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, check details stimulated emission will Napabucasin 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 TSA HDAC nmr state populations and lifetimes for Tm3+, which ensue after pumping the 3H4 state at 800 nm, depend on the competition between SPTLC1 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].