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MuonDiamagnetiShift

< From muon rates to London penetration | Index | Fitting Flux Lattices with a single Gaussian? >

Let us consider a muon superconducting sample in the shape of a slab. In an infinite slab the external induction $B_0=\mu_0 H$ must coincide with the average induction inside the slab.

Consider the simplified sketch on the right, showing the slab S (solid line), a red Gaussian surface P and the induction field lines. We neglect transverse field components, i.e we go far enough from the slab surface. Outside the induction is uniform. Inside there is a flux lattice, shown as dense and rarefied field line regions.The flux through the two large faces of P must be equal. The one outside, $P_{out}$ experiences a uniform field and $$\int_{P_{out}} \mathbf B \cdot d \mathbf a =AB_0$$

The one well inside the slab, $P_{in}$, experiences a non uniform field, distributed along $\hat z$ according to $p(B_z)$, and

$$\int_{P_{in}} \mathbf B \cdot d \mathbf a = A\overline{B} = A \int dB_z p(B_z)$$

therefore it must be $B_0=\overline{B}$. However experimentally it is not: the broadened precession line from muons implanted in the superconductor bulk is shifted diamagnetically, well beyond experimental error, to and average $B_d\ll B_0$.

The shift is due either from shape significant deviations from the infinite slab geometry or by the loss of the high field tail of the distribution, easily below noise level. A simple example of the first case is illustrated in the very qualitative sketch on the right. Since the demagnetization factor of the sphere is $N={1 \over 3}$ a corresponding fraction of flux is expelled from the sample. $$\int dB_z p(B_z)=A(1-N\chi)B_0,$$

Of course magnetic Gauss law is still valid (easy to see on Gaussian surface P), but one cannot neglect the transverse components of the induction and the measured $B_d$ is related, but not equal to the average $B_z$ value. A slightly more complex case arises for powders where the average $N$ over the grain shapes and orientations may need to be considered.

To estimate this effect, a typical TF experiment will observe a shift of order $\mu_0 \chi N_\text{eff}H$, proportional to the SQUID magnetization detected in the same magnetic field.

< From muon rates to London penetration | Index | Fitting Flux Lattices with a single Gaussian? >