Transverse-field SR spectra with approximately
muon decay events were taken under conditions of field cooling in applied
magnetic fields of 0.5T and 1.5T.
Figure 4.2(a) and Fig. 4.2(b) show the real and
imaginary asymmetry spectra
pertaining to the
sample resting in
an applied field of 0.5T and at a
temperature well above Tc (i.e.
For convenience the signals are displayed in a reference frame rotating
at a frequency 2.3MHz below the Larmor precession frequency of a
free muon. The average frequency of oscillation in Fig. 4.2(a) and
is determined by
the applied magnetic field of 0.5T. Consequently, the
corresponding frequency distribution [see Fig. 4.2(c)]
exhibits a single peak at 67.3MHz
related to the applied field through Eq. (3.14).
Visual inspection of
Fig. 4.2(a) and Fig. 4.2(b)
suggests that the real and imaginary asymmetry
spectra differ in phase by
consistent with the discussion
in the previous chapter. The solid curve passing through the data points is a
fit to the data using a polarization function in the form of
with a relaxation function resembling Eq. (3.33), so that
As one cools the sample below Tc, the relaxation rate of the muon precession signal increases due to the presence of the vortex lattice. The asymmetry spectra pertaining to a pair of counters for the crystals in an applied field of 0.5T is shown for three different temperatures below Tcin Fig. 4.3. The signals are shown in a rotating reference frame 3.3MHz below the Larmor precession frequency of a free muon. As the muons stop randomly on the length scale of the flux lattice, the muon spin precession signal provides a random sampling of the internal field distribution in the vortex state. The ensuing asymmetry spectrum is a superposition of a signal resembling Fig. 3.12 originating from muons which stop in the sample and an inseparable background signal resembling Fig. 4.2(c), due to muons which miss the sample, do not trigger the veto counter and whose positron also does not trigger the veto counter. The origin of the residual background signal is still uncertain, but may be due to those muons which scatter at wide angles after passing through the muon counter and are thus not vetoed by the V counter. Fig. 3.12 was obtained by choosing a rotating reference frame frequency equal to the background frequency, determined by fitting the data. Unfortunately, the broadening of the background signal is not necessarily identical to the field distribution above Tc and thus cannot be fixed in the fits below Tc.
The beat occurring in all three spectra of Fig. 4.3 is due to the
difference in the average precession frequency of a muon in the internal
field of the vortex lattice and the precession frequency of the muon
in the background field. The solid curves in Fig. 4.3
are fits to the
theoretical polarization function of Eq. (3.39).
An additional polarization
function in the form of Eq. (4.1)
was added to model the background signal
pertaining to the muons which missed the sample, so that the measured
asymmetry is of the form:
|A(t) = Asam (t) + Abkg (t)||(2)|
As the temperature is lowered, the vortices become better separated and the muon spin relaxes faster due to the presence of a broader distribution of internal magnetic fields. This is better displayed in Fig. 4.4 which shows the corresponding real Fourier transforms of Fig. 4.3. Recall from Eq. (3.6) that the width of the field distribution is proportional to . Thus it is clear from Fig. 4.4 that decreases with decreasing temperature. The sharp spike on the right side of each frequency distribution in Fig. 4.4 is attributed to the residual background signal. It has been determined to account for approximately of the total signal amplitude at 0.5T and at 1.5T.
Figure 4.5 shows the frequency distribution resulting from field cooling the sample in the 0.5T field [Fig. 4.5(a)] and then lowering the applied field by 11.3mT [Fig. 4.5(b)]. As shown the background signal shifts down by 1.5MHz and positions itself at the Larmor frequency corresponding to the new applied field. The signal originating from muons which stop in the sample does not appear to change under this small shift in field. This clearly demonstrates that at low temperatures the vortex lattice is strongly pinned. Furthermore, the absence of any background peak in the unshifted signal implies that the sample is free of any appreciable non-superconducting inclusions. The shifting of the background peak away from the sample signal has since been duplicated for higher temperatures and at other applied fields. Field shifts in excess of G were not attempted for fear that the crystals would shatter as a result of the strain exerted by the pinned vortex lattice.
The asymmetric frequency distribution shown isolated in Fig. 4.5(b) has the basic features one would anticipate for a triangular vortex lattice, but with the van Hove singularities shown in Fig. 3.1(b) smeared out. Structural defects in the vortex lattice, variations in the average field due to demagnetization effects and - anisotropy are all possible reasons for a lack of sharper features.