Detection of positrons

To reconstruct the muon spin direction, it is convenient to have two positron counters which are placed symmetrically with respect to the sample (muon) position, as shown in Fig.8.

  

Figure 8: A positron counter configuration to reconstruct muon spin polarization.

We define the muon spin polarization along the counter axis as $P_{\mu}(t)=\cos\Theta(t)$, where $\Theta(t)$ is the time evolution of the muon spin angle relative to the counter axis (see Fig.8). The positron count rate of each counter N_i(t) (i=1,2) is found by integrating the positron angular distribution of this tilted muon spin over the counter solid angle and positron energy. The answer becomes:

where

Here, $\tau_\mu(=2.2\,\mu{\rm s})$ is the muon lifetime, $\eta(\epsilon)$ is the detection efficiency of the positron counters and $\Omega_{i}$ is the solid angle of the counter i (=1,2). A_i is called the asymmetry of the individual counters, which is typically 0.2$\sim$0.3 in a conventional $\mu$SR setup.

The muon spin polarization $P_{\mu}(t)$, which is the information one would like to obtain from the $\mu$SR technique, is calculated by taking the corrected asymmetry (eq.12), which is essentially the count difference of the two counters normalized by the total count:  

$$A_{1}P_{\mu}(t) = \frac{\alpha N_{1}(t)-N_{2}(t)}{\alpha\beta N_{1}(t)+N_{2}(t)}$$ (3)

where the parameter $\alpha\equiv N_{2}^{0}/N_{1}^{0}=\Omega_{2}/\Omega_{1}\;(=1;\;{\rm ideally})$ corrects the deviation of the solid angles between the two counters, and $\beta\equiv A_{2}/A_{1}\;(=1;\;{\rm ideally})$ corrects the difference of the counter asymmetries.

In Fig.9, an example of the positron counts [N₁(t)] and the corrected asymmetry [$A_1P_\mu(t)$] is shown. A Larmor precession of the muon spin is seen, as an external magnetic field perpendicular to the initial muon spin direction was applied during this measurement.

  

Figure 9: (a) A typical time spectrum of one positron counter. (b) The corrected asymmetry (Eq.12) of the same measurement.

The next section presents more details of the experimental setup which is required for the $\mu$SR measurements.