Rotation and inversion symmetry properties of flux-line lattice in type II superconductors

Abstract

Magnetic flux lines are arranged regularly in the mixed state of a type II superconductor and form a two-dimensional triangular lattice. This is unaltered on rotation through an angle π/3, one of the flux lines being an axis of symmetry of the sixth order. It is also symmetric with respect to the inversion. The center of symmetry is any point on a flux line. The symmetry properties of the magnetic field, the order parameter, and Eilenberger's integrated Green's function are derived. The magnetic field is developed into Fourier series with the help of a two-dimensional reciprocal lattice. The normalized Fourier coefficient is the form factor in neutron diffraction and is shown to have a well-known property of the rotation symmetry. The Fourier transform of the normal Green's function satisfies a similar symmetry relation with an additional phase factor. When we expand the anomalous Green's function in terms of plane wave functions along the direction of one of the nearest-neighbor flux lines, the symmetry properties give useful conditions which the expansion coefficient satisfies as a function of the orthogonal coordinate and wave number vector. In addition to the plane wave expansion, the order parameter can be developed in terms of the harmonic oscillator eigenfunctions of the orthogonal coordinate with 6n quanta. The first term withn = 0 reproduces Abrikosov's solution. The symmetry properties help us effectively simplify the Eilenberger equation. An example of the simplification is given.