420 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



viously depends on the magnitude of Velo. When rel(T as calculated by 

 the Hines-Cutler method (with a lens correction added) remains greater 

 than about 2 throughout the range of interest, the difference between 

 the corresponding values obtained for rgs will be only a few per cent. 

 For these cases where rja does not get too small, the principal advan- 

 tages of this paper are in the inclusion of a correction to the anode lens 

 formula and in the comparative ease with which design parameters may 

 be obtained. In other cases r^la may become less than 1, and the theory 

 presented in this paper has extended the basic Hines-Cutler approach 

 so that one may make realistic predictions even under these less ideal 

 conditions where the departure from a laminar-type flow is quite severe. 



ACKNOWLEDGMENT 



We wish to thank members of the Mathematical Department at 

 B.T.L., particularly H. T. O'Neil and Mrs. L. R. Lee, for their help in 

 programming the problem on the analog computer and in obtaining the 

 large amount of computer data involved. In addition, we wish to thank 

 J. C. Irwin for his help in the electrolytic tank work and both Mr. Irwin 

 and W. A. L. Warne for their work on the beam analyzer. 



REFERENCES 



1. Pierce, J. R., Rectilinear Flow in Beams, J. App. Phys., 11, pp. 548-554, Aug., 



1940. 



2. Samuel, A. L., Some Notes on the Design of Electron Guns, Proc. I.R.E., 33, 



pp. 233-241, April, 1945. 



3. Field, L. M., High Current Electron Guns, Rev. Mod. Phys., 18, pp. 353-361, 



July, 1946. 



4. Davisson, C. J., and Calbick, C. J., Electron Lenses, Phys. Rev., 42, p. 580, 



Nov., 1932. 



5. Helm, R., Spangenburg, K., and Field, L. M., Cathode-Design Procedure for 



Electron Beam Tubes, Elec. Coram., 24, pp. 101-107, March, 1947. 



6. Cutler, C. C, and Hines, M. E., Thermal Velocity Effects in Electron Guns, 



Proc. I.R.E., 43, pp. 307-314, March, 1955. 



7. Cutler, C. C, and Saloom, J. A., Pin-hole Camera Investigation of Electron 



Beams, Proc. I.R.E., 43, pp. 299-306, March, 1955. 



8. Hines, M. E., Manuscript in preparation. 



9. Private communication. 



10. See for example, Zworykin, V. K., et al.. Electron Optics and the Electron 



Microscope, Chapter 13, Wiley and Sons, 1945, or Klemperer, O., Electron 

 Optics, Chapter 4, Cambridge Univ. Press, 1953. 



11. Brown, K. L., and Siisskind, C., The Effect of the Anode Aperature on Po- 



tential Distribution in a "Pierce" Electron Gun, Proc. I.R.E., 42, p. 598, 

 March, 1954. 



12. See, for example, Pierce, J. R., Theory and Design of Electron Beams, p. 147, 



Van Nostrand Co., 1949. 



13. See Reference 6, p. 5. 



14. Langmuir, I. L., and Blodgett, K., Currents Limited by Space Charge Be- 



tween Concentric Spheres, Phys. Rev., 24, p. 53, July, 1924. 



15. See Reference 12, p. 177. 



16. See Reference 12, Chap. X. 



