Translational Symmetry

Translational symmetry is the simplest and most fundamental type of symmetry in crystals. It is the repetition of a motif - which in a mineral represents a collection of atoms - by translation: sliding the pattern a fixed distance in a fixed direction, again and again, to produce a regularly spaced repeating array. ^[1] Everything else in crystallography - point symmetry, space groups, crystal systems, Bravais lattices - builds on top of this foundation.

Plane Lattices: Two Dimensions

In two dimensions, translating a spot parallel to one vector (a) produces a row of regularly spaced points. Translating that row parallel to a second vector (b) at some angle (γ) to the first produces a continuous, repeating two-dimensional array called a plane lattice, where the spots mark lattice nodes. ^[1]

Only five unique planar frameworks can emerge from linear repetition in 2D space: the square, rectangular, diamond, hexagonal, and oblique plane lattices. ^[1] These five frameworks rely entirely on just four distinct primary tile geometries - namely the square, the rectangle, the rhombus, and the parallelogram. ^[1] The unit mesh is the smallest repeat unit: its pattern is identical to every other unit mesh in the lattice, and every atom within it has a duplicate in adjacent unit meshes parallel to the two translation vectors. ^[1] This type of repeating pattern is exactly what governs how bricks are laid in a wall or tiles on a floor - brick walls and ceramic tile patterns are direct physical analogies for plane lattices. ^[1]

Space Lattices and Unit Cells: Three Dimensions

Extending translation into the third dimension produces a space lattice - a three-dimensional array of lattice nodes repeated in all directions. ^[1] The volume outlined by eight lattice nodes is the unit cell - the three-dimensional analogue of the unit mesh. ^[1]

The edges of the unit cell are parallel to crystal axes, labelled a, b, and c. These axes intersect at a point called the origin. The dimensions of the unit cell along those axes are also called a, b, and c, and the angles between the axes are α (between b and c), β (between a and c), and γ (between a and b). ^[1]

The unit cell is the fundamental building block of a crystal structure. Because the unit cell repeats by translation in all three directions to fill space, the entire crystal is - in principle - just one unit cell stacked in three dimensions an enormous number of times. Understanding the geometry of the unit cell is therefore the key to understanding the geometry of the whole crystal.

References

1. Nesse, W. D. (2017). Introduction to Mineralogy, 3rd ed. Oxford University Press.

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