EXCESS SKM /CONDI '(TON IIOI.E TRAXS/'OKT 427 



the diffusive spreading becomes sufficient to counterbalance it. It is not 

 necessary, for tlie production of a steep front of this kind, tliat the decrease 

 of 7e to zero be brought about with corresponding rapidity; even a gradual 

 decrease of jc will lead to a front which becomes steeper as it advances, 

 and if the decrease of y,, is not too gradual a "shock front" will have devel- 

 oped after a short distance. The order of magnitude of the "shock front 

 thickness" can be estimated by finding the value of the time A/ for which 

 the diflfusion distance A.vo = {2D A /) " equals the difference Axv between 

 the drift distances of the holes at the top and bottom of the front, i.e., 

 A.Vk = [V{0) — V(nu)]M, where V is given by (4) and ri/, is the height of 

 the front. For this value of A/, 



Axo = 2D/[V(0) - V(n,)] (48) 



and this is presumably of the order of magnitude of the thickness of the 

 front. If D is interpreted as D^ = kTmJe, which is good enough for the 

 present purpose, this gives 



2kT 1 



L 1 + Kl + W^e)J 



Of course, this extremely sharp front can be realized only when the condi- 

 tions of one-dimensional geometry are accurately fulfilled. When the geome- 

 try is made sufficiently ideal, observation of the thickness of the "shock 

 front" can provide a valuable check on the validity of the basic assumptions 

 of the theory such as the neglect of trapping.^ 



The author would like to express his indebtedness to many of his col- 

 leagues, and especially to J. Bardeen, J. R. Haynes, and W. van Roosbroeck, 

 for many illuminating discussions of the topics covered in this paper. 



^ The accompanying paper by W. Shockley, G. L. Pearson and J. R. Haynes describes 

 some observations of tliis shock wjve effect, though under conditions where y <IC I, so 

 that the thickness of the front as given by (49) is still fairly large. 



