The Muon fate in a µSR experiment

Muons are the products of the β decay of heavier particles, pions, which are in turn obtained from the “bombardment” of a graphite target with an accelerated proton beam. The weak interaction lead the pion to decay with a mean lifetime of 26 ns into a muon and a neutrino, following the scheme

π⁻ −→ µ⁻+ ¯νµ π⁺ −→ µ⁺+ νµ. (2.14)

A parity violation takes place and hence the emerging muons all have a negative he-licity h = −1, that is their spins are 100% polarized against the direction of motion¹⁰.

7An expression has become famous and is often reported to stress the surprise for muon discovery at the time: when Nobel laureate I.Rabi was told about the muon he exclaimedWho ordered that?.

8Further detail on the history of muons and µSR can be found here [121].

9Other Muons laboratories are TRIUMF, near Vancouver, in North America and KEK in Japan

10Helicity, indeed, is defined as the scalar product of the particle’s momentum and spin h = ~p · ~I.

Chapter 2. Solid-state spectroscopies applied to nanostructured materials

This is also one of the major advantages of µSR over NMR: no magnetic fields are needed to polarize the spins and therefore many experiments can be performed in zero external field (ZF). The polarization is maintained as the beam is transported to the muon spectrometers, while positive muons only are selected for experiments, since the negative ones may undergo a nuclear capture, which yields not negligible effects in case of nuclei with high atomic number.

Polarized muons are implanted in the sample, where they rapidly lose energy and in about 1 ns come at rest in some positions in the bulk sample. Here their spin polarization evolve as an effect of the local magnetic interactions, until the particles decay according to

µ −→ e⁺+ ν_e+ ¯ν_µ (2.15)

with a mean lifetime of approximately 2.2 µs, which makes muons suitable to probe magnetic interactions on the microseconds timescale. Once again the decay is led by the weak interaction and a parity violation occurs: this yields an anisotropic emission of the positron, more pronounced for higher positron energy.

muon spin μ⁺

e⁺ 53 MeV e⁺

26 MeV

Figure 33. Angular probability distribution for the positrons emitted by the decay of a µ⁺, for different positron energies. The positrons are preferentially emit-ted in the direction the muon spin had just before the

decay.

The angular distribution of the decay probability density is shown for differ-ent positron energies in figure 33 and is given by

W (θ) = 1 + a cos θ, (2.16) where θ is the angle between the muon spin and the direction of the positron emission and the factor 0 6 a 6 1, in-creases monotonically with the positron energy. Most of the positrons are emit-ted in the direction the muon spin was pointing at just before the decay and thus their detection allow to measure the time evolution of the muon spin polarization P(t).

2.2. µSR spectroscopy In summary, in a µSR experiment muons are implanted in the sample, their spin polarization evolve as an effect of the external and local magnetic fields and finally they decay emitting a detectable positron in the direction of their final polarization.

These positron are detected by means of scintillators, connected to photomultipliers for amplification, to obtain P(t) measurements.

Figure 34. Schematic representation of a ZF or LF µSR experiment. The detectors are arranged in two groups (forward and backward).

Because of geometrical constraints, generally two sets of detectors are employed: for ordinary longitudinal field measurements they are disposed as sketched in figure 34, in forward and backward directions with respect to the muon momentum. The counts of these two detector banks are separately summed up and the time evolution of the difference between forward (F) and backward (B) counts in principle describes the time evolution of the muon spin polarization. However, two important corrections must be taken into account:

• The exact difference of the F and B counts correctly describes the system only if the two sets of detectors are arranged in an ideal geometrical way, which is never the case. For this reason the counts of the backward detectors NB are considered as miscalibrated by a factor α, called the asymmetry of the detectors.

The effective backward counts are given by N_B^true = α · NB.

Chapter 2. Solid-state spectroscopies applied to nanostructured materials

• The number of counts exponentially reduces with time by the effect of the ra-dioactive decay of the muons. This reduction of the signal is intrinsic in the technique, but it contains no useful information and can be removed. The most relevant physical quantity is considered the decay asymmetry or polarization

P = N_F − αN_B

N_F + αN_B (2.17)

P represents the asymmetry of the forward and backward counts normalized point by point for the effect of the statistics of muon decay and therefore its time evolution reflects the true sample behaviour. The parameter α can be directly calibrated by a measurement in a low applied transverse field, typically of 20 or 100 G (these measurements are known as TF20 and TF100).

The two most common experimental configurations used in µSR experiments (actually we have already mentioned them) are the transverse field (TF) and longitudinal field geometry (LF or ZF if the magnetic field is set to zero). In the former the magnetic field is applied in a direction perpendicular to the muon polarization: the spins are thus forced to precess around this field at the frequency ω = γ_µH and an oscillating signal is observed¹¹. In the latter, on the contrary, the external field ~H is applied along the same direction of the muon spin. In general it will add to the local fields and, if it is large enough, it will keep the muon polarization fixed in the original direction.

The measurements performed on graphene and ball-milled graphite, shown in the ex-perimental sections, were mainly performed in the longitudinal ZF geometry, with the aim to observe the muons precession induced by the internal fields¹². Therefore it is worth discussing here the different behaviours that can be expected for the muon spin polarization in ZF, at least in the most significative physical situations. Next section takes care of this issue for the case of solid state powder samples, with a special focus on the dipolar interaction among nuclear and muon spins.

11A spin precession motion is seen as an oscillating signal because the detectors are disposed along one direction only (let’s say the z axis) an the projection of a circular motion onto an axis is a cosinusoidal oscillation.

12TF measurements were routinely used too, but only for calibrations and they will not be further discussed. In the experimental section, instead, minor experiments dedicated to the study of the signal changes under increasing LF are also reported.

2.2. µSR spectroscopy