I apologize if something like this has already been asked, but I don't have any ideas on search terms for this family of functions.
I'd like to know two things:
1) Is there a name for the family of functions of the form
$$f(t)= \exp{(-kt^n)}$$
Specifically, for $n=1$, it's just an exponential decay, and for $n=2$, it's a Gaussian (or Normal) distribution. What about situations where n is fractional (i.e. ${3}/{2}$)?
2) Is there any physical meaning for these types of functions? I realize this portion might belong on the Physics.SX site, so if no one has ideas, I can transfer part 2 over there. To elaborate on what I mean by "physical meaning", I work with NMR signals, and an exponential decay $(n=1)$ indicates a system with homogeneous broadening of lifetimes, and a Gaussian decay indicates inhomogenous broadening of lifetimes.
The function is (for my specific case) a compressed exponential function, and the general function family is the generalized normal distribution. For a physically relevant application of the compressed exponential, see the cross-posting here. Essentially, it comes from the observed relaxation curve being comprised of a distribution of Gaussian decay curves.