Exponential Experiments

It has been shown that variations in Ed with coverage can be used to explain the non-linearity of kinetic plots. However, such deviations can also be explained by pre-exponential variations and, in most cases, a shift of Ed or v values, or more rarely, a change in both together, can be used to fit experimental data successfully. A further problem with a simple analysis is that compensation effects between parameters can make the system appear simple, while give misleading values for kinetic terms. Most experiments assume a constant v, because of the difficulties of making accurate measurements of the pre-exponential. Experiments which have assumed a dependence of Ed on 6 can also be explained by applying a... 

Microscopic (Lattice) Reactor Theory. Wigner s version of the theory of the multiplication constant in the uranium lattice had been worked out by the time he arrived at Chicago in 1942. What remained to be done was its systematic application to the design of reactors, and to the interpretation of Fermi s exponential experiments. (I should stress that Fermi was his own. 

Wigner initiated and supervised the examination of as many combinations of fuel and moderator as the group could manage. By the end of 1945 he had explored lattices moderated with H2O, D2O, CO2, Be, BeO, and C, as well as homogenous mixtures of U, D2O and H2O. The constants such as diffusion length, resonance absorption, and cross-sections used in this exploration were based on experiments at Columbia, Princeton, and, later, on the exponential experiments in Chicago. 

One must understand that the uranium problem mobilized the best scientific brains of the time. Szilard, the brilliant visionary, and Fermi, the all-powerful experimental and theoretical genius, who actually achieved the first chain reaction, are unique and in a way unmatched. On the other hand, one must also recognize not that the chain reaction was so difficult to achieve, but that it was so easy. Lesser people than those who gathered at Chicago and Los Alamos could have succeeded - but could they have succeeded in only two years, the time from the first exponential experiment that showed k to exceed unity to the Hanford reactors and another year to Alamogordo ... 

As far as I know it was mainly the beryllium line that was pursued in Chicago with Mr. Allison being the leader of the work. This work was not dissimilar to that carried out by Fermi except in its emphasis on beryllium. They remeasured the diffusion length in graphite and were planning to make exponential experiments when the reorganization mentioned above took place. Also a table of nuclear cross section for thermal neutrons was prepared which proved to be very useful in later work. 

The discrepancy of 6 X 10 cm in the calculated and observed rod effect is well within experimental error since the results of two exponential experiments (with and without the rod) axe needed in the comparison, and each experiment may be off by 5 X 10 cm . Thus, if the no rod Laplacian were only 117 x 10 instead of 120 x 10 and the rod were observed to be 72.2 cm instead of 71.6, the agreement between theory and experiment would have been perfect. 

A Thorium-Uranium Exponential Experltnenti C. If. Skeen and W. W. Broum(AI). Because of uncertainties in the knowledge of t nuclear properties of thorium fuel and lattices containing this fuel, an experimental study was made of a thorium based fuel that is to be loaded into the Sodium Reactor Experiment (8RB) in the near future. An exponential experiment was performed with a square-celled lattice of 7-rod elements (l-in. diameter rods) spaced 9.5 in. apart in praphite. The fuel Is a Th-U-23S alloy containing 7.6% uranium by weight which is 93.13 atomic per cent U-235. The feel elements were 5 ft. long. The subcrltical lattice was placed On thd thermal column of a water boiler reactor which served as the source of neutrons for the assembly. 

Neutron Multiplication Measurements with Plutonium-Aluminum Alloy Rods in Light Water, V, I. Neeley, R. C. Uoyd, and E. D. Clayton (GE-HAPO). A series of near critical and exponential experiments have been performed with natural water moderated assemblies of Pu-Al alloy rods in order to establish bases for nuclear safety specifications lor handling and storage of this kind of fissile material. 

W. W. Brown, "Exponential Experiments with Organic Moderated Uranium Lattices, Proc. Second Geneva Conf. (Uidted Nations, New York), 12 (1958), p. 514. 

E. R. Cohen, Exponential Experiments on DjO-Uranium Lattices, P/605 Proc. 1st UN Cohf. on, Atomic Energy-Geneva (1955) 5, 268, United Nations New York., ... 

R. Persson et al., Exponential Experiments On ffeavy-Water-Natural Uranium Metal and Oxide Lattices, P/ 160, Proc. 2nd UN Conf. on Atomic Energy—Geneva (i9S8) 12, 364, United Nations, New York. 

Since Fermi s early work on exponential experiments the design of reactors has depended heavily on the experimental determination of nuclear properties of multiplying assemblies in subcritical Or critical experiments. In thermal assemblies the nuclear properties studied have included macroscopic flux traverses, from which the material buckling can be obtained, and ratios related to quantities which enter into the four-factor formula for k. One ratio measured in uranium fueled assemblies is ... 

D. E. Davenport, "Exponential Experiments in Graphite Systems, Proc, 1st VN Conf, on Atomic, Energy—Geneva 1955, 5, 309, United Nations, New York (1955),... 

Comparison between the results of the critical and exponential experiments are difficult, since there has not been too much overlap of the assemblies Investigated. The indirect use of the Integral data which, in the analysis, serves only as a check on the assumed multigroup constants, also leads to difficulties in the corapaarlson since the quantities measured in the two types of experiments are usually different. No systematic difference of accuracy has been observed in the extent of agreement between theory and measurement in the two types of experiment. The critical experiments have been far more useful, however, in that the precision of the experimental results is much greater. 

KOUTS, H. J. C., et al., "Exponential Experiments in Light Water, Proceedings of the International Cottfer ence on Peaceful Uses of Atomic Energy, S, United Nations, New York. 

R was impossible to make most of the solution/ring mixtures critical and, in fact, the source neidron multiplication was often too low to allow sensible extrapolation. Addition of only a few inches of solution above a very sub- critical mixture established a critical system that served as a source of neutrons distributed over the mixture and had spectral characteristics that allowed a measure of the neutron relaxation length in the mixture. The material buckling estimated for these exponential-experiment data is somewhat uncertain because values of the radial extrapolation distance were unavailable. It has been possible to show, however, that the material buckling of some of the mixtures studied is negative. 

A preliminary theoretical analysis of a series of exponential experiments with U -Th oxide rods in water has been conq>leted. The fuel rods are 0.034S in. thick Zlr-calloy tubes with an O.D. of 0.499 in. filled with vibratory compacted Thorium-Uranium-oxide powder. The fuel contains 3.0% of U Oa by weight and has an average density of 8.96 gm/cm. The fUel also contains about 9 parts/10 of boron as an Inunirlty and trace amounts of U .U ,andU . ... 

In a new test run on a uniform lattice with the new U-233 cross sections, METHUSELAH gave a k ff of 1.020 for a case in which the geometrical budding was assumed to be equal to the material budding that had been determined by an exponential experiment. No simple and Justifiable change could be made in the METHUSELAH library to improve the results. Consequently, it is expected that discrepancies of the order of several percent in keff can be expected for the cluster lattices. 

H. R.. WINDSOR, Buckling Measurements in Water Moderated and Reflected Exponential Experiments, Teclmical Ndte, liucl. St Eng. (in press). 

C. L. BROWN, R. C. LLOYD, S. R. BIE AN, and E. O. CLAYTON, Exponential Experiments and Neutron Multiplication Measurements with 1.25 w/o Enriched N Reactor Fuel Elements in Light Water," BNWL-52, Battelle-Northwest Labs. (March, 1969). 

Critical Approach and Exponential Experiments with Prototype FFTF Driver Fuel Pins in Water, C. X. Brown (PNL), C. A. Rogers (WADCO)... 

Lattice Pitch (In.) Water-to-Fuel Volume Ratio Critical Approach Experiments Exponential Experiments Material Buckling (m- ) Extrapolatton Distance". (cm)... 

Correlations with exponential experiments of fuel elements in water performed at the Savannah River Laboratory validate these calculations for this analysis ... 

Fig. 1. Vertfcal flux distribution obtained by exponential experiment.

For comparison, keff s were measured by a pulsed-neutron source technique on the same lattices as those of the exponential experiments. The prompt-neutron decay constant a is related to keff ... 

Another application of the exponential technique has been examined on the lattices. loaded with blade-type absorbers. The fuel rods were arrayed in a. 15 X 20 lattice, which was separated by an absorber into two 15X10 regions. The subcriticality was varied with the absorbers which had seven B4C concentrations. In this case, the keff from the exponential experiments showed good agreement with those from the pulsed-ncutron source experiments in the range of 0.97 > keff >0.90. 

In conclusion, the exponential experiment is an effective technique to determine keff of a hi ly subcritical system, when M of the system is known by some method and there is a sufficient region of uniform composition in at least one direction. 

The lattice mimtlon area deduced from an exponential experiment Is 6k6 cv (see Section 2.6 2) this Is separable Into a.Fermi age ( t and a diffusion area (L ) of l60 cm at room teoqperature At operating tempera-ture Increases by 109. ... 

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