Tetrahedral Point Research Articles

Exotic shape symmetries around the fourfold octupole magic number N=136 : Formulation of experimental identification criteria

We employ a realistic nuclear mean-field theory using the phenomenological, Woods-Saxon Hamiltonian with newly adjusted parameters containing no parametric correlations; originally present correlations are removed employing the Monte Carlo approach. We find very large neutron shell gaps at $N=136$ for all the four octupole deformations ${\ensuremath{\alpha}}_{3\ensuremath{\mu}=0,1,2,3}$. These shell gaps generate well-pronounced double potential-energy minima in the standard multipole $({\ensuremath{\alpha}}_{20},{\ensuremath{\alpha}}_{22},{\ensuremath{\alpha}}_{3\ensuremath{\mu}},{\ensuremath{\alpha}}_{40})$ representation, often at ${\ensuremath{\alpha}}_{20}=0$, which in turn generate exotic symmetries ${C}_{2v}$, ${D}_{2d}$, ${T}_{d}$, and ${D}_{3h}$, discussed in detail. The main goal of the article is to formulate spectroscopic criteria for experimental identification. Calculations employing macroscopic-microscopic method are performed for nuclei with $Z\ensuremath{\ge}82$ and $N\ensuremath{\ge}126$ in multidimensional deformation spaces to analyze the expected exotic symmetries and octupole shape instabilities in the mass table ``northeast'' of the doubly magic $^{208}\mathrm{Pb}$ nucleus. Whereas the proton-unperturbed properties of neutron-generated octupole shell effects are illustrated in detail for exotic ${}_{Z=82}{\mathrm{Pb}}_{N>126}$ nuclei, our discussion is extended into even-even $Z>82$ nuclei approaching the less exotic $Z/N$ ratios, to encourage experiments which could identify the predicted exotic symmetries. In addition to the tetrahedral point group symmetry, ${T}_{d}$, of which experimental evidence has recently been published, we present ${D}_{2d}$ symmetry resulting from a superposition of axially symmetric quadrupole and tetrahedral symmetries and two new point group symmetries, ${D}_{3h}$ and ${C}_{2v}$, associated with the octupole ${\ensuremath{\alpha}}_{33}$ and ${\ensuremath{\alpha}}_{31}$ energy minima, respectively. The multidimensional $n>2$ deformation spaces are treated as usual by projecting the total potential energies onto the $n=2$ subspace. Using the representation theory of point groups we formulate quantum mechanical criteria for experimental identification of exotic symmetries through analysis of the specific properties of the collective rotational bands generated by the symmetries. The resulting band structures happen to be markedly distinct from the structure of the bands generated by ellipsoidal symmetry quantum rotors; those various rotational properties are discussed in detail.

Response of the Skyrmion Lattice in MnSi to Cubic Magnetocrystalline Anisotropies.

We report high-precision small-angle neutron scattering of the orientation of the Skyrmion lattice in a spherical sample of MnSi under systematic changes of the magnetic field direction. For all field directions the Skyrmion lattice may be accurately described as a triple-Q[over →] state, where the modulus |Q[over →]| is constant and the wave vectors enclose rigid angles of 120°. Along a great circle across ⟨100⟩, ⟨110⟩, and ⟨111⟩ the normal to the Skyrmion-lattice plane varies systematically by ±3° with respect to the field direction, while the in-plane alignment displays a reorientation by 15° for magnetic field along ⟨100⟩. Our observations are qualitatively and quantitatively in excellent agreement with an effective potential, which is determined by the symmetries of the tetrahedral point group T and includes contributions up to sixth order in spin-orbit coupling, providing a full account of the effect of cubic magnetocrystalline anisotropies on the Skyrmion lattice in MnSi.

Geometry of the generalized Bloch sphere for qutrits

The geometry of the generalized Bloch sphere Ω3, the state space of a qutrit, is studied. Closed form expressions for Ω3, its boundary ∂Ω3, and the set of extremals are obtained by use of an elementary observation. These expressions and analytic methods are used to classify the 28 two-sections and the 56 three-sections of Ω3 into unitary equivalence classes, completing the works of earlier authors. It is shown, in particular, that there are families of two-sections and of three-sections which are equivalent geometrically but not unitarily, a feature that does not appear to have been appreciated earlier. A family of three-sections of obese-tetrahedral shape whose symmetry corresponds to the 24-element tetrahedral point group Td is examined in detail. This symmetry is traced to the natural reduction of the adjoint representation of SU(3), the symmetry underlying Ω3, into direct sum of the two-dimensional and the two (inequivalent) three-dimensional irreducible representations of Td.

A ten node tetrahedral Cosserat Point Element (CPE) for nonlinear isotropic elastic materials

It is known that the standard full integration ten node tetrahedral element is inaccurate for thin nearly incompressible structures. Also, the commercial code ABAQUS recommends replacing this standard element (C3D10) with an undocumented patented modified element (C3D10M) for contact problems. The objective of this work is to develop a ten node tetrahedral Cosserat Point Element (CPE) for nonlinear isotropic hyperelastic materials. Hyperelastic constitutive equations for the CPE are developed by treating the element as a structure with a strain energy function that is restricted to satisfy a nonlinear form of the patch test. A number of examples are considered which demonstrate that the resulting CPE is accurate and robust, it does not exhibit the numerical stiffness for nearly incompressible materials observed for (C3D10) nor the unphysical instabilities observed for (C3D10M). Moreover, the CPE can be used for thin structures and three-dimensional bodies with a smooth transition from compressible to nearly incompressible material behavior.

The topology of homophase misorientation spaces

Homophase misorientation spaces are investigated with a focus on the effect of symmetry operations on their topology and their minimum embedding dimensions in Euclidean space. Whereas the topology of rotation space is well established and requires a minimum of five variables for a one-to-one and continuous mapping, the spaces of orientations and misorientations are quotient spaces of the rotation space and are obtained by applying various equivalence relations. The equivalence relations for orientation spaces only involve the rotational symmetries of the underlying crystals. These spaces are classified under the three-dimensional manifolds called the spherical 3-manifolds, which have a non-trivial fundamental group, are not simply connected spaces, and do not embed in three-dimensional Euclidean space. In the case of homophase misorientation spaces, however, in addition to rotational symmetry operations there is a further ‘grain exchange symmetry’, which is shown to simplify the topology considerably. In some important cases this symmetry also reduces the number of Euclidean dimensions required to embed these misorientation spaces. The homophase misorientation spaces for the dihedral point groups D 2(222), D 4(422) and D 6(622), the tetrahedral point group T(23), and the octahedral group O(432) are all found to be embeddable in only three dimensions, two dimensions less than required for rotations. Hence, these misorientation systems can be represented using three variables in a one-to-one and continuous manner.

A 5 as a Homomorphic Image of a Subgroup of Picard Group

In this article we investigate that what actions of a subgroup G 1 of Picard group yield A 5 = PSL(2, 5). We draw coset diagrams (propounded by G. Higman) for the action of G 1 on finite fields PL(F 5 n ). The projective special linear groups PSL(2, p) when p = 5 can be seen as 5-multiples of tetrahedral rotation point group. It has applications in carbon chemistry and physics as PSL(2, 5) is the rotation group of the fullerene C 60.

Molecular crystal global phase diagrams. II. Reference lattices

In the first part of this series [Keith et al. (2004). Cryst. Growth Des. 4, 1009-1012; Mettes et al. (2004). Acta Cryst. A60, 621-636], a method was developed for constructing global phase diagrams (GPDs) for molecular crystals in which crystal structure is presented as a function of intermolecular potential parameters. In that work, a face-centered-cubic center-of-mass lattice was arbitrarily adopted as a reference state. In part two of the series, experimental crystal structures composed of tetrahedral point group molecules are classified to determine what fraction of structures are amenable to inclusion in the GPDs and the number of reference lattices necessary to span the observed structures. It is found that 60% of crystal structures composed of molecules with T(d) point-group symmetry are amenable and that eight reference lattices are sufficient to span the observed structures. Similar results are expected for other cubic point groups.

COLLECTIVE HAMILTONIANS WITH TETRAHEDRAL SYMMETRY: FORMALISM AND GENERAL FEATURES

Collective octupole variables invariant under tetrahedral point group transformations are used to construct tetrahedrally invariant collective Hamiltonians. The so-called intrinsic groups are considered as a tool for symmetry analysis of such Hamiltonians in intrinsic frames of a nuclei.

Nature of the hyper-Raman active vibrations of lithium borate glasses

Hyper-Raman spectra of two lithium borate glasses,4B2O3:Li2O and 2B2O3:Li2O, are compared to those of pure boron oxide glass,v-B2O3. A mode analysis is performed using a structural model based on thesymmetry of the elementary structural units (ESUs) constituting theglasses. Most spectral components arise from internal vibrations ofBO3 triangles,B3O3 boroxolrings, and BO4 tetrahedra. In particular, a mode associated with stretching motions ofBO4 units can be assignedto a vibration of F2 symmetry in the Td tetrahedral point group. The hyper-Raman scattering intensity of itstransverse optic component appears to be proportional to the number ofBO4 units in the glass. The boson peak observed in hyper-Raman scattering arises from externalmodes of rigid ESUs which correspond to librational motions coupled to their translations. Thescattering strength of these modes strongly decreases with increasing Li concentration. In purev-B2O3, external modes of boroxols presumably dominate this scattering. The decrease in theboroxol concentration in lithium borate glasses correlates with the apparent hardening ofthe boson peak.

Isotropic frequency selective surfaces made of cubic resonators

Isotropic frequency selective surface (FSS) made of cubic arrangements of split ring resonators (SRRs) is proposed and analyzed. For this purpose, a suitable isotropic modification of the SRR was used in the design of a cubic unit element invariant under the tetrahedral point symmetry group. It was experimentally demonstrated that the transmission through such a FSS is angle and polarization independent. For comparison, another FSS, whose unit elements do not satisfy necessary symmetries, was measured, showing clearly anisotropic behavior. We feel then that symmetries play an important role. Potential device applications are envisioned for antenna technology at microwave and terahertz frequencies.

Optical absorption spectra of substitutional Co 2+ ions in Mg x Cd 1– x Se alloys

Optical absorption spectra of substitutional Co2+ ions in Mgx Cd1–x Se alloys were investigated in the composition region of 0.0 ≤ x ≤ 0.4 and in the wavelength region of 300 to 2500 nm at 4.8 K and 290 K. We observed several absorption bands in the wavelength regions corresponding to the 4A2(4F) → 4T1(4P) transition and the 4A2(4F) → 4T1(4F) transition of Co2+ at a tetrahedral Td point symmetry point in the host crystals, as well as unknown absorption bands. The several absorption bands were analyzed in the framework of the crystal-field theory along with the second-order spin-orbit coupling. The unknown absorption bands were assigned as due to phonon-assisted absorption bands. We also investigated the variations of the crystal-field parameter Dq and the Racah parameter B with composition x in the Mgx Cd1–x Se system. The results showed that the crystal-field parameter (Dq ) increases, on the other hand, the Racah parameter (B ) decreases with increasing composition x, which may be connected with an increase in the covalency of the metal-ligand bond with increasing composition x in the Mgx Cd1–x Se system. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The Undecakisicosahedral Group and a 3-regular Carbon Network of Genus 26

Three projective special linear groups PSL(2,p), those with p = 5, 7 and 11, can be seen as p-multiples of tetrahedral, octahedral and icosahedral rotational point groups, respectively. The first two have already found applications in carbon chemistry and physics, as PSL(2,5) ≡ I is the rotation group of the fullerene C60 and dodecahedrane C20H20, and PSL(2,7) is the rotation group of the 56-vertex all-heptagon Klein map, an idealisation of the hypothetical genus-3 “plumber’s nightmare” allotrope of carbon. Here, we present an analysis of PSL(2,11) as the rotation group of a 220-vertex, all 11-gon, 3-regular map, which provides the basis for a more exotic hypothetical sp 2 framework of genus 26. The group structure and character table of PSL(2,11) are developed in chemical notation and a three dimensional (3D) geometrical realisation of the 220-vertex map is derived in terms of a punctured polyhedron model where each of 12 pentagons of the truncated icosahedron is connected by a tunnel to an interior void and the 20 hexagons are connected tetrahedrally in sets of 4.

Structure of incommensurate ammonium tetrafluoroberyllate studied by structure refinements and the maximum entropy method.

Incommensurately modulated ammonium tetrafluoroberyllate (AFB) occurs in a narrow temperature interval between the paraelectric room-temperature phase with space group Pnma (Ti = 178 K) and the ferroelectric low-temperature phase with space group Pna2(1) (Tc = 173 K). The structure is determined from accurate single-crystal X-ray diffraction data collected with synchrotron radiation at 175 K. The superspace group of the structure is Pnma(alpha00)0ss with alpha = 0.4796 (4). Both structure refinements and the maximum entropy method lead to the same structure model, which involves only single harmonic modulations. The building units of the structure are BeF4(2-) and NH4+ complex ions with approximately tetrahedral point symmetry. They are relatively rigid and the modulations consist mainly of translations of the tetrahedra and their rotations around a fixed axis. The modulation is related to changes in the network of the hydrogen bonds. The low-temperature superstructure can be described as a commensurately modulated structure with the same superspace symmetry. The first harmonic modulations of the low-temperature and incommensurate phases are related by a scale factor with a value of approximately two. In addition, the low-temperature phase exhibits a second harmonic modulation that is responsible for shifts along c and the ferroelectricity in this phase. The experimental data of the incommensurate phase do not contain any evidence for the presence of a second harmonic in the modulation functions. This suggests that the development of the second harmonic, i.e. the development of the spontaneous polarization, is responsible for the lock-in transition.

Density functional theory analysis of CaRg n+ complexes: (Rg=He, Ne, Ar; n=1–4)

CaRg n + (Rg=He, Ne, Ar) complexes with n=1–4, are investigated by performing using the B3LYP/6-311+G (3df) density functional theory calculations. The CaHe n + ( n=1–4) complexes are found to be stable. In the case of CaNe n + and CaAr n +, stable structures and stationary point were found only for n=1 and 2. For n=3 in the C 3 V and the D 3 h point group as well as for n=4 in the T d (tetrahedral) point group a saddle point (imaginary frequency) is observed and global minimum could be obtained along the potential energy surface.

Structure Determination of the TMA2HgCl4by X-Ray Diffraction and Maximum Entropie Method

The crystal structure of [N(CH 3 ) 4 ] 2 HgCl 4 has been studied by single crystal X-ray diffraction. the structure is orthorhombic Pnma with lattice parameters 12.419(7), b = 9.068(4), and c = 15.684(2) A at room temperature. Each TMA cation disorderly occupies two orientations (TMA is tetramethylammonium). Refinements using restrictions of the non-crystallographic tetrahedral point group for the TMA cations and refinements without restrictions are used to show that TMA has tetrahedral symmetry at all temperatures. The HgCl 4 anions are distorted, but the distortions are independent of the temperature.

Structural investigation of the phase transitions of [N(CH3)4]2HgCl4

Abstract The crystal structures of [N(CH3)4]2HgCl4 between 191 K and 350 K have been studied by single crystal X-ray diffraction. Below Tc 1 = 277 K the structure is monoclinic P21/a with lattice parameters a = 12.320 (7), b = 9.034 (4), c = 15.402 (4) Å, and β = 89.64 (1)° at T = 191 K. Above Tc 1 the structure is orthorhombic Pnma with lattice parameters 12.419 (7), b = 9.068 (4), and c = 15.684 (2) Å at room temperature, and with lattice parameters a = 12.421 (9), b = 9.125 (1), and c = 15.777(4) Å at T = 350 K. At room temperature each TMA cation disorderly occupies two orientations (TMA is tetramethylammonium). At low temperature the structure is fully ordered, with additional shifts and rotations of all ions, in accordance with the results of Asahi et al., J. Phys. Soc. Jpn. 60 (1991) 921. Refinements using restrictions of the non-crystallographic tetrahedral point group for the TMA cations and refinements without restrictions are used to show that TMA has tetrahedral symmetry at all temperatures. The HgCl4 anions are distorted, but the distortions are independent of the temperature. The phase transition at Tc 1 is described as an order/disorder transition accompanied by major rotations and shifts of the ions. New evidence for a previously reported phase transition at Tc 2 = 320 K (Zangar et al. Phase Transitions 12 (1988) 141) is obtained from NMR. This transition is shown to be a non symmetry breaking structural transition, that correspond to a gradual loss of the orientational order of the TMA cations.

A Tetrahedral Oligo(phenylenevinylene) Molecule of Intermediate Dimensions: Effect of Molecular Shape on the Morphology and Electroluminescence of Organic Glasses

Tetrakis[(4-(4′-(2″,5″-dioctyloxy-4″-(4‴-(2′‴,5′‴-dioctyloxy-4′‴-styryl)styryl)styryl)styryl)styryl)phenyl]methane (T-6R-OC8H17) is an organic chromophore that consists of four optoelectronic fragments (“arms”) connected to a tetrahedral point of convergence (carbon). Bulk samples are amorphous as determined by powder diffraction, while differential scanning calorimetry (DSC) is sometimes ambiguous. Film forming properties were studied by atomic force microscopy (AFM) and fluorescence microscopy as a function of casting solvent and heat treatment. The film forming qualities are useful for the fabrication of light-emitting diodes with low turn-on voltages. Device performance is also history dependent. The relationship between bulk morphology, film topology, photoluminescence (PL) properties, and light-emitting diode (LED) performance is discussed. A comparison of these compounds against the parent oligo(phenylenevinylene) arms, with respect to morphology, topology, and PL properties is also presented.

The crystal structure of Dps, a ferritin homolog that binds and protects DNA

The crystal structure of Dps, a DNA-binding protein from starved E. coli that protects DNA from oxidative damage, has been solved at 1.6 A resolution. The Dps monomer has essentially the same fold as ferritin, which forms a 24-mer with 432 symmetry, a hollow core and pores at the three-fold axes. Dps forms a dodecamer with 23 (tetrahedral) point group symmetry which also has a hollow core and pores at the three-folds. The structure suggests a novel DNA-binding motif and a mechanism for DNA protection based on the sequestration of Fe ions.

Structural Change and Relaxation Processes of Tetrahedral Point Defects

Structural Change and Relaxation Processes of Tetrahedral Point Defects

Effect of Habit Modification on Optical and X-ray Structures of Sodium Halate Mixed Crystals: The Etiology of Anomalous Double Refraction

Structures of mixed crystals of the isomorphous salts NaClO3 and NaBrO3 (sodium chlorate and sodium bromate, respectively) were reinvestigated by X-ray diffraction. Contrary to previous reports, NaCl x Br1−x O3 is not cubic. Data from adjacent {100} growth sectors of crystals of varying composition were refined in the triclinic space group P1; halate ions occupy nominally symmetry-related sites nonstatistically. Optical measurements showed that six asymmetric sectors in cubes are disposed to give an object with approximate tetrahedral point symmetry. We address forgotten anomalies, first observed almost 150 years ago, which could have been a sufficient basis for earlier structural reinvestigations. The mixed-crystal structure speaks to the general non-applicability of the Law of Isomorphism to solid solutions. A link between optical anisotropy and nonstatistical guest site occupancy was achieved by annealing crystals between 523 and 533 K. Na2S2O3 and Na2S2O6 were used as habit-modifying impurities to produce NaCl x Brl−x O3 crystals with {111} and {\overline 1\overline 1\overline 1} habits, respectively. Diffraction data from {111} and {\overline 1\overline 1\overline 1} growth sectors were refined in the trigonal space group R3. In each case the pyramidal halate ion that was located on the special position was depleted in BrO3 −. Conoscopic optical investigations nevertheless showed that the crystals are biaxial with a small 2V (10–15°), in marked contrast to the 90° angle in {100} crystals. We failed to reconcile the optical and X-ray structures by calculating the optical indicatrix with bond polarizability sum models. This led us to estimate the magnitude of other factors which contribute to the optical properties, including strain associated with dislocations which may exert its influence through combined piezoelectric and linear electro-optic effects.