The results of the seventh blind test of crystal structure prediction are presented, focusing on structure generation methods.

The results of the seventh blind test of crystal structure prediction are presented, focusing on structure ranking methods.

Incommensurate phase of potassium guaninate monohydrate is the first example of a modulation in purine derivatives and of a high-pressure incommensurate crystal structure to be solved for an organic compound.

A comparative description is presented of the symmetry and the magnetic structures found in the family of double perovskites A2MnB'O6 (mainly B' = Co and some Ni compounds for comparative purposes).

The structures of nine hexa­methyl-[1,1'-bi­phenyl]-4,4'-di­ammonium (HMB) salts are described

Tetragonal YxOs4B4 (x = 1.161) is an incommensurate composite of columns of Y atoms in a three-dimensional Os4B4 framework. The structure was refined using the superspace approach.

The hydro­thermal synthesis and structural characterization of four novel 3D pillared-layer metal–organic frameworks are studied, revealing how the malleability of copper coordination geometries drives diverse supramolecular isomerism. The findings provide new insights into designing advanced hybrid materials with tailored properties, emphasizing the significant role of reaction conditions and metal ion flexibility in determining network topologies.

Five multicomponent solid forms of an antifungal drug flucytosine are reported with a hygroscopic stability study. A detailed CSD search on the cocrystal/salts of flucytosine is evaluated and correlated the structures based on bond angles and bond distances.

YxOs4B4 (x = 1.161) crystallizes as a tetragonal incommensurate composite of columns of Y atoms extending along [001] in an Os4B4 framework. The structure was refined using the superspace approach. The basic structure of the Y subsystem can be idealized as having I4/mmm symmetry, with a crystallographically unique Y atom located on the 4/mmm position. The actual superspace symmetry is P42/nmc(00σ3)s0s0. The Y atoms feature only subtle positional modulation in the [001] direction. The Os4B4 subsystem [P42/ncm(00σ3)00ss superspace symmetry] is built of columns of edge-sharing Os4 tetrahedra extending along [001] and B2 dumbbells. The Os4 tetrahedra feature pronounced positional modulation with a distinct variation of the Os—Os bond lengths. Modulation of the B2 dumbbells is best described as a rotation about the [001] axis.

The crystal structure of the incommensurate modulated phase of potassium guaninate monohydrate has been solved on the basis of high-pressure single-crystal X-ray diffraction data. The modulated structure was described as a `mosaic' sequence of three different local configurations of two neighbouring guaninate rings. In contrast to known examples of incommensurate modulated organic compounds, the modulation functions of all atoms are discontinuous. This is the first example of the experimental detection of an incommensurate modulated crystal structure that can be modelled using the special `soliton mode' modulation function proposed by Aramburu et al. [(1995), J. Phys. Condens. Matter, 7, 6187–6196].

The seventh blind test of crystal structure prediction (CSP) methods substantially increased the level of complexity of the target compounds relative to the previous tests organized by the Cambridge Crystallographic Data Centre. In this work, the performance of density-functional methods is assessed using numerical atomic orbitals and the exchange-hole dipole moment dispersion correction (XDM) for the energy-ranking phase of the seventh blind test. Overall, excellent performance was seen for the two rigid molecules (XXVII, XXVIII) and for the organic salt (XXXIII). However, for the agrochemical (XXXI) and pharmaceutical (XXXII) targets, the experimental polymorphs were ranked fairly high in energy amongst the provided candidate structures and inclusion of thermal free-energy corrections from the lattice vibrations was found to be essential for compound XXXI. Based on these results, it is proposed that the importance of vibrational free-energy corrections increases with the number of rotatable bonds.

A seventh blind test of crystal structure prediction has been organized by the Cambridge Crystallographic Data Centre. The results are presented in two parts, with this second part focusing on methods for ranking crystal structures in order of stability. The exercise involved standardized sets of structures seeded from a range of structure generation methods. Participants from 22 groups applied several periodic DFT-D methods, machine learned potentials, force fields derived from empirical data or quantum chemical calculations, and various combinations of the above. In addition, one non-energy-based scoring function was used. Results showed that periodic DFT-D methods overall agreed with experimental data within expected error margins, while one machine learned model, applying system-specific AIMnet potentials, agreed with experiment in many cases demonstrating promise as an efficient alternative to DFT-based methods. For target XXXII, a consensus was reached across periodic DFT methods, with consistently high predicted energies of experimental forms relative to the global minimum (above 4 kJ mol−1 at both low and ambient temperatures) suggesting a more stable polymorph is likely not yet observed. The calculation of free energies at ambient temperatures offered improvement of predictions only in some cases (for targets XXVII and XXXI). Several avenues for future research have been suggested, highlighting the need for greater efficiency considering the vast amounts of resources utilized in many cases.

A seventh blind test of crystal structure prediction was organized by the Cambridge Crystallographic Data Centre featuring seven target systems of varying complexity: a silicon and iodine-containing molecule, a copper coordination complex, a near-rigid molecule, a cocrystal, a polymorphic small agrochemical, a highly flexible polymorphic drug candidate, and a polymorphic morpholine salt. In this first of two parts focusing on structure generation methods, many crystal structure prediction (CSP) methods performed well for the small but flexible agrochemical compound, successfully reproducing the experimentally observed crystal structures, while few groups were successful for the systems of higher complexity. A powder X-ray diffraction (PXRD) assisted exercise demonstrated the use of CSP in successfully determining a crystal structure from a low-quality PXRD pattern. The use of CSP in the prediction of likely cocrystal stoichiometry was also explored, demonstrating multiple possible approaches. Crystallographic disorder emerged as an important theme throughout the test as both a challenge for analysis and a major achievement where two groups blindly predicted the existence of disorder for the first time. Additionally, large-scale comparisons of the sets of predicted crystal structures also showed that some methods yield sets that largely contain the same crystal structures.

In electron diffraction, thermal atomic motion produces incoherent scattering over a relatively wide angular range, which appears as a diffuse background that is usually subtracted from measurements of Bragg spot intensities in structure solution methods. The transfer of electron flux from Bragg spots to diffuse scatter is modelled using complex scattering factors f + if' in the Bloch wave methodology. In a two-beam Einstein model the imaginary `absorptive' scattering factor f' can be obtained by the evaluation of an integral containing f over all possible scattering angles. While more sophisticated models of diffuse scatter are widely used in the electron microscopy community, it is argued in this paper that this simple model is appropriate for current structure solution and refinement methods. The two-beam model is a straightforward numerical calculation, but even this simplistic approach can become time consuming for simulations of materials with large numbers of atoms in the unit cell and/or many incident beam orientations. Here, a parameterized form of f' is provided for 103 elements as neutral, spherical atoms that reduces calculation time considerably.

The bulk solvent is a major component of biomacromolecular crystals that contributes significantly to the observed diffraction intensities. Accurate modelling of the bulk solvent has been recognized as important for many crystallographic calculations. Owing to its simplicity and modelling power, the flat (mask-based) bulk-solvent model is used by most modern crystallographic software packages to account for disordered solvent. In this model, the bulk-solvent contribution is defined by a binary mask and a scale (scattering) function. The mask is calculated on a regular grid using the atomic model coordinates and their chemical types. The grid step and two radii, solvent and shrinkage, are the three parameters that govern the mask calculation. They are highly correlated and their choice is a compromise between the computer time needed to calculate the mask and the accuracy of the mask. It is demonstrated here that this choice can be optimized using a unique value of 0.6 Å for the grid step irrespective of the data resolution, and the radii values adjusted correspondingly. The improved values were tested on a large sample of Protein Data Bank entries derived from X-ray diffraction data and are now used in the computational crystallography toolbox (CCTBX) and in Phenix as the default choice.

A crystal structure with N atoms in its unit cell can be solved starting from a model with atoms 1 to j − 1 being located. To locate the next atom j, the method uses a modified definition of the traditional R1 factor where its dependencies on the locations of atoms j + 1 to N are removed. This modified R1 is called the single-atom R1 (sR1), because the locations of atoms 1 to j − 1 in sR1 are the known parameters, and only the location of atom j is unknown. Finding the correct position of atom j translates thus into the optimization of the sR1 function, with respect to its fractional coordinates, xj, yj, zj. Using experimental data, it has been verified that an sR1 has a hole near each missing atom. Further, it has been verified that an algorithm based on sR1, hereby called the sR1 method, can solve crystal structures (with up to 156 non-hydrogen atoms in the unit cell). The strategy to carry out this calculation has also been optimized. The main feature of the sR1 method is that, starting from a single arbitrarily positioned atom, the structure is gradually revealed. With the user's help to delete poorly determined parts of the structure, the sR1 method can build the model to a high final quality. Thus, sR1 is a viable and useful tool for solving crystal structures.

In Part I of this series, all topologically possible 1-periodic infinite graphs (chain graphs) representing chains of tetrahedra with up to 6–8 vertices (tetrahedra) per repeat unit were generated. This paper examines possible restraints on embedding these chain graphs into Euclidean space such that they are compatible with the metrics of chains of tetrahedra in observed crystal structures. Chain-silicate minerals with T = Si4+ (plus P5+, V5+, As5+, Al3+, Fe3+, B3+, Be2+, Zn2+ and Mg2+) have a grand nearest-neighbour 〈T–T〉 distance of 3.06±0.15 Å and a minimum T⋯T separation of 3.71 Å between non-nearest-neighbour tetrahedra, and in order for embedded chain graphs (called unit-distance graphs) to be possible atomic arrangements in crystals, they must conform to these metrics, a process termed equalization. It is shown that equalization of all acyclic chain graphs is possible in 2D and 3D, and that equalization of most cyclic chain graphs is possible in 3D but not necessarily in 2D. All unique ways in which non-isomorphic vertices may be moved are designated modes of geometric modification. If a mode (m) is applied to an equalized unit-distance graph such that a new geometrically distinct unit-distance graph is produced without changing the lengths of any edges, the mode is designated as valid (mv); if a new geometrically distinct unit-distance graph cannot be produced, the mode is invalid (mi). The parameters mv and mi are used to define ranges of rigidity of the unit-distance graphs, and are related to the edge-to-vertex ratio, e/n, of the parent chain graph. The program GraphT–T was developed to embed any chain graph into Euclidean space subject to the metric restraints on T–T and T⋯T. Embedding a selection of chain graphs with differing e/n ratios shows that the principal reason why many topologically possible chains cannot occur in crystal structures is due to violation of the requirement that T⋯T > 3.71 Å. Such a restraint becomes increasingly restrictive as e/n increases and indicates why chains with stoichiometry TO<2.5 do not occur in crystal structures.

Analytical calculations of absorption corrections for X-ray powder diffraction experiments on non-ideal samples with surface roughness, porosity or absorption contrasts from multiple phases require complex mathematical models to represent their material distribution. In a computational approach to this problem, a practicable ray-tracing algorithm is formulated which is capable of simulating angle-dependent absorption corrections in reflection geometry for any given rasterized sample model. Single or multiphase systems with arbitrary surface roughness, porosity and spatial distribution of the phases in any combination can be modeled on a voxel grid by assigning respective values to each voxel. The absorption corrections are calculated by tracing the attenuation of X-rays along their individual paths via a modified shear-warp algorithm. The algorithm is presented in detail and the results of simulated absorption corrections on samples with various surface modulations are discussed in the context of published experimental results.

The atomic pair distribution function (PDF) is a real-space representation of the structure of a material. Experimental PDFs are obtained using a Fourier transform from total scattering data which may or may not have Bragg diffraction peaks. The determination of Bragg peak resolution in scattering data from the fundamental physical parameters of the diffractometer used is well established, but after the Fourier transform from reciprocal to direct space, these contributions are harder to identify. Starting from an existing definition of the resolution function of large-area detectors for X-ray diffraction, this approach is expanded into direct space. The effect of instrumental parameters on PDF peak resolution is developed mathematically, then studied with modelling and comparison with experimental PDFs of LaB6 from measurements made in different-sized capillaries.

Conjugation is the process by which plasmids, including those that carry antibiotic-resistance genes, are mobilized from one bacterium (the donor) to another (the recipient). The conjugation efficiency of IncF-like plasmids relies on the formation of mating-pair stabilization via intimate interactions between outer membrane proteins on the donor (a plasmid-encoded TraN isoform) and recipient bacteria. Conjugation of the R100-1 plasmid into Escherichia coli and Klebsiella pneumoniae (KP) recipients relies on pairing between the plasmid-encoded TraNα in the donor and OmpW in the recipient. Here, the crystal structure of K. pneumoniae OmpW (OmpWKP) is reported at 3.2 Å resolution. OmpWKP forms an eight-stranded β-barrel flanked by extracellular loops. The structures of E. coli OmpW (OmpWEC) and OmpWKP show high conservation despite sequence variability in the extracellular loops.

Human tRNA (uracil-5-)-methyltransferase 2 homolog A (TRMT2A) is the dedicated enzyme for the methylation of uridine 54 in transfer RNA (tRNA). Human TRMT2A has also been described as a modifier of polyglutamine (polyQ)-derived neuronal toxicity. The corresponding human polyQ pathologies include Huntington's disease and constitute a family of devastating neuro­degenerative diseases. A polyQ tract in the corresponding disease-linked protein causes neuronal death and symptoms such as impaired motor function, as well as cognitive impairment. In polyQ disease models, silencing of TRMT2A reduced polyQ-associated cell death and polyQ protein aggregation, suggesting this protein as a valid drug target against this class of disorders. In this paper, the 1.6 Å resolution crystal structure of the RNA-recognition motif (RRM) from Drosophila melanogaster, which is a homolog of human TRMT2A, is described and analysed.

Sorghum, a short-day tropical plant, has been adapted for temperate grain production, in particular through the selection of variants at the MATURITY loci (Ma1–Ma6) that reduce photoperiod sensitivity. Ma3 encodes phytochrome B (phyB), a red/far-red photochromic biliprotein photoreceptor. The multi-domain gene product, comprising 1178 amino acids, autocatalytically binds the phytochromobilin chromophore to form the photoactive holophytochrome (Sb.phyB). This study describes the development of an efficient heterologous overproduction system which allows the production of large quantities of various holoprotein constructs, along with purification and crystallization procedures. Crystals of the Pr (red-light-absorbing) forms of NPGP, PGP and PG (residues 1–655, 114–655 and 114–458, respectively), each C-terminally tagged with His6, were successfully produced. While NPGP crystals did not diffract, those of PGP and PG diffracted to 6 and 2.1 Å resolution, respectively. Moving the tag to the N-terminus and replacing phytochromobilin with phycocyanobilin as the ligand produced PG crystals that diffracted to 1.8 Å resolution. These results demonstrate that the diffraction quality of challenging protein crystals can be improved by removing flexible regions, shifting fusion tags and altering small-molecule ligands.

Preparation of biomacromolecules for structural biology studies is a complex and time-consuming process. The goal is to produce a highly concentrated, highly pure product that is often shipped to large facilities with tools to prepare the samples for crystallization trials or for measurements at synchrotrons and cryoEM centers. The aim of this article is to provide guidance and to discuss general considerations for shipping biomacromolecular samples. Details are also provided about shipping samples for specific experiment types, including solution- and cryogenic-based techniques. These guidelines are provided with the hope that the time and energy invested in sample preparation is not lost due to shipping logistics.

Crystallographic fragment screening has become a pivotal technique in structure-based drug design, particularly for bacterial targets with a crucial role in infectious disease mechanisms. The enzyme CdaA, which synthesizes an essential second messenger cyclic di-AMP (c-di-AMP) in many pathogenic bacteria, has emerged as a promising candidate for the development of novel antibiotics. To identify crystals suitable for fragment screening, CdaA enzymes from Streptococcus pneumoniae, Bacillus subtilis and Enterococcus faecium were purified and crystallized. Crystals of B. subtilis CdaA, which diffracted to the highest resolution of 1.1 Å, were used to perform the screening of 96 fragments, yielding data sets with resolutions spanning from 1.08 to 1.87 Å. A total of 24 structural hits across eight different sites were identified. Four fragments bind to regions that are highly conserved among pathogenic bacteria, specifically the active site (three fragments) and the dimerization interface (one fragment). The coordinates of the three active-site fragments were used to perform an in silico drug-repurposing screen using the OpenEye suite and the DrugBank database. This screen identified tenofovir, an approved drug, that is predicted to interact with the ATP-binding region of CdaA. Its inhibitory potential against pathogenic E. faecium CdaA has been confirmed by ITC measurements. These findings not only demonstrate the feasibility of this approach for identifying lead compounds for the design of novel antibacterial agents, but also pave the way for further fragment-based lead-optimization efforts targeting CdaA.

Animations, videos and 3D models have been designed to visualize the effects of symmetry operators on selected cases of crystal structures, pointing out the relationship with the diagrams published in International Tables for Crystallography, Vol. A.

This study validates the feasibility of applying a smearing method for the multi-slit very small angle neutron scattering instrument (MS-VSANS) at the China Spallation Neutron Source. Through analysis limited to a vertical range of 8 mm, the study demonstrates consistency between the predicted smearing function and experimental data, marking a significant milestone in utilizing real data from such instruments.

The growth quality of AlN single crystals was improved by optimizing the crucible structure for Al vapor transport with the help of finite element simulation.

New software capabilities in RMCProfile allow researchers to study the structure of materials by combining machine learning interatomic potentials and reverse Monte Carlo.

Rotations of small- and wide-angle X-ray scattering samples during acquisition are shown to give a drastic improvement in the reliability of the characterization of anisotropic precipitates in metallic alloys.

A symmetry guide for the secondary structural degrees of freedom and related physical properties generated by tilts of BX6 octahedra in perovskites is proposed.

The electric field responses of the average and local lattice strains and polar nanoregions of relaxor ferroelectric PMN-30PT single crystals were revealed by multi-scale and time-resolved X-ray diffraction under DC and AC electric fields.

This study examines the quality of charge density obtained by fitting the multipole model to wavefunctions in different basis sets. The complex analysis reveals that changing the basis set quality from double- to triple-zeta can notably improve the charge density related properties of a multipole model.

This paper describes real-time statistical analysis of liquid jet images for SFX experiments at the European XFEL. This analysis forms one part of the automated jet re-alignment system for SFX experiments at the SPB/SFX instrument of European XFEL.

A neutron far-field interferometer is under development at NIST with the aim of enabling a multi-scale measurement combining the best of small-angle neutron scattering (SANS) and neutron imaging and tomography. We use the close relationship between SANS, ultra-SANS, spin-echo SANS and dark-field imaging and measurements of monodisperse spheres as a validation metric, highlighting the strengths and weaknesses of each of these neutron techniques.

The symmetry groups of the weave patterns of the baskets, trays and mats of the Batak, an indigenous community in the Philippines, are discussed in this paper. The two-way twofold weaving technique is used by the Batak, and this study points to a total of 15 layer groups found in the Batak weaves out of the 80 layer groups known in crystallography.

The performance of a high-Z photon-counting detector for powder diffraction measurements at high (>50 keV) energies is characterized, and the appropriate corrections are described in order to obtain data of higher quality than have previously been obtained from 2D detectors in these energy ranges.

Combined 100 ms resolved grazing-incidence small/wide-angle X-ray scattering and optical interferometry reveal that the additive diiodooctane can significantly double the solvent evaporation rate, thereby effectively suppressing the rapid spinodal decomposition process in the early stage of spin-coasting, favouring slow phase segregation kinetics with nucleation and growth.

We explain analysis of RIXS, HERFD and XR-HERFD data to discover new physical processes in manganese and manganese-containing materials, by applying our new technique XR-HERFD, developed from high resolution RIXS and HERFD.

A chiral nickel(II) Schiff base complex derived from 2-hy­droxy­benzo­phenone and (1S,2S)-1,2-di­phenyl­ethyl­enedi­amine shows a λ conformation of the central di­amine chelate ring. The substituents on the C&z-dbnd;N carbon atoms significantly affect the circular dichroism spectra.

The crystal structure of a metabolite of the insecticide/acaricide etoxazole, designated R13 is presented along with a Hirshfeld surface analysis of inter­molecular inter­actions present in the crystal structure.

Crystallization of the title compound from CH2Cl2/n-pentane (1:5 v/v) at room temperature gave two polymorphs, which crystallize in monoclinic (P21/c; α form) and ortho­rhom­bic (Pna21; β form) space groups. The ReI complex mol­ecules in either polymorph adopt a six-coordinate octa­hedral geometry with three facially-oriented carbonyl ligands, one bromido ligand, and two nitro­gen atoms from one chelating ligand ppt-OMe. In the crystal, both polymorph α and β form di-periodic sheet-like architectures supported by multiple hydrogen bonds.

The mol­ecular and crystal structure of N-(4-meth­oxy­phen­yl)picolinamide were studied and Hirshfeld surfaces and fingerprint plots were generated to investigate various inter­molecular inter­actions.

A series of animations, videos and 3D models that were developed, filmed or built to teach the symmetry properties of crystals are described. At first, these resources were designed for graduate students taking a basic crystallography course, coming from different careers, at the National Autonomous University of Mexico. However, the COVID-19 pandemic had the effect of accelerating the generation of didactic material. Besides our experience with postgraduate students, we have noted that 3D models attract the attention of children, and therefore we believe that these models are particularly useful for teaching children about the assembled arrangements of crystal structures.

The continued advancement of complex materials often requires a deeper understanding of the structure–function relationship across many length scales, which quickly becomes an arduous task when multiple measurements are required to characterize hierarchical and inherently heterogeneous materials. Therefore, there are benefits in the simultaneous characterization of multiple length scales. At the National Institute of Standards and Technology, a new neutron far-field interferometer is under development that aims to enable a multi-scale measurement combining the best of small-angle neutron scattering (SANS) and neutron imaging and tomography. Spatially resolved structural information on the same length scales as SANS (0.001–1 µm) and ultra-small-angle neutron scattering (USANS, 0.1–10 µm) will be collected via dark-field imaging simultaneously with regular attenuation radiography (>10 µm). The dark field is analogous to the polarization loss measured in spin-echo SANS (SESANS) and is related to isotropic SANS through a Hankel transform. Therefore, we use this close relationship and analyze results from SANS, USANS, SESANS and dark-field imaging of monodisperse spheres as a validation metric for the interferometry measurements. The results also highlight the strengths and weaknesses of these neutron techniques for both steady-state and pulsed neutron sources. Finally, we present an example of the value added by the spatial resolution enabled by dark-field imaging in the study of more complex heterogeneous materials. This information would otherwise be lost in other small-angle scattering measurements averaged over the sample.

Nanometric precipitates in metallic alloys often have highly anisotropic shapes. Given the large grain size and non-random texture typical of these alloys, performing small- and wide-angle X-ray scattering (SAXS/WAXS) measurements on such samples for determining their characteristics (typically size and volume fraction) results in highly anisotropic and irreproducible data. Rotations of flat samples during SAXS/WAXS acquisitions are presented here as a solution to these anisotropy issues. Two aluminium alloys containing anisotropic precipitates are used as examples to validate the approach with a −45°/45° angular range. Clear improvements can be seen on the SAXS I(q) fitting and the consistency between the different SAXS/WAXS measurements. This methodology results in more reliable measurements of the precipitate's characteristics, and thus allows for time- and space-resolved measurements with higher accuracy.

Structure refinement with reverse Monte Carlo (RMC) is a powerful tool for interpreting experimental diffraction data. To ensure that the under-constrained RMC algorithm yields reasonable results, the hybrid RMC approach applies interatomic potentials to obtain solutions that are both physically sensible and in agreement with experiment. To expand the range of materials that can be studied with hybrid RMC, we have implemented a new interatomic potential constraint in RMCProfile that grants flexibility to apply potentials supported by the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) molecular dynamics code. This includes machine learning interatomic potentials, which provide a pathway to applying hybrid RMC to materials without currently available interatomic potentials. To this end, we present a methodology to use RMC to train machine learning interatomic potentials for hybrid RMC applications.

Besides traditional pinhole geometry, the multi-slit very small angle neutron scattering instrument (MS-VSANS) at the China Spallation Neutron Source also utilizes a multi-slit collimation system to focus neutrons. Using the special focusing structures, the minimum scattering vector magnitude (q) can reach 0.00028 Å−1. The special structures also lead to a significantly different smearing function. By comparing the results of theoretical calculations with experimental data, we have validated the feasibility of a smearing method based on a mature theory for slit smearing. We use the weight-averaged intensity of neutron wavelength as a representative to evaluate the effect from a broad wavelength distribution, concentrating on the effect from the geometry of the multi-slit structures and the detector. The consistency of the theoretical calculation of the smearing function with experimental VSANS scattering profiles for a series of polystyrene standards of different diameters proves the feasibility of the smearing method. This marks the inaugural use of real experimental data from an instrument employing a multi-slit collimation system.

In the conventional crucible structure for AlN crystal growth by physical vapor transport, owing to the long molecular transport path of Al vapor and the disruption of the gas flow by the presence of a deflector, the Al vapor easily forms polycrystals in the growth domain. The result is increased internal stress in the crystals and increased difficulty in growing large-sized crystals. On this basis, with the help of finite element simulations, a novel crucible structure is designed. This crucible not only optimizes the gas transport but also increases the radial gradient of the AlN crystal surface, making the enhanced growth rate in the central region more obvious. The thermal stresses between the deflector and the crystal are also reduced. High-quality AlN crystals with an FWHM of 79 arcsec were successfully grown with this structure, verifying the accuracy of finite element simulation of the growth of AlN crystals. Our work has important guiding significance for the growth of high-quality AlN crystals.

In the family of perovskite materials, the tilts of BX6 octahedra are the most common type of structural distortion. Conventionally, the formation of low-symmetry perovskite phases with tilted octahedra is analyzed by considering only primary order parameters. However, octahedral tilting also gives rise to secondary order parameters which contribute to additional atomic displacements, ordering and lattice distortions. Our study highlights the significant impact of secondary order parameters on the structural formation and emergent physical properties of perovskites. Through group-theoretical and crystallographic analyses, we have identified all secondary order parameters within Glazer-type tilt systems and clarified their physical manifestations. We explore the fundamental symmetry relationships among various structural degrees of freedom in perovskites, including tilt-induced ferroelasticity, correlations between displacements and ordering of atoms occupying different positions, and the potential for rigid unit rotations and unconventional octahedral tilts. Particular emphasis is placed on the emergence of secondary order parameters and their coupling with primary order parameters, as well as their symmetry-based hierarchy, illustrated through a modified Bärnighausen tree. We applied our theoretical insights to elucidate phase transitions in well known perovskites such as CaTiO3 and RMnO3 (where R = La and lanthanide ions), thereby demonstrating the significant influence of secondary order parameters on crystal structure formation. Our results serve as a symmetry-based guide for the design, identification and structural characterization of perovskites with tilted octahedra, and for understanding tilt-induced physical properties.