MAGNETIC EFFECTS ON NOISE IN A GERMANIUM FILAMENT 649 



is appreciably different from zero (that is, let 1// be much greater than 

 the lifetime). Then, in the integral (i), r{t — t') will have gone from 

 unity to zero long before J(f)e~^''^^^ has changed appreciably from its 

 value at t' = t. Therefore, for purposes of observation at frequencies 

 much smaller than the reciprocal lifetime, we can rewrite (1) as 



^g{t) oc J{t) [ r{t - t') dt' 



J— 00 



(2)* 



= J(t) / r{t) dt. 

 Jo 



The integral in (2) can be interpreted as the average lifetime of car- 

 riers. For, by definition, the rate of recombination at time t is — dr(t)/dtj 

 so that — {dr/dt)dt is the number of carriers recombining between time 

 t, t + dt. Hence the average lifetime is 



r = -f^ ^ ^ ^^ = -[^KO]^ + 1^ r(t) dt 



= f r{t)dt 



since tr{t) -^ as ^ -^ co . 



If 1// is not large compared with r one cannot simplify the integral 

 (1) in this way. One then has to consider separately each frequency 

 component Ag{f)e~^'''^' due to the current J{f)e'^'''^\ Then 



J— CO 



= J{f)e-'"" f r{f)<i 

 Jo 



e-"'f" dt' 



'"^'■dtr 



or 



Agif) = J(f)r(f) 

 where 



Jo 



r(^0^'"'^" dt\ 



The calculation of t(/) is more complicated than that of r = t(0), and 



* From here on the equality sign will replace the proportionality sign. The 

 resulting change of units is of no consequence in the final results which are only 

 concerned with ratios of conductivity modulations. 



