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Autocorrelation Due to Intersystem Crossing (Triplet State)

For multicomponent, three-dimensional diffusion the autocorrelation function including intersystem crossing is given by

Detailed Derivation

Fluorescence Reviewback to top

In Figure 1 we show a Jablonski diagram of a two state fluorescence system. The ground state is designated S₀ and the first excited state is designated S₁. The system starts off in the lowest vibrational energy level S₀ of the ground state.

A photon is absorbed and elevates the molecule to one of the vibrational energy levels of the first excited state. The forward rate constant for this transition is k₁₂. The molecule then decays extremely rapidly to the lowest vibrational energy level of the first excited state. We can treat this as instantaneous.

From the lowest energy level, the system then decays to one of the vibrational energy levels of the ground state with a rate constant of k₂₁. It should be noted that k₂₁ is the sum of a radiative decay rate, k_r and a nonradiative decay rate k_nr. Significantly, we note that the fluorescence is k_r times the concentration of the S₁ state, with additional extrinsic multipliers due to collection efficiency, etc.

In addition to their system of singlet states, fluorophores typically have a set of parallel states (excluding the ground state) referred to as triplet states. To go from a singlet to a triplet state is a so-called forbidden transition and therefore occurs with a small rate constant. In Figure 2, we have added the triplet state T₁, which is associated with the first singlet excited state S₁. Transitions from S₁→T₁ and from T₁→S₀ are forbidden.

These transitions represent a crossing between the system of singlet and the system of triplet states. The fluorescence effectively blinks off as molecules move into the triplet states and back on when they move into the singlet states. Effectively, because k₃₁ and k₂₃ are much smaller than k₁₂ and k₂₁, this blinking on and off is driven solely by the intersystem crossing.

Intersystem Crossing Transition Ratesback to top

From Figure 2 we can create a set of coupled differential equations for the three states which is given by

Equation (1) is subject to the boundary condition that at time zero all of the molecules are in the ground state. Thus,

We can, of course, solve Equation 1 and Equation 2 for all states. However, since we are interested in the fluorescence intensity which is proportional to the number of molecules in S₁, we will solve only for that state.

Considering Equation 3 and the fact that k₁₂ and k₂₁ are large compared to k₂₃ and k₃₁, the constant term represents the equilibrium value that the system attains at infinite time. The middle term has a rate constant which is dependent upon only k₁₂ and k₂₁ and is, therefore, large. This means that the middle term vanishes at a time scale which is much shorter than we are measuring. The third term is the intersystem crossing term about which we must concern ourselves. We may therefore rewrite equation (3) as

Where

Note that as t goes to infinity must go to 1-T.

Intersystem Crossing Rates in Autocorrelationback to top

We are here going to ignore the dependence of k₁₂ on position r. For a full treatment the reader is referred to to Widengren et al. (1995). Effectively intersystem crossing alters the quantum efficiency of the system. The molecule divide up into two fractions. There, is a fraction T which are now nonfluorescent and a fraction (1-T) + Te^-t/t_T which remain fluorescent. This fraction becomes a multiplier to the correlation function. Additionally the N_p in the correlation function must be replaced with (1-T)N_p. The correlation function becomes