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# FieldDistributionProbedByMuons

The reasons for measuring the field distribution of $p(B)$ are listed below;

The second moment of the line is directly connected with $\lambda$ for extreme type II superconductors. It is straightforward to compute the second moment of the distribution, having noted that the first moment, the average internal field, is $\frac {\Phi_0} {ab} = B$. As a matter of fact, since $\int d\mathbf{r} e^{i(\mathbf{Q}-\mathbf{Q}^\prime)*\cdot\mathbf{r}} = \delta(\mathbf{Q}-\mathbf{Q}^\prime}$, with $\mathbf{Q},\mathbf{Q}^\prime$ vectors of the reciprocal flux lattice, it is easy to work out from Eq. 3 that

$(1) \qquad\qquad \overline{\Delta B^2} = \,B^2 \sum_{\mathbf{Q}\ne0}\, \left[\frac {K_0(Q^2\xi^2)}{1+\lambda^2Q^2}\right]^2 ;$

For extreme type II $\xi$ is very much less than $\lambda$. For the smallest reciprocal vectors $Q^2\sim a^*b^*$, using the lattice spacing definition, one has $\xi^2Q^2\sim 2\pi$. Thus the Bessel function will cut-off the sum after few reciprocal lattice points in Eq. 1. Already for the first non vanishing reciprocal vector we may neglect unity in the denominator, since $\lambda^2Q^2\gg \xi^2Q^2$. Hence the standard deviation, the square root of the second moment, is

$\qquad\qquad \sigma \propto \frac 1 {\lambda^2}$.

This has lead to important considerations for high Tc cuprates, where the standard deviation of the distribution, proportional to

$\qquad\qquad\sigma \propto \frac 1 {\lambda^2} \, =\, \frac {\mu_0 n_se^2}{m }$

scales linearly with doping (Uemura plot), providing evidence that the disappearence of superconductivity in the underdoped regime is linked directly to the vanishing of the Fermi energy scale ($\varepsilon_F\propto n$ for a two dimensional metal like the cuprates}.

The first moment of the field distribution deserves a separate discussion.

Single crystal studies often allow a direct comparison of the full experimental lineshape with the field distribution, Eq. 3

Anisotropic superconductors may be investigated in a similar way, and the orientation dependence of the lineshape may give full access to the tensors that replace the scalars $\lambda$ and $\xi$.

The temperature dependence of $\lambda^{-2}$ provides information on the symmetry of the gap, a very important issue for exotic superconductors, such as cuprates, ruthenates , heavy fermions, and organic superconductors.

Flux lattice structure from µSR lineshape analyisis allows investigation of the rich soft matter properties of the fluxons, both for isotropic and anisotropic systems. This is complementary to Small Angle Neutron Scattering.

Two gap superconductivity introduces significative deviations in the field dependence of the lineshape and MgB2 was the prototypical case to show this.