A Bravais lattice has lattice points with the same environment such that the lattice has at least one of the rotational symmetries predicted by the crystallographic restriction theorem. In other words, lattices are Bravais lattices only if they meet the symmetry and ‘same environment’ criteria. Such lattices have patterns depicted in figures I to V in the diagram below.
Notice that a lattice with one-fold rotational symmetry is not included as one of the two-dimensional Bravais lattices. The reason is as follows:
The purpose of defining two-dimensional Bravais lattices is to use them to construct three-dimensional Bravais lattices and hence their associated unit cells, which are what chemists are ultimately interested in. As there are many possible one-fold rotational symmetry lattices in two dimensions, it is easier to employ the five specified two-dimensional lattices to develop all the possible unit cells, which include one that replicates to form lattices with 1-fold rotational symmetry (see following articles for details).
Question
Why is the honeycomb lattice (figure VI) not a Bravais lattice?
Answer
Even though the honeycomb lattice satisfies the rotational symmetry criterion, not all lattice points have the same environment. An easy way to determine whether all lattice points have the same environment is to see if their nearest neighbours are in the same directions. With reference to the diagram above, lattice point 1 of figure VI has nearest neighbours in the northeast, northwest and south directions while lattice point 2 has nearest neighbours in the southeast, southwest and north directions. The honeycomb lattice fails the ‘same environment’ criterion and therefore is not a Bravais lattice. Nonetheless, we can demarcate a unit cell for the lattice by combining two lattice points to establish a single basis (shaded pink). This method of pairing lattice points to form unit cells is commonly used in ionic crystals to determine their lattice types.