Self-quenching (also called cross-relaxation) occurs between two identical molecules (ions) when a first molecule (or ion) initially in an excited state exchanges energy with the second molecule (ion) that is initially in the ground state, resulting in both molecules (ions) simultaneously changing to excited states that are intermediate in energy between the two initial states. The drop in energy for the first molecule (ion) is equal to the increase in energy for the second molecule (ion), thereby conserving energy in self-quenching. In a common example of self-quenching, the intermediate states for the two molecules (ions) have the same energy and are energetically halfway between the initial excited state of the first molecule (ion) and the ground state of the second molecule (ion).
The example diagram below shows two sets of energy states for two identical Tm3+ ions (the energy states of each ion are enclosed within green rectangular boxes) and transformations of their current states under self-quenching. The pair of Tm3+ ions can undergo self-quenching into the 3F4 excited state (1) when one of the ions is initially excited into the upper 3H4 excited state (2) from the 3H6 ground state (0) by absorbing light at 780 nm (top left portion of the diagram) and thereby promoting an electron to the 3H4 state (shown in the bottom left portion of the diagram). This is depicted in the diagram by placing ‘e-‘ at the level of the 3H4 state. The second ion has its electron in the ground state 3H6. The two Tm3+ ions exchange energy by self-quenching (shown in the bottom right portion of the diagram). The electron on the left Tm3+ ion loses energy and transitions to the intermediate excited state 3F4 halfway between ground state 3H6 and excited state 3H4. Simultaneously the electron on the right Tm3+ ion gains energy and also transitions to the intermediate excited state 3H4. As the result, two ions are in the same excited energy state 3F4.
If one restricts the model to only three different states for Tm3+ ions, the rate equations describing cross-relaxation would have the following form:
∂N0 / ∂t = − c N0 N2
∂N1 / ∂t = 2 c N0 N2
∂N2 / ∂t = − c N0 N2
where c s a cross-relaxation rate for this type of ion, and N0, N1, N2 are population densities of ions that are in the 3H6 ground state (0), the 3F4 intermediate excited state (1) and the 3H4 upper excited state (2), respectively.
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