Sec. 5.6] NEUTRONS 131 



forms of moderators. If the source (or the fast neutron beam from a cyclo- 

 tron) is enclosed with materials such as water or paraffin for which the ratio 

 of the scattering to capture cross section is large, the fast neutrons are rapidly 

 reduced to thermal energies by elastic collisions. 



Once neutrons have been reduced to thermal velocities they continue for 

 the remainder of their lives to diffuse through the medium. On the average 

 they gain in subsequent collisions as much energy as they lose, and their 

 average velocity depends only on the temperature of the medium. Ulti- 

 mately each neutron is lost by capture, by decay into a proton and an elec- 

 tron, or by diffusion out of the medium. When equilibrium is reached, the 

 rate of formation of thermal neutrons just equals the rate of loss by all three 

 processes, and the neutron density throughout the medium remains constant 

 with time although not in space. If the diffusing medium is a hydrogenous 

 substance such as paraffin, most of the scattering nuclei have the same mass 

 number as the neutron and diffusion resembles in some respects the self- 

 diffusion of gases. Substances containing heavy nuclei, on the other hand, 

 allow very small momentum transfer per collision, and the neutrons scatter 

 elastically as from a solid, immovable object. The conditions here are more 

 like the diffusion of an electron gas in a conductor. 



In the stationary state, characterized by equal rates of thermal neutron 

 loss and gain, the energy distribution of neutrons is Maxwellian except for a 

 high-energy tail which varies as E~ ? '-. For the Maxwellian region the num- 

 ber of neutrons with energies lying in the interval E to E -f- dE is 



N dE = WW^'" dE 



where k = Boltzman constant 



T = absolute temperature 



Q = number of neutrons produced per sec 



t = mean life time for capture 

 The factor Qr is just the total number of neutrons with all energies. The 

 average velocity is v = (SkT/tmr) 1 ^, where m is the mass of the neutron. At 

 room temperature, 20°C, v is about 2.5 km per sec. 



Assuming that all neutrons within a diffusing medium possess thermal 

 velocities, or more exactly, have a Maxwellian distribution for a given 

 temperature, the neutron density in neutrons per cubic centimeter may then 

 be described by a diffusion equation in much the same way as for gases. It 

 is essentially an expression of the law of conservation of neutrons stating 

 that the rate of change in the number of neutrons per cubic centimeter in an 

 element of volume dV at the point (x,y,z) and time / equals the number of 

 neutrons produced per second plus the divergence of the neutron flux through 

 dV minus the number absorbed per cubic centimeter per second. 



