Showing papers on "Symmetry (geometry) published in 1972"

Group analysis of domains and domain pairs

Abstract: Basic symmetry properties of transformation twins and of ferroelectric or ferromagnetic domains are examined in terms of the abstract group theory. It is shown that the crystallographical relations between domains (twin components) and between domain pairs can be deduced from the decomposition of the symmetry group of the high symmetry phase into the left and double cosets of the group of the low symmetry phase. Expressions are derived for the numbers of proper and improper domains, for the number of crystallographically equivalent low symmetry phases, and for the number of crystallographically non-equivalent domain pairs. A classification of domain pairs according to their symmetry is proposed. The domain structure of the monoclinic phase in WO3 and the Dauphine twinning in quartz are analysed as illustrative examples.

Cayley maps and symmetrical maps

Abstract: In this paper we shall show how combinatorial methods can be applied to the study of maps on orientable surfaces. Our main concern is with maps which possess a certain kind of symmetry, called vertex-transitivity. We show how an extension of the well-known method of Cayley can be used to construct such maps, and we give conditions which suffice for the automorphism groups of these maps to have non trivial vertex-stabilizers. Finally, we investigate the special case when the skeleton of the map is a complete graph; a classical theorem of Frobenius then implies that all vertex-transitive maps are given by our extension of Cayley's construction.

Internal symmetry groups of non-rigid molecules

Abstract: Hougen has established, for quasi-rigid molecules, the relationship between permutationinversions acting on the molecular Hamiltonians written in Cartesian co-ordinates and permutation-rotations (perrotations) of symmetry acting on nuclear equilibrium configurations. We extend these relations to the case of non-rigid molecules. For this, we introduce kinetic perrotations which act on nuclear equilibrium configurations in the same way as do Altmann's isodynamic operators. We show that isodynamic operators do not always form a group. Moreover, their action cannot be extended to the electrons. They cannot be used for the classification of molecular wave functions. This classification is achieved by using the group of Longuet-Higgins and the group of the corresponding feasible perrotations.

Description of Orientation Distributions of Cubic Crystals by Means of 3-D Rotation Coordinates

Abstract: A method has been described to represent orientations and orientation distributions of cubic crystals by means of a 3-dimensional orientation space which is formed by the repeatedly discussed rotation coordinates (axis and angle of rotation). Special emphasis has been given to the problem of multi-valency of the representation due to the cubic symmetry, to the description of scattering around an ideal orientation and to the numerical evaluation of these orientation coordinates by means of rotation matrices.

Trigonal symmetry in electron diffraction patterns from faulted graphite

Abstract: Electron-diffraction patterns both with hexagonal and trigonal symmetry were obtained from graphite. The trigonal patterns are shown to arise from a stacking fault on the basal plane. This observation is a consequence of the dependence of the pattern symmetry on the symmetry of the whole crystal, not only the symmetry of the unit cell.

Relativistic dynamical symmetries and dilatations

Abstract: Characterizing the state of a relativistic particle by a pair (xμ,ξμ) of 4‐vectors, we are led, in a natural way, to a group H5 of canonical transformations which includes the Poincare group and dilatations. The structure of the group and its induced irreducible unitary representations are explored. It is shown that H5 has a semisimple noncompact subgroup which permits a systematic treatment of exact and of broken dilatation symmetry. The relevance of these ideas to scale dimension and to a new symmetry, scale conjugation, is discussed. As an application, a mass formula is derived from broken dilatation symmetry.

Symmetry of cubical and general polyominoes

Abstract: Publisher Summary This chapter discusses the symmetry of cubical and general polyominoes. The symmetry possessed by the square polynominoes is one of the 8 possible types which are cataloged and an observation is made about symmetry, which is of general configurations and in particularly to cubical polynominoes. It can be assured by hand construction of models of each symmetry type, for cubical polynominoes. However, the converse for one-dimensioanl polymoninoes is false, and there are two symmetry types I (no symmetry) and G (reflection in the center point). Polyominoes consist of connected segments of integral length, all of which are clearly of type G. In addition, there is no distinction among various possible positions of the center relative to the cells, where the center of a polyomino with given symmetry is the intersection of the centers of its symmetries.

Permutation symmetry and the n--electron problem.

Abstract: The techniques of tensor algebra customarily applied to exploit spatial symmetry are applied to exploit permutational symmetry of the $N$-electron problem. In the approximation of no spin-orbit coupling, the results are nontrivial and give a further reduction of what is normally regarded as the reduced matrix element with respect to spatial symmetry alone. The required $3\ensuremath{-}j$ coefficients of the permutation group are evaluated in an appendix so that intermediate group-theoretical indices that have no direct physical significance are eliminated from the formulation. The spin integral for any operator can always be reduced to known integrals of the fundamental Pauli operators. Thus all matrix elements can be reduced to a corresponding spin-free form with known weighting coefficients. An explicit expression is given for the matrix element of an operator suitable for evaluating spin-own-orbit coupling or spin density at the nucleus. A recursion relation for the Clebsch-Gordan coefficients of bipartition representations of ${S}_{N}$ in terms of its subgroups and the $9\ensuremath{-}j$ sympols of $\mathrm{SU}(2)$ is developed in the appendix. For one of the representations being the totally symmetric representation, the Clebsch-Gordan coefficient is known and the recursion relation (the group-orthogonality relation in this case) can be considered as giving nontrivial sum rules on the $9\ensuremath{-}j$ symbols of $\mathrm{SU}(2)$.

Irreducible tensor operators for finite groups

Abstract: Normalized tensor operators for a finite group G are defined by means of coefficients U which formalize the descent in symmetry from R3 to G. The properties of these coefficients are demonstrated and tables given. Some examples of application show their use and utility.

Controllability and observability in linear time-variable networks with arbitrary symmetry groups

Abstract: This paper presents a unified treatment of linear time-variable networks displaying arbitrary geometrical symmetries by incorporating group theory into an analysis scheme. Symmetric networks have their elements arranged so that certain permutations of the network edges result in a configuration which is identical with the original. These permutations lead to a group of monomial matrices which are shown to commute with the network A-matrix and the state transition matrix of the normal form equation. The representation theory of groups facilitates the study of those network properties which are determined solely by symmetry. By using group theory, a simple arithmetic condition is derived which, when satisfied, implies that the network is noncontrollable or nonobservable because of symmetry alone. The results allow the determination by inspection of linear combinations of the original state variables which result in noncontrollable variables. It is shown that networks displaying axial point group symmetry are generally only weakly controllable.

The method of ascent in symmetry. I. Theory and tables

Abstract: Supergroup tables are presented whereby a representation of a subgroup can be correlated with those representations of the supergroup which are obtained on ascent in symmetry. The method of derivation is explained and various orientations of the subgroup with respect to the supergroup considered. The tables also include the correlations between the double-valued representations of the corresponding double groups.

On the diffraction enhancement of symmetry

Abstract: Diffraction symmetry is in general specified by the point group of a crystal. However, there are some exceptional cases in which the diffraction symmetry becomes, other than as a result of Friedel's law, higher than the point-group symmetry (diffraction enchancement of symmetry). Using a general expression for the square of the structure amplitude, the necessary conditions for the diffraction enhancement have been systematically investigated for the following four kinds of structures: (1) crystals composed of essentially the same substructures, (2) different substructures which have the same symmetry, (3) substructures with the same point group and different space groups, and (4) substructures with isomorphic point groups. It has been shown that symmetry enhancement like 4/m, 4/mmm, {\bar 3}m1 ({\bar 3}1m), 6/m or 6/mmm does occur in addition to 2/m or mmm as suggested by Sadanaga & Takeda [Acta Cryst. (1968) B24, 144] and by Marumo & Saito [Acta Cryst. (1972) B28, 867].

Groups of linear operators defined by group characters

Abstract: Some of the recent work on invariance questions can be re- garded as follows: Characterize those linear operators on Hom(V, V) which preserve the character of a given representation of the full linear group. In this paper, for certain rational characters, necessary and sufficient conditions are described that ensure that the set of all such operators forms a group S. The structure of 2 is also determined. The proofs depend on recent results concerning derivations on symmetry classes of tensors. 1. Statements. Let G be any subgroup of the full linear group GL (n, C) over the complex numbers, and let U denote the linear closure of G in the total

The antisymmetrized squares and the symmetrized cubes of the irreducible corepresentations of the magnetic point groups

Abstract: The application of Landau's theory of continuous phase transitions to transitions between two different magnetically ordered phases with the same crystallographic structure involves the use of the symmetrized cubes and the antisymmetrized squares of the corepresentations of the group of the phase that has the higher symmetry. These symmetrized and antisymmetrized products are evaluated using a method described previously by Cracknell (1971), the results are tabulated.

Analysis of the Structure of Complex Proteins by Electron Microscopy

Abstract: Most multisubunit proteins and simple viruses are self-assembling systems, in that their structure is completely specified by that of their subunits. On thermodynamic grounds we would expect such limited structures to show point group symmetry, since this ensures that the maximum number of links of a given type [1], in this case of the most stable type, are formed. However, since proteins are not completely rigid structures and can undergo mutual distortions when they interact it would not be surprising if they sometimes deviate from the simple rules. Sufficient structural information is now available to provide an approximate estimate of the frequency of different types of symmetry and of deviations from them. It is of particular interest to have some idea of the frequency of such deviations since this will tell us how effectively we can apply arguments based on symmetry in our attempts to determine the structure of multisubunit proteins.