Data Fitting in the Time Domain

The $\mu$SR data are fitted to a theoretical model of the magnetic field B within a type II superconductor. The approximate field distribution n(B) yielded by taking the real amplitude of the Fourier transform of the measured muon spin polarisation  $\tilde{P}(t)$ is not useful for fitting, since the inherent finite time window introduces distortions in the form of ringing and broadening. To avoid these problems, all the results reported in this thesis come from fits in the time domain.

Fitting the recorded muon polarisation to a function calculated from a theoretical field B model forms the basis of a time domain analysis. The polarisation function
 $$\begin{multline}\tilde{P}(t) = A\exp(-\sigma_d^2t^2/2)\int_{-\infty}^{\infty}n(f . . . . . . )]\,df \\ + A_b\exp(-\sigma_b^2t^2/2)\exp[i(2\pi f_bt+\theta_b)] \end{multline}$$
utilised to fit the $\mu$SR data consists of a contribution from the muons that land in the superconducting sample and a term describing the background signal created by those that miss it. The parameters A and A_b reflect respectively the initial amplitudes of the superconducting and background asymmetries, or spin polarisations. The Gaussian damping factors $\exp(-\sigma_d^2t^2/2)$ and $\exp(-\sigma_b^2t^2/2)$ model the field Binhomogeneity which is additional to that of a regular array of vortices [49]. In the case of the superconducting signal the main sources of this are nuclear dipolar fields and vortex lattice disorder. The phase angles $\theta$ and $\theta_b$ account for the amount of spin precession that occurs before the muons trigger the muon counter. The vortex lattice field model described in the next section determines the distribution n(f) of Larmor precession frequencies f.