76 BELL SYSTEM TECHNICAL JOURNAL 



Similar analysis to points beyond the potential zero yields 



_ 1 ^1/* 



^^ ^ (2 - zyi^ "^ W^ ' ^^^^ 



Integration Constant Negative — Types C and D 



Introducing for the constant in equation (7) a negative value, say 

 — (aVi)^'-, will give a positive value of V (equal to ocVi) for dVjdx = 

 with d^V/dx^ > and must therefore lead to solutions of the C type. 



Integrating once more and introducing the unit So gives 



X = ± So{<p"'^ + 2^1/2) ^^1/2 _ ^1/2 + const. (22) 



Expressing distance from the first plane in units of ^o, we find two 

 possibilities: 



CD = -\- {<p^'^ + 2ai/2)^<pi/2 _ ^1/2 _ (1 + 2^1/2) Vl - ai/2 (23) 

 and 



(r_ = - (^1/2 _^ 2ai/2-)^^i/2 _ ^1/2 + (1 + 2ai/2)^i _ ^i/2_ (24) 



The first of these solutions gives a potential distribution rising con- 

 tinuously as a increases from zero, hence of type D as was anticipated 

 by the subscript. The second solution decreases to a minimum at 



Cmin. = (1 + 2al/2)Vl - al/2 (25) 



and then increases, the equation to the right of the minimum being 



tj^ = (^1/2 -f 2ai/2)V^i/2 - ai/2 + (1 + 2«i/2)^i _ ^i/2_ (26) 



Curves given by equations 23, 24 and 26 are drawn in Fig. 3. If 

 values of a and ^ corresponding to conditions on the boundary planes 

 are entered in the figure, a C solution is indicated only if the point 

 falls upon a curve of the (t+ type. This curve then gives the potential 

 distribution to the right of the minimum ; to the left of the minimum 

 the distribution is given by the a- curve with the same value of a, 

 which has the interpretation a = (fmin. for this case. Points entered 

 on the (T- or an curves will clearly give D type solutions. 



Three equations for limits of the C region can easily be written 

 down on the basis of the above equations: 



(b) a = 1 + ^3/4. (27) 



(d) <T = (1 + 2v?i/2)^i _ ^1/2^ (28) 



(e) <7 = {<p'i^ + 2)^1 <p'i^ - 1: (29) 



