314 HELL SYSTEM TECHNICAL JOURNAL 



this last quantity the expression prescribed h\ tlie spetiai relativity- 

 theorv '- is usefi, \iz. 



.( 1 

 V-v/1- 



in which m stands for the mass of the electron and cfi = v for its speed. 

 The equation of conservation of energy is then 



liv = lii'' + mc-( ,i-, -l). (8a) 



Conservation of momentum retiuires iliat ilu' numu-niuni of the 

 impinging quantum t)e ccjual to the suin of the momenta of the result- 

 ing quantum and the recoiHng electron. Momentum being a vector 

 quantity, this rule requires three scalar equations to express it, which 

 three may be reduced to two if we choose the .T-a.\is to coincide with 

 the direction in which the impinging quantum travels, and the y-axis 

 to lie in the plane common to the paths of the recoiling electron and 

 the resulting quantum. Designate b>' the angle between the paths 

 of the impinging quantum and the recoiling electron; by 6 the angle 

 between the paths of the two quanta. The magnitude of the momen- 

 tum-vector is, by the special relativity-theory, »/t'/\/l — /3^- Con- 

 ser\ation <>f nKiniciilum then recjuires: 



niv 



llV C = (llv' f ) cos 9H ; ^r : COS A, 



(8b) 

 ()= (hv ■ c) sin 0-\ , sin <^. 



Kliminating <t> and v between these three ecjuations, we arrive at 

 this relation between v and v' , the frequencies of the impinging quan- 

 tum and the recoiling quantum — or, as I shall hereafter say, between 

 the frequencies of the primary- X-ray and the scattered X-ray -and 

 the angle between the dircitinns (if liic piimar\' X-ra\' and the 

 scattered X-ra>- : 



. (9) 



" l-|--'^,(l-cos9) 

 mc- 



" If the reader prefers to use the familiar expressions \mv' for the kinetic energy 

 and »n< for the magnitude of the momentum of the electron, he will arrive at a 

 formula for v' which, while apparently dissimilar to (9) and not so elegant, is ap- 

 proximately identical with it when v is not too large — or, which comes practically 

 to the same thing, when hv is small in comparison with mc'; a condition which is 

 realized for all X-rays now being profluced. 



