PHENOMENA OF PHOSPHORESCENCE. 215 



While positive ions will occasionally break loose from the groups to 

 which they have attached themselves and return to the "free" condition, 

 this will occur only rarely on account of the sluggishness with which these 

 ions move. The small positive term due to this cause which should appear 

 in the right hand member of equation (3) is therefore negligible and may be 

 omitted. At high temperatures, however, and perhaps under other special 

 conditions, the omission of this term will no longer be permissible. If the 

 molecular movements are sufficiently violent, due to high temperature or 

 other causes, the formation of groups may even be prevented altogether. 

 Such cases will be considered later. 



Equation (3) may be written 



d dn d<p . ,. 



-j\ipn) = ip jT+n = a<pir-{ki+k2)<pn 

 at dt at 



Remembering that dn/dt = an' 1 this becomes 



=-(*, + ,), <p = <p Q e- mt 



where m = k\-\-k% and (po is the value of (p for / = o. 



In the case of the positive ions that are attached to neutral molecules of 

 the active substance recombinations are occurring with negative ions at 

 the rate a^;r, while new groups are being formed by the attachment of 

 free positive ions at the rate ki<pn 



.'. -(4>n)=ail/n 2 +ki(pn 

 at 



or 



dn d\j/ 

 (5) 4 (U + =-aif,n-+ki<pn 



Since 



dn 



, = an 1 

 dt 



we have 



d =k^ = k^e-' /.*=- *V-+ const. 



Putting \po for the value of \p when / = o 



(6) * = *o+^(i-e- w 



m 



For the intensities of the three constituents of the phosphorescent light 

 we have therefore 



.mi 



I l = p x aipn 1 = 



pia<poe~ 

 (1 ;/,,+a/) 2 



M*+JiW(i-""0] 



h = p3a(itpip)n z = 



(i/no+at) 2 

 pza\i ^0 knpa/m kjipghn c~ mt )} 



(j/tto+at) 



2 



