Fivefold Rotation Research Articles

Quasicrystal structures in submicron spheres of pure nickel and nickel-chromium alloys

The icosahedral quasicrystal with a fivefold rotation symmetry was first reported in Al-M (M = Cr, Mn, Fe) alloys. Later, quasicrystals with eightfold, tenfold, and twelvefold rotation symmetries were also reported. A twelvefold symmetry has been found in a Ni-70.6 at%Cr alloy and an eightfold symmetry has been found in a Cr-Ni-Si alloy. This indicates a potential for nickel-chromium alloys to form the quasicrystal phase.An intermediate state between crystalline and quasicrystalline states is also discussed by Li, Teng, et. al. in an Al-Cu-Mg alloy due to the coexistence of micro-quasicrystal and micro-crystal grains. When regions larger than 1 nm are measured, a coherent boundary is found between micro-quasicrystal and micro-crystal grains.A diffraction pattern with seemingly twelvefold symmetry is found in pure nickel and Ni-20at%Cr sphere of about 25 nm diameter as shown in Figure 1. The diffraction patterns do not have true twelvefold symmetry but the first diffraction ring is found to have twelve diffracted spots. Similar structures are found in Ni-10at%Cr and shown in Figure 2 with a series of diffraction patterns.

Growth of submillimeter copper dendrites exhibiting fivefold symmetry.

Cu or α-Cu-Zn dendrites which show the shape of pentamerous flower exhibiting fivefold symmetry are grown by zinc reduction of CuBr at 600°C or thereabouts. Since the morphology of dendrites reflects the symmetry of the growth axis, the growth axis is considered to be a fivefold rotation axis.The size of large dendrites reaches 120 μm. Such a large metallic crystal exhibiting fivefold symmetry in the morphology has not been observed.

Structure of F-Pili: Reassessment of the symmetry

Reassessment of the X-ray fibre diffraction patterns of F-pili using a more accurate subunit molecular weight suggests that subunits in F-pili are related by a fivefold rotation axis around the pilus axis. The identity of this fivefold symmetry with the fivefold rotation axis that relates the subunits in fd bacteriophage supports a simple model for tip-to-tip adsorption of bacteriophage to pili.

Symmetry, stability, and elastic properties of icosahedral incommensurate crystals.

The symmetry and stability of icosahedral incommensurate structures and generalized two-dimensional Penrose pentagonal structures are studied. The crystallographic properties of Penrose lattices are described by five-dimensional (5D) super space groups, and the icosahedral structures are described by 6D space groups, with or without improper translations. The density in real space is given as the density along a three-dimensional plane in this 6D space. The fivefold symmetry of the diffraction spectrum of Mn-Al alloys, which is inconsistent with three-dimensional translational invariance, reflects a fivefold rotation axis of the 6D space group. The six continuous degrees of freedom associated with the 6D space represent the usual three orthogonal rigid displacements of the crystal, plus three phase shifts associated with internal rearrangements, leading to three acoustic-phonon modes and three phason modes. There are two independent elastic constants, which is fewer than in any regular crystal, representing one-dimensional and five-dimensional irreducible strains, respectively. If the phase degrees of freedom are included, there are five generalized elastic constants. The stability of icosahedral structures and ``lyotropic'' Penrose structures can be understood from a phenomenological Landau theory. The ideal icosahedral crystal has perfect positional order, which is stable with respect to thermal fluctuations at low temperatures. The melting transition is first order.

Convergent-beam and small-area-parallel-beam electron diffraction of icosahedral quasicrystals of a melt-quenched Al-Mn alloy

Intensity distribution in convergent-beam electron diffraction (CBED) patterns obtained from icosahedral quasicrystas of a melt-quenched Al-Mn alloy reveal that the quasicrystals do not have fivefold, threefold and twofold rotation axes and have no inversion center, although ordinary diffraction patterns obtained thus far showed these rotation symmetries. CBED patterns taken from specimen areas of about 3 nm in diameter show a deviation in geometry in spot positions from the fivefold rotation symmetry. Ring patterns due to higher-order Laue zone reflections are not observed in CBED patterns. Kikuchi bands are composed of two sub-bands in the five equivalent directions, and each band has a different intensity profile. Parallel-beam (3 × 10 -5 rad) electron diffraction patterns obtained from specimen areas less than 100 nm in diameter also show a deviation from the fivefold symmetry in spot positions and make clear that each Bragg reflection consists of many fine spots which show no fivefold symmetry. It is proven experimentally that all the observed reflections occur already in the approximation of kinematical diffraction, although their intensities may be modified by dynamical diffraction effect.

On pseudo-rotation in (AsCH3)5 and (AsCF3)5

Two internal rearrangement processes can be deduced from the temperature dependences of the 1H and 19F nuclear magnetic resonance (n.m.r.) spectra of the asymmetric puckered five-membered ring compounds (AsCH3)5, (AsCF3)5, and (PCF3)5. The first is a fast "butterfly" motion of the ring puckering which yields a time-averaged plane of symmetry perpendicular to the ring through the puckered atom. This motion could not be stopped in any of the three compounds by cooling, even to −130 °C in the case of (AsCF3)5. The second is a slow pseudo-rotation, or a rotation of the perpendicular symmetry plane around the ring yielding a time-averaged fivefold rotation axis. There is a choice between two mechanisms for this second rearrangement, either a "clickstop" process in which the puckering progresses around the ring by one position at a time, or a "pancake" process in which the puckering drops randomly into any one of the five ring positions from a planar transition state. This choice can be resolved for this cyclic permutation process by appeal to the n.m.r. lineshapes. The present data for (AsCH3)5 favor the "clickstop" mechanism in spite of a competing intermolecular process at elevated temperatures with other components in the system which are in equilibrium with the ring compound.

Molecular Structure of Dicyclopentadienylberyllium (C5H5) 2Be

The molecular structure of gaseous (C5H5) 2Be has been determined from electron-scattering data. The molecule consists of two planar, symmetrical C5H5 rings. C1–C2 = 1.424±0.002 Å and C1–H1 = 1.070±0.005 Å. The rings are parallel and staggered with a vertical ring—ring distance h = 3.37±0.03 Å. The CH skeleton thus has point group symmetry D5d. The beryllium atom may occupy two alternative positions on the fivefold rotation axis h = 1.485±0.005 Å from one ring and h2 = 1.980±0.010 Å from the other. The complete molecule thus has point group symmetry C5v. It appears then that the potential energy curve of the beryllium atom has two minima. This is readily understood from an ionic binding model. The discrepancy between the values found for h and h1+h2 is believed to be due to intramolecular motion.