Fundamentals of crystallography

2013 ◽  
Author(s):  
Carmelo Giacovazzo
Keyword(s):  
Unit Cell ◽  
Generic Point ◽  
Cell Content ◽  
Daily Work ◽  
Space Groups ◽  
Crystallographic Symmetry ◽  
Unit Cells ◽  
Basic Concepts ◽  
Periodic Repetition ◽  
Unit Vectors

In this chapter we summarize the basic concepts, formulas and tables which constitute the essence of general crystallography. In Sections 1.2 to 1.5 we recall, without examples, definitions for unit cells, lattices, crystals, space groups, diffraction conditions, etc. and their main properties: reading these may constitute a useful reminder and support for daily work. In Section 1.6 we establish and discuss the basic postulate of structural crystallography: this was never formulated, but during any practical phasing process it is simply assumed to be true by default. We will also consider the consequences of such a postulate and the caution necessary in its use. We recall the main concepts and definitions concerning crystals and crystallographic symmetry. Crystal. This is the periodic repetition of a motif (e.g. a collection of molecules, see Fig. 1.1). An equivalent mathematical definition is: the crystal is the convolution between a lattice and the unit cell content (for this definition see (1.4) below in this section). Unit cell. This is the parallelepiped containing the motif periodically repeated in the crystal. It is defined by the unit vectors a, b, c, or, by the six scalar parameters a, b, c, α, β, γ (see Fig. 1.1). The generic point into the unit cell is defined by the vector . . . r = x a + y b + z c, . . . where x, y, z are fractional coordinates (dimensionless and lying between 0 and 1). The volume of the unit cell is given by (see Fig. 1.2) . . . V = a ∧ b · c = b ∧ c · a = c ∧ a · b. (1.1). . .

MCE2005– a new version of a program for fast interactive visualization of electron and similar density maps optimized for small molecules

2007 ◽  
Vol 40 (3) ◽  
pp. 600-601 ◽  
Author(s):  
Jan Rohlíček ◽  
Michal Hušák
Keyword(s):  
Small Molecules ◽  
Interactive Visualization ◽  
Unit Cell ◽  
Basic Unit ◽  
Space Groups ◽  
Density Maps ◽  
Unit Cells ◽  
Similar Density ◽  
Atom Position

A new version ofMCEis described. This new version supports atom-position generation based on space groups and can display more unit cells around the basic unit cell. The display settings have been improved to allow the colors of the background, atoms and maps to be changed.

Distribution of molecular centres in unit cells with respect to packing patterns

2004 ◽  
Vol 60 (5) ◽  
pp. 539-546 ◽  
Author(s):  
Elna Pidcock ◽  
W. D. Sam Motherwell
Keyword(s):  
Building Blocks ◽  
Unit Cell ◽  
Space Group ◽  
Space Groups ◽  
Molecular Building Blocks ◽  
Unit Cells ◽  
Symmetry Operators

Packing patterns, a new description of the limited number of possible arrangements of molecular building blocks in a unit cell, were assigned to many thousands of structures belonging to the space groups P21/c, P\bar 1, P212121, P21 and C2/c [Pidcock & Motherwell (2004). Cryst. Growth. Des. 4, 611–620]. The position of the molecular centre (in fractional coordinates) in the unit cell for these structures has been surveyed, with respect to the space group and the packing pattern. The results clearly show that the position at which the molecular centre is found in the unit cell is correlated with the packing pattern. The relationships between the orientation of the packing pattern in the unit cell and the symmetry operators of the space group are explored. Popular orientations of packing patterns within the unit cell are given.

Bayesian removal of noise for increased sensitivity in vector pattern recognition lattice imaging of interfaces

1993 ◽  
Vol 51 ◽  
pp. 994-995
Author(s):  
L. Fei ◽  
P. Fraundorf
Keyword(s):  
Pattern Recognition ◽  
Perfect Crystal ◽  
Unit Cell ◽  
Ion Milling ◽  
Beam Radiation ◽  
Major Interest ◽  
Unit Cells ◽  
Mapping Process ◽  
Increased Sensitivity ◽  
Electron Microscopes

Interface structure is of major interest in microscopy. With high resolution transmission electron microscopes (TEMs) and scanning probe microscopes, it is possible to reveal structure of interfaces in unit cells, in some cases with atomic resolution. A. Ourmazd et al. proposed quantifying such observations by using vector pattern recognition to map chemical composition changes across the interface in TEM images with unit cell resolution. The sensitivity of the mapping process, however, is limited by the repeatability of unit cell images of perfect crystal, and hence by the amount of delocalized noise, e.g. due to ion milling or beam radiation damage. Bayesian removal of noise, based on statistical inference, can be used to reduce the amount of non-periodic noise in images after acquisition. The basic principle of Bayesian phase-model background subtraction, according to our previous study, is that the optimum (rms error minimizing strategy) Fourier phases of the noise can be obtained provided the amplitudes of the noise is given, while the noise amplitude can often be estimated from the image itself.

scholarly journals Prediction of Effective Thermal Conductivities of Four-Directional Carbon/Carbon Composites by Unit Cells with Different Sizes

2021 ◽  
Vol 11 (3) ◽  
pp. 1171
Author(s):  
Chang Xu ◽  
Zhihong Sun ◽  
Guowei Shao
Keyword(s):  
Carbon Fiber ◽  
Longitudinal Direction ◽  
Unit Cell ◽  
Carbon Composites ◽  
Temperature Boundary ◽  
The Matrix ◽  
Unit Cells ◽  
Experimental Values ◽  
Different Diameters ◽  
Thermal Conductivities

Two-unit cells developed to predict the effective thermal conductivities of four-directional carbon/carbon composites with the finite element method are proposed in this paper. The smaller-size unit cell is formulated from the larger-size unit cell by two 180° rotational transformations. The temperature boundary conditions corresponding to the two-unit cells are derived, and the validity is verified by the temperature and heat flux distributions at specific positions of the larger-size unit cell and the smaller-size unit cell. The thermal conductivities of the carbon fiber bundles and carbon fiber rods are measured firstly. Then, combined with the properties of the matrix, the effective thermal conductivities of the four-directional carbon/carbon composites are numerically predicted. The results in transverse direction predicted by the larger-size unit cell and the smaller-size unit cell are both higher than experimental values, which are 5.8 to 6.2% and 7.3 to 8.2%, respectively. In longitudinal direction, the calculated thermal conductivities of the larger-size unit cell and the smaller-size unit cell are 6.8% and 6.2% higher than the experimental results, respectively. In addition, carbon fiber rods with different diameters are set to clarify the influence on the effective thermal conductivities of the four-directional carbon/carbon composites.

scholarly journals A Multiband Antenna Stacked with Novel Metamaterial SCSRR and CSSRR for WiMAX/WLAN Applications

Micromachines ◽  
2021 ◽  
Vol 12 (2) ◽  
pp. 113
Author(s):  
Rajiv Mohan David ◽  
Mohammad Saadh AW ◽  
Tanweer Ali ◽  
Pradeep Kumar
Keyword(s):  
Split Ring Resonator ◽  
Unit Cell ◽  
Ring Resonators ◽  
Split Ring ◽  
Medium Theory ◽  
High Frequency Structure Simulator ◽  
Dimension Analysis ◽  
Unit Cells ◽  
Coaxial Feed ◽  
Triple Band

This paper presents an innovative method for the design of a triple band meta-mode antenna. This unique design of antenna finds application in a particular frequency band of WLAN and WiMAX. This antenna comprises of a square complimentary split ring resonator (SCSRR), a coaxial feed, and two symmetrical comb shaped split ring resonators (CSSRR). The metamaterial unit cell SCSRR independently gains control in the band range 3.15–3.25 GHz (WiMAX), whereas two symmetrical CSSRR unit cell controls the band in the ranges 3.91–4.01 GHz and 5.79–5.94 GHz (WLAN). This design methodology and the study of the suggested unit cells structure are reviewed in classical waveguide medium theory. The antenna has a miniaturized size of only 0.213λ0 × 0.192λ0 × 0.0271λ0 (20 × 18 × 2.54 mm3, where λ0 is the free space wavelength at 3.2 GHz). The detailed dimension analysis of the proposed antenna and its radiation efficiency are also presented in this paper. All the necessary simulations are carried out in High Frequency Structure Simulator (HFSS) 13.0 tool.

scholarly journals Dynamic crushing behavior of closed-cell aluminum foams based on different space-filling unit cells

2021 ◽  
Vol 21 (3) ◽  
Author(s):  
S. Talebi ◽  
R. Hedayati ◽  
M. Sadighi
Keyword(s):  
Surface Area ◽  
Dynamic Behavior ◽  
Energy Absorption ◽  
Peak Stress ◽  
Absorption Capacity ◽  
Unit Cell ◽  
Lattice Structures ◽  
Energy Absorption Capacity ◽  
Closed Cell ◽  
Unit Cells

AbstractClosed-cell metal foams are cellular solids that show unique properties such as high strength to weight ratio, high energy absorption capacity, and low thermal conductivity. Due to being computation and cost effective, modeling the behavior of closed-cell foams using regular unit cells has attracted a lot of attention in this regard. Recent developments in additive manufacturing techniques which have made the production of rationally designed porous structures feasible has also contributed to recent increasing interest in studying the mechanical behavior of regular lattice structures. In this study, five different topologies namely Kelvin, Weaire–Phelan, rhombicuboctahedron, octahedral, and truncated cube are considered for constructing lattice structures. The effects of foam density and impact velocity on the stress–strain curves, first peak stress, and energy absorption capacity are investigated. The results showed that unit cell topology has a very significant effect on the stiffness, first peak stress, failure mode, and energy absorption capacity. Among all the unit cell types, the Kelvin unit cell demonstrated the most similar behavior to experimental test results. The Weaire–Phelan unit cell, while showing promising results in low and medium densities, demonstrated unstable behavior at high impact velocity. The lattice structures with high fractions of vertical walls (truncated cube and rhombicuboctahedron) showed higher stiffness and first peak stress values as compared to lattice structures with high ratio of oblique walls (Weaire–Phelan and Kelvin). However, as for the energy absorption capacity, other factors were important. The lattice structures with high cell wall surface area had higher energy absorption capacities as compared to lattice structures with low surface area. The results of this study are not only beneficial in determining the proper unit cell type in numerical modeling of dynamic behavior of closed-cell foams, but they are also advantageous in studying the dynamic behavior of additively manufactured lattice structures with different topologies.

scholarly journals A complicated quasicrystal approximant ∊16 predicted by the strong-reflections approach

2010 ◽  
Vol 66 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Mingrun Li ◽  
Junliang Sun ◽  
Peter Oleynikov ◽  
Sven Hovmöller ◽  
Xiaodong Zou ◽  
...  
Keyword(s):  
Electron Diffraction ◽  
Unit Cell ◽  
Space Groups ◽  
Inverse Fourier Transformation ◽  
Transmission Electron ◽  
Structure Factors ◽  
Precession Electron Diffraction ◽  
Density Map ◽  
Electron Density Map ◽  
Good Agreement

The structure of a complicated quasicrystal approximant ∊16 was predicted from a known and related quasicrystal approximant ∊6 by the strong-reflections approach. Electron-diffraction studies show that in reciprocal space, the positions of the strongest reflections and their intensity distributions are similar for both approximants. By applying the strong-reflections approach, the structure factors of ∊16 were deduced from those of the known ∊6 structure. Owing to the different space groups of the two structures, a shift of the phase origin had to be applied in order to obtain the phases of ∊16. An electron-density map of ∊16 was calculated by inverse Fourier transformation of the structure factors of the 256 strongest reflections. Similar to that of ∊6, the predicted structure of ∊16 contains eight layers in each unit cell, stacked along the b axis. Along the b axis, ∊16 is built by banana-shaped tiles and pentagonal tiles; this structure is confirmed by high-resolution transmission electron microscopy (HRTEM). The simulated precession electron-diffraction (PED) patterns from the structure model are in good agreement with the experimental ones. ∊16 with 153 unique atoms in the unit cell is the most complicated approximant structure ever solved or predicted.

Crystal chemistry of transition metal diarsenates M 2As2O7 (M = Mn, Co, Ni, Zn): variants of the thortveitite structure

2010 ◽  
Vol 66 (6) ◽  
pp. 603-614 ◽  
Author(s):  
Matthias Weil ◽  
Berthold Stöger
Keyword(s):  
Crystal Structure ◽  
Low Temperature ◽  
Transition Metal ◽  
Basic Structure ◽  
Structure Type ◽  
Ambient Conditions ◽  
Modulated Structure ◽  
Triclinic Crystal System ◽  
Space Groups ◽  
Unit Cells

The structures of the 3d divalent transition-metal diarsenates M 2As2O7 (M = Mn, Co, Ni, Zn) can be considered as variants of the monoclinic (C2/m) thortveitite [Sc2Si2O7] structure type with a ≃ 6.7, b ≃ 8.5, c ≃ 4.7 Å, α ≃ 90, β ≃ 102, γ ≃ 90° and Z = 2. Co2As2O7 and Ni2As2O7 are dimorphic. Their high-temperature (β) polymorphs adopt the thortveitite aristotype structure in C2/m, whereas their low-temperature (α) polymorphs are hettotypes and crystallize with larger unit cells in the triclinic crystal system in space groups P\bar 1 and P1, respectively. Mn2As2O7 undergoes no phase transition and likewise adopts the thortveitite structure type in C2/m. Zn2As2O7 has an incommensurately modulated crystal structure [C2/m(α,0,γ)0s] with q = [0.3190 (1), 0, 0.3717 (1)] at ambient conditions and transforms reversibly to a commensurately modulated structure with Z = 12 (I2/c) below 273 K. The Zn phase resembles the structures and phase transitions of Cr2P2O7. Besides descriptions of the low-temperature Co2As2O7, Ni2As2O7 and Zn2As2O7 structures as five-, three- and sixfold superstructures of the thortveitite-type basic structure, the superspace approach can also be applied to descriptions of all the commensurate structures. In addition to the ternary M 2As2O7 phases, the quaternary phase (Ni,Co)2As2O7 was prepared and structurally characterized. In contrast to the previously published crystal structure of the mineral petewilliamsite, which has the same idealized formula and has been described as a 15-fold superstructure of the thortveitite-type basic structure in space group C2, synthetic (Ni,Co)2As2O7 can be considered as a solid solution adopting the α-Ni2As2O7 structure type. Differences of the two structure models for (Ni,Co)2As2O7 are discussed.

The Influence of Multiple Inclusions on the Cauchy Stress of a Spherical Particle-Reinforced Composite Under Uniaxial Loading

2014 ◽  
Author(s):  
Ke Niu ◽  
Armin Abedini ◽  
Zengtao Chen
Keyword(s):  
Spherical Particle ◽  
Single Particle ◽  
Volume Fraction ◽  
Unit Cell ◽  
Spherical Particles ◽  
Uniaxial Tensile ◽  
Cauchy Stress ◽  
Particle Volume ◽  
Volume Fractions ◽  
Unit Cells

This paper investigates the influence of multiple inclusions on the Cauchy stress of a spherical particle-reinforced metal matrix composite (MMC) under uniaxial tensile loading condition. The approach of three-dimensional cubic multi-particle unit cell is used to investigate the 15 non-overlapping identical spherical particles which are randomly distributed in the unit cell. The coordinates of the center of each particle are calculated by using the Random Sequential Adsorption algorithm (RSA) to ensure its periodicity. The models with reinforcement volume fractions of 10%, 15%, 20% and 25% are evaluated by using the finite element method. The behaviour of Cauchy stress for each model is analyzed at a far-field strain of 5%. For each reinforcement volume fraction, four models with different particle spatial distributions are evaluated and averaged to achieve a more accurate result. At the same time, single-particle unit cell and analytical model were developed. The stress-strain curves of multi-particle unit cells are compared with single-particle unit cells and the tangent homogenization model coupled with the Mori-Tanaka method. Only little scatters were found between unit cells with the same particle volume fractions. Multi-particle unit cells predict higher response than single particle unit cells. As the volume fraction of reinforcements increases, the Cauchy stress of MMCs increases.

Absolute configurations of Emycin D, E and F; mimicry of centrosymmetric space groups by mixtures of chiral stereoisomers

1999 ◽  
Vol 55 (4) ◽  
pp. 607-616 ◽  
Author(s):  
Martina Walker ◽  
Ehmke Pohl ◽  
Regine Herbst-Irmer ◽  
Martin Gerlitz ◽  
Jürgen Rohr ◽  
...  
Keyword(s):  
Correct Choice ◽  
Unit Cell ◽  
Monoclinic Space Group ◽  
Space Group ◽  
Space Groups ◽  
Centrosymmetric Space Group ◽  
Distance Restraints ◽  
Similar Distance ◽  
Triclinic Unit Cell ◽  
R Factors

The crystal structures of Emycin E (1), di-o-bromobenzoyl-Emycin F (2) and o-bromobenzoyl-Emycin D (3) have been determined by X-ray analysis at low temperature. Emycin E and o-bromobenzoyl-Emycin D both crystallize with two molecules in a triclinic unit cell. These two structures can be solved and refined either in the centrosymmetric space group P\bar 1, with apparent disorder localized at or around the expected chiral centre, or in the non-centrosymmetric space group P1 as mixtures of two diastereomers without disorder. Only the latter interpretation is consistent with the chemical and spectroscopic evidence. Refinements in the centrosymmetric and non-centrosymmetric space groups are compared in this paper and are shown to favour the chemically correct interpretation, more decisively so in the case of the bromo derivative as a result of the anomalous dispersion of bromine. Structures (1) and (3) provide a dramatic warning of the dangers inherent in the conventional wisdom that if a structure can be refined satisfactorarily in both centrosymmetric and non-centrosymmetric space groups, the former should always be chosen. In these two cases, despite apparently acceptable intensity statistics and R factors (5.87 and 3.55%), the choice of the centrosymmetric space group leads to the serious chemical error that the triclinic unit cell contains a racemate rather than two chiral diastereomers! The weakest reflections are shown to be most sensitive to the correct choice of space group, underlining the importance of refining against all data rather than against intensities greater than a specified threshold. The use of similar-distance restraints is shown to be beneficial in both P1 refinements. Di-o-bromobenzoyl-Emycin F crystallizes in the monoclinic space group P21 with one molecule in the asymmetric unit and so does not give rise to these problems of interpretation. The absolute configuration of the two bromo derivatives, and hence the Emycins in general, was determined unambiguously as S at the chiral centre C3.

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