Flux slowing-down density relation

Neutrons of energy E are produced uniformly throughout an infinite medium. Subsequently, they slow down by elastic collisions until they reach Et i and there enter a one-velocity diffusion process. The medium is characterized by the macroscopic cross sections S, Za, S/, and S, which are all inversely proportional to the neutron speed at all speeds. Find the fraction of fissions caused by thermal neutrons. Assume the following slowing-down density and flux relation q E) = V t E) E)E. 

The presence of 2 instead of S, in (6.47) results from the particular choice of the relationship between the slowing-down density and the flux made in (6.6). It should be remembered that this relationship is justifiable only in a medium of light absorption, that is, one in which 2 2, [see (4.112) and related discussion]. Thus in most cases in which the present theory is expected to be useful, the two choices of r give nearly the same result however, for consistency, the definition (6.47) will be used here and in the remainder of this work. 

In this case the solution of Eq. (6.57) for the slowing-down density yields a source term for the thermal-diffusion equation (6.54c) which is proportional to the thermal flux. In general, the source term g(r,t th) is obtained from (6.72a). The factor Q(0) F(r) which appears in this result is given by (6.74) but, when 6=1 (i.e., no fast fissions) this expression reduces to the form F(r) Q(0) = 0th(r). If we use this relation in (6.72a),... 

The last relation follows by virtue of the fact that the slowing-down density is proportional to the fast flux through the coupling equation... 

The result (10.255) may also be used to obtain the criticality condition for the reactor with finite fuel lattice but with infinite reflector (composed of the same moderator material). It is necessary to impose the latter condition because otherwise expression (10.243) for the slowing-down density would no longer be valid and new functions satisfying boundary conditions at the surface of the reflector would be required. The appropriate neutron-balance relation in terms of the flux at the various fuel-rod locations is then given by... 

The slowing-down density given here is related to the flux (r,u) in the usual way, viz.,... 

This relation between the specific rate of photosynthesis and photon flux density is shown in Fig. 12. The term photosynthesis thus reflects only sugar production in the chloroplast and not biomass growth as a whole which will be covered hereafter. The specific rate of photosynthesis first increases rapidly with increasing photon flux density. At higher photon flux density the increase slows down and eventually the rate of photosynthesis reaches a... 

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