390 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



Boltzman's constant, T is the cathode temperature in degrees Kelvin, 

 and m is mass of the electron. For the case ixc = 0, but with arbitrary 

 initial transverse velocity, we will then have 



/^\ 



^^nl_ /kf ^^^'^ 



Tc y m 



Plence we can express a in terms of the thermal electron's radial po- 

 sition (r), and its initial transverse velocity, Vc , 



y m _ y 



. . - . /kT 



dt } f 



The quantity a can now be related to the radial spread of thermal 

 electrons (emitted from a given point on the cathode) with respect to 

 an electron with no initial velocity: By (11) we see that the number 

 of electrons leaving the cathode with dji/dt = Vc/ve is proportional to Vc 

 exp —Vcm/2kT. Suppose many experiments were conducted where all 

 electrons except one at the cathode center had zero emission velocity, 

 and suppose the number of times the initial transverse velocity of the 

 single thermal electron were chosen as Vc , is proportional to Vc exp 

 — Vcm/2kT. Then the probability, P{r), that the thermal electron 

 would have a radial position between r and r -\- dr when it arrived at the 

 transverse plane of interest would be proportional to Vc exp —Vc^(m/2kT). 

 Here Vc is the proper transverse velocity to cause arrival at radius r, and 

 by (17) we have 



a y m 

 so that the probability becomes 



Pir) = J.e-^^'''-'^ d (^Q (18) 



We therefore identify cr with the standard deviation in a normal or 

 Gaussian distribution of points in two dimensions. At the real cathod(\ 

 thermal electrons are simultaneously being emitted from the cathode 

 surface with a range of transverse velocities. However, if a as definml 

 above is small in comparison with r,. , the forces experienced by a ther- 

 mal electron when other thermal electrons are present will be very nearly 



