Wants to know the difference between the lifetime of Carriers, calculated from the TRPL and TCSPC.?
Anytime you have a transition from one state to another that depends only on the concentration of the first species and a kinetic rate, you'll fit the data with an exponential function. Atomic or molecular electronically excited states converting to ground states obey this. Separated electron-hole pairs recombining obey this.
Time Resolved Photoluminescence is a generic term covering any technology that can measure the time the system was excited and the time a luminescence photon was released. TCSPC is a specific design of TRPL involving a pulsed light source, two fast detectors, and a photon counting board. In TCSPC, it is almost always important to collect an instrument response function (IRF). That IRF depends on the pulse width of your light source as well as your collection electronics (detectors, wires, boards). Fitting involves convoluting that IRF with an exponential (or multiexponential) function.
If your TRPL system uses a pulsed light source, whether that be a femtosecond laser (ps) or a tungsten flash lamp (ms) to excite, and you see something that looks exponential as your data, this is the way you fit it.
If you have, for example, a sinusoidally modulated light source, you will ultimately get the same information, but the fit function will be quite different.
They're the same - or that is to say TCSPC is a way to do TRPL
In TCSPC (Time-correlated single photon counting) a pulsed laser is shot at the sample and the time between the pulse and a photon being detected is measured and binned in a histogram. Fitting this histogram to an exponential decay function can be used to find lifetime. This is an example of time resolved photoluminescence lifetime, it is measured in the time domain
Alternatively, you can measure lifetime in the frequency domain, though this is something I am not familiar with. I would recommend the textbook 'Principles of Fluorescence spectroscopy' by Lakowicz to gain a deeper understanding of these concepts
Jeremy Pronchiks answer is very good, I did not fully read before answering, apologies for repeated information
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