Editing Crystal structure (section)

== Unit cell ==
{{main|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.<ref>{{cite book|title=Basic Solid State Chemistry |first=Anthony R. |last=West |publisher=Wiley |edition=2nd |date=1999 |page=1 |isbn=978-0-471-98756-7}}</ref> 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]] (''x<sub>i</sub>'', ''y<sub>i</sub>'', ''z<sub>i</sub>'') 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.<ref name="Space-group symmetry">International Tables for Crystallography (2006). Volume A, Space-group symmetry.</ref>

<gallery class="center skin-invert-image">
Image:Lattic_simple_cubic.svg|Simple cubic (P)
Image:Lattice_body_centered_cubic.svg|Body-centered cubic (I)
Image:Lattice_face_centered_cubic.svg|Face-centered cubic (F)
</gallery>

=== Miller indices ===
[[File:Miller Indices Cubes.svg|class=skin-invert-image|thumb|upright=1.2|Planes with different Miller indices in cubic crystals]]
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.<ref name="Physics 1991">Encyclopedia of Physics (2nd Edition), [[Rita G. Lerner|R.G. Lerner]], G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3</ref>

By definition, the syntax (''hkℓ'') denotes a plane that intercepts the three points ''a''<sub>1</sub>/''h'', ''a''<sub>2</sub>/''k'', and ''a''<sub>3</sub>/''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 [[integer]]s with no common factors. Negative indices are indicated with horizontal bars, as in (1{{overbar|2}}3). 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
:<math>d = \frac{2\pi}  {|\mathbf{g}_{h k \ell}|}</math>

=== 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]].

===Interplanar spacing===
The spacing '''''d''''' between adjacent (''hkℓ'') lattice planes is given by:<ref>{{cite web |title=4. Direct and reciprocal lattices |website=CSIC Dept de Cristalografia y Biologia Estructural |url= http://www.xtal.iqfr.csic.es/Cristalografia/parte_04-en.html |date=6 Apr 2017 |access-date=18 May 2017 }}</ref><ref>{{Cite book|last=Edington|first=J. W.|date=1975|title=Electron Diffraction in the Electron Microscope|language=en-gb|doi=10.1007/978-1-349-02595-4|isbn=978-0-333-18292-5}}</ref>

*Cubic:
*:<math>\frac {1} {d^{2}}= \frac {h^2+k^2+\ell^2} {a^2}</math>
*Tetragonal:
*:<math>\frac {1} {d^{2}}= \frac {h^2+k^2} {a^2}+\frac{\ell^2}{c^2}</math>
*Hexagonal:
*:<math>\frac {1} {d^{2}}= \frac{4}{3}\left(\frac{h^2+hk+k^2}{a^2}\right)+\frac{\ell^2}{c^2}</math>
*Rhombohedral ([[Unit cell|primitive setting]]):
*:<math>\frac {1} {d^{2}}= \frac{(h^2+k^2+\ell^2)\sin^2\alpha+2(hk+k\ell+h\ell)(\cos^2\alpha-\cos\alpha)}{a^2(1-3\cos^2\alpha+2\cos^3\alpha)}</math>
*Orthorhombic:
*:<math>\frac {1} {d^{2}}= \frac{h^2}{a^2}+\frac{k^2}{b^2}+\frac{\ell^2}{c^2}</math>
*Monoclinic:
*:<math>\frac {1} {d^{2}}=\left(\frac{h^2}{a^2}+\frac{k^2\sin^2\beta}{b^2}+\frac{\ell^2}{c^2}-\frac{2h\ell\cos\beta}{ac}\right) \csc^2\beta</math>
*Triclinic:
*:<math>\frac {1} {d^{2}}= \frac{\frac{h^2}{a^2}\sin^2\alpha+\frac{k^2}{b^2}\sin^2\beta+\frac{\ell^2}{c^2}\sin^2\gamma+\frac{2k\ell}{bc}(\cos\beta\cos\gamma-\cos\alpha)+\frac{2h\ell}{ac}(\cos\gamma\cos\alpha-\cos\beta)+\frac{2hk}{ab}(\cos\alpha\cos\beta-\cos\gamma)}{1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha\cos\beta\cos\gamma}</math>