Description Of Symmetry Research Articles

Symmetry breaking and ionization from symmetry equivalent inner shells and lone pairs in Xα theory

The superiority of the localized, broken symmetry description for ionization from symmetry equivalent core holes and lone pairs in Xα theory is demonstrated by calculations on N 2 (1s) and by analysis of the literature results for O 3, OF 2, Ne, CF 4 and Cr(CO) 6. The differences between the Xα, HF, and experimental values, for localized ionization, can be traced directly to the approximate treatment of shell self-repulsion in Xα. When shell self-repulsion is properly taken into account via a simple perturbation treatment, the corrected Xα ionization energies are in excellent agreement with experiment. The strange result that the Xα delocalized solution has a lower total energy than the localized solution is similarly accounted for by correcting for the exact shell self-repulsion. The error in the Xα delocalized solution is much larger than the error for the localized solution with some important implications. The concepts of “Xα Koopmans energy” and Xα relaxation energy are shown to have practical importance is assessing whether the localized or delocalized solution is physically correct.

Recognition and fuzzy description of sides and symmetries of figures by computer

Abstract The detection of sides and corners is useful and important in the shape perception of visual picture patterns. Similarly, the recognition of symmetry in pictures is helpful in reducing the storage space, in retrieving a missing portion, or in correcting the orientation of its shape. In the present paper, algorithms are found to detect the aides and symmetry of simply closed two-dimensional man-made and machine-generated outlines. A description based on the fuzzy set theoretic approach is also developed for imperfect non-geometric figures and results of the execution of the algorithms are presented.

On the description of symmetry of OD structures (III). Short symbols for OD groupoid families

AbstractTwo kinds of short symbols are proposed: 1. The short symbol of any OD groupoid‐family consists of two parts: the symbol for the point group of the groupoid of partial symmetry operations and the symbol for the layer group. No information on the parameters is given. 2. The short symbol of any OD groupoid‐family consists of those parts that correspond to the lines of the full symbol. There is, however, only one of the partial symmetry operations given, which transform a layer into an adjacent layer. These short symbols are suitable for storage of the symmetry data of OD structures in computer operated data banks.

On the definition and properties of generalized 6-j symbols

Whipple’s work on the symmetries of well-poised 7F6(1) and Saalschürtzian 4F3(1) series with unit argument is applied to study the properties of 6-j symbols generalized to any arguments. For SU2, we obtain eleven different looking 4F3(1) series which can be used. Whipple’s parameter x s provide a good description of symmetries. We obtain quite simple recurrence relations, valid for any arguments, in terms of these parameters.

On the Description of Symmetry of OD Structure (II) The Parameters

AbstractFor the relative position of layers, the parameters are studied in detail. The number of independent parameters for an OD groupoid family may be determined by means of a table. Proposals are made for an extension of the symbols of some OD groupoid families, in order to give a full set of independent parameters. Rules for the linear dependence of parameters are examined.

On the Description of Symmetry of OD Structures (I) OD Groupoid Family, Parameters, Stacking

AbstractThe relations between the groupoid of partial symmetry operations of an OD structure and the description of the symmetry of this structure by an OD groupoid family as well as parameters and the stacking of layers are discussed. Proposals are suggested for some changes in the symbols of the OD groupoid families in order to simplify and unify these symbols.

Symmetry operations for displacively modulated structures

Displacive modulation is defined as periodic distortion with an incommensurable k vector. If the phase of the distortion is t, a symmetry operation can be expressed as a normal space-group operation combined with a sign reversal and/or a shift in the variable t. Earlier results obtained by de Wolff [Acta Cryst. (1974), A30, 777-785] are reformulated in terms of these operations. The periodic functions defining the displacements of two symmetry-related atoms are shown to be related by a simple equation. Applications to published structures are given. The validity of this equation depends on a symmetry-adapted choice of the vector k. It is shown by a two-dimensional example that there are cases for which that choice of k requires the introduction of 'improper symmetry translations' with an extra t shift of ½ or ½. The ensuing Bravais lattice types are similar to those used for the description of magnetic symmetry.

Wigner's theorem on symmetries in indefinite metric spaces

The description of symmetries in indefinite metric spaces is investigated. It is shown that ray transformations preserving the modulus of an indefinite scalar product can be implemented by linear or antilinear vector transformations which are generalized unitary or antiunitary operators with respect to the indefinite scalar product. A numbero f interesting features arise since such operators need not to be bounded.

A non-Lie algebraic framework and its possible merits for symmetry descriptions

A nonassociative algebraic construction is introduced which bears a relation to a Lie algebra ℒ paralleling the relation between an associative enveloping algebra and ℒ. The key ingredient of this algebraic construction is the presence of two parameters which relate it to the enveloping algebra of ℒ. The analog of the Poincaré–Birkhoff–Witt theorem is proved for the new algebra. Possibilities of physical relevance are also considered. It is noted that, if fully developed, the mathematical framework suggested by this new algebra should be non-Lie. Subsequently, a certain scheme resulting from specific considerations connected with this (non-Lie) algebraic structure is found to bear striking resemblance to a recent phenomenological theory proposed for explaining CP violation by the K0 system. Some relevant speculations are also made in view of certain recent trends of thought in elementary particle physics. Finally, in an appendix, a Gell-Mann–Okubo-like mass formula for the new algebra is derived for an SU (3) octet.

Ligand field distortion parameters

The factorization of ligand field parameters into angular and radial contributions is examined. The angular term breaks up into a product of geometric and chemical environment tensors each of which contributes to the magnitude of the distortion parameter. The purely geometric function may be expressed as axial projections which need not represent metal-ligand bond positions. This permits direct comparison of D ndand D nhgroup properties. This projection description involves no approximations and since the operators are Normalized Spherical Harmonic Hamiltonians the derived radial integrals are all scalars of the octahedron and comparable between groups in different subduction chains. The projection factorization aids the description of intermediate symmetries. The tetragonal antiprism, the trigonal prism and the trigonal bipyramid each are capable of intermediate symmetry derived from cubic figures. In each case the Hamiltonian contains only one independent parameter. Each of these special conditions can be used as the geometric origin in distortion space diagrams. The radial function r 4 is shown to be a function of n 8. Only a small increase in the principal number n of d Orbitals is needed to account for observed values of DQ and these are justified using a United Atom model of transition metal complexes. The ratio of radial functions A 1 in distortion parameters then represents slight variations in n induced by different ligands.

The Pseudo-Symmetry of Modulated Crystal Structures

A modulated structure can be depicted as a section through a four-dimensional periodic structure. In the latter, each atom is represented by a string continuing endlessly in the overall direction (e4) of the normal to R3, R3 being the hyperplane of the section. The strings have periodic bends or densifications for displacive and substitutional modulation respectively. Formulae for structure factors can be derived from this picture with little effort. The pseudo-symmetry of modulated structures can be described conveniently in this picture. Each four-dimensional space group to which the four-dimensional structure can belong is a possible MS3 (modulated three-dimensional structure) group of pseudo-symmetry, and is called an MS3 space group. It is shown that MS3 point groups are reducible in the form Q⊕∊. 1, where 1 is the unit 2 × 2 matrix, and ∊ = ± 1. A list is presented of these 31 groups written as black-and-white or colourless groups of three-dimensional symmetry. The MS3 space groups are discussed briefly. As an example of the peculiar differentiations caused by e4 being a unique direction, the 23 MS2 space groups are listed explicitly. Finally, it is shown that MS groups are essential for the description of MS symmetry, because very often the latter cannot be represented completely and unambiguously by the normal space group of an approximate superstructure.

Angular constraints and the Melosh transformation

The Melosh transformation between current and constituent quarks is discussed when formulated on a light-like plane, as is natural for the description of collinear symmetries like SU(6) W. There is no unique definition of the transformation in this context, but further angular constraints are introduced to ensure that the states coupled by SU(6) W transformations have the correct spin properties. It is then shown that there is no transformation generated by single quark operators which satisfies these requirements, save if only terms to first order in the transverse momentum are kept. Some of the resultant inconsistencies are illustrated by a discussion of baryon magnetic moments. It is concluded that there is no satisfactory explanation of μ p /μ n = − 3 2 from this approach to the Melosh transformation. A non-field theoretic version of the free quark model which solves these angular constraints is exhibited.

Phonon symmetries determined from space-group diagrams

The representations of space groups describe the symmetry of crystal lattice vibrations. These representations are presented in a diagrammatic way easily understood by those familiar with International Tables for Crystallography. A few diagrams are considered in detail in an effort to bring out the salient points. The relevance of plane groups and Shubnikov groups is emphasized, though it is clear that these groups in themselves are not sufficient for a complete description of phonon symmetries. The diagrammatic description must be considered as complementary to the analytic approach, but unlike the latter it is not capable of a thorough description. A table is given for certain space groups which indicates to what extent the diagrams can be found useful, encompassing one-dimensional and a limited number of two-dimensional representations. Although higher dimensions involve too many problems and must be left to the analytic approach, the insight gained by drawing diagrams may prove beneficial.

Нейтронография магнитных материалов

A review of neutron diffraction of magnetic materials is presented. The topics covered include: (1) A discussion of slow neutron scattering including nuclear and magnetic effects, with examples and typical experimental arrangements. (2) The determination of magnetic moments. In this section various models for Fe4N and Mn4N is discussed, as are magnetic moments of FeNi3, Fe3Al, and magnetic moments of transition metals and magnetic moments in the paramagnetic region. (3) Magnetic structure of transition metals. In this section are presented results concerning the antiferromagnetic structure of these metals, principally chromium; a description of magnetic symmetry; and the new class of spiral structures. (4) Magnetic structure of some compounds: the perovskite group of iron compounds; the perovskite group of rare earth metals; the antiferromagnetism of transitron metal carbonates; and trifluorides of transition metals. The breakdown of Hund's rule rs discussed, as is the magnetic anisotropy of some manganese compounds. (5) Magnetic critical and small angle scattering of neutrons. A qualitative discussion of Van Hove's theory is given. Results of small-angle scattering from iron at various temperatures are presented. Distant and close magnetic interactions are briefly noted, as is the dynamics of spin correlations. (6) Scattering of neutrons by spin waves. Thismore » section includes a brief discussion of magnons; the dispersion law of spin waves; and experimental results of scattering by spin waves. (7) Magnetic scattering of conduction electrons; principally the electrical resistance of disordered magnetic structures. Spin interactions are mentioned in connection with superconductivity.« less