A theory waiting for an experiment: pair-state formation in a nanocrystal

For certain dopants, luminescence measurements allow one to distinguish between single-ion and pair-state dopant emission. In a bulk crystal the concentration of each of these dopant-states can be calculated from the dopant fraction f present in the material, and is found to correlate with luminescence measurements. However, for a nanocrystalline host-lattice, these concentrations have not been calculated due to the difference in coordination numbers for ions at the surface (a substantial fraction in nanocrystals) and in the bulk. Simulations were recently published. Here, the exact solution is presented (though not derived), which allows one to calculate the fraction of dopants that are part of a pair-state for any dopant concentration, crystal size or crystal structure, without having to perform simulations. Several examples are presented and discussed.