Showing papers on "Symmetry (geometry) published in 1968"
TL;DR: In this paper, it is shown that to produce optical rotation in a chromophore, a potential function must have the symmetry properties of a pseudoscalar in the symmetry group of the unperturbed chromophores.
Abstract: It is shown that to produce optical rotation in a chromophore, a potential function must have the symmetry properties of a pseudoscalar in the symmetry group of the unperturbed chromophore It is thus possible to establish regional rules of optical rotation without recourse to particular models and assumptions of the electronic states of the chromophore Formulas and geometric representations of the resulting regional rules are given for a number of symmetries which are encountered in molecular problems The results are not applicable to coupled oscillator models of optical rotation
354 citations
TL;DR: A class of equal-energy codes for use on the Gaussian channel is defined and investigated, and some theorems on distances between words in group codes are demonstrated.
Abstract: A class of equal-energy codes for use on the Gaussian channel is defined and investigated. Members of the class are eared group codes because of the manner in which they can be generated from a group of orthogonal matrices. Group codes possess an important symmetry property. Roughly speaking, all words in such a code are on an equal footing: each has the same error probability (under the assumptions of the usual model) and each has the same disposition of neighbors. A number of theorems about such codes are proved. A decomposition theorem shows every group code to be equivalent to a direct sum of certain basic group codes generated by real-irreducible representations of a finite group associated with the code. Some theorems on distances between words in group codes are demonstrated. The difficult problem of finding group codes with large nearest neighbor distance is discussed in detail and formulated in several ways. It is noted that linear (or group) codes for the binary channel can be regarded as very speciM cases of the group codes discussed. A definition of a group code for the Gaussian channel follows. An equal-energy code C with parameters M and n for this channel is a collection of M distinct unit n -vectors, X_{1}, X_{2}, \cdots , X_{M} say, that span a Euclidean n -space. An n \times n orthogonal matrix 0 is said to be a symmetry of C if the M vectors Y_{i} = 0X_{i}, i = 1, 2, \cdots , M are again the collection C . The set of all symmetries of C , say 0_{1}, 0_{2}, \cdots , 0_{g} , forms a group \cal{G}(C) under matrix multiplication. If \cal{G}(C) contains M elements 0_{\alpha_{1}}, 0_{\alpha_{2}}, \cdots , 0_{\alpha M} such that X_{i} = 0_{\alpha i}X_{1}, i = 1, 2, \cdots , M , then C is called a group code.
106 citations
TL;DR: In this paper, the Ss were required to rate on scales of subjective complexity and affect two sets of figures, one comprising a series of random polygons varying in number of sides from four to 160, and the second, four series of symmetrical polygons ranging in size from 10 to 90.
Abstract: The Ss were required to rate on scales of subjective complexity and affect two sets of figures, one comprising a series of random polygons varying in number of sides from four to 160, and the second, four series of symmetrical polygons varying in number of sides from 10 to 90. Symmetry was found to be a significant determinant of rated complexity and affect, but did not affect these responses in the same way as did complexity.
56 citations
TL;DR: In this paper, the first part of a work of systematic studies of molecular vibrations of various models is presented, where some conventions are proposed as to the orientations of cartesian axes and degenerate symmetry coordinates, in addition to existing (but not universally adopted) conventions.
Abstract: This is the first part of a work of systematic studies of molecular vibrations of various models. Some conventions are proposed as to the orientations of cartesian axes and degenerate symmetry coordinates, in addition to already existing (but not universally adopted) conventions. Symmetry coordinates are specified in the present part for a number of three-atomic and four-atomic molecular models. The corresponding symmetrized G and Cα matrix elements have been worked out and are available on request to one of the authors (S. J. C.).
32 citations
TL;DR: In this paper, a set of coefficients for forming linear combinations of plane waves, both without and with spin, symmetrized with respect to all the necessary sub-groups of all 73 symmorphic space groups, is presented.
Abstract: A set of tables of coefficients for forming linear combinations of plane waves, both without and with spin, symmetrized with respect to all the necessary sub-groups of all 73 symmorphic space groups, is presented here. A symmetry analysis of all the Brillouin zone symmetry points of these 73 space groups is also included and serves as an index to the coefficient tables. A discussion of the method of calculation is included as well as a review of the theory which underlies the desirability of symmetrized functions, and the theory of the projection operator, by which these coefficients were calculated. Section 4, a guide to the use of the tables, may be read independently of the rest.
31 citations
17 citations
TL;DR: The 32 crystal classes were determined 137 years ago, in the same year in which group theory was born as discussed by the authors, and the determination of the 230 space groups, by Schonflies and by Fedorov (these are the discrete subgroups of the Euclidean group which contain three noncoplanar translations) was a masterpiece of analysis.
Abstract: Symmetry and invariance considerations have long played important roles in physics. The 32 crystal classes—that is, groups of rotations in three-dimensional space all the elements of which are of the order 2, 3, 4 or 6—were determined 137 years ago, in the same year in which group theory was born. The determination of the 230 space groups, by Schonflies and by Fedorov (these are the discrete subgroups of the Euclidean group which contain three noncoplanar translations) was a masterpiece of analysis and so was the determination by Groth of the possible properties of crystals with the symmetries of these space-groups.
16 citations
16 citations
TL;DR: In this paper, the same potentials were used to compute the eigenvalues for all irreducible representations of all points of symmetry, with the exception of the exceptional error being 0·0001 ryd.
Abstract: Cellular eigenvalues have been computed for copper metal on using exactly the same potential as Burdick and they agree with the augmented plane wave method results within 0·002 ryd. Empty-lattice eigenvalues for all irreducible representations of all points of symmetry are correct (with one exception) to 0·000 02 ryd, the exceptional error being 0·0001 ryd. For general k values most errors are below 0·005 ryd. These results have been obtained both for face-centred and body-centred cubic lattices.
12 citations
TL;DR: In this article, the Eckart-Wigner theorem is generalized to include nonunitary groups, based on the connection between corepresentations of a non-unitary group and the representations of its unitary part.
Abstract: The Eckart‐Wigner theorem is generalized to include nonunitary groups. The proof is based on the connection between corepresentations of a nonunitary group and the representations of its unitary part. All possible cases of the corepresentations have been considered, and general expressions for matrix elements of operators with given symmetry have been obtained. It has been shown that the antiunitary symmetry leads, in general, to additional connections between different matrix elements.
TL;DR: In this article, a method is presented for construction of symmetry coordinates for crystals with symmorphic space groups by using projection operators for both the translation group and the rotational part of the space group.
Abstract: In this paper a method is presented for construction of symmetry coordinates for crystals with symmorphic space groups. By using projection operators for both the translation group and the rotational part of the space group, formulas for symmetry coordinates are given, which reduce the vibrational problem into irreducible blocks under both the translational group and the little group Ĝk Further, a short discussion is given for selection rules for infrared and Raman transitions and the importance of the dispersion surface in analyzing empirical spectra is pointed out.
TL;DR: In this article, the irreducible representations of the double groups and time inversion symmetry of the groups of the k-vector are discussed. But the double group representations are not defined.
Abstract: In this paper magnetic space groups are defined. Their irreducible representations are given in terms of the irreducible representations of the group of the k-vector. The irreducible representations of the double groups and time inversion symmetry are discussed.
In der vorliegenden Arbeit werden magnetische Raumgruppen definiert. Ihre irreduziblen Darstellungen werden aus den irreduziblen Darstellungen der Gruppe des k-Vektors hergeleitet. Die Darstellungen der Doppelgruppen und die Zeitumkehrsymmetrie werden diskutiert.
TL;DR: In this paper, an analytical method for indexing powder patterns of polycrystalline substances having orthorhombic symmetry is discussed, which is an extension of Neskuchaev's method for systems of intermediate symmetry.
Abstract: An analytical method for indexing powder patterns of polycrystalline substances having orthorhombic symmetry is discussed. The technique is an extension of Neskuchaev's method for systems of intermediate symmetry. The procedure is applied to the indexing of the powder pattern for NiAl3.
TL;DR: The symmetrisor as mentioned in this paper is a 4-pole rotator with the property that the characteristic curve observed to one port is a characteristic curve of the element connected to the other port transformed by symmetry relative to a prescribed line through the origin.
Abstract: This letter presents a 4-pole, called a symmetrisor, by analogy with the rotator introduced by Chua. It has the property that the characteristic curve observed to one port is the characteristic curve of the element connected to the other port transformed by a symmetry relative to a prescribed line through the origin.
TL;DR: In this article, a theoretical analysis for the molecular vibrations of the following trigonal bipyramidal models: (iti) X3Y2 of D3h symmetry, (itii) XZ2)3(YW)2 with planar (XZ 2)3 structure (itD3h) symmetry, and (itiii) distorted XZ 2,3,YW 2 of D 3 symmetry.
Journal Article•
TL;DR: In this paper, a generalized treatment for combinations of infinitesimally and finitely separated coplanar positions is presented for four, five, and six positions lying symmetrically about an axis in the reference plane.
Abstract: A generalized treatment for combinations of infinitesimally and finitely separated coplanar positions is presented for four, five, and six positions lying symmetrically about an axis in the reference plane. Symmetry of the first and second kind are shown to be distinct because of the constraint system and not due to the form of the motion specification. The solution for the Burmester points is represented by the intersection of the Bottema conics. This development is supported by an algorithm for synthesizing five multiply separated positions in coplanar motion.
TL;DR: The helical parameters of translation plus rotation need not necessarily result in an integral number of subunits per turn of the helix; the α-helix has approximately 3.6 residues per turn as mentioned in this paper.
Abstract: Helical structures illustrate in a simple way the use of symmetry to assemble objects so that every object has an identical environment. Other ways in which such equivalence may be attained include crystallization, and the organization of the objects with the symmetry of solids such as the cube and tetrahedron. Whereas the latter can provide complete equivalence of the objects, in both crystals and helices there are end effects. For long helices the contribution of the non-equivalent objects at the ends of the helix to its gross properties may be insignificant.In homopolymers, residues within the helix may be perfectly equivalent. In biological macromolecules with an irregular sequence of side chains (or bases), the formal equivalence only extends to the backbone of the molecule.The helical parameters of translation plus rotation need not necessarily result in an integral number of subunits per turn of the helix; the α-helix has approximately 3.6 residues per turn. In this respect they differ from the helical screw' rotations found in crystal symmetry in which the three-dimensional lattice imposes a restriction resulting in integral repeats.Helices may be single or multistranded, and when they are multistranded further symmetry elements may he imposed on the helix symmetry in order to relate the strands. Thus DNA has a two-fold rotation axis perpendicular to the helix axis. Similarly, all the structures proposed for multistranded homopolynucleotides contain symmetry axes parallel to the helix axis. Examples of the use of helical symmetry based on molecules of biological interest are discussed.
TL;DR: In this paper, an 8-component spinor field carries with itself a large number of 4-vector currents and invariants, the relationships between which are analysed, and the vector densities can be grouped into a number of orthogonal frames, which describe tetrad fields.
Abstract: An 8-component spinor field carries with itself a large number of 4-vector currents and invariants, the relationships between which are analysed. The vector densities can be grouped into a number of orthogonal frames, which describe tetrad fields. Two of the tetrads, related to each other by charge conjugation, connect isospin transformations directly with local space-time rotations. The main tetrad planes determine geometrical configurations of considerable symmetry. The bilinear invariants define angles and hyperbolic angles which appear directly in the rotations and Lorentz transformations connecting the tetrads.
TL;DR: In this paper, the character tables of irreducible representations of discrete symmetry point groups of general order are arranged, and the characters of the point groups are described in terms of cardinality.
Abstract: In this paper the character tables of irreducible representations of discrete symmetry point groups of general order are arranged.