## 3 Experimental Details and Results
Figure 2 shows the \(\mu\)-X ray spectra of the muonic K-series, measured with all the twelve coaxial Ge detectors, for the six different muon momenta, in the energy range \(900<E<2150\) keV. At the lowest momentum (\(p\) = 28.6 MeV/c) (Figure 2a), all the muons stop in the first sample layer and one is able to observe the entire K-series of the \(\mu\)-X rays from Fe. In addition, the \(2p_{3/2}-1s_{1/2}\) (K\({}_{\alpha 1}\)) and the \(2p_{1/2}-1s_{1/2}\) (K\({}_{\alpha 2}\)) transitions at 1257.2 keV and 1253.1 keV are resolved, as shown in the inset of Figure 2a. Upon increasing the muon momentum, one detects first the \(\mu\)-X rays from both the Fe and Ti layers (\(p\) = 35.0 and 37.5 MeV/c, see Figure 2b,c); then the ones from all the three Fe, Ti and Cu layers (\(p\) = 40.0, see Figure 2d). At \(p\) = 42.5 MeV/c (Figure 2e), one looses the Fe-signal and only Ti and Cu \(\mu\)-X rays are observed. At \(p\) = 45.0 MeV/c (Figure 2f), only the Cu muonic X-rays are observed. It should be noted that the photopeak signal-to-noise ratio (SNR) is \(\sim\)20 for the \(K_{\alpha}\) line of Cu (see Figure 2f). To perform the analysis and estimate where the muons are stopping, we have chosen the characteristic \(\mu\)-X ray peaks of the three layers as follows: (i) K\({}_{\alpha 1}\) (1253.1 keV) and K\({}_{\alpha 2}\) (1257.2 keV) lines of Fe, (ii) K\({}_{\alpha 3}\) (931.8 keV) of Ti, and (iii) K\({}_{\alpha 1}\) (1507.7 keV) and K\({}_{\alpha 2}\) (1514.2 keV) lines of Cu. The intensity of these peaks was obtained by dividing the area under these peaks (after a proper background subtraction, as described in Ref. [56]) by the efficiency at the corresponding energies. Since at \(p\) = 28.6 MeV/c, we see only the Fe lines, the total intensity of the K\({}_{\alpha 1}\) (1253.1 keV) and K\({}_{\alpha 2}\) (1257.2 keV) peaks is normalized to 100. At \(p\) = 35.0 MeV/c, as both Fe and Ti lines are present, the intensities of the peaks of Fe and Ti are normalized in such a manner that both sum up to 100. The same procedure is followed for the rest of the muon momenta. By analyzing the relative intensities of the different peaks, we are thus able to estimate experimentally the fraction of the muon ensemble stopping in the different layers at a given muon momentum. To validate our approach, we compared our results with stopping profile simulations obtained with the Particle and Heavy Ion Transport code System (PHITS) simulation tool (ver. 2.88) [57; 58]. The PHITS simulations were used to represent the interaction of the negative muon beam, with a given muon momentum (\(p\)) and a Gaussian momentum distribution with standard deviation \(\sigma=\Delta p\) (relative width \(\Delta p/p=2\%\)), on the three-layered sandwich sample (Fe,Ti, and Cu) placed in vacuum, including the 200-\(\mu\)m-thick muon entrance detector. The simulations were carried out from \(p\) = 28 to 46 MeV/c, at intervals of 1 MeV/c to obtain the percentage of muons stopping in a given layer and the penetration depths of the muons. The interpolated simulated data points (using Hermite interpolation of order three) are shown as solid lines (blue for Fe, green for Ti, and red for Cu) in Figure 3a. The efficiency corrected and normalized intensities (as described in the previous paragraph) of the muonic-\(K_{\alpha}\) lines of Fe, Ti, and Cu were used to estimate the percentage of muons stopping in each layer. They are shown in the figure as blue filled circles, green filled squares and red filled diamonds, respectively. The difference between the experimental and the simulated points, called the residuals, at the six different muon momenta is shown in Figure 3 (bottom panel). From this plot, one can see that we obtain a very good agreement, within \(\sim\)10%. The high rates, low pile-up and excellent SNR (see Figure 2) allows the analysis of elemental composition in a short time scale (30-60 min).