Hexagonal vortex lattice

In addition to the hexagonal-square transition, a reorientation transition of the hexagonal vortex lattice from a state with the diagonal of the rhombic unit cell along [110] direction to [100] direction has been observed for YNi2B2C (Paul et al. 1998). Figure 51b shows that for H applied perpendicular to c the transition to a (nearly) square lattice occurs at a field of about 1 Tesla and at 0.8 Tesla a reorientation transition of the diagonal of the rhombic cell takes place (Sakata et al. 2000). 

An interesting question is whether the subtle effects of non-locality and, in particular, the hexagonal-square transition of the vortex lattice would be preserved in the superconducting state of magnetic superconductors such as / Ni2B2C with R = Er and Tm. 

The development of patterns is not necessarily a manifestation of a nonequilibrium process. A spatially non-uniform state can correspond to the minimum of the free-energy functional of a system in thermodynamic equihbrium, as Abrikosov vortex lattices, stripe ferromagnetic phases and periodic diblock-copolymer phases mentioned above. In the latter, a hnear chain molecule of a diblock-copolymer consists of two blocks, say, A and B. Above the critical temperature Tc, there is a mixture of both types of blocks. Below Tc, the copolymer melt undergoes phase separation that leads to the formation of A-rich and B-rich microdomains. In the bulk, these microdomains typically have the shape of lamellae, hexagonally ordered cylinders or body-centered cubic (bee) ordered spheres. On the surface, one again observes stripes or hexagonally ordered spots. 

Next came the likewise phenomenological Ginzburg-Landau theory of superconductivity, based on the Landau theory of a second-order phase transition (see also Appendix B) that predicted the coherence length and penetration depth as two characteristic parameters of a superconductor (Ginzburg and Landau, 1950). Based on this theory, Abrikosov derived the notion that the magnetic field penetrates type II superconductors in quantized flux tubes, commonly in the form of a hexagonal network (Abrikosov, 1957). The existence of this vortex lattice was... 

Fig. 26. Parameters describing diffraction from vortex lattice in superconducting UPtj as a function of field H applied along a, (a) Internal field deduced from the unit cell size. The line with slope unity is expected for singly quantized vortices, (b) Crystallographic angle a defined in the inset. For a conventional hexagonal lattice, a = 30°. (c) Scattering length, in magnetic field units, for the (1,0) Bragg reflection from the vortex lattice. (From Klerman ct al. 1992.)...

The crystal structure of UPts is hexagonal (see fig. 138, left) which gives rise to strong anisotropy of physical parameters (Shivaram et al. 1986, Zhao et al. 1991, Keller et al. 1994, Lussier et al. 1994, Fleddeijohann and Hirschfeld 1995). Direct detection of the vortex flux lattice by neutrons was reported by Kleiman et al. (1992). Magnetic neutron scattering and related work will be mentioned in context below. 

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