To determine the low temperature behaviour of several assumptions were made in the fitting procedure to reduce the number of independent variables. To start with, the Ginzburg-Landau parameter was assumed to be independent of temperature. Although this is strictly valid only for weak coupling s-wave superconductors away from T_{c}, the lineshapes are not very sensitive to in the low-field region being considered here. Determining a value for was accomplished by first fitting the recorded asymmetry spectra with as a fixed quantity. The value of which minimized the sum of the 's for each temperature considered was then taken to be the best value for . The value gave the best overall fit to both the 0.5T and 1.5T data. Increasing to 73 was found to change by less than 0.3nm. The value is close to the value determined from previous lineshape measurements on similar crystals in higher magnetic fields [77].
Fits to the early part of the signal (i.e. the first s)
for data below T_{c} using an
equation in the form of Eq. (4.1) with a single gaussian
relaxation function of the form
and
pertaining to the average internal field, provides a simplified
visual display of the dependence of the lineshape width on temperature.
It is straightforward to use the polarization function in
Eq. (4.1) to relate the relaxation parameter
to the
second moment
.
The relationship is
[68]
(4) |
(5) |
A more precise treatment of the data using the phenomenological model
of Eq. (3.39), holds the two parameters
and
accountable for the width of the measured field
distribution in the sample. Clearly these two parameters must combine
to mimic the behaviour in Fig. 4.6. Since
and
both contribute to the linewidth and both are
expected to be temperature dependent quantities, the two parameters
cannot be treated as independent of one another when analyzing the
data. Indeed, fits to the data in which both parameters
were free to vary have
and
playing off one another as
in Fig. 4.7. A temperature point which appears locally high in the
vs. T plot,
appears locally low in the
vs. T plot and vice versa. A plot of
vs.
suggests a linear correlation between the two parameters
as shown in Fig. 4.8.
The solid curve through the points in Fig. 4.8
has the following form:
Figure 4.9 shows the total arising from global fits of the 0.5T and 1.5T data for various choices of the constant C. The proportionality constant C was found to be and for the 0.5T and 1.5T data, respectively. The depolarization rate was approximately and for the 0.5T and 1.5T fields, respectively.
In the first type of analysis, the total asymmetry amplitude for signals recorded below T_{c} was fixed to the value of the precession amplitude obtained from fitting data above the transition temperature, prior to determining C in Eq. (4.6). Below T_{c} the asymmetry amplitude of the measured signal is the sum of the precession amplitude of the background signal ( ) and the precession amplitude of the signal originating from within the sample ( ). Thus, here we are assuming that the total precession amplitude of the resultant signal is independent of temperature, but dependent upon the applied magnetic field. The asymmetry amplitude above T_{c}at fields of 0.5T and 1.5T were found to be and , respectively. The field dependence is primarily attributed to the finite timing resolution of the counters, which causes the observed precession amplitude to decrease as the period of the muon precession becomes comparable to the timing resolution.
In the final step of this
analysis, the status of the fitting parameters
was then as follows:
1. Sample Signal [refer to Eq. (3.39)]:
Variable parameters:
i) The amplitude
ii)
iii) The average internal field
iv) The initial phase
Fixed parameters:
i)
ii)
fixed to
according to Eq. (4.6)
2. Background Signal [refer to Eq. (4.3)]:
Variable parameters:
i) The field
ii)
iii) The initial phase
(same as for sample signal)
Fixed parameters:
i) The amplitude,
Thus in the final fit of the data there were six independent parameters.
Figure 4.10 shows the variation with temperature
of the initial phase ,
the
average field
,the amplitude
and
relaxation rate
of the background precession
signal, obtained from
fits of the 0.5T data. As indicated in Fig. 4.10(a),
the phase of the
initial muon spin polarization remains nearly constant throughout the
temperature scan (i.e.
rad).
This implies that there were no appreciable fluctuations
in the applied field or electronics.
The nearly constant field
in Fig. 4.10(b)
is a further indication of a highly stable applied
magnetic field.
The 1.5T data is not shown because there was a significant
change in the applied field after 40K.
Figure 4.10(d) shows a significant drop in the relaxation rate of the background signal at higher temperatures, indicating some temperature dependence for . However, at lower temperatures (K) the background relaxation rate and hence the contribution of the background signal to the second moment exhibits no obvious correlation with temperature. This suggests that plays little role in the temperature dependence of in Fig. 4.6. The fact that suggests that the background is caused by a material with a large nuclear dipolar interaction such as Cu, or is in a region of fairly large field inhomogeneity.
Figure 4.11 and Fig. 4.12 show the temperature dependence of , and arising from the same fits which produced the results in Fig. 4.10. Together these parameters constitute three of the four variable parameters (the other being ) which pertain to the signal originating from the sample. The sample amplitude depicted in Fig. 4.11(a) shows some scatter and a slight decrease at higher temperatures. The scatter in the asymmetry amplitude is not all that surprising considering that the data was recorded over a period of 5 days, through which time, small fluctuations in experimental conditions were unavoidable. For instance, one such experimental variation was the rate at which ^{4}He was pumped through the cryostat. At higher temperatures (where the required cooling power is low) the amount of ^{4}He flowing into the cryostat and the corresponding pumping rate were minimized in an effort to keep the heater voltage small to preserve the supply of ^{4}He and to reduce thermal gradients between the thermometers and the sample. However, to maintain low temperatures a much larger flow of ^{4}He was required. The increased density of helium atoms in the cryostat increases the probability of scattering the incoming muons before they can reach the sample, thus increasing the background signal and decreasing the magnitude of . To minimize this effect, the cryostat sample space was pumped on hard, but the choice of a specific combination of ^{4}He-flow rate and the pumping rate was purely judgemental. This is a possible explanation for the scatter observed in Fig. 4.11(a). However, the downward trend of as one increases the temperature may be purely statistical, as a similar behaviour was not observed in more recently recorded data fitted with the same procedure. Recall that since the total asymmetry amplitude was fixed, the variation of with temperature in Fig. 4.10(c) appears as a mirror image of Fig. 4.11(a).
Figure 4.11(b) shows the temperature variation of the average
internal field
experienced by muons implanted in the
sample. For comparison,
the background field
is also plotted in Fig. 4.11(b).
In general, the field at any point in the sample is
the sum of the local fields in Eq. (3.13).
For all temperatures,
is less than
,
but
appears to approach
at both
ends of the temperature scan. In the high-temperature regime the vortex
cores begin to overlap with the internal field distribution approaching
full penetration of the applied field. Thus it is not surprising to see
the average internal field
approach
as one increases the temperature towards T_{c}.
The rise in average field
at low temperatures, however, is
more difficult to understand.
Such an increase has also been reported in previous
work by Riseman [77]
and observed in more recent data taken at different
fields. The cause for such behaviour is puzzling indeed.
However, Fig. 4.11(b) is consistent with the time spectrum
shown in Fig. 4.3
which shows a more distinct beat in the muon spin precession signal at the
intermediate temperature T=35.5K, corresponding to a greater
separation between the average precession frequency of muons subjected
to the internal field distribution
and the average precession frequency of muons in the background field.
This suggests that the
increase in
at low
temperatures may be due to some intrinsic phenomenon of the
sample itself.
Figure 4.12 shows the
temperature dependence of
(which
in the phenomenological London Model is directly
proportional to the superfluid density n_{s}) for the applied field
of 0.5T . Since the relaxation rate
is assumed
proportional to
[see Eq. (4.6)], the
variation of
with temperature resembles the behaviour
in Fig. 4.12.
Figure 4.13 shows the
low-temperature dependence of
for both 0.5T and 1.5T applied fields.
As shown,
the presence of a linear term (i.e.
)
in the low-temperature region is unmistakeable
for both 0.5T and 1.5T fields,
with the latter showing a weaker linear dependence
on T. A fit to the low-temperature data (i.e. below 55K), with an
equation of the form:
In Fig. 4.14 the temperature dependence of at 0.5T and 1.5T is shown. The solid curves represent microwave measurements of the change in penetration depth taken in zero static magnetic field by Hardy et al. [6]. For the purpose of comparison, for each field is chosen to be the value obtained from fitting the SR low-temperature data with Eq. (4.7). Surprisingly, the microwave data shows a much better agreement with the SR data at the higher magnetic field of 1.5T.
There was some concern after completion of the above analysis that fixing the total asymmetry amplitude to the value above T_{c}may introduce systematic errors by constraining the fits. The large fluctuation in the amplitude of the muon spin precession signal originating from the sample [see Fig. 4.11(a)] was the source of such concerns. Intuitively, we expect that should scale with the percentage of muons striking the sample. The fluctuations in this percentage during the actual experiment were probably not large enough to account for the large scatter in Fig. 4.11(a). If one dismisses the previous explanation for the large fluctuations in , it is worth investigating this matter further.
Since is not expected to change significantly over the temperature scan, the data was refitted first by designating and as variable parameters. As in the previous analysis, was assumed to be proportional to through Eq. (4.6). The proportionality constant C was determined to be 0.0250(10) and for the 0.5T and 1.5T data respectively. The variable parameters pertaining to the background precession signal varied with temperature according to Fig. 4.15. Comparing with Fig. 4.10, the phase shifts down rad, while shifts upward mT. The degree of fluctuation in both these parameters appears similar to that of the previous analysis, so again it seems apparent that there were negligible fluctuations in the applied field.
The amplitude and the relaxation rate [see Fig. 4.15(c) and Fig. 4.15(d)] show almost no change from the results depicted in Fig. 4.10. Even the size of the statistical error bars are comparable. These results indicate that the fitting program is capable of clearly separating the unwanted background signal from the sample signal.
Figure 4.16(a) shows the temperature dependence of the amplitude corresponding to the muon spin precession signal originating from the sample. The downward trend with increasing T appears slightly more prominent than in Fig. 4.11(a). The temperature dependence of in Fig. 4.16(b) is significantly different from that in the previous analysis. The average field is greater than at the lowest of temperatures and does not dip as far below as in Fig. 4.11(b) for temperatures beyond this. At the high-temperature end in Fig. 4.11(b), recovers to approximately the same value as in Fig. 4.11(b). Again the rise in at low temperatures is surprising. It is possible that this is an effect due to - anisotropy. The presence of -anisotropy would distort the vortex lattice into isoceles triangles as shown in Fig. 3.9. If this lattice were to be modelled by one consisting of equilateral triangles as assumed in our analysis, then there would be some error in the determination of the average field . This would be a greater problem at low temperatures where the cores are further apart and errors in spectral weighting are more pronounced.
The low-temperature dependence of is shown in Fig. 4.17. Surprisingly, the scatter in the data points is not significantly greater than in Fig. 4.13. Noticeably different however, is an increase in the linear term (see Table 4.1). Furthermore, fits to Eq. (4.6) yield Å and Å for the 0.5T and 1.5T data, respectively. A comparison to the microwave measurements of Hardy et al., assuming these values for is shown in Fig. 4.18. There appears to be even less agreement at 0.5T than previously noted in Fig. 4.14(a). However, the agreement at 1.5T in Fig. 4.18(b) is comparable to that in Fig. 4.14(b) despite the significant difference in .
As a final step in the analysis, the data was refitted with the amplitude fixed to the average value of the data below 55K in Fig. 4.16(a). This results in a noticeable reduction in the scatter for the parameters and (see Fig. 4.19). Fixing in this way significantly shifts the data points above 40K. This is not surprising since was fixed to the low-temperature average. The phase shows a slight decrease at high temperatures [see Fig. 4.19(a)] and levels off above 40K [see Fig. 4.20(b)]. These results suggest that fixing to the low-temperature average reduces the scatter in the low-temperature data, but it is not yet clear whether or not we are introducing non-physical deviations in the high-temperature region.
The reduction in scatter is most noticeable in Fig. 4.21 which shows the temperature dependence of . From Eq. (4.6) we find Å and Å for the 0.5T and 1.5T data, respectively. A plot of the temperature dependence of over the full temperature scan is shown in Fig. 4.22. The two fields appear to converge well before T_{c}, but the crossover is difficult to determine. As shown in Fig. 4.23, there is improved agreement between the microwave measurements and the 0.5T SR data, while the agreement with the 1.5T data is comparable to that of the previous two fitting methods. The total asymmetry amplitude of the muon spin precession signal as determined from all three fitting procedures is shown in Fig. 4.24. It appears as though one is justified in fixing the total asymmetry amplitude, as the average values are comparable.
The results from all three types of analysis are summarized in Table 4.1. Methods (ii) and (iii) give comparable results, but differ substantially from method (i). The difference appears to be related to the proportionality constant C of Eq. (4.6). As C increases, so does .
It should be noted that for method (i) in Table 4.1, the total asymmetry amplitude was fixed prior to the determination of C. This may in fact be the most significant difference between method (i) and the other fitting procedures, in which C was determined before fixing any additional parameters. To see if this is the case, the 0.5T data was refit, by first determining the proportionality constant C and then fixing the total asymmetry amplitude to the average value for the data below 55K (see method (iv) in Table 4.1). Remarkably, the total asymmetry amplitude was found to be the same as in method (i) (i.e. ). The linear coefficient and the quadratic coefficient [determined by fitting the low-temperature data to Eq. (4.7)] are virtually the same for methods (i) and (iv), but the values obtained for are very different. Moreover, the value of from method (iv) is comparable to (ii) and (iii). All of this implies that is significantly influenced by changes in C, but is little affected by the manner in which the amplitude of the precession signal is treated in the fitting procedure. Also, the deviations in the linear term from one method to the next are likely not significant enough to suggest that there is any difference in the behaviour of at low temperatures.