Rectangular, primitive

A rectangular primitive plane lattice has lattice parameters ... 

Figure 3.5 Symmetry of the plane lattices (a, b) oblique primitive, mp, 2 (c, d) rectangular primitive, op, 2mm (e, f) rectangular centred, oc, 2mm (g, h) square, tp, 4mm (i, j) hexagonal primitive, hp, 6mm...

In Fig. 2.24, m represents a mirror line. Let T be a primitive translation and T the mirror image of T. (A translation T is primitive if 1/2T is not a translation.) T + T and T — T are perpendicular and define a rectangular cell. If T + T and T — T are primitive translations, the rectangular cell is centered because there is a lattice point in the middle. We would thus choose the diamond (T,T) as the primitive cell. In contrast, if 1/2(T + T ) and 1/2(T — T ) are translations, we then obtain a rectangular primitive cell. These two rectangular planar lattices, primitive, on the one hand and centered or diamond, on the other, are representative of two types of lattice that it is important to differentiate. It is impossible to find a primitive diamond-shaped cell for the first, or a primitive rectangular cell for the second. These considerations lead us to an operational definition for the Bravais lattices (or classes). A Bravais lattice is characterized by ... 

Ribbon phases have been the most comprehensively studied of the intermediate phases. They occur when the surfactant molecules aggregate to form long flat ribbons with an aspect ratio of about 0.5 located on two dimensional lattices of oblique, rectangular (primitive or centred), or hexagonal symmetry. Ribbon phases were first proposed by Luzzati [68,69] in aqueous sur-... 

Figure A.2 The five surface Bravais lattices square, primitive rectangular, centered rectangular, hexagonal, and oblique.

Figure 3.5. A bi-dimensional lattice of points is shown which can be built on the basis of the translation units a (the shortest one) and the corresponding unit cell. The origin of the cell is arbitrary (for inst. (a) or (b)) it contains 1 point. A more symmetric cell (c) may be built with the edges A and B, however it is double primitive centred rectangular, containing two equivalent points.

Figure 1.6. Plan and side views of the structure of the Cu(410)-O surface phase. The full and dashed lines show respectively the primitive and centred rectangular surface unit meshes (that are unchanged from those of the clean surface by the adsorption).

Figure 11.3. The five distinct plane (2D) lattices (a) oblique, (b) primitive rectangular, (c) square, (d) and (e) are both centered rectangular but show alternative choices of unit cell, (/) hexagonal.

If we introduce one restriction, namely, y = 90°, we get a rectangular lattice. This is the second of the five types, Figure 11.36. For a reason to be made clear shortly we describe this more precisely as a primitive rectangular lattice. 

Rotational Symmetry of 2D Lattices. Each of the five lattices has rotational symmetry about axes perpendicular to the plane of the lattice. For the oblique lattice and both the primitive and centered rectangular lattices these are twofold axes, but there are several types in each case. The standard symbol for a twofold rotation axis perpendicular to the plane of projection is . In the case of the square lattice there are fourfold as well as twofold axes. The symbol for a fourfold axis seen end-on is For the hexagonal lattice there are two-, three-, and sixfold axes the latter two are represented by a and , respectively. In Figure 11.4 are shown all of the rotation axes possessed by each lattice. 

Additional Comments on Centering. In working out systematically the existence of just five distinct planar lattices, we arrived at the centered rectangular lattice by regarding it as a centered alternative to a primitive, non-... 

The position 0, marked above, represents our starting point (the rectangular potential well). A considerable time elapsed between the primitive confined rotator model described in 1981 (VIG, GT, [18]) and the hat-curved model. 

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