Series Solution of the Ginzburg-Landau Equations for the Abrikosov Mixed State

Periodic solutions of the Ginzburg-Landau (GL) equations in the form of a power series in the quantity $\frac{(H_{c2\mathrm{}}-B)}{B}$ are found. The lowest order term of the series satisfies the linearized GL equation. The order parameter, the free energy, and the magnetic moment are explicitly evaluated to the next higher order for the regular triangular lattice and the square lattice. These results evaluate the error in using the solutions of the linearized equation and extend the range of magnetic field for which the mixed-state configuration is known. An orthonormal set of functions in which the order parameter may be expanded is generated. The mixed-state solutions of the linearized equation with fluxoid quantum numbers greater than unity are determined and shown to have higher free energy than the unit-fluxoid solutions of the same symmetry.