6.3.1 Zero Field $\mu {\cal SR}$ in Rb₁C₆₀ & Cs₁C₆₀

The ZF muon spin relaxation at low temperature for A = Rb and Cs is shown in figure 6.40 (together, for comparison, with similar spectra for A = K). The quenching by LF of the ZF relaxation is also shown. The low field required to completely quench the relaxation indicates that in each case the relaxation is due to static fields. The characteristic features of the relaxation in the magnetic state (Fig.6.40a and 6.40b) are that there appear to be two well defined components, one fast-relaxing and one slow-relaxing. We note that this qualitative behaviour has been observed in all the other ZF $\mu {\cal SR}$ measurements in these systems; furthermore, it is not very different from the relaxation in other fulleride magnets [206,132].

A closer look at the fast relaxing component for A = Rb (see 6.41) reveals that the relaxation is quite exponential (and otherwise featureless) back to the earliest times accessible in the data. Note that the data (for A = Rb) contains an oscillating signal at $\approx 320$MHz which is due to the TDC (described in section 2.1.3). This oscillation is binned over in the figure, but its existence effectively limits the timing resolution to a few periods of the clock cycle (3ns). More recent data (stars in Fig.6.41) does not suffer from this problem. At the top of Fig.6.41, the relaxation at 50K (above the magnetic transition) is shown for comparison, and at the bottom the analogous TF relaxation at 1.5T is shown in the rotating reference frame, rotating at the average precession frequency. The TF data for A = Rb (in the following section) also contains the clock signal mentioned above. The spectrum at 2K (stars) is from a different sample than the rest of the data, and indicates sample independence of the qualitative features of the ZF relaxation at early time. We note that another ZF $\mu {\cal SR}$ measurement on a sample of Rb₁C₆₀ from a different source finds that the fast component at low temperatures is quite flat at early times, in contrast to the linear behaviour apparent in Fig.6.41.

It has recently been found that the sample corresponding to the the stars in Fig.6.41 likely contains an impurity phase of Rb₃C₆₀ (perhaps as large as 20% [207]). It is not known whether the previous sample contains a similar impurity phase, but it is certainly possible. The low temperature ZF spectrum in Rb₃C₆₀ is shown in Fig. 4.27. It is clear that the small Mu component of such a signal would be too small in amplitude to make a significant contribution to the relaxation, and the large slow gaussian relaxation could not be confused with the fast relaxing signal.

In contrast to A = Rb, for A = Cs at the lowest temperature, there appears to be a rapidly damped oscillation (Fig.6.42). A subsequent measurement on the same sample in the dilution fridge (DR) at 1.0K is shown for comparison. The overall asymmetry in the DR is lower, but the data at 1.0K and 1.9K are consistent. Note that, except for the data in the DR, all the data for A = Cs was taken in a separate specta apparatus, and because of the absence of the ``t₀'' stright-through peaks in the sample spectra in such an apparatus, earlier times are accessible. This is the reason that the DR data begins at 50ns instead of closer to zero time. Moreover, no clock signal is present in any of the A = Cs data. The oscillation is of relatively small amplitude and cannot be resolved at temperatures higher than 2K. The oscillation frequency at 1.9K and 1.0K is approximately 6 MHz.

The presence of such an oscillation indicates that there is some longe range magnetic order. The amplitude of the oscillating signal suggests that the fraction of the sample which possesses this order is relatively small (less than $\sim 40$%). From this data, it is not possible to determine what form the magnetic ordering takes (SDW, local AFM etc.). The frequency of an oscillation in the ZF spectrum (due to magnetic ordering) is determined by the size of the field (averaged over all the muon sites). This field is in turn determined by the size of the ordered moment, the distance to the muon site, and the effective form of the coupling (e.g. dipolar). Without knowledge of the muon site, it is not possible to estimate the size of the ordered moment from the oscillation frequency, however, note that the oscillation frequency in the TMTSF SDW systems[23] is much smaller (< 0.6MHz). Thus if the ordering is of a SDW type, the magnitude of the SDW and/or the coupling between the $\mu^+$ and the SDW must be considerably larger in Cs₁C₆₀. The presence of any sort of oscillation implies that at least a fraction of the muons is stopping in a set of one or a few well-defined sites. The field at the $\mu^+$, given by the oscillation frequency $6\mbox{MHz}/\gamma_\mu \approx 450$G. The magnitude of the internal fields is thus consistent (in order of magnitude) with the observed NMR broadening of $\sim 2000$ppm at 7T [64].

With the exception of the low temperature data in Cs₁C₆₀, no oscillatory behaviour is observed, so we must introduce a model for the relaxation. Unlike the frequency of ZF oscillations observed in other magnetic systems, the relaxation rate in such a model is probably not directly related to the magnetic order parameter, but, in order to at least summarize the temperature and field dependence, such a model is still useful. It must, however, be considered entirely phenomenological.

A reasonable description of the low temperature data is that it is the sum of three terms:

where the subscripts F and S refer to fast and slow, and A_NR is a non-relaxing component which accounts for muons stopping in the silver mask around the sample. For ZF, we fix A_NR at some reasonable value, corresponding to about 20% of the muons, as it should be approximately temperature independent. One can use, for the two relaxing components, a stretched exponential relaxation, i.e.

which interpolates between an exponential and a gaussian, but such a model contains too many parameters for a satisfactory description of the data. Note that fitting the entire relaxation to a stretched exponential yields an unreasonably low power $\beta < 0.5$, indicating that the relaxation is highly two component, as opposed to a smooth distribution of relaxation rates.

It is found that the relaxation for A = Rb can be well described with exponential relaxations for both the fast and slow components; whereas, for A = Cs, a better choice is gaussian relaxation for both components. Once an appropriate form for the relaxation is selected, there are still two possible assumptions regarding the behaviour of the two components.

1.

If the two components come from two inequivalent muon sites, then we may assume that the relative amplitudes of the two sites remains constant with temperature, neglecting any muon site-changing dynamics (which would seem to be inconsistent with the static nature of the relaxation).

2.

On the other hand, if the two components are due to some kind of inhomogeneity in the magnetic state, then we may assume the relative amplitudes can vary. An example of this kind of behaviour is found in spin-glasses, where frozen and dynamic regions co-exist, and their relative fractions change with temperature.

It is found that, if the assumption of the first case is pursued, that the relaxation rate of the fast component increases to unrealistically large values as the temperature increases, indicating that the constant fraction assumption is inappropriate. This is also apparent in the data: the fast component appears to change with temperature in amplitude more strongly than in relaxation rate.

Using the second assumption, we find the following behaviour for the temperature dependence of the ZF relaxation (see Fig.6.43 for A = Cs and Fig.6.44 for A = Rb). The fast component seems to decrease continuously in amplitude, with roughly constant relaxation rate as the temperature is increased from 2K. Above about 6K (Rb) and 10K (Cs), the two components become difficult to distinguish. In the case of Rb, the amplitude of the fast component was constrained to approach zero roughly continuously, while for Cs, the two components were allowed to mix. Above 15K (Rb) and 35K (Cs) the relaxations were fit to a single relaxing component. The transition temperature appears to be about 20K for both systems, but there is a broad range of temperatures above this where some sort of onset behaviour is apparent. Even below the transition, the relaxation varies rather gradually with temperature. Not that the difference in total amplitude in the DR is the result of differences in the counter geometry, thus the two low temperature points for Cs (triangles and nablas in Fig.6.43b) should not be taken as continuations of the higher temperature data. However, the relative magnitudes of the two components (i.e. they are nearly equal) is consistent with the higher temperature data (as expected). Note that the form of the relaxation function indicates that the field distribution is broad (enough to simulate an expoential for A = Rb) and centred at or very near zero field. In addition the distribution, for A = Cs, has a broad small peak centred at about 450G.