650 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1953 



at the present time the experunental situation does not call for refine- 

 ments of this kind. Therefore we shall restrict ourselves to the calcula- 

 tion of r. 



To evaluate r it is not necessary to consider a time-dependent case 

 at all. In our experiment, it is the mean square conductivity fluctuation 

 which is actually observed. Hence from (2) 



<^g'> = <J^(0> r\ 



If the emission processes are stationary in time, <J^{t)> is time in- 

 dependent : 



<A/> = <Jl> r. 



Now r can be written as 



I 



r{t - t') dt', 



which is simply the total concentration at the present time t due to a 

 constant injection from time — oo to the present. 



Therefore the problem is reduced to finding the total carrier concen- 

 tration in the filament due to a distribution of sources of constant 



strength V<«/o> • 



Let w{x^ y, z; Xi , yi , Zi) denote the carrier concentration at x, y, z 



due to a steady unit source at o^i , yi , Zi . Then the total carrier con- 

 centration is 



r{xi ,yi,Zi) = j w(x, y, z; Xi , yi , Zi) dx dy dz. 



The reason for the dependence on xi , yi , Zi , is that the recombination 

 process takes place largely on the surface. Therefore a source near the 

 surface will yield a smaller concentration than one well inside the fila- 

 ment. (Volume recombination will be neglected throughout this paper.) 

 The mean square conductivity modulation due to many statistically 

 independent sources at Xr , yr , Zr (r = 1, 2 • • • ) is then 



< {Agf> = i:T\Xr , yr , Zr) <j\Xr , yr , Zr)> . 



The behavior of w is governed by the diffusion equation, subject to 

 the boundary conditions expressing the recombination process, and sub- 

 ject to a suitable singularity at Xr , yr , Zr , expressing the injection of a 

 unit current. J^ut in two and three dimensions the solution is not avail- 

 able in closed form, or at any rate not in terms of the elementary trans- 

 cendental functions. The infinite series for the solution is not easy to 



