Σ = 5 CSL and O-Lattice

Here is a large scale illustration of a Σ = 5 relationship in the O-lattice.

There are more O-lattice points than lattice coincidence points - we have one extra equivalence point (equivalence coordinates (1/2, 1/2) in addition to the O-point with equivalence coordinates (0, 0).

The lattice constant (and therefore the unit vectors) of the O-lattice are smaller by a factor of 2^1/2.

In just looking at the picture, it is tempting to identify more O-points, the green point, e.g., looks very much like an O-point. Well, it is not, because:

The green point, while marking middle positions on both lattices, is not an O-point, because its internal coordinates are (1/2, 0) in lattice 1 and (0, 1/2) in lattice 2. And while, yes, this marks a point midway between to lattice points in both lattices, it is still not an equivalence point!

The picture contains a new, very important feature:

The yellow triangles denote pattern elements. While they indicate the rotation, they may simply be taken as symbols representing the specific arrangement of atoms at specific equivalence points. Equal symbols indicate equal arrangements, and identical equivalence points have identical pattern elements.

The unit vectors of the three lattices are also shown; this will be important in some future context.

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7.3.1 The Basic Concept