The Relaxation Function

In the vortex state the muons experience a spatially varying field strength $B(\vec{r})$. Consequently the x-component of the $\mu ^{+}$polarization may be written:

$$P_{x}(t) = \frac{1}{N} \sum_{i=1}^{N} \cos \left[ \gamma_{\mu} B(\vec{r}_{i})t + \theta \right]$$ (31)

where $\theta$ is the initial phase. In a real superconductor there are additional contributions to the relaxation rate, so that a more appropriate description of the $\mu ^{+}$ polarization is:

$$P_{x}(t) = G_{x}(t) \frac{1}{N} \sum_{i=1}^{N} \cos \left[ \gamma_{\mu} B(\vec{r}_{i})t + \theta \right]$$ (32)

where G_x(t) is a relaxation function (or depolarization function), which describes the damping of the precession signal. The precise form of the relaxation function G_x(t) depends on the origin of the depolarization. In many cases one assumes a gaussian form which is appropriate when the depolarization originates from the muon-nuclear-dipolar interaction [79].

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 $\lambda$. As $\lambda$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.

  

Figure 3.12: Asymmetry spectra plotted in a rotating reference frame due primarily to the vortex lattice in single-crystal $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{6.95}$ }$. The data is for an applied transverse field of 15kG at the temperatures (a) T=75K where $\lambda \approx 2040$Å and (b) T=2.9K where $\lambda \approx 1526$Å.

The interaction of the $\mu ^{+}$-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 $\mu ^{+}$-site is assumed. Above T_c, this leads to a Gaussian relaxation function:

$$G_{x}(t) = e^{- \sigma_{d}^{2} t^{2} / 2}$$ (33)

where $\sigma_{d}$ is the muon spin depolarization rate due to the nuclear-dipolar fields. The value of $\sigma_{d}$ determined from data taken above T_c, is assumed to be the same in the superconducting state.

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:

$$G_{x}(t) = e^{- \left( \sigma_{d}^{2} + \sigma_{f}^{2} \right) t^{2} /2} = e^{- \sigma_{eff}^{2} t^{2} /2}$$ (34)

where $\sigma_{f}$ is the muon spin depolarization rate due to lattice disorder and any additional depolarizing phenomena and $\sigma_{eff}$is the effective depolarization rate such that $\sigma_{eff}^{2} = \sigma_{d}^{2} + \sigma_{f}^{2}$. The influence of $\sigma_{eff}$ is to further broaden the field distribution n(B) beyond that due to $\lambda$.

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.