Unit cell
The crystal structure of a material (the arrangement of atoms within a given type of crystal) can be described in terms of its unit cell. The unit cell is a box containing one or more atoms arranged in three dimensions. The unit cells
stacked in
threedimensional space describe the bulk arrangement of atoms of the crystal. The unit cell is represented in terms of its
lattice parameters, which are the lengths of the cell edges (a, b and c) and the angles between them (alpha, beta and gamma), while the positions of the atoms inside the unit cell are described by the set of atomic positions (x_{i}, y_{i}, z_{i}) measured from a reference lattice point. Commonly, atomic positions are represented in terms of
fractional coordinates, relative to the unit cell lengths.
The atom positions within the unit cell can be calculated through application of symmetry operations to the asymmetric unit. The asymmetric unit refers to the smallest possible occupation of space within the unit cell. This does not, however imply that the entirety of the asymmetric unit must lie within the boundaries of the unit cell. Symmetric transformations of atom positions are calculated from the
space group of the crystal structure, and this is usually a
black box operation performed by computer programs. However, manual calculation of the atomic positions within the unit cell can be performed from the asymmetric unit, through the application of the symmetry operators described within the International Tables for Crystallography: Volume A.^{
[4]}
Miller indices
Planes with different Miller indices in cubic crystals
Vectors and planes in a crystal lattice are described by the threevalue Miller index notation. It uses the indices ℓ, m, and n as directional parameters, which are separated by 90°, and are thus orthogonal.^{
[5]}
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
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 (ℓmn) 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
 $d={\frac {2\pi }{\mathbf {g} _{\ell mn}}}$
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 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 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.
Dense crystallographic planes
 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, rhombohedral, tetragonal, and trigonal/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):
 $d_{\ell mn}={\frac {a}{\sqrt {\ell ^{2}+m^{2}+n^{2}}}}$
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:
 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
facecentered cubic (fcc) and
bodycentered cubic (bcc) 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 spacing d between adjacent (hkl) lattice planes is given by:^{
[6]}
 Cubic:
 ${\frac {1}{d^{2}}}={\frac {h^{2}+k^{2}+l^{2}}{a^{2}}}$
 Tetragonal:
 ${\frac {1}{d^{2}}}={\frac {h^{2}+k^{2}}{a^{2}}}+{\frac {l^{2}}{c^{2}}}$
 Hexagonal:
 ${\frac {1}{d^{2}}}={\frac {4}{3}}\left({\frac {h^{2}+hk+k^{2}}{a^{2}}}\right)+{\frac {l^{2}}{c^{2}}}$
 Rhombohedral:
 ${\frac {1}{d^{2}}}={\frac {(h^{2}+k^{2}+l^{2})\sin ^{2}\alpha +2(hk+kl+hl)(\cos ^{2}\alpha \cos \alpha )}{a^{2}(13\cos ^{2}\alpha +2\cos ^{3}\alpha )}}$
 Orthorhombic:
 ${\frac {1}{d^{2}}}={\frac {h^{2}}{a^{2}}}+{\frac {k^{2}}{b^{2}}}+{\frac {l^{2}}{c^{2}}}$
 Monoclinic:
 ${\frac {1}{d^{2}}}=\left({\frac {h^{2}}{a^{2}}}+{\frac {k^{2}\sin ^{2}\beta }{b^{2}}}+{\frac {l^{2}}{c^{2}}}{\frac {2hl\cos \beta }{ac}}\right)\csc ^{2}\beta$
 Triclinic:
 ${\frac {1}{d^{2}}}={\frac$