Lattice point symmetry

Finally, the hexagonal primitive (hp) lattice, (Figure 3.5i), has a hexad rotation axis at each lattice point. This generates diads and triads as shown. In addition, there are six mirror lines through each lattice point. In other parts of the unit cell, two mirror lines intersect at diads and three mirror lines intersect at triads, (Figure 3.5j). The lattice point symmetry is described by the symbol 6mm. 

Tablel.3-b Crystal families, crystal systems, crystallographic point groups, conventional coordinate systems, and Bravais lattices in three dimensions. Lattice point symmetries (holohedries) are given in bold...

As seen in Table 2, many of the chiral tubules with d = 1 have large values for M for example, for the (6,5) tubule, M = 149, while for the (7,4) tubule, M = 17. Thus, many 2tt rotations around the tubule axis are needed in some cases to reach a lattice point of the ID lattice. A more detailed discussion of the symmetry properties of the non-symmorphic chiral groups is given elsewhere in this volume[8. ... 

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.

The properties of the periodic surfaces studied in the previous sections do not depend on the discretization procedure in the hmit of small distance between the lattice points. Also, the symmetry of the lattice does not seem to influence the minimization, at least in the limit of large N and small h. In the computer simulations the quantities which vary on the scale larger than the lattice size should have a well-defined value for large N. However, in reality we work with a lattice of a finite size, usually small, and the lattice spacing is rather large. Therefore we find that typical simulations of the same model may give diffferent quantitative results although quahtatively one obtains the same results. Here we compare in detail two different discretization... 

If we now apply rotadonal nnmetxy (Factor II given in 2.2.1) to the 14 Bravais lattices, we obtain the 32 Point-Groups which have the factor of symmetry imposed upon the 14 Bravais lattices. The symmetry elements that have been used are ... 

Now examine the symmetry elements for the cubic lattice. It is easy to seethat the number of rotation elements, plus horizontal and vertical symmetry elements is quite high. This is the reason why the Cubic Structure is placed at the top of 2.2.3. E)ven though the lattice points of 2.2.1. are deceptively simple for the cubic structure, the symmetry elements are not... 

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry relationships between the molecules. Common symmetry operations are two- or three-fold screw (rotation) axes, mirror planes, inversion centers (centers of symmetry), and rotation followed by inversion. There are 230 different ways to combine allowed symmetry operations in a crystal leading to 230 space groups.12 Not all of these are allowed for protein crystals because of amino acid asymmetry (only L-amino acids are found in proteins). Only those space groups without symmetry (triclinic) or with rotation or screw axes are allowed. However, mirror lines and inversion centers may occur in protein structures along an axis. 

Crystal lattices can be depicted not only by the lattice translation defined in Eq. (7.2), but also by the performance of various point symmetry operations. A symmetry operation is defined as an operation that moves the system into a new configuration that is equivalent to and indistinguishable from the original one. A symmetry element is a point, line, or plane with respect to which a symmetry operation is performed. The complete ensemble of symmetry operations that define the spatial properties of a molecule or its crystal are referred to as its group. In addition to the fundamental symmetry operations associated with molecular species that define the point group of the molecule, there are additional symmetry operations necessary to define the space group of its crystal. These will only be briefly outlined here, but additional information on molecular symmetry [10] and solid-state symmetry [11] is available. 

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes, a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry... 

The environment of an ion in a solid or complex ion corresponds to symmetry transformations under which the environment is unchanged. These symmetry transformations constitute a group. In a crystalline lattice these symmetry transformations are the crystallographic point groups. In three-dimensional space there are 32 point groups. 

Fig. 6.5. Close-packed surface with tetragonal symmetry, (a) The square lattice in real space. There is an atom on each lattice point, (b) The reciprocal space.

The unit cells for the two-dimensional lattices are parallelograms with their corners at equivalent positions in the array (i.e., the corners of a unit cell are lattice points). In Figure 1.17, we show a square array with several different unit cells depicted. All of these, if repeated, would reproduce the array it is conventional to choose the smallest cell that fully represents the symmetry of the structure. Both unit cells (la) and (lb) are the same size but clearly (la) shows that it is a square array, and this would be the conventional choice. Figure 1.18 demonstrates the same principles but for a centred rectangular array, where (a) would be the conventional choice because it includes information on the centring the smaller unit cell (b) loses this information. It is always possible to define a non-centred oblique unit cell, but doing so may lose information about the symmetry of the lattice. 

The true unit cell is not necessarily the smallest unit that will account for all the reciprocal lattice points it is also necessary that the cell chosen should conform to the crystal symmetry. The reflections of crystals with face-centred or body-centred lattices can be accounted for by unit cells which have only a fraction of the volume of the true unit cell, but the smallest unit cells for such crystals are rejected in favour of the smallest that conforms to the crystal symmetry. The... 

But this does nob end the tale of possible arrangements. Hitherto we have considered only those symmetry operations which carry us from one atom in the crystal to another associated with the same lattice point—the symmetry operations (rotation, reflection, or inversion through a point) which by continued repetition always bring us back to the atom from which we started. These are the point-group symmetries which were already familiar to us in crystal shapes. Now in "many space-patterns two additional types of symmetry operations can be discerned--types which involve translation and therefore do not occur in point-groups or crystal shapes. 

The second condition is that there must be, at the centre of symmetry, an atom whose diffracting power is very much greater than that of any of the other atoms in the cell. The third is that there must be only one such atom per lattice point in the projected unit cell j this atom is conveniently taken as the origin of the projected cell. In these circumstances, it is certain that the phase angles of all the reflections with respect to the origin are 0°, since the wave from the heavy atom at... 

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