Resonance-escape Probability

The resonance escape probability is defined as the probability that a fission neutron escaping a fuel element for the first time algua down to thexmal energies without resonance capture Ii a redomlnantly system it Is sufficient to consider resonance escape in U only The definition further excludes any l/v-captures in tte materials considered The fo.Uowing general formula may be used  

Six e the contribution to the resonance absorption in a fuel element from neutrons native to the fuel element in question is much less than the contributions from other fuel elements the function is very close to zero, m the H Reactor CX)872. 

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Nd. From the high isotope ratio " Nd/ " Nd and the low isotope ratio it must be concluded that chain reactions have occurred. The age of the Oklo deposits was found by Rb/ Sr analysis to be a 1.7 10 y. At that time the concentration of in natural uranium was 3%. The presence of water in the sedimentary ore deposits led to high values of the resonance escape probability p (eq. (11.1)) and to criticality of the systems ( eff > ) ... 

Because has a shorter half-life than " U, all uranium ores were richer in in the past. From Rb— Sr analysis the age of the Oklo dqxisit is known to be 1.74 X 10 y at that time the content of natural uranium was 3%. Although the fission factor i rapidly increases with content (about 1.8 for 3 % U), conditions in the natural Oklo deposit were such (e 1.0, p 0.4,/ 1.0) that < 1. The deposit is sedimentary and was formed in the presence of water, which greatly increases the resonance escape probability factor p for an atomic ratio H20 U of 3 1, p 0.8, and > 1. Thus conditions existed in the past for a spontaneous, continuing chain reaction to occur in the Oklo deposit. 

Abstract. The resonance escape probability, p, in a lattice is expressed in terms of the resonance absorption integral (f era dEjE)efi and the spatial distribution of resonance neutrons. It is shown that the dependence on the spatial distribution is very nearly the same whether the slowing down in the moderator is Gaussian or exponential. The dependence on (f a a dEjE)ef( is rigorous only for a hydrogen moderator for a heavy moderator the formula overestimates the resonance escape. 

Equations (6) or (4) would permit the calculation of the resonance escape probability, assuming that a and P are known, if d and i.e., the density distribution of resonance neutrons were known. Unfortunately, this is really... 

We shall now outline a calculation of the resonance escape probability based on a somewhat more complicated model in which the slowing down in the moderator is Gaussian (instead of exponential as in section 4), and the resonance absorption in the metal consists of a surface and volume term, both of which are constant over the entire resonance energy range. If we assume no energy loss in the metal, then the equation for qo, the number of neutrons x velocity per cc and unit In E in the metal is the same as equation 13 ... 

The resonance escape probability is the number of neutrons leaving the bottom of the resonance energy region per second divided by the number which enter the top of the region per second. This quantity is therefore (ti =lnEo/Ei)... 

In these and the above equations, the a are cross sections per imit volume, the a in (8) is scattering cross section, the average loss in r per collision. The are used because the material may contain different types of atoms. The (Ta is the thermal absorption cross section r(r) the resonance absorption cross section per unit volume. The = qef is the multiplication constant divided by the resonance escape probability. The product of thermal utilization / and (Ta is the effective cross section of uranium per unit volume, i.e., its cross section per unit volume multiplied by the thermal neutron density in it and divided by the average thermal neutron density. One can write, therefore, (Tu for f(Ta- If one multiplies this with rj the result is the same as crfU where fission cross section for thermal neutrons per unit volume, p the number of fast neutrons per fission. As a result, the third term in (7) can be written also as e is the multiplication by fast effect)... 

The factor p is the resonance escape probability for the inclusion of which the reason was given above. Equations (10), (10 a) go over into (5 a), (5 b) of CP-1461, if one restricts oneself to the stationary case, i.e., sets the right sides of (10), (10 a) equal zero, except that the present equations apply also if the diffusion coefficients are not constant throughout the pile. The purpose of rederiving (10) and (10 a) was not to obtain the latter, rather trivial generalization but to obtain the proper factor for the right side of (10). Cf. also CP-1554. 

The effectiveness of the moderators and the optimum moderator/fuel ratios previously presented have been based only on the use of as a fuel. On the basis of cross-sectional data for the fissile nuclides, it is to be expected that the optimum moderator/fuel ratio would be larger for Pu fuel and smaller for fuel. In addition, for heterogeneous, low-enrichment, or natural-uranium reactors it is desirable to achieve a large resonance escape probability, and, consequently, the optimum moderator/ fuel ratio is even larger than that previously indicated. Typical initial values for the C/U ratios in several gas-cooled reactors can be found in Table I. 

The resonance escape probability p therefore amounts to p = 0.8, leaving (179 x 0.8 ) 143 neutrons. After reaching thermal energy the neutrons diffuse through the reactor core. During this time different effects can take place ... 

R. J. French, The Resonance Escape Probability in Clustered Fuel Rod Lattices, Trans-> Am. Nuclear Soc., Vol. 2, No. 2. 

The resonance escape probability (actually the probability that neutrons escape capture in the Th-232 present while slowing down to cadmium cut-off) was measured directly by activating Th-234 foils in the fuel of the central cell. Sets Of bare and cadmium-covered foils were irradiated at a distance of 18 in. aWe the bottom of the assembly. The activity was determined by counting the 89.5-kev internal conversion x-ray from Pa-233 which follows the Th-232 (n, y) reaction. The resonance escape probability p was then obtained from the equation... 

Parameters relating to resonance escape probability, fast effect, and conversion ratio, namely p", S", 6 , were... 

D. Kein, A. Z. Kranz, G. G. Smith, W.Baer, and J. De-Juren, Measurements of Thermal Utilization Resonance Escape Probability and Fast Effect in Water-Moeferated, Slightly Enriched Uranium and Uraniuni Oxide Lattice, J. Nuc, Sci, and Eng., 3 403. ... 

The measurement of the resonance escape probability p involved measurements of the 104 kev neptunium X-ray according to techniques described by Kranz. The ratio of epicadmlum to subcadmium U-238 absorptions was corrected for both leakage and cpicadmium 1/v absorptions. 

A. Z. kranz, Measurements of Thermal Utilization, Resonance Escape Probability, and Fast Fission Factor of Water Moderated Slightly Enriched Uranium Lattices, WAPD-134 (Sept. 1955). 

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