For the process C2 that feeds both the 3F4 and 3H5 levels, the energy gap is a deficit of -641 cm-1. This process must absorb three phonons from the lattice to complete. However, phonon absorption processes have much stronger temperature dependence than phonon-emitting processes. At low temperatures,

any relaxation process that emits phonons, such as cross-relaxation or multi-phonon relaxation, can proceed through spontaneous emission. At high temperatures, stimulated emission will BMS-907351 purchase occur as phonon occupation increases, which increases the relaxation rate. Therefore, the temperature dependence of the rate for a phonon emission process W e is given by (4) where N e is the number of phonons (ΔE/ħω) emitted to fill the energy gap ΔE that have energy ħω and n is the phonon occupation number [35]. However, phonon absorption processes must have occupied phonon states in order to proceed. The temperature dependence of the rate W a for a phonon absorption process is given by (5)

where N a is the number of phonons absorbed. The temperature dependencies of Equations 4 and 5 arise because the phonon occupation number n follows a Bose-Einstein distribution given by (6) where ħω is the maximum phonon energy (260 cm-1 for YCl3) [36]. Therefore, the maximum phonon energy is the most important parameter in controlling

the temperature and energy gap dependence of all phonon-assisted relaxation processes, including cross-relaxation and multi-phonon relaxation. Excited selleck products state populations and lifetimes for Tm3+, which ensue after pumping the 3H4 state at 800 nm, depend on the competition between Fenbendazole spontaneous emissions of radiation, cross-relaxation, multi-phonon relaxation, and up-conversion. At temperatures greater than 500 K, multi-phonon relaxation is the dominant process, which results in quenching of the fluorescence from all levels. At room temperature, near 300 K, multi-phonon relaxation is reduced and cross-relaxation can proceed. However, at 300 K, the occupation of phonon states is still substantial, which allows the endothermic process C2 to compete with the exothermic process C1. A macroscopic model of the populations of the four lowest levels of Tm3+ was constructed using coupled time-dependent rate equations [33]. Rate constants for spontaneous emission, cross-relaxation, and up-conversion were determined by fitting the model to fluorescence lifetime data at 300 K, a temperature at which multi-phonon relaxation can be neglected. Rate constants for multi-phonon relaxation were determined by fitting the model to lifetime data above 400 K, temperatures at which multi-phonon relaxation is significant [33].