# Meaning of generalized normal distribution

I asked a version of this question over on Math.SX, and never received a response… perhaps it will be more appropriate here.

I'm looking at spectroscopic data (specifically a $T_2$ coherence decay curve of some NMR data). Normally, this data is fit to a single or multi-exponential decay to account for multiple components. However, I have a data set that fits best to a function with a power in the exponent near 1.4 (in between a Gaussian and single exponential decay).

Is there any physical meaning for generalized normal distribution functions? To elaborate on what I mean by "physical meaning", when working with spectroscopic absorptions, an an exponential decay (n=1) indicates a system with homogeneous broadening of lifetimes, while a Gaussian decay indicates inhomogenous broadening of lifetimes. What does a power between these two values indicate? Is there a precedent for using this sort of peak shape (or decay function in this case) in spectroscopic analysis?

### --EDIT--

To demonstrate the phenomenon, here are a couple of sample curves with some data. The depressed points at the start may be an experimental artifact, but I'd still be curious to know if there is any physical precedent for the exponential power between 1 and 2.

$e^{-t/T_2}$ $e^{-(t/T_2)^{1.6}}$ • You may be looking at a sum of several transition lifetimes (i.e. several events make up an observable decay), so you get a non-exponential distribution as in my answer here, but, on the other hand, not enough random variables are being summed, so the sum-of-lifetimes doesn't quite behave as the Gaussian limit of the Central Limit Theorem. See the section "Proof of classical CLT" in the Central Limit Theorem Wikipage and reflect on the proof a bit and I think you'll see what's going on. – Selene Routley Feb 17 '14 at 5:06
• Would Cross Validated be a better home for this question? – Qmechanic Feb 17 '14 at 5:58
• @DustinWheeler: I've never heard of a Gaussian decay occurring in NMR coherence. Do you have a reference for that? Typical decoherence processes are self-similar and thus exponential. – DumpsterDoofus Feb 17 '14 at 22:49
• @DustinWheeler: The only reference I was able to find for the existence of a Gaussian decay in NMR is by artificially introducing it through spectral postprocessing procedures like "Lorentzian-to-Gaussian", which isn't actually caused by a physical process, but rather is a computational technique. – DumpsterDoofus Feb 17 '14 at 23:49
• @DumpsterDoofus: The Gaussian comment was a bit off-the-cuff, but Gaussian peaks are common spectral observations in the field of solid-state NMR, where inhomogeneous broadening is common due to static nuclear positions and orientations, and inhomogeneous dipolar fields about a nucleus. They are not visible in a $T_2$ type experiment such as the one I've described though. – Dustin Wheeler Feb 18 '14 at 21:50

## 1 Answer

Thanks to @user12262 for pointing me in the direction of the KWW function. After perusing that link and searching SciFinder for stretched and compressed exponential functions in relation to NMR, I ran across this paper (subscription required, sorry).

To (briefly) summarize the paper, the compressed exponential function, $e^{-kt^q}$, with $1 < q < 2$ can be represented as a distribution of Gaussian functions with different relaxation rates,

$$R_C (t) = \frac{1}{\pi} \int^∞_0 P_C(s; q)\, e^{−(sr^*t)^2} d\textrm{s},$$

where $R_C(t)$ is the observed decay curve, $P_C(s; q)$ is the probability distribution of Gaussian decays, $r^*$ represents some average value of the rate, and $s = r/r^*$. As the value of $q$ approaches 2, the distribution function approaches a delta spike (as one would expect).

In the case of NMR $T_2$ decays, this most likely represents a distribution of relaxation couplings (e.g. interactions with 1, 2, 3, etc. other nearby spins).

• I see now the reason for your response to my comment. The above is a little different from what I meant by "sum of lifetimes": apologies for my lack of clarity. What I meant was that a particle has to go through one process, with a certain lifetime, before it begins the next process, with another lifetime, and so on, so the pdfs convolve. If you get enough of them convolving, the overall effect is Gaussian, which you can understand by looking at the Taylor series for the characteristic function: this is how the classical proof of the central limit theorem works. – Selene Routley Feb 27 '14 at 22:54