474 BELL SYSTEM TECHNICAL JOURNAL 



TABLE VI 

 Values of fi. 



(i,,/j). The values listed in the table were selected from a more 

 extensive table given by Langmuir.^'' From equation (87) it follows 

 that space charge acts as if the work function of the surface were in- 

 creased by Vm- 



An expression for i as a function of V could be obtained by eliminat- 

 ing Vm and Xrn. between equations (86), (87) and (88). Because of the 

 nature of these equations an analytical expression for i cannot be 

 given. However, for any temperature and electrode spacing it is 

 possible to calculate i as a function of V. The effect of introducing the 

 Maxwellian distribution of velocities can be seen by comparing curves 

 calculated from equations (86), (87) and (88) with equation (81). 

 Such a comparison is made in Fig. 24. Equation (81) gives a straight 

 line with a slope of 3/2 on such a plot and is represented by curve 1. 

 Curves 2 and 3 were calculated from equations (86), (87) and (88) for 

 parallel plates of tungsten spaced 1 cm. apart at 2000 and 3000° K., 

 respectively. The introduction of the Maxwellian velocity distribu- 

 tion causes the currents to be somewhat higher than predicted by the 

 simple 3/2 power law, especially at low applied potentials. Further- 

 more, at applied potentials considerably less than necessary for 

 saturation, the slope of log i vs. log V is less than 3/2. Near the break 

 point, the slope is practically 3/2. 



For long coaxial cylinders, Schottky ^^ and Langmuir ^^' *^ have 

 ]:)ointed out that the effect of introducing the IVIaxwellian velocity 

 distribution is less important than its introduction in the plane parallel 

 case. Langmuir ^^ has discussed an approximate formula for this case. 



Effect of Fermi-Dirac Velocity Distribution 



The introduction of the newer theory that the free electrons in the 

 metal have a F'ermi-Dirac velocity distribution requires no modification 

 of the space charge equations deduced on the assumption of a Maxwel- 

 lian distribution. This is because of the fact that the electrons which 

 escape across the potential barrier at the surface have a Maxwellian 

 distribution, as was shown in connection with Fig. 1. In this con- 

 nection it is of interest to compare some calculations by Bartlett ''* as- 



