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Alloy broadening is a mechanism by which thespectral lines of an alloy are broadened by the random distribution of atoms within the alloy.^[1] It is one of a number ofspectral line broadening mechanisms.

Alloy broadening occurs because the random distribution ofatoms in an alloy causes a different material composition at different positions. Insemiconductors and insulators the different material composition leads to differentband gap energies. This gives differentexciton recombination energies. Therefore, depending on the position where an exciton recombines the emitted light has a different energy. The alloy broadening is aninhomogeneous line broadening, meaning that its shape is Gaussian.^{[citation needed]}

In the mathematical description it is assumed that no clustering occurs within the alloy. Then, for a binary alloy of the form${\displaystyle {\text{A}}_{1-\text{x}}^{}{\text{B}}_{\text{x}}^{}}$, e.g.${\displaystyle {\text{Si}}_{1-\text{x}}^{}{\text{Ge}}_{\text{x}}^{}}$, thestandard deviation of the composition is given by:^[2]

${\displaystyle \mathrm{\Delta }x=\sqrt{\frac{x\cdot (1-x)}{N}}}$,

where${\displaystyle N}$ is the number of atoms within the excitons' volume, i.e.${\displaystyle N=V_{exc}\cdot n}$ with${\displaystyle n}$ being the atoms per volume. In general, the band gap energy${\displaystyle E_g}$ of a semiconducting alloy depends on the composition, i.e.${\displaystyle E_g}$. The band gap energy can be considered to be thefluorescence energy. Therefore, the standard deviation in fluorescence is:^[3]

${\displaystyle \mathrm{\Delta }E=\frac{\mathrm{d}{E}_g}{\mathrm{d}x}\cdot \sqrt{x\cdot \frac{1-x}{N}}}$

As the alloy broadening belongs to the group of inhomogeneous broadenings theline shape of the fluorescence intensity${\displaystyle I(E)}$ is Gaussian:^{[why?][citation needed]}

${\displaystyle I(E)\sim \exp \left(-\frac{(E-E_0{)}^2}{2\cdot \mathrm{\Delta }{E}^2}\right)}$

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