[DOUGLAS] /3-RAYS FROM RADIUM 119 



completely explains the apparent difference in range between the 

 primary and scattered rays, and if so, it may be said that to a first 

 approximation there is no loss of energy due to scattering. 



(3) This conclusion is confirmed by the following theoretical 

 considerations: 



In the Phil. Mag., Vol. 2^, 1914, p. 499, C. G. Darwin gives the 

 calculations regarding the collisions of a-particles with light atoms. 

 In the Phil. Mag., Vol. 21, 1911, p. 684, Sir E. Rutherford states that 

 collisions with light atoms by a and by jS-particles obey the same 

 general laws; the main difference being that the probability of a 

 large deflection is much greater in the case of the (8-particle due to its 

 mass and its momentum being so much less than the mass and momen- 

 tum of the a-particle. 



It seems reasonable, then, to employ Darwin's method of ap- 

 proach, extending his reasoning to the problem of energy loss. 



Consider the deflection of a |ô-particle of mass M and velocity V 

 due to collision with the nucleus of an atom of mass m at rest. Let 

 be the deflection of the /3-particle and v its resultant velocity; and 

 let the atom be set in motion in a direction 6 with a velocity u. 



The equations of motion are: 



M V= Mv COS0 -\-m u cos d 

 O = Mv sin (j) — mil sin f 

 MV' = Mv~ + in it^ 

 V 



and hence v= t^~; — (If cos <6 d= Vm"— M'sin^ci) 



M-\-ni 



The energy of the /5-p'article before collision was 3^ MV^. Its 

 energy after collission is 



^Mv'' = y2M (j^J—^ (M cos± Vm~-AP sin'à) Y 

 Hence the loss in energy is given by: 



^MV^ n-777^ — To (Mcoscjy ± Vw^-M^ sin^^)^ 



In the particular case of scattering through an angle of 180°, this loss 



(/-M±m\ -2 \ 

 1— ( ■ — j j 



The lower sign gives zero, while the upper sign gives: 



1 



In the case of jS-particles scattered by hydrogen M= ^ ■ m = 



