Fit Quality

Figure 5.5 displays a typical fitted muon precession signal  $\tilde{P}(t)$ in a rotating reference frame.

  

Figure 5.5: Fitted muon polarisation  $\tilde{P}(t)$ in LuNi₂B₂C under an applied field $H = 1.2\,\mathrm{T}$ at temperature $T = 2.6\,\mathrm{K}$. The real and imaginary polarisation components are perpendicular to the field. The squares represent data while the solid lines are the fitted function (5.1). The insets show the residuals  $\tilde{R}(t)$ formed by subtracting the fitted function from the data.

The fitted function passes through most of the data points. The insets to this figure contain residual plots which exhibit a clear oscillation, indicating that the fitted function describes the data imperfectly. Such plots offer a useful qualitative assessment of the accuracy of a fit.

Chi-squared $\chi^2$ provides a quantitative evaluation of the fit quality. It is defined as

$$\chi^2 = \sum_{i = 1}^n \frac{\left(f(x_i) - y_i\right)^2}{\left(\delta y_i\right)^2}$$ (7.8)

where f(x) is the function fitted to the n measurements $y_i \pm \delta y_i$. A decrease in $\chi^2$ of at least one reveals a significantly improved fit, while a reduced chi-squared $\chi^2/n \le 1$means the fit is perfect. For the results reported in this thesis, the minimised reduced chi-squareds range between $\chi^2/n = 1.53$ and $\chi^2/n = 1.84$. These fits are comparable in quality to those obtained for other superconductors through analysis of $\mu$SR data in the time domain. Use of a more accurate cutoff factor [50] than the simple Gaussian $\exp( - k^2\xi^2/2)$ in expression (5.2) might supply better fits for the LuNi₂B₂C data discussed in this thesis. Figure 5.6 shows the dependence of $\chi ^2/n$ on the penetration depth $\lambda$ in LuNi₂B₂C at several temperatures T, and Table 5.1 lists the number of degrees of freedom n for each of these temperatures. (The number of degrees of freedom n for fits at other temperatures is similar.)
 

 

Table 5.1: Number of data points n analysed at each temperature T in Figures 5.6, 5.7 and 5.8.

Temperature T ( K)	Number of data points n
2.6	556
5.5	550
8.5	547

For each set value of the penetration depth $\lambda$ the parameters vary, in the way described in Section 5.3, until $\chi^2$ is minimised. The error bars for fitted parameters reported in this thesis are the amount by which the parameter must change to raise $\chi^2$ by one while the other parameters vary normally. At each temperature there is a clear minimum in $\chi ^2/n$ as a function of penetration depth $\lambda$, which moves to higher values of the penetration depth $\lambda$ at warmer temperatures. The shallowness of the minima is at least partly a consequence of compensation by the other free parameters, especially the nonlocality parameter C. Parameter C drops monotonically by two to three orders of magnitude over the displayed interval of increasing penetration depth $\lambda$. This substantial playoff comes from the factor $\lambda^4 C$ appearing in the internal spatial field profile (5.2). The widening minima at higher temperatures T reflects the diminishing asymmetry of the internal field distribution n(B) as it approaches a Gaussian form. Figure 5.7 reveals how $\chi ^2/n$ varies with the nonlocality parameter C.
 
At each temperature $\chi ^2/n$ grows rapidly as Cdecreases below the location of the minimum $\chi ^2/n$. This clearly demonstrates the substantial improvement in fit quality achieved by incorporating nonlocal corrections into the London model. The slow growth in $\chi ^2/n$ as the nonlocality parameter C increases away from the minimum further evidences the playoff between C and the penetration depth $\lambda$. The penetration depth $\lambda$ falls monotonically as the nonlocality parameter C rises over the plotted interval. The shallower minima at warmer temperatures Tagain stem from the more Gaussian-like internal field distribution n(B). Figure 5.8 displays $\chi ^2/n$ for a range of vortex core radii $\rho$ at the same temperatures T.
 
Since the core radius $\rho$ is a calculated rather than fitted parameter, this graph is constructed by minimising $\chi^2$ at fixed values of the coherence length $\xi$ and noting the corresponding core radii $\rho$. The strongly linear relationship between the coherence length $\xi$ and the core radius $\rho$ makes this process possible. Similarly, altering the coherence length $\xi$ by its uncertainty $\delta\xi$ and observing the consequent change in the core radius $\rho$ produces error bars  $\delta\rho$ for the core radius $\rho$. This means of computation of the uncertainty $\delta\rho$in the core radius $\rho$ is far simpler and more tractable than one based on combining the uncertainties of the fitted parameters according to equations (5.2) and (5.10). At each temperature T the $\chi ^2/n$ curve has a single minimum, which moves to larger core radii $\rho$ as the temperature T rises. This behaviour of the core radius $\rho$, and also the penetration depth $\lambda$, with temperature T is explored in more detail in the next chapter.