MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 253 



do contain the mean free time yield to judicious treatment. For example, 

 we have the hard sphere formula (100) for the drift velocity of an ion. 

 This formula happens to be limited to the high field case and mass ratio 

 unity. On the other hand we have formula (136) which holds for all fields 

 and all mass ratios, but assumes constant mean free time. We now adopt 

 this formula as a general guess for the hard sphere model, interpreting 

 r as previously as the mean free time between collisions; this quantity 

 is now no longer a constant, but should be taken as 



" = VWTW) ^^^^^ 



The denominator is the root mean square relative velocity which is 

 familiar from other applications. The interpretation (166) yields a tenta- 

 tive formula for the drift for all mass ratios and for all fields. Specializing 

 to the high field case, we may neglect {C ) in (166) and then substitute 

 for (c) from (55). This yields the high field formula 



This is indeed a very successful formula. For ions in the parent gas it 

 differs from (100) by only 4 per cent. For electrons it checks the result 

 of Dru3rvesteyn^ to within 12 per cent. Finally, for heavy ions in a light 

 gas, we find exact agreement with equation (71). As a second specializa- 

 tion we msiy apply (166) to the low field case. We must then set 



and get from (136) and (166) 



1/1 IV'' eE\ ,_„, 



All dimensional factors in this formula are correct. Numerically (168) 

 is somewhat inferior to (167) ; for the factor differs from the correct one 

 by 20 per cent. Nevertheless, the combination of (136) and (166) gives 

 results which are semiquantitatively correct in all relevant limiting cases. 

 This makes it a reliable interpolation formula for intermediate field con- 

 ditions; for this case (c) would have to be substituted from (122) and 

 the resultant quadratic equation solved for (cz). 



From the examples given we may conclude that the mean free time 

 formulas contain in essence information applicable to other types of 

 elastic scattering. 



3^ See Reference 2, page 40, second equation. 



