Since the discovery of quasicrystals in 1982, there has been a lot of progress in finding sufficient and necessary conditions for their mathematical counterparts such as point sets and measures to have pure point diffraction. A comparatively less tackled issue is the presence of absolutely continuous components. An inherent stochasticity or randomness of the underlying structure guarantees the presence of absolutely continuous spectrum, but is not the only mechanism that does so. There are deterministic examples arising from aperiodic tilings of the d-dimensional Euclidean space which also share this feature [3,4].
In this talk, we will present the mathematical requirements for such objects to give rise to absolutely continuous diffraction components [1,5], and we will give some deterministic examples which satisfy them. We will also briefly mention several sufficient criteria to conclude the singularity of the spectrum (i.e., absence of absolutely continuous components) [1,2].

This is based on joint works with Michael Baake, Natalie Priebe Frank, Franz Gaehler and Uwe Grimm.

1. M. Baake, F. Gaehler, N. Mañibo, Renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction, Commun. Math. Phys. 370 (2019) 591--635.

2. M. Baake, U. Grimm, N. Mañibo, Spectral analysis of a family of binary inflation rules, Lett. Math. Phys. 108 (2018) 1783-1805.

3. L. Chan, U. Grimm, I. Short, Substitution-based structures with absolutely continuous spectrum, Indag. Math. 29 (2018) 1072-1086.

4. N. P. Frank, Substitution sequences in Z^d with a non-simple Lebesgue component in the spectrum, Ergodic Th. & Dynam. Syst. 23 (2003) 519-532.

5. N. Strungaru, On the Fourier analysis of measures with Meyer set support, J. Func. Anal. 278 (2020) 108404.

Correlated disorder in crystalline materials gives rise to single crystal diffuse scattering. While the average structure determination via Bragg data analysis is considered a standard procedure, disorder analysis is thought of as a lengthy and complicated process. We present a mean field approximation to model single crystal diffuse scattering in molecular materials from a simple pair-interaction Hamiltonian.

Mean filed theory is a self-consistent field theory, which is widely used in statistical physics to model high-dimensional random systems. It has proven a valuable tool in the analysis of magnetic diffuse scattering data [1]. Here, the formalism is applied to describe orientationally disordered molecular crystals,

We present a computational study based on the mean field model suggested by Naya [2] and proof its applicability to strongly correlated disorder, where the local building block geometry dictates allowed and prohibited local configurations. The system that will be analysed in detail is a two-dimensional analogue of Hg(NH₃)₂Cl₂ as depicted in Figure 1 (a) [3]. The Hg atoms are disordered over the cubic face centres to form [H₃N - Hg - NH₃]²⁺ molecules. The local arrangement is strictly dictated by these building rules.

We compare the results of the diffuse scattering analysis using the mean field model as introduced by Naya [2] to the results of RMC modelling and ΔPDF models based on a Warren-Cowley short range order parameter refinement (see Figure 1 (b)). Finally, the stability of the mean field analysis on limited data availability is demonstrated: Diffraction experiments under pressure or electric field yield a limited reciprocal space coverage. Here, we demonstrate the robustness of the proposed method against incomplete data sets.

Figure 1. (a) Disordered Hg(NH₃)₂Cl₂ [3], where the Hg is disordered over the cubic face centres. (b) Simulated data for a two dimensional analogue compared to refinements using mean field theory (MF), DPDF analysis for the Warren-Cowley short-range order parameters (WC) and reverse Monte Carlo modelling (RMC).

[1] Paddison, J.A.M., Stewart, J.R. et al. (2013). Phys. Rev. Letters. 110, 267207.

[2] Naya S. (1974) J. Phys. Soc. Jap. 37, 340-347.

[3] Lipscomb, W. N. (1953) Analytical Chemistry 25, 737-739.

Kossel lines and X-ray localized conical modes

V.A.Belyakov

Landau institute for Theoretical Physics, Kosygin str.2 , 119334 Moscow,

Russiabel@landau.ac.ru

An alternative way to describe the X-ray Kossel lines [1] based at the localized conical X-Ray modes existing in perfect crystals is proposed. A theory of the X-ray Kossel lines is presented in the framework of two-wave dynamical diffraction approximation for the conical modes [2]. The theoretical results compared with the known experimental results show a good general agreement with the main experimental observation as for the X-ray [3], so for the optical [4] Kossel line patterns. The influence of crucial parameters of the crystal (absorption, perfection, sample size, the Borrmann Effect etc.) on the shape of Kossel lines are discussed. For confirming a direct connection of Kossel lines with the localized conical X-Ray modes is proposed to apply a time-delayed techniques in studying the Kossel lines.

[1] Kossel, W., Loeck, V. & Voges, H. (1935). Z. Fur Phys. 94, 139.

[2] Belyakov, V. A. (2019). Diffraction Optics of Complex Structured Periodic Media, 2nd. Ed. Springer, Chapts. 5-8.

[3] Belyakov, V. A. (2021). JETP, 132, 323.

[4] Belyakov, V. A. (2020). Crystals, 132, 323.

Keywords: localized X-ray modes; Kossel line patterns; optical Kossel lines

3DED (three-dimensional electron diffraction) is currently already routinely used for the characterization of the average structure from Bragg reflections, and recently its use for quantifying correlated disorder from electron diffuse scattering is also taken off [1,2,3].

In this work, we used a combination of 3DED, HAADF-STEM and STEM-EDX to quantify the correlated disorder in Ge₄Bi₂Te₇. Ge₄Bi₂Te₇ is reported to contain vacancy-layers along <11-1>_Fm3m and <1-1-1>_Fm3m with a higher Bi-concentration neighbouring these layers, which leads to the occurrence of streaks of diffuse scattering [4, 5].

Using the combination of advanced TEM techniques, we have not only confirmed the defects previously found by single crystal X-ray Diffraction but also observed and characterized a plethora of other forms of correlated disorder not reported before in literature for this material, including domains with locally different structure and composition, interstitial atoms and local periodicity between Ge and Bi. This diversity in correlated disorder results in 3D diffuse scattering and superstructure reflections that previously passed unnoticed to other techniques.

This work illustrates the large potential of TEM in characterizing correlated disorder from the analysis of diffuse scattering.

1. Zhao, Haishuang, et al. "Elucidating structural order and disorder phenomena in mullite-type Al₄B₂O₉ by automated electron diffraction tomography." Journal of Solid State Chemistry 249 (2017): 114-123.

2. Brázda, Petr, et al. "Mapping of reciprocal space of La_{0. 30}CoO₂ in 3D: Analysis of superstructure diffractions and intergrowths with Co₃O₄." Journal of Solid State Chemistry 227 (2015): 30-34.

3. Brázda, Petr, et al. "Calcium-induced cation ordering and large resistivity decrease in Pr_{0. 3}CoO₂." Journal of Physics and Chemistry of Solids 96 (2016): 10-16.

4. Urban, Philipp, et al. "Real structure of Ge₄Bi₂Te₇: refinement on diffuse scattering data with the 3D-ΔPDF method." Journal of Applied Crystallography 48.1 (2015): 200-211. 5. Callaert, Carolien. "Characterization of defects, modulations and surface layers in topological insulators and structurally related compounds." PhD thesis, University of Antwerp, 2020.

We acknowledge the financial support of the Research Foundation-Flanders (FWO), project G035619N, “Quantification of 3D correlated disorder in materials from electron diffraction diffuse scattering with application to lithium battery materials”.

Disorder is commonly used in chemistry for designing functional materials. For instance, preparation of solid solutions is nothing else than the introduction of a controlled number of point defects in a crystal. Disordered systems, though, provide more degrees of freedom: not only the number of defects, but also their distribution can be used to optimise the functional properties of materials, however up until now, defect distribution was hard to control and thus was rarely used in practice.

In this talk we will show how to precisely tune distribution of point defects by changing various chemical parameters during crystal growth and characterise it with the single crystal diffuse scattering.

We will use Prussian Blue Analogues (PBAs) as our model system. PBAs is a class of cyanide materials with the general formula M[M’(CN)₆]_1-δ * xH₂O where M and M’ are transition metals. Depending on the nature of transition metals, PBAs can accommodate a large number of vacancies on the M’(CN)₆ site (for instance δ=0.33 for M=Mn and M’=Co) which makes them highly porous and, as a result, attractive for hydrogen storage applications. Distribution of M’(CN)₆ vacancies is important for the performance of this material, since more disordered vacancy configurations provide more diffusion pathways through the structure, larger accessible volume, and easier transport.

[1] Simonov, Arkadiy, et al. "Hidden diversity of vacancy networks in Prussian blue analogues." Nature 578.7794 (2020): 256-260.

Reverse Monte Carlo (RMC) model is a powerful tool based on supercell approach, targeting at the structure model that explains comprehensive experimental datasets. Typically, the RMCProfile package can incorporate neutron/X-ray total scattering, Bragg and extended X-ray absorption fine structure (EXAFS) data. For practical implementation, apart from theoretical pattern calculation and structure model adjustment based on metropolis algorithm, there are various effects under certain circumstances that one needs to take into account to avoid artificial effects. Here we are going to introduce several different types of correction that we recently developed and implemented, in the framework of RMCProfile, namely, 1) the correction for nano-size effect concerning total scattering modelling for nano-systems from 0D nanoparticles to 2D nanosheets [1]. 2) the implementation of arbitrary Bragg peak profile in a tabulated manner, through interacting with Topas software [2]. 3) the correction for finite instrument resolution effect going beyond the conventionally used analytical approach based on Gaussian assumption for peak shape [2]. Through such development and implementation, we hope to extend the scope of application of RMCProfile package for solving structural problems from local perspective. Typically, the implementation of resolution correction enables the modelling to an otherwise-unreachable super-large length scale, e.g., 100 Å, following the supercell approach.