## Crystal structure |

In **crystal structure** is a description of the ordered arrangement of ^{[3]} Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of

The smallest group of particles in the material that constitutes the repeating pattern is the **unit cell** of the structure. The unit cell completely defines the symmetry and structure of the entire crystal lattice, which is built up by repetitive

The lengths of the principal axes, or edges, of the unit cell and the angles between them are the *lattice parameters*. The ^{[3]} All possible symmetric arrangements of particles in three-dimensional space may be described by the 230

The crystal structure and symmetry play a critical role in determining many physical properties, such as

- unit cell
- classification by symmetry
- atomic coordination
- grain boundaries
- defects and impurities
- prediction of structure
- polymorphism
- physical properties
- see also
- references
- external links

Crystal structure is described in terms of the geometry of arrangement of particles in the unit cell. The unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure.^{[4]} The geometry of the unit cell is defined as a *a*, *b*, *c*) and the angles between them (α, β, γ). The positions of particles inside the unit cell are described by the *x _{i}*,

Vectors and planes in a crystal lattice are described by the three-value *ℓ*, *m*, and *n* as directional orthogonal parameters, which are separated by 90°.^{[6]}

By definition, the syntax (*ℓmn*) denotes a plane that intercepts the three points *a*_{1}/*ℓ*, *a*_{2}/*m*, and *a*_{3}/*n*, 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, it means that the planes do not intersect that axis (i.e., the intercept is "at infinity"). A plane containing a coordinate axis is translated so that it no longer contains that axis before its Miller indices are determined. The Miller indices for a plane are

Considering only (*ℓmn*) planes intersecting one or more lattice points (the *lattice planes*), the distance *d* between adjacent lattice planes is related to the (shortest)

The crystallographic directions are geometric *planes* linking nodes. Some directions and planes have a higher density of nodes. These high density planes have an influence on the behavior of the crystal as follows:^{[3]}

Optical properties :Refractive index is directly related to density (or periodic density fluctuations).Adsorption andreactivity : 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 andcrystallites 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, rhombohedral, tetragonal, and trigonal/hexagonal systems there is one unique axis (sometimes called the **principal axis**) which has higher **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.

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 *a*, the spacing *d* between adjacent (ℓmn) lattice planes is (from above):

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:

- Coordinates in
*angle brackets*such as ⟨100⟩ denote a*family*of directions that are equivalent due to symmetry operations, such as [100], [010], [001] or the negative of any of those directions. - Coordinates in
*curly brackets*or*braces*such as {100} denote a family of plane normals that are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.

For

The spacing * d* between adjacent (

- Cubic:
- Tetragonal:
- Hexagonal:
- Rhombohedral:
- Orthorhombic:
- Monoclinic:
- Triclinic:

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