206 STUDIES IN LUMINESCENCE. 



Assuming that ions are produced at a rate proportional to E and that 

 excitation has proceeded for a sufficient time to produce a steady condi- 

 tion, the number of ions per cubic centimeter at any depth x will be deter- 

 mined by the equation 



^ =o = //-ao 2 = //o^ v -a 3 o= J~ e" w * 



rfJ \ a 



If w is the number of ions per cubic centimeter at the time / we have 



i 



n = 



i/no+at 



and the light emitted per unit volume, denoted by i, is 



. . 2 pa- 



i = paw = - - 



(i;>io+at)- 



p being the light emitted as the result of one recombination. 



Since the emitted light suffers absorption, the amount contributed to the 

 total observed intensity by a layer of thickness dx will be 



idxe yx = -r~. r- 

 (i/no+aty 



and the total intensity is 



e~ yx dx C* e~ yx e~ ffx dx 



.co e ~ y *dx /'" e r *e "*dx 



where 



Putting 

 so that 



a 



a = \ - 



a/e~^ /2 = 3 



apt -^ /\* 



d z = e 2 dx and e yx 



()* 



_ 2pa f z m+l dz 

 ~ p(at) m+2t ' at {a+zY 



where w = 2/7/3. 



Successive partial integration gives 



2 S m+:? 



2pa f i z m+i 

 I== ~ p(at) m+2 L^+^ (a+z) 2 + w + 2-m+3 (a+z) 3 



W+2-W+3- w+4(a+s) 4 "j,, 



r m-M -i0 



Upon putting in the limits this becomes 

 2 pa 



1 = 



{m+2 



)/3'(a+a/) 2 L I ^w+3'a + af w+ 3 - m+ 4 \a+at) J 



