On the Theory of Space Charge Between Parallel 

 Plane Electrodes 



By C. E. FAY, A. L. SAMUEL and W. SHOCKLEY 



The problem of the potential distribution, current, and electron 

 transit time resulting from the perpendicular injection of electrons 

 into the space between parallel planes is considered. The elec- 

 trons are assumed to be injected uniformly with velocities cor- 

 responding to the potential of the plane through which they are 

 injected. Consideration of all possible solutions of the basic equa- 

 tion shows that four general types of potential distribution are 

 possible. Curves are given which enable the easy calculation of 

 transmitted current and transit time and show the complete po- 

 tential distribution for any concrete example. The case for cur- 

 rent injected through both planes is also considered. 



The complete mathematical treatment is given in the appendix. 



SPACE charge has been studied extensively since the publication of 

 the first papers on the subject by Child ^ in 1911 and Langmuir^ 

 in 1913. These papers in common with many which followed ^' ^' *■ ^' ^ 

 dealt for the most part with potential distributions which occur when 

 electrons are injected into a region with relatively small initial velocities. 

 The problem of space charge between parallel planes when the 

 electrons possess arbitrary initial velocities, although contained 

 implicitly in some of this early work, was first considered in detail by 

 Gill ^ in 1925. Gill appears to have clearly understood the phe- 

 nomenon but did not publish a complete analysis. Other workers, 

 beginning with Tonks ^ in 1927, have considered various aspects of 

 the problem, !"• " and recently Plato, Kleen, and Rothe ^^^ ^^ published 

 an extensive analysis. They did not include transit time calculations 

 and their published curves are not easily adaptable for numerical 



1 C. D. Child, Phys Rev., 32, 492 (1911). 



2 1. Langmuir, Phys. Rev., 2, 450-486 (1913); also Phys. Zeits, 15, 348 (1914). 

 3 W. Schottky, Phys. Zeits, 15, 526 and 624 (1914). 

 ^ P. S. Epstein, Vehr. d.D. Phys. Ges., 21, 85 (1919). 

 *T. C. Frv, Phys. Rev., 17, 441 (1921); Phys. Rev., 22, 445 (1923). 

 « I. Langmuir, Phys. Rev., 21, 419 (1923). 



' I. Langmuir and K. B. Blodgett, Phys. Rev., 22, 347 (1923); Phys. Rev., 24, 49 

 (1924). 



« E. W. B. Gill, Phil. Mag., 49, 993 (1925); Phil. Mag., 10, 134 (1930). 



^ L. Tonks, Phys. Rev., 30, 501 (1927). 



'» H. C. Calpine, Wireless Engifteer^ 13, 473 (1936). 



" Myers, Hartree and Porter, Proc. Roy. Soc. 158, 23 (Jan. 1937). 



12 G. Plato, W. Keen, and H. Rothe, Zeit.f. Phys., 101, 509 (July 1936). 



13 H. Rothe and W. Keen, Telejunken Rohre No. 9 (April 1937). 



49 



