Crystallographic Theory Research Articles

Nontrivial nanostructure, stress relaxation mechanisms, and crystallography for pressure-induced Si-I\u2009\u2192\u2009Si-II phase transformation

Crystallographic theory based on energy minimization suggests austenite-twinned martensite interfaces with specific orientation, which are confirmed experimentally for various materials. Pressure-induced phase transformation (PT) from semiconducting Si-I to metallic Si-II, due to very large and anisotropic transformation strain, may challenge this theory. Here, unexpected nanostructure evolution during Si-I → Si-II PT is revealed by combining molecular dynamics (MD), crystallographic theory, generalized for strained crystals, and in situ real-time Laue X-ray diffraction (XRD). Twinned Si-II, consisting of two martensitic variants, and unexpected nanobands, consisting of alternating strongly deformed and rotated residual Si-I and third variant of Si-II, form {111} interface with Si-I and produce almost self-accommodated nanostructure despite the large transformation volumetric strain of -0.237. The interfacial bands arrest the {111} interfaces, leading to repeating nucleation-growth-arrest process and to growth by propagating {110} interface, which (as well as {111} interface) do not appear in traditional crystallographic theory.

The Fuglede conjecture for convex domains is true in all dimensions

A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $\Omega$ is a spectral set if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it has long been known that for a convex body $\Omega \subset \mathbb{R}^d$ the "tiling implies spectral" part of the conjecture is in fact true. To the contrary, the "spectral implies tiling" direction of the conjecture for convex bodies was proved only in $\mathbb{R}^2$, and also in $\mathbb{R}^3$ under the a priori assumption that $\Omega$ is a convex polytope. In higher dimensions, this direction of the conjecture remained completely open (even in the case when $\Omega$ is a polytope) and could not be treated using the previously developed techniques. In this paper we fully settle Fuglede's conjecture for convex bodies affirmatively in all dimensions, i.e. we prove that if a convex body $\Omega \subset \mathbb{R}^d$ is a spectral set then $\Omega$ is a convex polytope which can tile the space by translations. To prove this we introduce a new technique, involving a construction from crystallographic diffraction theory, which allows us to establish a geometric "weak tiling" condition necessary for a set $\Omega \subset \mathbb{R}^d$ to be spectral.

Editorial

Editorial

Comment on “Crystallographic, spectroscopic, thermal, optical investigations and density functional theory calculations for novel Co(II) and Mn(II) complexes [Applied Physics A (2021) 127:132

Comment on “Crystallographic, spectroscopic, thermal, optical investigations and density functional theory calculations for novel Co(II) and Mn(II) complexes [Applied Physics A (2021) 127:132

Michael Mark Woolfson. 9 January 1927—23 December 2019

Michael Woolfson contributed enormously to the theory and practice of X-ray crystallography for almost 60 years. He extended the theory of ‘direct methods’, which provided a general solution of the crystallographic phase problem from the measured diffraction data alone, and provided tools to exploit this theory. Software developed in his laboratory, such as the computer package MULTAN, was responsible for about half of the structures determined around the world in the 1970s and 1980s. His parallel work on the origin of the Solar System renewed interest in the Capture theory, and he contributed numerous stimulating ideas to the general study of the origin and dynamical evolution of the Solar System. Michael was a great educator and mentor for both undergraduates and young researchers, and many of these went on to make significant scientific contributions.

On intergranular mechanical interactions and the theory of deformation crystallography of metals

The equilibrium of intergranular stress and strain can be realized simultaneously, whereas five independent slip systems of the Taylor principle and the criterion of minimal internal work are unnecessary. In fact, the Taylor principle applied in current theories is incorrect both in practice and theory, in which the activation mechanism of plastic deformation systems must violate the Schmid’s law and deviate from the elastic-plastic characteristics of deformed matrix. The intergranular reaction stress (RS) during deformation can be calculated according to Hooke’s law and elastic limit without additional subjective presupposition, therefore the RS theory is established intuitively. Under the combination of the RS (the intergranular elastic effect) and the external loading the slips penetrating grains are activated and produce deformation texture, but certain non-penetrating slips near grain boundaries will become active (the intergranular plastic effect) and produce some random texture when a RS reaches the yield strength of grains. The RS theory is simple, intuitive and reasonable, based on which the texture simulation can well reproduce the texture formation of various metals under different external loadings and under different crystallographic mechanisms.

Stereoselective synthesis, structure and DFT studies on fluoro- and nitro- substituted spirooxindole-pyrrolidine heterocyclic hybrids

Structurally diverse highly functionalized flouro- and nitro - substituted spirooxindole-pyrrolidine heterocyclic hybrids have been achieved in good yields. The molecular structure of these diverse spiroheterocyclic hybrids was elucidated by spectroscopic analysis and further confirmed by single crystal X-ray crystallographic studies and density functional theory (DFT) calculations. Analysis of molecular packing for these hybrids was performed using Hirshfeld analysis. Geometric parameters calculated by DFT are in good agreement with the experimental data obtained from X-ray crystallography.

Crystallographic, spectroscopic, thermal, optical investigations and density functional theory calculations for novel Co(II) and Mn(II) complexes

Two new picolinate complexes, namely diaquabis(6-chloropyridine-2-carboxylato-N,O)cobalt(II) (1) and diaquabis(6-chloropyridine-2-carboxylato-N,O)manganese(II) (2) were synthesized. These complexes were all characterized using FT-IR and UV–Vis spectra, single crystal X-ray diffraction method and thermal analysis. Both complexes have similar molecular structure characterized as distorted octahedral geometry. B3LYP level was also executed to provide a deeper insight to the structural, electronic and nonlinear optical properties of 1 and 2. The detailed vibrational analysis was performed for 1 and 2 on the basis of potential energy distribution (PED) analysis. The natural bond orbital (NBO) energies between the lone pair electrons of donor N/O atoms and anti-lone pair electrons of Co(II) and Mn(II) ions also proved the distorted octahedral geometry for both complexes. The band gap energy values for the allowed direct transition are found 4.29 and 4.26 eV by using Tauc model for complex 1 and 2, respectively. Finally, the optical conductivity (σopt), electrical conductivity (σelec), the third-order nonlinear optical susceptibility χ(3), the molar polarizability αp and dielectric property are determined for the title complexes in the UV–Vis region. The first-order hyperpolarizability (β) parameter for 2 was obtained as higher than that of 1 due to the high spin electronic configuration of Mn(II) central ions

A Homological Invariant of Certain Torsion Free Crystallographic Groups

Several homological invariants namely the nonabelian tensor square, the exterior square and the Schur multiplier of groups have been of research interests by group theorists over the years. Besides, there are also some other homological invariants which can be deduced from these invariants, as example, the central subgroup of the nonabelian tensor square of a group G, known as ()G?. The computations of the homological invariants of crystallographic groups strengthen the link between group theory with crystallography theory. In this paper, ()G?is determined for certain torsion free crystallographic groups focusing on those with cyclic point groups of order three and five.

P,N-Chelated Gold(III) Complexes: Structure and Reactivity.

Gold(III) complexes are versatile catalysts offering a growing number of new synthetic transformations. Our current understanding of the mechanism of homogeneous gold(III) catalysis is, however, limited, with that of phosphorus-containing complexes being hitherto underexplored. The ease of phosphorus oxidation by gold(III) has so far hindered the use of phosphorus ligands in the context of gold(III) catalysis. We present a method for the generation of P,N-chelated gold(III) complexes that circumvents ligand oxidation and offers full counterion control, avoiding the unwanted formation of AuCl4–. On the basis of NMR spectroscopic, X-ray crystallographic, and density functional theory analyses, we assess the mechanism of formation of the active catalyst and of gold(III)-mediated styrene cyclopropanation with propargyl ester and intramolecular alkoxycyclization of 1,6-enyne. P,N-chelated gold(III) complexes are demonstrated to be straightforward to generate and be catalytically active in synthetically useful transformations of complex molecules.

Characterization and retrieval method of equivalent electromagnetic parameters for carbon-fiber-reinforced polymer laminas

In order to describe the electrical properties of the anisotropic carbon-fiber-reinforced polymer (CFRP) laminas accurately, a characterization and retrieval method of equivalent electromagnetic (EM) parameters is proposed in this paper. According to the classical theory of Crystallography, the equivalent EM parameters of each layer of CFRP laminas are characterized as the anisotropic tensor. On this basis, a single-layer structure model of CFRP laminas based on micro-scale is established, by using the periodic structure finite-difference time-domain method and the S-parameter retrieval method, the equivalent EM parameter tensor of the single-layer CFRP lamina is retrieved. By constructing the global and local coordinate systems, the transformational relations between tensors in different layers are established, thus the equivalent EM parameter tensors of the rest layers are obtained accordingly. Finally, the rationality and validity of the proposed method are proved by numerical examples.

Crystallographic Analysis and Mechanism of Martensitic Transformation in Fe Alloys

A new version of the crystallographic theory of martensitic transformation has been developed, where instead of Bain deformation, the deformation of the lattice was performed by a shear along the twinning system of crystals and an additional change in dimensions in three mutually perpendicular directions. In the proposed variant, the angle of relaxation rotation of martensite crystal was about 1.8°; in the standard phenomenological theory, it was 10°. The new version made it possible to establish the mechanism of lattice deformation upon martensitic transformation, to determine the angle of relaxation rotation, and to carry out crystallographic analysis of martensitic transformation in various alloys. The formation of crystals (laths) of dislocation martensite in medium-carbon steel was found to occur with redistribution of carbon.

On the grain boundary network characteristics in a martensitic Ti–6Al–4V alloy

The characteristics of the intervariant boundary network that resulted from the $$\beta \to \alpha^{\prime}$$ martensitic phase transformation in a Ti–6Al–4V alloy were studied using the crystallographic theories of displacive transformations, five-parameter grain boundary analysis and triple junction analysis. The microstructure of Ti–6Al–4V martensite consisted of fine laths containing dislocations and fine twins. The misorientation angle distribution revealed four distinct peaks consistent with the intervariant boundaries expected from the Burgers orientation relationship. The phenomenological theory of martensite predicted four-variant clustering to have the lowest transformation strain among different variant clustering combinations. This configuration was consistent with the observed Ti–6Al–4V martensitic microstructure, where four-variant clusters consisted of two pairs of distinct V-shape variants. The $$63.26^\circ /[\overline{10}\, 5\, 5\, \overline{3}]_{{\alpha^{\prime}}}$$ and $$60^\circ /[1\, 1\, \overline{2}\, 0]_{{\alpha^{\prime}}}$$ intervariant boundaries accounted for ~ 38% and 33% of the total population, respectively. The five-parameter boundary analysis showed that the former had a twist character, being terminated on the $$(\overline{3}\, 2\, 1\, 0)_{{\alpha^{\prime}}}$$ plane, and the latter revealed a symmetric tilt $$(1\, 0\, \overline{1}\, 1)_{{\alpha^{\prime}}}$$ boundary plane. The $$63.26^\circ /[\overline{10}\, 5\, 5\, \overline{3}]_{{\alpha^{\prime}}}$$ and $$60^\circ /[1\, 1\, \overline{2}\, 0]_{{\alpha^{\prime}}}$$ had the highest connectivity at triple junctions among other intervariant boundaries. Interestingly, the boundary network in Ti–6Al–4V martensite was significantly different from the commercially pure Ti martensite, where only $$60^\circ /[1\, 1\, \overline{2}\, 0]_{{\alpha^{\prime}}}$$ intervariant boundaries largely were found at triple junctions due to the formation of three-variant clustering to minimize the transformation strain. This difference is thought to result from a change in the martensitic transformation mechanism (slip vs twinning) caused by the alloy composition.

Optimization Scheme of the Orientation Relationship from Crystallographic Statistics of Variants and its Application to Lath Martensite

In crystallographic theory of martensitic transformation, a slightly change of orientation relationship between lath martensite and prior austenite may indicate a different transformation mechanism and a change of mechanical property. Nevertheless, when the retained austenite can be hardly found in specimen, it is a great challenge to establish an accurate determination method for orientation relationship from the variant orientations only. Although some methods have been proposed, a reasonable evaluation of the solution accuracy is still waiting to be performed, and two further attentions should be addressed clearly: 1) significant improvement of calculation efficiency without sacrificing any solution accuracy; 2) effective elimination of the influence of the random cross-sections on the calculation. After an in-depth analysis of these issues, a new approach was developed involving two key steps in this paper. Firstly, an accurate clustering algorithm was developed to perform a statistic of all variants, which can give the most probable orientation and area fraction for each variant. Different from previous methods, the second step is to perform an optimization scheme established in this work, which takes variant fractions into consideration, to determine the optimal orientation relationship and orientation of prior austenite. In addition, the influence of the random cross-sections can be effectively eliminated by inputting the equally-weighted variants into the optimization scheme. Furthermore, in-situ high temperature EBSD examinations were first performed to validate solution accuracy, the corresponding results clearly indicated that the orientation of austenite obtained from this approach exhibited a high accuracy (with an error less than 1°), exceeding the accuracy achieved by the previous methods in addition to avoiding the possible deviation of using distorted retained austenite at room temperature. Indeed, the solution accuracy is contributed by the robust optimization scheme which strictly satisfies the criterion of a minimum average deviation between the measured and predicted orientation of martensitic variants. Another contributor to the accuracy is the good agreement between the clustered variant orientations and the measured data. Under the high accuracy, the variant-level calculation, which can save much computational time compared with pixel/domain-level calculation, is beneficial for the efficiency promotion.

Topological carbon allotropes: knotted molecules, carbon-nano-chain, chainmails, and Hopfene

Carbon allotropes such as diamond, nano-tube, Fullerene, and Graphene were discovered and revolutionised material sciences. These structures have unique translational and rotational symmetries, described by a crystallographic group theory, and the atoms are arranged at specific rigid positions in 3-dimensional (D) space. Regardless of these exotic molecular structures, the structures of materials are topologically trivial in a mathematical sense, that their bonds are connected without a link nor a knot. These days, the progress on the synthetic chemistry is significant to make various topologically non-trivial molecular structures. Topological molecules (0D) including Trefoil knots, a Hopf-link, a Möbius strip, and Borromean rings, were already realised. However, their potentially exotic electronic properties have not been sufficiently explored. Here, we propose a new 3D carbon allotrope, named Hopfene, which has periodic arrays of Hopf-links to knit horizontal Graphene sheets into vertical ones without connecting by σ bonds. We conducted an ab inito band structure calculation using a Density-Functional-Theory (DFT) for Hopfene, and found that it is well-described by a tight-binding model. We confirmed the original Dirac points of 2D Graphene were topologically protected upon the introduction of the Hopf links, and low-energy excitations are described by 1D, 2D, and 3D gapless Fermions.

Finite strain phase-field microelasticity theory for modeling microstructural evolution

Implementing the phase-field model at finite strains is usually considered unattainable through the spectral method, largely because of the nonlinearity in the transformation-induced microelasticity. Here we present a phase-field microelasticity (PFM) theory at finite strains with a representation in the reference configuration, allowing the spectral method to be readily incorporated. Following the spirit of Khachaturyan’s PFM theory at small strains, the elastic energy is formulated as a functional of microstructure (order parameters) solely, which should automatically satisfy the mechanical equilibrium. Thermodynamic consistency of the current theory under multiplicative decomposition of the total deformation gradient (into elastic and inelastic parts) and in conjunction with hyperelasticity and the time-dependent Ginzburg-Landau equation is shown rigorously. The new theory is first applied to the classical Eshelby’s inclusion problem, where shear-dilation coupling due to geometric nonlinearity is shown and a convergence study between small strain and finite strain theories is also carried out. The effects of geometric nonlinearity on the co-evolution of micromechanics and microstructure is further studied through modeling the growth of {101¯2}〈1¯011〉 deformation twins in magnesium. The simulation results suggest significant differences in terms of the shape of and the stress field around the deformation twin. In particular, the current finite strain PFM theory predicts a deviation of the twin boundary plane from the theoretical K1 plane, which is not captured in the small strain theory nor in the crystallographic theory. A parametric study further reveals that the observed deviation is caused by the tip effect of the finite-sized twin plate when the aspect ratio is relatively small. The symmetry of the stress field distribution around the twin tip is also found to be drastically different between the small strain and finite strain based phase-field modeling. The sharp twin tip observed in experiments is also shown to be likely related to the anisotropy in twin/matrix interface mobility.

Selective Growth of Stacking Fault Free ⟨100⟩ Nanowires on a Polycrystalline Substrate for Energy Conversion Application.

Cubic semiconductor nanowires grown along ⟨100⟩ directions have been reported to be promising for optoelectronics and energy conversion applications, owing to their pure zinc-blende structure without any stacking fault. But, until date, only limited success has been achieved in growing ⟨100⟩ oriented nanowires. Here we report the selective growth of stacking fault free ⟨100⟩ nanowires on a commercial transparent conductive polycrystalline fluorine-doped SnO2 (FTO) glass substrate via a simple and cost-effective chemical vapor deposition (CVD) method. By means of crystallographic analysis and density functional theory calculation, we prove that the orientation relationship between the Au catalyst and the FTO substrate play a vital role in inducing the selective growth of ⟨100⟩ nanowires, which opens a new pathway for controlling the growth directions of nanowires via the elaborate selection of the catalyst and substrate couples during the vapor-solid-liquid (VLS) growth process. The ZnSe nanowires grown on the FTO substrate are further applied as a photoanode in photoelectrochemical (PEC) water splitting. It exhibits a higher photocurrent than the ZnSe nanowires do without preferential orientations on a Sn-doped In2O3 (ITO) glass substrate, which we believe to be correlated with the smooth transport of charge carriers in ZnSe ⟨100⟩ nanowires with pure zinc-blende structures, in distinct contrast with the severe electron scattering happened at the stacking faults in ZnSe nanowires on the ITO substrate, as well as the efficient charge transfer across the intensively interacting nanowire-substrate interfaces.

A facile microwave assisted synthesis and structure elucidation of (3R)-3-alkyl-4,1-benzoxazepine-2,5-diones by crystallographic, spectroscopic and DFT studies

The use of microwave (MW) irradiation in organic synthesis has become increasingly popular within the pharmaceutical and academic arenas because it is a new enabling technology for drug discovery and development. It is a rapid way of synthesis, which involves faster reaction rates and high selectivity to conventional heating method of syntheses. The MW-assisted 7-exo-tet cyclization of N-acylanthranilic acids afforded (3R)-3-alkyl-4,1-benzoxazepines-2,5-diones in very short duration (20min) with extraordinary high yields in comparison to conventional heating mode of synthesis. The method development, comparative yields of MW-assisted and thermal method of syntheses, crystallographic, spectroscopic and density functional theory (DFT) studies are reported herein. Four novel compounds with chemical formulas C10H9BrClNO35m, C19H19NO36e, C13H14ClNO36h and C12H11Br2NO36h were synthesized, validated by 1HNMR, 13CNMR, FT-IR, UVVis, EIMS spectroscopic techniques and confirmed by using single crystal X-ray diffraction (SC-XRD) study. The DFT and TDDFT calculations at B3LYP/6-311+G(d,p) level of theory were performed for comparative analysis of spectroscopic data, optimized geometries, frontier molecular orbitals (FMOs), natural bond orbital (NBO) analysis and nonlinear optical (NLO) properties of 5m, 6e, 6h and 6o. Overall, experimental findings were supported nicely by corresponding DFT computed results. The NBO analysis confirmed that the presence of non-covalent interactions, hydrogen bonding and hyper- conjugative interactions are pivotal cause for the existence of 5m, 6e, 6h and 6o in the solid-state. NLO analysis showed that 5m, 6e, 6h and 6o have significant NLO properties as compared to prototype standard compound which disclosed their potential for technology related applications.

Equivalent method of evaluating mechanical properties of perforated Ni-based single crystal plates using artificial neural networks

Creep experiments were performed to study the mechanical characteristics of Ni-based single crystal superalloy samples with densely arranged air film holes. An equivalent model based on crystallographic theory coupled with continuous damage mechanics was developed. The equivalent method was proposed, combining artificial neural networks and the equivalent solid material concept. A series of experiments and calculation results on plates with diverse arrangement patterns and ligament efficiencies were compared to validate the accuracy of the proposed method. The results indicate that the creep curves of the simplified models agree well with the experimental results. The microstructural evolution observed by scanning electron microscopy can be evaluated using the evolution of the maximum resolved shear stress with the creep time. The damage distribution of the simplified models is consistent with the crack initiation location and propagation region of samples. Additionally, the slip system actuation of the simplified models is the same as that of the experimental results. The equivalent errors of the creep time, maximum resolved shear stress and creep damage are less than 15%, except for individual points, indicating that the equivalent method proposed in this study is feasible and accurate.

A review of basic crystallography and x-ray diffraction applications

Although various researched works have been carried out in x-ray crystallography and its applications, but there are still limited number of researches on crystallographic theories and industrial application of x-ray diffraction. The present study reviewed and provided detailed discussion on atomic arrangement of single crystals, mathematical concept of Bravais, reciprocal lattice, and application of x-ray diffraction. Determination of phase identification, crystal structure, dislocation density, crystallographic orientation, and gran size using x-ray diffraction peak intensity, peak position, and peak width were discussed. The detailed review of crystallographic theories and x-ray diffraction application would benefit majorly engineers and specialists in chemical, mining, iron, and steel industries.