Entropic Comparison of Atomic-Resolution Electron Tomography of Crystals and Amorphous Materials

Electron tomography bears promise for widespread determination of the three-dimensional arrangement of atoms in solids. However, it remains unclear whether methods successful for crystals are optimal for amorphous solids. Here, we explore the relative difficulty encountered in atomic-resolution tomography of crystalline and amorphous nanoparticles. We define an informational entropy to reveal the inherent importance of low-entropy zone-axis projections in the reconstruction of crystals. In turn, we propose considerations for optimal sampling for tomography of ordered and disordered materials.

Figure 1

(a) Normalized informational entropy for three models plotted as a function of particle rotation (about the vertical axis). Plots are mirror symmetric about the 90° abscissa, and just over half of the range of each plot is shown. (Left side) Projection entropies, $S(I_1^{(2)})$. (Right side) Joint entropies $S(I_1^{(2)}|I_2^{(2)})$. Five views of the crystalline model and one view of the amorphous model are shown. (i),(ii) Crystals rotated 13.6° from the [010] projection about the [001] and [101] axes, respectively. (iii) Random amorphous view. (iv),(v),(vi) Crystals viewed down the $[010]$, $[110]$, and $[11\overline{1}]$ zone axes, respectively. (b) Representation of the 3D voxel volume, showing three perpendicular projections of a particle. The intensity trace along $i$ in the view $I_{ik}^{(2)}$ is normalized to its maximum value.

Figure 2

Reconstruction quality of simulated crystalline and amorphous nanoparticles as a function of image resolution. (Left axis) Root-mean-square error of reconstructed atomic coordinates. (Right axis) The fraction of atoms recovered. Solid markers are for crystal reconstructions, open markers for the amorphous reconstruction.

Figure 3

Two-dimensional $x\text{−}y$ planes (“orthoslices”) from the centers of the reconstruction volumes for (a) crystalline and (b) amorphous samples using 48 “non-zone-axis” projections (only projections at angles corresponding to high-entropy crystalline projections). (c),(d) Orthoslices from the centers of the reconstruction volumes for (c) crystalline and (d) amorphous samples using only 12 “zone-axis” projections (only angles corresponding to low-entropy crystalline projections).