Recombination of Ions in Gases, 243 



will enter such a sphere in time dt is independent of the 

 radius of tii8 sphere and of the external field, and is given by 



Aarfa + k^Vedt i 



where k h k 2 are the velocities of the two ions under unit field, 

 P is the number of positive ions per c.c, and e the charge on 

 an ion. 



If we call each case of an ion crossing one of these surfaces 

 a " collision," and suppose that each collision is equivalent 

 to recombination, we get for the law of recombination 



dp tin . ,, , N 



where p, n are the densities of positive and negative electri- 

 fication respectively. Since recombination does not necessarily 

 result every time an ion comes within the sphere of action of 

 another ion, the above expression only gives an upper limit 

 to its amount. Langevin puts 



where e is a function of the pressure whose possible values 

 range from zero to unity. The coefficient of recombination 

 a is thus equal to 47r(A 1 + &2) 6 - 



We shall now proceed to calculate a value of e. To do 

 this it will be necessary to consider what happens to an ion 

 when it enters the region in the immediate neighbourhood of 

 an ion of opposite sign where the electric force is very intense. 

 If we describe round each negative ion a concentric sphere 

 whose radius is 4x 10~ 6 cms., then by the time that a positive 

 ion reaches one of these spheres the work done on it by the 

 attracting forces will be about one two-hundredth of the 

 total work done during recombination. It is evident that 

 inside this region the ion will be greatly accelerated and its 

 behaviour will be quite different from what it was outside. 

 To fix our ideas, we may for the moment regard the negative 

 ion as fixed and the positive as projected through the spherical 

 boundary. The positive ion will then describe an orbit 

 round the negative and in general will return to the boundary 

 at some other point. If this is the case it will have returned 

 to the region where the kinetic energy, which it gains in 

 virtue of the work done by the attractive forces, during its 

 passage through a distance equal to the mean free path, is 

 not great compared with the mean heat energy ; in fact it 



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