878 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



the presence of thermal motions and beam ripples, the ac power is found 

 to be concentrated chiefly at the rim of the beam. Occasionally, several 

 concentric rings of noise maxima are found within the beam, possibly 

 due to unusual cathode conditions. 



When the ripple wavelength is very long, two maxima of noise power 

 are observed to flank each beam waist. A fir.st-order calculation of wave 

 propagation along a rippled laminar-flow beam accounts for this pat- 

 tern by showing that space-charge waves grow at the expense of dc 

 kinetic energy in the radial charge motion. In rippled-beam amplifica- 

 tion of noise, the product /max^min has been found to increase, and the 

 the ratio /max/-^min remain nearly constant, because both fast and slow 

 waves are amplified, the former less than the latter, and because the 

 wave coupling is not lossless. 



Positive ions tend to collect at the waist of rippled beams, thereby 

 extending the region in which electrons pass close to the axis, instead 

 of circling about it. The overlap of their orbits leads to net cancellation 

 of radial charge motion, and hence a reduction in rippled-beam ampli- 

 fication. This may explain why positive ions tend to inhibit the "grow- 

 ing noise" phenomenon. 



REFERENCES 



1. W. W. Rigrod, Noise Spectrum of Electron Beam in Longitudinal Magnetic 



Field. Part I — The Growing Noise Phenomenon, p. 831 of this issue. 



2. C. C. Cutler, Spurious Modulation of Electron Beams, Proc. I.R.PL, 44, p. 61, 



Jan., 1956. 



3. K. G. Hernquist, Plasma Ion Oscillations in Electron Beams, J. Appl. Phys. 



26, p. 544, May, 1955. 



4. F. B. Llewellyn and A. E. Bowen, The Production of UHF Oscillations by 



Diodes, B. S.T.J. , 18, p. 280, April, 1939. 



5. O. Klemperer, Lifluence of Space Charge on Thermionic Emission Velocities, 



Proc. Royal Soc. (London) (A) 190, p. 376, 1947. 



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Charge of some Electron Emission Systems, Journal of Electronics, 1, p. 601 

 May, 1956. 

 7 C.N. Smyth, Total Emission Damping with Space-Charge-Lmiited Cathodes, 

 Nature, 157, p. 841, June 22, 1946. 



8. W. Veith, Electron Energy Distribution in Space-Charge-Limited Electron 



Streams, Zeit. f. angew. Physik, 7, No. 9, p. 437, 1955. 



9. W. W. Rigrod and J. A. Lewis, Wave Propagation Along a Magnetically- 



Focused Cylindrical Electron Beam, B. S.T.J. , 33, p. 399, March, 1954. 



10. R. W. Peter, S. Bloom, and J. A. Ruetz, Space-Charge-Wave Amplification 



Along an Electron Beam by Periodic Change of the Beam Impedance, RCA 

 Rev., 15, p. 113, March, 1954. 



11. J. R. Pierce, The Wave Picture of Microwave Tubes, B. S.T.J. , 33, p. 1343, 



Nov., 1954. 



12. L. J. Chu, 1951 I.R.E. Electron Tube Conference on Electron Devices. 



13. H. A. Haus and D. L. Bobroff, Small Signal Power Theorem for Electron 



Beams (to be published). 



14. K. T. Dolder and O. Klemperer, Space-Charge Effects in Electron Optical 



Systems, J. App. Phys., 26, p. 1461, Dec, 1955. 



15. S. J. Buchsbaum and E. Gordon, Highly Ionized Microwave Plasma, M.I.T. 



R.L.E. Quarterly Prog. Rep., p. 11, Oct. 15, 1956. 



