PRINCIPLES OF RADIOLOGICAL PHYSICS 43 



The following formula summarizes these results. Consider an incident particle 

 having a mass m, a velocity v, and an electric charge ei which traverses a thin 

 layer of matter of thickness t containing A^ scatterer particles of charge 62 per 

 unit volume. Assume that the scatterer particles are fixed in space or that they 

 have an exceedingly large mass so that they do not recoil at all. The probability 

 for the particle to be deflected by an angle between i} and ?? + 5?? is given by 



e\el 27r sin ??5i? 

 ^^'^^^^ = '^'' -(^^^ sin^ (^/2) 



= ^'^ ri; . !n'/^xn. 27r sin t?5?? (13) 



[2mv sm (t^/2)]^ ^ ^ 



The expression 2'mv sin (t?/2) represents the momentum change, according to 

 the diagram in Fig. 1-33. 



In practice the scatterer particles recoil as a result of the collision. When 

 this is considered, Rutherford's analysis still 



applies to the motion of the particles with / 



respect to each other rather than with re- / RECOIL 



\ . fi , r f f rr, /MOMENTUM 



spect to a fixed irame of reference. Ihe OF scatterer 



, r M r XI XX X- , , INITIAL MOMENTUM 



speed of recoil oi the scatterer particle and -^— *-j momentum 



the energy which it acquires depend on the '^'AM^^"^--.,,^^^ /--'^'change OF 



ratio of the masses of the two particles. ^'^^^^^vT-^r parVicle^ 



If the incident particle is much lighter -^ , „o t^- ^ .1 



, ,, ,, , , , 1<IG. 1-33. Diagram of the mo- 



than the scatterer, as, for example, when an ^^^^^^ ^^^^^^^ .^ Rutherford 



electron is scattered by a nucleus or when scattering 



an a particle is scattered by a heavy 



nucleus, the scatterer recoils hardly at all and takes up hardly any energy. 



If the incident particle is much heavier than the scatterer, as, for example, 

 when a proton hits an electron, the scatterer offers little resistance to the inci- 

 dent particle and is unable to deflect it greatly or to take up much of its energy. 

 The transfer of some energy to the scatterer is comparatively likely, but the 

 amount of energy transferred cannot exceed a rather low limit. 



If the two particles have comparable masses, the speed and energy of recoil 

 may be comparable to the speed and energy of the incident particle. 



The probability that the scatterer will receive a recoil energy comprised 

 between Q and Q + 5Q is given by a rather simple formula, in which the sym- 

 bols are the same as in Eq. (13) except that m indicates the mass of the scatterer 

 instead of the mass of the incident particle: 



27re?e'2 8Q 

 P(Q) dQ = m ^ ~ (14) 



Notice that this probability depends on the velocity v but not on the mass of the 

 incident particle. 



These results are modified when the speed of approach of the particles is com- 

 parable to the speed of light and also when the two particles are identical. 



2-2d. Magnetic Effects. Electrically charged particles in motion generate 

 magnetic forces and are themselves subject to magnetic forces produced by other 



