Nonleakage probability

The fraction of the fast neutrons that do not escape from the reactor as they degrade from fission to resonance energy depends on the size and moderating properties of the reactor. This fraction is denoted as Pi, the fission-to-resdrmnce nonleakage probability. Hence, the rate at which fast neutrons degrade into the resonance region is er) N a [Pg.127]

Of the latter, some diffuse to outer surfaces and escape, but the fraction Fj remains in the reactor as thermal neutrons Fj is called the resonance-to-thermal nonleakage probability. Finally, the neutrons complete an energy cycle as et]m o PipP2 neutrons reach thermal energy per unit volume per unit time. The product P1P2 is the fission-to-thermal nonleakage probability, which we shall denote as F,h. 

Neutron Age ond Nonleokoge Probobility for a 2% U-235 Enriched UF4-Paroffin Mixture With a H U>235 Ratio of A04, J. T, Mihalczo (ORNL). The neutron age and nonleakage probability for a mixture of 2% U-235 enriched UF and paraffin with a H U-23S atomic ratio of 404 have been determined from critical experiments at the Oak Ridge National Laboratory Critical Facility. These experiments are a continuation of a program reported previously. The material is a homogeneous mixture which contains 85 w/o UF. and 15 w/o paraffin and has a density of 3.45 g/cc. 

Kp(B) = nonleakage probability for prompt neutrons from fission, and Kd(B) = nonleakage probability for delayed neutrons, ... 

The results of the calculations for keff are shown in Table I, along with the nonleakage probabilities P (B and P((B ) calculated for the fast and thernaal ranges in the moments miethod. The estimated uncertainties in these quantities are also shown. Table n compares the results of calculations and experiments for cadmium ratios and activation of various foils. 

Again note that B is obtained from (5.172). The multiplication constant k for the finite medium is customarily called the effective multiplication constant, since it is the infinite-medium multiplication constant k multiplied by a reduction factor (1 + L B ) this factor is the one-velocity nonleakage probability and will be discussed in detail later. We summarize the above results by combining the information from (5.151) and (5.183) ... 

We showed previously that the multiplication constant k for the finite medium could be written in the form (5.183) and identified the factor (1 + L B ) as the nonleakage probability for the neutrons based on the one-velocity approximation. In the case that the diffusion equation is used to describe the distribution of thermal neutrons in a reactor, this factor gives the nonleakage probability for the thermals. We demonstrate this relationship by applying two separate lines of reasoning According to our usual definition of the multiplication constant of a system, we can write... 

In our present situation, ni and no represent the total neutron populations in the first and second generations for a finite system. Thus if no neutrons are absorbed by a finite medium, these neutrons produce a certain number of fissions which yield further neutrons. Not all these second-generation neutrons, however, are available to the chain reaction a certain fraction leak out through the boundaries. Therefore, if we call nf the number of neutrons produced by the reactions because of the no neutrons if the system had been infinite and let psL be the nonleakage probability for a neutron in the finite system, then it follows that nfp L is the number of second-generation neutrons available to the finite system for maintaining the chain. Thus, rii = n p L, and... 

We can derive this result by an alternate approach. By definition, the nonleakage probability must be given by the fraction of neutrons absorbed in the system. Let us compute, therefore, the ratio of the total number of neutrons absorbed per unit time in the reactor to the total... 

Note that g denotes the fast nonleakage probability and p the resonance-escape probability in the usual way [see Eqs. (4.260)]. Clearly, the probability that a neutron will not be absorbed in traversing the lethargy interval (0,w) is the product of the probability that it will not be absorbed in the interval (0,w ) and the probability that it will not be absorbed in (u, w) thus for u < u,... 

Equation (6.103) applies since the resonance-escape probability for any particular interval is independent of the past history of the neutron. The fast nonleakage probabilities are also independent, and it follows that... 

The fast nonleakage probability based on these data may be computed from Eq. (6.79) it is found to be th = 0.7364. 

We demonstrate the calculation of a temperature coefficient by considering the initial startup condition of the CP-5 and show how this coefficient may be used to predict the excess reacftivity (or multiplication) of the reactor at room temperature. Specifically, we compute the temperature coefficient of the hot clean reactor from the temperature derivatives of its thermal utilization, fast nonleakage probability, thermal nonleakage probability, and resonance-escape probability. The change in fc for a given change ST in temperature is then easily computed from the relation (6.142). 

The temperature coefficient of the fast nonleakage probability is obtained from (6.169), which requires an estimate of ath- Now... 

The temperature coefficient of the thermal nonleakage probability is computed next. The appropriate relation is given in (6.165)... 

Equation (8.190), along with the determinant (8.188), may be regarded as the criticality requirements for this model. [Note that the last two factors on the left side of (8.190) give the fast and thermal nonleakage probabilities, respectively.] As in the bare-reactor calculation, (8.190) is solved for m (and X ) and the critical size (or concentration) obtained from the determinant. 

If the resonance-escape probability has been obtained by some other means, then a knowledge of Xjoo yields g. It is interesting to note that, for very large specimens, the value of g obtained in this way should agree well with the Fermi age expression for the fast nonleakage probability, that is,... 

The quantities pth and g h in (9.84) denote the resonance-escape and fast nonleakage probabilities to thermal in the usual way [(4.260) and (6.79)). Penally, note that since the reactor described by (9.84) is not critical the multiplication constant will be different from unity and may be obtained from the relation... 

However, it seems clear that, in the case in which the unit-cell dimensions are small in comparison with the diffusion length, the form (1 H-would be a good approximation to the thermal nonleakage probability, and it would also be acceptable to use the general form L = D/S. Then, if we use... 

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