## 4 Discussion  
This study is built on conclusions from previous MeV-STEM simulation trends in amorphous ice. It was observed that for MeV-STEM in materials heavier and denser than amorphous ice, the high-energy electron beam was able to penetrate significantly more while maintaining nanoscale resolution when compared to a lower-energy beam. From the beam width plots and electron intensity counts, an incident beam focused deeper in the sample as opposed to the top yields a narrower beam at higher convergence semi-angles. This may be more accessible in a laboratory setting, as small convergence semi-angles are more challenging to maintain, making defocus depth an important factor for controlling the image resolution. It was also observed that in general, a smaller convergence semi-angle resulted in a smaller beam width in thick samples across defocus depth values and resulted in higher quality images with higher spatial resolution, although small convergence semi-angles are limited by experimental accessibility. It is worth noting that at a higher convergence semi-angle, more electrons were collected within 10 mrad (Table 2) as a percentage of initial electrons detected in this range, which indicates a more cylindrical beam profile than for smaller angles, so the beam includes information from more of the sample and may produce a more complete image. Balancing these effects will be critical in future experimental trials of MeV-STEM. Variation across materials should be interpreted as a function of both nuclear charge (Z) and material number density. It is clear from simulation results that with increasing Z, beam width increases and signal intensity in small collection angles decreases, both an indication that electrons are scattered more as one would expect from stronger Coulomb interactions with nuclei. However, despite carbon's lower nuclear charge in comparison to an oxygen-dominated ice sample, simulations revealed that carbon has a wider beam and less beam intensity collected in small collection angles. This is because of carbon's high number density relative to amorphous ice, as the ratio is 100.33 to 30.77 (expressed in \begin{table}
\begin{tabular}{c c} \hline \hline
**Material** & **Linear Depth (\(\upmu\)m)** \\ \hline Ice & 17.9 \(\pm\) 0.9 \\ Carbon (a) & 9.1 \(\pm\) 0.5 \\ Silicon Dioxide & 3.85 \(\pm\) 0.19 \\ Silicon & 3.35 \(\pm\) 0.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Maximum linear depth of Beer–Lambert plots for varying material. Simulations used a 3 MeV beam focused at 10 \(\upmu\)m and a convergence semi-angle of 1 mrad. number per nm\({}^{3}\)), so there are more potential \"targets\" for electrons to scatter off, which outweighs the difference in Z. We can expect to see comparable effects with higher nuclear charge and higher density materials. To return to the observation of signal intensities in Table 1, these variations did not result in significant differences in beam intensity collected in the 10-50 mrad range, which corresponds to annular dark field imaging techniques. However, clear variation was shown across materials in the 0-10 mrad range, corresponding to bright field imaging techniques. As different substances have demonstrated significantly different beam attenuation in this range, bright field imaging has been demonstrated to be viable for use in amplitude contrast imaging and substance identification in MeV-STEM with the relatively low-Z substances examined in this study. The portion of this study focused on Beer-Lambert law linearity revealed a clear dependence on initial beam energy as expected and a noticeable change with nuclear charge and density. The effects of forward scattering introduced narrow windows of nonlinear behavior at small depths for lighter materials, but maximum linear depths were confirmed to extend beyond 15 \(\upmu\)m for a material such as ice and at least on the order of a few \(\upmu\)m for heavier materials.