Noncrystallographic symmetry and the sampling theorem

1974 ◽  
Vol 140 (5-6) ◽  
Author(s):  
P. M. Colman
Keyword(s):  
Crystal Structure ◽  
Simple Model ◽  
Local Symmetry ◽  
Real Space ◽  
Sampling Theorem ◽  
Model Structure ◽  
Asymmetric Unit ◽  
Structure Factors ◽  
Shannon Sampling Theorem ◽  
Space Method

AbstractWhen the asymmetric unit of a crystal structure expresses some noncrystallographic symmetry it is often found that these local symmetry elements involve only proper rotations. Under these conditions the relations between the structure factors resulting from this repetition within the crystallographic asymmetric unit reduce to a statement of the Whittaker-Shannon sampling theorem, and thereby suggest the real space method for solution. A simple model structure is solved as an example.

The Crystal Structure of cis-Inositol Monohydrate—Un Objet Retrouvé

1996 ◽  
Vol 49 (3) ◽  
pp. 413 ◽  
Author(s):  
HC Freeman ◽  
DA Langs ◽  
CE Nockolds ◽  
YL Oh
Keyword(s):  
Crystal Structure ◽  
Least Squares ◽  
Direct Methods ◽  
Monoclinic Space Group ◽  
High Symmetry ◽  
Asymmetric Unit ◽  
Fourier Methods ◽  
Full Matrix ◽  
Structure Factors ◽  
Independent Molecules

cis-Inositol monohydrate, C6H12O6.H2O, crystallizes in the monoclinic space group P 21/n [a 9.900(8), b 9.296(8), c 17.795(15) Ǻ, β 90.5(1)°, Z 8]. The normalized structure factors Eh have an atypical statistical distribution, and attempts to solve the structure by direct methods (triplet relationships) were unsuccessful. The structure was ultimately solved by Patterson and Fourier methods, and was refined by full-matrix least squares [Rw = 0.047 for 1665 independent reflections ≥2σ(Imin)]. The cis-inositol molecules have approximately trigonal symmetry, as expected. The difficulties encountered during the structure analysis are explained by the presence of two nearly identical molecules of high symmetry in the asymmetric unit. The independent molecules are related by translational pseudosymmetry, and their orientations are such that all the C-C and C-O bonds in the structure are approximately parallel to a small number of directions.

Phase determination and structure refinement by density and shift functions

1970 ◽  
Vol 26 (2) ◽  
pp. 230-234 ◽  
Author(s):  
H. Jagodzinski ◽  
D. Philipp
Keyword(s):  
Crystal Structure ◽  
Structure Refinement ◽  
Three Dimensional ◽  
Complete Information ◽  
Shift Function ◽  
Model Structure ◽  
Phase Determination ◽  
One Dimensional ◽  
Structure Factors ◽  
The Right

Any crystal structure may be described in terms of a sublattice of points, each of which represents a certain fraction of the electron density. Multiplying this sublattice by a density function f(x) and applying a shift function s(x), which brings the atoms into the right positions, the correct crystal structure can be given in many different ways. It is shown that the shift function s(x) yields phase relations between the structure factors F(h), which may be evaluated directly, if the coefficients of the Fourier representation of s(x) converge rapidly. This behaviour is demonstrated for the case of a one-dimensional acentric model structure consisting of 50 atoms. Complete information on the structure may be obtained by routine methods with the aid of 5 given phases of the structure factor. This procedure may also be applied to three-dimensional structures, if the corresponding computer programs are available.

The molecular replacement method

1990 ◽  
Vol 46 (2) ◽  
pp. 73-82 ◽  
Author(s):  
M. G. Rossmann
Keyword(s):  
Real Space ◽  
Molecular Replacement ◽  
Replacement Method ◽  
Structure Factors ◽  
Phase Extension ◽  
Molecular Replacement Method ◽  
Unknown Structure ◽  
The Right ◽  
Insight Into ◽  
Space Method

Molecular replacement can be used for obtaining approximate phasing of an unknown structure from a known related molecule and for phase improvement as well as extension in the presence of noncrystallographic symmetry. Emphasis is placed on the latter procedure. It is shown that the real-space method of iterative electron density averaging and Fourier back transformation corresponds to iterative phase substitution in the right-hand side of expressions to give a set of improved phases. Analysis of these expressions (the 'molecular replacement equations') provides insight into the limits of possible phase extension, and the implications for the use of calculated structure factors when there are no observed amplitudes. It is shown that the percentage of observed data and inaccuracy of the observed amplitudes available for phase extension are compensated by the extent of noncrystallographic redundancy and the fraction of crystal cell volume that may be flattened because it is outside the control of noncrystallographic symmetry.

Solving crystal structures without Fourier mapping. II. Non-centrosymmetric case

2000 ◽  
Vol 215 (11) ◽  
Author(s):  
K. Pilz ◽  
K.F. Fischer
Keyword(s):  
Crystal Structure ◽  
Ab Initio ◽  
Reciprocal Lattice ◽  
Three Dimensional ◽  
Algebraic Method ◽  
Asymmetric Unit ◽  
One Dimensional ◽  
Partial Structures ◽  
Structure Factors

An algebraic method for the determination of one-dimensional centrosymmetric structures (or projections) of equal atoms has been extended to crystals without a centre of symmetry. The technique is an (almost) ab initio one, provides higher resolution than a Fourier map using the same number of structure factors, and permits either a unique solution compatible with the data employed or provides all possible solutions. It appears suitable in particular for partial structures with not too many atoms per asymmetric unit (e.g. anomalous scatterers in a “biological“ crystal structure). It only needs rather selected data: from central reciprocal lattice rows for one-dimensional projections plus from (at least) two rows parallel to the first ones for three-dimensional structures.

Alkyl-chain disorder in tetraisohexylammonium bromide

2012 ◽  
Vol 68 (4) ◽  
pp. o152-o155 ◽  
Author(s):  
Malcolm A. Kelland ◽  
Amber L. Thompson
Keyword(s):  
Crystal Structure ◽  
Crystal Growth ◽  
Alkyl Chain ◽  
Growth Inhibitor ◽  
Ammonium Cation ◽  
Asymmetric Unit ◽  
Bromide Anion ◽  
Alkyl Chains ◽  
Hydrate Crystal ◽  
Structure Ii Clathrate Hydrate

Tetraisohexylammonium bromide [systematic name: tetrakis(4-methylpentyl)azanium bromide], C24H52N+·Br−, is a powerful structure II clathrate hydrate crystal-growth inhibitor. The crystal structure, in the space groupP3221, contains one ammonium cation and one bromide anion in the asymmetric unit, both on general positions. At 100 K, the ammonium cation exhibits one ordered isohexyl chain and three disordered isohexyl chains. At 250 K, all four isohexyl chains are disordered. In an effort to reduce the disorder in the alkyl chains, the crystal was thermally cycled, but the disorder remained, indicating that it is dynamic in nature.

2,4-Diamino-6-phenyl-1,3,5-triazin-1-ium nitrate: intriguing crystal structure with high Z′/Z′′ and hydrogen bond numbers and Hirshfeld surface analysis of intermolecular interactions

CrystEngComm ◽  
2021 ◽  
Author(s):  
Nicoleta Caimac ◽  
Elena Melnic ◽  
Diana Chisca ◽  
Marina S. Fonari
Keyword(s):  
Crystal Structure ◽  
Hydrogen Bond ◽  
Surface Analysis ◽  
Intermolecular Interactions ◽  
Title Compound ◽  
Ionic Species ◽  
Hirshfeld Surface ◽  
Hirshfeld Surface Analysis ◽  
Asymmetric Unit ◽  
Space Group

The title compound crystallises in the triclinic centrosymmetric space group P1̄ with an intriguing high number of crystallographically unique binary salt-like adducts (Z′ = 8) and a total number of ionic species (Z′′ = 16) in the asymmetric unit.

scholarly journals Enabling propagation of anisotropic polaritons along forbidden directions via a topological transition

2021 ◽  
Vol 7 (14) ◽  
pp. eabf2690
Author(s):  
J. Duan ◽  
G. Álvarez-Pérez ◽  
K. V. Voronin ◽  
I. Prieto ◽  
J. Taboada-Gutiérrez ◽  
...  
Keyword(s):  
Crystal Structure ◽  
Silicon Carbide ◽  
Molybdenum Trioxide ◽  
Real Space ◽  
Negative Permittivity ◽  
Topological Transition ◽  
Α Phase ◽  
Direct Light ◽  
Gap Opening ◽  
Phonon Polaritons

Polaritons with directional in-plane propagation and ultralow losses in van der Waals (vdW) crystals promise unprecedented manipulation of light at the nanoscale. However, these polaritons present a crucial limitation: their directional propagation is intrinsically determined by the crystal structure of the host material, imposing forbidden directions of propagation. Here, we demonstrate that directional polaritons (in-plane hyperbolic phonon polaritons) in a vdW crystal (α-phase molybdenum trioxide) can be directed along forbidden directions by inducing an optical topological transition, which emerges when the slab is placed on a substrate with a given negative permittivity (4H–silicon carbide). By visualizing the transition in real space, we observe exotic polaritonic states between mutually orthogonal hyperbolic regimes, which unveil the topological origin of the transition: a gap opening in the dispersion. This work provides insights into optical topological transitions in vdW crystals, which introduce a route to direct light at the nanoscale.

scholarly journals Powder study of 3-azabicyclo[3.3.1]nonane-2,4-dione form 2

2006 ◽  
Vol 62 (7) ◽  
pp. o3046-o3048 ◽  
Author(s):  
Ashley T Hulme ◽  
Philippe Fernandes ◽  
Alastair Florence ◽  
Andrea Johnston ◽  
Kenneth Shankland
Keyword(s):  
Crystal Structure ◽  
Simulated Annealing ◽  
Hydrogen Bonds ◽  
Powder Diffraction ◽  
Title Compound ◽  
Variable Temperature ◽  
Polycrystalline Sample ◽  
Asymmetric Unit ◽  
X Ray ◽  
Compound C

A polycrystalline sample of a new polymorph of the title compound, C8H11NO2, was produced during a variable-temperature X-ray powder diffraction study. The crystal structure was solved at 1.67 Å resolution by simulated annealing from laboratory powder data collected at 250 K. Subsequent Rietveld refinement yielded an R wp of 0.070 to 1.54 Å resolution. The structure contains two molecules in the asymmetric unit, which form a C 2 2(8) chain motif via N—H...O hydrogen bonds.

Sign in / Sign up

Export Citation Format

Share Document