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Fit Quality
Figure 5.5 displays a typical fitted muon precession
signal
in a rotating reference frame.
Figure 5.5:
Fitted muon polarisation
in LuNi_{2}B_{2}C under an applied field
at temperature
.
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
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.
Chisquared
provides a quantitative evaluation of the fit quality.
It is defined as

(7.8) 
where f(x) is the function fitted to the n measurements
.
A decrease in
of at least one reveals
a significantly improved fit, while a reduced chisquared
means the fit is perfect. For the results reported in this thesis, the
minimised reduced chisquareds range between
and
.
These fits are comparable in quality to those obtained
for other superconductors through analysis of SR data in the time domain.
Use of a more accurate cutoff factor [50]
than the simple Gaussian
in expression (5.2)
might supply better fits for the LuNi_{2}B_{2}C data discussed in this thesis.
Figure 5.6 shows the dependence of
on the penetration
depth
in LuNi_{2}B_{2}C 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
the parameters vary,
in the way described in Section 5.3, until
is minimised. The error bars for fitted parameters reported
in this thesis are the amount by which the parameter must change to
raise
by one while the other parameters vary normally.
At each temperature there is a clear minimum in
as a
function of penetration depth ,
which
moves to higher values of the penetration depth
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 .
This substantial playoff comes from the factor
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
varies with the nonlocality parameter C.
At each temperature
grows rapidly as Cdecreases below the location of the minimum .
This clearly
demonstrates the substantial improvement in fit quality achieved by
incorporating nonlocal corrections into the London model. The slow growth
in
as the nonlocality parameter C increases away from the
minimum further evidences the playoff between C and the
penetration depth .
The penetration depth
falls
monotonically as the nonlocality parameter C rises
over the plotted interval. The shallower minima at warmer temperatures Tagain stem from the more Gaussianlike internal field distribution n(B).
Figure 5.8 displays
for a
range of vortex core radii
at the same temperatures T.
Since the core radius
is a calculated rather than fitted parameter,
this graph is constructed by minimising
at fixed values of
the coherence length
and noting the corresponding core radii .
The
strongly linear relationship between the coherence length
and the
core radius
makes this process possible. Similarly, altering the
coherence length
by its uncertainty
and observing the
consequent change in the core radius
produces error bars
for the
core radius .
This means of computation of the uncertainty
in the core radius
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
curve
has a single minimum, which moves to larger core radii
as the
temperature T rises. This behaviour of the core radius ,
and also the
penetration depth ,
with temperature T is explored in more detail in
the next chapter.
Next: Results and Discussion
Up: Analysis
Previous: Calculation of the Core Radius
Jess H. Brewer
20011031