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The figure below is Onnes' original data.

The transition region is very narrow. It suggests a first order phase transition, but still not so conclusive.

What is the latest understanding of the transition behavior? Is it continuous or discontinuous? What is the scale behavior?

enter image description here

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The transition shown in Onnes picture is a second order phase transition (the data is for Hg). The order of the transition refers to the continuity of the derivatives of thermodynamical potentials not of every measurable quantity.

In particular, following Landau's theory, the thermodynamic potential is written in terms of a so-called "order parameter". This is a quantity that is zero in the disordered state (generally high temperature) and finite in the ordered state. In a second order phase transition the order parameter goes continuously to zero in the disordered state.

For a superconducting transition, the order parameter is roughly the superconducting gap (it is a bit more complicated than that, but for this discussion the gap is enough). The gap is finite in the superconducting state and zero in the normal state. It does vanish continuously and behaves as $1 - (T/T_c)⁴$ close to $T_c$.

As a side note, the discontinuity in the resistivity is not a measurement of the order of the transition. Actually, the resistivity would be a very bad measurement for that. If you take a very disordered inhomogenous system, you still have may have a sharp transition if you get a quick percolation path between electrodes.

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