MuonProduction

As we have seen muon beams are produced from charged pions, via the decay:

{$$ \begin{eqnarray} \pi^+ &\rightarrow& \overline\mu^+ + \nu_\mu \\ \pi^- &\rightarrow& \mu^- + \overline{\nu}_\mu \end{eqnarray}$$},

the bar indicating a leptonic antiparticle (the decay conserves the number of leptons).

In order to have such pions, a proton beam of sufficent energy is aimed at a solid target and a transport channel for charged particles is tuned to focus one open surface of the target, at one end, onto the sample, at the opposite end.

The energy of the primary proton beam must be above the threshold for pion production.

Pion production

To determine this threshold, consider that the dominant channel for positive pions is

{$$\begin{equation} p+p,n \rightarrow n+p,n+\pi \end{equation}$$}

where the second particle on the left (proton or neutron) is at rest.

Assuming the proton and neutron mass both equal to {$M=938 \mbox{MeV/c^2}$} and studying the system in the center of mass, where the two particles have equal and opposite momentum {$q$} we suppose that the threshold momentum for pion production (mass {$m=139.57 \mbox{MeV/c^2}$}) is for the head-on collision, where the proton turns into a neutron and bounces off with momentum {$p$} in the reverse direction, whereas the pion momentum {$x$} is opposite to {$p$} and so is the final momentum of the original particle at rest ({$-p+x$}).

Energy conservation imposes that

{$$\begin{equation} \tag{1}2\sqrt{M^2+q^2}=\sqrt{M^2+p^2}+\sqrt{M^2+(p-x)^2}+\sqrt{m^2+x^2} \end{equation}$$}

One can see that the minimum {$q$} is that for which {$p=x=0$}, i.e. {$q\ge \sqrt{Mm+m^2/4}=361.93 \mbox{MeV/c}$}¸ which means {$q^\prime\ge 790.6 \mbox{MeV/c}$} in the laboratory frame ({$q^\prime=2q\,\sqrt{M^2+q^2}/M$}). This means that the threshold kinetic energy of the proton beam in the laboratory frame is {$\sqrt{M^2+q^{\prime^2}}-M=281.5$} MeV.

In practice, looking at experimental pp data one sees that the cross section increases rapidly up to {$q^\prime=1.1 \mbox{GeV/c}$}, corresponding to a beam energy of 500 MeV, representing an optimum in some sense.

Transport

Pions of a well defined linear momentum are selected with a transport channel from the primary target on which the proton beam is impinging. The target is often pyrolitic carbon, with special care for cooling, since typical proton beams at muon facilities are from hundreds to thousands of µA.

A transport channel for charged particles is roughly equivalent to an optical beam and it is composed of magnetic lenses and magnetic prisms. The function of a (chromatic) lens is provided by two or three quadrupole magnets at short distance from each other. Quadrupoles have two opposed north poles and two opposed south poles each. The function of the prism is produced by a dipole magnet, a bender, which acts as a momentum selector. Muons, pions, electrons and any other particle of the same momentum and charge are trasported down such a beam.

An additional important element is the electrostatic separator, with crossed electric and magnetic fields. This device acts on a straight line as a velocity selector, since the two fields can be tuned to compensate each other for a specific mass. Hence it removes the other particles from the muon beam.

At best, the cross section of the beam at one of its subsequent foci is equal to cross section of the pion source on the primary target, which, in turns, is determined by the proton beam cross section and, possibly, by the target geometry (its thickness, for muon beams extracted at right angles to the protons) .