Editing Crystal structure (section)

=== Planes and directions ===
The crystallographic directions are geometric [[line (mathematics)|line]]s linking nodes ([[atom]]s, [[ion]]s or [[molecule]]s) of a crystal. Likewise, the crystallographic [[plane (mathematics)|plane]]s 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:<ref name="Solid State Physics 2010"/>

*[[optics|Optical properties]]: [[Refractive index]] is directly related to density (or periodic density fluctuations).
*[[Adsorption]] and [[reactivity (chemistry)|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.

[[File:Cristal densite surface.svg|class=skin-invert-image|thumb|Dense crystallographic planes]]

*Microstructural [[Crystallographic defect|defects]]: [[sintering|Pores]] and [[crystallite]]s tend to have straight grain boundaries following higher density planes.
*[[cleavage (crystal)|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):
:<math>d_{\ell mn}= \frac {a} { \sqrt{\ell ^2 + m^2 + n^2} }</math>
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 {{angbr|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 [[face-centered cubic]] (fcc) and [[body-centered 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 (crystal)|supercell]] and hence are again simply the [[Cartesian coordinates|Cartesian directions]].