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Superconductors

µSR owes its user comminity mostly to physicists and material scientists involved in magnetism, but superconductivity is the second field of election. The reason is quite apparent: muons can map the internal magnetic field distribution, which is very drastically modified in a superconductor.

The field-temperature phase diagram of superconductors may be very complicated and a few fundamental properties and quantities of the superconducting state are directly accessible from the field distribution inside the material.

This type of application historically follows that of NMR, which has a very similar capability.

 A. Abrikosov (1928 - )

However nuclei with spin larger than $\frac 1 2$ suffer from the complications of additional quadrupolar interactions and often the time scale of dynamical relaxations interferes with the detection of the mere static field distribution, whereas the measure with muons is at first glance quite straightforward.

 The typical measure is that of the second moment of the field distribution in a type II superconductor. In a nut-shell: a large enough external magnetic field $\mu_0\mbox{\it\bf H}$ the magnetic flux penetrates in a regular array of quantized flux tubes, the fluxons of the Abrikosov lattice, the red rings in the cross-sectional view of the figure. This implies that the magnetic field $\mbox{\it\bf B}(\mbox{\it\bf r})$ varies continuously from a maximum, at the center of the fluxon cores, to a minimum, at the centers of the flux lattice cells, giving rise to a peculiar field distribution with a peak at an intermediate most frequent value. Since muons stop at interstitial sites of the crystal lattice, which is incommensurate with the Abrikosov lattice, their Larmor frequencies sample the latter at random positions. The frequency distribution maps the field intensity distribution and their second moment is mostly determined by the penetration depth $\lambda$, which in turns is a measure of the superconducting carrier density. Sketch of the Abrikosov and crystal lattices, in a cross section orthogonal to $\mu_0\mbox{\it\bf H}$. Red rings show the fluxon cores for an arbitrary large field and black dots the atoms.

For a start we shall describe very simplified models: the London model of a superconductor, the Pippard corrections due to the finite mean free path of electrons, a hint at the Ginzburg-Landau equations, and a description of how to get the field distribution in the simplest cases with examples of matlab programs.