7.3: Polar Lattice
The polar lattice is a lattice dual of the direct lattice, which is the ancestor of the reciprocal lattice. It was introduced by Auguste Bravais in a " mémoire" presented to the Académie de Sciences de Paris on 11 December 1848.
The construction of the polar lattice is essentially the same as that of the reciprocal lattice, but the parameter along a row of the polar lattice is V 2 /3 / d ( hkl ) instead of 1/ d ( hkl ). The polar lattice has thus the same dimensions as the direct lattice, namely Ångstroms, instead of Ångstroms -1 , like the reciprocal lattice.
- The unit cell of the polar lattice has the same volume as that of the direct lattice.
- The scalar product of the basis vectors of the direct and polar lattice is V 2 / 3 δ ij , where δ is Kroneker's tensor and the indices i and j point to the basis vectors.
The polar lattice was introduced to facilitate the morphological study of crystals.