Excited level, steady-state rate equation

For excitation from level 1 to 2 the steady state rate equations become... 

In the absence of the SLR processes, the value of this ratio would depend on the mode of preparation of the triplet state. Three extreme cases could be realized, in the first case, the system is being continuously excited and decays continuously, i.e., the system is in a steady state. By equating the rates of pumping of any zf level with its rate of decay, one obtains the following ratios ... 

Under direct Lnm excitation and assuming that the population of the ground state (Ng) of the emitting 4f ion decays rapidly to populate the first excited state (Ne), the rate equations reduce to those of a two-level system, and further assuming steady state, one gets... 

Depending on the mode of excitation, n /n would be related to the rate constants involving the and Tj levels as given by eqs. (1) - (3). Thus for a system which exists in a steady state before sweeping the microwaves, the following equation is applicable ... 

Equation (90) can be modelled by a 4-level system, involving 2G9/2 (f), 4F9/2, 4I9/2 (/) and 4I 5/2 [380], similar to Eqs. (88)a,b, except that level/is fed by branching from 4F9/2, instead of being pumped directly from the ground state. Whereas the solution Eq. (89) refers to the steady state, the solution for the decay of 2G9/2 luminescence following pulsed excitation is rather more complex. Although the experimental decay curve can be well-modelled, the upconversion rate constant U is not well-determined [380]. 

The Bloch equations (Eq. 5) can be solved under different conditions. The transient solution yields an expression for 0-22 (0> time-dependent population of the excited singlet state S. It will be discussed in detail in Section 1.2.4.3 in connection with the fluorescence intensity autocorrelation function. Here we are interested in the steady state solution (an = 0-22 = < 33 = di2 = 0) which allows to compute the line-shape and saturation effects. A detailed description of the steady state solution for a three level system can be found in [35]. From those the appropriate equations for the intensity dependence of the excitation linewidth Avfwhm (FWHM full width at half maximum) and the fluorescence emission rate R for a single absorber can be easily derived [10] ... 

When a coherent laser field with average incident energy density W and frequency co interacts with a collection of N two-level atoms in ordinary vacuum, the steady state behavior of the system is governed by the well-known Einstein rate equations. These equations implicitly make use of the smooth nature of the vacuum density of states = o l 7t c ) in the vicinity of the atomic transition frequency co coq. In steady state equilibrium, the ratio of the number of excited atoms N2 to the total number of atoms is given by (Laudon, 1983)... 

Unfortunately the excitation rate to the lower level can never be zero since even in the absence of collisions with atoms and electrons, the lower laser level is populated by spontaneous emission from the upper level k at the rate k ki" minimum requirement which must be satisfied for a steady state inversion to be possible may be obtained in this case by substituting S =0 into equation (11.15) giving... 

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