Energy transfer models

The energy transfer process can be modelled for a particular configuration of donor and acceptor ions with the following exponential function here $t$ is time, ${\gamma }_r$ is the radiative decay rate, and ${\gamma }_{tr,j}$ is the energy transfer rate for this configuration. Assuming a single-step multipole-multipole interaction (currently, the only model implemented), ${\gamma }_{tr,j}$ may be modelled as a sum over the transfer rates to all acceptors ${\gamma }_{tr,j}$ takes the form:

$${\gamma }_{tr,j}=C_{cr}\sum _i{\left(\frac{R_0}{R_i}\right)}^s$$ $I_j(t)=e^{-({\gamma }_r+{\gamma }_{tr,j})t}$ Here $R_i$ is the distance between donor and acceptor, $C_{cr}$ is the energy-transfer rate to an acceptor at $R_0$ the nearest-neighbour distance, and $s$ depends on the type of multipole-multipole interaction ($s$ = 6, 8, 10 for dipole-dipole, dipole-quadrupole and quadrupole-quadrupole interactions respectively). The term ${\sum }_i{\left(\frac{R_0}{R_i}\right)}^s$ forms the basis of our Monte Carlo simulation; we will refer to this as our interaction component. The lifetime for an ensemble of donors can be modelled by averaging over many possible configurations of donors and acceptors:

$$I(t)=\frac{1}{n}\sum _{j=1}^n{e}^{-({\gamma }_r+{\gamma }_{tr,j})t}$$

Where $n$ is the number of unique random configurations. We can also define an average energy transfer rate ${\gamma }_{av}$ defined as: $$\frac{1}{n}\sum _{j=1}^n{\gamma }_{tr,j}$$ This is a useful value for comparing hosts at a similar concentration.