Alloy broadening

In the mathematical description it is assumed that no clustering occurs within the alloy. Then, for a binary alloy of the form ${\displaystyle {\ce {A_{1-x}B_{x}}}}$ , e.g. ${\displaystyle {\ce {Si_{1-x}Ge_{x}}}}$ , the standard deviation of the composition is given by:^[2]

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

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 the fluorescence energy. Therefore, the standard deviation in fluorescence is:^[3]

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

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

${\displaystyle I(E)\sim \exp \left(-{\frac {(E-E_{0})^{2}}{2\cdot \Delta E^{2}}}\right)}$