# Fluorescence intensity decay models

Excited state decay of organic fluorophores

After electronic excitation the molecule typically decays to the lowest vibrational level of the first excited singlet state, S_{1}. In far most cases, this happens non-radiatively in the order of pico-seconds by a combination of internal conversion (IC) and vibrational relaxation. From S_{1} a number of decay processes can occur to the electronic ground state, S_{0}. If the oscillator strength of the S_{0}-S_{1} transition is high the molecule may decay to the ground state via emission of radiation (fluorescence) and the molecule is thus termed a fluorophore.

The fluorescence, however, "competes" with non-radiative decay processes such as IC, intersystem crossing (ISC), or a bimolecular quenching process such as FRET and collisional quenching. Each process is associated with a rate constant, *k*_{i}. The process with the largest value of *k*_{i} dominates the decay. The total rate of decay from the excited state, then, is typically given by that of a unimolecular process:

where the sum is over all decay processes from the excited state. The solution to this 1. order differential equation is

where [S_{1}] is the concentration of fluorophores in S_{1} at time *t* and [S_{1}]_{0} is the concentration of fluorophores in S_{1} at time *t* = 0. The lifetime of the fluorophore, *τ*, is defined as the time it takes a population of excited fluorophores to reach 1/*e* of [S_{1}]_{0}. Inserting this in the equation above and isolating *t* we get

where again the sum is over all decay processes from S_{1}. The lifetime also corresponds to the average time that the molecule spends in the excited state.

Assuming that the fluorescence intensity is proportional to the excited state population, [S_{1}], the intensity decay is exponential and given by:

where *I*_{0} is the intensity at *t* = 0. If the sample possess more than one lifetime the intensity as a function of time is the sum of intensities from each decay:

which is called a multi-exponential decay. *α*_{j} is the pre-exponential factor and can be used to represent the fraction of fluorophores with lifetime *τ*_{j}. The amplitude-weighted lifetime of the fluorophore is then given by:

**Fluorescence decay models in ****DecayFit**

In the DecayFit freeware the implemented intensity decay models are listed in the Fit models listbox:

The intensity decay expression of the fit model selected in the listbox is shown below the listbox:

A description of each of the fit models is provided below.

1. **single_exp**

A single-exponential decay corresponding to a single lifetime. Only the lifetime is provided as fit parameter.

2. **double_exp**

A double-exponential decay corresponding to two lifetimes. The sum of the pre-exponential factors is constrained to 1 which means that only one pre-factor, a_{1}, is provided as fitting parameter while the other is set to be 1-a_{1}.

3. **triple_exp**

A triple-exponential decay corresponding to three lifetimes. The sum of the pre-exponential factors are constrained to be a_{1}+a_{2}+a_{3} = 1.

4. **four_exp**

A four-exponential decay corresponding to four lifetimes. The sum of pre-exponential factors are constrained to be a_{1}+a_{2}+a_{3}+a_{4} = 1.

5. **FRET**

A decay model for analyzing FRET data. The model implements a Gaussian distance distribution between the donor (D) and acceptor (A) with the distribution center (**R-mean**) and Full Width at Half Maximum (**FWHM**) as fitting parameters. The critical distance, **R**_{0}, must be supplied as input and is not meant to be a fitting parameter (although this is possible by varying the *min *and *max *values).

Up to three lifetimes can be specified for the donor in absence of acceptor (**tauD**_{1}, **tauD**_{2}, **tauD**_{3}), each with a pre-exponential factor (**a**_{1}, **a**_{2}, **a**_{3}). These are not meant as fitting parameters but should be determined on a donor-only reference sample. If the donor has a single lifetime only set the *min *and *max *values of **a2** and **a3** to 0 (default).

The fraction of donors not coupled to an acceptor, **f**, can also be included as a parameter, either constrained or loosely fitted.

An additional component not involved in FRET with a lifetime '**tauB**' and relative fraction '**b**' can be included in the model. The fraction of the FRET-component in the decay, then, is 1-b.

*Note*: When using dynamic averaging, the D-A distance is modelled using a single distance - the mean of the Gaussian distribution (R-mean).

6. **lifetime_dist**

A Gaussian lifetime distribution model. A mixture of two independent Gaussians can be used, each with a lifetime center and a lifetime distribution width (FWHM) as fitting parameters.

7. **double_exp_uncon**

A double exponential decay in which there is no constraint on the sum of pre-exponential factors.

8. **triple_exp_uncon**

A triple exponential decay in which there is no constraint on the sum of pre-exponential factors.