244 Wheelock — Nature of lojiization Produced hy a Rays. 



III. A I'heoretical Expression for the Saturation Curve. 



A theoretical expression for the saturation curve may be 

 obtained as follows, on the hypothesis of the ions beinj; formed 

 in columns along the path of the a particle and a uniform dis- 

 tribution of ions in the columns. 



Let N be the number of positive ions 

 s^enerated by one a particle 

 in the chamber. 

 • n^ be the number of positive ions per c.c. 

 a be radius of a column. 

 b be depth of ionization chamber. 



The time for an ion to go from any position, x, to the top 

 plate is given by i 



^ _ ^ ^ where Jc = mobility of the ion 

 k X and X ^ electric force. 



Limiting time for any ion to have a chance to recombine is 



, b since half of the ions are 



k X going each way. 



Limiting time during which an ion at x can recombine is 



1/2^^ 

 k X 



Since the a particle passes through the chamber almost 

 instantaneously, the ionization is over immediately 



and 



(In 



— a??" 



dt 

 1 

 n 



n. 



whence 



Therefore, for an ion which starts at x, the final density nl, 

 that is the number of ions per c.c. which will reach the plate, 

 will be given by : 



1 _ 1 2_ b — x 



n^ n^ 2 k% 



The charge per column, reaching the plate, will be : 



eiro' I n^ dx 



= Tra'e / 



' _ ^^^»^^ dx 



{2hX + an^b) — an^x 



