Editing Field electron emission (section)

=== Correction factor for the Schottky–Nordheim barrier ===
[[File:Sn barrier.svg|thumb|Schottky–Nordheim barrier for Fowler–Nordheim field emission (and [[Schottky effect|enhanced thermionic emission]])]]
The Schottky–Nordheim barrier, which is the barrier model used in deriving the standard Fowler–Nordheim-type equation,<ref name=fd07>{{cite journal|doi=10.1098/rspa.2007.0030|title=Reformulation of the standard theory of Fowler–Nordheim tunnelling and cold field electron emission|year=2007|last1=Forbes|first1=Richard G.|last2=Deane|first2=Jonathan H.B.|journal=[[Proceedings of the Royal Society A]]|volume=463|pages=2907–2927|bibcode = 2007RSPSA.463.2907F|issue=2087 |s2cid=121328308}}</ref> is a special case. In this case, it is known that the correction factor <math> \it{\nu} </math> is a function of a single variable ''f<sub>h</sub>'', defined by ''f<sub>h</sub>''&nbsp;=&nbsp;''F''/''F<sub>h</sub>'', where ''F<sub>h</sub>'' is the field necessary to reduce the height of a Schottky&ndash;Nordheim barrier from ''h'' to 0. This field is given by

{{NumBlk|:|<math> \, F_h = (4\pi \epsilon_0/e^3) h^2 = (0.6944617 \; \mathrm{V}\; {\mathrm{nm}}^{-1})(h/{\rm{eV}})^2. </math>|{{EquationRef|10}}}}

The parameter ''f<sub>h</sub>'' runs from 0 to 1, and may be called the ''scaled barrier field'', for a Schottky–Nordheim barrier of zero-field height ''h''.

For the Schottky&ndash;Nordheim barrier, {{nowrap|''ν''(''h'', ''F'')}} is given by the particular value ''ν''(''f<sub>h</sub>'') of a function ''ν''(''ℓ''{{prime}}). The latter is a function of mathematical physics in its own right with explicit series expansion<ref name="DF08">{{cite journal |last1=Deane |first1=Jonathan H B |last2=Forbes |first2=Richard G |year=2008 |title=The formal derivation of an exact series expansion for the principal Schottky–Nordheim barrier function, using the Gauss hypergeometric differential equation |journal=Journal of Physics A: Mathematical and Theoretical |volume=41 |issue=39 |page=395301 |bibcode=2008JPhA...41M5301D |doi=10.1088/1751-8113/41/39/395301 |s2cid=122711134}}</ref> and has been called the ''principal Schottky&ndash;Nordheim barrier function''. The following good simple approximation for ''ν''(''f<sub>h</sub>'') has been found:<ref name=fd07/>

{{NumBlk|:|<math>v(f_h) \approx 1 - f_h + \tfrac{1}{6} f_h\ln f_h. </math>|{{EquationRef|11}}}}