Magnetic structures in type-II superconductors with impurities

It is theoretically shown that in type-II superconductors with an arbitrary concentration of impurities a periodic lattice structure of flux lines exists in the whole region of the mixed state. This is a generalization of Usadel's remark, given in the dirty limit. Eilenberger's transport-like equations are used. The flux is shown to be always quantized. Symmetry properties make it possible to expand Eilenberger's functions, the order parameter and the magnetic field in a Fourier series with respect to spatial variation. Equations for the Fourier coefficients are derived. They are solved in two cases in the dirty limit. First, in the region close to Hc2 where the order parameter is so small that the power series expansion is applicable. The well-known results by Maki are reproduced. Second, in a wider range of the mixed state, where one can derive Takayama and Maki's result by introducing an additional approximation. The approximation is found to be applicable except at very low temperatures.