paper by A. A. Abrikosov, generalizes these ideas. In the limit of very short coherence length the vortex solution is identical to London's fluxoid, where the vortex core is approximated by a sharp cutoff rather than a gradual vanishing of superconducting condensate near the vortex center. Abrikosov found that the vortices arrange themselves into a regular array known as a vortex lattice. Near a so-called upper critical magnetic field, the problem of a superconductor in an external field is equivalent to the problem of vortex state in a rotating superfluid, discussed by Lars Onsager and Richard Feynman. Flux pinning In the vortex state, a phenomenon known as flux pinning becomes possible. This is not possible with type-I superconductors, since they cannot be penetrated by magnetic fields. If a superconductor is cooled in a field, the field can be trapped, which can allow the superconductor to be suspended over a magnet, with the potential for a frictionless joint or bearing. The worth of flux pinning is seen through many implementations such as lifts, frictionless joints, and transportation. The thinner the superconducting layer, the stronger the pinning that occurs when exposed to magnetic fields. Materials Type-II superconductors are usually made of metal alloys or complex oxide ceramics. All high temperature superconductors are type-II superconductors. While most elemental superconductors are type-I, niobium, vanadium, and technetium are elemental type-II superconductors. Boron-doped diamond and silicon are also type-II superconductors. Metal alloy superconductors can also exhibit type-II behavior (e.g. niobium-titanium, one of the most common superconductors in applied superconductivity), as well as intermetallic compounds like niobium-tin. Other type-II examples are the cuprate-perovskite ceramic materials which have achieved the highest superconducting critical temperatures. These include La1.85Ba0.15CuO4, BSCCO, and YBCO (Yttrium-Barium-Copper-Oxide), which is famous as the first material to achieve superconductivity above the boiling point of liquid nitrogen (77 K). Due to strong vortex pinning, the cuprates are close to ideally hard superconductors. Important uses Strong superconducting electromagnets (used in MRI scanners, NMR machines, and particle accelerators) often use coils wound of niobium-titanium wires or, for higher fields, niobium-tin wires. These materials are type-II superconductors with substantial upper critical field Hc2, and in contrast to, for example, the cuprate superconductors with even higher Hc2, they can be easily machined into wires. Recently, however, 2nd generation superconducting tapes are allowing replacement of cheaper niobium-based wires with much more expensive, but superconductive at much higher temperatures and magnetic fields "2nd