In the vortex state the muons experience a spatially varying field strength
.
Consequently the *x*-component of the polarization may be written:

where is the initial phase. In a real superconductor there are additional contributions to the relaxation rate, so that a more appropriate description of the polarization is:

where

There is no significant loss of polarization during the short time over which the muons thermalize. This is because the primary interactions by which the muons rapidly lose their initial kinetic energy are electrostatic in nature and hence do not affect the muon spin [80]. Loss of the muon spin polarization in the vortex state is primarily due to the inhomogeneous field distribution, which in turn can be related to the magnetic penetration depth . As decreases, the spatial variation in the magnetic field becomes greater and there is a corresponding increase in the relaxation rate of the muon spin polarization (see Fig. 3.12). As Fig. 3.1(b) indicates, the field distribution for a perfect vortex lattice is far from being gaussian, but rather is highly asymmetric.

The interaction of the -spin with nuclear-dipolar fields
in the sample leads to further damping of the precession signal
and a corresponding broadening of the field distribution. Normally,
a gaussian distribution of the dipolar fields at the -site
is assumed. Above *T*_{c}, this leads to a Gaussian relaxation function:

where is the

For a real sample in the vortex state, decoration experiments have
shown that the vortex lattice is not perfect. Deviations from the
ideal flux-line lattice lead to a further relaxation of the
precession signal [72]. Consequently we can redefine the
relaxation function as:

where is the muon spin depolarization rate due to lattice disorder and any additional depolarizing phenomena and is the

It should be noted that the additional broadening of the field distribution due to flux-line lattice disorder is difficult to define. Because the field distribution corresponding to an ideal flux-line lattice is highly asymmetric, one would anticipate distortions of the lattice to also be asymmetric in nature. In Eq. (3.34) we are assuming the distortions are gaussian distributed, but it can be shown analytically that convoluting with a gaussian does not change the average field of the distribution.

2001-09-28