912 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



face can be calculated if the anode is a larger paraboloid whose equation is 

 given by 



y = xyiild + r) (2) 



provided the origin is at its vertex, d is the distance between the two ver- 

 tices, and the axes of the two paraboloids are the same. The field Fh for 

 points at which 3; = A, is given by* 



Fn = KkV = ^(1 ^ 2h/r)Hn{\ + 2d/r) ^^^ 



where h is distance along the axis of the small paraboloid. At the tip or ver- 

 tex of the W point, /? = and 



Hence F^ = P,/U + ^V 



(5) 



For an angle of 60° with the axis, h/r = .47 and 



Fn = .72 Fo 



For clean W, this predicts that the emission density at an angle of 60° 

 with the axis should be .008 of the emission density along the axis. For 

 angles less than 10° the field and emission densities should differ only slightly 

 from that for the axis values. Experiment shows that these predictions are 

 qualitatively fulfilled. 



Subsequent photographs will show that for clean W most of the emission 

 comes from regions which surround the 100 plane. For the 611 plane (p = 

 4.4 volts.^ The area of these highly emitting regions corresponds to about 

 \ of the area of the screen which in turn corresponds to about wr^ crn^ on the 

 W point. Hence we have taken the highly emitting area to be r^ cm^. The 

 highest emitting areas make an angle of about 25° with the axis of the W 

 point or with the 110 direction. From Eq. (3) we calculate that the field is 

 about .924 that at the tip of the point. 



In order to obtain a value of r, the radius of curvature of our W point, we 

 proceeded as follows: We observed the emission current i as a function of the 

 applied voltage V and plotted log i — 2 log V vs \/V. Straight lines were 

 obtained whose slopes and intercepts for clean W depended on the highest 

 temperature and time at which the W loop was glowed. We then plotted a 

 similar family of theoretical lines for various assumed values of r. The ex- 



* We are indebted to our colleague S. P. Morgan for Eqs. (1) to (4). 



