## 2 The MIXE Technique  
The muon is a lepton of mass \(m_{\mu}=105.66\) MeV/c\({}^{2}\), and thus \(\sim\)207 times heavier than the electron (\(m_{e}=0.511\) MeV/c\({}^{2}\)). There are two kinds of muons, namely positive muons (\(\mu^{+}\), antiparticle) and negative muons (\(\mu^{-}\), particle). For research on condensed matter and material science, spin polarized \(\mu^{+}\) are routinely used in different experiments such as muon spin rotation, relaxation and resonance studies (\(\mu\)SR) [35]. The advantage of \(\mu^{+}\) is that these stop at interstitial sites in a sample, acting as localized magnetic probes by making use of the asymmetric emission of positrons when they decay. The \(\mu^{-}\) can also be used for \(\mu^{-}\)SR studies [36]. However, as the negative muons do not stop at an interstitial site but are rather captured by the atoms of the sample, due to the electromagnetic interaction with the nucleus, they form the so-called muonic atoms. Hence, the interpretation of \(\mu^{-}\)SR data can be quite difficult, hampering a large use of this technique. An alternative use of \(\mu^{-}\) is to study the formation process of muonic atoms. When a \(\mu^{-}\) is captured by an atom, the resulting muonic atom is typically created in an excited state, with the muon in an muonic orbit principal quantum number \(n_{\mu}\sim\)14 [37]. Subsequently, the muon relaxes in a time-scale of the order of \(10^{-13}\)s to the lowest \(n_{\mu}=1\) muonic orbit by emitting a series of so called muonic X-rays (\(\mu\)-X rays) [38]. The energy of the \(\mu\)-X rays, which can be used to investigate the charge radius of the nucleus [39; 40; 41], is a fingerprint of the type of atom having captured the muon. Hence, the determination of the \(\mu\)-X rayenergies provides information about the presence of the different atomic species. Correcting the intensities of the observed \(\mu\)-X rays, for various processes, allows to quantify the weight percentages of the elements present in the sample under study. The processes to take into account are the branching ratio of the \(\mu\)-X rays, competing Coulomb capture probabilities of the different elements (generally heavier atoms have a higher capture probability) and X-ray absorption effects inside the sample. Due to their higher mass, muons exhibit a different stopping profile in matter than X-rays or electrons. Their energy loss distribution is similar to that of protons, a Bragg curve. The mean penetration depth of muons depends on the density of the target material and the incoming muon momentum [42]. This allows to deduce the elemental composition of a material deep below the surface in the range from tens of \(\mu\)m to mm and profiling with step sizes down to 10 \(\mu\)m to 100 \(\mu\)m, respectively. As an example, for a muon beam with a mean momentum of 30 MeV/c (corresponding to a kinetic energy of 4.18 MeV) and a momentum distribution characterized by a momentum bite \(\Delta p/p=2\%\) (\(\sigma=\Delta p\)), the stopping depth is 260 \(\pm\) 20 \(\mu\)m in a copper target. Also owing to the heavier mass, a \(\mu\)-X ray has an energy \(\sim\)207 times higher than that associated with the corresponding electronic transition (neglecting finite size effects). This much higher \(\mu\)-X ray energy and the correspondingly smaller mass attenuation coefficient has the consequence that the created \(\mu\)-X rays have a much higher probability to pass through the target material from much deeper regions than the characteristic X-rays of electronic transitions.