The Concept of a Lattice—In Two Dimensions

vectors a and b can be considered as two edges of a parallogram called a unit cell. The lattice can also be thought of as an infinite array of unit cells, all neatly fitted together so as to leave no gaps. 

To summarize, a lattice is an infinite array of identical points (i.e., each one has exactly the same environment of other points) and the points are obtained one from another by translations only (no rotations or reflections being required). 

The Number of Distinct 2D Lattices. We shall now show that there are only five different types of lattice in a plane. The one we have already con- 

In considering the hexagonal lattice, our attention is strongly drawn to the question of the symmetry of lattices. It is a question that must eventually be addressed in more detail for this as well as the other four plane lattices and we shall do so shortly. However, we shall first deal with a geometrical aspect of plane lattices that hinges on just one of their possible symmetry properties, namely, rotational symmetry. When we have done this it will be clear why the five lattices just described are the only ones possible. We shall understand why it is that we need not look for some special value of y that would allow for fivefold or sevenfold, eightfold, and so on rotational symmetry. 

This means that only those values of 1 - 2 cos 2n/n that are integers are allowed. For what angles, 2nln, is the value of the cosine equal to 0, or an integral multiple thereof. Clearly, only those listed in Table 11.1 have the appropriate values. We therefore conclude that the only permissible axes of rotational symmetry for a lattice or for any array of real objects distributed 

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