424 HKLI. SYSTKM TKCIIMCAI. JOl RXAL 



idealized case wliere holes are injected on the plane .v = and no electrons 

 are removed there, the equations to be satisfied are 



A (^^^ - Dn {^-^ + (».M. + nnn,)E- = jje (46) 



(44) 

 (45) 



where subscripts + and — refer to conditions just to the right of x = 

 and just to the left, respectively. Using the neutrality condition tie = «o + "a 



these are three equations for the five unknowns ( — ) , E± ,)ih .To complete 



\ dx /i 



the determination of these quantities the differential equation (22) must be 

 solved and the boundary conditions imposed that ;/;, -^ as .v -^ ± ^. 

 Actually the problem of estimating conditions at .v = may not be quite 

 as formidable as the preceding paragraph suggests, at least if the diffusion 

 parameter J /J is reasonably small and ii jc/Ja is also not too large. For such 

 cases the "upstream diffusion" of holes into the region of negative x will 

 probably reach a steady state in a very short time. Solutions of the steady 

 state differential equation in this region have been obtained numerically by 

 W. van Roosbroeck (unpublished). Such solutions will give one relation be- 

 tween )ih and I — - J ; another relation, in the form of a fairly narrow range 



of limits, is provided by the fact that ( — -^' I will under these conditions be 



m. 



« ( — ) , being in fact probably somewhere between zero and the value 



for the diffusionless case with the same value of Uhi . 



Of course, if the mathematical solution for this one-dimensional idealiza- 

 tion is to be applied to a case where holes are injected into a filament by a 

 I)ointed electrode on its boundary, little meaning can be attached to vari- 

 ations in the ;//. of the mathematical solution within a range of .v values 

 smaller than the diameter of the filament. 



6. SUM.MAKV AND DiSCUSSIO.V 



There are three principal factors which limit the range of conditions 

 within which the present theory provides a useful approximation to the 

 transient behavior of )ii. as a function of / and .v. These are diffusion, trap- 

 ])ing, and departure from one-dimensional geometry. If the geometry is 

 sufficiently nearly one-dimensional and trapping is negligible, the discussion 



