The direct product structure of atomic orbital manifolds: extension to non-Euclidean permutation groups

Atomic orbital manifolds are described as Γ × Γ direct products of degenerate irreducible representations (irreps) of odd dimension in suitable groups, which may be either the polyhedral symmetry point groups (T, O, I) or larger simple non-Euclidean permutation groups containing the polyhedral symmetry point groups as subgroups. Such larger non-Euclidean groups include the simple pollakispolyhedral groups ⁷ O [≈L ₂(7)] and ¹¹ I [≈L ₂(11)] of orders 168 and 660, respectively, and the simple alternating groups A ₆ and A ₇ of orders 360 and 2520, respectively. Thus the 9-dimensional (9D) direct product T × T of a 3D irrep of the polyhedral or larger point groups leads to the nine-orbital sp³d⁵ manifold. In icosahedral symmetry I this 9D direct product splits into separate irreps for the s, p and d orbitals, whereas in the larger non-Euclidean ⁷ O group this direct product splits only into a 6D irrep for the gerade orbitals (s + d) and a 3D irrep for the ungerade p orbitals. A similar method can be used to study the 25-orbital sp³d⁵f⁷g⁹ manifold as H × H direct products in the icosahedral group I and the two larger simple non-Euclidean groups A ₆ and ¹¹ I, in which I is a subgroup of index 6 and 11, respectively. The mathematics implicit in these direct products appears to have some connections with a few areas of chemical physics including icosahedral quasicrystals and unusual degeneracies in the Coulomb energies of terms in transition metal atomic spectra.     

Three-Dimensional Reconstruction of Icosahedral Virus by Symmetry-Adapted Functions

A three-dimensional （3D） reconstruction of icosahedral virus is carried out by the icosahedral symmetry-adapted function （ISAF） method in spherical coordinates. In order to reduce the influence of noise, it is better to use the basis functions that have identical symmetry with the object reconstructed. It is verified that the ISAF method has stronger ability to reduce the influence of noise to grain the resolution better than that of the conventional method by the simulation of 3D reconstruction.     

Electronic states of the icosahedral quasiperiodic systems in magnetic fields

The electronic properties of the icosahedral quasiperiodic system in a tile-dependent or uniform magnetic field is studied by quasi-Bloch scheme and the general solutions are obtained for the electronic states. The behavior of the three-dimensional (3D) non-interacting electrons in an icosahedral quasiperiodic system may be treated as the projection of that of the non-interacting pseudo-electrons in 6D. In the presence of the tile-dependent magnetic field, the non-interacting electrons are quasi-Bloch electrons, while they are partial quasi-Bloch electrons when the magnetic field is uniform.     

Computing methods for icosahedral and symmetry-mismatch reconstruction of viruses by cryo-electron microscopy

Three-dimensional(3 D) reconstruction of icosahedral viruses has played a crucial role in the development of cryoelectron microscopy single-particle reconstruction, with many cryo-electron microscopy techniques first established for structural studies of icosahedral viruses, owing to their high symmetry and large mass. This review summarizes the computational methods for icosahedral and symmetry-mismatch reconstruction of viruses, as well as the likely challenges and bottlenecks in virus reconstruction, such as symmetry mismatch reconstruction, contrast transformation function(CTF)correction, and particle distortion.     

Formation and stability of large B 6 O clusters with icosahedral structure

In this paper, the formation and the stability of large B₆O icosahedral particles was discussed on the basis of elastic deformation theory. Our calculation illustrate the stability of macroscopic Mackay packing B₆O icosahedral particles at high pressure. The transition pressure from rhombohedral structure of B₆O particles to macroscopic B₆O icosahedral ones was calculated to be 6 GPa, which is in good agreement with the experimental data (4.0-5.52 GPa). The maximum diameter of B₆O icosahedral particles at low pressure is estimated to be 200-300 nm. Received 30 November 2000     

Mössbauer studies of Al75Cu15−xFexV10 icosahedral alloys

⁵⁷Fe Mössbauer effect measurements have been performed at room temperature on the alloy series Al₇₅Cu_15–xFe_xV₁₀ withx=0.15 (amorphous and icosahedral phases), 3 and 6 (icosahedral phases). There is more disorder in the icosahedral phase of thex=0.15 alloy than in its amorphous phase. The bimodal character of the distribution of quadrupole splittings becomes less pronounced with increasingx and disappears forx6. This is interpreted in terms of two classes of transition metal sites in Al-transition metal-based icosahedral alloys.     

Elastic waves in carbon 2D supracrystals

The elastic moduli and propagation velocities of elastic waves in 2D supracrystalline nanoallotropes of carbon have been calculated. It has been shown that these velocities in sp ² nanoallotropes are close to those in graphene and exceed the propagation velocities of elastic waves in single-crystal diamond by a factor of 2. The propagation velocities of both longitudinal and transverse elastic waves in carbon 2D supracrystalline sp ³ nanoallotropes are several times lower than those in sp ² nanoallotropes.     

Approximate indexing of icosahedral polyhedra with fibonacci numbers

In general, indexing faces of icosahedral face-forms requires irrational numbers. However, for many practical purposes an approximate indexing based on triplets of integer numbers can be used. Two possible approaches called, respectively, “Fibonacci Matrix Methods” (FMM) and the “Linear Combination Method” (LCM) are described. FMM relies on the use of “auxiliary” matrices F_n, F² _n, F³ _n and F⁴ _n which have Fibonacci numbers as their elements. These matrices allow good approximation of the results usually obtained using the standard five-fold rotation matrices which are typical of icosahedral symmetry. LCM is based on the use of a classical crystallographic rule i.e. the so-called “Goldschmidt Complication Law” which is just a particular case of linear combination of triplets of face indices, with integers as coefficients. The occurrence of large integer indices is remarked.     

Experimental evaluation of phonon–phason coupling in icosahedral quasicrystals

By measuring phonon strain introduced in crystal approximants, the sign and magnitude of the phonon–phason coupling constant have been evaluated for icosahedral quasicrystals of Mg–Ga–Al–Zn and Al–Cu–Fe systems. The evaluated coupling constants are approximately ?0.04μ and 0.004μ (μ?=?shear modulus) for the former and the latter, respectively. They are in good agreement with the results of a previously reported theoretical calculation. Possible effects of phonon–phason coupling on the onset of phasonic elastic instability in icosahedral quasicrystals are discussed.     

Elastic Green's function of icosahedral quasicrystals

The elastic theory of quasicrystals considers, in addition to the “normal” displacement field, three “phason” degrees of freedom. We present an approximative solution for the elastic Green's function of icosahedral quasicrystals, assuming that the coupling between the phonons and phasons is small. Received: 18 December 1997 / Accepted: 6 March 1998     

Structural transformation from the AlCuFe icosahedral phase to the 1/1 cubic (AlSi)CuFe approximant phase; three dimensional models of translation defects

Unusual results are obtained by High Resolution Electron Microscopy (HREM) studies of the 1/1 cubic approximant phase of the (AlSi)CuFe icosahedral phase. We study, in a 3D model of AlCuFe icosahedral phase, the possibilities of transformation of this structure into a periodic cubic structure. We propose a model of transformation coherent with the experimental results. Received 31 May 1999 and Received in final form 23 December 1999     

Symmetry-dependent Mn-magnetism in Al<Subscript>69.8</Subscript>Pd<Subscript>12.1</Subscript>Mn<Subscript>18.1</Subscript>

We investigated the stability of magnetic moments in Al_69.8Pd_12.1Mn_18.1. This alloy exists in both, the icosahedral (i) and the decagonal (d) quasicrystalline form. The transition from the i- to the d-phase is achieved by a simple heat treatment. We present the results of measurements of the ²⁷Al NMR-response, the dc magnetic susceptibility, and the low-temperature specific heat of both phases. In the icosahedral compound, the majority of the Mn ions carries a magnetic moment. Their number is reduced by approximately a factor of two by transforming the alloy to its decagonal variety. For both compounds, we have indications for two different local environments of the Al nuclei. The first reflects a low density of states of conduction electrons and a weak coupling of the Al nuclei to the Mn-moments. The second type of environment implies a large d-electron density of states at the Fermi level and a strong coupling to the magnetic Mn moments. Spin-glass freezing transitions are observed at T^deca_f=12 K for the decagonal, and T^ico_f=19 K for the icosahedral phase.     

Formation of two-dimensional crystals of icosahedral RNA viruses

The formation of 2D arrays of three small icosahedral RNA viruses with known 3D structures (tomato bushy stunt virus, turnip yellow mosaic virus and bromegrass mosaic virus) has been investigated to determine the role of each component of a negative staining solution containing ammonium molybdate and polyethylene glycol. Virion association was monitored by dynamic light scattering (DLS) and virus array formation was visualised by conventional transmission electron microscopy and cryo-electron microscopy after negative staining. The structural properties of viral arrays prepared in vitro were compared to those of microcrystals found in the leaves of infected plants. A novel form of macroscopic 3D crystals of turnip yellow mosaic virus has been grown in the negative staining solution. On the basis of the experimental results, the hypothesis is advanced that microscopic arrays might be planar crystallisation nuclei. The formation of 2D crystals and the enhancing effect of polyethylene glycol on the self-organisation of virions at the air/water interface are discussed. SYNOPSIS: The formation of 2D arrays of icosahedral viruses was investigated by spectroscopic and transmission electron microscopic methods.     

Basics of NMR line shape in quasicrystals

The measurement and analysis of broad nuclear magnetic resonance (NMR) spectra of quasicrystals require experimental methods and theoretical interpretations different from NMR investigations of regular periodic crystals. Frequency- and field-sweep methods for recording quasicrystalline NMR spectra are described and compared with the measurement of²⁷Al NMR spectra of icosahedral AlPdMn and decagonal AlNiCo quasicrystals. The nuclear spin interactions that determine the NMR line shape are the same for both types of the above Al-based quasicrystals, where the electric quadrupolar interaction with the broad distribution of its electric field gradient parameters predominantly determines the shape of the broad satellite “background” intensity. The essential observations are an almost isotropic²⁷Al NMR spectrum of the icosahedral quasicrystals and a strong angular dependence of the spectrum of decagonal quasicrystals.     

Study of background processes with the formation of neutrons in nuclear reactions in the energy range of 26–32 kev

The nature of background processes accompanying astrophysical nuclear reactions induced by hydrogen, helium, and neon ions in deuterated targets with small cross sections has been studied in calculations and experiments. The experiments have been performed at a Hall pulsed plasma accelerator in the ion energy range of 26–32 keV. The yield of background neutrons and γ-quanta with energies below 4 MeV in the proton-induced D(p, γ)³He reaction is primarily due to the presence of a natural impurity of gaseous deuterium in gaseous hydrogen and the chain of D(D, ³He)n → (n, γ) or (n, n'γ) reactions. A small contribution comes from the chain of D(¹H, ¹H)D → D(D, ³He)n → (n, γ) or (n, n'γ) reactions. It has been shown that background neutrons and γ-quanta from the D(4He, γ)6Li reaction are entirely due to the chain of D(⁴He, ⁴He)D → D(D, ³He)n → (n, γ) or (n, n'γ) reactions. It has been shown that the yield of neutrons and γ-ray photons detected at the interaction of neon ions with deuterated targets is also entirely due to the chain of elastic- scattering reactions of neon ions on deuterons in the target and to subsequent inelastic processes of interaction of deuterons accelerated at elastic scattering with other deuterons of the target. The main contribution to the yields of background neutrons and γ-quanta comes from doubly charged neon ions. The main conclusion is that the explanation of the yield of neutrons and γ-quanta at the interaction of hydrogen, helium, and neon ions with deuterated targets does not require “exotic” theoretical models.     

Structural evolution of Cu during rapid quenching by ab initio molecular dynamics

The structural transitions of Cu during two distinct quenching processes (Q1: 4.0×¹⁰¹³ K/s, Q2: 2.0×¹⁰¹⁴ K/s) were investigated by ab initio molecular dynamics simulation. The variations with temperature of internal energy, pair correlation functions g(r) and bond pairs have been characterized in both quenching processes. It is shown that liquid Cu transforms to fcc phase at the temperature about 600 K under the quenching condition Q1. The investigation of atomic diffusion by mean square displacement further demonstrates this result. When quenched under Q2, however, the liquid Cu is frozen into glass state at the temperature about 800 K. This work also reveals that icosahedral and tetrahedral clusters are predominant in the liquid state, while the icosahedral, bcc and tetrahedral clusters predominate in the glass state. The icosahedral and bcc short range ordering (SRO) are largely enhanced during the liquid-glass quenching process, whereas the tetrahedral SRO is slightly decreased.     

Colloidal dipolar interactions in 2D smectic-C films

We use a two-dimensional (2D) elastic free energy to calculate the effective interaction between two circular disks immersed in smectic-C films. For strong homeotropic anchoring, the distortion of the director field caused by the disks generates topological defects that induce an effective interaction between the disks. We use finite elements, with adaptive meshing, to minimize the 2D elastic free energy. The method is shown to be accurate and efficient for inhomogeneities on the length scales set by the disks and the defects, that differ by up to 3 orders of magnitude. We compute the effective interaction between two disk-defect pairs in a simple (linear) configuration. For large disk separations, D, the elastic free energy scales as ∼D ^-2, confirming the dipolar character of the long-range effective interaction. For small D the energy exhibits a pronounced minimum. The lowest energy corresponds to a symmetrical configuration of the disk-defect pairs, with the inner defect at the mid-point between the disks. The disks are separated by a distance that is twice the distance of the outer defect from the nearest disk. The latter is identical to the equilibrium distance of a defect nucleated by an isolated disk. Received 26 October 2001 and Received in final form 14 December 2001     

The bond force constants and elastic properties of boron nitride nanosheets and nanoribbons using a hierarchical modeling approach

A hierarchical approach bridging the atomistic and nanometric scales is used to compute the elastic properties of boron nitride nanosheets and nanoribbons, examining the effect of sheet size, aspect ratio and anisotropy. The approach consists in obtaining the bond force (force field) constants by dedicated computations based on density functional theory (DFT) and using such constants as input for larger scale structural models solved by finite element analysis (FEA). The bond force constants calculated by DFT are 616.9 N/m for stretching, 6.27×10⁻¹⁹ Nm/rad² for in-plane rotation and 1.32×10⁻¹⁹ Nm/rad² for dihedral rotation. Using these constants, the elastic properties of boron nitride nanosheets and nanoribbons predicted by FEA are almost independent of the sheet size, but strongly dependent on their aspect ratio. The sheet anisotropy increases with increased aspect ratio, with nanoribbons of aspect ratios of 10 exhibiting a ratio of elastic moduli along both in-plane directions of 1.7.     

Coherent spectroscopy of Mandelstam-Brillouin scattering in spatially inhomogeneous media

Mandelstam-Brillouin (MB) steady-state scattering in an elastic medium with a dense local zone inhomogeneity is considered in the 1D approximation. It is shown that for a certain size of inhomogeneity, the scattered radiation spectrum contains individual resonances whose frequencies depend on the elastic properties of microscopic inclusions. Experiments were performed using coherent four-photon scattering spectroscopy in the range 0–1 cm⁻¹ with a resolution of 0.06 cm⁻¹ in specially processed distilled water and in an aqueous solution of α-chymotrypsin albumin. In both media, the presence of MB resonances displaced is detected relative to the water resonance (≈0.25 cm⁻¹) in different directions and corresponding to different types of microinclusions.