214 STUDIES IN LUMINESCENCE. 



modes of recombination will probably differ in violence, in frequency, and 

 in radiating power. 



Of the total number ;/ of positive ions at any time / let <pn be free, and 

 \pn attached to neutral molecules of the active substance ; then (i <p \p) n 

 will be attached to molecules of the solvent. <p and \p are proper fractions 

 whose sum is less than unity, and which, in general, are functions of I; 

 for the distribution of the positive ions will be subject to alteration during 

 the decay of phosphorescence as well as during excitation. The number 

 of recombinations between a free positive ion and an electron in unit time 

 will be proportional to the number of free positive ions <pn, and to the 

 number of free negative ions //, and may therefore be put equal to anpu 2 , 

 where ai is the coefficient of recombination. If the energy radiated as 

 light due to one recombination is pi the intensity of the phosphorescence 

 due to this mode of recombination will be 



1 1 piOLnpll 2 



Similarly 



1 2 = p-id-2^ n- I?, = p 3 a 3 {\ <p ip) n? 



for the other two types of recombination. 



The total phosphorescent light will thus consist of three parts. If the 

 different modes of recombination give rise to vibrations of different fre- 

 quency the phosphorescence spectrum will consist of three bands which 

 decay at different rates, and which may differ widely in intensity. 



The determination of the law of decay of phosphorescence upon the basis 

 of the theory just outlined is thus seen to involve the determination of 

 n, ip, and \p as functions of /. In the general case the solution presents 

 difficulties that are wellnigh insurmountable. The problem may be sim- 

 plified, however, by an assumption which, while doubtless not exactly 

 true, probably gives in the majority of cases a close approximation to the 

 actual conditions. It is assumed, namely, that ai = a 2 = a.- ! ; in other words, 

 the probability that a collision will result in recombination is the same for 

 the three types of collision. In this case 



{i) = a<pu 2 a^ir ad ^ \^)n 2 = an 2 



at 



(2) n 



i/wo+a/ 



where Vo is the number of positive (or negative) ions when excitation ceases. 

 To determine <p and \p as functions of / it is necessary to make some 

 assumption regarding the rate at which the free positive ions attach them- 

 selves to neutral molecules. Since the number of neutral molecules is 

 presumably large as compared with the number of ions, it is reasonable 

 to assume that the rate at which new groups are formed is proportional 

 to the number of ions that are still free to enter into such combination. 

 In other words, we may assume knpn and k<npn as the rates of formation of 

 new groups of the two possible types. Since the positive ions, whether 

 free or attached, are also recombining with the negative ions we have 



(3) - <>") = -a-ipn 2 - (ki-\-k 2 )<pn 



at 



