# Distorted hexagonal lattices

When a vortex lattice forms in a Type-II superconductor, it usually forms a hexagonal lattice, although this can be distorted, for example by some sort of coupling to the underlying crystal lattice, or by the directions of any nodes in the superconducting gap.  In some cases this can give rise to a square lattice.  Here, we are going to focus on small distortions of the hexagonal latiice.  In the perfect hexagonal case, the triangles that make up the hexagon are equilateral.  The easiest distortion to consider is that these equilateral triangles become isosceles triangles.  If they become scalene triangles, as in niobium, this is a bit trickier, and we will leave that for later.

Each flux line in the vortex lattice carries one quantum of magnetic flux, Φ0 = 2.067833848 x 10-15 Wb, and so as the external magnetic field is increased, the number of flux lines passing through the material must increase, until the superconductivity collapses. If we assume that the vortex lattice is perfect, with no defects, then each unit cell of the hexagonal lattice contains one flux line, and the area of that unit cell is a function of the external magnetic field, B.

This area should remain unchanged irrespective of the distortion of the hexagonal lattice, and this obviously imposes known constraints on the unit cell parameters. Vortex lattices can be studied in reciprocal space as well as real space, which is the way that I usually investigate them. The Bragg reflections in reciprocal space are determined by the plane spacings (or d-spacing) in the real-space lattice. The end result is that in reciprocal space, the Bragg reflections are arranged hexagonally, but that hexagon is rotated by 90o. Then we can look at how the same distortion will appear, while maintaining a fixed unit cell area (which is determined by the applied magnetic field). In the example below, the angle γ has increased from 120o to 130o. What we typically measure in a diffraction study of a vortex lattice is the position of the Bragg reflections, which enables us to measure the angles between the spots, like ξ and η in the pictures above. We can also measure the momentum transfer wavevector associated with given peaks, from which we can eventually get the d-spacing.  In the picture above, the red spots should have one magnitude of wavevector, and the blue spots should have a different one.

We can see how these wavevectors vary as a function of the real space opening angle γ, as well as the two opening angles, ξ and η, in reciprocal space.  The example below is calculated for B = 1.6 T, and the pictures at the top are for γ = 135o.  The lowest frame in the picture shows the ratio of the largest real-space unit cell parameters (a and b) over the smallest.  This is sometimes used as a means to assess the amount of anisotropy from real-space images.  This particular factor is field independent. The opening angles ξ and η also depend only on γ, and so are field independent.  This means that the information about the actual field is only encoded in the observed wavevectors.  The ratio of minimum and maximum measured wavevectors will also be field-independent, and give the same answers as the ratio of maximum and minimum unit cell parameters.

Finally, an animated illustration. 