Sec. 5.3] NEUTRONS 125 



been established [4]. For isotropic scattering, it may be expressed as the 

 sum of two distinct processes. The first arises from potential scattering for 

 which the cross section is identical to that obtained for the collision of two 

 hard spheres and is just equal to the total effective surface area of the nucleus. 

 The second or resonance term becomes important when the energy of the 

 incident neutron is nearly equal to that of a quantum state of the nucleus. 

 Resonance scattering may then make a large contribution to the cross 

 section. Breit and Wigner [3] have shown that when the De Broglie wave- 

 length of the neutron is large compared to the nuclear radius and the levels 

 are widely spaced so that only one level is affected, the cross section near 

 resonance can be represented by an expression similar to that for the dis- 

 persion of light. The total cross section, therefore, may be written as the 

 sum of potential scattering and the Breit- Wigner "one-level" formula [3,4]. 



= ^ + l( 1± 2iTl)^ 



4R(E - E r ) + X r T n 



— p— cm- 



(E - E r y + Z- 



where R = effective nuclear radius 

 E = neutron energy 

 E r = resonance energy 



X r = neutron wavelength at resonance ( = h 2 / Sir 2 /jlE)^ 

 n = reduced mass = mM /(m + M) 

 Y n = neutron width of level 

 T = total level width 



i = angular momentum of nucleus before collision, + sign if spin of 

 resonance level is i + } £; — sign if i — 1 •_>• If i = (even-even 

 nuclei) + sign is used 

 The shape of the curve for the second term near resonance is similar to the dis- 

 persion curve for light. A maximum occurs near exact resonance where 

 o-max = 47r7t 2 + lirXlTn/T, and a minimum value is found at 



E r — -Emin = h r T n /2R 



where cr mi n = 2tR 2 . 



The relative magnitude of resonance and potential scattering is made 

 apparent by the ratio of their cross sections at exact resonance 



<r p 2\ 



XrTnV 



RT J 



For most nuclei the neutron width T n is a small fraction (~ 10 -3 ) of the 

 total width T, and since K ~ R, the quantity KTjRr « 1. Hence, in 

 most cases resonance scattering is unimportant compared with potential 

 scattering. There are however a few elements such as silver for which 



