Conference Agenda

Although amorphous materials lack long-range translation order, they are strongly ordered at the length scale of single interatomic bonds, and retain some order into more distant coordination shells. One form of order is local, approximate rotational symmetry. For example, local five-fold rotational symmetry has been shown in simulations to correlate to slower dynamics in metallic liquids and greater strength in metallic glasses [1]. Local structural symmetry gives rise to symmetry in electron nanodiffraction speckle patterns acquired with sub-nanometer probes, but evaluating symmetries in patterns from amorphous materials is challenging. Only small clusters of atoms have symmetry, their symmetry is often distorted or imperfect, and the symmetric cluster is embedded in more atoms which are disordered. One method to assess symmetries in nanodiffraction is the angular power spectrum. We have used angular power spectrum data to demonstrate that Zr₆₅Cu_27.5Al_7.5 glasses exhibit increasing 4-fold and 5-fold structural symmetry with increasing stability in the glassy state and increasing hardness [2].

However, angular symmetry is subject to three artifacts that give rise to power that does not correspond to symmetries in the structure. First, aliasing transfers power from lower n to higher n. Second, electron nanodiffraction patterns without Friedel symmetry give rise to non-structural odd-order power. Third, the extra atoms surrounding the ordered cluster can create speckles which cause the power in rotational orders that are not present in the structure. We have proposed a different method to assess symmetries in amorphous nanodiffraction inspired by the Symmetry STEM method for crystals [3]. This method defines symmetry coefficients S_n which sample only the discrete angles associated with n-fold symmetry [4]. The discrete sampling avoids the aliasing and Friedel breakdown artifacts entirely, and it reduces the incidence of the structure overlap artifact. Electron scattering simulations in Figure 1(a) show that S_n is sensitive to nearest-neighbour icosahedral order in metallic glass atomic models. Experiments on Pd₄₃Ni₁₀Cu₂₇P₂₀ in Figure 1(b) result in symmetries consistent with dodecahedral structure found in previous studies.

Figure 1(c) demonstrates the use of symmetry coefficients for spatial mapping of high-symmetry clusters. It is derived from a 4D STEM data set acquired with a high sensitivity, high-speed direct electron detection camera recently developed by the Wisconsin Materials Research Science and Engineering Center and Direct Electron, Inc. These data were acquired at 7,000 frames per second (fps). The current maximum speed of the camera is 24,000 fps, and the ultimate design speed is in excess of 100,000 fps. These extremely high speed will enable time-resolved, in situ experiments using symmetry coefficients to track the evolution of local symmetries in supercooled liquids as they cool through the glass transition.

[1] Cheng, Y. Q. Q., Ma, E. (2011) Prog. Mater. Sci. 56, 379.

[2] Muley, S. V., Cao, C., Chatterjee, D., Francis, C., Lu, F. P., Ediger, M. D., Perepezko, J. H., Voyles, P. M. (2021). Phys. Rev. Mater. 5, 033602.

[3] Krajnak, M. & Etheridge, J. A (2020) Proc. Natl. Acad. Sci. 117, 27805.

[4] Shuoyuan, H., Francis, C., Ketkaew J., Schroers J., Voyles P. M. (in preparation)

The pair-distribution function (PDF) method is widely used to obtain structural information beyond the typical diffraction techniques together with standard structure refinement utilizing Bragg reflections [1]. PDF analysis using x-ray and neutron powder diffraction data is well established. Electron-based PDF (ePDF) analysis has drawn considerable attention in recent years [2,3,4]. In addition to the very strong electron-matter interaction (~10⁵ than x-ray), the main advantages of the electron-based over x-ray and neutron-based is to utilize modern electron microscopes, which offer (sub)nano-sized probe of the electron beam and various imaging and spectroscopy techniques simultaneously. Therefore, ePDF is particularly good for study nano-materials, disordered materials and specific region of interest in the specimens.

Although the procedures of ePDF analysis is similar to those obtained by x-ray and neutron, several parameters and steps, which are due to electron scattering and TEM practice, are crucial in the processing. For instance, Q_max is always important in all PDF experiments. However, Q_min, which is not considered in x-ray and neutron, should be carefully determined in ePDF. On the other hand, the shape factor can influence the ePDF results. The smaller the particles the stronger effect can be seen. Different stepwise atomic layer arrangement of the crystal surface can contribute significantly in the analysis [5]. In addition to experimental and analysis procedures different to those for x-ray or neutron-based, the uniqueness and possibility of ePDF analyses of some amorphous materials and nanostructures will be discussed.

[1] Egami, T. & Billinge, S. J. L. (2002). Underneath the Bragg Peaks: Structural Analysis of Complex Materials. Amsterdam: Elsevier Science.

[2] Abeykoon, M., Malliakas, C. D., Juhás, D., Božin, E. S., Kanatzidis, M. G. & Billinge, S. J. L. (2012). Z. Kristallogr. 227 248.

[3] Tran, D.-T., Svensson, G. & Tai, C.-W. (2017). J. Appl. Crystallogr. 50, 304.

[4] Gorelik, T. E., Neder, R., Terban, M. W., Lee, Z., Mu, X., Jung, C., Jacob T. & Kaiser, U. (2019). Acta Crystallogr. B 75, 532.

[5] Tran, D.-T., Svensson, G. & Tai, C.-W. (2016). arXiv:1602.08078.

The Knut and Alice Wallenberg (KAW) Foundation is acknowledged for providing the electron microscopy facilities and financial support under the project 3DEM-NATUR for the initial development. Swedish Research Council (project no. 2018-05260) is also acknowledged.

Correlated disorder is any type of deviation from the average crystal structure that is correlated over the range of a few unit cells only. As correlated disorder lies at the origin of the physical properties of a compound, many open questions in materials science are related to it. Unfortunately, the diffuse scattering analysis from single-crystal X-ray and neutron diffraction needs large crystals which are often not available. In the case of submicron sized crystals, pair distribution function analysis on powder samples could be applied. However, as an alternative we suggest to turn to single-crystal electron diffraction. While the quantitative analysis of diffuse X-ray and neutron scattering has already been done for different types of correlated disorder, we will present for the first time the quantitative analysis of diffuse electron scattering using an evolutionary algorithm in DISCUS [1].

In the electron diffraction patterns of Li_1.2 Ni_0.13Mn_0.54Co_0.13O₂ diffuse streaks are present, which are caused by stacking faults (i.e. variations in the stacking of subsequent Li-, O- and transition metal -layers). An evolutionary refinement algorithm in DISCUS was used to determine the stacking fault probability as well as the twin ratio in Li_1.2Ni_0.13Mn_0.54Co_0.13O₂ by a refinement of the intensity profile of the diffuse streaks_. The refinement algorithm was first tested on simulated data, after which it was applied to experimental electron diffraction data obtained by three-dimensional electron diffraction (3D ED).

Funding information

The research leading to these results has received funding from the Research Foundation Flanders (FWO Vlaanderen) (grant No. G035619N)

[1] Proffen, T., & Neder, R. B. (1997). J. Appl. Crystallogr. 30, 171-175.

Metal-organic frameworks (MOFs) are known for their versatile combination of inorganic building units and organic linkers, which offers immense opportunities in a wide range of applications. However, many MOFs are typically synthesized as multiphasic polycrystalline powders, which are challenging for studies by X-ray diffraction. Therefore, developing new structural characterization techniques is highly desired in order to accelerate discoveries of new materials. Here, we report a high-throughput approach for structural analysis of MOF nano- and sub-microcrystals by three-dimensional electron diffraction (3DED). A new zeolitic-imidazolate framework (ZIF), denoted ZIF-EC1, was first discovered in a trace amount during the study of a known ZIF-CO₃-1 material by 3DED. The structures of both ZIFs were solved and refined using 3DED data. ZIF-EC1 has a dense 3D framework structure, which is built by linking mono- and bi-nuclear Zn clusters and 2-methylimidazolates (mIm^-). The discovery of this new MOF highlights the power of 3DED in developing new materials.

In recent years, Electron Diffraction, and especially the 4D-STEM [2] is growingly becoming a routine part of structural characterizations of materials at the nano-scale. Its un-matched spatial resolution (down to sub-nm) enables the exploration of local variations within a sample, which alternatively is averaged over the entire irradiated sampled area, when explored, for example, by x-rays. As often shown in electron microscope, samples are often heterogeneous, and consequently their local properties, which then reflect on the average behavior of the material, composite, or device. Besides morphology and composition, the local structural order can vary, especially in evolving systems. In this study, we explore how far we can take electron diffraction when the interest is in the evolution of materials.

We challenge ourselves with mapping the local structure in a composite of crystalline Ni and amorphous Zr-Cu-Ni-Al Bulk Metallic Glass (BMG) that was fused into a composite via hot-rolling [2]. Using a fast camera looking through the fluorescence screen, we captured diffraction patterns in a 4D-STEM modality, where we captured diffraction patterns coupled with beam precession with 3 nm step size in a Ni/BMG/Ni cross-section sample that was cut from the composite - in total 131x289 diffraction patterns. Using the collected diffraction patterns and tailoring automated data reduction and analysis pipelines, such as auto-masking, azimuthal-integration, Fourier-transformation to get the electron Pair Distribution Function (ePDF) and various fittings of the PDF, we were efficiently deriving a large set of physically meaningful scalars, which we generalize as Quantities of Interest, or QoI of a scanning nano-structure electron microscopy (SNEM).

Using different QoI's (see Figure)- those derived directly from the images, such as virtual-dark-field and mostly from ePDF's, we could map out most clearly Ni/BMG boundaries, visualize inter-diffusion between Ni and BMG, extract regions of formed nano-crystals within the BMG, follow compositional changes via the average bond-distance (verified with EELS from the same area) and extract the distribution of atomic pairs. We were also able to map the deviation of each pixel within the BMG from an expected structural model, which exposed the BMG/Ni inter-diffusion front. Finally, we could estimate the effective local structure of the nano-crystalline inclusions within the BMG as FCC structures.

Using an assembly of these results we could learn that nano-crystalline inclusions within the BMG are located in regions where Cu is deficient and Zr is in excess. Most importantly, we learn the origin of success in forming the Ni/BMG composite via hot-rolling, which is nonetheless, a challenging goal. Hot rolling is found to be a challenging process due to the excessive formation of nano-crystallites at the Ni/BMG interface. Here, however, the assembly of SNEM results suggested that the metallic Ni amorphized instead of the BMG--Ni crystallize at the BMG/Ni interface. These results emphasize the richness SNEM experiments hide, and with the assistance of automated (and in the future - autonomated) pipelines can expose the story of evolving systems.

[1] Xiaoke Mu, Andrey Mazilkin, Christian Sprau, Alexander Colsmann, Christian Kübel, (2019). Microscopy. 3554, 301
[2] Sina Shahrezaei, Douglas C. Hofmann, Suveen N. Mathaudhu (2019). JOM. 71, 2

An information theory based method [1,2] for the classification of more or less 2D periodic images from real-world crystals with atomic resolution [3] into non-disjoint plane symmetry groups is briefly described and applied to three pairs of synthetic images. One image of these three pairs is free of noise by design. Gaussian noise of mean zero has been added to the other three images of these pairs. All three image pairs are also highly pseudo-symmetric so that it is, for human beings on the basis of a visual inspection alone, very difficult to identify the underlying plane symmetry of the noisy image correctly. This is because all genuine symmetries and pseudo-symmetries are broken by the added noise so that their differences are diminished. The new information theory based classification method, on the other hand, overcomes such challenges [4].

The method enables the objective, i.e. researcher independent, identification of the plane symmetry group that provides the best known separation of structure and generalized noise at the given noise level of a processed image. This identification enables the most meaningful averaging of the image in the spatial frequency domain in support of subsequent crystallographic analyses. This kind of averaging is over all correctly identified asymmetric units in the image and removes noise more effectively than traditional Fourier filtering. Ratios of numerically obtained geometric Akaike Information Criterion values, i.e. first-order geometric-bias corrected sums of squared residuals between the complex-valued Fourier coefficients of the raw image intensity and their counterparts from the applicable symmetry models of this data, are utilized for plane symmetry classifications within the appropriate symmetry hierarchy branch or intersecting branches. The symmetrized model of the noisy raw data that is the best representation of the 2D periodic structure in the Kullback-Leibler divergence sense will be found within such a single branch or at the crossing with other such branches in cases of more elaborate symmetries.

Numerically obtained confidence levels are assigned to the classification of noisy images into minimal supergroups over their translationengleiche maximal subgroups. The resulting plane symmetry classifications are always generalized noise level dependent, which allows for better classification results as noise decreases with future improvements to the imaging and image-processing procedures. The information theory based method delivers only probabilistic classifications as it is fundamentally unsound to assign an abstract mathematical concept, such as a plane symmetry group, with 100 % certainty to the record of a noisy real-world imaging experiment of an imperfect real-world crystal.

[1] P. Moeck, Symmetry 10, paper 133 (46 pages) (2018), open access, DOI: 10.3390/sym10050133

[2] P. Moeck, IEEE Transactions on Nanotechnology 18, 1166-1173 (2019), DOI: 10.1109/TNANO.2019.2946597, see also http://arxiv.org/abs/1902.04155, August 31, 2019 for an expanded version of this review

[3] P. Moeck, A. Dempsey, and C. Shu, Microscopy and Microanalysis 25 (Suppl. 2), 184–185 (2019), DOI: 10.1017/S143192761900165X

[4] P. Moeck and A. Dempsey, Microscopy and Microanalysis 25 (Suppl. 2), 1936–1937 (2019), DOI: 10.1017/S1431927619010419