Reciprocal Lattice

Planes are a fundamental concept in the discussion of crystals. In fact, the crystal can be specified in terms of just its lattice planes and the distances between them, instead of in terms of all of its its elementary sites (or equivalently, its primitive vectors). Connected to this idea is the notion of the reciprocal lattice. As the name implies, the reciprocal lattice is indeed a lattice, and it can be specified by its own set of primitive vectors. The significance of the reciprocal lattice is that each of the vectors it defines (by joining any site on on the lattice to another on it) corresponds to a plane in the original, or direct lattice---each reciprocal-lattice vector is a normal to a direct-lattice plane. Moreover, the distance between the direct-lattice planes will correspond to the size of the reciprocal-lattice vector---the larger the magnitude of the vector, the smaller the distance between the corresponding planes. It is also worth noting that the reciprocal lattice of the reciprocal lattice is the original direct lattice.

This one-to-one correspondence between a lattice (which defines the sites) and its reciprocal (which describes its planes) indicates that the lattice planes are no less elementary a description of the lattice than are the lattice sites themselves. We denote the primitive vectors of the reciprocal lattice as a*, b*, and c*, and they are given uniquely in terms of the direct-lattice primitives a, b, and c according to

$a^\ast = \frac{2\pi b \times c}{a \cdot b \times c}$

$b^\ast = \frac{2\pi c \times a}{a \cdot b \times c}$

$c^\ast = \frac{2\pi a \times b}{a \cdot b \times c}$

For cubic crystals, each direct-lattice primitive vector has the same direction as its reciprocal-lattice counterpart.