Plane lattices unit cells

A primitive unit cell is a lattice unit cell that contains only one lattice point. The four primitive plane lattice unit cells are labelled p oblique, imp), rectangular, op), square, (tp) and hexagonal, (hp). They are normally drawn with a lattice... 

In this diagram, a series of hexagon-shaped planes are shown which are orthogonal, or 90 degrees, to each of the corners of the cubic cell. Each plane connects to cuiother plane (here shown as a rectangle) on each fiace of the unit-cell. Thus, the faces of the lattice unit-cell and those of the reciprocal unit-cell can be seen to lie on the same pltme while those at the corners lie at right angles to the corners. 

In Fig. 4.3, an additional set of planes, and thus an additional source of diffraction, is indicated. The lattice (dark lines) is shown in section parallel to the ab faces or the xy plane. The dashed lines represent the intersection of a set of equivalent, parallel planes that are perpendicular to the xy plane of the paper. Note that the planes cut each a edge into two parts and each b edge into one part, so these planes have indices 210. Because all (210) planes are parallel to the z axis (which is perpendicular to the plane of the paper), the / index is zero. [Or equivalently, because the planes are infinite in extent, and are coincident with c edges, and thus do not cut edges parallel to the z axis, there are zero (210) planes per unit cell in the z direction.] As another example, for any plane in the set shown in Fig. 4.4, the first plane encountered from any lattice point cuts that unit cell at a/2 and b 3, so the indices are 230. 

Figure 3.9 The glide operation (a) reflection across a mirror line (b) translation parallel to the mirror plane by a vector t, which is constrained to be equal to T/2, where T is the lattice repeat vector parallel to the glide line. The lattice unit cell is shaded...

Similar measurements were performed using freeze-fracture electron microscopy. Two major sets of fracture planes were identified. One set, parallel to the (2 2 0) planes, yielded a square lattice similar to that seen in the thin-section study. The other set, parallel to the (1 1 1) planes, yielded an hexagonal lattice. Unit cell sizes, calculated on the basis of the diamond lattice, were very similar to those obtained from the thin-section electronmicrographs. 

Figure 11.12 Model for a (100) surface of an fee lattice, (a) A symmetric slab model shows two Identical surfaces (dashed planes at top and bottom). A possible parent lattice unit cell is shown as a box and is chosen in such a way that - in the periodic continuation of the unit cell-the top and the bottom surfaces are separated by several layers of vacuum V and cannot interact across V. In this example, the lattice provides four surface layers (S) on each side of the slab and three bulk layers (B) in the middle of the slab. As the latter model the bulk structure of a system, the B layers have fixed occupancy and hence do not contribute to the configuration space. The different...

Fig. XVII-18. Contours of constant adsorption energy for a krypton atom over the basal plane of graphite. The carbon atoms are at the centers of the dotted triangular regions. The rhombuses show the unit cells for the graphite lattice and for the commensurate adatom lattice. (From Ref. 8. Reprinted with permission from American Chemical Society, copyright 1993.)...

Amphiboles. The crystalline stmcture common to amphibole minerals consists of two ribbons of siUcate tetrahedra placed back to back as shown in Figure 5. The plane of anionic valency sites created by this double ribbon arrangement is neutralized by the metal cations. The unit cell has seven cationic sites of three different types these sites can host a large variety of metal cations without substantial dismption of the lattice. 

If atoms, molecules, or ions of a unit cell are treated as points, the lattice stmcture of the entire crystal can be shown to be a multiplication ia three dimensions of the unit cell. Only 14 possible lattices (called Bravais lattices) can be drawn in three dimensions. These can be classified into seven groups based on their elements of symmetry. Moreover, examination of the elements of symmetry (about a point, a line, or a plane) for a crystal shows that there are 32 different combinations (classes) that can be grouped into seven systems. The correspondence of these seven systems to the seven lattice groups is shown in Table 1. 

Fig. 8. Lattice distortions in a graphite sheet. For an in-plane distortion (left), the bond denoted by a thin line becomes shorter and that denoted by a thick line becomes longer, leading to a unit cell three times as large as the original. For an out-of-plane distortion (right), an atom denoted by a black dot is shifted down and that denoted by a white circle moves up.

A specific attribute of unit cell building is the inclination of the axis of macromolecule chains, in relation to the normal, to the plane of the base of the cell (ab). According to Yamashita [11] this inclination is within the range of 25-35° (Fig. 4). Against the background of space lattices of other types of fibers, the lattice of crystalline regions in PET fibers is characterized by a number of specific features. These are ... 

The translational direction of the lattice (c), which is the direction that the crystallite axis is not, because of the triclinic crystallographic system, is perpendicular to the plane of the unit cell base (ab). 

Figure 16-16. Molecular packing of Oocl-OPV5 in the crystal lattice. Lett oblique view of (he unit cell of Oocl-OPV5 right projection of the unit cell on a plane perpendicular to the ci-axis.

It is, however, more revealing in the context of monodromy to allow/(s, ) to pass from one Riemann sheet to the next, at the branch cut, a procedure that leads to the construction in Fig. 4, due to Sadovskii and Zhilinskii [2], by which a unit cell of the quantum lattice, with sides defined here by unit changes in k and v, is transported from one cell to the next on a path around the critical point at the center of the lattice. Note, in particular, that the lattice is locally regular in any region of the [k, s) plane that excludes the critical point and that any vector in the unit cell such as the base vector, marked by arrows, rotates as the cell is transported around the cycle. Consequently, the transported dashed cell differs from that of the original quantized lattice. 

The easiest ciystal lattice to visualize is the simple cubic stracture. In a simple cubic crystal, layers of atoms stack one directly above another, so that all atoms lie along straight lines at right angles, as Figure 11-26 shows. Each atom in this structure touches six other atoms four within the same plane, one above the plane, and one below the plane. Within one layer of the crystal, any set of four atoms forms a square. Adding four atoms directly above or below the first four forms a cube, for which the lattice is named. The unit cell of the simple cubic lattice, shown in... 

Fig. 4.12 (a) CdSe wurtzite unit cell (b) schematic illustration of a hexagonal (wurtzite) CdSe basal plane on a (111) section of the gold lattice, emphasizing the 2 3 lattice match. Note the [111] Au//(0001)CdSe orientation, with the CdSe a-directions aligned along the (llO)Au. The outlined rhombus indicates the projection of a CdSe unit cell. (Adapted from [112])... 

In this case, we can show that the planes of the cubic lattice are defined by moving various distances in the lattice from the origin. What is meant by this terminology is that as we move from the (0,0,0) origin just 1.00 unit-cell disteuice in the x" direction to the (1,0,0) point in the cubic lattice, we have defined the 100 plane (Note that we are using the ni... 

Thus, the planes of the lattice are found to be important and can be defined by moving along one or more of the lattice directions of the unitcell to define them. Also important are the symmetry operations that can be performed within the unit-cell, as we have illustrated in the preceding diagram. These give rise to a total of 14 different lattices as we will show below. But first, let us confine our discussion to just the simple cubic lattice. 

Thus, these intercepts are given in terms of the actual unit-cell length found for the specific structure, and not the lattice itself. The Miller Indices are thus the indices of a stack of planes within the lattice. Planes are important in solids because, as we will see, they are used to locate atom positions within the lattice structure. 

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