Magnetic flux density and the critical field in the intermediate state of type-I superconductors

To address unsolved fundamental problems of the intermediate state (IS), the equilibrium magnetic flux structure and the critical field in a high purity type-I superconductor (indium film) are investigated using magneto-optical imaging with a 3D vector magnet and electrical transport measurements. The least expected observation is that the critical field in the IS can be as small as nearly 40% of the thermodynamic critical field $H_c$. This indicates that the flux density in the $\mathit{\text{bulk}}$ of normal domains can be $\mathit{\text{considerably}}$ less than $H_c$, in apparent contradiction with the long established paradigm, stating that the normal phase is unstable below $H_c$. Here we present a novel theoretical model consistently describing this and $\mathit{\text{all}}$ other properties of the IS. Moreover, our model, based the rigorous thermodynamic treatment of observed laminar flux structure in a tilted field, allows for a $\mathit{\text{quantitative}}$ determination of the domain-wall parameter and the coherence length, and provides new insight into the properties of all superconductors.