## 3 Results
### Beer-Lambert Law for Varying Materials at 3 MeV  
Figure 5 demonstrates the log ratio plots used to evaluate the validity of the Beer-Lambert law in the sample for the four materials used in this study. Each simulation that produced data for these plots was performed with a convergence semi-angle of 1 mrad, an incident energy of 3 MeV, a collection angle of 8.8 mrad, a camera distance of 10 \(\upmu\)m, and a focal depth of 10 \(\upmu\)m. Because maximum linear depth is a function of convergence semi-angle and camera distance (see Section 3.2), these values are used for consistency across materials. The red line on each plot shows the linear fit used to evaluate linearity of the data, and the vertical line marks the maximum linear depth. Table 3 summarizes these results.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Defocus Depth and** & **0 \(\upmu\)m** & **1 \(\upmu\)m** & **2 \(\upmu\)m** & **10 \(\upmu\)m** \\
**Convergence Semi-Angle** & & & & \\ \hline
10/1 & 10,000 & 7565 & 5667 & 1567 \\
1/10 & 7411 & 5513 & 4299 & 1533 \\
0/1 & 10,000 & 7493 & 5559 & 1575 \\
0/10 & 6928 & 5174 & 4124 & 1428 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Number of electrons collected within a 10 mrad collection angle for a 3 MeV electron beam in silicon dioxide. Imaging mode and convergence semi-angle were varied and shown in the leftmost column in the format “x/y,” where x is the defocus depth in \(\upmu\)m, representing imaging mode, and y is the convergence semi-angle in mrad. The total number of electrons collected at various depths is shown.  
Figure 5: Relation between log ratio of intensities and thickness in various materials collected in an 8.8 mrad range with a 3 MeV incident beam. Vertical lines mark the maximum linear depth, listed in Table 3. Blue lines show simulation results, while red lines show the linear fit. (**A**) Crystalline silicon. (**B**) Silicon dioxide. (**C**) Amorphous carbon. (**D**) Ice. The vertical axis is unitless.  
As explained in Section 2.4, the analysis for each material used a different initial linear range, and the lighter materials required some of the low-depth data to be omitted due to forward scattering, which is identifiable in Figure 5C,D by the slight concave-up curve at low x-values. Initial linear ranges used are as follows: crystalline silicon, 1.0 \(\upmu\)m; silicon dioxide, 1.1 \(\upmu\)m; amorphous carbon, 2.0 \(\upmu\)m; ice, 6.0 \(\upmu\)m. For amorphous carbon, depths less than 0.5 \(\upmu\)m were omitted, and for ice, depths less than 1.2 \(\upmu\)m were omitted. Because of this dependence on initial range, a slight error may be introduced proportional to the maximum linear depth, which has been estimated as \(\pm\)5% and is included in Table 3. This error is a product of this study's use of Hayashida & Malac's method of linearity determination as explained in Section 2.4.  
The maximum linear depth of silicon dioxide at 3 MeV is 3.85 \(\upmu\)m, approximately 6 times higher than at 300 keV, similar to silicon. The maximum linear depth of ice at 3 MeV is 17.9 \(\upmu\)m, approximately 11.5 times higher than at 300 keV. This shows a very clear dependence of maximum linear depth on beam energy, which is even more pronounced in low atomic number samples.