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BELL SYSTEM TECHNICAL JOURNAL 



sizes and varying work functions above and below some mean value. 

 To treat this case would obviously require very complex expressions. 

 We can simplify the problem without departing too far from actual 

 conditions by postulating a surface which is divided up into a large 

 number of squares arranged in a checkerboard fashion. We might 

 suppose that all black squares have the same a and all white squares 

 are bare or else have a smaller a. It turns out, however, that the 

 formulas and the computations are much simpler if we suppose a is 

 largest at the center of each black square and is least at the center of 

 each white square; between the centers a is given by a cosine law. In 

 other words on the black squares we have a hill of charge while on the 

 white squares we have a valley of charge. It will be found that such 

 a charge distribution predicts changes in emission with applied 

 fields, which agree rather well with experiment if the size of the 

 squares is comparable to the crystal size and the difference in contact 

 potential between the hills and valleys corresponds to several tenths 

 of a volt. 



(A) 



1b 



2b 3b 

 X 



4b 5b 



(B) 



SUB CHECKERS 



(c) 



Fig. 11 — A. Checkerboard array. B. Charge distribution for hill and valley checker- 

 board. C. Subdivision of checkers. 



To represent such a charge distribution, choose the origin of coordi- 

 nates at the center of a covered square; let x be measured parallel to 

 one edge of the squares while y is measured perpendicular to this 

 edge as indicated in Pig. 11 A. Let the length of each square be b. 



