Slowing-down density with absorption

The collision density in an absorbing medium F u) differs from the expression (4.89) by the addition of a term Sa(i ) 0(u) to the left-hand side. Thus, 

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. 

The substitution of this expression into the integral of (4.1046) yields, upon integration, 

This equation may be solved by using the relation (4.106) for q u). The result is 

0 both and y approach unity, and this approximate expression for the slowing-down density approaches the result for pure hydrogen [see Eq. (4.98)]. 

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Here, r = ln(JB/ thermai) has a somewhat different definition from the usual one it is zero for neutrons, the energy E of which is thermal, (t t) is the transport cross section which may depend on the energy but does not depend on the position thermal neutrons the average logarithmic energy loss (independent of position) <7, the absorption cross section for thermal neutrons which depends on the position. q(r) is the density of fast neutrons per unit r (it is not Fermi s slowing down density Q), multiplied with the velocity, n the density of thermal neutrons times their velocity, /(r) dr is the number of fission neutrons per slow neutron captured in U, for which r is between r and r + dr. Finally pi is the chance of escaping resonance absorption and p2 the thermal utilization. The multiplication constant here... 

At this point it is interesting to note that, if q(E) is a solution of Eq. (1) for the case of no absorption, then the corresponding slowing-down density q (E) for the case with weak absorption is merely q(E) times the resonance escape probability. 

Let us consider the data on the dependence of the kinetics of et decay at 77 K on the radiation dose. As seen from Fig. 11, over the dose range 3 x 1019 - 3.6 x 102° eV cm 3, the kinetics of et decay is virtually independent of the dose. At the same time, at lower doses, the decay of et is significantly slowed down. For example, for a dose of 1019 eV cm-3, the change in optical density of y-irradiated samples at the maximum (585 nm) of the et absorption spectrum with time is also described by eqn. (5), but the slope of the kinetic curve the coefficient M in eqn. (5)] is smaller by almost a factor of two [28] than for the curve of Fig. 11. Further investigations by pulse radiolysis technique with spectrophotometric recording of et showed that, at a still lower dose (6 x 1017 eV cm"3) no decay of et in water-alkaline matrices is observed at all [43] while at high doses (5 x 1021 eV cm"3) for the same samples, the decay of efr does occur [43]. A decrease in the rate of etr decay via the reaction with O at small doses was also reported in ref. 44. This behaviour of the kinetic curves seems to reflect special features of the spatial distribution of etr and 0 particles in samples irradiated with different doses. 

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