Fractional coordinates
Fractional coordinates, also known as crystal coordinates, constitute a specialized coordinate system in crystallography for specifying the positions of atoms or points within a crystal's unit cell as dimensionless fractions of the lattice parameters a, b, and c.[1] These coordinates are expressed as a triplet (x, y, z), where x, y, and z range typically from 0 to 1 (or equivalently -0.5 to 0.5), representing the proportional distances along the respective unit cell edges from the origin at (0, 0, 0).[2] For instance, the center of the unit cell is always at (0.5, 0.5, 0.5), independent of the cell's dimensions or shape.[1] This system leverages the periodic nature of crystal lattices, where adding or subtracting integer values to any coordinate (e.g., (x+1, y, z)) yields an equivalent position in an adjacent unit cell due to translational symmetry.[3] Unlike orthogonal Cartesian coordinates, which measure absolute distances in angstroms along perpendicular axes, fractional coordinates align with the non-orthogonal crystallographic axes separated by angles α, β, and γ, making them essential for handling the anisotropy of real crystals.[4] Conversion between fractional and Cartesian coordinates is achieved via a transformation matrix derived from the unit cell parameters, facilitating calculations such as interatomic distances using the metric tensor.[2] Fractional coordinates play a pivotal role in structural analysis, enabling the concise representation of atomic arrangements in crystallographic databases such as the Inorganic Crystal Structure Database (ICSD) and the Cambridge Structural Database (CSD) and in computational tools for symmetry operations, Fourier transforms, and molecular modeling.[1][5][6] They simplify the description of complex crystal structures across infinite lattices and are particularly valuable in fields such as materials science and biochemistry for interpreting X-ray diffraction data and simulating periodic systems.[4]Crystal Structure Basics
Crystal Lattice
A crystal lattice is defined as an infinite array of discrete points in space, where each point has identical surroundings, generated by integer combinations of three discrete translation vectors. This arrangement forms the mathematical framework underlying the periodic structure of crystalline solids, allowing the positions of lattice points to be reached by translating the structure along these vectors without rotation or reflection.[7] The concept of the crystal lattice was formalized by Auguste Bravais in his 1850 work, which systematically classified possible lattice arrangements.[8] Bravais lattices represent the 14 unique three-dimensional configurations of such points, categorized into seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.[9] These lattices differ in their symmetry and the relationships between their translation vectors, encompassing primitive, body-centered, face-centered, and base-centered variants where applicable, ensuring all possible periodic point arrays are covered without redundancy.[10] The repeating unit of the lattice is defined by three non-coplanar lattice vectors, conventionally denoted as a, b, and c, which serve as the basis for generating all lattice points via linear combinations with integer coefficients.[11] These vectors determine the periodicity and geometry of the lattice, with their lengths and angles specifying the crystal system's parameters. The unit cell, as the smallest volume containing the full lattice symmetry, is constructed from these vectors.[12]Unit Cell
The unit cell serves as the fundamental repeating unit in a crystal lattice, representing the smallest volume that encapsulates all essential structural information of the crystal; translating this volume by integer multiples of the lattice vectors generates the entire periodic structure. It forms a parallelepiped bounded by the three lattice vectors \mathbf{a}, \mathbf{b}, and \mathbf{c}.[13] The infinite crystal lattice arises from the translational repetition of these unit cells in three dimensions.[14] The geometry of the unit cell is characterized by six lattice parameters: the edge lengths a = |\mathbf{a}|, b = |\mathbf{b}|, c = |\mathbf{c}|, and the interaxial angles \alpha (between \mathbf{b} and \mathbf{c}), \beta (between \mathbf{a} and \mathbf{c}), and \gamma (between \mathbf{a} and \mathbf{b}).[14] These parameters are constrained by the symmetry of the seven crystal systems, which classify all possible Bravais lattices. In the triclinic system, the lowest symmetry, all lengths and angles are unequal (a \neq b \neq c, \alpha \neq \beta \neq \gamma), allowing maximal asymmetry.[15] Higher-symmetry systems impose equalities, such as the cubic system where a = b = c and \alpha = \beta = \gamma = 90^\circ, resulting in a regular cube.[15] The full set of relations is summarized below:| Crystal System | Edge Lengths | Angles |
|---|---|---|
| Triclinic | a \neq b \neq c | \alpha \neq \beta \neq \gamma |
| Monoclinic | a \neq b \neq c | \alpha = \gamma = 90^\circ, \beta \neq 90^\circ |
| Orthorhombic | a \neq b \neq c | \alpha = \beta = \gamma = 90^\circ |
| Tetragonal | a = b \neq c | \alpha = \beta = \gamma = 90^\circ |
| Trigonal (rhombohedral) | a = b = c | \alpha = \beta = \gamma \neq 90^\circ |
| Hexagonal | a = b \neq c | \alpha = \beta = 90^\circ, \gamma = 120^\circ |
| Cubic | a = b = c | \alpha = \beta = \gamma = 90^\circ |