Related Experiment Video
Updated: Jul 14, 2026
06:35
Construction and Systematical Symmetric Studies of a Series of Supramolecular Clusters with Binary or Ternary Ammonium Triphenylacetates
Published on: February 15, 2016
Orthorhombic sphere packings. II. Bivariant lattice complexes.
Heidrun Sowa1, Elke Koch, Werner Fischer
1Institut für Mineralogie, Petrologie und Kristallographie der Philipps-Universität Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany. heidrun.sowa@geo.uni-goettingen.de
Summary
Researchers identified 90 types of sphere packings within orthorhombic lattice complexes, advancing crystal structure analysis. This work details their symmetries and applications in describing crystal structures.
Area of Science:
- Crystallography
- Materials Science
- Geometry
Background:
- Sphere packings are fundamental to understanding crystal structures.
- Orthorhombic bivariant lattice complexes represent a significant class of crystal structures.
- Previous classifications may not have exhaustively covered all possible sphere packings within these complexes.
Purpose of the Study:
- To systematically derive and classify all homogeneous sphere packings and interpenetrating sphere layers.
- To analyze the symmetries of these derived sphere packings.
- To demonstrate the utility of these sphere packings in crystal structure analysis.
Main Methods:
- Systematic derivation of sphere packings based on 22 orthorhombic bivariant lattice complexes.
- Analysis of the maximal inherent symmetry for each sphere packing type.
- Comparative analysis of crystal structures using the derived sphere packings.
Main Results:
- Identification of 90 distinct types of sphere packings.
- Determination that only 47 of these types exhibit orthorhombic maximal inherent symmetry.
- Demonstration of sphere packings' applicability in comparing and describing crystal structures.
Conclusions:
- The study provides a comprehensive catalog of sphere packings for orthorhombic lattice complexes.
- The findings offer new insights into the symmetry properties of sphere packings.
- The derived sphere packings serve as a valuable tool for crystallographic research and structure description.
Related Concept Videos
01:24The Seven Crystal Systems: Overview
Crystals with various point group symmetries belong to different crystal classes, which are synonymous terms. Despite being in the same class, crystals may have distinct shapes, like cubes and octahedra. There are 32 three-dimensional point groups, all of which are systematically divided into seven crystal systems.The basic cubic crystal system, exemplified by NaCl, features orthogonal vectors (α = β = �� = 90°) of equal lengths (a = b = c). When specific requirements are not imposed on the...
02:46Crystal Field Theory - Tetrahedral and Square Planar Complexes
Tetrahedral Complexes
Crystal field theory (CFT) is applicable to molecules in geometries other than octahedral. In octahedral complexes, the lobes of the dx2−y2 and dz2 orbitals point directly at the ligands. For tetrahedral complexes, the d orbitals remain in place, but with only four ligands located between the axes. None of the orbitals points directly at the tetrahedral ligands. However, the dx2−y2 and dz2 orbitals (along the Cartesian axes) overlap with the ligands less than the dxy,...
Crystal field theory (CFT) is applicable to molecules in geometries other than octahedral. In octahedral complexes, the lobes of the dx2−y2 and dz2 orbitals point directly at the ligands. For tetrahedral complexes, the d orbitals remain in place, but with only four ligands located between the axes. None of the orbitals points directly at the tetrahedral ligands. However, the dx2−y2 and dz2 orbitals (along the Cartesian axes) overlap with the ligands less than the dxy,...
02:42Valence Bond Theory
Coordination compounds and complexes exhibit different colors, geometries, and magnetic behavior, depending on the metal atom/ion and ligands from which they are composed. In an attempt to explain the bonding and structure of coordination complexes, Linus Pauling proposed the valence bond theory, or VBT, using the concepts of hybridization and the overlapping of the atomic orbitals. According to VBT, the central metal atom or ion (Lewis acid) hybridizes to provide empty orbitals of suitable...
02:33Lattice Centering and Coordination Number
The structure of a crystalline solid, whether a metal or not, is best described by considering its simplest repeating unit, which is referred to as its unit cell. The unit cell consists of lattice points that represent the locations of atoms or ions. The entire structure then consists of this unit cell repeating in three dimensions. The three different types of unit cells present in the cubic lattice are illustrated in Figure 1.
Types of Unit Cells
Imagine taking a large number of identical...
Types of Unit Cells
Imagine taking a large number of identical...
02:58Crystal Field Theory - Octahedral Complexes
Crystal Field Theory
To explain the observed behavior of transition metal complexes (such as colors), a model involving electrostatic interactions between the electrons from the ligands and the electrons in the unhybridized d orbitals of the central metal atom has been developed. This electrostatic model is crystal field theory (CFT). It helps to understand, interpret, and predict the colors, magnetic behavior, and some structures of coordination compounds of transition metals.
CFT focuses on...
To explain the observed behavior of transition metal complexes (such as colors), a model involving electrostatic interactions between the electrons from the ligands and the electrons in the unhybridized d orbitals of the central metal atom has been developed. This electrostatic model is crystal field theory (CFT). It helps to understand, interpret, and predict the colors, magnetic behavior, and some structures of coordination compounds of transition metals.
CFT focuses on...
01:29Crystallographic Point Groups
Crystallographic point groups represent the various symmetry operations that can occur within crystals. They are unique in that at least one point will always remain unchanged during these actions. For instance, consider the triclinic system. This system, devoid of any axis or plane of symmetry, aligns with the C1 and Ci point groups.where Cᵢ is characterized solely by a center of inversion.Contrastingly, the monoclinic system introduces an element of symmetry. This system with one plane and...