FCSXpert Solutions: Fluorescence Correlation Spectroscopy Simplified!.
## FCS Classroom

#### Autocorrelation Due to Intersystem Crossing (Triplet State)

### Detailed Derivation

#### Fluorescence Reviewback to top

#### Intersystem Crossing Transition Ratesback to top

#### Intersystem Crossing Rates in Autocorrelationback to top

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

In Figure 1 we show a Jablonski diagram of a two state fluorescence system. The ground state is designated S_{0} and the first excited state is designated S_{1}.
The system starts off in the lowest vibrational energy level S_{0} 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 k_{12}. 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 k_{21}.
It should be noted that k_{21} 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 S_{1} 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 T_{1}, which is associated with the first singlet excited state S_{1}.
Transitions from S_{1}→T_{1} and from T_{1}→S_{0} 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 k_{31} and k_{23} are much smaller than k_{12} and k_{21}, this blinking on and off is driven solely by the intersystem crossing.

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

. (1)

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

. (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 S_{1}, we will solve only for that state.

. (3)

Considering Equation 3 and the fact that k_{12} and k_{21} are large compared to k_{23} and k_{31}, 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 k_{12} and k_{21} 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

. (4)

Where

. (5)

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

We are here going to ignore the dependence of k_{12} 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 N_{p} in the correlation function must be replaced with (1-T)N_{p}.
The correlation function becomes

. (6)