Symmetry Relationships Research Articles

Observed wurtzite derivatives and related dipolar tetrahedral structures

Eight lower-symmetry ordered derivatives of the wurtzite type and an additional six related dipolar tetrahedral structure types were observed. All these structures are based on hexagonal closest packing of anions. Their mutual symmetry relationships can be summarized in a Bärnighausen diagram. Dipolar tetrahedral structures are characterized by centrosymmetric space groups, thus resulting in a dipolar distribution of the coordination tetrahedra. Dipolar tetrahedral structures often exhibit disorder and random statistical occupations and several good ionic conductors are found among them. The wurtzite-type derivatives are characterized by polar space groups and all coordination tetrahedra in them are pointing in the same direction. Some simple, geometrically possible wurtzite derivatives have never been observed. Their absence can be rationalized on the basis of a poor geometrical fit between tetrahedral chains as demonstrated by computer simulations of their hypothetical structures. The ratio ϵ (quotient of the thickness of one hexagonal layer in [001] over one measure of the mean diameter of one atom within this layer) is useful for ordering the wurtzite derivatives on the basis of their chemical composition, since ϵ increases from the oxides, over the nitrides to the sulfides. The ϵ ratio also distinguishes clearly between the oxidic wurtzite types (0.79) and the tetrahedral rutile types of β-BeO derivatives (0.86), while the dipolar tetrahedral types have ϵ values around 0.82, that is, close to the value expected for hexagonal closest packing.

Cross-Bispectrum Computation and Variance Estimation

A method for the estimation of cross-bispectra of discrete real time series is developed. The asymptotic variance properties of the bispectrum are reviewed, and a method for the direct estimation of bispectral variance is given. The symmetry properties are described which minimize the computations necessary to obtain a complete estimate of the cross-bispectrum in the right-half-plane. A procedure is given for computing the cross-bispectrum by subdividing the domain into rectangular averaging regions which help reduce the variance of the estimates and allow easy application of the symmetry relationships to minimize the computational effort. As an example of the procedure, the cross-bispectrum of a numerically generated, exponentially distributed time series is computed and compared with theory.

Connections between finger print patterns and the form of terminal phalanges

The present paper resumes the theme taken up by Bonnevie in her studies of the embryonic development of dermatoglyphic systems. Relicts of the embryonic finger balls remain in later life (Fleischhacker) and influence the symmetry relationships of the terminal phalanges. Attention was, therefore, focussed on the connection between the asymmetry of terminal phalanges and that of the patterns. All loops and whorls with a different ridge count on the radial and on the ulnar side are asymmetrical. In order to determine the asymmetry of the terminal phalanges, casts were made of the fingers of 200 schoolchildren. Then, contour drawings were laid into various planes using the phalangograph (constructed on the same lines as the perigraph), and the plane of intersection of the two sides of the symmetry axis was determined planimetrically. A high correlation of r = 0·954 was established between the degree of asymmetry of the patterns and that of the terminal phalanges (cf. Figure 3).

Uniform compounds of uniform polyhedra

AbstractThe work of Coxeter, Longuet-Higgins and Miller (1953) and of Skilling (1975) is extended to give a complete list of uniform compounds of uniform polyhedra. Symmetry relationships between polyhedra and their compounds are discussed.

A Description of the Basic Mechanisms of Electrical Excitation Techniques for Simulating System-Generated EMP

This paper extends previous studies of electrical excitation techniques for SGEMP simulation by considering simulation in terms of basic electromagnetic response parameters. Thus, fundamental properties of Maxwell's equations and basic symmetry relationships are emphasized rather than detailed computer calculations or specific experimental data.

Order-of-scattering of partially polarized radiation in inhomogeneous, anisotropically scattering atmospheres

Reciprocity and symmetry relationships, representing local invariants for the scattering phase-matrix, are derived for twelve cases of particle assemblies studied by van de Hulst (1957) including situations of scattering in an arbitrary direction, in the near forward and near backward directions. These relations are used to generate corresponding relations representing global invariants for the scattering and transmission matrices of atmospheres consisting of such assemblies. The latter relations are obtained from the matrix integro-differential equations for scattering and transmission; they apply to single scattering, any finite order of scattering, and after an arbitrary cumulation of scattering orders (finite or infinite). Our results are summarized in Tables I and II for general inhomogeneous atmospheres and for particular inhomogeneous atmospheres that are symmetrical with respect to their central level. The latter case includes homogeneous atmospheres as a special case. The largest set of local relations obtained contains three independent relations (called universal, reversal, exchange) which can further be combined to yield four additional dependent relations. This circumstance happens in three out of the above twelve cases. In the remaining cases fewer relations (both independent and dependent) remain valid. Likewise, a maximal set of three independent global relations is obtained for general inhomogeneous atmospheres; they too can be linearly combined to yield seven other dependent relations. For the symmetrically inhomogeneous atmospheres, three independent and seven dependent additional relations are obtained. On the basis of these tables, it becomes a trivial matter to provide the local and global invariants (both the independent and the dependent relations) for any assembly of particles and atmospheric inhomogeneity. A mixture of Rayleigh-Cabannes scattering by anisotropic molecules or extremely small particles and Mie scattering by large isotropic particles is considered for illustration. Lastly, the group properties of these invariants are studied.

Ice Fabrics and Petrography, Meserve Glacier, Antarctica

Results of petrographic and fabric analysis of fine-grained cold ice from the tongue of Meserve Glacier, Antarctica, are described. Most of the basal ice is remarkably uniform in texture and shows an optic-axis fabric with a single strong maximum, which is consistent with the steady-state conditions of flow. Within 0.5 m of the ice–rock interface, irregularities in the bed cause flow perturbations which are correlated with recrystallization and changes in fabric of the ice. Optic-axis fabrics in the basal ice show close symmetry relationships with dimensional fabric and deformation symmetry. Grain-size of the ice increases towards the surface of the glacier and the single maximum of the optic-axis fabric undergoes a rotation about the flow vector. In the near surface, where strain-rates are relatively much lower, the optic-axis fabric symmetry is not closely related to either deformation symmetry or the dimensional fabric. Syntectonic recrystallization of ice throughout the glacier tongue characteristically produces a strong single-maximum fabric, the orientation of which in relation to the stress field is apparently determined by stress level. Under steady-state conditions of flow, the strength of the maximum also appears to be a function of stress level.

Ice Fabrics and Petrography, Meserve Glacier, Antarctica

Results of petrographic and fabric analysis of fine-grained cold ice from the tongue of Meserve Glacier, Antarctica, are described. Most of the basal ice is remarkably uniform in texture and shows an optic-axis fabric with a single strong maximum, which is consistent with the steady-state conditions of flow. Within 0.5 m of the ice–rock interface, irregularities in the bed cause flow perturbations which are correlated with recrystallization and changes in fabric of the ice. Optic-axis fabrics in the basal ice show close symmetry relationships with dimensional fabric and deformation symmetry. Grain-size of the ice increases towards the surface of the glacier and the single maximum of the optic-axis fabric undergoes a rotation about the flow vector. In the near surface, where strain-rates are relatively much lower, the optic-axis fabric symmetry is not closely related to either deformation symmetry or the dimensional fabric. Syntectonic recrystallization of ice throughout the glacier tongue characteristically produces a strong single-maximum fabric, the orientation of which in relation to the stress field is apparently determined by stress level. Under steady-state conditions of flow, the strength of the maximum also appears to be a function of stress level.

Morphogenetic experiments in facial asymmetry: the nasal cavity.

AbstractAn experiment was designed to test the response of the nasal cavity and associated structures to maxillary deformity. Forty young M. mulatta were surgically produced in 20 animals, and the small maxillary segments moved medialward. Intrapair observation tests were applied to selected measurements and indices of symmetry relationships. Deformity of the surgically undisturbed nasal septum occurred in response to the maxillary deformation. The lateral walls were moved medially with the maxilla, but in six months symmetry relationships were similar to those found in the control animals. The lateral walls of the nasal cavity appeared to be relatively independent of the shape and position of the tooth carrying part of the maxilla. The development and use of primate models can contribute to understanding the extent of the adaptational response systems in facial morphogenesis.

Magnetic Exchange in Transition Metal Complexes. VII. Spin Coupling in μ4-Oxohexa-μ-halotetrakis [(Triphenylphosphine Oxide or Pyridine) Copper(II)]: Evidence for Antisymmetric Exchange

Magnetic susceptibility measurements are reported for a group of tetrameric Cu(II) complexes Cu4OX6L4 with X = Cl, L = (C6H5)3PO and X = Br, L = (C6H5)3PO or C5H5N. All samples show an effective magnetic moment μeff per copper ion which exhibits a maximum in the 40–60°K temperature range and which, in general, cannot be quantitatively accounted for by simple Heisenberg theory. The results contrast with those observed for the closely related anionic cluster [Cu4OCl10]4− which exhibits a monotonic variation of μeff with temperature and is compatible with simple Heisenberg theory. It is suggested that the difference may be due to a different relative arrangement of 3d Cu(II) energy levels in the anionic cluster, leaving an orbital singlet state lowest in the [Cu4OCl10]4− environment but an orbitally degenerate level lowest in the other complexes. To test the hypothesis the copper-pair exchange Hamiltonian is derived for the orbitally degenerate case and shown, for the tetramer environment, to contain a large antisymmetric component. Using this orbitally degenerate spin-pair Hamiltonian, the statistical problem for a tetramer of interacting spins is solved to give an effective magnetic moment per copper ion as a function of temperature. The resulting theory gives excellent agreement with the experimental results for the Cu4OX6L4 complexes. In addition, the required values for the exchange parameters within the theory are physically realistic and may be qualitatively understood in terms of orbital symmetry relationships.

Better perception of geometric figures through folding and cutting

Folding and cutting are very popular activities in the elementary school because children love to cut and create things. A book by Johnson1 has clearly shown how paper folding can be used as a concrete approach for understanding geometric concepts, line and angle relationships, symmetry, and circle relationships among other things. For children, however, these relationships can be made more concrete through folding and cutting activities.

Symmetry Relationships for Scattering of Polarized Light in a Slab of Randomly Oriented Particles

Abstract In this paper we discuss symmetry properties of the matrices describing the reflection and transmission of polarized radiation by a slab of randomly oriented particles. A complete set of such symmetry relations valid in the common case in which there is no birefringence or dichroism is given. The derivation proceeds via 1) the symmetry properties of the phase matrix describing the scattering in a volume element and 2) the symmetry properties of the reflection and transmission matrices based on single scattering only. Birefringence and dichroism may occur if the particles do not have a plane of symmetry. The study of symmetry relations for this case is not carried beyond the stage of the phase matrix. Possible applications and some errors in the literature are pointed out.

Correction of Imperfectly Tuned Wide-Line NMR Signals

During the course of investigations at this laboratory of native and modified cotton celluloses by means of wide-line NMR spectroscopy, it is frequently necessary to record the proton signal of dry cotton cellulose. The peak-to-peak line width of this signal, as measured on the first derivative of the absorption curve, which is the signal recorded, is about 12 G. The instrument employed in this work is a Varian VF-16 B NMR spectrometer. The normal mode of tuning the instrument is to display the input signal on the oscilloscope, tune out both the V-mode and U-mode signals by means of the probe paddles, and reintroduce leakage of the signal desired, in this case the V-mode. The method of detection of the signal used in the instrument results in the first derivative of the absorption being presented on the recorder when tuned to the V-mode. Alternatively, tuning to the U-mode results in the presentation of the first derivative of the dispersion being recorded. If one assumes that the V-mode and U-mode signal shapes may be adequately described as the derivatives of curves with the forms of the Bloch susceptibilities, then the following observations concerning the symmetry of the signals may be made. (Many wide-line NMR signals may not possess these properties, owing to chemical shifts or other phenomena. However, for a number of spectra frequently investigated, such as proton signals from polymeric materials, these symmetry relationships appear to be generally applicable.) The first derivative of the absorption g'(h) is ungerade, i.e., g'(−h) = − g'(h). The first derivative of the dispersion f' (h) is gerade, f'(h) = f' (−h). Here h denotes the deviation of the magnetic field from the resonance peak value. Unfortunately, the broad line width of dry cotton permits only a small segment of the input signal to be displayed on the oscilloscope and consequently the technique outlined above cannot be employed in tuning the signal. One may tune the instrument with some convenient narrow signal such as doped water and then insert the cotton sample. The wide-line signal will then normally be very close to being tuned but will have some of the U-mode signal present. Since the signal strength is normally small it is necessary to scan such signals slowly. One scan may take from 10 to 100 min. Thus, the final tuning of the signal, which can be done only by reference to the recorder presentation, may be tedious and time consuming. The purpose of this communication is to outline a method of correcting an NMR signal which may have a component of the undesirable U-mode signal present.

Orbital symmetry relationships for thermal and photochemical concerted cycloadditions to the benzene ring

Orbital symmetry analysis of various modes of concerted cycloaddition to the benzene ring, including dimerisation, indicates that the Woodward–Hoffmann rules apply, when appropriately viewed, to the thermal reactions, but not to the photochemical ones: the derived relationships accord well with existing experimental observations, and permit various predictions.

Structure of viruses of the papilloma-polyoma type IV. Analysis of tilting experiments in the electron microscope

Earlier papers in this series on the electron microscopy of negatively-stained human and rabbit papilloma viruses have shown both particles to be composed of 72 morphological units arranged on the T = 7 icosahedral surface lattice. This surface lattice is skew and can exist in either right- or left-handed forms: the hand of the human virus is dextro ( T = 7 d) and that of the rabbit virus is laevo ( T = 7 l). The absolute hand of the structure was determined in each case from images of particles which were stained dominantly on one side, but these “one-side” images form only a small proportion of the total. This paper presents the results of experiments in which “two-side” electron microscope images of the same virus particles obtained before and after tilting the supporting grid are compared with simulated images, i.e. superposition patterns from an appropriate model tilted through the same angle. The electron microscope images used were taken from special fields of view—close-packed arrays of particles in a sheet of negative stain—in which the particle structure is well preserved. The superposition patterns were computed and photographed from a display screen, the parameters of the model being varied until the detail in certain distinctive patterns agreed best with that in the corresponding electron microscope images (Plate I). Once the best model was obtained, a gallery of superposition patterns was calculated as a function of the spherical co-ordinates of the axis of view (e.g. Plate IV). When two superposition patterns are selected from the gallery which individually give the best fit for the pair of electron microscope images obtained from the same virus particle before and after tilting the grid, the axis and the angle of rotation to get from one pattern to the other, is, within experimental error, exactly the same as the axis and angle of tilt used in the experiment (Plate V). Moreover, the observed sense of rotation is consistent only with a dextro structure for the human virus and the laevo structure for the rabbit virus. There is very good correlation in detail between the electron microscope images and the simulated images computed for the model. This correspondence extends right to the periphery of the computed superposition patterns (Plate VI) and shows the faint outermost detail in the images to be meaningful, attesting to the faithfulness with which the negative stain preserves and outlines the structure. In the Appendix, the symmetry relationships between the superposition patterns obtained along different axes of view are discussed with the aid of a stereogram.

An algebra for the point groups

A concept of composite numbers to denote coordinates of points in a cartesian frame of reference is developed in order to describe certain symmetry relationships between points in a crystal lattice. The laws of an algebra for use in the study of the crystallographic point groups are stated and are established on the basis of conventional matrix algebra.

Symmetry relationships for the rotational collective model

Davydov's sum rule for the energy levels of angular momenta two and three of a hydrodynamic rotor is shown to be a special case of a more general sum rule which is valid for any quantal rotor. This sum rule and other energy relationships which may be useful are shown to follow from symmetry considerations. Deviations from the simple energy relationships which arise when one considers the rotation-vibration model of even nuclei are also discussed.

A General Theory of Passive Linear Electroacoustic Transducers and the Electroacoustic Reciprocity Theorem. I

A theory of the operation of passive linear electroacoustic transducers is developed on the basis of the most general linear equations (in this case, integral equations) relating the pressure and normal velocity at each point on the transducer surface and the voltage and current at the transducer's electrical terminals. These, together with the appropriate solutions of the wave equation expressed through the use of Green's functions for the medium in which the transducer is immersed and the equations defining the electrical termination of the transducer, completely characterize the behavior of the transducer and allow explicit calculation of such quantities as impedances, responses, etc., in terms of four parameters entering the fundamental equations. On the basis of this theory, a proof of the reciprocity theorem for electroacoustic transducers relating their speaker and microphone responses is presented embodying the conditions necessary for its validity. These conditions are essentially the existence of certain symmetry relationships among the transducer parameters. When these symmetry relationships may be expected to hold is to be discussed in Part II of this paper to appear later. Some applications of the theory are presented and others are outlined.