Quantitative Modeling of High-Energy Electron Scattering in Thick Samples Using Monte Carlo Techniques

Abstract

Cryo-electron microscopy (cryo-EM) is a powerful tool for imaging biological samples but is typically limited by sample thickness, which is restricted to a few hundred nanometers depending on the electron energy. However, there is a growing need for imaging techniques capable of studying biological samples up to 10 µm in thickness while maintaining nanoscale resolution. This need motivates the use of mega-electron-volt scanning transmission electron microscopy (MeV-STEM), which leverages the high penetration power of MeV electrons to generate high-resolution images of thicker samples. In this study, we employ Monte Carlo simulations to model electron–sample interactions and explore the signal decay of imaging electrons through thick specimens. By incorporating material properties, interaction cross-sections for energy loss, and experimental parameters, we investigate the relationship between the incident and transmitted beam intensities. Key factors such as detector collection angle, convergence semi-angle, and the material properties of samples were analyzed. Our results demonstrate that the relationship between incident and transmitted beam intensities follows the Beer–Lambert law over thicknesses ranging from a few microns to several tens of microns, depending on material composition, electron energy, and collection angles. The linear depth of silicon dioxide reaches 3.9 µm at 3 MeV, about 6 times higher than that at 300 keV. Meanwhile, the linear depth of amorphous ice reaches 17.9 µm at 3 MeV, approximately 11.5 times higher than that at 300 keV. These findings are crucial for advancing the study of thick biological and semiconductor samples using MeV-STEM.

1. Introduction

Understanding the building blocks of living organisms is critical for advancing fields such as medicine, biochemistry, and material science. One way to study these structures is by observing them in three dimensions (3D) with nanoscale resolution, a process that requires advanced imaging techniques. Such high-resolution imaging enables researchers to understand cell behavior and its implications for environmental studies, health, energy, and other fields.

While exciting imaging techniques such as single particle cryogenic electron microscopy (cryo-EM) allow researchers to determine the structure of biological macromolecules, the rapidly advancing technologies of volume electron microscopy (vEM) are being used to visualize 3D volumes with nanoscale resolution [1,2]: serial block-face scanning electron microscopy (SBF-SEM) [3], focused ion beam scanning electron microscopy (FIB-SEM) [4], array tomography [5], cryo-FIB SEM [6,7,8,9,10], and cryo-electron tomography (cryo-ET) [11]. Despite the impressive discoveries made with vEM, it is fundamentally constrained by sample thickness. This limitation arises from the low penetration depth of the imaging electrons and the extensive time required for thinning the samples to achieve nanometer resolution. To visualize thick samples up to 10 µm with nanometer resolution, mega-electron-volt scanning transmission electron microscopy (MeV-STEM) has been proposed [12]. STEM combines SEM and TEM techniques by directly focusing the electron beam on the sample like SEM but creating an image from transmitted electrons computationally. STEM has already demonstrated the ability to adequately image samples up to 1 µm [13,14], which is a significant improvement over cryo-EM. In STEM, image resolution depends on electron probe broadening and intensity, both measured by the number of electrons captured within a collection angle. Single and multiple elastic and inelastic scattering events with varying beam focal depth and collection angles have been previously investigated [15], revealing that increased sample thickness leads to greater beam widening and multiple scattering. However, simulations demonstrated that through a 10 µm thick sample of amorphous ice, 42% of MeV electrons were collected within 10 mrad, confirming the viability of MeV-STEM. These simulations utilize the Monte Carlo (MC) technique, which uses random sampling with weighted probabilities to address complex systems that are difficult to solve analytically.

To quantitatively model the interactions between high-energy electrons and various materials and systematically investigate the possibility and performance of MeV-STEM over materials other than amorphous ice, we carried out extensive MC simulations. In this study, a previously developed MC program was employed to study the effect of sample material, beam convergence semi-angle (angular spread of beam on incidence), collection angle, and beam energy. The beam width increases from amorphous ice to carbon, silicon dioxide, and crystalline silicon. The beam profiles are also significantly affected by the beam focusing position. The applicability of the Beer–Lambert law in predicting signal attenuation in MeV-STEM versus lower energy incident electrons is also investigated. The results further suggest that MeV-STEM is a promising technique for producing high-resolution images of thicker biological samples, capable of increasing the rate and scope of scientific discovery.

2. Methods

2.1. Interaction Cross-Sections

When electrons impinge on a material, they may be scattered elastically, with no energy loss, or inelastically, where energy is transferred to the material. Scattering occurs with probabilities determined by the cross-section and the sample density. Alternatively, electrons may pass through without scattering, maintaining their initial trajectories. The total cross-section expresses the probability of each scattering event, with the combined probabilities of all scattering events plus the non-scattering event summing to unity. The densities of the materials used in this study are summarized here, in g/cm³: ice, 0.92 [16]; amorphous carbon, 2.00 [17]; crystalline silicon, 2.33 [18]; silicon dioxide, 2.65 [19]. Note that amorphous silicon dioxide has multiple potential structures with comparable densities; the alpha quartz structure was assumed in this study. These materials were chosen to represent a variety of material densities and atomic numbers from low- to mid-range and support a variety of applications. Understanding scattering in ice is critical for imaging vitrified samples, and amorphous carbon is regularly used as a substrate in biological sample imaging, while silicon and silicon dioxide are critical to the microelectronics and semiconductor industries and are important imaging subjects. Real biological and electrical samples are heterogeneous, so understanding electron scattering from these fundamental substances is critical to recognizing them in more complex samples.

The differential scattering cross-section describes the angular distribution of scattered electrons. For elastic scattering events, the Rutherford formula for the screened Coulomb potential of sample nuclear charge is used. Utilizing the Wentzel approximation to describe screening, the first-order perturbative Born approximation is as follows [13]:

$${\displaystyle \frac{d{\mathsf{\sigma }}_{el}}{d\mathsf{\Omega }}}={\left[{\displaystyle \frac{2Z{R}^2\left(1+E/E_0\right)}{a_H\left(1+{\left(\mathsf{\theta }/{\mathsf{\theta }}_0\right)}^2\right)}}\right]}^2,{\mathsf{\theta }}_0={\displaystyle \frac{\mathsf{\lambda }}{2\mathsf{\pi }R}},R=a_H{Z}^{-1/3}$$

(1)

In this equation, σel is the elastic scattering cross-section, Ω is the solid angle, Z is the atomic number, E is the electron energy, E₀ is the rest energy of the electron, aH is the Bohr radius (0.0529 nm), θ is the scattering angle, θ₀ is the characteristic scattering angle below which 50% of the electrons are scattered into, and λ is the electron wavelength. Integrating this equation with respect to solid angle yields the total elastic scattering cross-section [13]:

$${\mathsf{\sigma }}_{el}\approx {\displaystyle \frac{1.4\times {10}^{-6}{Z}^{3/2}}{{\mathsf{\beta }}^2}}\left(1-{\displaystyle \frac{0.26Z}{137\mathsf{\beta }}}\right),{\mathsf{\beta }}^2=1-{\left[{\displaystyle \frac{E_0}{E+E_0}}\right]}^2$$

(2)

Inelastic scattering, which is concentrated in a much smaller angular range than elastic scattering, is approximated by a Bethe model, producing the following differential cross-section [13]:

$${\displaystyle \frac{d{\mathsf{\sigma }}_{inel}}{d\mathsf{\Omega }}}\approx {\displaystyle \frac{Z{\mathsf{\lambda }}^4{\left(1+\frac{E}{E_0}\right)}^2}{4{\mathsf{\pi }}^2{a}_H^2}}\left[{\displaystyle \frac{1-{\left(1+\frac{{\mathsf{\theta }}^2+{\mathsf{\theta }}_0^2}{{\mathsf{\theta }}_0^2}\right)}^{-2}}{{\left({\mathsf{\theta }}^2+{\mathsf{\theta }}_E^2\right)}^2}}\right],{\mathsf{\theta }}_E={\displaystyle \frac{\mathsf{\Delta }E}{E}}{\displaystyle \frac{E+E_0}{E+2E_0}}$$

(3)

Here, σinel is the inelastic scattering cross-section, θE is the characteristic angle that is responsible for the decay of the inelastic scattering, and ∆E is the mean energy loss from a single inelastic scattering event. This parameter will be further discussed in Section 2.2. Now, the total inelastic cross-section may be approximated [20]:

$${\mathsf{\sigma }}_{inel}\approx {\displaystyle \frac{1.5\times {10}^{-6}{Z}^{1/2}}{{\mathsf{\beta }}^2}}ln\left(2/{\mathsf{\theta }}_C\right),{\mathsf{\theta }}_C={\displaystyle \frac{⟨\mathsf{\Delta }E⟩}{{\mathsf{\beta }}^2\left(E+E_0\right)}}$$

(4)

For molecules composed of multiple elements, such as ice and silicon dioxide, the cross-section must be calculated for an atom of each constituent element, multiplied by relative frequency (e.g., two hydrogen atoms to one oxygen in ice), and then summed. In this study, cross-sections at each angle were calculated using differential cross-sections and stepwise intervals. The total cross-section for both elastic and inelastic was defined as the sum of the cross-section at each angle normalized by the total cross-sections calculated by the respective Equation (2) or (4).

2.2. Mean Energy Loss

Mean energy loss, also referred to as stopping power, is a quantity associated with the amount of energy lost during an inelastic scattering event. This quantity is experimentally determined and necessary to calculate inelastic cross-sections. For ice, this value is experimentally known to be 39.3 eV [20]. However, such values are not available in the literature for amorphous carbon, crystalline silicon, and amorphous silicon dioxide. Therefore, they must be calculated using available spectra from electron energy loss spectroscopy (EELS). The following equation describes how mean energy loss can be calculated from EELS spectra in the low energy loss range, which was defined in this study as approximately 10–90 eV, beginning immediately after the zero-loss peak centered at 0 eV.

$$⟨\mathsf{\Delta }E⟩={\displaystyle \frac{{\int }_0^{\mathrm{\infty }}\left(\mathsf{\Delta }E\right)I_{inel}d\mathsf{\Delta }E}{{\int }_0^{\mathrm{\infty }}{I}_{inel}d\mathsf{\Delta }E}}$$

(5)

where I_inel is the signal intensity at energy loss ΔE.

The 200 kV crystalline silicon and amorphous silicon dioxide EELS data were provided by Marie Cheynet with Université Grenoble Alpes. The 200 kV amorphous carbon EELS data were provided by Graham Carpenter with Natural Resources Canada. Note that the spectral data for carbon were complete up to approximately 45 eV, so to remain consistent with integration for other materials, this curve was extended using a power series fit to 90 eV. Using Equation (5), the following values were calculated for mean energy loss: 54.65 eV for crystalline silicon, 52.55 eV for amorphous silicon dioxide, and 32.55 eV for amorphous carbon.

2.3. Monte Carlo Simulation

The simulated beam parameters were defined according to proposed experimental designs for MeV-STEM [12,15]. The following parameters were employed: 2 pm geometrical emittance, 3 × 10⁻⁵ energy spread, optimized chromatic (1.8 cm) and spherical (16 cm) aberrations of the STEM column, 1 nm of the minimum root-mean-square (RMS) beam size, and 1–10 mrad of semi-convergence angle at the focal position.

In a single MC simulation, 10,000 electrons were each randomly assigned an event at each depth with the probability determined by the cross-section of each event type (elastic, inelastic, and unscattered). A simulation (i.e., 10,000 electrons traced) was performed for each variation in experimental conditions. It is worth noting that there are discrepancies between the calculated and theoretical ratios of σ_inel/σ_el. It is predicted to be ~26/Z [21] and ~20/Z [13]. In this study, this ratio was defined as lying between these predictions but remains an approximation and may be considered a potential source of error: 3 for amorphous ice, 3.8 for amorphous carbon, 1.6 for crystalline silicon, and 3 for the oxygen atoms and 1.6 for the silicon atoms in silicon dioxide. The total inelastic cross-section was calculated as a multiple of the total elastic cross-section based on these ratios. Sample thickness dt was set at 0.5 nm. For ice, the inelastic cross-section of hydrogen was demonstrated to be inaccurate [20]; thus, the inelastic cross-section (total and incremental) of ice was set to 110% of that of oxygen-only ice. The scattering angle for each event was probabilistically determined using differential cross-sections calculated using Equations (1) and (3). The angular intervals were set to 0.001 mrad up to 0.6 mrad, then 0.01 mrad up to 6.3 mrad, followed by 0.1 mrad to 67 mrad, 1 mrad to 600 mrad, and 10 mrad for remaining angles. The position and angle were recorded for each event as the electron traveled through the sample, so they could be plurally scattered with each event type recorded accordingly.

Since the incoming electron beam is circularly symmetric about the incident axis, only a cross-section parallel to and across the incident axis was studied in the simulations. The beam width at a selected depth t was defined as the width containing 68% of the electrons. Note that for a defined collection angle that excludes some number of electrons, the beam width refers to the beam size within which 68% of the collected electrons are contained.

2.4. Beer–Lambert Law

By measuring the intensity drop of an incident electron beam, it is possible to estimate the thickness of the sample. This method depends on proportionality between thickness and signal decay described by the Beer–Lambert attenuation law:

$$I_{tr}=I_0{e}^{-t/{\mathsf{\lambda }}_{total}}$$

(6)

In this equation, I_tr refers to the transmitted beam intensity, I₀ is the incoming primary electron beam intensity, and λ_total is the total mean free path, or the average distance an electron travels in the sample before a collision. Using the number of electrons as a measure of beam intensity, plotting the natural logarithm of primary over transmitted intensity versus thickness reveals a linear dependence under this condition.

In this study, the range of linearity is determined as described in Hayashida & Malac (2022) [22]. For the energy scale in that study, the authors note that the range of 0–180 nm was consistently linear, so in this study a short initial range approximated as linear is used in a regression to determine a slope and y-intercept for the plot. Using these parameters and extending the regression line indefinitely, when the R² value describing the relationship of the data to the fit was ≥0.95, the data were said to be linear, so the maximum thickness was defined as the largest value of t for which this condition was met. For higher energies where linearity is extended, initial linear ranges were first estimated via visual inspection and then tested by slightly increasing and decreasing the range to ensure that no notable change in maximum thickness was observed, and then the same method was used. Variations in initial linear range and different thresholds for linear behavior would change maximum linear depth measurements, but this study utilized methods established by Hayashida & Malac for consistency.

It is worth noting that for higher energies and lighter materials, some regions of nonlinearity were identified at the top of the sample, which is likely a result of forward scattering. This effect means that high-energy beams through thin samples display less attenuation, which limits the applicability of the Beer–Lambert law in such situations and can interfere with image interpretation, prompting the need for consideration of beam energy for different sample materials and thicknesses (i.e., lower energies for thinner samples). Forward scattering is important to understand when imaging with high-energy beams but is not the focus of this study, so to avoid producing inaccurate linear fits, these regions were omitted from the regression and therefore did not contribute to identifying the maximum linear depth. For these instances, the omitted range is reported in the Results section below.

3. Results

3.1. Electron Beam Profiles in Materials

Figure 1 shows the beam profiles of 10,000 electrons at 3 MeV in crystalline silicon (Figure 1A,B) and amorphous carbon (Figure 1C,D) by tracing the path of each individual electron in a simulation run using its associated event types and angles of deflection. These plots are included to demonstrate the scattering process and the occurrence of out- and back-scattering. The beam was focused on the bottom of the 10 µm sample, using a convergence semi-angle of 1 mrad. All electrons are shown, as the heatmap plots (Figure 1B,D) reveal beam intensity. For carbon, most electrons are found within 100 nm of the center at 10 µm. For the silicon sample, the beam intensity is significantly less at 10 µm than in carbon but is still largely concentrated within 150 nm of the center. Matching predictions for higher nuclear charge, the silicon simulation also clearly displays more instances of out- and back-scattering. Silicon dioxide and ice plots can be found in Appendix A.

3.2. Beer–Lambert Law at Low and High Energies

Figure 2 demonstrates the log ratio plots generated to evaluate the validity of the Beer–Lambert law in the sample. To compare results with experimental measurements of Beer–Lambert behavior in STEM as demonstrated by Hayashida et al. [22], simulations to produce the plots in Figure 2 were performed with a top-focused beam on silicon dioxide, with a collection angle of 8.8 mrad. The 300 keV beam (Figure 2A) produced a maximum linear thickness on the order of 350–550 nm, slightly underestimating the experimental value of 650 nm. Error may have been introduced since Hayashida et al. employed an energy filter while this simulation did not, so these maximum thicknesses should be treated as order-of-magnitude estimates in comparison to experimental results. With this stipulation, the correct order of magnitude was achieved for lower energy STEM, so qualitative conclusions about the effect of incident energy may be drawn based on results from MeV-STEM.

The 3 MeV beam (Figure 2B) produced a maximum linear thickness on the order of 4 µm. For the same reason as the lower energy simulation, an exact value was not calculated, as a single camera distance and convergence semi-angle were not chosen. From the order-of-magnitude estimate, keeping all other factors equal, increasing incident energy from 300 keV to 3 MeV resulted in the applicability of the Beer–Lambert law to nearly 10 times the thickness range.

The maximum linear depth of other materials at 300 keV was also estimated. For amorphous ice and a bottom-focused beam, this value was found to be 1.52 µm. Compared to silicon dioxide, silicon had a maximum linear depth on the order of 300–400 nm with a top-focused beam. These results are compared to the maximum linear depth with 3 MeV beams in Section 3.5.

3.3. Effect of Sample Material on Beam Profile and Decay

The effect of various materials on beam width was investigated in Figure 3, where the only changing variable between simulations was the sample material used. Beam width was represented by R_0.68; the perpendicular distance within which 68% of the electrons detected were located was calculated using each electron’s horizontal spread from the center, which was then plotted at each thickness. At 10 µm, this width was found to be 27.0 nm at 10 µm for amorphous ice, 35.5 nm for amorphous carbon, 55.7 nm for silicon dioxide, and 60.0 nm for crystalline silicon. Note that the R_0.68 measurement of beam width in this simulation only accounts for electrons detected within the specified collection angle, which was set to 10 mrad in Figure 3.

The number of electrons detected at a certain depth is used to quantify electron beam intensity.

Table 1. Electron beam intensity expressed as a percentage of the incident beam detected at 10 µm. Simulations used a bottom-focused 3 MeV beam and a convergence semi-angle of 1 mrad.

Material	Collection Angle (0–10 mrad)	Collection Angle (10–50 mrad)
Ice	41.43%	53.18%
Carbon (a)	27.87%	58.96%
Silicon Dioxide	15.67%	51.14%
Silicon	14.08%	48.49%

shows beam intensity for the same simulations plotted in Figure 3 in the form of a percentage of the incident beam, calculated for the bright field imaging angular range (0–10 mrad) and the annular dark field imaging range (10–50 mrad). Any remaining electrons are scattered outside 50 mrad, in the high-angle annular dark field imaging range that was not considered in this study. Materials with increasing beam width in the bright field range shown in Figure 3 display fewer electrons detected in that angular range. This indicates the viability of bright field amplitude contrast imaging, motivating much of the analysis in this study to be performed with a collection angle of 10 mrad.

3.4. Effects of Convergence Semi-Angle and Imaging Modes

Figure 4 demonstrates the results of the investigation into the effects of convergence semi-angle (α) and imaging mode via beam focal depth on beam width through silicon dioxide. Beam width represented by R_0.68 collected within 10 mrad at 10 µm depth was calculated and compared:

• When the beam is focused on the top surface, a larger convergence semi-angle produces wider beams, thus larger beam width: top-focused (at 0 µm) with α = 1 mrad was 55.0 nm, and top-focused with α = 10 mrad was 63.1 nm (pink and cyan curves in Figure 4).
• When the beam convergence semi-angle is set at 1 mrad, a deeper focus produces a wider beam in the top region of the sample, but the beam width is similar at locations deeper than 3 µm (pink and red curves in Figure 4). This observation is similar to what has been observed for a thinner sample [15].
• When the beam width is fixed at the top surface by either focusing the α = 10 mrad beam at 1 µm or focusing the α = 1 mrad beam at 10 µm (blue and red curves in Figure 4), the more diverged beam produces a narrower beam up to 2 µm in depth. This might be because the α = 10 mrad beam continues to focus after entering the sample. After that, the more diverged beam becomes wider than the less diverged beam as expected.

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 µm, representing imaging mode, and y is the convergence semi-angle in mrad. The total number of electrons collected at various depths is shown.

Defocus Depth and Convergence Semi-Angle	0 µm	1 µm	2 µm	10 µm
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

lists the number of electrons collected within a 10 mrad collection angle for each combination of imaging mode and convergence semi-angle in Figure 4, evaluated at various depths for reference. The number of collected electrons is significantly fewer for the more diverged beam (α = 10 mrad).

3.5. 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 µm, and a focal depth of 10 µm. 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. Maximum linear depth of Beer–Lambert plots for varying material. Simulations used a 3 MeV beam focused at 10 µm and a convergence semi-angle of 1 mrad.

Material	Linear Depth (µm)
Ice	17.9 ± 0.9
Carbon (a)	9.1 ± 0.5
Silicon Dioxide	3.85 ± 0.19
Silicon	3.35 ± 0.17

summarizes these results.

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 µm; silicon dioxide, 1.1 µm; amorphous carbon, 2.0 µm; ice, 6.0 µm. For amorphous carbon, depths less than 0.5 µm were omitted, and for ice, depths less than 1.2 µm 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 ±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 µm, approximately 6 times higher than at 300 keV, similar to silicon. The maximum linear depth of ice at 3 MeV is 17.9 µm, 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.

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 number per nm³), 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 µm for a material such as ice and at least on the order of a few µm for heavier materials. These conclusions are significant, as the accuracy of Beer–Lambert attenuation behavior in high-energy imaging can be used to predict signal strength in various materials and ensure strong signals through thicker samples. Future studies are needed to test Beer–Lambert attenuation under certain experimental conditions, realized with MeV-STEM technology based on proposed experimental designs such as those outlined by Yang & Wang et al. [12]. It is also important to experimentally examine the Beer–Lambert law in heterogeneous samples and consider more complex interactions as would be present in real-world applications. As MeV-STEM is a recently proposed technique, the purpose of simulations such as this is to demonstrate the viability of nanoscale resolution imaging on the order of 10 µm thick samples, which was successfully carried out. These results are also intended to serve as a reference for optimizing experimental control for increased resolution and penetration.