## 2 Muons
### Muon Production and Muon Beamlines  
In nature, muons are produced by the interaction of cosmic rays with the particles present in the atmosphere: there is about 1 muon/second striking on your palm as you read this article. However, to conduct negative muon experiments for elemental analysis requires a greater data rate. These increased data rates are achieved by using a proton accelerator in which the protons (with energies greater than 500 MeV) strike a target, often carbon. If any of these protons collide with a proton/neutron from the carbon target, it will result in the production of pions. In this case, the negative pions decay into negative muons, after about \(\sim\)26 ns. In the muon beamline, initially, the pions are extracted from the carbon target: these decay in-flight into muons and the muons are transported to an area where the instrument and material of interest is placed. An example beamline is shown in Figure 4. The beam of muons is focused by the quadrupole magnets and the momentum (or energy) of the muons is controlled by the bending (or dipole) magnets. By controlling the momentum/energy, the penetration depth of the muons can be controlled. The momentum can be varied from 15 MeV/c to 120 MeV/c at ISIS, these of course will vary slightly from facility to facility. The penetration depth, R, can be determined by:  
\[\text{R}=ap^{3.5}\]  
where \(a\) is inversely proportional to the stopping power (which only slightly depends on the element) and \(p\) is the momentum.  
There are a number of programs that can model the stopping profile, e.g., GEANT4, G4 beamline, SRIM/TRIM [17; 18; 19]. A popular and easy-to-use program is SRIM/TRIM, which can simulate these stopping profiles, and may give a more accurate result, by including contributions from many physical processes. A typical maximum penetration depth is about 1 cm in a metal (e.g., copper, silver, iron) and more in less dense materials (e.g., carbon \(\sim\)2.8 cm and water \(\sim\)6.1 cm). These calculations have been confirmed experimentally by some studies [7; 20; 21], which also show the depth selectivity (Figure 5).  
Figure 3: A typical pure silver spectrum. (**a**) With low energy detectors (up to 1 MeV), low energy transitions are detected. Since the muon travels to the nucleus across the energy states of the muonic atom, most of the transition can be detected; (**b**) high energy transition of silver. Data was taken and adapted from [16].  
Figure 4: A schematic of the RIKEN-RAL beamline based at ISIS. Elemental analysis is currently performed at Port 4. The figure was taken and adapted, with permission from Ref. [7] © 2016 Elsevier Ltd.  
Figure 5: Muon depth distribution over a set of different momentum: (**a**) lower momentum; (**b**) higher momentum; (**c**) characteristic X-ray energy distribution. (**a,b**) The stopping depth was calculated by  
taking copper as reference material, taking into account a momentum spread of 4%. The expected locations of the sample material boundary in this reference scale are shown by the dotted vertical lines, while the accumulative thickness is reported on the top axis. For a more detailed survey on particle ranges in material, see Section 34 \"Passage of particles through matter\" (Figure 34.4) in [22]. (**c**) The 3D graph shows the distribution of the characteristic muonic X-ray intensity of the elements present in the different layers, with the penetration depth (in g/cm\({}^{2}\)) reported on the top axis. The data and the figure were taken and adapted from Ref. [23].