Sec. 5.6] NEUTRONS 133 



The source strength q, the number of thermal neutrons formed per cubic 

 centimeter per unit time, is in general a function of the coordinates x,y,z. 

 Most sources provide only fast neutrons which require 20 to 30 collisions to be 

 reduced to thermal velocities. Hence, for a point source of fast neutrons the 

 virtual source of slow neutrons is some function of the radial distance. In 

 many instances, however, slow neutrons are obtained either from a moderator, 

 used to slow down neutrons from a cyclotron, or as a collimated beam from a 

 nuclear reactor. The source can then often be regarded as a plane with a 

 strength of n v, where n is the number of neutrons per cubic centimeter of 

 beam and v is the average velocity. 



The mean life r of a neutron with respect to capture by a nucleus is related 

 to the absorption cross section by the expression r = l/N<r a v. 



The quantity tD in the diffusion equation is referred to as the diffusion 

 length L. It often makes its appearance in solutions of the equation in terms 

 of the form e~ x/L , where x is the distance from the source. If scattering is 

 isotropic and if a a « a s , the diffusion length is 



T 1 



If scattering is not isotropic and a a is not small compared with a s , a more 

 exact expression for L is 



1 



N[\ - (2a a /5a s )](3a a( r s y- Cm 



Examples of solutions of the diffusion equation may be illustrated by two 

 different and simple diffusion geometries which are of practical interest. 



The first is that of a point source of neutrons placed at the center of a 

 sphere of scattering material of radius R. The density of neutrons at any 

 radial distance r from the center is 



n = 



10 e K/L . . R- r 



sinh - 



2w\vr (e 2R/L + 1) L 



where Q is the source strength in neutrons produced per unit time and v is 

 the average velocity. 



The second geometry considered here is a semi-infinite medium with a 

 plane boundary. It is assumed that the source is also a plane with a strength 

 of Q neutrons per square centimeter per second and placed deep within the 

 medium. It is apparent that this geometry may be used to obtain a rough 

 approximation of the distribution in a thick slab of material into which 

 a beam of slow neutrons is directed. From the plane side opposite that 

 which the beam enters the neutron density is a linearly increasing function of 



