Table 1 Formulas for stretched exponential population and luminescence decays. The approximate solution H(λ) is shown in ref. [9]
From: Reappraising the Luminescence Lifetime Distributions in Silicon Nanocrystals
Population decay ct/c0 | Intensity decay g(t) | Mean time constant | Mean decay time | Rate distribution |
|---|---|---|---|---|
exp[−(λSEt)β] | \( {\beta \lambda}_{SE}^{\beta }{t}^{\beta -1}\exp \left[-{\left({\lambda}_{SE}t\right)}^{\beta}\right] \) | \( \frac{\tau_{SE}}{\beta}\Gamma \left[\frac{1}{\beta}\right] \) | \( {\tau}_{SE}\frac{\Gamma \left(2/\beta \right)}{\Gamma \left(1/\beta \right)} \) | H(λ) |
\( \frac{1}{\Gamma \left(1/\beta \right)}\Gamma \left[1/\beta, {\left({\lambda}_{SE}t\right)}^{\beta}\right] \) | \( \frac{\lambda_{SE}\beta }{\Gamma \left(1/\beta \right)}\exp \left[-{\left({\lambda}_{SE}t\right)}^{\beta}\right] \) | \( {\tau}_{SE}\frac{\Gamma \left(2/\beta \right)}{\Gamma \left(1/\beta \right)} \) | \( {\tau}_{SE}\frac{\Gamma \left(3/\beta \right)}{2\Gamma \left(2/\beta \right)} \) | H(λ)/λ |