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Su ?oLASSIFIED UNCLASSIFIED BNL1986 Subject Category: PHYSICS UNITED STATES ATOMIC ENERGY COMMISSION A MEASUREMENT OF NEUTRON TEMPERATURE IN A URANIUM RODWATER MODERATED LATTICE By H. Kouts K. Downes G. Price R. Sher August 16, 1954 Brookhaven National Laboratory Upton, New York Technical Information Service, Oak Ridge, Tennessee L 7'~, .d t3: Z"r Work performed under Contract No. AT(302)Gen16. Date Declassified: October 27, 1955. This report has been reproduced directly from the best available copy. Issuance of this document does not constitute authority for declassification of classified material of the same or similar content and title by the same authors. Printed in USA, Price 15 cents. Available from the Office of Technical Services, Department of Commerce, Wash ington 25, D. C. GPO 987852 This report was prepared as a scientific account of Govern mentsponsored work and is made available without review or examination by the Government. Neither the United States, nor the Commission, nor any person acting on behalf of the Commis sion makes any warranty or representation, express or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this report, or that the use of any infor mation, apparatus, method, or process disclosed in this report may not infringe privately owned rights. The Commission assumes no liability with respect to the use of, or for damages resulting with respect to the use of any information, apparatus, method, or proc ess disclosed in this report. A MEASUREMENT OF NEUTRON TEMPERATURE IN A URANIUM RODWATER 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 crosssections of the two poisons, and since the two have very different crosssection 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. C7605 (Safety of Subcritical Loadings in T526), 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 1 2 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 & chromelalamel 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 off 3  change because of the effect on f. For sufficiently small increments, the change in keff is proportional to the macroscopic crosssection of the poison in the moderator: SkoffO Zpoison 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  1MC1k k 1 2 1 k2 =1 k1 h1 e. 1 R2 R1 od with the same constants of proportionality in both relations. Therefore 1 1 R2 7cd 1 1 7 R1 Ro Again, the macroscopic crosssections are proportional to the mole fractions M times the microscopic crosssections, with Avogodro's number the constant of 4 proportionality. Thus R2 R1 d cd I I MB 1B W, R The crosssections 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 (kT)2 dE 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 crosssection we take as 1/v, with cl = 11P.6. This corresponds to the latest crosssection 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 crosssection of 7700 barns at the peak of the resonance. Accordingly, e2 is assumed to be 10.680. The boron crosseection can be integrated immediately, to give oB 105 barns Integration of the expression for the cadmium crosseection 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. 5 cd = 0 .2 O2AT 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 a a e 2sin b v d 2 cos b vd v 0 o 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. 6770 (1951). 6  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. Thus .09332 "cd .9332 cEd= .790 a .022 .5587 an %ed = 4.73 .17 OR 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 crosssections. e estimate that this uncertainty could increase the error by at most .OWV6 electron volts, or 7 K. BORON CROSSSECTION AVERAGED OVER A MAXWELL DISTRIBUTION WITH CHARACTERISTIC ENERGY kT (ASSUMING o = 750 BARNS AT .025e.v.) 700 650 S600 m 550 500 .025 .030 .035 kT (ev.) Fig. I J x 1 cr. w z r. Lw w cr zo 0 O0 cn z 0a II p z SNUV9 8 U) z O 4D > 0 (0 LO o * w O 3: 1 9 UNIVERSITY OF FLORIDA 3 1262 08229 990 9 
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