The role of absorption in three-dimensional electron diffraction dynamical structure refinement

TL;DR

This work systematically assesses the role of absorption in 3D electron diffraction dynamical refinement. By combining analytical two-beam theory, Bloch-wave simulations, and experimental refinements on CsPbBr$_3$, quartz, and borane, it shows that absorption induces a uniform thickness-dependent decay with a mean length set by $U_0'$, while many-beam coupling introduces reflection-specific, orientation-dependent deviations that grow near zone axes. For the high-Z CsPbBr$_3$ material, including absorption reduces the refinement residual $R_{\mathrm{obs}}$ from $6.4\%$ to $5.3\%$, whereas quartz and borane exhibit negligible improvements, indicating absorption is generally negligible for routine integrated-intensity refinements unless thickness approaches $\xi_g$ in high-$Z$ systems. The findings explain longstanding observations of large residuals near zone axes and provide a framework for incorporating absorption into refinements, while highlighting the need for more complete phonon treatments and temperature effects in future work.

Abstract

The role of absorption in 3D electron diffraction is established through analytical theory, simulation, and dynamical refinement. A two-beam expression for the absorbed integrated intensity is derived, showing that for $t/ξ_g \ll 1$ reflections follow a uniform exponential decay set by the mean absorptive potential $U_0'$. Many-beam simulations demonstrate that neglecting absorption in dynamical refinement of integrated intensities incurs a residual that increases linearly with thickness and diverges near zone axes. Dynamical refinements were performed on CsPbBr$_3$, quartz, and borane, with the inclusion of absorption yielding an improvement in $R_{\mathrm{obs}}$ from $6.4$ to $5.3$ \% for CsPbBr$_3$ and negligible changes for quartz and borane. Absorption is therefore deemed negligible for routine refinement of integrated intensities except in high-$Z$ materials at thicknesses approaching $ξ_g$.

Figures (5)

• Figure 1: Integrated intensity vs thickness for ${CsPbBr_3}$, 200 keV, $[uvw]=[-0.43, 1.00, 0.93]$, without (top) and with absorption (bottom), for (a) two-beam, (b) many-beam model. The six most intense reflections are highlighted in colour, with all others shown in grey for clarity.
• Figure 2: Ratio of integrated intensity ($I^{\mathrm{int}}_{\mathrm{abs}}/I^{\mathrm{int}}_{\mathrm{no\,abs}}$) as a function of thickness for $\mathrm{CsPbBr_3}$ with $[uvw]=[-0.43,\,1.00,\,0.93]$. (a) Two-beam model; (b) many-beam model. A decaying exponential was fitted to each $\textit{hkl}$ curve, yielding absorption parameters $\bar{\lambda}=89.9\,\mathrm{nm}$, $\sigma_{\lambda}=2.0\,\mathrm{nm}$ (two-beam) and $\bar{\lambda}=93.2\,\mathrm{nm}$, $\sigma_{\lambda}=10.1\,\mathrm{nm}$ (many-beam).
• Figure 3: Thickness dependence of residual error $R_1$ (%), between simulated $I_\mathrm{abs}$ and $I_{\mathrm{no\,abs}}$. (a) Integrated intensities (dark blue) compared with per-tilt intensities (light blue) for $\mathrm{CsPbBr_3}$ at $[uvw]=[-0.43, 1.00, 0.93]$; light blue points are $R_1$ averages across $\pm1.5^\circ$ tilt-series with error bars showing standard deviation around the mean. Best-fit slopes were determined, yielding $0.048~\%~\text{nm}^{-1}$ and $0.058~\%~\text{nm}^{-1}$ for the integrated intensities and per-tilt values, respectively. (b) Integrated-intensity $R_1$ curves for CsPbBr$_3$, $\alpha$-quartz, and borane, each shown with three representative orientations (best-fit slopes and $[uvw]$ indices given in Supplementary Table 1.)
• Figure 4: Orientation dependence of absorption effects in $\mathrm{CsPbBr_3}$. (a) Thickness dependence of $R_1$ between simulated $I_\mathrm{abs}^{\mathrm{int}}$ and $I^{\mathrm{int}}_{\mathrm{no\,abs}}$ for orientations 1, 8, and 19. (b) Corresponding simulated diffraction patterns, each enclosed by a dashed box in the colour of its respective curve in (a).
• Figure 5: Results of dynamical refinements performed with and without the inclusion of absorption for CsPbBr$_3$. Residuals $R_{obs}$ (%) are compared across rotation indices for refinements carried out on identical datasets and parameters. Supplementary Fig. S4 shows the corresponding $wR_\mathrm{all}$ results for CsPbBr$_3$ and the full orientation-dependent residuals for $\alpha$-quartz and borane.