Fundamental translations

The points on a lattice are defined by three fundamental translation vectors, a, b, and c, such that the atomic arrangement looks the same in every respect when viewed from any point r as it does when viewed at point r ... 

The pattern points associated with a particular lattice are referred to as the basis so that the description of a crystal pattern requires the specification of the space lattice by ai a2 a3 and the specification of the basis by giving the location of the pattern points in one unit cell by K, i= 1,2,. .., (Figure 16.1(b), (c)). The choice of the fundamental translations is a matter of convenience. For example, in a face-centred cubic fee) lattice we could choose orthogonal fundamental translation vectors along OX, OY, OZ, in which case the unit cell contains (Vg)8 + (l/2)6 = 4 lattice points (Figure 16.2(a)). Alternatively, we might choose a primitive unit cell with the fundamental translations... 

Similarly, for the body-centered cubic (bcc) lattice one might choose an orthogonal set for the fundamental translations giving a non-primitive unit cell with two lattice points per cell (Figure 16.2(b)) or one could choose a primitive unit cell with the fundamental translations... 

A crystal pattern may possess rotational symmetry as well as translational symmetry, although the existence of translational symmetry imposes restrictions on the order of the axes. The fundamental translations (a in eq. (1) are the basis vectors of a linear vector... 

Figure 16.5. A hexagonal planar net is generated by the fundamental translations a1 a2 (each of length a) and a12 — 2n/3. To generate a space lattice with three-fold rotational symmetry, the second and third layers must be translated so that Pi lies over the points marked P2 and P3, respectively, that is at (1/3 2/3 1/3) and (2/3 1/3 2/3). If using hexagonal coordinates a3 is normal to the plane of a1 a2 and lies along e3, so that this unit cell (3R) contains three lattice points (Figure 16.4).

Crystal system Unit cell Lattice Fundamental translations Point groups... 

Figure 16.6. Primitive unit cell of the rhombohedral lattice 3 R. The three fundamental translations a1 a2, a3 are of equal length and make equal angles with e3. Hexagonal nets in four successive layers show how the rhombohedral cell may be constructed.

We now remove the inconvenience of the translation subgroup, and consequently the Bravais lattice, being infinite by supposing that the crystal is a parallelepiped of sides Aja,-where ay, j 1,2,3, are the fundamental translations. The number of lattice points, N1N2N3, is equal to the number of unit cells in the crystal, N. To eliminate surface effects we imagine the crystal to be one of an infinite number of replicas, which together constitute an infinite system. Then... 

Notation The fundamental translations are denoted in this book by ai, a2, a3. Superscript tc denotes tetragonal and cubic systems only. 

The reciprocal lattice is generated from the fundamental translations bi b2 b3 defined by... 

The primitive rhombohedral cell in Figure 16.6 can be specified by giving the length a ai and the angle a between any pair of the fundamental translation vectors ab a2, 83. Choose ei along the projection of ai in the xy plane 0 is the angle made by ai with e3. 

The alternative explanation of the x-ray data is that the cylindrical molecule is formed of three chains, which are coiled about one another. The structure that we propose is a three-chain structure, each chain being a helix with fundamental translation equal to 3.4 A, and the three chains being related to one another (except for differences in the nitrogen bases) by the operations of a threefold axis. 

Aperiodic crystal is described by two entities, the lattice and the basis. The (translational) lattice is a perfect geometrical array of points. All lattice points are equivalent and have identical surroundings. This lattice is defined by three fundamental translation vectors a,b,c. Starting from an arbitrarily chosen origin of the lattice, any other lattice point can be reached by a translation vector r that satisfies... 

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