< How to produce a net spin polarization | Index | Longitudinal field >
Parity violation is also the key to the detection of the muon spin evolution in time. As in the case of the muon production it is implemented in angular momentum conservation with the constraint that the positron and the neutrinos:
{$ (1) \qquad\qquad \overline \mu^+ \rightarrow \overline e^+ + \nu_e + \overline \nu_\mu $}
distinguish between left and right. They must have well defined helicities (negative for the particle {$\nu_e$} and positive for the antiparticles {$\overline e^+$} and {$\overline\nu_\mu$}).
|
Since (1) is a three body decay, there is a continuum of possible geometries, as shown in the sketch on the left) and, correspondingly, a range of energies for the emitted positron ({$E\le 52.83 \mbox{MeV}$}). The correlation imposed on the directions of the muon spin and of the positron momentum is however simple at maximum positron energy, which corresponds to the two neutrino being emitted both in the opposite direction to the positron. In this particular collinear case:
- the two neutrinos do not contribute to angular momentum, since they are collinear and of opposite helicity
- the positron must have positive helicity, hence it must come out in the direction of the muon spin.
|
|
|
This correlation allows the experimental determination of the muon spin direction from the identification of the positron momentum direction. The correlation is is reduced for lower energies,
|
|
More precisely, the probability distribution function for the positron emission is correlated to the instantaneous direction of the muon spin by:
{$ (2) \qquad\qquad P(\theta) \quad \propto \quad 1+A(E)\cos\theta $}
where {$A(E)$} is an asymmetry factor that depends on the energy of the emitted positron, equal to 1 for {$E=E_{max}$}.
The polar plot on the right shows the probability distribution lobe fro this maximum asymmetry, such that the probability in each direction is proportional to the length of the blue segment along that direction.
|
|
|
|
The distribution vs. both angle and energy is given in terms of {$x=E/E_{max}$} as:
{$ W(x,\theta)=\frac {E(x)} {4\pi} [1+a(x) cos\theta] $}
with {$ E(x)=2x^2(3-2x)$} and {$a(x)=(2x-1)/(3-2x)$} shown in the plot on the left. The dashed line is the weighted asymmetry spectrum, {$E(x)a(x)/2$}.
|
|
The average over all energy of the function {$A(E)$} is equal to 0.33, which is the ideal experimental asymmetry value when no positron electron discrimination is performed. This is shown by the second polar plot on the right.
|
|
< How to produce a net spin polarization | Index | Longitudinal field >
