A new method for lattice reduction using directional and hyperplanar shearing.

A geometric method of lattice reduction based on cycles of directional and hyperplanar shears is presented. The deviation from cubicity at each step of the reduction is evaluated by a parameter called `basis rhombicity' which is the sum of the absolute values of the elements of the metric tensor associated with the basis. The levels of reduction are quite similar to those obtained with the Lenstra-Lenstra-Lovász (LLL) algorithm, at least up to the moderate dimensions that have been tested (lower than 20). The method can be used to reduce unit cells attached to given hyperplanes.

A fast algorithm to find reduced hyperplane unit cells and solve N-dimensional Bézout's identities.

Deformation twinning on a plane is a simple shear that transforms a unit cell attached to the plane into another unit cell equivalent by mirror symmetry or 180° rotation. Thus, crystallographic models of twinning require the determination of the short unit cells attached to the planes, or hyperplanes for dimensions higher than 3. Here, a method is presented to find them. Equivalently, it gives the solutions of the N-dimensional Bézout's identity associated with the Miller indices of the hyperplane.

The transformation matrices (distortion, orientation, correspondence), their continuous forms and their variants. Corrigenda.

Appendices B4 and B5 of Cayron [Acta Cryst. (2019), A75, 411-437] contain equations involving the point group and the metric tensor in which the equality symbol should be substituted by the inclusion symbol.

The transformation matrices (distortion, orientation, correspondence), their continuous forms and their variants.

The crystallography of displacive/martensitic phase transformations can be described with three types of matrix: the lattice distortion matrix, the orientation relationship matrix and the correspondence matrix. Given here are some formulae to express them in crystallographic, orthonormal and reciprocal bases, and an explanation is offered of how to deduce the matrices of inverse transformation. In the case of the hard-sphere assumption, a continuous form of distortion matrix can be determined, and its derivative is identified to the velocity gradient used in continuum mechanics. The distortion, orientation and correspondence variants are determined by coset decomposition with intersection groups that depend on the point groups of the phases and on the type of transformation matrix. The stretch variants required in the phenomenological theory of martensitic transformation should be distinguished from the correspondence variants. The orientation and correspondence variants are also different; they are defined from the geometric symmetries and algebraic symmetries, respectively. The concept of orientation (ir)reversibility during thermal cycling is briefly and partially treated by generalizing the orientation variants with n-cosets and graphs. Some simple examples are given to show that there is no general relation between the numbers of distortion, orientation and correspondence variants, and to illustrate the concept of orientation variants formed by thermal cycling.

A one-step mechanism for new twinning modes in magnesium and titanium alloys modelled by the obliquity correction of a (58°, a + 2b) prototype stretch twin.

The \{ 11{\overline 2}2\} and \{ 11{\overline 2}6\} twinning modes were recently discovered by Ostapovets et al. [Philos. Mag. (2017), 97, 1088-1101] and interpreted as \{ {10{\overline 1}2} \}-\{ {10{\overline 1}2} \} double twins formed by the simultaneous action of two twinning shears. Another interpretation is proposed here in which the two conjugate twinning modes result from a one-step mechanism based on a (58°, a + 2b) prototype stretch twin and differ from each other only by their obliquity correction. The results are also compared with the classical theory of twinning and with the Westlake-Rosenbaum model.

Variant selection in surface martensite.

A significant variant selection is reported in isothermal martensite formed on the surface of an Fe-30% Ni sample. The selection phenomenon is modelled using different descriptions of the martensitic phase transformation. In particular, matrices based on the phenomenological theory of martensite crystallography, the Jaswon and Wheeler distortion, and the continuous face centred cubic-body centred cubic distortion are compared. All descriptions allow good predictions of the variant selection. However, the Jaswon and Wheeler distortion and the continuous distortion better account for other features of the surface martensite, such as the {225}γ habit plane and the accommodation mechanism by twin-related variant pairing.

Hard-sphere displacive model of deformation twinning in hexagonal close-packed metals. Revisiting the case of the (56°, a) contraction twins in magnesium.

Contraction twinning in magnesium alloys leads to new grains that are misoriented from the parent grain by a rotation (56°, a). The classical shear theory of deformation twinning does not specify the atomic displacements and does not explain why contraction twinning is less frequent than extension twinning. The paper proposes a new displacive model in line with our previous work on martensitic transformations and extension twinning. A continuous angular distortion matrix that transforms the initial hexagonal close-packed (h.c.p.) crystal into a final h.c.p. crystal is determined such that the atoms move as hard spheres and reach the final positions expected by the orientation relationship. The calculations prove that the distortion is not a simple shear when it is considered in its continuity. The ({0{\overline 1}1}) plane is untilted and restored, but it is not fully invariant because some interatomic distances in this plane evolve during the distortion process; the unit volume also increases up to 5% before coming back to its initial value when the twinning distortion is complete. Then, the distortion takes the form of a simple shear on the ({0{\overline 1}1}) plane with a shear along the direction [{18,{\overline 5},{\overline 5}}] of amplitude 0.358. Experiments are proposed to validate or disprove the model.

{225}γ habit planes in martensitic steels: from the PTMC to a continuous model.

Fine twinned microstructures with {225}γ habit planes are commonly observed in martensitic steels. The present study shows that an equibalanced combination of twin-related variants associated to the Kurdjumov-Sachs orientation relationship is equivalent to the Bowles and Mackenzie's version of the PTMC for this specific {225}γ case. The distortion associated to the Kurdjumov-Sachs orientation relationship results from a continuous modeling of the FCC-BCC transformation. Thus, for the first time, an atomic path can be associated to the PTMC.

Multiscale phase mapping of LiFePO4-based electrodes by transmission electron microscopy and electron forward scattering diffraction.

LiFePO4 and FePO4 phase distributions of entire cross-sectioned electrodes with various Li content are investigated from nanoscale to mesoscale, by transmission electron microscopy and by the new electron forward scattering diffraction technique. The distributions of the fully delithiated (FePO4) or lithiated particles (LiFePO4) are mapped on large fields of view (>100 × 100 µm(2)). Heterogeneities in thin and thick electrodes are highlighted at different scales. At the nanoscale, the statistical analysis of 64 000 particles unambiguously shows that the small particles delithiate first. At the mesoscale, the phase maps reveal a core-shell mechanism at the scale of the agglomerates with a preferential pathway along the electrode porosities. At larger scale, lithiation occurs in thick electrodes "stratum by stratum" from the surface in contact with electrolyte toward the current collector.

Embedded layer of Ag nanoparticles prepared by a combined PECVD/PVD process producing SiOxCy-Ag nanocomposite thin films.

Structural properties of SiO(x)C(y)-Ag nanocomposite thin films prepared by a dual process PVD-PECVD in the same reactor have been investigated. The experimental results have demonstrated the influence of a PECVD process carried out at room temperature for the growth of a dielectric matrix on the size and the distribution density of Ag nanoparticles (NPs) deposited beforehand by magnetron sputtering. The plasma during the growth of the encapsulation SiO(x)C(y) layer caused a diffusion of silver from NPs through the SiO(x)C(y) matrix associated with a decrease in the average size of nanoparticles and an increase of their distribution density. Silver diffusion is blocked at a barrier interface to form a buried layer of individual Ag NPs which, for instance, can find plasmonic applications. Silver also diffuses toward the outer surface inducing antibacterial properties. In both cases initial Ag NPs act as reservoirs for multifunctional properties of advanced nanostructured films.

Odd electron diffraction patterns in silicon nanowires and silicon thin films explained by microtwins and nanotwins.

Odd electron diffraction patterns (EDPs) have been obtained by transmission electron microscopy (TEM) on silicon nanowires grown via the vapour-liquid-solid method and on silicon thin films deposited by electron beam evaporation. Many explanations have been given in the past, without consensus among the scientific community: size artifacts, twinning artifacts or, more widely accepted, the existence of new hexagonal Si phases. In order to resolve this issue, the microstructures of Si nanowires and Si thin films have been characterized by TEM, high-resolution transmission electron microscopy (HRTEM) and high-resolution scanning transmission electron microscopy. Despite the differences in the geometries and elaboration processes, the EDPs of the materials show great similarities. The different hypotheses reported in the literature have been investigated. It was found that the positions of the diffraction spots in the EDPs could be reproduced by simulating a hexagonal structure with c/a = 12(2/3)(1/2), but the intensities in many EDPs remained unexplained. Finally, it was established that all the experimental data, i.e. EDPs and HRTEM images, agree with a classical cubic silicon structure containing two microstructural defects: (i) overlapping Σ3 microtwins which induce extra spots by double diffraction, and (ii) nanotwins which induce extra spots as a result of streaking effects. It is concluded that there is no hexagonal phase in the Si nanowires and the Si thin films presented in this work.

Growth of one-dimensional Si/SiGe heterostructures by thermal CVD.

The first results on a simple new process for the direct fabrication of one-dimensional superlattices using common CVD chambers are presented. The experiments were carried out in a 200 mm industrial Centura reactor (Applied Materials). Low dimensionality and superlattices allow a significant increase in the figure of merit of thermoelectrics by controlling the transport of phonons and electrons. The monocrystalline nanowires produced according to this process are both one-dimensional and present heterostructures, with very thin layers (40 nm) of Si and SiGe. Concentrations up to 30 at.% Ge were obtained in the SiGe parts. Complementary techniques including transmission electronic microscopy (TEM), selected area electron diffraction (SAED), energy dispersive x-ray spectroscopy (EDS), scanning transmission electron microscopy (STEM) in bright field and high angle annular dark field (HAADF STEM), and energy-filtered transmission electron microscopy (EF-TEM) were used to characterize the nanoheterostructures.

Multiple twinning in cubic crystals: geometric/algebraic study and its application for the identification of the Sigma3(n) grain boundaries.

Multiple twinning in cubic crystals is represented geometrically by a three-dimensional fractal and algebraically by a groupoid. In this groupoid, the variant crystals are the objects, the misorientations between the variants are the operations, and the Sigma3(n) operators are the different types of operations (expressed by sets of equivalent operations). A general formula gives the number of variants and the number of Sigma3(n) operators for any twinning order. Different substructures of this groupoid (free group, semigroup) can be equivalently introduced to encode the operations with strings. For any coding substructure, the operators are expressed by sets of equivalent strings. The composition of two operators is determined without any matrix calculation by string concatenations. It is multivalued due to the groupoid structure. The composition table of the operators is used to identify the Sigma3(n) grain boundaries and to reconstruct the twin related domains in the electron back-scattered diffraction maps.

GenOVa: a computer program to generate orientational variants.

A computer program called GenOVa, written in Python, calculates the orientational variants, the operators (special types of misorientations between variants) and the composition table associated with a groupoid structure. The variants can be represented by three-dimensional shapes or by pole figures.

ARPGE: a computer program to automatically reconstruct the parent grains from electron backscatter diffraction data.

A computer program called ARPGE written in Python uses the theoretical results generated by the computer program GenOVa to automatically reconstruct the parent grains from electron backscatter diffraction data obtained on phase transition materials with or without residual parent phase. The misorientations between daughter grains are identified with operators, the daughter grains are identified with indexed variants, the orientations of the parent grains are determined, and some statistics on the variants and operators are established. Some examples with martensitic transformations in iron and titanium alloys were treated. Variant selection phenomena were revealed.

Groupoid of orientational variants.

Daughter crystals in orientation relationship with a parent crystal are called variants. They can be created by a structural phase transition (Landau or reconstructive), by twinning or by precipitation. Internal and external classes of transformations defined from the point groups of the parent and daughter phases and from a transformation matrix allow the orientations of the distinct variants to be determined. These are algebraically identified with left cosets and their number is given by the Lagrange formula. A simple equation links the numbers of variants of the direct and inverse transitions. The equivalence classes on the transformations between variants are isomorphic to the double cosets (operators) and their number is given by the Burnside formula. The orientational variants and the operators constitute a groupoid whose composition table acts as a crystallographic signature of the transition. A general method that determines if two daughter variants can be inherited from more than one parent crystal is also described. A computer program has been written to calculate all these properties for any structural transition; some results are given for Burgers transitions and for martensitic transitions in steels. The complexity, irreversibility and entropy of fractal systems constituted by orientational variants generated by thermal cycling are briefly discussed.