MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 239 



angle of deflection x due to a potential of the type (123). The result is 



du 



TT - 2 / 



/I _ e'PjM + m) 4V^' (124) 



Here b is the ''impact parameter", and Ui is the lower of the two positive 

 roots of the polynomial in the denominator; if the polynomial has no 

 real root, the integration goes from to oo . The question whether the 

 denominator has a real root or not is tied up with the nature of the orbit. 

 If b is sufficiently large a root exists and the orbit looks like a hyperbola, 

 Fig. 2(a); for small b no root exists and the two particles are ''sucked" 

 towards each other in a spiralling orbit as shown in Fig. 2(b). The two 

 regimes are separated by a limiting orbit in which the particles spiral 

 asymptotically into a circular orbit. This limiting orbit is found by 

 setting the discriminant of the square root in (124) equal to 0. We find 



From this value of bum a cross section and a mean free time r« 

 for spiralling collisions can be derived. We find 



"' = 2^^ t(K+1)p} ^^^^^ 



This is indeed a constant mean free time as stated, the speed of encounter 

 7 having dropped out. 



1/r is the dimensional quantity entering into the averages {<p(x)/t) 

 which occur in the Sections IIB and IIIA. In working them out in detail 

 as was done by Hasse^ one has to take into account hyperbolic collisions 

 also; for them a r cannot be defined or comes out to be zero in the mean. 

 This is due to small angle deflections which are infinitely probable. 

 However, any quantity (^(x)/r to be averaged in a physical problem con- 

 tains a <p(x) which vanishes for such impacts. Hence finite averages 

 result which do not give overdue weight to these types of collisions. 

 Following Hasse^, we do this in the following way for the present case. 

 We write (124) in the form 



r^i dv 

 X = TT - 2 / 7 -Tyj2 (127) 



dv _ 



4/3^J 



