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# Data Analysis

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 Tc, 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 Tc 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)

Comparing to Eqs. (3.6), (3.7) and (3.8), one has

 (5)

Because of the asymmetric shape of the true field distribution, using a gaussian distribution of internal fields gives poor fits to the data. However, the fits are sufficient to provide a crude estimate of the temperature dependence of the second moment. Fig. 4.6 shows the variation of the broadening parameter for single gaussian fits of the 0.5T and 1.5T data. Both sets of data suggest that the width of the field distribution varies linearly with temperature below 20K. Furthermore, the broadening of the lineshape appears larger at these low temperatures for the applied field of 0.5T. The linear term appears to weaken slightly at 1.5T. Above 20K the data for the two fields are nearly identical. The single gaussian fits of course cannot determine explicitly, but are useful nonetheless in giving an approximate temperature dependence of . They also help facilitate a comparison with previous studies where only single gaussian fits were possible.

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:

 (6)

where the relaxation parameter [see Eq. (4.1)] is determined by a run taken above Tc in the normal state and C is a constant (i.e. the slope) chosen so as to minimize the total for all runs in which K. In other words, only runs where K were considered in determining C since it is the low temperature regime which is of primary importance.

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 Tc 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 Tc 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 Tcat 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 4He was pumped through the cryostat. At higher temperatures (where the required cooling power is low) the amount of 4He 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 4He and to reduce thermal gradients between the thermometers and the sample. However, to maintain low temperatures a much larger flow of 4He 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 4He-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 Tc. 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 ns) 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:

 (7)

gives Å and Å for the 0.5T and 1.5T data respectively, where the quoted uncertainties are purely statistical [87]. It should be noted that this equation is purely phenomenological and cannot be extended to include the higher-temperature data. Both curves suggest that pair breaking persists at the lowest of temperatures in , which is inconsistent with conventional s-wave pairing of carriers. This low-temperature behaviour indicates a more complicated gap function characteristic of the presence of nodes in the energy gap.

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 Tcmay 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 Tc, 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 .

Table 4.1: Comparison of the fitting procedures.
 Fitting Applied C'' from Procedure Field Eq. (3.6) (T) ( ) (Å) (K-1) (K-2) Method (i): 0.5 0.0293 1451(2) 0 is fixed before'' 1.5 0.0258 1496(1) determining C Method (ii): and 0.5 0.0250 1350(2) are free'' 1.5 0.0243 1437(1) Method (iii): is fixed'' 0.5 0.0250 1362(2) 1.5 0.0243 1445(1) Method (iv): 0.5 0.0250 1347(2) 0 is fixed after'' determining C

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.

Next: Conclusion Up: Experimental Determination of Previous: The Measured Asymmetry
Jess H. Brewer
2001-09-28