Apparatus Setup

The experiment reported in this thesis used the M20 beamline at TRIUMF in conjunction with the Helios $\mu$SR spectrometer to study the LuNi₂B₂C sample described in Section 3.2. The M20 beamline delivers muons of mean momentum  $28\,\mathrm{MeV}/\mathrm{c}$, so they stop throughout the bulk of the sample. Figure 4.1
 
illustrates the path of the muon beam through the experimental apparatus. The collimator restricts the diameter d of the muon beam to $d \approx 1\,\mathrm{cm}$ before the beam exits the vacuum of the beamline through a thin plastic window. Each muon then triggers the muon counter (M), starting the time to digital converter (TDC). The muon then travels through two more thin plastic windows, and the intervening vacuum of a helium gas flow cryostat. This vacuum thermally insulates the sample space from the warm bore of the Helios magnet. The muon finally comes to rest in the sample, which is attached with a little Apiezon N grease to a thin Mylar film stretched over the end of an aluminium sample holder tube. The crystal $\mathbf{\hat{c}}$ axis lies parallel to the applied field  H. An average of $2.197 \,\mu\mathrm{s}$ later the muon decays, emitting a positron preferentially along the muon spin direction. When this positron is detected, the TDC is stopped. The TDC therefore records the elapsed time between the arrival of a muon and the detection of the subsequent decay positron. Sometimes a muon misses the sample, and instead it or its decay positron arrives at the veto (V) counter. Electronic logic modules reject such decay events, as well as ambiguous ones where the detectors register more than one muon or more than one positron within specified time periods of about  $10\,\mu\mathrm{s}$.

Histograms are constructed for the number N_i of positrons detected in the ith positron counter during each time interval $\Delta t$ after the TDC starts, from which the muon spin polarisation  P(t) is computed. Figure 4.2 depicts the arrangement of the positron

  

Figure 4.2: Cross-sectional view of forward positron counters F_i and F_ij (i = T or B and j = L or R) as seen by the muon beam (represented as travelling into the page). The valid detection of a decay positron requires both the F_i and F_ij counters to trigger simultaneously ( $F_i \cdot F_{ij}$).

detectors. The positron counters form two concentric rings about the muon beam, with four counters in the inner layer and two in the outer. The inner detectors cover approximately equal solid angles around the sample, and the outer counters each completely overlap two of the inner ones. For a positron detection to be valid, both the inner and outer counters in a given direction must trigger simultaneously. The appropriate time bin of the histogram associated with that inner counter then increases by one. The number N_i of positrons registered per time bin $\Delta t$ in the ith inner counter follows the form [7]

$$N_i(t) = N_i^0 \exp (-t/ \tau_{\mu})[1 + P_i(t)] + B_i^0$$ (6.3)

where N_i⁰ is a normalisation factor and B_i⁰ is the random background signal. In the simple situation where all muons experience the same local field  B, the precession signal P_i(t) recorded by the ith inner detector is $P_i(t) = A_i \cos (\gamma_\mu B + \theta_i)$, where A_i is the initial precession amplitude and $\theta_i$ is the initial phase of the muon spin polarisation  P(0) relative to the centre of the ith counter. Generally, these precession signals P_i(t) are extracted numerically from equation (4.3). The single-counter functions P_i(t) belonging to each pair of opposing inner detectors combine to produce a component of the complex muon spin polarisation $\tilde{P}(t) = P_x(t) + i P_y(t)$. Half the difference of the single-counter functions P_i(t) of an opposing pair constitutes a polarisation component P_x(t) or P_y(t), depending on the choice of detector pair. The complex polarisation  $\tilde{P}(t)$ so obtained is then fitted to a theoretical model to extract the superconducting parameters of interest. In the experiment reported in this thesis, the four histograms N_i(t) together typically contained around  $2.7 \times 10^7$ decay events for a given applied magnetic field H and temperature T, which were gathered over about two hours.

Feedback control systems stabilise the applied field H and temperature T. Current carrying copper coils on the outside of the Helios superconducting magnet compensate for field drifts detected with a Hall probe situated near the sample. This keeps external field Hfluctuations below  $0.05\,\mathrm{mT}$. Helium streaming past the sample from the nearby diffuser supplies cooling power, while a Lakeshore 330autotuning temperature controller heats the diffuser. The heating varies in such a way as to maintain the temperature of the diffuser at the set value. One calibrated GaAlAs diode monitors the diffuser temperature, and two others provide independent readings of the sample temperature. The analysis of the collected data is explained in the next chapter.