Unit Cell

A unit cell is a region of a periodic crystal structure whose repetition by translations reproduces the structure throughout space. In three dimensions, it is conventionally represented as a parallelepiped defined by three independent lattice vectors. Its geometry describes the repeat distances and directions; its contents specify the arrangement of atoms, ions, or molecules. A unit cell need not be the smallest possible repeating region, because a larger conventional cell may express the crystal’s symmetry more clearly. (journals.iucr.org)

Lattice and atomic contents

The distinction between a lattice and a structure is fundamental in crystallography. A Bravais lattice is an abstract periodic array of equivalent points. A crystal structure is obtained by associating the same atomic arrangement, often called a motif or basis, with every lattice point. Lattice points therefore do not necessarily correspond to individual atoms: a motif may contain several atoms of different chemical species. (it.iucr.org)

For a primitive lattice basis $\mathbf a,\mathbf b,\mathbf c$, every lattice translation has the form

The coefficients are integers, distinguishing lattice translations from arbitrary displacements. Translating every constituent by $\mathbf T$ leaves the ideal periodic structure unchanged. Cell boundaries are bookkeeping devices rather than physical walls: an atom or molecule may straddle a boundary, and changing the cell origin changes its coordinates without changing the structure. (it.iucr.org)

Cell geometry and coordinates

Six lattice parameters specify the cell’s size and shape: the edge lengths $a,b,c$, and the angles $\alpha,\beta,\gamma$. By convention, $\alpha$ lies between $\mathbf b$ and $\mathbf c$, $\beta$ between $\mathbf a$ and $\mathbf c$, and $\gamma$ between $\mathbf a$ and $\mathbf b$. Lengths are commonly reported in ångströms, with $1\ \text{Å}=10^{-10}$ metre. The axes need not be perpendicular. (iucr.org)

The volume is

Here the determinant is calculated from the matrix whose columns are the cell vectors in Cartesian coordinates. For an orthogonal cell, this reduces to $V=abc$. (iucr.org)

An atomic position is conveniently written using fractional coordinates:

Thus $(1/2,1/2,1/2)$ denotes the cell centre. Adding integers to these coordinates selects translationally equivalent positions. Coordinates can be reduced to $0\leq x,y,z<1$, although doing so may split a molecule across cell boundaries. In a nonorthogonal cell, coordinate differences alone do not give physical distances; the cell geometry must also enter the calculation. (iucr.org)

Primitive and conventional cells

A primitive cell contains exactly one lattice point when shared boundary points are counted appropriately. It has the minimum volume for a cell based on the full translation lattice. Different primitive cells can describe the same lattice, so “primitive” does not identify a unique shape or orientation. It also does not mean that the cell contains only one atom. (it.iucr.org)

A conventional cell follows crystallographic conventions chosen to display lattice symmetry conveniently. It may be primitive or centred. A body-centred conventional cell contains two lattice points; a face-centred cell contains four. Accordingly, their volumes are two and four times their respective primitive-cell volumes. Centring describes additional lattice translations relative to the selected cell axes, not additional chemical species. (iucr.org)

This distinction prevents confusion between lattice-point counts and atomic counts. In sodium chloride, the conventional cubic cell contains four sodium ions and four chloride ions, corresponding to four NaCl formula units. The smaller primitive cell contains one formula unit, or two ions, although it represents the same structure. (dictionary.iucr.org)

Symmetry and the asymmetric unit

The seven crystal systems are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Their classification concerns crystal symmetry, not merely cell dimensions. For example, cubic symmetry requires equal conventional cell edges and right angles, but those geometric relations alone do not prove that an atomic structure has cubic symmetry. Accidental metric equalities can occur in structures of lower symmetry. (journals.iucr.org)

A space group describes the structure’s translations together with its other symmetry operations. The asymmetric unit is a region from which those operations generate the complete structure. It is distinct from the unit cell, which repeats by translations; an asymmetric unit can occupy only part of a cell. (journals.iucr.org)

The symbol $Z$ denotes the number of chemical formula units in the specified unit cell. Its value depends on both the formula convention and the cell choice. It is normally a positive integer, although crystallographic nomenclature permits carefully justified fractional values in rare circumstances. (dictionary.iucr.org)

Determination by diffraction

In X-ray crystallography, diffraction measurements connect lattice geometry with atomic arrangement. The reciprocal lattice provides the framework for indexing reflections, while Bragg’s law relates the spacing of lattice planes to wavelength and diffraction angle. (dictionary.iucr.org)

Reflection intensities carry information about the cell contents through the structure factor. In a simplified expression,

where $h,k,l$ are Miller indices, $f_j$ describes an atom’s scattering contribution, and $x_j,y_j,z_j$ are its fractional coordinates. The sum extends over the atoms in one unit cell. Structural refinement compares calculated and observed diffraction data to adjust atomic positions and other model parameters. (dictionary.iucr.org)