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µSR

Chapters:

  1. Introduction
  2. The muon
  3. Muon production
  4. Spin polarization
  5. Detect the µ spin
  6. Implantation
  7. Paramagnetic species
  8. A special case: a muon with few nuclei
  9. Magnetic materials
  10. Relaxation functions
  11. Superconductors
  12. Mujpy
  13. Mulab
  14. Musite?
  15. More details

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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 B0=μ0H 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, Pout experiences a uniform field and

PoutBda=AB0

The one well inside the slab, Pin, experiences a non uniform field, distributed along ˆz according to p(Bz), and

PinBda=A¯B=AdBzp(Bz)

therefore it must be B0=¯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 BdB0.


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=13 a corresponding fraction of flux is expelled from the sample.

dBzp(Bz)=A(1Nχ)B0,

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 Bd is related, but not equal to the average Bz 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 μ0χNeffH, 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? >

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