  Equilibrium magnetic properties of the mixed state in type-II superconductors
were measured with high purity bulk and film niobium samples in parallel and
perpendicular magnetic fields using dc magnetometry and scanning Hall-probe
microscopy. Equilibrium magnetization data for the perpendicular geometry were
obtained for the first time. It was found that none of the existing theories is
consistent with these new data. To address this problem, a theoretical model is
developed and experimentally validated. The new model describes the mixed state
in an averaged limit, i.e. %without detailing the samples' magnetic structure
and therefore ignoring interactions between vortices. It is quantitatively
consistent with the data obtained in a perpendicular field and provides new
insights on properties of vortices. % and the entire mixed state. At low values
of the Ginzburg-Landau parameter, the model converts to that of Peierls and
London for the intermediate state in type-I superconductors. It is shown that
description of the vortex matter in superconductors in terms of a 2D gas is
more appropriate than the frequently used crystal- and glass-like scenarios.
