Past exam of the condensed matter physics major option course of the University of Oxford / 2018 / 5 / b / Solution

For an ideal long specimen with negligible demagnetizing field, the Meissner state has $B={\mu }_0(H+M)=0$, so $M=-H$. A type-I superconductor retains this response up to its thermodynamic critical field $H_c(T)$ and then enters the normal state, where the small normal magnetic response can be neglected. Nonzero demagnetizing fields can create an intermediate state and broaden the measured transition.

A type-II superconductor is fully screening only below $H_{c1}(T)$. Between $H_{c1}$ and $H_{c2}$, magnetic flux enters as quantized vortices and the reversible magnetization remains diamagnetic but decreases in magnitude towards zero at $H_{c2}$. Above $H_{c2}$ the bulk is normal. Ginzburg–Landau theory distinguishes the types at $\kappa =1/\sqrt{2}$, not at $\kappa =1$; the later condition in the source merely restricts its estimate.

The $H$–$T$ sketch has two increasing-width regions below $T_c$: Meissner below the lower boundary, mixed vortex state between the two boundaries, and normal state above the upper boundary. Both critical fields vanish at $T_c$. The phase-diagram panel of the original figure in solution shows this topology; its normalized field curves are illustrative, not a fit to the source magnetization loop.