Editing Field electron emission (section)

=== Motive energy ===
For an electron, the one-dimensional [[Schrödinger equation]] can be written in the form

{{NumBlk|:|<math>\frac{\hbar^2}{2 m} \frac{\mathrm{d}^2 \Psi(x)}{\mathrm{d}x^2} = \left[U(x)-E_{\mathrm{n}}\right]\Psi(x) = M(x)\Psi(x), </math>|{{EquationRef|1}}}}

where Ψ(''x'') is the electron [[wave function|wave-function]], expressed as a function of distance ''x'' measured from the emitter's electrical surface,<ref name=F99>{{cite journal|doi=10.1016/S0304-3991(99)00098-4|title=The electrical surface as centroid of the surface-induced charge|year=1999|last1=Forbes|first1=R|journal=Ultramicroscopy|volume=79|issue=1–4|pages=25–34}}</ref> ''ħ'' is the [[reduced Planck constant]], ''m'' is the electron mass, ''U''(''x'') is the [[potential energy|electron potential energy]], ''E''<sub>n</sub> is the [[total energy|total electron energy]] associated with motion in the ''x''-direction, and ''M''(''x'') {{nowrap| {{=}} [''U''(''x'') &minus; ''E''<sub>n</sub>]}} is called the electron motive energy.<ref name=HN49>{{cite journal|doi=10.1103/RevModPhys.21.185|title=Thermionic Emission|year=1949|last1=Herring|first1=Conyers|journal=Reviews of Modern Physics|volume=21|pages=185–270|last2=Nichols|first2=M.|bibcode=1949RvMP...21..185H|issue=2}}</ref> ''M''(''x'') can be interpreted as the negative of the electron kinetic energy associated with the motion of a hypothetical classical point electron in the ''x''-direction, and is positive in the barrier.

The shape of a tunneling barrier is determined by how ''M''(''x'') varies with position in the region where {{nowrap|''M''(''x'') > 0}}. Two models have special status in field emission theory: the ''exact triangular (ET) barrier'', given in ({{EquationNote|2}}); and  the ''Schottky&ndash;Nordheim (SN) barrier'', given in ({{EquationNote|3}}).<ref>{{cite journal|author=W. Schottky|journal=Phys. Z.|volume=15|year=1914|page=872}}</ref><ref name=n28b>{{cite journal|author=L.W. Nordheim|journal=[[Proceedings of the Royal Society A]] | volume=121|year=1928|pages=626–639|doi=10.1098/rspa.1928.0222|title=The Effect of the Image Force on the Emission and Reflexion of Electrons by Metals|bibcode = 1928RSPSA.121..626N|issue=788 |doi-access=free}}</ref> 

{{NumBlk|:|<math>M^{\mathrm{ET}}(x) = h - eFx </math>|{{EquationRef|2}}}}
{{NumBlk|:|<math>M^{\rm{SN} }(x) = h - eFx -\frac{e^2}{16\pi\varepsilon_0 x}, </math>|{{EquationRef|3}}}}

Here ''h'' is the zero-field height (or ''unreduced height'') of the barrier, ''e'' is the [[elementary charge|elementary positive charge]], ''F'' is the barrier field, and ''ε''<sub>0</sub> is the [[vacuum permittivity|electric constant]]. By convention, ''F'' is taken as positive, even though the [[electric field|classical electrostatic field]] would be negative.  The SN equation uses the classical image potential energy to represent the physical effect "correlation and exchange".