Nanoclusters play key roles in a wide range of materials and devices because of their unique physical and chemical properties. These properties are determined by the specific three‐dimensional (3D) morphology, structure and composition. It is well known that extremely small changes in their local structure may result in significant changes of their properties. Therefore, development of techniques to measure the atomic arrangement of individual atoms down to (sub)‐picometer precision is important. This allows one to fully understand and greatly enhance the properties of the resulting materials, increasing the number of applications. Electron tomography using aberration‐corrected scanning transmission electron microscopy (STEM) is considered as one of the most promising techniques to achieve atomic resolution in 3D. Although this is not yet a standard possibility for all structures, significant progress has recently been achieved using different approaches [1,2]. Once the atoms can be resolved in 3D, the next challenge is to refine the atom positions in order to locate them as precisely as possible. However, the answer to the question how precise these measurements are, is still open. Here, we investigate the theoretical limits with which atoms of a nanocluster can be located in 3D based on the acquisition of a tilt series of annular dark field (ADF) STEM images. A parametric model, describing the expectations of the intensities observed when recording a tilt series of ADF STEM images, is needed in order to derive an expression for the highest attainable precision [3,4]. Although the multislice method is more accurate to describe the electron‐object interaction, it is very time‐consuming, especially when simulating a tilt series of images. Therefore, a Gaussian approximation model has been used as well in order to perform fast, albeit approximate simulations that allow us to get insight into the precision that can be attained to locate atoms in 3D. The precision has been computed for locating the central atom of four gold nanoclusters of different sizes with a Mackay icosahedral morphology. A cross‐section of such a nanoparticle is shown in Fig. 1(a) indicating the x‐, y‐, and z‐axis. In Fig. 1(b), the attainable precision is shown for the x‐, y‐ and z‐coordinate of the central atom computed taking  all the atoms into account, the atoms of the central plane (orange atoms and red atom in Fig. 1), or the central atom only (red atom in Fig. 1(a)) based on simulations using the Gaussian approximation model. From this figure, it can be seen that the precision is not significantly affected by neighbouring atoms, and therefore, it is allowed to use only the central atom to evaluate the attainable precision. In figure 2(a), 2(b) and 2(c) the attainable precision is illustrated as a function of the number of projections, the tilt range, and the incident electron dose. The precision increases with increasing number of projections, tilt range, and incident electron dose. Using optimal parameters for the number of projections, the tilt range and electron dose determined based on the calculation of the precision using the Gaussian approximation model, realistic STEM simulations have been performed using the multislice method. The precision has been evaluated for a dose of 8680 e − /Å 2  as a function of the inner detector radius of the annular STEM detector (Fig. 3(a)). The optimal inner angle equals the semi‐convergence angle. Next, the precision to locate the central atom is determined for the different cluster sizes using all optimised settings (Fig. 3(b)). Here, it is shown that theoretically, a precision of a few picometers can be attained for locating atoms in 3D using a tilt series of ADF STEM images. In conclusion, it is shown that the attainable precision for locating atoms in 3D can be optimized as a function of the number of projections, tilt range, electron dose, and inner radius of the STEM detector. It is demonstrated that a precision in the picometer range for positioning atoms in 3D is feasible.

Quantitative scanning transmission electron microscopy (STEM) using an annular dark field (ADF) detector has become a widely used technique for the characterization of materials at the atomic level. The quantification process involves the comparison of experimental data with image simulations, the use of statistical tools in a parameter estimation framework or a combination of both [1]. These methods have been developed using different measures for comparison, like peak intensities at the atom column position [2], image contrast variations [3] or so‐called scattering cross‐sections [4, 5]. The latter correspond to the total scattered intensity integrated over the atom column area. They have been shown to be very sensitive to the number of atoms in a column and its composition [1, 4, 6, 7]. Figure 1a shows the increase in peak intensity (green axis) and cross‐section (black axis) versus increase in number of atoms for a Pt column in [110] zone axis. As it can be observed, the peak intensity saturates after around 8 atoms meanwhile the cross‐section monotonically increases. In this work, we perform an analysis of how the electron wave propagates inside the crystal for the probe positions that conform the scattering cross‐section. With this, we analyse how the signal is generated for different detector collection angle regimes. Then, the analysis allows to identify the origin of the scattered signal and why scattering cross‐sections are more sensitive for composition and number of atoms as compared to peak intensities. In Figure 1b, we show a simulated image of a unit cell of a Pt crystal in [110] zone axis with a color‐edited version indicating the labels of the probe positions that form the scattering cross‐section and their respective distance to the atom column position. Figure 2 shows the probability amplitude of the electron wave as it propagates through the crystal for probe position r0 (a), which corresponds to the peak intensity, and for the sum of all the probe positions that conform the scattering cross‐section (b). From this, we observe that the atom column is excited deeper in the column when analyzing the cross‐sections. The off‐column probe positions carry more rich information about the scattering process for different thickness and collection regimes, which explains the increased sensitivity of this measure to the number of atoms and its composition. We then discuss the contribution to the scattered intensity for different detector collection angle regimes, such as LAADF, MAADF and HAADF.