248 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



IVB. LONGITUDINAL DIFFUSION FOR THE HARD SPHERE MODEL 



Whenever the mean free time condition for collisions is not fulfilled, 

 then the computation of diffusion coefficients requires a procedure analo- 

 gous to that of Section HE. Since this entails some numerical work the 

 calculation was only carried out for a case which was thought to be of 

 experimental interest, namely for longitudinal diffusion of ions in the 

 parent gas. In other words, we are extending the numerical computation 

 at the end of Section HE to include longitudinal diffusion. The computa- 

 tion to provide us with the undetermined constant of equation (28b) 

 for the special case when m and M are equal ; it also offers, incidentally, 

 a good test case for applying the method of Section HE outside the 

 area for which it was designed originally. 



Since the equation is only to be solved in the high field case we may 

 apply to (20) the reduction method of Section HA. If we introduce also 

 the specialization warranted by the hard sphere model and unit mass 

 ratio then, in analogy to equation (40), we get the following starting 

 equation 



dg{w) , , . 1 f" w'^ sin x dx 



, , V 1 r wsmxdx f f f^ . 



dWz 47r Jo w^ Jo (153) 



= —\k{wz — {wz)}h(w) 



Here the dimensionless variable w defined by (85) has been employed 

 instead of c. 



Equation (153) is the fundamental equation of our problem; it is an 

 inhomogeneous version of equation (40). We solve the equation in the 

 same way as we did previously, namely by decomposing ^(w) into spheri- 

 cal harmonics and forming moments. In other words we follow step by 

 step the procedure of Section HA, the only difference being the presence 

 of an inhomogeneous term. We shall not enumerate all these steps again. 

 We shall only note in passing the inhomogeneous form of (47) which is 



74 



2 Jo 



— g,{w')Py (cos k) sin xdx - wg^w) 

 dgy-i 



2v — 1 \ dw w 



g^iiw) > 



w J 



V -f 1 jdg^i ,y-{-2 . S\ 



