122 JSOTOPIC TRACERS AND NUCLEAR RADIATIONS [Chap. 5 



tions, when equilibrium is established between the rates of formation and loss 

 of thermal neutrons, the properties of the neutron gas may be described in 

 terms of the spatial density and distribution of kinetic energies, together with 

 associated quantities such as neutron flux, mean free path, and diffusion 

 length. 



Associated with each neutron as with all particles is a De Broglie wave for 

 which the wavelength in angstrom units is X = h/mv = 0.2S6E~ 1/2 where E is 

 the kinetic energy in electron volts. Thermal neutrons have wavelengths 

 comparable to interatomic distances, and because of the absence of charge 

 they are diffracted and reflected by crystals in much the same way as x-rays 

 of the same wavelength. 



5.2. Neutron Processes. In accordance with Heisenberg's uncertainty 

 principle the probability per unit time that an excited nucleus, formed by 

 capture of a neutron, undergoes a transition to a lower excited state 

 is related to the uncertainty in the energy, AE, of the initial quantum 

 level of the compound nucleus. This is expressed as the probability of a 

 transition per unit time or l/At = AE/h, where h = Planck's constant/27r 

 and At is the uncertainty in time which is taken as the mean life of the state. 

 Each quantum state has a most probable energy, its exact resonance energy, 

 but because of the uncertainty principle the energy of the corresponding 

 level in individual nuclei of the same species falls on a distribution curve 

 forming a peak for which the maximum is the most probable value of the 

 level energy. The uncertainty in energy of a particular level, designated by 

 T and expressed in electron volts, is taken as the width of the associated 

 resonance peak at one-half its maximum value. It is referred to as the total 

 width of the level. When the width of a level is small, the mean life of the 

 excited state is large. 



The lowest excited levels of nuclei are, in general, widely spaced, usually 

 at intervals of the order of several kev. The resonance peaks representing 

 the energies of the levels consequently are well-defined and widely spaced 

 compared with their widths. At higher excitation energies, however, the 

 density of levels increases until, for energies between 5 to 8 mev, the spacing 

 between levels is reduced to only 10 to 20 ev. Discrete resonances then no 

 longer exist; the levels are broad and tend to overlap because of their small 

 separation. Consequently, when nuclei are raised to excited states of high 

 energy, for example by capture of fast neutrons, many levels may be affected 

 which then contribute to the nuclear processes by which the nucleus returns 

 to lower lying levels or to the ground state. 



In general the total width T is the sum of the partial widths of various proc- 

 esses that may occur in the transition. This includes the emission of gamma 

 rays, neutrons, charged particles and, in the special case of elements beyond 

 actinium, also fission. The total width therefore is V = T y + r„ + T g + T/. 





