= Exciton optical transition
{title2=$E_{\rm opt}\simeq E_g-E_B$}
For the ideal <Wannier-Mott exciton> in <energy bands> whose edges occur at the same <crystal momentum>, the pair excitation <energy>, measured from the unexcited crystal, is
$$
E_{X,n}(\mathbf K)=E_g-\frac{R^*}{n^2}+\frac{\hbar^2K^2}{2M}.
$$
Here $\hbar\mathbf K$ is the pair's total <crystal momentum>; the last term is <center of mass> <kinetic energy>. For a permitted <photon absorption> transition with negligible recoil <energy>, $\hbar\omega\simeq E_g-E_{B,n}$. An <electric-dipole selection rule> and a nonzero transition <matrix element> are still needed: a <bound state> need not produce an observable <spectral line>. If the relevant <energy band> edges have different values of <crystal momentum>, a <phonon> can also be needed to conserve <crystal momentum>; subtracting a binding scale from a <band gap> alone does not specify the optical threshold.
\Image[/exciton-binding-optical-transition.png]
{title=<Exciton binding energy> and hypothetical <exciton optical transition> positions}
{description=Original three-dimensional <Wannier-Mott exciton> model with the inputs given above. Left: internal <energy eigenvalues> relative to the unbound continuum. Right: corresponding optical <energy> positions for hypothetical permitted transitions, with recoil neglected. Equal stick heights encode positions only, not predicted strengths or widths. The continuum threshold is $1.500\,\mathrm{eV}$; these sticks are not an empirical <optical spectrum> and are not the <copper(I) oxide> experiment below.}
{source=exciton-binding-optical-transition.py}
For example, the lowest internal <bound state> in this hypothetical model costs $1.489115\,\mathrm{eV}$ to create optically, yet needs only $0.010885\,\mathrm{eV}$ more to dissociate. The large creation <energy> and the small <exciton binding energy> answer different physical questions. An <electron-hole recombination> event can instead release excitation <energy> by <photon emission> or to other degrees of freedom; an emitted <photon> need not reproduce the original absorption <energy> if relaxation intervened.
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