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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Crystal structure | 3/5 | https://en.wikipedia.org/wiki/Crystal_structure | reference | science, encyclopedia | 2026-05-05T10:52:12.114890+00:00 | kb-cron |
A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case the crystal system and lattice system both have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both lattice systems exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system.
In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.
==== Point groups ==== The crystallographic point group or crystal class is the mathematical group comprising the symmetry operations that leave at least one point unmoved and that leave the appearance of the crystal structure unchanged. These symmetry operations include
Reflection, which reflects the structure across a reflection plane Rotation, which rotates the structure a specified portion of a circle about a rotation axis Inversion, which changes the sign of the coordinate of each point with respect to a center of symmetry or inversion point Improper rotation, which consists of a rotation about an axis followed by an inversion. Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively called symmetry elements. There are 32 possible crystal classes. Each one can be classified into one of the seven crystal systems.
=== Space groups === In addition to the operations of the point group, the space group of the crystal structure contains translational symmetry operations. These include:
Pure translations, which move a point along a vector Screw axes, which rotate a point around an axis while translating parallel to the axis. Glide planes, which reflect a point through a plane while translating it parallel to the plane. There are 230 distinct space groups.
== Atomic coordination == By considering the arrangement of atoms relative to each other, their coordination numbers, interatomic distances, types of bonding, etc., it is possible to form a general view of the structures and alternative ways of visualizing them.
=== Close packing ===
The principles involved can be understood by considering the most efficient way of packing together equal-sized spheres and stacking close-packed atomic planes in three dimensions. For example, if plane A lies beneath plane B, there are two possible ways of placing an additional atom on top of layer B. If an additional layer were placed directly over plane A, this would give rise to the following series:
...ABABABAB... This arrangement of atoms in a crystal structure is known as hexagonal close packing (hcp). If, however, all three planes are staggered relative to each other and it is not until the fourth layer is positioned directly over plane A that the sequence is repeated, then the following sequence arises:
...ABCABCABC... This type of structural arrangement is known as cubic close packing (ccp) or face-centered cubic (fcc). The unit cell of a ccp arrangement of atoms is the face-centered cubic (fcc) unit cell. This is not immediately obvious, as the closely packed layers are parallel to the {111} planes of the fcc unit cell. There are four different orientations of the close-packed layers.
=== APF and CN ===
One important characteristic of a crystalline structure is its atomic packing factor (APF). This is calculated by assuming that all the atoms are identical spheres, with a radius large enough that each sphere abuts on the next. The atomic packing factor is the proportion of space filled by these spheres which can be worked out by calculating the total volume of the spheres and dividing by the volume of the cell as follows:
A
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{\displaystyle \mathrm {APF} ={\frac {N_{\mathrm {particle} }V_{\mathrm {particle} }}{V_{\text{unit cell}}}}}
Another important characteristic of a crystalline structure is its coordination number (CN). This is the number of nearest neighbours of a central atom in the structure. The APFs and CNs of the most common crystal structures are shown below:
The 74% packing efficiency of the FCC and HCP is the maximum density possible in unit cells constructed of spheres of only one size.
=== Interstitial sites ===
Interstitial sites refer to the empty spaces in between the atoms in the crystal lattice. These spaces can be filled by oppositely charged ions to form multi-element structures. They can also be filled by impurity atoms or self-interstitials to form interstitial defects.
== Defects and impurities ==
Real crystals feature defects or irregularities in the ideal arrangements described above and it is these defects that critically determine many of the electrical and mechanical properties of real materials.
=== Impurities ===
When one atom substitutes for one of the principal atomic components within the crystal structure, alteration in the electrical and thermal properties of the material may ensue. Impurities may also manifest as electron spin impurities in certain materials. Research on magnetic impurities demonstrates that substantial alteration of certain properties such as specific heat may be affected by small concentrations of an impurity, as for example impurities in semiconducting ferromagnetic alloys may lead to different properties as first predicted in the late 1960s.
=== Dislocations ===
Dislocations in a crystal lattice are line defects that are associated with local stress fields. Dislocations allow shear at lower stress than that needed for a perfect crystal structure. The local stress fields result in interactions between the dislocations which then result in strain hardening or cold working.
=== Grain boundaries ===