“…The rate equations for the population densities n 1 , n 2 , n 3 , n 4 (normalized to the concentration of Tm 3+ ions in the core N Tm = 5·10 19 cm −3 ) are given by 29 :where τ 4 , τ 4 R and τ 4 NR are the total, radiative and non-radiative lifetimes of level 4, respectively ( τ 4 = 0.3 ms 20,34 , τ 4 R = 0.4 ms 29 , τ 4 NR = (1/ τ 4 − 1/ τ 4 R ) −1 = 1.2 ms); τ 3 is the total (non-radiative) lifetime of level 3 ( τ 3 = 0.13 μs 20,34 ); τ 2 is the total lifetime of level 2 ( τ 2 = 3 ms 20,24,29 ); β 4y is the branching ratio from level 4 to level y = 1, 2, 3 ( β 41 = 0.9, β 42 = 0.07, β 43 = 0.03 29 ); K CR is the coefficient of cross-relaxation ( K CR = 1000 s –1 29 ), W xy are the stimulated rates. Stimulated rates by hydroxyl groups were neglected due to their extremely low concentration in the produced sample.…”