2.2.1 Time-Differential $\mu{\cal SR}$

By far the most common form of $\mu{\cal SR}$experiment operates in the time differential mode. An arrangement of detectors surrounding a sample is shown schematically in Fig. 2.4.

  

Figure 2.4: Schematic diagram showing an arrangment of a sample and five scintillation particle detectors. Each muon triggers the thin muon (TM) detector on entering the experiment. Later, the decay positron triggers one of the positron detectors.

The positron detectors are usually mounted in pairs on opposite sides of the sample; Back-Front, Left-Right and Up-Down, a geometry well suited to extracting the signal, which we will come to shortly. Clever variations on this scheme have been used for special applications, to accomodate very small samples or very high magnetic fields for example, but all of these perform essentially the same function of detecting the angular distribution of positrons. We include in our example a magnetic field along the ${\hat y}$-axis, transverse to the initial muon polarization so that the spins undergo Larmor precession in the ${\hat x}-{\hat z}$ plane until the muons decay. Such experiments are termed transverse-field (TF) experiments, giving a signal that demonstrates in an intuitive way the analogy between $\mu{\cal SR}$ and traditional NMR. One can also perform experiments with no (zero) external magnetic field (ZF) or with a longitudinal field (LF) parallel to the initial muon spin, with no other change to the experimental apparatus.

Each muon enters the experiment with a velocity of about 1/4 of the speed of light, passing through a thin (0.125 mm) plastic muon scintillation detector (TM) which provides a ``start" pulse to the data acquisition electronics. The muon then enters the sample a few cm further downstream, where it comes to a stop and eventually decays. When one of the positron detectors intercepts the outgoing positron, a ``stop" pulse is generated and the elapsed time is determined by a time-to-digital converter (TDC) - essentially a stopwatch able to measure time intervals with a resolution of about 1 ns. The histogram for that positron detector is then incremented in the bin corresponding to the measured time interval. In most experiments the histogram time range extends up to 5 or 6 muon lifetimes.

The high speed front-end logic of the data acquisition system, implemented in standard NIM electronics, is shown schematically in Fig. 2.5. The rest of the data acquisition electronics consists of a histogramming memory, various support systems such as event scalars and detector selection registers and a computer which controls the experiment and reads out the data.

  

Figure 2.5: Schematic logic diagram of the basic fast front-end electronics and signal timing for time-differential $\mu{\cal SR}$ experiments. Two decay positron detectors are shown; more may be added as required by the detector geometry.

In order to measure the elapsed time we must know from which muon a given decay positron originated. The easiest way to do this is to veto any event for which a positron cannot be unambiguously paired with a muon. Each muon is given a fixed time interval (``pile-up" gate) to decay, during which the arrival of another muon would veto the entire sequence and both muons would be discarded. At meson facilities that produce pions continuously (as opposed to those with pulsed primary proton beams), the muons arrive at random time intervals. This and the dead-time of the TDC place a practical limit on the mean rate at which one can take muons to about 50,000/s. Events in which two or more positrons are detected within the data gate are also rejected by the TDC. Delays are introduced into the positron ``stop" signals so that the early part of each histogram contains counts from uncorrelated start and stop signals. This allows the background rates to be estimated from the part of each histogram before t=0, the moment when start and stop signals happen simultaneously.

Histograms are accumulated for each positron detector i=(B,F,L,R,U,D), each having the form

$$N_i(t) = N_0 \left\{ b_i + {e}^{-t/\tau_{\mu}}[1+ a_i(t)] \right\},$$ (5)

where N₀ is the normalization, a_i(t) is the experimental asymmetry and b_i is a time-independent background. The single-histogram asymmetry is easily obtained from the histogram by rearranging this;  

$$a_i(t) = \left[ \frac{N_i(t)}{N_0} - b_i \right] {e}^{t/\tau_{\mu}} -1,$$ (6)

which can be fitted numerically to a model function to obtain estimates of N₀ and b_i. An example of raw histogram data from the Back counter is shown in Fig. 2.6a. In this experiment a transverse magnetic field of 51.5 G caused the muons' spins to precess at a frequency $\omega_{\mu} = \gamma_{\mu} B$ = $2\pi \times$ 0.699 MHz, which is apparent in the sinusiodal oscillations in the asymmetry. Figure 2.6b is the asymmetry $a_{\rm B}(t)$extracted from the raw data using Eq. (2.8); the solid line is a theoretical asymmetry representing the sum of two signals with different (exponential) relaxation rates.

  

Figure 2.6: (a) A histogram from the ``Back" positron detector of a time-differential experiment on a sample of liquid Ne. (b) The asymmetry $a_{\rm B}(t)$ extracted from the same histogram according to Eq. (2.8). The oscillations are due to the Larmor precession of the muons' magnetic moments in an externally applied transverse magnetic field of 51.5 G.