THERMIONIC ELECTRON EMISSION 447 



Then the surface charge density cr is given by 



a — p -{- IJ- COS {irxlb) cos ijylb) = p + mi8. (70) 



in which p is the mean value of cr, p + /x is the maximum value of a, 

 p — ju is the minimum a, and /3 = cos (wxlb) cos (Tylb) ; /3 has values 

 between + 1 and — 1. It readily follows that along the edges of the 

 squares /3 = and a = p. Figure IIB shows o- as a function of x 

 when y — 0, b, 2b or nb. 



The great advantage of this particular charge distribution is that 

 we can represent the potential due to this charge and its image at any 

 point above the surface by means of a comparatively simple formula, 

 viz., 



Pale = - 300 X 47r/[p + m/3 exp (- \27rz/6)], (71)* 



P„\s the potential energy of an electron due to the charge distribution 

 at a point which is z cm. above the surface over a region at which the 

 charge density is c. The charge distribution is located in a plane 

 which is / cm. above the surface. This charge distribution induces a 

 corresponding negative charge distribution at z = — /, i.e., / cm. below 

 the surface, p and p. are in e.s.u. of charge per cm.^ Sometimes it 

 will be convenient to treat p and p as if they were expressed in volts, 

 i.e., as if p and p stood for 12007rp/ or 12007rp/, respectively. The 

 total potential energy of an electron at z cm. from the surface is given 

 by P in 



P = Pi+ Pa- Fez 



= P^- 300e/4z - 12007r/[p + pl3 exp (- zv^vr/^)] - Fez, (72) 



where Pi= P^- 300e/4z and e = 4.774 X lO"!". 



In Fig. 12, curve 1 shows Pi z;^. z; curve 2 shows Pa for various values 

 of |8; the curve for (8 = + 1 is for the normal taken at the center of a 

 hill; |S = — 1 is for the center of a valley; |3 = is for the edge of the 

 squares; all other curves must lie between those for /3 = + 1 and 

 /3 = - 1. Curves 3 show Pi + Pa for /3 = 1, and - 1. 



For all values of ^ between 1 and the curves have the same maxi- 

 mum value which occurs when Zc = <» . The value of this maximum 

 is P^ — llOOwlp. This means that for all points of a hill checker 

 the work function is reduced by the same amount, namely, 12007r/p; 



* For the derivation of this and several other formulas I am indebted to Professor 

 V. Rojansky now at Union College, who worked with me on this problem in the 

 summer of 1930. 



