Time Differential $\mu$SR (TD-$\mu$SR)

In a time differential $\mu$SR experiment (Fig. 2.2), the muons pass through a plastic scintillator (muon counter M) placed in front of the sample (S), which starts an electronic clock (TDC $\equiv$ time-to-digital converter). Muons which miss the sample strike a scintillator detector placed behind the sample (veto counter V) and are rejected. A positron emitted from a muon stopping in the sample is detected by one of the surrounding counters (U, D, L, R, B, F). When this happens, the electronic clock is stopped and the event is recorded in a time histogram. The number of decay positrons that are detected per time bin $\Delta t$ in the $i^{th}$ counter is given by

$$N_{i}(t) = N^{0}_{i}e^{-t/\tau_{\mu}}[1+A^{0}_{i}P_{i}(t)]+B_{i}\, ,$$ (3.4)

where $N^{0}_{i}$ is a normalization constant, $A^{0}_{i}$ is the maximum asymmetry and $B_{i}$ is the time-independent background associated with uncorrelated muon decay events. The single-counter ``asymmetry'' function is defined as $A_{i}(t)$ = $A^{0}_{i}P_{i}(t)$. One could also form the ``two-counter asymmetry'' by combining the time spectra from two detectors from the opposite side of the sample, e.g. U, D in Fig. 2.2 in the following way:

$$A_{U,D}(t) = \frac{N_{U} - N_{D}}{N_{U} + N_{D}}\, .$$ (3.5)

This eliminate operation the exponential contribution of the muon life time.

Fig. 2.2: Schematic diagram for a typical TD-$\mu$SR experiment. The $\mu$SR coordinate system convention is also shown at the lower left.

A good event in a TD-$\mu$SR experiment is defined as M $\cdot\overline{\bf V}\cdot$P, where P $\equiv$ U, D, L, R, B, F. Depending on the directions of the external magnetic field and the muon-spin polarization, two or four of the P counters are employed. Often the initial muon-spin polarization is rotated by 90$^{\circ}$ with respect to the beam momentum direction using a Wien filter (i.e. mutually perpendicular electric and magnetic fields) [18]. One reason for this is to avoid positron contamination from the beam in the F counter. The electronics is configurated to allow only one muon at a time in the sample, so that it is clear which muon a decay positron originates from. The ``muon gate'', which dictates how long one waits for a decay positron, is typically set to $\sim 10$ $\mu$s. If the muon counter is triggered during this $10$ $\mu$s window or the decay positron is not detected, no stop counts are recorded.