Editing Field electron emission (section)

===Decay width===
The ''decay width'' (in energy), ''d<sub>h</sub>'', measures how fast the escape probability ''D'' decreases as the barrier height ''h'' increases; ''d<sub>h</sub>'' is defined by:

{{NumBlk|:|<math>\frac{1}{d_h} = -\frac{\mathrm{d}(\ln D)}{\mathrm{d}h}. </math>|{{EquationRef|12}}}}

When ''h'' increases by ''d<sub>h</sub>'' then the escape probability ''D'' decreases by a factor close to e (&nbsp;≈&nbsp;2.718282). For an elementary model, based on the exact triangular barrier, where we put ''ν''&nbsp;=&nbsp;1 and ''P''&nbsp;≈&nbsp;1, we get
: <math>d_h^{\mathrm{(el)}} = \frac{2F}{3b\sqrt{h}} = \frac{e F}{g \sqrt{h}}. </math>

The decay width ''d<sub>h</sub>'' derived from the more general expression ({{EquationNote|12}}) differs from this by a "decay-width correction factor" ''λ<sub>d</sub>'', so:

{{NumBlk|:|<math>d_h=  \lambda_d d_h^{\mathrm{(el)}} = \frac{\lambda_d e F}{g \sqrt{h}}. </math>|{{EquationRef|13}}}}

Usually, the correction factor can be approximated as unity.

The decay-width ''d''<sub>F</sub> for a barrier with ''h'' equal to the local work-function ''φ'' is of special interest. Numerically this is given by:

{{NumBlk|:|<math> d_{\mathrm{F}}= \frac{\lambda_d e F}{g \sqrt{\phi}}  \approx \frac{e F}{g \sqrt{\phi}} \approx 0.09759678 \; \mathrm{eV} \, \cdot \sqrt{\frac{1\ \mathrm{eV}}{\phi}} \cdot \frac{F}{1\ \mathrm{V}\ \mathrm{nm}^{-1}}. </math>|{{EquationRef|14}}}}

For metals, the value of ''d''<sub>F</sub> is typically of order 0.2&nbsp;eV, but varies with barrier-field ''F''.