Editing Bravais lattice (section)

==Unit cell==
{{Main|Unit cell}}
In crystallography, there is the concept of a unit cell which comprises the space between adjacent lattice points as well as any atoms in that space. A unit cell is defined as a space that, when translated through a subset of all vectors described by <math>\mathbf{R} = n_{1}\mathbf{a}_{1} + n_{2}\mathbf{a}_{2} + n_{3}\mathbf{a}_{3}</math>, fills the lattice space without overlapping or voids. (I.e., a lattice space is a multiple of a unit cell.)<ref name=":0">{{Cite book|last1=Ashcroft|first1=Neil|title=Solid State Physics|last2=Mermin|first2=Nathaniel|publisher=Saunders College Publishing|year=1976|isbn=0030839939|pages=71–72}}</ref> 

There are mainly two types of unit cells: primitive unit cells and conventional unit cells. A primitive cell is the very smallest component of a lattice (or crystal) which, when stacked together with lattice translation operations, reproduces the whole lattice (or crystal).<ref>{{Cite web|title=Materials & Solid State Chemistry (course notes)|author= Peidong Yang|url=http://nanowires.berkeley.edu/teaching/253a/2016/253A-2016-01.pdf|year=2016|department=UC Berkeley|series=Chem 253}}</ref> Note that the translations must be lattice translation operations that cause the lattice to appear unchanged after the translation. If arbitrary translations were allowed, one could make a primitive cell half the size of the true one, and translate twice as often, as an example. 

Another way of defining the size of a primitive cell that avoids invoking lattice translation operations, is to say that the primitive cell is the smallest possible component of a lattice (or crystal) that can be repeated to reproduce the whole lattice (or crystal), ''and'' that contains exactly one lattice point. In either definition, the primitive cell is characterized by its small size. 

There are clearly many choices of cell that can reproduce the whole lattice when stacked (two lattice halves, for instance), and the minimum size requirement distinguishes the primitive cell from all these other valid repeating units. If the lattice or crystal is 2-dimensional, the primitive cell has a minimum area; likewise in 3 dimensions the primitive cell has a minimum volume. 

Despite this rigid minimum-size requirement, there is not one unique choice of primitive unit cell. In fact, all cells whose borders are primitive translation vectors will be primitive unit cells. The fact that there is not a unique choice of primitive translation vectors for a given lattice leads to the multiplicity of possible primitive unit cells. Conventional unit cells, on the other hand, are not necessarily minimum-size cells. They are chosen purely for convenience and are often used for illustration purposes. They are loosely defined.

===Primitive unit cell===
Primitive unit cells are defined as unit cells with the smallest volume for a given crystal. (A crystal is a lattice and a basis at every lattice point.) To have the smallest cell volume, a primitive unit cell must contain (1) only one lattice point and (2) the minimum amount of basis constituents (e.g., the minimum number of atoms in a basis). For the former requirement, counting the number of lattice points in a unit cell is such that, if a lattice point is shared by ''m'' adjacent unit cells around that lattice point, then the point is counted as 1/''m''. The latter requirement is necessary since there are crystals that can be described by more than one combination of a lattice and a basis. For example, a crystal, viewed as a lattice with a single kind of atom located at every lattice point (the simplest basis form), may also be viewed as a lattice with a basis of two atoms. In this case, a primitive unit cell is a unit cell having only one lattice point in the first way of describing the crystal in order to ensure the smallest unit cell volume.

There can be more than one way to choose a primitive cell for a given crystal and each choice will have a different primitive cell shape, but the primitive cell volume is the same for every choice and each choice will have the property that a one-to-one correspondence can be established between primitive unit cells and discrete lattice points over the associated lattice. All primitive unit cells with different shapes for a given crystal have the same volume by definition; For a given crystal, if ''n'' is the density of lattice points in a lattice ensuring the minimum amount of basis constituents and ''v'' is the volume of a chosen primitive cell, then ''nv'' = 1 resulting in ''v'' = 1/''n'', so every primitive cell has the same volume of 1/''n''.<ref name=":0" />

Among all possible primitive cells for a given crystal, an obvious primitive cell may be the [[parallelepiped]] formed by a chosen set of primitive translation vectors. (Again, these vectors must make a lattice with the minimum amount of basis constituents.)<ref name=":0" /> That is, the set of all points <math>\mathbf{r} = x_{1}\mathbf{a}_{1} + x_{2}\mathbf{a}_{2} + x_{3}\mathbf{a}_{3}</math> where <math>0 \le x_{i} < 1</math> and <math>\mathbf{a}_{i}</math> is the chosen primitive vector. This primitive cell does not always show the clear symmetry of a given crystal. In this case, a conventional unit cell easily displaying the crystal symmetry is often used. The conventional unit cell volume will be an integer-multiple of the primitive unit cell volume.