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Exact analysis of surface field reduction due to field-emitted vacuum space charge, in parallel-plane geometry, using simple dimensionless equations

### Abstract

This paper reports (a) a simple dimensionless equation relating to field-emitted vacuum space charge (FEVSC) in parallel-plane geometry, namely , where is the FEVSC “strength” and is the reduction in emitter surface field (-with/field-without FEVSC), and (b) the formula , where is the ratio of emitted current density to that predicted by Child’s law. These equations apply to any charged particle, positive or negative, emitted with near-zero kinetic energy. They yield existing and additional basic formulas in planar FEVSC theory. The first equation also yields the well-known cubic equation describing the relationship between and applied voltage; a method of analytical solution is described. Illustrative FEVSC effects in a liquid metal ion source and in field electron emission are discussed. For Fowler–Nordheim plots, a “turn-over” effect is predicted in the high FEVSC limit. The higher the voltage-to-local-field conversion factor for the emitter concerned, then the higher is the field at which turn over occurs. Past experiments have not found complete turn over; possible reasons are noted. For real field emitters, planar theory is a worst-case limit; however, adjusting on the basis of Monte Carlo calculations might yield formulae adequate for real situations.

© 2008 American Institute of Physics

Received 08 May 2008
Accepted 20 August 2008
Published online 20 October 2008

Article outline:

I. INTRODUCTION
II. THEORETICAL DERIVATIONS
A. Definitions and conventions
B. Planar space-charge equation
C. Derivation of a dimensionless equation
D. General solutions
III. MATHEMATICAL SPACE-CHARGE REGIMES
A. Negligible-space-charge regime
B. Small-space-charge regime
C. Branch-point neighborhood
D. Child’s law regime
E. Partial space-charge equations
F. Discussion
IV. RELATIONSHIP BETWEEN and
V. ILLUSTRATIVE APPLICATIONS
A. Field reduction at the LMIS apex
B. Field-stress reduction at the LMIS apex
C. Effect of FEVSC on Fowler–Nordheim plot shape
D. Criterion for onset of space-charge effects
E. Criterion for approach to the Child’s law regime
VI. A MODIFIED PLANAR-GEOMETRY MODEL
VII. RELATIVISTIC EFFECTS
VIII. SUMMARY

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2008-10-20

2016-02-09

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