Time-dependent spectral transport: A Monte Carlo study

An inhomogeneously broadened spectral line, for a spatially random site distribution with dipole-dipole coupling to first-neighbor sites, is studied using a Monte Carlo Technique. The site-site coupling is chosen at random according to the probability density function for first-neighbor dipolar interactions $\frac{\left[\exp \right(-\frac{1}{X^{\frac{1}{2}}}\left)\right]}{X^{\frac{3}{2}}}$; where $X$ is the transfer rate proportional to $\frac{1}{r^6}$, and $r$ is the spatial separation of the interacting sites. This form is appropriate for a dilute concentration of sites on a three-dimensional random lattice. We arrange the sites in a linear array, and the spectral diffusion equation (including back transfer) is solved iteratively. We show that this is an adequate description of the three dimensional system at short and intermediate times for the dilute concentrations considered in this study. A group of sites with a specific energy is excited at (time) $t=0\mathrm{}$. The time development of the excitation profile is then studied. For very short times, the decay of the initial excitation is linear in time, appropriate to the largest value of $X$ (near neighbor site). For longer times, but over a fairly narrow time interval, the decay proceeds as $\exp (-a{t}^{\frac{1}{2}})$, as calculated by Inokuti and Hirayama for dipolar coupling in the absence of back transfer. At the longest times, the decay follows the one-dimensional diffusion result, $t^{-\frac{1}{2}}$. The intermediate-time regime, covering most of the decay of the initial excitation profile, does not seem to follow any simple time dependence.