Upon implantation into a material a muon will rapidly thermalise, whilst preserving its initial spin direction, before Larmor precessing around the local magnetic field, Bloc, at a frequency given by ωµ = γµBloc, where
γµ=2π×135.5M HzT−1 is the muon gyromagnetic ratio. Since the muon is
itself unstable it will decay, with a lifetime ofτµ= 2.197 µs, into a positron.
tary muon spin direction is governed by the probability distribution given by equation 2.2 where a is an energy dependent asymmetry factor taking an average value of 13:
W(φ) = 1 +acos (φ) . (2.2)
W(φ)has its maximum where positron emission takes place along the muon spin direction at the moment of decay [87]. A schematic diagram of the de- cay process, and subsequent positron detection, is shown in figure 2.2. A muon enters the experiment at t0 and with an initial spin angle θ0. The
muon spin will then begin to process about the local field to a new angle θ
where θ(t) = ωµt+θ0. At any given time, t, the chance the muon will de-
cay into a positron is given by τ1
µe
−t/τµ. Through detection of the resultant
positron and knowledge of the time span of the event, the evolution of the muon spin direction can be tracked and information about the local field experienced established. This requires knowledge of a statistically signifi- cant number of single muon events each with an identical starting condition. Hence the importance of a spin-polarised monochromatic muon beam [90].
LEFT POSITRON DETECTOR
RIGHT POSITRON DETECTOR +
t0, θ0 t, θ
ωμ
Figure 2.2: A schematic diagram of the basic principle of µSR. A muon enters the apparatus at t0 with an initial spin angle θ0 (grey arrow). At
some later time, t, the muon spin has rotated about the local field to an angle θ (purple arrow) and decays emitting a positron preferentially along
θ.
In order to monitor the spin rotation of the muon ensemble two positron de- tectors are positioned either side, in this case in the left and right directions, of the sample space. The number of positron events detected on either side can then be counted as a function of time. The resultant time spectraNL(t)
evolution of the muon spin polarisation along the detection axis [86]. NL(t) = N0,Lexp − t τµ [1 +A0P(t)] NR(t) = N0,Rexp − t τµ [1−A0P(t)] (2.3)
N0 is the total number of muon events andA(t) = A0P(t)is the asymmetry
spectrum which is defined as the normalised difference between the two detector signals as in equation 2.4 whereα is an efficiency parameter with a value close to unity.11
A(t) = A0P(t) =
αNL(t)−NR(t)
αNL(t) +NR(t)
(2.4) The maximum observable asymmetry between the two detectors is parame- terised byA0 the upper limit of which is set at 13 by the intrinsic asymmetry
of the muon decay. In practiceA0 is often lower and depends on the precise
details of the experimental configuration and the energy distribution of the emitted positrons [90]. Typically, for the measurements presented in this thesis,A0 is of the order of 0.20.
The main panel of figure 2.3 shows an example rawµSR time spectrum where the signals from the left and right detectors have been plotted sepa- rately in blue and red respectively. The signal shows periodic maxima and minima as the muon spin is directed towards or away from the relevant detector. This oscillatory component is then modulated by an exponen- tial decay associated with the muon lifetime. The inset of figure 2.3 shows the corresponding asymmetry spectrum, A(t), plotted in green. Here the oscillation frequency scales with the local field that the muon ensemble ex- perienced. At t0 the maximum asymmetry is measured, in the example
shown this is about 20%, but for t > t0 there is some loss of the asymmetry
amplitude. This damping is caused by depolarisation due to the presence of a distribution of internal fields and is represented in the inset by the
11In practice this depends on the precise experimental configuration and is related to
the beam alignment with respect to the two detectors. Please see section 2.1.4 for further details.
NL NR
~ 2.2 s
Figure 2.3: Main panel: an example µSR time spectrum with the counts from each detector plotted separately where NL and NR are shown in blue
and red respectively. The exponential decay in the signal is due to the muon lifetime τµ. Inset: the corresponding asymmetry spectrum is plotted
in green. The oscillation period in the signal scales with the average field that the muon ensemble experienced and the damping, represented by the magenta envelope, scales with the muon depolarisation.
magenta envelope. A(t) contains all of the information pertaining to the local magnetic environment experienced by the implanted muons. In or- der to extract this information, one is required to fit the spectrum with an appropriate functional form which will depend on the precise experimental configuration [86].