Crystal structure


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 and the angles between them. The positions of particles inside the unit cell are described by the fractional coordinates 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 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. If one or more of the indices is zero, the planes do not intersect that axis. 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. 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 planes intersecting one or more lattice points, the distance d between adjacent lattice planes is related to the reciprocal lattice vector orthogonal to the planes by the formula

Planes and directions

The crystallographic directions are geometric lines linking nodes 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.
  • 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 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 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 ; similarly for the reciprocal lattice. So, in this common case, the Miller indices and both simply denote normals/directions in Cartesian coordinates. For cubic crystals with lattice constant a, the spacing d between adjacent lattice planes is :
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 denote a family of directions that are equivalent due to symmetry operations, such as , , or the negative of any of those directions.
  • Coordinates in curly brackets or braces such as denote a family of plane normals that are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.
For face-centered cubic and body-centered cubic lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions.

Interplanar spacing

The perpendicular spacing d between adjacent lattice planes is given by:
  • Cubic:
  • :
  • Tetragonal:
  • :
  • Hexagonal:
  • :
  • Rhombohedral :
  • :
  • Orthorhombic:
  • :
  • Monoclinic:
  • :
  • Triclinic:
  • :

    Classification by symmetry

The defining property of a crystal is its inherent symmetry. Performing certain symmetry operations on the crystal lattice leaves it unchanged. All crystals have translational symmetry in three directions, but some have other symmetry elements as well. For example, rotating the crystal 180° about a certain axis may result in an atomic configuration that is identical to the original configuration; the crystal has twofold rotational symmetry about this axis. In addition to rotational symmetry, a crystal may have symmetry in the form of mirror planes, and also the so-called compound symmetries, which are a combination of translation and rotation or mirror symmetries. A full classification of a crystal is achieved when all inherent symmetries of the crystal are identified.

Lattice systems

Lattice systems are a grouping of crystal structures according to the point groups of their lattice. All crystals fall into one of seven lattice systems. They are related to, but not the same as the seven crystal systems.
The most symmetric, the cubic or isometric system, has the symmetry of a cube, that is, it exhibits four threefold rotational axes oriented at 109.5° with respect to each other. These threefold axes lie along the body diagonals of the cube. The other six lattice systems, are hexagonal, tetragonal, rhombohedral, orthorhombic, monoclinic and triclinic which is the least symmetrical as it possess only identity.

Bravais lattices

s, also referred to as space lattices, describe the geometric arrangement of the lattice points, and therefore the translational symmetry of the crystal. The three dimensions of space afford 14 distinct Bravais lattices describing the translational symmetry. All crystalline materials recognized today, not including quasicrystals, fit in one of these arrangements. The fourteen three-dimensional lattices, classified by lattice system, are shown above.
The crystal structure consists of the same group of atoms, the basis, positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the Bravais lattices. The characteristic rotation and mirror symmetries of the unit cell is described by its crystallographic point group.

Crystal systems

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, 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.