
9.
PUROHIT S N,
Time-dependent neutron thermalization in liquid moderators
H2O and D2O.
IAEA Symp. on Pulsed Neutron Research, Karlsruhe 1965,
Vol. 1, Vienna 1965, p. 273.
10.
POOLE M J,
Private communication.
11.
MÖLLER E,
Neutron moderation studied by the time-dependent reaction
rate method.
IAEA Symp. on Pulsed Neutron Research, Karlsruhe 1965,
Vol. 1, Vienna 1965, p. 155.
- 16 -
12.
Neutron cross sections. 1958.
(BNL 325, 2 ed.)
13.
SOKOLOWSKI E, PEKAREK H and JONSSON E,
Cross section measurements for some elements suited as thermal
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Nukleonik 6_ (1964) 245.
14.
BJERRUM-MJÖLLER H, SHORE F J and SAILOR V L,
Low-energy neutron resonances in erbium and gadolinium.
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Neutron cross sections. 1964.
(BNL 325 2 ed. , Suppl. No. 2.)
16.
WESTCOTT C H,
Effective cross section values for well-mode rated thermal reactor
spectra. 3 corr. ed. I960.
(AECL-1101.)
17.
CALAME G P,
Non-l/v thermalization parameters for the mass-1 and Nelkin scattering kernels.
Nucl. Sci. Eng. 2O_(1964) 352.
18.
MEADOWS J W and WHALEN J F,
Thermal neutron absorption cross sections by the pulsed source
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FOSTER G D J r . ,
Age of Na-Be neutrons in water and kerosene.
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PÖNITZ W and WATTECAMPS E,
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HAYWOOD B C,
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- 17 -
APPENDIX
Calculation of detector efficiency
The detector, which is assumed to be spherical,e. g. have no
directional dependence, is placed outside the moderator, according
to Fig. 6, where the dimensions of the calculations are defined. The
symmetry of the arrangement immediately indicates that there will
be no cp-dependence in the efficiency.
Supposing that the gamma rays from the capture are emitted
isotropically and that the attenuation of the gamma rays is given by
the total attenuation coefficient [i, then the efficiency e(r, 0) is
given by
where 0(r, 9) is the solid angle subtended by the detector from the
point (r, 9).
Simple trigonometric arguments give
t2 = r 2 + D2 - 2rD cos 9
2
2
2
p = r + R - 2rR cosa
sina
p
'
R
sin(9+Y)
1
sin 9
sin(9+Y)
D
and these give in turn
p (K
2
2 D 2 sin 9
+ r2
1
1
AX
- 2— cos 9
r
2D £>in 9
e 2 rfR2
1 +
,
•; f 2
D2
2 " 2— cos 9
r
r
2D
1
sin29
- 2— cos 9
r
]*]
V s
- 1
The double sign in front of the last parenthesis gives two solutions; the
- sign is valid in the region 0 ^ 9 ^ arccos = , and the.+ sign in the
region arccos — <, 9 £• rr.
The solid angle element dQ is in spherical coordinates (9' , cp1 )
given by dQ = d(cos 9' )dcp' .
Thus
.
•å
=
2TT
é;l
arccos 9
dc
P'
I
.
d(cos9')=§(l -arccos 9^=^(1
arccos 92
i
^2
+
2 2
'
2 2
If -L / B » 1, series expansion in B /t
gives the well known
formula for the solid angle
2
2
valid if B « I .
With t as above we get
Q
4n
if,
I
e
f
V
B
2
r + D - 2rDcos
+ r 2 + D 2 - 2rDcos9
Thus the efficiency
e(r, 9) is given by
r 2 + D2 -
B 2 + r 2 + D 2 - 2rDcos9
• exp
D2
R +L r 2
^
2D 2^ sin 2 9
1 + D . 2^2^-cos
2
r
± 2
T(R2 -
2
ZP/sin'e
)(r2
nR
1 +D
i%2 - 2Rr- cos 9
r
->~ 2 •
n2
n
2
1 + —2 - 2 -r cos 9
r
Integration of e(r, 9) over d(cos 9) gives 4TT e(r).
f
- 19 -
Table 1.
Group structure for the calculations
Group boundaries, eV
Nelkin model
Gas model
1.020
1. 000
0.850
0.950
0.625
0. 850
0.400
0. 750
0. 320
0. 700
0.280
0.650
0.250
0.600
0.1 80
0.550
0.140
0. 500
0.100
0.450
0. 080
0.400
0.067
0. 350
0.058
0. 300
0. 050
0.250
0.042
0.200
0.035
0.150
0.030
0. 100
0.025
0. 050
0.020
0.040
0.015
0.030
0. 010
0. 020
0.005
6.5 • 10"
0.010
5
0
- 20 -
Table 2.
(*)
Parameters of reaction rate curves for cadmium in water
Concentration of
Cd atoms,
10 19 • cm"3
Nelkin
Gas
oM
3.6
1.64
5.57
13.4
27.4
3. 4
3. 5
3. 3
2.9
2.8
2.6
2.5
2. 2
2. 1
t max
peat
extr
X 104
Nelkin G a s
Nelkin
1.96
2.03
2.20
2. 34
2. 71
g"
1
Gas
1. 84
1. 83
1. 70
1. 69
1. 97
4 42
4. 37
2. 08
9. 63
9. 35
2 . 41
18 1
17. 3
For the concentration 0, the parameters refer to calculations
with time-dependent weighting of the cross sections.
1
1
i
i
I
i
i rr i
T T I
I t
f
l
l
l
l
l
'
*
'
1
i
i
i
19.-3
10
c
•- 9
8
§7
a:
cm
15
10
I,
Fig. i.
20
25
30
}JS
Reaction r*t« curves for different concentrations of cadmium
in water, calculated with scattering kernels for the Nelkin model (N)
and the gas model (G).
-
. 2B=5"
D = 18cm
. 28*3"
D = 16cm
. 2B«2"
D=18cm
i
j
i
j
i
i
-1
10
D=18cm
UJ
10
2B=2"
D=27cm
. 2 B = 5"
D= 78cm
-2
_2B=2
2B =3" Ds78cm
2B=2"
10
D = 78c m
-3
I
I
r, cm
Fig.
2.
t
I
10
I
I
I
I
Radial dependence of the detector efficiency,
1
/
\
I
15
with the gamma ray attenuation coefficient 0. 04 cm"
0. 03 c m -
I
calculated
(—• •••»• •.«) and
s
Plexiglass
\Li
Light
pipe
Lead
shield
^Water
Plastic
scintillator
0
target
10 20 30
cm
Fig. 3.
Moderator assembly with gamma ray detector.
tube
0.5
0
10
15
r, cm
Fig. 4.
Measured detector efficiency for different radii before
and after adjustment with the lead shield.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
1 00 _
0
Fig. 5.
Reaction rate curves for cadmium in water. The solid line
curves represent calculations for the gas (G), Nelkin (N) and
Haywood (H) kernels respectively. The broken line curve is the r e sult of the experiment.
Detector
Fig. 6.
Moderator
Spherical moderator and detector with notations, used
in the efficiency calculations.
LIST OF PUBLISHED AE-REPORTS
1—145. (See the back cover earlier reports.)
146. Concentration of 24 trace elements in human heart tissue determined
by neutron activation analysis. By P. O . Wester. 1964. 33 p. Sw. cr. 8:—.
147. Report on the personnel Dosiraetry at AB Atomenergi during 1963. By
K.-A. Edvardsson and S. Hagsgård. 1964. 16 p. Sw. cr. 8:—.
148. A calculation of the angular moments of the kernel for a monatomic gas
scatterer. By R. Håkansson. 1964. 16 p. Sw. cr. 8:—.
149. An anion-exchange method for the separation of P-32 activity in neuIron-irradited biological material. By K. Samsahl. 1964. 10 p. Sw. cr
150. Inelastic neutron scattering cross sections of Cu 6 ' and Cuss in the energy
region 0.7 to 1.4 MeV. By B. Holmqvist and T. W i e d l i n g . 1964. 30 p.
Sw. cr. 8:—.
151. Determination of magnesium in needle biopsy samples of muscle tissue
by means of neutron activation analysis. By D. Brune and H. E. Sjöberg.
1964. 8 p. Sw. cr. 8.—.
152. Absolute El transition probabilities in the dofermed nuclei Y b 1 " and
H P " . By Sven G . Malmskog. 1964. 21 p. Sw. cr. 8:—.
153. Measurements of burnout conditions for flow of boiling water in vertical
3-rod and 7-rod clusters. By K. M . Becker, G . Hernborg and J. E. Flinla.
1964. 54 p. Sw. c i . 8:—
154. Integral parameters of the thermal neutron scattering law. By S. N .
Purohit. 1964. 48 p. Sw. cr. 8:—.
155. Tests of neutron spectrum calculations with the help of foil measurments
in a D2O and in an HiO-moderated reactor and in reactor shields of
concrete and iron. By R. Nilsson and E. A a l t o . 1964. 23 p. Sw. cr. 8:—.
156. Hydrodynamic instability and dynamic burnout in natural circulation
two-phase f l o w . An experimental and theoretical study. By K. M . Becker. S. Jahnberg, 1. Haga, P. T. Hansson and R. P. Mathisen. 1964. 41 p.
Sw. cr. 8:—.
157. Measurements of neutron and gamma attenuation in massive laminated
shields of concrete and a study of the accuracy of some methods of
calculation. By E. Aalto and R. Nilsson. 1964. 110 p. Sw. cr. 10:—.
158. A study of the angular distributions of neutrons from the Be' (p,n) B'
reaction at low proton energies. By. B. Antolkovic', B. Holmqvist and
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» (n, p) Mns< reaction cross section in the neutron
energy range 2.3—3.8 MeV. By A . Lauber and S. Malmskog. 1964. 13 p.
Sw. cr. 8;—.
161. Comparisons of measured and calculated neutron fluxes in laminated
iron and heavy water. By. E. A a l t o . 1964. 15 p. Sw. cr. 8:—.
162. A needle-type p-i-n junction semiconductor detector for in-vivo measurement of beta tracer activity. By A . Lauber and B. Rosencranlz. 1964. 12 p.
Sw. cr. 8:—.
163. Flame spectro photometric determination of strontium in water and
biological material. By G. Jönsson. 1964. 12 p. Sw. cr. 8:—.
164. The solution of a velocity-dependent slowing-down problem using case's
eigenfunction expansion. By A. Claesson. 1964. 16 p. Sw. cr. 8:—.
165. Measurements of the effects of spacers on the burnout conditions for
flow of boiling water in a vertical annulus and a vertical 7-rod cluster.
By K. M. Becker and G . Hemberg. 1964. 15 p. Sw. cr. 8:—.
166. The transmission of thermal and fast neutrons in air filled annular ducts
through slabs of iron and heavy water. By J. Nilsson and R. Sandlin.
1964. 33 p. Sw. cr. 8:—.
167. The radio-thermoluminescense of CaSO*: Sm and its use in dosimetry.
By B. Bjärngard. 1964. 31 p. Sw. cr. 8:—.
168. A fast radiochemical method for the determination of some essential
trace elements in biology and medicine. By K. Samsahl. 1964. 12 p. Sw.
cr. 81—.
169. Concentration of T7 elements in subcellular fractions of beef heart tissue
determined by neutron activation analysis. By P. O. Wester. 1964. 29 p.
Sw. cr. 8:—.
170. Formation of nitrogen-13, fluorine-17, and fluorine-18 in reactor-irradiated
HjO and D2O and applications to activation analysis and fast neutron
flux monitoring. By L. Hammar and S. Forsen. 1964. 25 p. Sw. cr. 8:—.
171. Measurements on background and fall-out radioactivity in samples from
the Baltic bay of Tvären, 1957—1963. By P. O. Agnedat. 1965. 48 p. Sw.
cr. 8:—
172. Recoil reactions in neutron-activation analysis. By D. Brune. 1965. 24 p.
Sw. cr. 8:—.
173. A parametric study of a constant-Mach-number MHD generator with
nuclear ionization. By J. Braun. 1965. 23 p. Sw. cr. 8:—.
174. Improvements in applied gamma-ray spectrometry with germanium semiconductor dector. By D. Brune, J. Dubois and S. Hellström. 1965. 17 p.
Sw. cr. 8:—.
175. Analysis of linear MHD power generators. By E. A . Witalis. 1965. 37 p.
Sw. cr. 8:—.
176. Effect of buoyancy on forced convection heat transfer in vertical channels — a literature survey. By A. Bhattacharyya. 1965. 27 p. Sw. cr. 8:—.
177. Burnout data for flow of boiling water in vertical round ducts, annuli
and rod clusters. By K. M. Becker, G . Hernborg, M . Bode and O. Erikson. 1965. 109 p. Sw. cr. 8:—.
178. An analytical and experimental study of burnout conditions in vertical
round ducts. By K. M . Becker. 1965. 161 p. Sw. cr. 8:—.
179. Hindered El transitions in Eu'« and Tb 1 ' 1 . By S. G. Malmskog. 1965. 19 p.
Sw. cr. 8.—.
v
180. Photomultiplier tubes for low level Cerenkov detectors. By O. Slrindehag. 1965. 25 p. Sw. cr. 8:—.
181. Studies of the fission integrals of U235 and Pu239 with cadmium and
boron filters. By E. Hellstrand. 1965. 32 p. Sw. cr. 8:—.
182. The handling of liquid waste at the research station of Studsvik,Sweden.
By S. Undhe and P. Linder. 1965. 18 p. Sw. cr. 8:—.
183. Mechanical and instrumental experiences from the erection, commissioning and operation of a small pilot plant for development work on
aqueous reprocessing of nuclear fuels. By K. Jönsson. 1965. 21 p. Sw.
cr. 8:—.
184. Energy dependent removal cross-sections in fast neutron shielding
theory. By H. Grönroos. 1965. 75 p. Sw. cr. 8:—.
185. A new method for predicting the penetration and slowing-down of
neutrons in reactor shields. By L. Hjärne and M . Leimdörfer. 1965. 21 p.
Sw. cr. 8:—.
186. An electron microscope study of the thermal neutron induced loss in
high temperature tensile ductility of N b stabilized 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
MeV gamma activity. By R. S. Forsyth and W . H. Blackadder. 1965.
22 p. Sw. cr. 8:—.
188. Trace elements in human myocardial infarction determined by neutron
activation analysis. By P. O. Wester. 1965. 34 p. Sw. cr. 8:—.
189. An electromagnet for precession of the polarization of fast-neutrons.
By O . Aspelund, J. Björkman and G . Trumpy. 1965. 28 p. Sw. cr. 8:—.
190. On the use of importance sampling in particle transport problems. By
B. Eriksson. 1965. 27 p. Sw. cr. 8:—.
191. Trace elements in the conductive tissue of beef heart determined by
neutron activation analysis. By P. O . Wester. 1965. 19 p. Sw. cr. 8:—.
192. Radiolysis of aqueous benzene solutions in the presence of inorganic
oxides. By H. Christensen. 12 p. 1965. Sw. cr. 8:—.
193. Radiolysis of aqueous benzene solutions at higher temperatures. By
H. Christensen. 1965. 14 p. Sw. cr. 8:—.
194. Theoretical work for the fast zero-power reactor FR-0. By H. Häggblom.
1965. 46 p. Sw. cr. 8:—.
195. Experimental studies on assemblies 1 and 2 of the fast reactor FRO.
Part 1. By T. L. Andersson, E. Hellstrand, S-O. Londen and L. I. Tirén.
1965. 45 p. Sw. cr. 8:—.
196. Measured and predicted variations in fast neutron spectrum when penetrating laminated Fe-DjO. By E. A a l t o , R. Sandlin and R. Fräki. 1965.
20 p. Sw. cr. 8:—.
197. Measured and predicted variations in fast neutron spectrum in massive
shields of water and concrete. By E. Aalto, R. Fräki and R. Sandlin. 1965.
27 p. Sw. cr. 8:—.
198. Measured and predicted neutron fluxes in, and leakage through, a configuration of perforated Fe plates in D 2 O. By E. A a l t o . 1965. 23 p. Sw.
cr. 8:—.
199. Mixed convection heat transfer on the outside of a vertical cylinder.
By A. Bhattacharyya. 1965. 42 p. Sw. cr. 8:—.
200. An experimental study of natural circulation in a loop with parallel
flow test sections. By R. P. Mathisen and O . Eklind. 1965. 47 p. Sw.
cr. 8:—.
201. Heat transfer analogies. By A. Bhattacharyya. 1965. 55 p. Sw. cr. 8:—.
202. A study of the "384" KeV complex gamma emission from plutonium-239.
By R. S. Forsyth and N . Ronqvist. 1965. 14 p. Sw. cr. 8:—.
203. A scintillometer assembly for geological survey. By E. Dissing and O .
Landström. 1965. 16 p. Sw. cr. 8:—.
204. Neutron-activation analysis of natural water applied to hydrogeology.
By O . Landström and C. G . Wenner. 1965. 28 p. Sw. cr. 8:—.
205. Systematics of absolute gamma ray transition probabilities in deformed
odd-A nuclei. By S. G . Malmskog. 1965. 60 p. Sw. cr. 8:—.
206. Radiation induced removal of stacking faults in quenched aluminium.
By U. Bergenlid. 1965. 11 p. Sw. cr. 8;—.
207. Experimental studies on assemblies 1 and 2 of the fast reactor FRO.
Part 2. By E. Helistrand, T. L. Andersson, B. Brunfelter, J. Kockum, S-O.
Londen and L. I. Tirén. 1965. 50 p. Sw. cr. 8:—.
208. Measurement of the neutron slowing-down time distribution at 1.46 eV
and its space dependence in water. By E. Möller. 1965. 29. p.Sw.cr.8:—.
209. Incompressible steady flow with tensor conductivity leaving a transverse
magnetic f i e l d . By E. A. Witalis. 1965. 17 p. Sw. cr. 8:—.
210. Methods for the determination of currents and fields in steady twodimensional MHD flow with tensor conductivity. By E. A. Witalis. 1965.
13 p. Sw. cr. 8:—.
211. Report on the personnel dosimetry at AB Atomenergi during 1964. By
K. A. Edvardsson. 1966. 15 p. Sw. cr. 8:—.
212. Central reactivity measurements on assemblies 1 and 3 of the fast
reactor FRO. By S-O. Londen. 1966. 58 p. Sw. cr. 8:—.
213. Low temperature irradiation applied to neutron activation analysis of
mercury in human whole blood. By D. Brune. 1966. 7 p. Sw. cr. 8:—.
214. Characteristics of linear MHD generators with one or a few loads. By
E. A. Witalis. 1966. 16 p. Sw. cr. 8:—.
215. An automated anion-exchange method for the selective sorption of five
groups of trace elements in neutron-irradiated biological material.
By K. Samsahl. 1966. 14 p. Sw. cr. 8:—.
216. Measurement of the time dependence of neutron slowing-down and
thermalization in heavy water. Sy E. Möller. 1966. 34 p. Sv/. cr. 8:—.
217. Electrodeposition of acHnide and lanthanide elements. By N-E. Barring.
1966. 21 p. Sw. cr. 8:—.
218. Measurement of the electrical conductivity of He3 plasma induced by
neutron irradiation. By J. Braun and K. N y g a a r d . 1966. 0 p. Sw. cr. 8:—.
219. Phytoplankton from Lake Magelungen, Central Sweden 1960—1963. By T.
W i l l é n . 1966. U p. Sw. cr. 8:—.
220. Measured and predicted neutron flux distributions in a material surrounding av cylindrical duct. By J. Nilsson and R. Sandlin. 1966. 37 p.
Sw. cr. 8:—.
221. Swedish work on brittle-fracture problems in nuclear reactor pressure
vessels. By M. Grounes. 1966. 34 p. Sw. cr. 8:—.
222. Total cross-sections of U, UOj and ThOj for thermal and subthermal
neutrons. By S. F. Beshai. 1966. 14 p. Sw. cr. 8:—.
223. Neutron scattering in hydrogenous moderators, studied by time. Dependent reaction rate method. By L.G. Larsson, E. Möller and S . N . Purohit.
1966. 26 p. Sw. cr. 8:—.
Förteckning över publicerade AES-rapporter
1. Analys medelst gamma-spektrometri. Av D. Brune. 1961. 10 s. Kr 6:—.
2. Bestrålningsförändringar och neutronatmosfär i reaktortrycktankar —
några synpunkter. Av M . Grounes. 1962. 33 s. Kr 6:—.
3. Studium av sträckgränsen i mjukt stål. Av G . Östberg och R. Attermo.
1963. 17 s. Kr 6:—.
4. Teknisk upphandling inom reaktorområdet. Av Erik Jonson. 1963. 64 s.
Kr 8:—.
5. Ågesta Kraftvärmeverk. Sammanställning av tekniska data, beskrivningar
m. m. för reaktordelen. Av B. Lilliehöök. 1964. 336 s. Kr 15:—.
Additional copies available at the library of AB Atomenergi, Studsvik,
Nyköping, Sweden. Transparent microcards of the reports are obtainable
through the International Documentation Center, Tumba, Sweden.
EOS-tryckerierna, Stockholm 1966