132 ISOTOPIC TRACERS AND NUCLEAR RADIATIONS [Chap. 5 



— = div (D grad n) + q 



dt t 



where D = diffusion constant 



q = source strength; number of thermal neutrons produced per unit 



time as a function of position 

 t = mean life of neutron before capture 

 In the steady state dn/dt = 0; the density distribution does not change 

 with time. Furthermore, if the medium is homogeneous and isotropic, the 

 diffusion equation then reduces to 



fl 



Dy 2 n + q - - = 



r 



or 



The diffusion constant D is found from ordinary kinetic theory to be given 

 by D = Xt r f/3, where X tr is the transport mean free path and v is the neutron 

 velocity. This relation is found to hold for neutron diffusion provided that 

 the scattering is isotropic, scattering and absorption cross sections are con- 

 stant, and the rate of change of n is small within a distance of one X tr . The 

 quantity X tr is the mean distance a neutron travels through the medium in 

 the direction of its initial motion after a great number of collisions. This is 

 related to the transport cross section by X tr = 1/Natr, where N is the number 

 of nuclei per cubic centimeter. The quantity o- tr is, in turn, defined by 

 o\ r = as (I — cos 6), where a s is the scattering cross section and cos 6 is the 

 average value of the cosine of the angle of deflection from the initial direction 

 in each collision. If the scattering is strictly isotropic, i.e., all angles are 

 equally probable, then cos = 0, cr tr = as and X tr = X, where X is the mean 

 free path or X = l/Na. The value of cos 6 for anisotropic scattering depends 

 on both the mechanism of scattering and the mass number of the scattering 

 nuclei. The first effect must be determined empirically, but the latter 

 effect leads to a u = a s (l — 2/3M), where M is the mass number of the 

 scattering nucleus. In the case of hydrogen, for example, scattering is pre- 

 dominantly forward since a neutron cannot be scattered more than 90 deg in 

 a single collision. The effect in hydrogen is large, whereas in heavy elements 

 it is entirely negligible. 



In some media the absorption cross section <r a is comparable to the scatter- 

 ing cross section and cannot be neglected. In such instances a more accurate 

 expression for the diffusion constant, taking a a into account, is 



D = 



3Na tT [l - (%)(*„/,„)] 



