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What is decay associated spectra?

Suppose we measure the fluorescence intensity over different wavelengths and over time, we get:

$$I(\lambda,t) = \sum_i^n \alpha_i(\lambda) \exp(\frac{-t}{\tau_i}).$$

The assumption is that there are n component,species, in the $I(\lambda,t)$. If we fit the right hand to the experimentally obtained $I(\lambda,t)$, and get $\alpha_i$ and $\tau_i$, then people call $\alpha_i$ the decay associated spectra.

Now, if we integrate over time we get the steady state emission spectra. The thing that I cannot understand is the decay associated spectra. What does it mean? If it is the steady state spectra of species i, then why does it become negative sometimes? People, say that when it becomes negative, it indicates energy transfer between species. Could someone please elaborate this concept more?

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Are you saying that sometimes $a_i$ is negative? –  garyp Feb 24 at 22:20
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@garyp: Yes, he is. It's a result of incorrect use of decay associated spectra in a situation where a species associated spectra model is necessary. –  DumpsterDoofus Feb 24 at 22:32
    
@DumpsterDoofus Negative values appear after a curve-fitting routine? –  garyp Feb 24 at 22:54
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@garyp: Yes, see the answer I posted below. It's a modeling anomaly, indicating that the model is incorrect. –  DumpsterDoofus Feb 24 at 22:57

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A bit of clarification: when a component of a decay associated spectrum (DAS) which takes on negative values for certain values of $\lambda$ is observed, that isn't actually meant to say that the component actually has a negative spectrum in physical reality. Rather, it is a sign that the modeling technique used, $$I(\lambda,t) = \sum_i^n \alpha_i(\lambda) \exp\left(-\frac{t}{\tau_i}\right),$$ is failing to correctly fit the decay process. In other words, negative values are a sign that the model is incorrect and is giving nonsense results that are incorrect.

A decay associated spectrum is defined as the spectra of the components that are calculated when fitting the experimental decay spectrum $I_e(\lambda,t)$ to a model in which there are $N$ chemical components whose decay transfer matrix $\mathbf{K}$ is diagonal, ie $$\mathbf{K}=\text{diag}(k_1,k_2,...,k_N).$$ In this situation, you obviously have individual exponential decays for each of the $N$ components, with time constant $\tau_i=k_i^{-1}$.

However, very often real photophysical systems are more intertwined, and their decay transfer matrix $\mathbf{K}$ is more generally a real symmetric matrix. By the spectral theorem, this can be diagonalized and solved for the decays; when fitting experimental data to this model, the spectra of the components calculated are referred to as species associated spectra (SAS).

In short, if you fit to a DAS model and you get negative DAS values, you're doing it wrong, and the system is more complicated than multiexponential decays, so you need to do an SAS fit.

Here is a decent article on the subject, which explains things in more detail.

Apologies in advance if anything that I wrote above is wrong, I just learned it all 10 minutes ago.

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