Space groups and crystallographic symmetry: writing a multi-featured tutorial in a new style

Knowledge of space groups and the implications of space group symmetry on the physical and chemical properties of solids are pivotal factors in all areas of crystalline solids. As Jerry Jasinski and I met to bring our ideas in teaching this subject to life, we both felt that `early and often' – teaching the concepts with textual and visual reinforcement, is the key to providing a sound basis for students in this subject. The tutorial contains > 200 PowerPoint `slides', in five modules, arranged by crystal class; a sixth module covers special topics. A `credits' module gives the direct addresses of all embedded links. Space-group diagrams appear in International Tables format. The triclinic and monoclinic groups (2 + 13) are built from `scratch', and are derived from the Hermann–Mauguin symbol. An additional section provides practice on many (but not all) of the orthorhombic groups in crystal class 222. Finally, a `Special Topics' section on enantiomorphous space groups features space groups P41 and P43. In the tutorial, lattice points build iteratively and interactively via keyclick, and the coordinates of points `pop up' as the unit cell is filled. We trust that the elements of guidance, inquiry and occasional humor will make the learning process enjoyable.

Optimized Voltage-Induced Control of Magnetic Domain-Wall Propagation in Hybrid Piezoelectric/Magnetostrictive Devices

A theory of voltage-induced control of magnetic domain walls propagating along the major axis of a magnetostrictive nanostrip, tightly coupled with a ceramic piezoelectric, is developed in the framework of the Landau–Lifshitz–Gilbert equation. It is assumed that the strains undergone by the piezoelectric actuator, subject to an electric field generated by a dc bias voltage applied through a couple of lateral electrodes, are fully transferred to the magnetostrictive layer. Taking into account these piezo-induced strains and considering a magnetostrictive linear elastic material belonging to the cubic crystal class, the magnetoelastic field is analytically determined. Therefore, by using the classical traveling-wave formalism, the explicit expressions of the most important features characterizing the two dynamical regimes of domain-wall propagation have been deduced, and their dependence on the electric field strength has been highlighted. Moreover, some strategies to optimize such a voltage-induced control, based on the choice of the ceramic piezoelectric material and the orientation of dielectric poling and electric field with respect to the reference axes, have been proposed.

Crystal symmetry for incommensurate helical and cycloidal modulations

A classification of magnetic superspace groups compatible with the helical and cycloidal magnetic modulations is presented. Helical modulations are compatible with groups from crystal classes 1, 2, 222, 4, 422, 3, 32, 6 and 622, while cycloidal modulations are compatible with groups from crystal classes 1, 2, m and mm2. For each magnetic crystal class, the directions of the symmetry-allowed (non-modulated) net ferromagnetic moment and electric polarization are given. The proposed classification of superspace groups is tested on experimental studies of type-II multiferroics published in the literature.

Structures of Pt(0)P3, Pt(0)P4 and Pt(II)P4 – Distortion isomers

In this review are analyzed and classified crystallographic and structural parameters of P(0)P3, Pt(0) P4 and Pt(II)P4 derivatives – distortion isomers. Some of the isomers are differing not only by degree of distortion but also by crystal class. There are three types of organo-phosphines which build up the respective geometry about the platinum atoms. In Pt(0)P3 a distorted trigonal planar geometry is build up by three monodentate PPh3 ligands. In Pt(0)P4 a tetrahedral geometry with various degree of distortion is build up by a pair of homo-bidentate ligands. In Pt(II)P4 isomers a square-planar geometries with various degree of distortion are build up by bidentate-P,P’donor ligands, (except one example of isomers, where a tetradentate is involved). The bidentate-P,P’-donor ligands form: four-(PNP,PCP), five-(PC2P) and six-(PC3P) metallocyclic rings. The tetradentate forms five-(PC2P). There are some cooperative effects between Pt–P bond distances and the metallocyclic rings, and at the same time a distortion of the respective geometry increases.

Normally supportive sublattices of crystallographic space groups

The tabulation of normal subgroups of 3D crystallographic space groups that are themselves 3D crystallographic space groups (csg's) is an ambitious goal, but would have a variety of applications. For convenience, such subgroups are referred to as `csg-normal' while normal subgroups of the crystallographic point group (cpg) of a crystallographic space group are referred to as `cpg-normal'. The point group of a csg-normal subgroup must be a cpg-normal subgroup. The present work takes a significant step towards that goal by tabulating the translational subgroups (a.k.a. sublattices) that are capable of supporting csg-normal subgroups. Two necessary conditions are identified on the relative sublattice basis that must be met in order for the sublattice to support csg-normal subgroups: one depends on the operations of the point group of the space group, while the other depends on the operations of the cpg-normal subgroup. Sublattices that meet these conditions are referred to as `normally supportive'. For each cpg-normal subgroup (excluding the identity subgroup 1) of each of the arithmetic crystal classes of 3D space groups, all of the normally supportive sublattices have been tabulated in symbolic form, such that most of the entries in the table contain one or more integer variables of infinite range; thus it could be more accurately described as a table of the infinite families of normally supportive sublattices. For a given pair of cpg-normal subgroup and normally supportive sublattice, csg-normal subgroups of the space groups of the parent arithmetic crystal class can be constructed via group extension, though in general such a pair does not guarantee the existence of a corresponding csg-normal subgroup.

Generalized Dynamic Analytical Model of Piezoelectric Materials for Characterization Using Electrical Impedance Spectroscopy

Piezoelectric materials have the intrinsic reversible ability to convert a mechanical strain into an electric field and their applications touch our daily lives. However, the complex physical mechanisms linking mechanical and electrical properties make these materials hard to understand. Computationally onerous models have historically been unable to adequately describe dynamic phenomena inside real piezoelectric materials, and are often limited to over-simplified first-order analytical, quasi-static, or unsatisfying phenomenological numerical approaches. We present a generalized dynamic analytical model based on first-principles that is efficiently computable and better describes these exciting materials, including higher-order coupling effects. We illustrate the significance of this model by applying it to the important 3m crystal symmetry class of piezoelectric materials that includes lithium niobate, and show that the model accurately predicts the experimentally observed impedance spectrum. This dynamic behavior is a function of almost all intrinsic properties of the piezoelectric material, so that material properties, including mechanical, electrical, and dielectric coefficients, can be readily and simultaneously extracted for any size crystal, including at the nanoscale; the only prior knowledge required is the crystal class of the material system. In addition, the model’s analytical approach is general in nature, and can increase our understanding of traditional and novel ferroelectric and piezoelectric materials, regardless of crystal size or orientation.

Synthesis, Characterization, Crystal and Molecular Structure Analysis of 1-(2-Chlorophenyl)-3-methyl-4-(p-tolylthio)-1H-pyrazol-5-ol

The synthesis of a novel tolylthiopyrazol bearing methyl group has been achieved by transition metal free N-chlorosuccinimide mediated direct sulfenylation of 1-aryl pyrazolones at room temperature. The product obtained was characterized by spectroscopic techniques and finally confirmed by X-ray diffraction studies. The compound 1-(2-chlorophenyl)-3-methyl-4-(p-tolylthio)-1H-pyrazol-5-ol (m.f. C17H15N2OSCl) crystallizes in monoclinic crystal class in space group P21/c with cell parameters a = 9.6479(5) Å, b = 15.1233(8) Å, c = 11.4852(6) Å, β = 108.374(2)°, V=1590.4(2) Å3 and Z = 4. The final residual factor R1 = 0.0499.