Oblique rectangular, centered

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

Figure 1.6. (a) The five two-dimensional Bravais lattices. Clockwise from upper left square, rectangular, oblique, hexagonal, and centered-rectangular (center), (b) The 14 three-dimensional Bravais lattices. 

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.

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. 

In two dimensions the unit cell is a parallelogram whose size and shape are defined by two lattice vectors (a and b). There are four primitive lattices, lattices where the lattice points are located only at the corners of the unit cell square, hexagonal, rectangular, and oblique. In three dimensions the unit cell is a parallelepiped whose size and shape are defined by three lattice vectors (a, b and c), and there are seven primitive lattices cubic, tetragonal, hexagonal, rhombohedral, orthorhombic, monoclinic, and triclinic. Placing an additional lattice point at the center of a cubic unit cell leads to a body-centered cubic lattice, while placing an additional point at the center of each face of the unit cell leads to a face-centered cubic iattice. 

Fig. 3.7 Sketch of the modulated smectic phases. For the sake of clarity, the sinusoidal modulations are drawn in an exaggerated way. a Shows the SmA phase, which is described with a centered rectangular lattice and b shows the SmC phase in which the mesogens are found on an oblique lattice...

Figure 4. Plan views of two-dimensional lattices of columnar phases ellipses denote disks that are tilled with respect to the column axis (a) hexagonal (P6 2/m 2/ra) (b) rectangular (P 2 /a) (c) oblique (P,) (d) rectangular (P-Ja)-, (e) rectangular, face-centered, tilted columns (C2Jm). (From Levelut [12], reproduced by permission of the division des Publications Societe de Chimie Physique).

Face-centered rectangular lattice in 2D. All the lattice points could be generated by translating along the oblique primitive lattice vectors a and b, or by the nonprimitive vectors a and b with a basis of (0,0) and (l/2,l/2). 

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