On representing rotations by Rodrigues parameters in non-orthonormal reference systems.

A Rodrigues vector is a triplet of real numbers used for parameterizing rotations or orientations in three-dimensional space. Because of its properties (e.g. simplicity of fundamental regions for misorientations) this parameterization is frequently applied in analysis of orientation maps of polycrystalline materials. By conventional definition, the Rodrigues parameters are specified in orthonormal coordinate systems, whereas the bases of crystal lattices are generally non-orthogonal. Therefore, the definition of Rodrigues parameters is extended so they can be directly linked to non-Cartesian bases of a crystal. The new parameters are co- or contravariant components of vectors specified with respect to the same basis as atomic positions in a unit cell. The generalized formalism allows for redundant crystallographic axes. The formulas for rotation composition and the relationship to the rotation matrix are similar to those used in the Cartesian case, but they have a wider range of applicability: calculations can be performed with an arbitrary metric tensor of the crystal lattice. The parameterization in oblique coordinate frames of lattices is convenient for crystallographic applications because the generalized parameters are directly related to indices of rotation-invariant lattice directions and rotation-invariant lattice planes.

Orientation mapping by transmission-SEM with an on-axis detector.

Conventional orientation mapping in a scanning electron microscope (SEM) is a valuable technique for characterizing crystalline materials, but its application to ultrafine or nano-grain materials is limited by its spatial resolution. The resolution can be increased by collecting transmission diffraction patterns in SEM. In previous works, such patterns were collected using off-axis detectors in nearly vertical position. To avoid some drawbacks of such arrangement, a new configuration was devised in which the scintillator is located underneath the thin foil on the optical axis of the microscope, and the light is reflected towards the camera by a mirror. This simple configuration gives intense patterns even at very low probe currents, and can be potentially used for collecting maps of relatively high spatial resolution. Example maps reveal details with dimensions of about 5nm. Because of its resolution and geometric simplicity, the proposed configuration will open new opportunities in SEM-based characterization of nanocrystalline materials.

Orientation precision of TEM-based orientation mapping techniques.

Automatic orientation mapping is an important addition to standard capabilities of conventional transmission electron microscopy (TEM) as it facilitates investigation of crystalline materials. A number of different such mapping systems have been implemented. One of their crucial characteristics is the orientation resolution. The precision in determination of orientations and misorientations reached in practice by TEM-based automatic mapping systems is the main subject of the paper. The analysis is focused on two methods: first, using spot diffraction patterns and 'template matching', and second, using Kikuchi patterns and detection of reflections. In simple terms, for typical mapping conditions, their precisions in orientation determination with the confidence of 95% are, respectively, 1.1 ° and 0.3 °. The results are illustrated by example maps of cellular structure in deformed Al, the case for which high orientation sensitivity matters. For more direct comparison, a novel approach to mapping is used: the same patterns are solved by each of the two methods. Proceeding from a classification of the mapping systems, the obtained results may serve as indicators of precisions of other TEM-based orientation mapping methods. The findings are of significance for selection of methods adequate to investigated materials.

Volume of intersection of two balls in orientation space.

Orientations deviating from an ideal orientation by angles not exceeding a given limit constitute a ball in the metric space of orientations. Such balls arise in crystallographic computations, and in some cases intersections of the balls are of interest. This paper provides an analytical formula for calculation of the volume of the intersection of two balls from their radii and the distance between their centers. The formula is used to determine the volumes of the intersections of balls corresponding to misorientations of coincident lattices.

Shear banding phenomenon in a Cu-8 at.% Al alloy analysed by orientation imaging microscopy.

The microstructure and texture of Cu-8 at.% Al alloy single crystal with (112)[11 1] orientation plane strain compressed at 77 K were characterized by scanning electron microscope and transmission electron microscope orientation mappings in order to investigate the influence of twins and shear bands on slip propagation across a structure of twin-matrix layers and the resulting texture evolution. It was shown that the strong, initial texture changes are due to deformation twinning at low deformations. At larger deformations, twin-matrix bending within some narrow areas led to the formation of kink-type bands, which became the precursors of shear bands. It is shown with scanning electron microscope and transmission electron microscope orientation mappings how the structure of twins and matrix was incorporated into the shear band, and what kind of the dislocation mechanism was responsible for strain accommodation at the macroscale. It was found that the reorientation of the main (111) twinning planes towards the shear band plane facilitates further dislocation slip in the shear direction.

Determination of lattice parameters from multiple CBED patterns: a statistical approach.

This study deals with the uncertainty of the measurement of lattice parameters by CBED using the kinematic approximation. The analysis of a large number of diffraction patterns acquired on a silicon sample at 93 K with a LaB(6) TEM without energy filter shows the presence of both the systematic and the random parts of errors. It is established that random errors follow the normal statistical distribution and that the precision quantified by the relative standard deviation is about 3-4 x 10(-4) for lattice parameter measurements made from single pattern. The error sources are analyzed, different ways of enhancement are reviewed, and a new approach is proposed. It is shown that both accuracy and precision can be simply improved by taking into account multiple patterns analysis for the determination of the actual voltage, the single lattice parameter "a" or the complete set of lattice parameters. The precision of about 1.5-2 x 10(-4) can be reached using a minimum of three HOLZ line patterns for the single "a" parameter and about 5 x 10(-4) for the complete set of lattice parameters using six diffraction patterns. The use of multiple patterns also allows overcoming the non-uniqueness of solution linked to the CBED studies.

An algorithm for refinement of lattice parameters using CBED patterns.

A new algorithm for calculation of lattice parameters from convergent beam electron diffraction (CBED) patterns has been developed. Like most of the previous approaches to the problem, it is an optimization procedure matching geometric elements of high order Laue zone (HOLZ) lines in experimental patterns to corresponding elements of kinematically simulated patterns. The procedure uses an original objective function based directly on the underlying algebraic equation of the HOLZ lines. Although the new approach requires crystal orientation parameters to be fitted alongside the strain components, it is easier to implement than methods used previously. It is also straightforward to apply to strain determination from multiple patterns. Numerical tests on dynamically simulated patterns show that in the case of one or two patterns, the new procedure gives results that are more reliable than the established method based on HOLZ line intersections. As an example application, the a and c parameters of a TiAl alloy are determined.

Shear banding in twinned structure of copper deformed at 77 K.

The local crystallography and microstructure within shear bands has been examined in single crystals of {112}<111> orientation of pure copper deformed at 77 K by channel-die compression to true strains of about 1. Setting up a system for making high-resolution orientation maps using transmission electron microscopy has provided advantageous circumstances for the analysis of orientation changes within shear bands. The present work shows that, despite the plane strain deformation mode, the mechanism of lattice rotation within emerging shear bands may lead to Goss {110}< 001> and Brass{110}<112> texture components.

Polycrystal orientation maps from TEM.

Determination of topography of crystallite orientations is an important technique of investigation of polycrystalline materials. A system for creating orientation maps using transmission electron microscope (TEM) Kikuchi patterns and Convergent beam electron diffraction patterns is presented. The orientation maps are obtained using a step-by-step beam scan on a computer-controlled TEM equipped with a CCD camera. At each step, acquired diffraction patterns are indexed and orientations are determined. Although, the approach used is similar to that applied in SEM/electron back scattered diffraction (EBSD) orientation imaging setups, the TEM-based system considerably differs from its SEM counterpart. The main differences appear due to specific features of TEM and SEM diffraction patterns. Also, the resulting maps are not equivalent. On these generated by TEM, the accuracy of orientation determination can be better than 0.1 degrees. The spatial resolution is estimated to be about 10nm. The latter feature makes the TEM orientation mapping system an important tool for studies at fine scale unreachable by SEM/EBSD systems. The automatic orientation mapping is expected to be a useful complement of the conventional TEM contrast images. The new technique will be essential for characterization of fine structure materials. To illustrate that, example maps of an aluminum sample produced by severe plastic deformation are included.

Mapping the mesoscale interface structure in polycrystalline materials.

A new experimental approach to the quantitative characterization of polycrystalline microstructure by scanning electron microscopy is described. Combining automated electron backscattering diffraction with conventional scanning contrast imaging and with calibrated serial sectioning, the new method (mesoscale interface mapping system) recovers precision estimates of the 3D idealized aggregate function G(x). This function embodies a description of lattice phase and orientation (limiting resolution approximately 1 degree) at each point x (limiting spatial resolution approximately 100 nm), and, therefore, contains a complete mesoscale description of the interfacial network. The principal challenges of the method, achieving precise spatial registry between adjacent images and adequate distortion correction, are described. A description algorithm for control of the various components of the system is also provided.

[A "double flap" technique in stabilization of the prostheses of the PORP and TORP type]. / Technika "podwójnego plata" w stabilizacji protezek typu PORP i TORP.

The authors described the stabilization method of prostheses type PORP and TORP in one and two steps hearing improvement surgery. The method is very simple and prevents the secondary tympanic membrane perforations.