Transverse Field $\mu$SR

Transverse field $\mu$SR provides an effective means of measuring the distribution n(B) of internal magnetic fields B within a superconductor. This technique employs a beam of muons polarised so that their ensemble averaged spin direction is perpendicular to their momentum, which is itself parallel to the applied field  H. The muons enter the sample one at a time and stop at random locations within the vortex lattice. There the spin of each muon precesses with an angular Larmor frequency  $\omega = \gamma_\mu B$, directly proportional to the local magnetic field B. The magnetogyric ratio for muons is $\gamma_\mu = 851.6 \,\mathrm{M}\mathrm{rad}\mathrm{s}^{-1} \mathrm{T}^{-1}$. The implanted muon decays after a mean lifetime of $\tau_\mu = 2.197 \,\mu\mathrm{s}$, emitting a positron preferentially in the muon spin direction. The detection of many such positrons reveals the ensemble averaged muon spin polarisation  P(t), also called the precession signal. The polarisation amplitude  $\left\vert\mathbf{P}(t)\right\vert$ attenuates over time as the muon spins dephase due to the spatial variation of the magnetic field B within the vortex lattice. For this reason the spin precession signal  P(t) constitutes a sensitive measure of the distribution n(B) of magnetic fields B within a flux line lattice.

TRIUMF generates nearly 100% polarised muon beams through the parity violating decay of pions. The pions arise from protons, accelerated to about $500\,\mathrm{MeV}$, striking a production target. Those pions decaying at rest near the target surface supply the muons employed in most modern $\mu$SR experiments. Almost all of these positive pions ($\pi^+$) disintegrate into a positive muon ($\mu^+$) and a muon neutrino ($\nu_\mu$) via the weak interaction

$$\pi^+ \rightarrow \mu^+ + \nu_\mu$$ (6.1)

In the rest frame of these pions, the spin and momentum of neutrinos are oppositely directed. Therefore, in this frame, the spin and momentum of each muon created through process (4.1) must also be oppositely directed in order to conserve linear and angular momenta. This means that muons emitted in a given direction by pions decaying at rest are automatically highly polarised. In this way muon beams suitable for $\mu$SR spectroscopy are produced.

The preferential emission of a positron (e⁺) in the muon spin direction when a positive muon ($\mu^+$) decays also stems from parity violation in the weak interaction. This weak decay nearly always produces an electron neutrino ($\nu_e$) and a muon antineutrino ( $\overline{\nu}_\mu$) as follows:

$$\mu^+ \rightarrow e^+ + \nu_e + \overline{\nu}_\mu$$ (6.2)

In the rest frame of the muon, the spin and momentum of the neutrino are antiparallel, while those of the antineutrino are parallel. When the neutrino and antineutrino are emitted in the same direction, the positron has maximum energy, and its spin direction is the same as that of the muon so as to conserve angular momentum. Consequently, the momentum of the positron reveals the spin direction of the muon at the time it decayed, since weak interactions like this one will only create a highly relativistic positron whose spin and momentum are parallel. Overall, taking into account all possible momenta of the emitted neutrino and antineutrino, the positron tends to be emitted in the muon spin direction. This enables the ensemble averaged muon spin polarisation  P(t) to be measured.