FCSXpert Solutions: Fluorescence Correlation Spectroscopy Simplified!.
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Appendices

FCS Classroom

Autocorrelation Due to Intersystem Crossing (Triplet State)

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

equation

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.

Jablonski Diagram Singlet
Figure 1 : Jablonski Diagram of simple two singlet state fluorescence

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, kr and a nonradiative decay rate knr. Significantly, we note that the fluorescence is kr times the concentration of the S1 state, with additional extrinsic multipliers due to collection efficiency, etc.

Jablonski Diagram Triplet
Figure 2 : Jablonski Diagram of two singlet states and triplet state

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)

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

equation
. (2)

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.

equation
. (3)

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

equation
. (4)

Where

equation
. (5)

Note that as t goes to infinity equation 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/tT which remain fluorescent. This fraction becomes a multiplier to the correlation function. Additionally the Np in the correlation function must be replaced with (1-T)Np. The correlation function becomes

equation
. (6)