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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Crystal structure | 1/5 | https://en.wikipedia.org/wiki/Crystal_structure | reference | science, encyclopedia | 2026-05-05T10:52:12.114890+00:00 | kb-cron |
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles in a material that constitutes this repeating pattern is the unit cell of the structure. The unit cell completely reflects the symmetry and structure of the entire crystal, which is built up by repetitive translation of the unit cell along its principal axes. The translation vectors define the nodes of the Bravais lattice. The lengths of principal axes/edges, of the unit cell and angles between them are lattice constants, also called lattice parameters or cell parameters. The symmetry properties of a crystal are described by the concept of space groups. All possible symmetric arrangements of particles in three-dimensional space may be described by 230 space groups. The crystal structure and symmetry play a critical role in determining many physical properties, such as cleavage, electronic band structure, and optical transparency.
== Unit cell ==
Crystal structure is described in terms of the geometry of the arrangement of particles in the unit cells. The unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure. The geometry of the unit cell is defined as a parallelepiped, providing six lattice parameters taken as the lengths of the cell edges (a, b, c) and the angles between them (α, β, γ). The positions of particles inside the unit cell are described by the fractional coordinates (xi, yi, zi) along the cell edges, measured from a reference point. It is thus only necessary to report the coordinates of a smallest asymmetric subset of particles, called the crystallographic asymmetric unit. The asymmetric unit may be chosen so that it occupies the smallest physical space, which means that not all particles need to be physically located inside the boundaries given by the lattice parameters. All other particles of the unit cell are generated by the symmetry operations that characterize the symmetry of the unit cell. The collection of symmetry operations of the unit cell is expressed formally as the space group of the crystal structure.
=== Miller indices ===
Vectors and planes in a crystal lattice are described by the three-value Miller index notation. This syntax uses the indices h, k, and ℓ as directional parameters. By definition, the syntax (hkℓ) denotes a plane that intercepts the three points a1/h, a2/k, and a3/ℓ, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell (in the basis of the lattice vectors). If one or more of the indices is zero, the planes do not intersect that axis (i.e., the intercept is "at infinity"). A plane containing a coordinate axis is translated to no longer contain that axis before its Miller indices are determined. The Miller indices for a plane are integers with no common factors. Negative indices are indicated with horizontal bars, as in (123). In an orthogonal coordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components of a vector normal to the plane. Considering only (hkℓ) planes intersecting one or more lattice points (the lattice planes), the distance d between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula
d
=
2
π
|
g
h
k
ℓ
|
{\displaystyle d={\frac {2\pi }{|\mathbf {g} _{hk\ell }|}}}
=== Planes and directions === The crystallographic directions are geometric lines linking nodes (atoms, ions or molecules) of a crystal. Likewise, the crystallographic planes are geometric planes linking nodes. Some directions and planes have a higher density of nodes. These high-density planes influence the behaviour of the crystal as follows:
Optical properties: Refractive index is directly related to density (or periodic density fluctuations). Adsorption and reactivity: Physical adsorption and chemical reactions occur at or near surface atoms or molecules. These phenomena are thus sensitive to the density of nodes. Surface tension: The condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species. The surface tension of an interface thus varies according to the density on the surface.
Microstructural defects: Pores and crystallites tend to have straight grain boundaries following higher density planes. Cleavage: This typically occurs preferentially parallel to higher density planes. Plastic deformation: Dislocation glide occurs preferentially parallel to higher density planes. The perturbation carried by the dislocation (Burgers vector) is along a dense direction. The shift of one node in a more dense direction requires a lesser distortion of the crystal lattice. Some directions and planes are defined by symmetry of the crystal system. In monoclinic, trigonal, tetragonal, and hexagonal systems there is one unique axis (sometimes called the principal axis) which has higher rotational symmetry than the other two axes. The basal plane is the plane perpendicular to the principal axis in these crystal systems. For triclinic, orthorhombic, and cubic crystal systems the axis designation is arbitrary and there is no principal axis.
==== Cubic structures ==== For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted a); similarly for the reciprocal lattice. So, in this common case, the Miller indices (ℓmn) and [ℓmn] both simply denote normals/directions in Cartesian coordinates. For cubic crystals with lattice constant a, the spacing d between adjacent (ℓmn) lattice planes is (from above):
d
ℓ
m
n
=
a
ℓ
2
+
m
2
+
n
2
{\displaystyle d_{\ell mn}={\frac {a}{\sqrt {\ell ^{2}+m^{2}+n^{2}}}}}
Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes: