# Annick De Backer University of Antwerp, Antwerpen

University of Antwerp, Antwerpen

## Materials Science, Statistics, Probability Theory

PhD

## Publications

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**ABSTRACT:**Quantitative atomic resolution annular dark field scanning transmission electron microscopy (ADF STEM) has become a powerful technique for nanoparticle atom-counting. However, a lot of nanoparticles provide a severe characterisation challenge because of their limited size and beam sensitivity. Therefore, quantitative ADF STEM may greatly benefit from statistical detection theory in order to optimise the instrumental microscope settings such that the incoming electron dose can be kept as low as possible whilst still retaining single-atom precision. The principles of detection theory are used to quantify the probability of error for atom-counting. This enables us to decide between different image performance measures and to optimise the experimental detector settings for atom-counting in ADF STEM in an objective manner. To demonstrate this, ADF STEM imaging of an industrial catalyst has been conducted using the near-optimal detector settings. For this experiment, we discussed the limits for atomcounting diagnosed by combining a thorough statistical method and detailed image simulations. - [Show abstract] [Hide abstract]

**ABSTRACT:**The three dimensional (3D) atomic structure of nanomaterials, including strain, is crucial to understand their properties. Here, we investigate lattice strain in Au nanodecahedra using electron tomography. Although different electron tomography techniques enabled 3D characterizations of nanostructures at the atomic level, a reliable determination of lattice strain is not straightforward. We therefore propose a novel model-based approach from which atomic coordinates are measured. Our findings demonstrate the importance of investigating lattice strain in 3D. - [Show abstract] [Hide abstract]

**ABSTRACT:**Annular dark-field (ADF) scanning transmission electron microscopy (STEM) has become widely used in quantitative studies based on the opportunity to directly compare experimental and simulated images. This comparison merely requires the experimental data to be normalised and expressed in units of 'fractional beam-current'. However, inhomogeneities in the response of electron detectors can complicate this normalisation. The quantification procedure becomes both experiment and instrument specific, requiring new simulations for the particular response of each instrument's detector, and for every camera-length used. This not only impedes the comparison between different instruments and research groups, but can also be computationally very time consuming. Furthermore, not all image simulation methods allow for the inclusion of an inhomogeneous detector response. In this work, we propose an alternative method for normalising experimental data in order to compare these with simulations that consider a homogeneous detector response. To achieve this, we determine the electron flux distribution reaching the detector by means of a camera-length series or a so-called atomic column cross-section averaged convergent beam electron diffraction (XSACBED) pattern. The result is then used to determine the relative weighting of the detector response. Here we show that the results obtained by this new electron flux weighted (EFW) method are comparable to the currently used method, while considerably simplifying the needed simulation libraries. The proposed method also allows one to obtain a metric that describes the quality of the detector response in comparison with the 'ideal' detector response. Copyright © 2015 Elsevier B.V. All rights reserved. - [Show abstract] [Hide abstract]

**ABSTRACT:**Quantitative annular dark field scanning transmission electron microscopy (ADF STEM) has become a powerful technique to characterise nano-particles on an atomic scale. Because of their limited size and beam sensitivity, the atomic structure of such particles may become extremely challenging to determine. Therefore keeping the incoming electron dose to a minimum is important. However, this may reduce the reliability of quantitative ADF STEM which will here be demonstrated for nano-particle atom-counting. Based on experimental ADF STEM images of a real industrial catalyst, we discuss the limits for counting the number of atoms in a projected atomic column with single atom sensitivity. We diagnose these limits by combining a thorough statistical method and detailed image simulations. Copyright © 2014 Elsevier B.V. All rights reserved. -
##### Article: Optimal experimental design for nano-particle atom-counting from high-resolution STEM images

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**ABSTRACT:**In the present paper, the principles of detection theory are used to quantify the probability of error for atom-counting from high resolution scanning transmission electron microscopy (HR STEM) images. Binary and multiple hypothesis testing have been investigated in order to determine the limits to the precision with which the number of atoms in a projected atomic column can be estimated. The probability of error has been calculated when using STEM images, scattering cross-sections or peak intensities as a criterion to count atoms. Based on this analysis, we conclude that scattering cross-sections perform almost equally well as images and perform better than peak intensities. Furthermore, the optimal STEM detector design can be derived for atom-counting using the expression for the probability of error. We show that for very thin objects LAADF is optimal and that for thicker objects the optimal inner detector angle increases. - [Show abstract] [Hide abstract]

**ABSTRACT:**We report an innovative method to explore the optimal experimental settings to detect light atoms from scanning transmission electron microscopy (STEM) images. Since light elements play a key role in many technologically important materials, such as lithium-battery devices or hydrogen storage applications, much effort has been made to optimize the STEM technique in order to detect light elements. Therefore, classical performance criteria, such as contrast or signal-to-noise ratio, are often discussed hereby aiming at improvements of the direct visual interpretability. However, when images are interpreted quantitatively, one needs an alternative criterion, which we derive based on statistical detection theory. Using realistic simulations of technologically important materials, we demonstrate the benefits of the proposed method and compare the results with existing approaches. - [Show abstract] [Hide abstract]

**ABSTRACT:**High angle annular dark field scanning transmission electron microscopy (HAADF STEM) images provide sample information which is sensitive to the chemical composition. The image intensities indeed scale with the mean atomic number Z. To some extent, chemically different atomic column types can therefore be visually distinguished. However, in order to quantify the atomic column composition with high accuracy and precision, model-based methods are necessary. Therefore, an empirical incoherent parametric imaging model can be used of which the unknown parameters are determined using statistical parameter estimation theory (Van Aert et al., 2009, [1]). In this paper, it will be shown how this method can be combined with frozen lattice multislice simulations in order to evolve from a relative toward an absolute quantification of the composition of single atomic columns with mixed atom types. Furthermore, the validity of the model assumptions are explored and discussed. -
##### Conference Paper: Atomic number estimation from STEM images: what are the limits?

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**ABSTRACT:**The principles of detection theory [1] have been used to explore the limits with which two atom types of only slightly different atomic number can be distinguished from STEM images [2]. If one has to choose between two hypotheses corresponding to the presence of atom type 1 or 2, there is a probability of assigning the wrong hypothesis. Using repetitive image simulations with induced Poisson noise this probability of error, P e , can be computed and minimised as a function of the experimental design. Here the inner detector radius of an annular detector is considered as a tunable microscope parameter. The Kullback-Leibler divergence is an alternative performance measure to this probability of error which does not require repetitive simulations. The results of both measures will be compared in the case where one has to choose between the presence of an Al or Ti atom. - [Show abstract] [Hide abstract]

**ABSTRACT:**A thorough understanding of the three dimensional (3D) atomic structure and composition of core-shell nanostructures is indispensable to obtain a deeper insight on their physical behaviour. Such 3D information can be reconstructed from two dimensional (2D) projection images using electron tomography. Recently, different electron tomography techniques have enabled the 3D characterization of a variety of nanostructures down to the atomic level. However, these methods have all focused on the investigation of nanomaterials containing only one type of atom type. Here, we combine statistical parameter estimation theory with compressive sensing based tomography to determine the positions and chemical nature of each atom in hetero-nanostructures. The approach is applied here to investigate the interface in core-shell Au@Ag nanorods but it is of great interest in the investigation of a broad range of nanostructures. -
##### Article: Estimation of unknown structure parameters from high-resolution (S)TEM images: What are the limits?

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**ABSTRACT:**Statistical parameter estimation theory is proposed as a quantitative method to measure unknown structure parameters from electron microscopy images. Images are then purely considered as data planes from which structure parameters have to be determined as accurately and precisely as possible using a parametric statistical model of the observations. For this purpose, an efficient algorithm is proposed for the estimation of atomic column positions and intensities from high angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) images. Furthermore, the so-called Cramér-Rao lower bound (CRLB) is reviewed to determine the limits to the precision with which continuous parameters such as atomic column positions and intensities can be estimated. Since this lower bound can only be derived for continuous parameters, alternative measures using the principles of detection theory are introduced for problems concerning the estimation of discrete parameters such as atomic numbers. An experimental case study is presented to show the practical use of these measures for the optimization of the experiment design if the purpose is to decide between the presence of specific atom types using STEM images. - [Show abstract] [Hide abstract]

**ABSTRACT:**In the present paper, a statistical model-based method to count the number of atoms of monotype crystalline nanostructures from high resolution high-angle annular dark-field (HAADF) scanning transmission electron microscopy (STEM) images is discussed in detail together with a thorough study on the possibilities and inherent limitations. In order to count the number of atoms, it is assumed that the total scattered intensity scales with the number of atoms per atom column. These intensities are quantitatively determined using model-based statistical parameter estimation theory. The distribution describing the probability that intensity values are generated by atomic columns containing a specific number of atoms is inferred on the basis of the experimental scattered intensities. Finally, the number of atoms per atom column is quantified using this estimated probability distribution. The number of atom columns available in the observed STEM image, the number of components in the estimated probability distribution, the width of the components of the probability distribution, and the typical shape of a criterion to assess the number of components in the probability distribution directly affect the accuracy and precision with which the number of atoms in a particular atom column can be estimated. It is shown that single atom sensitivity is feasible taking the latter aspects into consideration. - [Show abstract] [Hide abstract]

**ABSTRACT:**We report a method to reliably count the number of atoms from high-angle annular dark field scanning transmission electron microscopy images. A model-based analysis of the experimental images is used to measure scattering cross sections at the atomic level. The high sensitivity of these measurements in combination with a thorough statistical analysis enables us to count atoms with single-atom sensitivity. The validity of the results is confirmed by means of detailed image simulations. We will show that the method can be applied to nanocrystals of arbitrary shape, size, and atom type without the need for a priori knowledge about the atomic structure. - [Show abstract] [Hide abstract]

**ABSTRACT:**Quantitative structural and chemical information can be obtained from high angle annular dark field scanning transmission electron microscopy (HAADF STEM) images when using statistical parameter estimation theory. In this approach, we assume an empirical parametrized imaging model for which the total scattered intensities of the atomic columns are estimated. These intensities can be related to the material structure or composition. Since the experimental probe profile is assumed to be known in the description of the imaging model, we will explore how the uncertainties in the probe profile affect the estimation of the total scattered intensities. Using multislice image simulations, we analyze this effect for Cs corrected and non-Cs corrected microscopes as a function of inaccuracies in cylindrically symmetric aberrations, such as defocus and spherical aberration of third and fifth order, and non-cylindrically symmetric aberrations, such as 2-fold and 3-fold astigmatism and coma. -
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**ABSTRACT:**A new tungsten bronze in the Sb-W-O system has been prepared in a solid state reaction from Sb2O3, WO3 and W metal powder. The average structure was determined by single crystal X-ray diffraction. SbxWO3+y (x similar to 0.11) crystallizes in the orthorhombic space group Pm2(1)n (no. 31), a = 27.8135(9) angstrom, b = 7.3659(2) angstrom and c = 3.8672(1) angstrom. The structure belongs to the (n)-ITB class of intergrowth tungsten bronzes. It contains slabs of hexagonal channels formed by six WO6 octahedra. These slabs are separated by three layers of WO6 octahedra that are arranged in a WO3-type fashion. The WO6 octahedra share all vertices to build up a three-dimensional framework. The hexagonal channels are filled with Sb atoms to similar to 80% and additional 0 atoms. The atoms are shifted out of the center of the channels. Exit-wave reconstruction of focal series of high resolution-transmission-electron-microscope (HRTEM) images combined with statistical parameter estimation techniques allowed to study local ordering in the channels. Sb atoms in neighbouring channels tend to be displaced in the same direction, which is in agreement with total energy calculations on ordered structure models, but the ratio of the occupation of the two possible Sb sites varies from channel to channel. The structure of SbxWO3+y exhibits pronounced local modulations. -
##### Article: High precision measurements of atom column positions using model-based exit wave reconstruction

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**ABSTRACT:**In this paper, it has been investigated how to measure atom column positions as accurately and precisely as possible using a focal series of images. In theory, it is expected that the precision would considerably improve using a maximum likelihood estimator based on the full series of focal images. As such, the theoretical lower bound on the variances of the unknown atom column positions can be attained. However, this approach is numerically demanding. Therefore, maximum likelihood estimation has been compared with the results obtained by fitting a model to a reconstructed exit wave rather than to the full series of focal images. Hence, a real space model-based exit wave reconstruction technique based on the channelling theory is introduced. Simulations show that the reconstructed complex exit wave contains the same amount of information concerning the atom column positions as the full series of focal images. Only for thin samples, which act as weak phase objects, this information can be retrieved from the phase of the reconstructed complex exit wave. - [Show abstract] [Hide abstract]

**ABSTRACT:**Quantification of the chemical composition from High Angle Annular Dark Field Scanning Transmission Electron Microscopy (HAADF STEM) images is in principle possible since the image intensities scale with the atomic number Z [1]. However, direct visual interpretation is inadequate when the difference in atomic number of distinct atomic column types is small or when there is intermixing of different atomic types in the same column. Therefore, quantitative methods to extract structural and chemical information are needed. Using statistical parameter estimation theory, the total scattered intensity of each atomic column can be measured [2]. Consequently, an empirical incoherent imaging model of the HAADF signal is proposed for quantitative measurement of atomic column types and positions. This model considers a parameterized object function peaked at the atomic column positions that is convoluted with the probe intensity profile. The unknown parameters of this model are estimated using a criterion of goodness of fit, which quantifies the similarity between an experimental image and this model. By minimizing the least squares sum, the volumes under the peak of the object function are estimated for each atomic column. These volumes represent the total scattered intensity of each atomic column and they are related to the chemical composition of that specific column. As such, the local structural and chemical composition can be obtained with good accuracy and precision. In order to investigate the capabilities of this approach, a known structure has been quantified and the results have been compared using standard simulation methods. It has been shown that, in order to compare experimental images with simulations, a calibration of the HAADF detector must be performed [3,4]. Under these experimental conditions, images of a Pb 1.2 Sr 0.8 Fe 2 O 5 compound were obtained in two major zone axes, [100] and [010], using a FEI Tecnai F20 microscope operating at 200kV high tension. From these images, quantification of the Sr -Pb mixed columns was carried out. Figure 1 represents a scheme of the crystal structure showing the atomic column positions with mixed composition (red circles). According to structure refinement [5], the composition for these columns is Sr 0.8 -Pb 0.2 . This composition was used as a starting condition for frozen phonon simulations using the STEMSim program [6]. Next, the scattered intensities have been estimated by fitting the empirical incoherent imaging model with respect to the experimental and simulated images. In order to calculate the percentage of Pb content in the mixed columns, the estimated intensities obtained from experimental data were compared with the ones obtained from simulations. By assuming a linear increase with increasing Pb content in the column, the percentage of Pb content in the unknown columns has been refined. Then, a new frozen phonon simulation with the calculated composition for each atomic column was performed. Figure 2 shows the obtained results for this analysis from which it follows that the agreement between experimental, fitted model and simulation is very good. This example thus shows that the composition of each atomic column can be determined using this quantitative model-based method [7]. References [1] S J Pennycook, D E Jesson, Ultramicroscopy 37 (1991) p.14 [2] S Van Aert, et al. Ultramicroscopy 109 (2009) p.1236 [3] J M Lebeau, et al. Physical Review Letters 100 (2008), 206101 [4] A Rosenauer, et al. Ultramicroscopy 109 (2009) p.1171 [5] M Batuk, et al. to be pusblished [6] A Rosenauer, M Schowalter, in: Springer Proceedings in Physics: Microscopy of Semiconducting materials (MSM) Conference, Cambridge, 2007 [7] This work was supported by the Flemish Fund for Scientific Research (FWO Vlaanderen). J. V. acknowledges funding from the European Research Council (ERC grant nr.278510-VORTEX). Figure 1. Scheme of the crystal structure of Pb 1.2 Sr 0.8 Fe 2 O 5 in [100] and [010] zone axis. Color code of atomic columns: Pb (blue), Mixed Sr-Pb (red), Fe (yellow), Oxygen (green).

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