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In a transversefield muon spin rotation experiment (Fig. 2.3), a magnetic field is applied
perpendicular to the initial muonspin polarization direction. A muon stopping in the sample Larmor precesses about the
local magnetic field (which in general is different than the external field) at an angular frequency
=
, where
= MHzT is the muon gyromagnetic ratio.
Fig. 2.3:
Schematic diagram of a TFSR experiment. The initial muon spin
polarization
, indicated by the open arrow at , is rotated by 90 so that it is perpendicular
to the direction of the external magnetic field . The open arrows at the sample S illustrate the Larmor
precession of the muon spin.

In a superconductor the muons stop at well defined sites in the crystal lattice. However, in the vortex state of a typeII
superconductor, the muons stop randomly on the length scale of the vortex lattice (which is typically two to three orders of
magnitude larger than that of the crystal lattice). Consequently, SR is an effective local probe of the spatial
variation of internal magnetic fields due to the periodic arrangement of vortices. For the case where the external field is
directed along , the component of the muonspin polarization function is

(3.6) 
where is the probability that a muon sees a local magnetic field between and +, and is the initial
phase. There are several sources of the local magnetic field sensed by the muons. First there is the magnetic field
inhomogeneity associated with the vortex lattice. For a typeII superconductor, the spatial profile of the magnetic field
in the  plane due to an applied field along the axis ( direction) is reasonably
described by the phenomenological model [19]

(3.7) 
where is a reciprocal lattice vector, =
is the reduced field, is a modified Bessel
function and =
. Equation (2.7), which is derived from GinzburgLandau
theory, assumes that
, where
is the magnitude of the smallest nonzero
reciprocallattice vector in the summation. This condition is satisfied for an extreme typeII superconductor like the
high cuprates, where
. This selfconsistent analytic function, agrees extremely well with
the exact numerical solutions of the GinzburgLandau equations at low reduced fields . Random vortex pinning and thermal
fluctuations modify the field distribution associated with Eq. (2.7), which assumes a perfect periodic
arrangement of vortices.
The muon is also sensitive to both nuclear and electronic dipole moments. The nuclear moments are randomly oriented at
temperatures reachable in a SR experiment (i.e. mK), whereas electronic magnetic moments may order.
In general, the magnetic moments that are static on the SR time scale broaden the measured internal magnetic field
distribution. To account for the effects of this additional source of field inhomomgeneity on the muonspin polarization
function, Eq. (2.6) can be multiplied by a depolarization function such that

(3.8) 
The precise functional form of depends on the nature of the additional sources of magnetic field at the muon site.
For example, a Gaussian function =
with a depolarization rate
, characterizes the damping of the muonspin precession signal due to dense static dipole moments [20].
Next: Fast Fourier Transform (FFT)
Up: SR Spectroscopy
Previous: Time Differential SR (TDSR)
Contents
Jess H. Brewer
20030701