Unit translational vectors

These crystal systems some times are also named as crystal classes. Now as the unit cells (Fig. 4.2) are the building block of the two three-dimensional patterns, that is, space lattices, they also show the same symmetry as that of the space lattices and they are characterized by relations between the axial lengths known as unit translational vectors and the angles between them. It is more convenient to express these edge distances of the unit cells as vectors and so they will henceforth be noted as unit translational vectors a, b, and c and the angles between them as a, and 7. Table 4.3 lists those unit cells and the relations between the unit translational vectors. 

Fig. 5.1. a is the unit translational vector. The motif suffers a reflection on the mirror plane and undergoes a translation half the way. g the mirror perpendicular to the diagram and is known as the glide plane... 

The relations between the unit translational vector of direct lattice and reciprocal lattice can be derived as follows. 

Here, lA is the radius of the sphere of reflection, and thus mounting the crystal in two other directions as rotation axis, the other two unit translational vectors a and b can be determined [2,3]. 

The unit cell of the carbon nanotube is shown in Fig. 1 as the rectangle bounded by the vectors Q and T, where T is the ID translation vector of the nanotube. The vector T is normal to C, and extends from... 

Fig. 4. The relation between the fundamental symmetry vector R = p3] -1- qa2 and the two vectors of the tubule unit cell for a carbon nanotube specified by (n,m) which, in turn, determine the chiral vector C, and the translation vector T. The projection of R on the C, and T axes, respectively, yield (or x) and t (see text). After N/d) translations, R reaches a lattice point B". The dashed vertical lines denote normals to the vector C/, at distances of L/d, IL/d, 3L/d,..., L from the origin.

In general, we use only the lattice constants to define the solid structure (unless we are attempting to determine its S5nnmetry). We can then define a structure factor known as the translation vector. It is a element related to the unit cell and defines the basic unit of the structure. We will call it T. It is defined according to the following equation ... 

Vector notation is being used here because this is the easiest way to define the unit-cell. The reason for using both unit lattice vectors and translation vectors lies in the fact that we can now specify unit-cell parameters in terms of a, b, and c (which are the intercepts of the translation vectors on the lattice). These cell parameters are very useful since they specify the actual length eind size of the unit cell, usually in A., as we shall see. 

The infinite slab is a monolayer limited by two (010) planes (model 1). It is built with a unit cell M02O6 and two translation vectors in the a and c directions, all the atoms having their usual coordination number as in the bulk. 

As vectors a, b and c we choose the three basis vectors that also serve to define the unit cell (Section 2.2). Any translation vector t in the crystal can be expressed as the vectorial sum of three basis vectors, t = ua + vb + wc, where u, v and w are positive or negative integers. 

Translationengleiche subgroups have an unaltered translation lattice, i.e. the translation vectors and therefore the size of the primitive unit cells of group and subgroup coincide. The symmetry reduction in this case is accomplished by the loss of other symmetry operations, for example by the reduction of the multiplicity of symmetry axes. This implies a transition to a different crystal class. The example on the right in Fig. 18.1 shows how a fourfold rotation axis is converted to a twofold rotation axis when four symmetry-equivalent atoms are replaced by two pairs of different atoms the translation vectors are not affected. 

Figure 9.3 Cluster of unit cells of the cesium chloride crystal structure. This figure shows that ions of the same sign in this structure line up along the 100 directions. Thus the three rows are orthogonal to one another. Translation of a (100) plane of ions over its nearest (100) neighboring plane keeps ions of opposite sign adjacent to one another. This is also the case on the (110) planes, but the translation vector is V2 larger than for the the (100) planes.

High polymer calculations can be performed on polysaccharides. Calculation of unit cell translation vectors (15), heats of polymerization (15), and elastic moduli (16) can readily be done. The accuracy of such calculations is the same as that of equivalent molecular species. A limitation of elastic moduli calculations is that the polymer is assumed to be 100% ordered, a state not commonly found in polysaccharides. 

Calculate the volume of a unit cell from the lattice translation vectors. 

In the second half of this article, we discuss dynamic properties of stiff-chain liquid-crystalline polymers in solution. If the position and orientation of a stiff or semiflexible chain in a solution is specified by its center of mass and end-to-end vector, respectively, the translational and rotational motions of the whole chain can be described in terms of the time-dependent single-particle distribution function f(r, a t), where r and a are the position vector of the center of mass and the unit vector parallel to the end-to-end vector of the chain, respectively, and t is time, (a should be distinguished from the unit tangent vector to the chain contour appearing in the previous sections, except for rodlike polymers.) Since this distribution function cannot describe internal motions of the chain, our discussion below is restricted to such global chain dynamics as translational and rotational diffusion and zero-shear viscosity. 

In this section, we extend consideration from the Lorentz to the Poincare group within the structure of 0(3) electrodynamics, by introducing the generator of spacetime translations along the axis of propagation in the normalized (unit 12-vector) form ... 

The unit 12-vector acts essentially as a normalized spacetime translation on the classical level. The concept of spacetime translation operator was introduced by Wigner, thus extending [100] the Lorentz group to the Poincare group. The PL vector is essential for a self-consistent description of particle spin. 

The nature of the dual vector ( ) can be deduced without using any equation of motion, but the dual 4-vector is a fundamental geometric property in the four dimensions of spacetime. The complete description of the electromagnetic field in 0(3) electrodynamics must therefore involve boosts, rotations, and spacetime translations, meaning that is a fundamental geometric property of spacetime. The unit 4-vector i M is orthogonal to the unit 4-vector... 

In some circumstances the magnitudes of the translation vectors must be taken into account. Let us demonstrate this with the example of the trirutile structure. If we triplicate the unit cell of rutile in the c direction, we can occupy the metal atom positions with two kinds of metals in a ratio of 1 2, such as is shown in Fig. 3.10. This structure type is known for several oxides and fluorides, e.g. ZnSb20g. Both the rutile and tlie trirutile structure belong to the same space-group type PAjmnm. Due to the triplicated translation vector in the c direction, the density of the symmetry elements in trirutile is less than in rutile. The total number of symmetry operations (including the translations) is reduced to... 

Shown in Figure 11.2c are several pairs of vectors, each pair providing a way to generate the entire lattice from one point. Clearly, there is an infinite number of other ways to choose such pairs of vectors. Each pair defines two directions and the pertinent unit translations. For practical reasons the pre-... 

Suppose we were to center the oblique lattice of Figure 11.3a. This does not in any way improve its symmetry. All we have is another, denser oblique lattice, which would be properly defined as having a smaller set of defining translation vectors and a unit cell with half the area, as shown in Figure 11.6a. 

Again, our first concern must be to see how many ways there are in which the translation vectors can be related to one another (relative lengths, angles between them) to give distinct, space-filling patterns of equivalent points. We have seen (Section 11.2) that in 2D there were only 5 distinct lattices. We shall now see that in 3D there are 14. These are often designated eponymously as the Bravais lattices and are shown in Figure 11.11, in the form of one unit cell of each. 

Figure 11.12. (a) Conventional labeling of translation vectors and angles used to generate a 3D lattice from an initial point P. (b) Conventional labeling of unit cell edges and faces. 

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... 

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