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Conf. 661019-6
FROMPT-NEUTRON TIME BEHAVIOR IN
DELAYED-CRITICAI, COUPLED URANIUM-METAL CYLINDERS* H.C. $ 1.MN 20
MV 57
J. T. Mihalczo
Oak Ridge National Laboratory
Oak Ridge, Tennessee 37830
ABSTRACT
Pairs of identical coaxial 27.94-cm-diam cylinders of unmoderated
and unreflected uranium metal have been assembled to delayed criticality
with large spaces between the flat surfaces.
The uranium metal was en-
riched to 93.15 wt% in the 2350 isotope. Prompt-neutron decay constants
of the assemblies were measured by the Rossi-a technique, and the results
.
:
agreed with those predicted by a method in which a v ir of equations
represented the kinetics of a coupled-core assembly. The equations
neglected delayed neutrons and assumed space-independent kinetics for
each cylinder with a time lag to represent the flight time of neutrons
from one cylinder to the other. Multiplication factors calculated for
the assemblies using Se transport theory also agreed with the experi-
mental values.
RAREASED FOR ANNOUNCEMINI
LEGAL NOTICE
This report was prepared as an account of Government sponsored work. Neither the United
States, zor the Commission, nor any person acting on behalf of the Commission:
A. Makes any warranty or representation, expressed or implied, with respect to the accu.
Troy, completeness, or usefaloss of the information contained in this report, or what the we'
of any information, apparatue: method, or procons dsoloved in this report may not latring
privatly owned rights or
B. Asnimo may Vabilities with respeot to the use of, or for dumugon resulting from the
un of any information, apparatus, method, or prooesn'discloned in this report.
As used in the above, "person soting on behalf of the Commission" inoiudas any ome
ploys or contrinotor of the Commission, or employds of such contractor, to the extent that
fuok omployos or contractor of the Commission, or employee of such contractor preparos,
disseminates, or provides access to, any Information pursuant to blo employment or contract
with the Commission, or his employment with such contractor.

IS NUCLUIR SCIENCE ABSTRACTS
TXT
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AAP
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*Research sponsored by the U.S. Atomic Energy Commission under contract
with the Union Carbide Corporation.
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INTRODUCTION
In order to understand the neutron interaction between the two com-
ponents of a coupled-core assembly that has a single large gap, it 18
desirable to study the properties of the assembly that depend directly
on the neutrons traveling across the gap from one component to the other.
One such property is the delayed-critical configuration. Another is the
prompt-neutron decay constant at delayed critical, which is equal to the
effective delayed-neutron fraction divided by the prompt-neutron life-
time.
In the investigation reported here, these properties were studied in
delayed-critical experiments with pairs of unmoderated and unreflected
cylinders of uranium metal (93.15 wt% <330) assembled so that their axes
were coincident and their flat surfaces were separated by large gaps.
In such assemblies the neutron exchange is principally between the flat
surfaces, and the prompt-neutron decay constant is a strong function of
the size of the gap between these surfaces and the fraction of the neutron
population crossing the gap. This results from the fact that the prompt-
neutron lifetime, on which the decay constant depends, 18 strongly af-
fected by the gap. In a single unmoderated delayed-critical cylinder
of uranium metal (0 = 18.75 g/cm) the prompt-neutron lifetime 18 6.2
nsec. In a two-component system, however, the lifetime is increased
by the time it takes a neutron to travel across the air gap, an increase
which is considerable since the neutrons leaking from a single delayed-
critical cylinder have an average speed of about 0.7 cm/nsec.
------**
In the experiments performed in this study prompt-neutron decay
constants were determined by Rossi-e measurements on a number of two-
component delayed-critical assemblies and some of the results were
compared with calculated decay constants. The theory on which the cal-
culations are based has been discussed by Baldwin and others.? A coupled
core experiment using the pulse neutron technique has been performed pre-
viously by Helmick, Chezem, and Seale" et Los Alamos.
Transport calculations using Sg theory were also performed to
obtain multiplication factors for comparison with the experiments.
EXPERIMENTAL RESULTS
As shown in Fig. 1, the upper cylinder of each assembly was built
on a 0.025-cm-thick stainless steel diephragm held in position by a
76-cm-ID, 2.54-cm-thick aluminun clamping ring, and the lower cylinder
was supported on a low-mass aluminum stand.
The assembly of the system
to delayed critical with the desired spacing was completed when the
lower section was raised remotely by a piston having a maximum stroke of
61 cm. (This apparatus has been described previously by Rohrer et al.")
The thickness of the cylinders and the spacing between the flat faces
were the variables that were adjusted until criticality was achieved,
the diameters of all the cylinders remaining constant at 27.94 cm. The
uranium-metal density in the assembled cores varied from 18.71 to 18.73
g/cm3.
All reactivities were obtained from stable reactor periods. The
reactivity contribution of the support structure, including the diaphragm,
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was accounted for by measuring the effects of appropriate equivalent
materials on the reactivity of each assembly and converting the resu. 3
to an equivalent change in sering of the cylinders. The conversion
was based on measured values of the change in reactivity per unit change
in spacing. A plot of the critical separation as a function of the thick-
ness of one cylinder in each assembly is shown in Fig. 2. Tie maximum
separation distance, 36.96 cm, was determined for an assembly in which each
cylinder was 8.25 cm thick.
The prompt-neutron decay constant for each critical configuration was
measured by the Fossi-a technique, using instrimentation employing two
detectors. One was a plastic scintillator (NE-102) which triggered a time-
of-flight type analyzer, and the other was a spiral 2350 fission counter
which provided the count signal. (This instrumentation has been described
in Ref. 1.) The effect of the detector locations on the results obtained
was investigated by making successive measurements with the f'ission counter
centered immediately below and above the lower cylinder (Positions A and
B) and immediately below and above the upper cylinder (Positions C and D).
During these measurements the plastic scintillator was located at a point
0
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which was about 1 in. above the top surface of the upper cylinder and 4 in.
from its center. For the cases in which the fission coir .es was in flux-
symmetric positions (A and D, or B and C), the data collected were indisting-
uishable; 1.e., the collection time, the number of triggers required to
collect a given amount of data, and the background level were identical.
Typical decay curves for the 8.25-cm-diam cylinders are given in Fig. 3 where
the error listed for the prompt-neutron decay constant is one standard deviation, as
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obtained from a least-squares analysis of the data. The agreement of
the prompt-neutron decay constants indicates the insensitivity of the
measurement to the detector position.
Figure 4 is a plot of the prompt-neutron decay constants of all the
coupled-core assemblies as a function of the thickness of one of the
cylinders in each assembly. The decay constant decreases rapidly
from 1.07 isec- to about 0.85 usec- at a separation of about 8 cm
and then increases, returning to a value of 1.07 usec -- when the cylinders
are infinitely separated. This behavior can be qualitatively explained
by an expression obtained previously for the prompt-neutron lifetime:
1 = 4x + +(x)
HURT
where x is the spacing of the cores, la is the prompt-neutron lifetime
In the assembly if the neutron flight time across the gap were zero,
*(x) is the probability that a neutron born in one cylinder leaks into
the other, t(x) is the flight time from one cylinder to the other, and
R(x) is the probability that a neutron born in one cylinder will be re-
flected back by the other. The term (1 - R(x)] in the denominator of this
expression accounts for the multiple reflections of the neutrons back
the lifetime is that of a single critical cylinder. As the space between
the cylinders is as infinite, P(x) approaches zero faster than t(x).
approaches Infinity, since P(x) decreases as 1/x+, and the lifetime again
approaches that in a single-component assembly. Thus, since the prompt-
neutron decay constant is inversely proportional to the prompt-neutron
lifetime, it should decrease and then increase as the spacing increases.
1
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CALCULATIONS OF PROMPT-NBUIRON TDI BUATTRO BY COUPIXD-CORE DETICS
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11
1.
.
Calculated values of the prompt-neutron decay constants for the
·assemblies were obtained by a method in which a pair of equations repre-
sented the kinetics of a coupled-core assembly. If delayed neutrons are
neglected, these equations can be written as follows:
11
an, (t)
at
e a N2(t) + y N (t + 7) + 6(t),
are . Q Ny(+) + Y NA (t = 1) ,
· where
No neutron population in core 1,2 at time t,
a = prompt-neutron decay constant for one core when it is
isolated from the other,
my s el(1/8),
• • reactivity contribution of one core to the other,
1. prompt-neutron lifetime in the isolated core,
B = effective delayed neutron fraction,
T - flight time of the neutrons traveling between cores.
The first term on the right represents the decay of the neutron population
in one core, while the second term represents the neutron source from the
other core delayed a time 7. The third term in the first equation 18 the
source tern. This term 18 a delta function in time and represents the neutron
which Initiates the fission chain process that 18 studied by the Rossi-a
technique. Since the initiation of a single flosion chain can take place
only in one core at a time and since the cores are identical, the source
term appears in one equation only.
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Calculations vere performed on the tax 7090 to solve the above
equations for xy(t) and X (t) for some of the coupled-core assemblies,
The value of a used for each configuration was that obtained from measure-
ments of the prompt-neutron decay constant for an isolated (subcritical)
cylinder having the same dimensions as those used in the assembly. The
value of ¢ was also obtained from the measurements of the prompt-neutron
decay constant for the isolated cylinders. Since the measurements on the
two-component assemblies were all made at delayed critical, the value of €
is just that reactivity by which the isolated cyl.inder 18 subcritical. The
prompt-neutron 11fetime was obtainsd using the lifetime inferred from measure-
merits of the prompt-neutron decay constant at delayed critical and from
calculated changes in the lifetime for subcritical cylinders. The value
used for B, the effective delayed neutron fraction, was 0.0067.
The values of _(t) and N,(t) so obtained were then used to compute
the Rob8i-a correlation probabilities, which in turn were used to calculate
the prompt-neutron decay constants. The calculated and experimental prompt-
neutron decay constants are compared in Table I. The sensitivity of these
values to errors in the constants appearing in the equations has been in-
vestigated by varying the values of the constants used. These resulting
changes in the prompt-neutron decay constant are also given in Dable I.
The large sensitivity of the calculations to changes in a org is due to
the fact that these assemblies are delayed critical and the decay in one
cylinder 18 almost compensated for by the delayed source from the other (1.6.,
the absolute magnitude of a 18 just slightly larger than that of y, but differs in
· dze). The sensitivity of the solutions to the values of a and sy is larger, the larg
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Table I. Comparison of Calculated and Measured Prompt-Neutron Decay
Constants and Sensitivity of Calculations to Variations in
a and a
Prompt-Neutron
Percent Change in
Calculated Prompt-
Neutron Decay Con-
stant Resulting from ..
1% Change in:
Decay Constant (usec-2)
Cylinder
Thickness
(cm)
Measured
Calculated
6.357
6.672
7.015
- 0.872 $ 0.009
- 0.855 + 0.008
- 0.874 + 0.007
- 0.878. 0.008
- 0.900 + 0.008
- 0.920 $ 0.008
- 0.932 $ 0.008
- 0.94
- 0.92
- 0.91
- 0.90
- 0.88
- 0.88
• 25.0
> 21.8
17.8
- 14.4
- 11.3
• 8.0
24.7
21.4
17.5
14.0
10.7
7.309
.
7.627
7.946
8.255
7.3
- 0.91
- 4.5
3.9

coupling between cylinders. The solutions are not very sensitive to the
value of the flight time, T. A 1% increase in a will change the solution
about 0.2% in all cases.
The largest disagreement between measured and calculated prompt-neutron
decay constants is 8% and in this case calculation would agree with experi-
ment if either a or y were changed only by 0.4%. The agreement between the
measurements and the calculations is quite good considering the sensitivity
or the calculation to the value of the decay constant, a, of the isolated
cylinder or to the coupling coefficient, y, and indicates that the model
used satisfactorily predicts the prompt-neutron time behavior.
TRANSPORT THEORY CALCULATIONS OF MULTIPLICATION FACTORS
The multiplication factors of the experimental configurations were
calculated by multigroup transport theory using the son method' in the Sg
approximation and the six-group cross-section values of Hansen and Roachº
for the 2350 and 238y isotopes. Fission of 234 and 2350 was included and
the absorption and scattering properties of these isotopes were assumed to
be the same as those of 230. The calculated values are all between 0.989
and 1.002, as is shown in Table II. The good agreement with experiment
indicates the ability of the transport method for calculation of assemblies
irith large gaps.
The dependence of the calculated multiplication constants on the order
of the so calculations was investigated in one case. For the 6.039-cm-thick
cylinders, the calculated multiplication factors for 6th, 8th, and 10th
order sy calculetions were 0.991, 0.998, and 1.002, respectively. These
results indicate that t
pproximation is adequate.
-H
Y
--
-
Table II. Sa Transport Theory Calculations of the
Múltiplication Factor
Thickness of Fuel
in Each Core
(cm)
Calculated Multiplication
Factor
4.331
4.445
4.768
5.081
5.399
5.721
6.039
6.357
6.672
7.015
7.309
0.995
0.989
0.993
0.998
0.990
0.994
0.998
1.002
0.998
0.998
0.996

15
.. ACKNOWLEDGEMENTS
• The author wishes to acknowledge the assistance of the staff of the
ORNL Critical Experiments Facility, in particular the work of J. J. Lynn,
J. R. Taylor, and J. F. Ellis in the performance of the experiment. Appre-
ciation 18 also expressed to P. Stewart, B Whitesides, and L. Stewart of
the computer Science Center who performed the calculations and least-squares
analysis.
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REFERENCES
J. T. Mihalczo, Nucl. sci. Eng. 20, 60 (1964).
H. H. Helmick, C. G. Chezem, and R. L. Seale, Trans. Am. Nucl. Soc.
8, 222 (June 1965).
G. C. Baldwin, Nucl. Sci. Eng. 6, 320 (1959); R. Avery, "Theory of
Coupled Reactors," Proceedings of the Second International Conference
on Peaceful Uses of Atomic Energy, Geneva 1958, Vol. 12, p.182 (1959);
S. J. Geuge, F. T. Adler, and P. N. Powers, "Neutron Dynamics and Control."
Proceedings of a Symposium on Nuclear Engineering at the University of
Arizona, p. 45 (May 1960); F. R. Pluta, "Neutron Dynamics and Control,"
Proceedings of a Symposium on Nuclear Engineering at the University of
Arizona, p. 544 (May 1960).
E. R. Rohrer et al., "Neutron Physics Division Annual Progress Report,
Sept. 2, 1961," ORNI-3193, p. 168 (1961).
J. T. Mihalczo, Trans. Am. Nucl. Soc. Ý, 60 (June 1963).
J. T. Mihalczo, Trans. Am. Nucl. Soc. 9, 175 (June 1966).
Clarence E. Lee, "The Discrete So Approximation to Transport Theory,"
LA-2595 (March 1962).
G. E. Hansen and W. H. Roach, "Six and Sixteen Group Cross Sections
for Fast and Intermediate Critical Assemblies," LAMS 2543 (1961).
4.
5.
6.
7.
8.
.......
.........
............
..
..., de se ....
LIST OF FIGURES
Fig. 1. Typical Uranium-Metal Coupled-Core Assembly. Upper cylinder
18 on a stainless steel diaphragm and lower core is on aluminum support
stand.
Fig. 2. Critical Separation Distance as a function of the Thickness
of One Cylinder in Each Coupled-Core Assembly.
Fig. 3. Prompt-Neutron Decay in Coupled-Core Assembly for various
Detector Locations. The assembly consisted of two 27.94-cm-diam x 8.25-
cm-thick cylinders spaced with 36.96 cm between their flat surfaces.
Fig. 4. Prompi-Neutron Decay Constant as a Function of the Thickness
of One Cylinder in Each Coupled-Core Assembly.
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DATE FILMEI
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