  The importance of accounting for the inhomogeneity of the magnetic field
distribution and roundness of domain walls near the surface of type-I
superconductors in the intermediate state for forming the equilibrium flux
structure was predicted by Landau eight decades ago. Further studies confirmed
this prediction and extended it to all equilibrium properties of this state.
Here we report on direct depth-resolved measurements of the field distribution
and shape of domains near the surface of high-purity type-I (indium) films in a
perpendicular field using Low-Energy Muon Spin Rotation spectroscopy. We find
that at low applied fields (in about half of the field range of the
intermediate state) the field distribution and domains' shape agrees with that
proposed by Tinkham. However, for high fields our data suggest that reality
differs from theoretical expectations. In particular, the width of the
superconducting laminae can expand near the surface leading to formation of a
maximum in the static magnetic field in the current-free space outside the
sample. A possible interpretation of these experimental results is discussed.
