Improper Rotations Research Articles

Characterization of coincidence site lattices generated by the composition of simple reflections of cubic lattices

This work provides a comprehensive characterization of coincidence site lattices that arise from the compositions of the reflections of cubic lattices. For this purpose, we apply Cartan's theorem and use reflections as basic isometries, which allows the use of Clifford algebra as a powerful tool to handle reflections and products of reflections, leading to explicit expressions for coincidence indices and the basis of coincidence lattices. The three possible cases were studied: simple reflections, the product of two reflections (rotations), and the product of three reflections, where a factorization that produces improper rotations was provided. In the case of rotations, it is formally proven that Ranganathan's well‐known generating function of coincidence indices for cubic systems under rotations yields all possible coincidence lattices. Overall, this study provides valuable insights into the mathematical properties of coincidence site lattices and the use of Clifford algebras to describe orthogonal transformations.

Spin 1/2 one‐ and two‐particle systems in physical space without eigen‐algebra or tensor product

Under the spin–position decoupling approximation, a vector with a phase in 3D orientation space endowed with geometric algebra substitutes the vector–matrix spin model built on the Pauli spin operator. The standard quantum operator‐state spin formalism is replaced with vectors transforming by proper and improper rotations in the same 3D space—isomorphic to the space of Pauli matrices. In the single‐spin case, the novel spin 1/2 representation (1) is Hermitian, (2) shows handedness, (3) yields all the standard results and its modulus equals the total spin angular momentum , (4) formalizes irreversibility in measurement, and (5) permits adaptive imbedding of the 2D spin space in 3D. Maximally entangled spin pairs (1) are in phase and have opposite handedness, (2) relate by one of the four basic improper rotations in 3D: plane reflections (triplets) and inversion (singlet), (3) yield the standard total angular momentum, and (4) all standard expectation values for bipartite and partial observations follow. Depending on whether proper and improper rotors act one—or two—sided, the formalism appears in two complementary forms, the “spinor” or the “vector” form, respectively. The proposed scheme provides a clear geometric picture of spin correlations and transformations entirely in the 3D physical orientation space.

GESUS, an Interactive Computer Application for Teaching and Learning the Space Groups of Symmetry

This paper describes a digital application designed for learning Space Groups of Symmetry (SGS). It teaches how to recognize the operations performed by the symmetry elements, both point (or 2D) operators (proper and improper rotations, including mirroring, inversion, and other rotoinversions), and space (or 3D) operators (screw axes and glide planes), as well as their combinations with the translations of the lattice. The software applies a 3D space vision to identify the elements of symmetry compatible with the proposed structural models. The symbols of internationally accepted representation are used. The system, class, and space group of the crystal that agree with the proposed model are solved. Two settings are taken into account in the Monoclinic system. The application self-evaluates and assesses the knowledge acquired, allowing each exercise to be re-done until it is correctly completed with the appropriate recommendations. This application constitutes a useful and easy-to-use tool for SGS learning. It is aimed at beginner students of crystallography, with elementary knowledge about elements of symmetry, Bravais lattices, crystal classes, and wallpaper groups.

Symmetry-Consistent Expansion of Interaction Kernels Between Rigid Molecules

We discuss the expansion of interaction kernels between anisotropic rigid molecules. The expansion decouples the correlated orientational variables so that it can be utilized to derive macroscopic models. Symmetries of two types are considered. First, we examine the symmetry of the interacting cluster, including the translation and rotation of the whole cluster, and label permutation within the cluster. The expansion is expressed by symmetric traceless tensors, and the linearly independent terms are identified. Then, we study the molecular symmetry characterized by a point group in $O(3)$. The proper rotations determine what symmetric traceless tensors can appear. The improper rotations decompose these tensors into two subspaces and determine how the tensors in the two subspaces are coupled. For each point group, we identify the two subspaces, so that the expansion consistent with the point group is established.

Prevalence of musculoskeletal disorders and their association with ergonomic physical risk factors among women working in tea gardens of Darjeeling district of West Bengal, India

Introduction: Work related musculoskeletal disorders (WRMSDs) is common among tea-plucking folks and may be attributed to the long duration of work in same posture, the load they carry while plucking and transporting the leaves to the nearest depot, improper job rotations and the difficult hilly terrains.
 Objective: To determine the prevalence and pattern of musculoskeletal disorders among tea garden workers and to ascertain the risk factors for the same.
 Methods: A cross-sectional study was conducted among 210 female tea garden workers employed in tea gardens of Darjeeling district using 30 cluster sampling design. Data on musculoskeletal morbidities were collected using Standard Nordic Musculoskeletal Questionnaire and ergonomic risk was assessed by Rapid Entire Body Assessment (REBA) worksheet.
 Results: Proportion of musculoskeletal disorders in past 12 months was 92.4% and in past 7 days was 71.4%. Upper extremities were the most commonly affected body part. Higher altitude, age, BMI, more duration in present job, history of work related injuries and increased REBA risk assessment score were found to be significantly associated with musculoskeletal disorders.
 Conclusion: Musculoskeletal morbidities among female tea garden workers of Darjeeling were found to be considerably high. Ergonomic mechanised assistance in tea plucking can reduce this proportion of MSD and thus may go a long way in reducing sickness absenteeism and increasing productivity.
 Financial support and sponsorship - Nil.
 Conflicts of interest - There are no conflicts of interest.

Recursive evaluation and iterative contraction of N-body equivariant features

Mapping an atomistic configuration to a symmetrized N-point correlation of a field associated with the atomic positions (e.g., an atomic density) has emerged as an elegant and effective solution to represent structures as the input of machine-learning algorithms. While it has become clear that low-order density correlations do not provide a complete representation of an atomic environment, the exponential increase in the number of possible N-body invariants makes it difficult to design a concise and effective representation. We discuss how to exploit recursion relations between equivariant features of different order (generalizations of N-body invariants that provide a complete representation of the symmetries of improper rotations) to compute high-order terms efficiently. In combination with the automatic selection of the most expressive combination of features at each order, this approach provides a conceptual and practical framework to generate systematically improvable, symmetry adapted representations for atomistic machine learning.

Computer-Aided Identification of Symmetry Relating Groups of Molecules

Molecular symmetry plays an important role in many areas of chemistry, and students are generally taught to identify intramolecular symmetries at the undergraduate level. Intermolecular symmetry, the symmetry among groups of molecules, is of great importance in topics such as X-ray crystallography, but it receives substantially less attention. Even when familiar with such symmetries as translations, screw rotations, and glide reflections, students rarely obtain the same level of proficiency in identifying these symmetries via inspection as they do with the proper and improper rotations involved in point group symmetry. We describe a tool designed to scaffold the student’s ability to mentally manipulate molecules and identify the presence and nature of intermolecular symmetry. The tool consists of an algorithm that the student follows to generate centroids between atoms in putatively symmetry-related molecules. The easily recognized patterns formed by the centroids provide key information about the presence and nature of any intermolecular symmetry relating the molecules. The theoretical basis for the patterns formed by the centroids is provided along with worked examples, an example problem set, and a prescription for generating new exercises.

The H2S dimer revisited – Insights from wave-function and density functional theory methods. Ab initio molecular dynamics simulations of liquid H2S

We performed an investigation of the hydrogen bonding in H2S dimers in gas-phase using HF, MP2, PBE, PBE0 and BLYP. A series of symmetry operations such as translations and proper and improper rotations were performed to the donor and acceptor molecules, and the energetic dependency of the H-bond as a function of these transformations was obtained. We found that the DFT methods predict a repulsive region around 5 to 6 times more repulsive that that obtained with the wave-function methods. Additionally, we determined the Lennard-Jones potential for the translations and derived a Morse potential for the different rotations and ab initio MD simulations were performed for liquid H2S. We found that remarkably, the length of the H-bonds and height of rotational barriers in liquid H2S are similar to the ones of the dimer. This is because of the weak and highly directional H-bonds in both the liquid and H2S dimer.

On the constraints of electromagnetic multipoles for symmetric scatterers: eigenmode analysis.

The scattering and resonant properties of optical scatterers/resonators are determined by the relative ratios among the associated multipole components, the calculation of which usually is analytically tedious and numerically complicated for complex structures. Here we identify the constraints as well as the relative relations among electromagnetic multipoles for the eigenmodes of symmetric scatterers/resonators. By reducing the symmetry properties of the vector spherical harmonic waves to those of the modified generating functions, we systematically study the required conditions for electromagnetic multipoles under several fundamental symmetry operations, i.e., 2D rotation and reflection operations and 3D proper and improper rotations. Taking a 2D scatterer with C4v as an example, we show that each irreducible representation of C4v can be assigned to corresponding electromagnetic multipoles, and consequently the constraints of the electromagnetic multipoles can be easily extracted. Such group approach can easily be extended to more complex 3D scatterers with higher symmetry group. Subsequently, we use the same procedure to map out the complete relation and constraint on the electromagnetic multipoles of a 3D scatterer imposed by D3h symmetry. Our theoretical analyses are in perfect agreements with the fullwave finite element calculations of the eigenmodes of the symmetric scatters.

Rotation covariant local tensor valuations on convex bodies

For valuations on convex bodies in Euclidean spaces, there is by now a long series of characterization and classification theorems. The classical template is Hadwiger’s theorem, saying that every rigid motion invariant, continuous, real-valued valuation on convex bodies in [Formula: see text] is a linear combination of the intrinsic volumes. For tensor-valued valuations, under the assumptions of isometry covariance and continuity, there is a similar classification theorem, due to Alesker. Also for the local extensions of the intrinsic volumes, the support, curvature and area measures, there are analogous characterization results, with continuity replaced by weak continuity, and involving an additional assumption of local determination. The present authors have recently obtained a corresponding characterization result for local tensor valuations, or tensor-valued support measures (generalized curvature measures), of convex bodies in [Formula: see text]. The covariance assumed there was with respect to the group [Formula: see text] of orthogonal transformations. This was suggested by Alesker’s observation, according to which in dimensions [Formula: see text], the weaker assumption of [Formula: see text] covariance does not yield more tensor valuations. However, for tensor-valued support measures, the distinction between proper and improper rotations does make a difference. This paper considers, therefore, the local tensor valuations sharing the previously assumed properties, but with [Formula: see text] covariance replaced by [Formula: see text] covariance, and provides a complete classification. New tensor-valued support measures appear only in dimensions 2 and 3.

Tri-Dirac surface modes in topological superconductors

We propose a new type of topological surface modes having cubic dispersion in three-dimensional topological superconductors. Lower order dispersions are prohibited by the threefold rotational symmetry and time-reversal symmetry. Cooper pairing in the bulk changes sign under improper rotations, akin to$^{3}$He-B. The surface manifestations are a divergent surface density of states at the Fermi level and isospins that rotate three times as they circle the origin in momentum space. We propose that Heusler alloys with band inversion are candidate materials to harbor the novel topological superconductivity.

Eight types of symmetrically distinct vectorlike physical quantities.

The Letter draws the attention to the spatiotemporal symmetry of various vectorlike physical quantities. The symmetry is specified by their invariance under the action of symmetry operations of the nonrelativistic space-time rotation group O(3)×(1,1') = O'(3), where 1' is a time-reversal operation, the symbol × stands for the group direct product, and O(3) is a group of proper and improper rotations. It is argued that along with the canonical polar vector, there are another seven symmetrically distinct classes of stationary physical quantities, which can be--and often are--denoted as standard three-component vectors, even though they do not transform as a static polar vector under all operations of O'(3). The octet of symmetrically distinct "directional quantities" can be exemplified by two kinds of polar vectors (electric dipole moment P and magnetic toroidal moment T), two kinds of axial vectors (magnetization M and electric toroidal moment G), two kinds of chiral "bidirectors" C and F (associated with the so-called true and false chirality, respectively) and still another two bidirectors N and L, achiral ones, transforming as the nematic liquid crystal order parameter and as the antiferromagnetic order parameter of the hematite crystal α-Fe(2)O(3), respectively.

Combined atomic ordering in the A and B sublattices of perovskite structure

The simultaneous ordering of cations in the A and B sublattices of cubic perovskite structure has been investigated by the group-theoretical method. It is found that 147 ordered phases may exist. Among them, there may be 121 phases with simultaneous cation ordering in the 1(a)- and 1(c) positions of perovskite structure. It is shown that 53 phases are described by improper order parameters related to cation ordering, the formation of 56 phases is related to improper rotations of octahedra, and 15 phases are improper ferroelectrics. Calculated structures of some types of ordered low-symmetry perovskite modifications are presented.

Algorithms for computer detection of symmetry elements in molecular systems

Simple procedures for the location of proper and improper rotations and reflexion planes are presented. The search is performed with a molecule divided into subsets of symmetrically equivalent atoms (SEA) which are analyzed separately as if they were a single molecule. This approach is advantageous in many aspects. For instance, in those molecules that are symmetric rotors, the number of atoms and the inertia tensor of the SEA provide one straight way to find proper rotations of any order. The algorithms are invariant to the molecular orientation and their computational cost is low, because the main information required to find symmetry elements is interatomic distances and the principal moments of the SEA. For example, our Fortran implementation, running on a single processor, took only a few seconds to locate all 120 symmetry operations of the large and highly symmetrical fullerene C720, belonging to the Ih point group. Finally, we show how the interatomic distances matrix of a slightly unsymmetrical molecule is used to symmetrize its geometry.

Impropers-wave symmetry of the electronic pairing in iron-based superconductors by first-principles calculations

By means of space-group symmetry arguments, we argue that the electronic pairing in iron-based high temperature superconductors shows a structure which is a linear combination of planar s-wave and d-wave symmetry channels, both preserving the 3-dimensional A_1g irreducible representation of the corresponding crystal point-group. We demonstrate that the s- and d-wave channels are determined by the parity under reflection of the electronic orbitals through the iron planes, and by improper rotations around the iron sites. We provide evidence of these general properties by performing accurate quantum Monte Carlo ab-initio calculations of the pairing function, for a FeSe lattice with tetragonal experimental geometry at ambient pressure. We find that this picture survives even in the FeSe under pressure and at low temperatures, when the tetragonal point-group symmetry is slightly broken. In order to achieve a higher resolution in momentum space we introduce a BCS model that faithfully describes our QMC variational pairing function on the simulated 4x4 FeSe unit cell. This allows us to provide a k-resolved image of the pairing function, and show that non-isotropic contributions in the BCS gap function are related to the improper s-wave symmetry. Our theory can rationalize and explain a series of contradictory experimental findings, such as the observation of twofold symmetry in the FeSe superconducting phase, the anomalous drop of T_c with Co-impurity in LaFeAsO_(1-x)F_x, the s-to-d-wave gap transition in BaFe_2As_2 under K doping, and the nodes appearing in the LiFeAs superconducting gap upon P isovalent substitution.

Symmetry and group theory throughout physics

Symmetry and group theory throughout physics

Geometry of Entanglement Witnesses for Two Qutrits

We characterize a convex subset of entanglement witnesses for two qutrits. Equivalently, we provide a characterization of the set of positive maps in the matrix algebra of 3 × 3 complex matrices. It turns out that boundary of this set displays elegant representation in terms of SO(2) rotations. We conjecture that maps parametrized by rotations are optimal, i.e. they provide the strongest tool for detecting quantum entanglement. As a byproduct we found a new class of decomposable entanglement witnesses parametrized by improper rotations from the orthogonal group O(2).

THE GEOMETRIC PICTURE OF GENERALIZED-CP AND HIGGS-FAMILY TRANSFORMATIONS IN THE TWO-HIGGS-DOUBLET MODEL

In the two-Higgs-doublet model (THDM), generalized-CP transformations ([Formula: see text] where X is unitary) and unitary Higgs-family transformations (φi→Uijφj) have recently been examined in a series of papers. In terms of gauge-invariant bilinear functions of the Higgs fields φi, the Higgs-family transformations and the generalized-CP transformations possess a simple geometric description. Namely, these transformations correspond in the space of scalar-field bilinears to proper and improper rotations, respectively. In this formalism, recent results relating generalized CP transformations with Higgs-family transformations have a clear geometric interpretation. We will review what is known regarding THDM symmetries, as well as derive new results concerning those symmetries, namely how they can be interpreted geometrically as applications of several CP transformations.

A Fast Inverse Kinematics Algorithm for Joint Animation

The cyclic coordinate descent(CCD) is a well-known algorithm used for inverse kinematics solutions in multi-joint chains. CCD algorithm can be easily implemented, but it can take a series of iterations before converging to a solution and also generate improper joint rotations. This paper presents a novel Target Triangle algorithm that can fast decides orientation and angle of joint rotation, and eliminates problems associated with improper and large angle rotations. Experimental results are presented to show the performance benefits of the proposed algorithm over CCD methods.

A robust inverse kinematics algorithm for animating a joint chain

The cyclic coordinate descent (CCD) is a well-known algorithm used for inverse kinematics solutions in applications involving joint chains and moving targets. Even though a CCD algorithm can be easily implemented, it can take a series of iterations before converging to a solution and also generate undesirable joint rotations. This paper presents a novel single-pass algorithm that is fast and eliminates problems associated with improper and large angle rotations. Experimental results are presented to show the performance benefits of the proposed algorithm over CCD and the 'triangulation' methods, using different types of cost functions.