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While the microscopic mechanism leading to type II superconductivity is still debated, do scientists at least have a handle on the phenomonology?

Specifically, are the experimental properties of type II superconductivity understood well enough that it is possible in principle for the properties of a newly discovered material to lead to declaring a type III superconductor? Or is type II so poorly understood, that it is best described as "not type I" superconductivity, and thus leaving no room for a type III superconductor?

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There is a type "III". Its actually known as type I.5 because it displays characteristics of both type I and II: – user346 Apr 1 '11 at 4:36
This question is not quite right. Type I and II are descriptions only of phenomenology: whether vortices penetrate the bulk or not. There are classic (i.e. BCS/Eliashberg) superconductors in both. Whilst materials such as cuprates tend to be type II, the converse is simply not true. – genneth Apr 1 '11 at 9:26

There are clear distinctions between type I and type II superconductors. The first is the ability of the superconductor to support a mixed state region of magnetic flux, which is isolated to type II superconductors. The second is the coherence length of the superconducting pairs.

The breakdown in description of superconducting behavior lies in the ability to describe the superconducting mechanism. The BCS description of superconductivity describes well the superconductivity phenomena in type I and some type II materials. The type of superconductivity observed in high-Tc cuprates is not well described by BCS theory, but these materials are definitely classified as type II superconductors.

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