You can find it any book about time resolved spectroscopy. Simply use tau (av)={(alpha1.tau1)+(alpha2.tau2)}/(alpha1+alpha2) where tau's are decay components and alphas respective amplitudes.

So, to calculate average lifetimes for multi-exponential fits, you would need the fitting parameters such as Ai and (tau)i, where Ai is amplitude of ith lifetime component and (tau)i is the respective lifetime value. Now, it is important whether you acquired time resolved decays using transient absorption or time-resolved emission measurements. For transient absorption measurements, you have change in absorbance on y-axis which is directly proportional to excited state population decaying, so amplitude Ai is actually the number of excited state population decaying with (tau)i lifetime and your average lifetime would be:

= {Sigma (Ai* taui)}/{Sigma (Ai)}

However, if you acquired your decay profiles using time-resolved emission measurements (such as TCSPC), then the y-axis is actually the number of photons and the x-axis is arrival time. So, here Ai itself is not the number of photons with lifetime (tau)i, but it is equal to (Ai*(tau)i). So your average lifetime would be:

= {Sigma (Ai* taui)2}/{Sigma (Ai*taui}

Now, since you have mentioned in the question that these are fitting parameters from fluorescence experiment, you should use equation (2) to calculate avergae liftetimes. I hope this helps.

- Saurabh

You can use the Amplitude weighted average lifetime or the Intensity weighted average lifetime. In most cases you will need the intensity weight one, but for FLIM-FRET you should use the amplitude weighted lifetime. See a guide: http://www.picoquant.com/images/uploads/downloads/flim-fret-calculation_for_multi-exponential_donors_step_by_step.pdf

In the double exponential decay of the emitted photoflux from an excited material the photoflux phi can be expressed by:

phi(t)= ph1 exp -t/ Tau1 + phi2 exp -t/Tau2,

One can fit the decay curve with such an equation getting all the four parameters including the average lifetimes Tau1 andTau2.

In case of Taui is much smaller than Tau2 , one can observe at first the decay of Taue and then followed Tau2. It seems that this is the case of your material.

Best wishes

Hello Saurabh Chauhan ,

I like your answer but just one question: for the TCSPC measurements, you wrote = {Sigma ( (Ai* taui)^2 ) }/{Sigma (Ai*taui)}

but it should be = {Sigma ( Ai*( taui)^2 )}/{Sigma (Ai*taui)} ?

Jiawen, you are correct. Thanks. Saurabh

Saurabh Chauhan can you plz add reference for the second formula?

Thanks in advance.

Tamal Dey - if you are looking for some references, I recommend two papers:

(1) Article The Correct Use of “Average” Fluorescence Parameters

(2) Article On the choice of proper average lifetime formula for an ense...

If you carefully read those two papers you will find out that it is not that obvious which average lifetime formula should be used in a particular situation. Also, both formulas mentioned by Saurabh Chauhan can be written in the form of integrals and in such situation you do not need to fit the experimental data.