Editing Field electron emission (section)

==== Recommended form for simple Fowler–Nordheim-type calculations ====
Explicitly, this recommended ''simplified standard Fowler–Nordheim-type equation'', and associated formulae, are:

{{NumBlk|:|<math> J = \; a {\phi^{-1}} F^2 \exp[- v(f) \;b \phi^{3/2} / F ], </math>|{{EquationRef|30a}}}}

{{NumBlk|:|<math> a \approx \; 1.541434 \times 10^{-6} \; \mathrm{A \; eV} \; {\mathrm{V}}^{-2};\;\;\;\;\; b \approx 6.830890 \; {\mathrm{eV}}^{-3/2} \; \mathrm{V} \; {\mathrm{nm}}^{-1}, </math>|{{EquationRef|30b}}}}

{{NumBlk|:|<math> v(f) \approx 1 - f + (1/6) f \ln f </math>|{{EquationRef|30c}}}}

{{NumBlk|:|<math> f = \; F/F_{\phi} = (e^3 / 4 \pi \epsilon_0) (F/ {\phi}^2) = (1.439964 \; {\mathrm{eV}}^2 \; {\mathrm{V}}^{-1} \; \mathrm{nm}) (F/ {\phi}^2). </math>|{{EquationRef|30d}}}}

where ''F''<sub>''φ''</sub> here is the field needed to reduce to zero a Schottky–Nordheim barrier of unreduced height equal to the local work-function ''φ'', and ''f'' is the scaled barrier field for a Schottky–Nordheim barrier of unreduced height ''φ''. [This quantity ''f'' could have been written more exactly as ''f''<sub>''φ''</sub><sup>SN</sup>, but it makes this Fowler–Nordheim-type equation look less cluttered if the convention is adopted that simple ''f'' means the quantity denoted by ''f''<sub>''φ''</sub><sup>SN</sup> in,<ref name=fd07/> eq. (2.16).] For the example case ({{nowrap|1=''φ'' = 4.5 eV}}, {{nowrap|1=''F'' = 5 V/nm}}), {{nowrap|1=''f'' ≈ 0.36}} and {{nowrap|''v''(''f'') ≈ 0.58}}; practical ranges for these parameters are discussed further in.<ref name=F08a>{{cite journal|doi=10.1116/1.2834563|title=Description of field emission current/voltage characteristics in terms of scaled barrier field values (f-values)|year=2008|last1=Forbes|first1=Richard G.|journal=Journal of Vacuum Science and Technology B|volume=26|issue=1|page=209|bibcode = 2008JVSTB..26..209F }}</ref>

Note that the variable ''f'' (the scaled barrier field) is not the same as the variable ''y'' (the Nordheim parameter) extensively used in past field emission literature, and that "''v''(''f'')" does NOT have the same mathematical meaning and values as the quantity "''v''(''y'')" that appears in field emission literature. In the context of the revised theory described here, formulae for ''v''(''y''), and tables of values for ''v''(''y'') should be disregarded, or treated as values of ''v''(''f''<sup>1/2</sup>). If more exact values for ''v''(''f'') are required, then<ref name=fd07/> provides formulae that give values for ''v''(''f'') to an absolute mathematical accuracy of better than 8×10<sup>−10</sup>. However, approximation formula (30c) above, which yields values correct to within an absolute mathematical accuracy of better 0.0025, should gives values sufficiently accurate for all technological purposes.<ref name=fd07/>