## 2 The MIXE Technique
In analogy to the Moseley law [43] for the electrons, and by assuming a point-charge nucleus, one can express the Bohr model energy of the \(\mu\)-X ray created between transitions between states with principal quantum numbers \(n_{i}\) (initial) and \(n_{f}\) (final) as:  
\[E_{i\to f,\mu}=\frac{m_{\mu}}{m_{e}}R_{y}(Z-S_{scr,\mu})^{2}(\frac{1}{n_{f}^ {2}}-\frac{1}{n_{i}^{2}})\simeq\frac{m_{\mu}}{m_{e}}E_{i\to f,e}\simeq 207 \times E_{i\to f,e}\, \tag{1}\]  
where \(m_{\mu}\) is the reduced mass of the system formed by the muon and the nucleus and \(R_{y}\) is the Rydberg constant. Note that by analogy with the Moseley law, we replace the atomic number \(Z\) by \(Z_{eff}=(Z-S_{scr,\mu})\). We have \(S_{scr,\mu}=0\) since there is no screening of the charge of the nucleus seen by the muon involved in the transition, as solely one \(\mu^{-}\) is present at a given time in the sample. Though the \(S_{scr,\mu}=0\), the electron screening also affects the energy of the \(\mu\)-X rays. While this effect can lead to energy shifts ranging from a fraction of eV to few hundreds of eV (depending on the muonic transition and the element), these are of the order of permille or less [44]. The situation is however different for the case of electrons where for K\({}_{\alpha}\) transitions (2\(p\to 1s\)), for example, one has \(S_{scr,e}=1\) as already one electron is present in the \(1s\) level. This difference explains the first approximation in Equation (1). Note that the Bohr model ignores the fine structure of the levels, which is determined by relativistic effects and the spin-orbit coupling. Hence, all states with the same principal quantum number \(n\) and different azimuthal quantum number \(\ell\) have the same binding energy in this model. Note also that for heavy nuclei, the \(\mu\)-X ray energy will be smaller than the one predicted by Equation (1), as the size of the nucleus cannot be neglected anymore with respect to the characteristic radius of the muonic state. This deviation will be smaller for large values of \(n_{f}\), corresponding to a large radius of the muonic state. Concluding, by measuring the \(\mu\)-X ray energy one can identify the element (similar to the XRF technique), which had captured the muon. However, the fundamental differences compared to the XRF technique are, that the muon momentum can be tuned to implant the muons at controlled depths deep inside the material (from tens of micrometers to several millimeters), and that the created \(\mu\)-X rays have enough energy to escape the material and being detected (We note here that for pure iron an X-ray created by the muonic transition K\({}_{\alpha 2}\) will have a range three orders of magnitude larger than the corresponding X-ray created by an electronic transition). This technique is referred to as the Muon Induced X-rayEmission (MIXE) technique. MIXE can be applied to carry out non-destructive elemental analysis deep inside a material, which is not possible by XRF or PIXE. Once the muon has reached its ground state, i.e., \(n_{f}=1\), it either decays or gets captured by the nucleus. The respective branching probabilities will depend on the overlap between the muonic and nuclear wavefunctions. The probability of the capture by the nucleus increases with \(Z\) through the reaction  
\[\mu^{-}+p\longrightarrow n+\
u_{\mu}\;. \tag{2}\]  
This leads to a decrease of the muon lifetime, as the muon lifetime \(\tau\) is given by  
\[\frac{1}{\tau}=\frac{1}{\tau_{\text{decay}}}+\frac{1}{\tau_{\text{nuclear  capture}}}\;. \tag{3}\]  
For low \(Z\) muonic atoms the muon lifetime is essentially identical to that of the free muon (\(\sim\)2.2 \(\upmu\)s), whereas for heavy atoms it can be lower than 100 ns [45]. The capture of the \(\mu^{-}\) by the nucleus (Equation (2)) results in the formation of a \(Z-1\) nucleus in an excited state, which will relax through different possible processes, for which some are connected with the emission of characteristic gamma-rays.