2.1 The basics

  The $\mu$SR technique is based on the two properties of muons:

(a)

muons are 100% spin-polarized, when produced from the $\pi\mu$ decay:  

$$\pi^{+}\rightarrow \mu^{+} + \nu_{\mu} \nonumber$$

(b)

muons decay into one positron and two neutrinos with a lifetime of $\tau_{\mu}$=2.2$\mu$s:  

$$\mu^{+}\rightarrow e^{+} + \nu_{e} + \bar{\nu}_{\mu} \nonumber$$

The feature (a) results from the `parity violation' of the weak decays, namely, the fact that only left-handed neutrinos exist. As shown in Fig.6, the muon from the $\pi\mu$ decay has to be spin-polarized, in order to conserve the total spin zero of the pion.

  

Figure 6: Schematic view of the $\pi\mu$ decay. The muon is spin polarized, because pion has spin 0 and neutrino is spin-polarized.

The feature (b) yields an asymmetric angular distribution of the decay-positrons relative to the muon spin direction. The angular distribution of decay positrons depends on their kinetic energy, which ranges from 0 to $E_{\rm max}$$\approx$$m_{\mu}c^{2}/2$=53 MeV. The angular distribution is expressed as [29]:  

$$dN= \frac{1+a(\epsilon)\cos\theta}{4\pi}d\Omega\times\rho(\epsilon)d\epsilon$$ (2)

where

and $\epsilon=E/E_{\rm max}$ is the normalized positron energy, $\theta$is the angle of positron emission measured from the muon spin direction, and $d\Omega$ is a small solid angle. After integrating the positron energy, the angular distribution behaves as shown in Fig.7.

  

Figure 7: Angular distribution of the positron after integrating over its energy. The radial distance represents the relative probability that a positron is emitted in a given direction.

The basic idea of the $\mu$SR technique follows the next three steps:

(1)

a muon is implanted in the sample with its spin polarized.

(2)

the muon spin changes its direction because of the magnetic environment around.

(3)

the muon decays into a positron and two neutrinos. The muon spin polarization at the time of the decay is reconstructed by measuring the positron directions for many incident muons.