scholarly journals Systematic Enumeration of High Symmetry Molecules by Means of Unit Subduced Cycle Indices with and without Chirality Fittingness

1990 ◽  
Vol 63 (1) ◽  
pp. 203-215 ◽  
Author(s):  
Shinsaku Fujita
Keyword(s):  
High Symmetry ◽  
Systematic Enumeration ◽  
Chirality Fittingness ◽  
Cycle Indices

scholarly journals Combinatorics of Edge Symmetry: Chiral and Achiral Edge Colorings of Icosahedral Giant Fullerenes: C80, C180, and C240

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1308
Author(s):  
Krishnan Balasubramanian
Keyword(s):  
Equivalence Classes ◽  
Group Symmetry ◽  
Irreducible Representations ◽  
Inversion Method ◽  
High Symmetry ◽  
Edge Colorings ◽  
Edge Group ◽  
Resonance Energies ◽  
Conjugated Circuits ◽  
Cycle Indices

We develop the combinatorics of edge symmetry and edge colorings under the action of the edge group for icosahedral giant fullerenes from C80 to C240. We use computational symmetry techniques that employ Sheehan’s modification of Pόlya’s theorem and the Möbius inversion method together with generalized character cycle indices. These techniques are applied to generate edge group symmetry comprised of induced edge permutations and thus colorings of giant fullerenes under the edge symmetry action for all irreducible representations. We primarily consider high-symmetry icosahedral fullerenes such as C80 with a chamfered dodecahedron structure, icosahedral C180, and C240 with a chamfered truncated icosahedron geometry. These symmetry-based combinatorial techniques enumerate both achiral and chiral edge colorings of such giant fullerenes with or without constraints. Our computed results show that there are several equivalence classes of edge colorings for giant fullerenes, most of which are chiral. The techniques can be applied to superaromaticity, sextet polynomials, the rapid computation of conjugated circuits and resonance energies, chirality measures, etc., through the enumeration of equivalence classes of edge colorings.

scholarly journals Standard mark table and standard USCI-CF table of the point group D6h. Unification of candidates derived from different D6h-skeletons

2021 ◽  
Vol 87 (3) ◽  
pp. 481-525
Author(s):  
Shinsaku Fujita ◽  
Keyword(s):  
Point Group ◽  
Gap Function ◽  
Combinatorial Enumeration ◽  
Cycle Index ◽  
Single Standard ◽  
Partial Cycle ◽  
Unit Subduced Cycle Index ◽  
Chirality Fittingness ◽  
Usci Approach ◽  
Cycle Indices

Combined-permutation representations (CPRs) for characterizing -skeletons (a benzene skeleton, a Haworth-projected skeleton, a superphane skeleton, and a coronene skeleton) are constructed by starting from respective sets of generators, where the permutation of each generator is combined with a mirror-permutation of 2-cycle to treat both achiral and chiral substituents under the GAP system. Thereby, the CPR of degree 8 (= 6 + 2) for the benzene skeleton, the CPR of degree 14 (= 12 + 2) for the Haworth-projected skeleton, the CPR of degree 14 (= 12 + 2) for the superphane skeleton, the CPR of degree 14 (= 12 + 2) for the coronene skeleton are generated to give primary mark tables (tables of marks) based on these CPRs. These primary mark tables generated by the GAP system are different in the sequence of subgroups from each other, although they stem from the same point group . They are unified into a single standard mark table by means of a newly-devised GAP function MarkTableforUSCI. Moreover, another newly-devised GAP function constructUSCITable is employed to construct a standard USCI-CF (unit-subduced-cycle-index-with-chirality-fittingness) table concordantly. After a set of PCI-CFs (partial cycle indices with chirality fittingness) is calculated for each skeleton, symmetry-itemized combinatorial enumeration is conducted by means of the PCI method of Fujita’s USCI approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin-Heidelberg, 1991).

The usefulness of high index low symmetry zone axes for holz line analysis

1987 ◽  
Vol 45 ◽  
pp. 384-385
Author(s):  
C. M. Sung ◽  
D. B. Williams
Keyword(s):  
Lattice Parameter ◽  
High Order ◽  
Zone Axis ◽  
High Index ◽  
High Symmetry ◽  
Second Phases ◽  
Line Patterns ◽  
Low Symmetry ◽  
Line Analysis

Researchers have tended to use high symmetry zone axes (e.g. <111> <114>) for High Order Laue Zone (HOLZ) line analysis since Jones et al reported the origin of HOLZ lines and described some of their applications. But it is not always easy to find HOLZ lines from a specific high symmetry zone axis during microscope operation, especially from second phases on a scale of tens of nanometers. Therefore it would be very convenient if we can use HOLZ lines from low symmetry zone axes and simulate these patterns in order to measure lattice parameter changes through HOLZ line shifts. HOLZ patterns of high index low symmetry zone axes are shown in Fig. 1, which were obtained from pure Al at -186°C using a double tilt cooling holder. Their corresponding simulated HOLZ line patterns are shown along with ten other low symmetry orientations in Fig. 2. The simulations were based upon kinematical diffraction conditions.

Systematic zone-Axis orientation of NM-scale particles for microdiffraction

1990 ◽  
Vol 48 (4) ◽  
pp. 262-263
Author(s):  
E D Boyes ◽  
L Hanna
Keyword(s):  
Real Space ◽  
Dark Field ◽  
Zone Axis ◽  
High Symmetry ◽  
Lattice Imaging ◽  
Cell Level ◽  
Bright Field ◽  
Axis Orientation ◽  
Annular Dark Field ◽  
Target Designation

A VG HB501 FEG STEM has been modified to provide track whilst tilt [TWIT] facilities for controllably tilting selected and initially randomly aligned nanometer-sized particles into the high symmetry zone-axis orientations required for microdiffraction, lattice imaging and chemical microanalysis at the unit cell level. New electronics display in alternate TV fields and effectively in parallel on split [+VTR] or adjacent externally synchronized screens, the micro-diffraction pattern from a selected area down to <1nm2 in size, together with the bright field and high angle annular dark field [HADF] STEM images of a much wider [˜1μm] area centered on the same spot. The new system makes it possible to tilt each selected and initially randomly aligned small particle into a zone axis orientation for microdiffraction, or away from it to minimize orientation effects in chemical microanalysis. Tracking of the inevitable specimen movement with tilt is controlled by the operator, with realtime [60Hz] update of the target designation in real space and the diffraction data in reciprocal space. The spot mode micro-DP and images of the surrounding area are displayed continuously. The regular motorized goniometer stage for the HB501STEM is a top entry design but the new control facilities are almost equivalent to having a stage which is eucentric with nanometric precision about both tilt axes.

Using crystallographic symmetry in diffraction and contrast analysis

1989 ◽  
Vol 47 ◽  
pp. 504-505
Author(s):  
U. Dahmen ◽  
K.H. Westmacott
Keyword(s):  
Electron Microscopy ◽  
High Symmetry ◽  
Two Phase ◽  
Tem Analysis ◽  
Crystallographic Symmetry ◽  
The Matrix ◽  
Contrast Analysis ◽  
Practical Tool ◽  
Convergent Beam ◽  
Beam Diffraction

Despite the increased use of convergent beam diffraction, symmetry concepts in their more general form are not commonly applied as a practical tool in electron microscopy. Crystal symmetry provides an abundance of information that can be used to facilitate and improve the TEM analysis of crystalline solids. This paper draws attention to some aspects of symmetry that can be put to practical use in the analysis of structures and morphologies of two-phase materials.It has been shown that the symmetry of the matrix that relates different variants of a precipitate can be used to determine the axis of needle- or lath-shaped precipitates or the habit plane of plate-shaped precipitates. By tilting to a special high symmetry orientation of the matrix and by measuring angles between symmetry-related variants of the precipitate it is possible to find their habit from a single micrograph.

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