## 2 Methods  
### 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/cm\({}^{3}\): 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]:  
\[\frac{d\sigma_{el}}{d\Omega}=\left[\frac{2 ZR^{2}(1+E/E_{0})}{a_{H}\Big{(}1+ \left(\theta/\theta_{0}\right)^{2}\Big{)}}\right]^{2},\theta_{0}=\frac{ \lambda}{2\pi R},R=a_{H}Z^{-1/3} \tag{1}\]  
In this equation, \(\sigma el\) is the elastic scattering cross-section, \(\Omega\) is the solid angle, \(Z\) is the atomic number, \(E\) is the electron energy, \(E_{0}\) is the rest energy of the electron, \(aH\) is the Bohr radius (0.0529 nm), \(\theta\) is the scattering angle, \(\theta_{0}\) is the characteristic scattering angle below which 50% of the electrons are scattered into, and \(\lambda\) is the electron wavelength. Integrating this equation with respect to solid angle yields the total elastic scattering cross-section [13]:  
\[\sigma_{el}\approx\frac{1.4\times 10^{-6}Z^{3/2}}{\beta^{2}}\left(1-\frac{0.26Z }{137\beta}\right),\,\beta^{2}=1-\left[\frac{E_{0}}{E+E_{0}}\right]^{2} \tag{2}\]  
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]:  
\[\frac{d\sigma_{inel}}{d\Omega}\approx\frac{Z\Lambda^{4}\Big{(}1+\frac{E}{E_{0 }}\Big{)}^{2}}{4\pi^{2}a_{H}^{2}}\Bigg{[}\frac{1-\left(1+\frac{\theta^{2}+ \theta_{0}^{2}}{\theta_{0}^{2}}\right)^{-2}}{\left(\theta^{2}+\theta_{E}^{2} \right)^{2}}\Bigg{]},\theta_{E}=\frac{\Delta E}{E}\frac{E+E_{0}}{E+2E_{0}} \tag{3}\]  
Here, _oitel_ is the inelastic scattering cross-section, \(\theta E\) is the characteristic angle that is responsible for the decay of the inelastic scattering, and \(\Delta E\) is the mean energy loss from a single inelastic scattering event. This parameter will be further discussed in Section 2.2.