Lethargy, neutron

The proposed calculation for the slowing-down density, in an infinite medium, due to a uniform line source of zero lethargy neutrons may be computed from (6.45), using the source (6.46) ... 

Isotropic sources of zero lethargy neutrons are distributed uniformly over a spherical surface of radius a. 

Since the lethargy interval occupied by neutrons is so small, it is often a surprisingly good approximation to assume in fast reactors that all the neutrons have the same energy, Eq, This corresponds to assuming that both the fission spectrum and the kernels H and K are 8-functions in E ... 

Analytical formulation. Call /(r, co. A) the fiux of photons or neutrons at the point r = (x,y,z), traveling in a direction co = (wxyOjy.coz) = (0,cp) with an energy represented by a suitable parameter A (wavelength for photons or lethargy for neutrons). This flux obeys the transport equation... 

For the ith species, define 2,to be the probabihty that a scattered neutron has a lethargy between u and u du and a direction lying in the solid angle rf 2 about 2 when its initial lethargy and direction are Then the number of neutrons scattered in unit time into the differential element drdudSl is... 

The quantity q defined above is approximately equal to the slowing down density Q (defined as the number of neutrons slowed down per unit volume per unit time from lethargies less than u to lethargies greater than u). That q and Q are not the same under the hypotheses already made is seen by evaluating Q as follows. 

To grasp the general principles involved, it seems appropriate to approximate reactors by finite models, dividing the neutron population into a finite number of subclasses, according to their position, lethargy, and (in transport models) direction of motion. Such a subdivision of the neutron phase-space into a finite number m of cells, corresponds to what must be done to solve reactor problems on digital computing machines. 

From the definitions of f and w, it is clear that ( is the average gain in the lethargy of a neutron per collision. From (4.42) we have... 

It is important to note that this result gives the probability that a neutron scattered at uo emerges with lethargy [Pg.85]

The distinction arises from the definition of these two types of functions. The fluxes and densities are defined per unit energy (or lethargy, or speed) and therefore require that the width of the interval in question be specified. The cross sections, on the other hand, specify the interaction characteristics between nuclei and neutrons of a particular energy. Figure 4.9 shows a typical cross-section curve given as a function of neutron energy. Note that all three cross sections [Pg.86]

The slowing-down density may also be defined in terms of the neutron speed or lethargy. As in the case of the neutron cross sections [cf. Eq. (4.59)],... 

This quantity may also be expressed in terms of the slowing-down density g(u). According to (4.47), du/ is the average number of scatterings made by a neutron in traversing du. But, q u) neutrons per unit volume per unit time slow past lethargy u therefore. 

It is of interest to compare the expression (4.65) with the corresponding relation given in terms of the neutron energy. The symbol f denotes the average gain in lethargy per collision its counterpart in energy space is (1 — a)E/2 [see Eq. (4.40)]. Thus if dE corresponds to du, then... 

It can be demonstrated by means of a crude probability argument that these results (4.99) and (4.100) are indeed the nonabsorption probabilities defined above. For convenience, we will carry out the calculation in terms of the lethargy variable. Suppose that the interval 0 to w (which corresponds to the interval Eo to E) is divided into N subintervals Alii which are sufficiently small so that the cross sections may be given one particular value [2a(ui),2,(w<),. . . ] over each subinterval (see Fig. 4.17). Then let pi denote the probability that a neutron will pass through Aui without being absorbed. But, each must be an independent probability therefore,... 

The term at the left gives the rate at which neutrons are removed from the unit lethargy interval about u by both scatterings and absorptions. The terms at the right give the rate at which neutrons are added to this interval by scattering in from above. 

Equation (4.118) is a statement of the physical fact that each neutron which slows down past lethargy u has made its last scattering collision with some particular nuclear type at some lethargy u < u. The prob-... 

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