Showing papers on "Symmetry (geometry) published in 1969"

Scattering matrices for the Raman effect in magnetic crystals

Abstract: The symmetry of a magnetic crystal can be described by one of the black and white Shubnikov point groups containing both unitary and anti-unitary elements. The theory is reviewed that enables a tensor to be symmetry adapted to any one of the irreducible corepresentations of a non-unitary group and it is applied to the determination of scattering matrices for the Raman effect in magnetically ordered crystals. An example is given in detail for the magnetic point group of antiferromagnetic MnF2 and complete tables are presented for all the fifty-eight black and white magnetic point groups.

Symmetry requirements for the stabilization of one class of diradicals

Abstract: A TRIPLET ground state can exist for a molecule whenever there are only two electrons to be placed in a pair of degenerate or nearly degenerate orbitals. Such degeneracy may occur for both molecules which have normal Kekul6 structures (e.g. cyclobutadienel) and for formal diradicals (e.g. trimethylenemethane) . The stability of the lowest singlet state may be increased by breaking as seriously as possible the degeneracy of the nonbonding orbitals. Where geometry allows it, a very effective procedure for accomplishing this is the distortion from maximum molecular symmetry to a less symmetrical structure. This is probably what occurs in the ground state of cyclobutadiene.2 Symmetry breaking may also be accomplished by replacing carbon by hetero-atoms, or by substitution or annelation by groups with low-lying unoccupied molecular orbitals (good acceptors in the normal terminology) or high-lying occupied molecular orbitals (good donors) .3 Indiscriminate replacement or substitution may often result in simultaneous lowering or raising of both components of the degenerate pair, with no net stabilization of the singlet. For maximum symmetry-breaking effect one should perform the replacement or substitution so as to destabilize or stabilize only one level of the degenerate pair, or, better, to destabilize one and stabilize the other. The symmetry arguments for devising the best strategy are described below for cyclobu tadiene. Consider a cyclobutadiene substituted by two groups X with low-lying unoccupied n* orbitals and two groups Y with high-lying occupied n levels. There are two possible substitution patterns, (1) and (2). The degenerate orbital

Symmetries of links

Abstract: In this paper certain properties called symmetries are defined for links, and the problem of determining those links admitting a particular symmetry is attacked. The problems of symmetry are the generalization to links of the problems of amphichaerality and invertibility of knots which, because of Trotter's proof that there are noninvertible knots [13], are now fairly well under control. A link of two components is called interchangeable if it possesses a special type of symmetry, and certain invariants of interchangeability of a link are given and examined.

Symmetries of type N empty space-times possessing twisting geodesic rays

Abstract: It is shown that if the geodesic ray congruence associated with an empty space-time of type N is not hypersurface orthogonal (that is the geodesic rays are twisting) then the space-time has at most one symmetry.

DS-spaces of Robinson and Trautman: topological identifications and asymptotic symmetry

Abstract: Some topological identifications are introduced into the DS-spaces of Robinson and Trautman, and the symmetry and asymptotic symmetry groups of the resulting spaces are calculated. The latter have structures similar to the GBM group, each being the semi-direct product of a group of supertranslations by a group of classical type.

Spectral properties of higher derivations on symmetry classes of tensors

Abstract: when ƒ>> 1, and Qi(r) = 0 if p = 0, O ^ D = Pif p = 1 (Greub [2, p. 67]). In this announcement we outline an extension of this definition to higher order derivations in general symmetry classes of tensors. We also state some of the eigenvalue properties of higher derivations and show how these may be applied to some classical matrix problems. Let H be a subgroup of Sp and let % be a character of degree 1 on H. (We assume that the order of H exceeds the characteristic of P.) If U is a vector space over R and (vi, • • • , vp) is a ^-multilinear function on the Cartesian product X\\V to U, then is said to be symmetric with respect to H and x if

Magnetic symmetry and limiting groups

Abstract: Using the method of construction of the magnetic symmetry groups already developed by the authors, the magnetic symmetry groups associated with the limiting groups have been derived. The number of constants required to describe the three magnetic properties studied for each one of the derived magnetic symmetry groups is also enumerated.

Relations entre la symétrie des diagrammes de diffraction des électrons lents et celle du cristal

Abstract: The very strong absorption of low energy electrons by matter has some consequences on the symmetry of diffraction patterns. The symmetry of a pattern is that of one or several structural planes according to the depth reached by the electrons. It may differ from the symmetry of the crystal unit cell. When the symmetry of the crystal is lower than that of the structural planes, the symmetry of the pattern corresponds to that of the diffracting medium whereas if the symmetry of the crystal is higher than that of the structural planes, the symmetry of the pattern is lower than that of the diffracting medium.

A Symmetry Property of Transfer Matrices

Abstract: The transfer matrix method of vibration analysis is widely used for the calculation of the natural frequencies of turbo-generator rotors, and of propeller blades, axial compressor blades, turbine blades, and helicopter rotors. A comprehensive account of the method has been given by Pestel and Leckie. The basis of the method is to deine a state vector, which gives all the generalised displacements and forces at a given cross-section of the system. The state vector at one cross-section is then related to the state vector at a neighbouring cross-section by means of a transfer matrix. It is found that, by a suitable choice of the order and signs of the elements of the state vector, the transfer matrix can in many cases be made symmetrical about the secondary diagonal; that is the diagonal running from the top right corner to the bottom left corner of the matrix and is also often called the “cross diagonal”.

New rectangles for dynamic symmetry

Abstract: Dynamic symmetry has been applied to artistic design ever since its development in 1920 by Jay Hambidge. Unfortunately, there has been a certain amount of prejudice against the system because of its debatable association with Greek vases. As a practical tool, also, it has been hindered by lack of a sufficient number of rectangles that are dynamically divisible. This limitation is removed by the present paper, which exhibits over 80 new dynamic rectangles.

Symmetry in Chemistry

Abstract: Chemists have somehow neglected the concept of molecular symmetry, and it would be to their advantage to become more fully cognizant of it. Knowledge of the symmetry concepts in chemistry affords one a better understanding of the ever-increasing battery of tools available for the solution of structural problems. Raman, infrared, and ultraviolet spectroscopy, X-ray, electron, and neutron diffraction methods are some of the powerful tools that are based on symmetry considerations. Knowing the symmetry of a molecule one can predict the infrared or Raman spectrum, and knowing the spectrum one can arrive at the symmetry or structure of the molecule.

Symmetry and configuration in words and patterns of similar structure

Abstract: Three structural variables, class, symmetry, and configuration, were investigated tachistoscopically Identification thresholds were significantly lower for ascending configura-tional stimuli except in the case of symmetrical words An even stronger relationahip for symmetrical stimuli to have lower thresholds than asymmetrical stimuli was evident Errors towards symmetry were also found during prethreshold trials being interpreted as the general tendency towards “good form”