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Subject Category: PHYSICS
UNITED STATES ATOMIC ENERGY COMMISSION
A MEASUREMENT OF NEUTRON
TEMPERATURE IN A URANIUM
ROD-WATER MODERATED LATTICE
August 16, 1954
Brookhaven National Laboratory
Upton, New York
Technical Information Service, Oak Ridge, Tennessee
7'~, .d t3: Z"r
Work performed under Contract No. AT(30-2)-Gen-16.
Date Declassified: October 27, 1955.
This report has been reproduced directly from the best
Issuance of this document does not constitute authority
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ington 25, D. C.
This report was prepared as a scientific account of Govern-
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ess disclosed in this report.
A MEASUREMENT OF NEUTRON TEMPERATURE IN A URANIUM ROD-WATER MODERATED LATTICE
By H. Kouts, K. Downes, G. Price, and R. Sher
Abstract: The relative danger coefficients of boron and cadmium have been
measured p an assembly of 1.15% enriched uranium rods in ordinary water, at
a water to uranium volume ratio of 3. The experiment consisted in finding the
relative effects on neutron multiplication of measured amounts of boron and
cadmium in the water moderator. The ratio of the observed danger coefficients
is a measure of the ratio of the cross-sections of the two poisons, and since
the two have very different cross-section curves in the thermal range, a basis
for the estimation of a neutron temperature exists. The measurement resembles
somewhat one done by G. P. Gavin (KAPL 112).
Under the aswBmption that the thermal neutrons have a Maxwell distribution of
velocities, the measurement imnlies a characteristic temperature of .0262
.001 volts (304 160 O). The water temperature at the time of the measure-
ment was 297 OK.
rperinmental Methods: A lattice of 1.15% enriched uranium rods, .600" diameter,
as loaded in a light water moderator until a multiplication of about 500 was
reached. The method of loading, and the safety precautions were as described
in BNL Log No. C-7605 (Safety of Subcritical Loadings in T-526), except that
now the console and instrumentation have been greatly improved. For instance,
during the measurement described in this report the flux levels were monitored
by six separate detectors and channels of instrumentation, and the safety rod
we set to trip on any one of four flux level monitors. In addition, a manual
rod trip and a manual water dump were available,
After this predetermined stopping point had been reached, the source was re-
noved, and the count rate from multiplication of spontaneous fission neutrons
was measured. A measured amount of boric acid solution was added to the water
moderator, after the equivalent volume of rure water was removed, and the count
rate was again measured. A measured amount of cadmium sulphate solution was
then added (again after removal of the same volume of moderator), and the count
rate was measured a third time.
Three counters were used to measure the flux levels at each stage. One was a
small BF3 counter at the center of a triangular lattice cell; the second was
a mall enriched uranium fission chamber at the center of a triangular lattice
cell, and the third was a fission chamber located inside a fuel rod. Three de-
tectors were used in order that independent checks on the results might be had;
they also increased the statistical accuracy, in that the determination of
relative flux level would be based on adding up the count rates from all counters.
While the flux level measurementa were made, the water temperature was monitored
by means of & chromel-alamel thermocouple.
Analysis: The reciprocal of the observed count rate is, for loadings sufficiently
near critical, linear in keff:
1 ff 1 the operator, as done her, ll
If a anall amnunt of poison is added to the moderator, as uwas done here, ke will
change because of the effect on f. For sufficiently small increments, the change
in keff is proportional to the macroscopic cross-section of the poison in the
and we suppose this relation to hold for the poison concentrations used in this
measuremat. We let the subscripts 0, 1, 2 refer respectively to situations
where the moderator ws pure water, where it contained boron, and where it con-
tained boron and cadmiu. Farther, we let the changes in keyf produced by the
addition of the boron and the cadmium be respectively -$and -A Then
Il-( -ko = 1 -1 -
1 2 1 -k2 =1 k1
h1 e. 1
R2 R1 od
with the same constants of proportionality in both relations.
1 1 7
Again, the macroscopic cross-sections are proportional to the mole fractions M
times the microscopic cross-sections, with Avogodro's number the constant of
R2 R1 d cd
I I MB 1B
The cross-sections to be used in this expression are averages over the energy
distribution of the themal neutrons in the lattice. If we assume the flwc to
have a Maxwell energy distribution
% (E) dE = 1 EK
then the averages are so
o = dE $ (E) ao (E) = eldB E1/2e-
o = dE (E) () = C2 dE 1/2 .-ET
S(E- E)2 (+ /4
The boron cross-section we take as 1/v, with cl = 11P.6. This corresponds to
the latest cross-section value of 750 barns at .025 e.v. The cadmium cross-
section is by recent data* characterized by BE = 0.176 e.v.,r = 0.115 e.v., and
a cross-section of 7700 barns at the peak of the resonance. Accordingly, e2 is
assumed to be 10.680.
The boron cross-eection can be integrated immediately, to give
oB 105 barns
Integration of the expression for the cadmium cross-eection is more involved.
It us accomplished in two ways by numerical integration for several values of
kT, and by an analytic approximation. This latter method involves transforming
the integral to
* J. Harvey, Private canmnnication.
cd = 0 .2 O-2AT da
Q (U2 E,)2 +*2
than (using Paarseval's theorem on the convolution of two Fourier transforms)
into the form o
a = e2 dt F (eS2) u 2
-d 2 (U. 0g)2 +
The symbol "P" means here "Fourier transform of". This latter form can again be
transformed to lead to error functions of complex argument. These in turn lead
to expressions of the form
e 2sin b v d 2 cos b vd v
These integrals can be evaluated by means of an approximation valid over the
entire range of the argument to better than one part in 105:
,2 -, d -n2
e 11t +2 Z2e cosh 2n
\(I n = 1)
Finally, all integrals involve only tabulated functions trigonometricc, hyper-
bolic, and error functions of real argument).
The calculation is only sketched, because of its length. The two independent
evaluations of oed (nuerical and analytic) agree to better than 1%.
The curves of ao, ocd, and ced are given as figures 1, 2, and 3.
Prior to the poisoning of the moderator, a curve of the spontaneous fission
multiplication was run as a function of rod loading. From this the critical
* See H. E. Salser, Mathenatical Tables and other aids to computation, pp. 67-70
mass was determined. This permitted rough estimation of the change in kef
caused by the poison. We found that adding the boron changed kef by about
3.6 x 10 ; the effect of adding cadmium uws to reduce keff again by about
2.8 x 10 3. These changes are certainly small enough to permit assuming linear-
ity between keff and Z poison"
Results: The observed count rates are given in table I. By the methods of the
preceding section, we calculate for the three cases:
BF3 counter: .780 .035
Fission chamber a
in moderator: .840 .053
Fission chamber 754 09
in fuel rod: = .754 + .049
Counters summed: .790 .022
The mole fractions used were respectively .09332 of cadmium and .5587 of boron.
.9332 cEd-= .790 a .022
%ed = 4.73 .17
From figure 3 we get, then,
kT = .0262 .0014 ev
= 304 a* 160
The water temperature during the measurement us 2970 K. Thus the neutron
temperature was 70 160 above the moderator temperature.
The probable error cited is based solely on statistical considerations. There is
a possible additional source of error, from incorrect cross-sections. e estimate
that this uncertainty could increase the error by at most .OWV6 electron volts,
or 7 K.
BORON CROSS-SECTION AVERAGED OVER A MAXWELL
DISTRIBUTION WITH CHARACTERISTIC ENERGY kT
(ASSUMING o = 750 BARNS AT .025e.v.)
.025 .030 .035
O 4D >
0 (0 LO
UNIVERSITY OF FLORIDA
3 1262 08229 990 9
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