2.1.3 $\mu {\cal SR}$ Experimental Setup and Data

The experiments described in this thesis were conducted on the M15 and M20 beamlines at TRIUMF, which provide high intensity beams of $\sim$100% spin polarized positive surface muons. The kinetic energy of surface muons ($\sim$4.1 MeV) gives them a mean stopping range of 140 mg/cm²; consequently, $\mu {\cal SR}$ is essentially a bulk probe.

One can follow the spin-polarization of an ensemble of implanted muons via detection of their high energy decay positrons which, due to the asymmetry of the weak decay of the muon, are emitted preferentially along the direction of the parent muon's spin. The muon and its decay positron are detected in fast plastic scintillation detectors. The muon detector is thin enough ($\approx$0.25mm) that muons will pass through it. Typically there is an array of between 1 and 4 positron detectors in well defined directions relative to the magnetic field at the sample position. The histogram of the time differences between muon implantation and decay positron detection in counter i is of the form:

where N_i0 is an overall normalization, B_i is a time-independent background, A_i is the experimental asymmetry typically in the range 0.2-0.3, $\hat{i}$ is a unit vector along the direction joining the centre of the sample to the centre of the solid angle subtended by the counter i, $\tau_\mu \approx 2.2 \mu$s is the muon lifetime which sets the practical upper limit for the timescale of observable variations in ${\bf P}_\mu(t)$, the muon polarization. Through a variety of methods, one extracts $A_i {\bf P}_\mu(t)\cdot\hat{i}$and fits the time dependence to an appropriate model.

Practically, the histogramming is accomplished with a Time-to-Digital (TDC) Convertor (LeCroy, model 4204). A good start event is defined logically by $S = M \cdot \bar{P}$, where M indicates that the muon counter has triggered, and $\bar{P}$ indicates that there isn't already a muon in the sample. The condition $\bar{P}$ ensures that subsequent positron detection can be associated unambiguously with the muon that starts the TDC. A good stop event is defined by $E = \Pi \cdot \bar{P}$, where $\Pi$ indicates that a positron detector has been triggered. In addition, the time range which is measured is always limited by a data gate of arbitrary length, but typically it is set at about 5$\tau_\mu$.The time between the S and E is measured by the TDC, and the result is routed to the section of a histogramming memory unit for the detecting positron counter.

There is a known flaw in the LeCroy 4204 TDC. The unit has an internal or gate which takes as inputs the stop pulses from the various positron counters. The output of this internal or is contaminated by a high frequency clock signal (usually above 300MHz), so using the output of the internal or as the input to the TDC stop yields a sharp high frequency to the data (e.g. see Fig. 4.14b). The solution to this problem is straightforward. One simply uses a reliable high speed or gate in place of the internal or. The output of such an or gate can be timed so that the stop pulses from it arrive at the stop input of the TDC at the same time they would have in using the internal or. The clock signal contamination exists in some of the data of chapter 6, but for all recent data, we use the above ``fix'' to avoid the problem.

Another technique that is used in some of the data reported in this thesis is the newly developed ``Separate Spectra Method''[96]. In this method a second thin muon counter is placed in the cryostat immediately in front of the sample. A high purity silver mask is placed in front of this muon counter, so that muons passing through the mask only stop in the sample. The standard outer muon counter is used for the starts, but the inner muon counter routes the stop events to one of two histograms for each counter, i.e. it separates events from muons that stop in the sample from those from muons stopped in the mask. Thus, a calibration experiment is done in situ, under the same conditions of field and temperature. In addition, the amplitude of the sample signal is maximized by eliminating contribution due to background. Furthermore, it is found that the peak at the zero time of the histograms is eliminated in the sample spectra. The ``t₀'' peak is due to straigt-through events, mainly of positron contamination in the beam. The coincidence counting of the two muon counters eliminates the peak, since the positrons have little probability of triggering the thin muon counter (because they deposit little energy), and the probability for a positron triggering both muon counters is negligibly small. Practically the routing in the Separate Spectra Technique is accomplished by initiating a data gate D for the inner muon counter, i.e. start D when $M_{1} \cdot M_{2} \cdot \bar{P}$, where M_i are the muon counter pulses. The stop condition is modified for the routing by demanding, for the sample spectra a coincidence with the gate D, and, for the reference spectra, coincidence with $\bar{D}$.

Such time-differential $\mu$SR measurements (in which ${\bf P}_\mu(t)$rather than its integral is measured) fall into three geometric categories: longitudinal (LF), transverse (TF), and zero (ZF) field, depending on the direction of the applied magnetic field relative to the direction of the initial muon spin polarization (Fig. 2.6).

In the LF situation, the left (L) and right (R) counters play no role. The muons enter from the left, pass through the thin muon (TM) detector and, via the aperture in the ``backward'' (B) positron counter, pass into the sample. The initial muon spin polarization ${\bf P}_\mu(0)$points backwards, and consequently, if the detection characteristics of the two symmetric counters are otherwise balanced, the B counter will initially detect more positrons on average than its ``forward'' (F) counterpart. After implantation, if the muon spin depolarizes in times shorter than $\sim 50\tau_\mu$, then the asymmetry in the count rates will decay observably with time. Often, ${\bf P}_\mu(t)$ simply decays exponentially , and the LF relaxation rate is exactly analogous to T₁^-1 in NMR. In ZF, both T₁ processes and inhomogeneous static internal fields (for example, nuclear dipolar fields) contribute to the relaxation of ${\bf P}_\mu(t)$; whereas, in longitudinal fields exceeding the magnitude of any static internal fields, ${\bf P}_\mu(t)$ relaxes only by T₁. ZF $\mu {\cal SR}$ is thus a very sensitive site-based probe of static magnetism. In the TF geometry, ${\bf P}_\mu(0)$ is perpendicular to ${\bf H}$, and ${\bf P}_\mu(t)$ exhibits oscillations at the Larmor frequency determined by the value of the magnetic field at the muon and the gyromagnetic ratio, $\gamma_\mu = 135.54$ MHz/T. The TF experiment is analogous to the free induction decay of NMR with the TF relaxation rate being identified with T₂^-1. An example of the time histogram of a single counter in a TF experiment (following Eq. (2.4)) is given in Fig. 2.7a.

A schematic diagram (approximately to scale) of the typical setup is shown in Fig. 2.8. The four side counters were used in the TF measurements, and the cup shaped F counter and annular B counters were used in the LF and ZF measurements. In the original experiment[83], the sample was suspended on a thin sheet of mylar, and the vessel had windows on both sides of the sample. The apparatus could then be used in a low background mode[84] with the F cup playing the role of a veto counter. In this situation, the definition of a good start is modified to start only if the muon has landed in the sample, i.e. $S = M \cdot \bar{P} \cdot \bar{V}$, where V indicates that the veto counter has triggered (i.e. the muon has gone straight through and stopped in the cup shaped counter. In the TF situation, one can also use the veto counter to ``shade'' the side positron counters by defining a good stop as $S = \Pi \cdot \bar{P} \cdot \bar{V}$. This relies on the shape of the cup and the relative geometry of the cup and the side counters. With the availability of larger quantities of material this mode of operation was no longer necessary. The sample cell, F counter, and sample thermometers were mounted on the end of a lucite lightguide sample rod in the He space of a helium gas-flow cryostat. For the standard sample cell, a high purity annular silver mask was placed immediately in front of the cell so that muons that did not enter the sample cavity would stop in the silver and contribute only a benign temperature independent background. Between the beamline vacuum and the sample, the muons passed through 4 Kapton windows, the muon counter, a small air-gap, a thin aluminized mylar heat shield, and a small gap in cold helium gas. The total stopping density that these intervening obstacles presented to the muons was about 63 mg/cm². Precautions were taken to keep this density as small and constant as possible by preventing condensation on the outer cryostat window and limiting the pressure of the He gas at low temperature. At high fields, the helical positron paths have curvature on the scale of the detectors, and the effective solid angles of the counters consequently change. For example, the count rate in the B counters shown in the figure fall off significantly above about 2T. Subsequent improvements to the B counters reduced this problem. The initial[83] data on R₃C₆₀ and some of the data presented here used only the F counter, while some of the data used both F and B.