Detecting Unit Cells

The oblique unit cell

The arrangement of repeated units of periodic patterns can be described by a Bravais lattice. The units may be individual atoms, molecules or, for example, artificially created calibration structures. The two dimensional Bravais lattice summarizes the geometry of the periodic structure, and consists of all points with position vectors S of the form:

where a and b are any two dimensional vectors of different and k, and l range through all integer values. An area on the surface that, when translated through all the vectors in a Bravais lattice just fills all of the surface without either overlapping itself or leaving voids, is called a primitive unit cell of the lattice, and the a and b vectors are then called primitive vectors. There is no unique way of choosing a primitive unit cell for a Bravais lattice and they can have different forms: square, hexagonal (rhombus), rectangular or oblique.

The oblique unit cell is general and we will use this in the following. Note also that even though the calibration structure is quadratic the distortions are most likely to create images with oblique unit cells. By Fourier transform of a Bravais lattice a reciprocal lattice K is created with Fourier peaks that can be described by the reciprocal unit cell vectors A and B:

where k, and l range through all integer values. The important property of the Fourier transform is that the reciprocal unit cell vectors can be found directly from the positions of the peaks. Having found the A and B reciprocal unit cell vectors the spatial unit cell vectors a and b can be found as