The “changeover field” in thermal-field shaping

  • R. G. ForbesEmail author
Part of the Condensed Matter book series (volume 45B)


This chapter provides the condition for field changeover from blunting to sharpening.

At sufficiently high temperatures, a field emitter will change shape, as a result of the migration of surface atoms. As discussed in [14Mil], the direction of change, as indicated by the relevant electrical Gibbs function, depends in principle on the applied voltage. If the voltage is sufficiently small, then the surface-energy term dominates, and the emitter will tend to become blunter and to “ball up”; if the voltage is sufficiently large, then the energy term relating to the capacitance between the emitter and its surroundings (and to the work done by the high-voltage generator) dominates, and the emitter will tend to “reach out toward its surroundings,” by becoming sharper or by the formation of nanoprotrusions.

The condition for changeover from blunting to sharpening is often formulating as the stress condition
$$ {\varepsilon}_0{F}^2>2{\gamma}^0\left(\frac{1}{r_1}+\frac{1}{r_2}\right), $$
where r1 and r2 are the principal local radii of curvature. There is reason to think that this is not a general result (see [09F1], Sect. 2.4.2), but in practice it seems to work adequately for field emitters ([09S1], p. 18). For a field emitter apex of radius ra, Eq. 160.1 produces a formula for the changeover field (from blunting to sharpening) Fb→s:
$$ {F}^{\mathrm{b}\to \mathrm{s}}=2{\left(\frac{\gamma^0}{\varepsilon_0{r}_{\mathrm{a}}}\right)}^{1/2} $$

Values of this changeover field for a tip radius of 100 nm are shown in the last column of Table  158.2[] and are typically a few V/nm or less. Obviously, for a tip radius of 1 nm, the changeover fields would be higher by a factor of 10. Comparisons with predicted evaporation field values indicate that, for all except very small tip apex radii, in the vicinity of 1 nm, emitter sharpening is the expected TF-shaping tendency in atom-probe operating conditions.

Symbols and abbreviations

Short form

Full form


thermal field


  1. [09F1]
    Forbes, R.G., Mair, G.L.R.: Liquid metal ion sources, Chapter 2 in [09O]Google Scholar
  2. [09O]
    Orloff, J. (ed) Handbook of Charged Particle Optics (2nd Edition). Boca Raton, CRC Press (2009)Google Scholar
  3. [09S1]
    Swanson, L.W., Schwind, G.A.: Chapter 1 in [09O]Google Scholar
  4. [14Mil]
    Miller, M.K., Forbes, R.G.: Atom Probe Tomography: The Local Electrode Atom Probe. Springer, New York (2014)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  1. 1.Advanced Technology InstituteUniversity of SurreyGuildfordUK

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