The reciprocal lattice in two dimensions

Many of the physical properties of crystals, as well as the geometry of the three-dimensional patterns of radiation diffracted by crystals, (see Chapter 6) are most easily described by using the reciprocal lattice. The two-dimensional (plane) lattices, sometimes called the direct lattices, are said to occupy real space, and the reciprocal lattice occupies reciprocal space. The concept of the reciprocal lattice is straightforward. (Remember, the reciprocal lattice is simply another lattice.) It is defined in terms of two basis vectors labelled a and b.  

The direction of these vectors is perpendicular to the end faces of the direct lattice unit cell. The lengths of the basis vectors of the reciprocal lattice are the inverse of the perpendicular distance from the lattice origin to the end faces of the direct lattice unit cell. For the square and rectan- 

For the oblique and hexagonal plane lattices, these are given by  

It will be seen that the angle between the reciprocal axes, a and b, is (180 — y) = y, when the angle between the direct axes, a and b, is y. It is thus simple to construct the reciprocal lattice by drawing a and c at an angle of (180 — y) and marking out the lattice with the appropriate spacing a and b.  

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