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Chapters:
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MuSR /
FirstMomentThe magnetic induction inside a superconductor can be described appropriately with the use of the demagnetizing factor, as in the case of magnetic materials. In the general case the sample magnetization, {$M$}, the induction {$B$} and the magnetic field {$H$} will not be uniform inside the sample, even at a macroscopic scale, averaging over distances greater than {$\lambda,\xi$}, due to the presence of magnetic poles on the sample surface (equivalently, of surface supercurrents). If we consider the special case of a sample of ellipsoidal shape, with field parallel to one of the ellipsoid principal axes, the macroscopic fields will be uniform. This is the case for instance when the sample itself is placed in a uniform region of the applied field (call it {$H_0=\frac {B_0}{\mu_0} $}). In a µSR experiment quite often the sample approximates the shape of a slab (be it a pellet or a mosaic of platelets) and the field is applied perpendicular to the slab, along the muon beam, with the initial muon spin rotated to approach the transverse geometry (e.g. GPS or LTF). In this case the first moment of the flux lattice lineshape should be equal, in first approximation, to the applied external field, with no diamagnetic shift. As a matter of fact, since the demagnetizing factor is {$N=1$}, the internal induction will be {$B_m=\mu_0[H_0+(1-N)]M=\mu_0H_0$}. This implies that one should not detect a diamagnetic shift in the first moment of the flux lattice line on GPS-LTF, and that the observation of such a shift will be due to the departure from the ideal case, {$N=1$}. In identical conditions the magnetization, such as that measured in a SQUID magnetometer for the same applied field {$H_0$}, is {$ M=\chi H_m = \chi(H_0 - NM)$}, i.e. {$M=\frac {\chi} {1+N\chi} H_0 $} This equation means that {$H=H_0/(1+N\chi)$} inside the slab. Notice that this condition is unphysical for the Meissner state: if one assumes {$N=1, \chi=-1$} the equation dictates {$H,M \rightarrow\infty$}. The superconducting Meissner state therefore breaks down since either {$H>H_c$}, or {$H>H_{c1}$}. When the latter is the case one has instead {$B_m=\mu_0(H+M)=\mu_0(\frac {H_0} {1+\chi} + \frac{\chi H_0} {1+\chi})=\mu_0H_0=B_0$} If the field is applied in the plane of the slab (as for moderate transverse fields on MUSR) one has {$N=0$}. Since then {$M=\chi H_o$} one has {$B_m=\mu_0(1+\chi)H_0$} hence for {$H_0\gt H_{c1}$} a large diamagnetic shift ({$ -1\ll\chi<0$}) should be apparent in the µSR data. In the more general case of {$0\le N\le 1$} one has {$B_m= \mu_0 \,\frac {1+\chi} {1+N\chi}\, H_0 \approx \mu_0[1+(1-N)\chi)]H_0$} with the approximation valid for {$|\chi|\ll 1$}. Note that all the above discussion applies to the first moment of the line: if a single Gaussian fit is performed on the data, the best fit is likely to adapt the first moment of the Gaussian to the saddle point singularity of the flux lattice lineshape, providing a shift even when the first moment vanishes. |