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raul_l
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Hi
I wonder if anyone could help me with this.
I'm studying a crystal whose luminescence is excitonic in nature. Since the excitation density is high (with femtosecond laser pulses) exciton-exciton interactions have to be taken into account. The following kinetic equations are used to describe the situation:
\frac{\partial n(\vec{r},t)}{\partial t} - D \triangledown ^2 n(\vec{r},t)= \frac{n(\vec{r},t)}{\tau} - n^2(\vec{r},t) \int{w(r) g(r, t) dV} (1)
\frac{\partial g(r,t)}{\partial t} - D \triangledown ^2 g(r,t) = -w(r) g(r,t) (2)
where n(r,t) is the exciton concentration, D is the diffusion coefficient, tau is the luminescence decay time, g(r,t) is the correlation function, w(r) is the energy transfer rate between excitons (here it's the Förster model with w(r)=\frac{1}{\tau}\frac{R_0}{r} but it doesn't matter) and r is the distance between excitons. The last term in Eq. (1) is the bimolecular term that describes excitonic interactions.
The second equation is the kinetic equation of the correlation function. This is the part I don't understand. Where does it come from? A colleague told me that it is generally known that the correlation function follows the same laws as the physical quantity it is connected with (in this case n(r,t)) and therefore has a similar kinetic equation. But that doesn't help much. I haven't been able to find any derivations or explanations for why Eq. (2) holds.
For example, here A. N. Vasil'ev, IEEE Trans. Nucl. Sci. 55, 1054 (2008) Eq. (3) it is simply stated that that's the case.
I've tried googling this but I'm not even sure what the right keywords would be. Correlation dynamics? Bimolecular kinetic equations?
P.S. Sorry if this is in the wrong section. It isn't homework but I could still use some help.
I wonder if anyone could help me with this.
I'm studying a crystal whose luminescence is excitonic in nature. Since the excitation density is high (with femtosecond laser pulses) exciton-exciton interactions have to be taken into account. The following kinetic equations are used to describe the situation:
\frac{\partial n(\vec{r},t)}{\partial t} - D \triangledown ^2 n(\vec{r},t)= \frac{n(\vec{r},t)}{\tau} - n^2(\vec{r},t) \int{w(r) g(r, t) dV} (1)
\frac{\partial g(r,t)}{\partial t} - D \triangledown ^2 g(r,t) = -w(r) g(r,t) (2)
where n(r,t) is the exciton concentration, D is the diffusion coefficient, tau is the luminescence decay time, g(r,t) is the correlation function, w(r) is the energy transfer rate between excitons (here it's the Förster model with w(r)=\frac{1}{\tau}\frac{R_0}{r} but it doesn't matter) and r is the distance between excitons. The last term in Eq. (1) is the bimolecular term that describes excitonic interactions.
The second equation is the kinetic equation of the correlation function. This is the part I don't understand. Where does it come from? A colleague told me that it is generally known that the correlation function follows the same laws as the physical quantity it is connected with (in this case n(r,t)) and therefore has a similar kinetic equation. But that doesn't help much. I haven't been able to find any derivations or explanations for why Eq. (2) holds.
For example, here A. N. Vasil'ev, IEEE Trans. Nucl. Sci. 55, 1054 (2008) Eq. (3) it is simply stated that that's the case.
I've tried googling this but I'm not even sure what the right keywords would be. Correlation dynamics? Bimolecular kinetic equations?
P.S. Sorry if this is in the wrong section. It isn't homework but I could still use some help.