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Simple derivation of the formula for Sommerfeld supply density used in electron-emission physics and limitations on its use

### Abstract

In a free-electron model, an electron crossing a mathematical plane inside a conductor can be characterized by the energy components associated with its motion normal and parallel to the plane. These components define a two-dimensional “energy-space.” The “supply density” is defined as the electron current crossing the plane, per unit area of the plane, per unit area in energy-space, when the relevant electron states are fully occupied. For a bulk free-electron conductor, the supply density is the same at all points in energy-space and has been called the “Sommerfeld supply density” . This is given by , where is the elementary positive charge, is the electron mass, and is Planck’s constant. This result is often a convenient starting point for developing basic theories of electron emission. A simple proof of it is recorded here. For small electron emitters, it can be a poor approximation to assume that the supply density is constant in energy-space. Consequently, if an emitter is sufficiently small, then the emission will not be well described by the usual basic emission equations. Criteria for assessing what counts as “sufficiently small” are discussed.

© 2010 American Vacuum Society

Received 25 March 2010
Accepted 21 September 2010
Published online 02 December 2010

Article outline:

I. INTRODUCTION
II. DERIVATION OF THE FORMULA FOR
III. FUNDAMENTAL NATURE OF THE SUPPLY-DENSITY STATEMENT
IV. LIMITATIONS OF USE

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2010-12-02

2016-08-24

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