Bravais hexagonal

The lattice shown in Fig. 20.1(b), which is usually imaged by STM or AFM, is formed by only three instead of six carbon atoms. The corresponding nearest neighbor carbon—carbon atom distance is that of the Bravais hexagonal lattice referred to above. The scheme depicted in Fig. 20.1 (a, b) accounts for the appearance of this image. 

The overall symmetry of the stmcture is related to the value of n in the series formula. In cases where n is a [(multiple of 3)-1], the stmcture has a Bravais hexagonal lattice, while if n is given by a [(multiple of 3)] or a [(multiple of 3)+1], the structure conforms to a Bravais rhombohedral lattice. 

Figure Bl.21.4. Direct lattices (at left) and reciprocal lattices (middle) for the five two-dimensional Bravais lattices. The reciprocal lattice corresponds directly to the diffraction pattern observed on a standard LEED display. Note that other choices of unit cells are possible e.g., for hexagonal lattices, one often chooses vectors a and b that are subtended by an angle y of 120° rather than 60°. Then the reciprocal unit cell vectors also change in the hexagonal case, the angle between a and b becomes 60° rather than 120°.

Figure BT2T4 illustrates the direct-space and reciprocal-space lattices for the five two-dimensional Bravais lattices allowed at surfaces. It is usefiil to realize that the vector a is always perpendicular to the vector b and that is always perpendicular to a. It is also usefiil to notice that the length of a is inversely proportional to the length of a, and likewise for b and b. Thus, a large unit cell in direct space gives a small unit cell in reciprocal space, and a wide rectangular unit cell in direct space produces a tall rectangular unit cell in reciprocal space. Also, the hexagonal direct-space lattice gives rise to another hexagonal lattice in reciprocal space, but rotated by 90° with respect to the direct-space lattice.

Fig. 3.8 Some basic Bravais lattices (a) simple cubic, (b) body-centred cubic, (c) face-centred cubic and (d) simple hexagonal close-packed. (Figure adapted in part from Ashcroft N V and Mermin N D 1976. Solid State Physics.

In two dimensions, five different lattices exist, see Fig. 5.6. One recognizes the hexagonal Bravais lattice as the unit cell of the cubic (111) and hep (001) surfaces, the centered rectangular cell as the unit cell of the bcc and fee (110) surfaces, and... 

In a three-dimensional lattice, we have observed planes of atoms (or ions) composing the lattice. Up to now, we have assumed that these planes maintain a certain relation to one another. That is. we have shown that there are a set of planes as defined by the hkl values, which in turn depends upon the type of Bravais lattice that is present. However, we find that it is possible for these rows of atoms to "slip" from their equilibrium positions. Hiis gives rise to another type of lattice defect called "line defects". In the following diagram, we present a hexagonal lattice in which a line defect is present ... 

The Miller indices of planes in crystals with a hexagonal unit cell can be ambiguous. In order to eliminate this ambiguity, four indices, (hkil), are often used. These are called Miller-Bravais indices and are only used in the hexagonal system. The index i is given by... 

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

We consider the case where the sites form a Bravais lattice, which means that the position Rt of an / s-site i can be expressed by the primitive vectors a, b, and ns-c of the hexagonal lattice and the integers na[Pg.21]

Figure 3.4. The crystal systems and the Bravais lattices illustrated by a unit cell of each. All the points which, within a unit cell, are equivalent to each other and to the cell origin are shown. Notice that, in the primitive lattices the unit cell edges are coincident with the smallest equivalence distances. For the rhombohedral lattice, described in terms of hexagonal axis, the symbol hR is used instead of a symbol such as rP. In the construction of the so-called Pearson symbol ( 3.6.3), oS and mS will be used instead of oC and mC.

Figure A.1.2. Miller-Bravais axes in the hexagonal system.

Since a subsidiary axis is assumed in addition to the and axes in the hexagonal system (and in the hexagonal expression of the trigonal system), the index is expressed by four indices for a general face (hktl). This is called the Miller-Bravais index. For example, in Fig. A.3.1(b) a face ACB cuts the + and + axes at 1 (= OA, OB), and the - axis at 0C = 0D = OB/2. Therefore this face is indexed as (1120). From geometry, h(l) + k(l) = i(2). 

In Section 11.4 the fourteen 3D lattices (Bravais lattices) were derived and it was shown that they could be grouped into the six crystal systems. For each crystal system the point symmetry of the lattice was determined (there being one point symmetry for each, except the hexagonal system that can have either one of two). These seven point symmetries are the highest possible symmetries for crystals of each lattice type they are not the only ones. 

Figure 6. TEM images of STAC-1 viewed down the a axis of a hexagonal unit cell (indicated by [M/]h) or the [110] direction of a cubic unit cell (indicated by [M/]c). The crystal is dominated by ABCABC close packing (indicated on (a)) with one stacking fault (marked by a horizontal line). A Fourier transform optical diffraction pattern with both Miller-Bravais indices to the hexagonal unit cell and Miller indices (in parentheses) to the cubic unit cell is inserted in (b). Simulated images based on a proposed model (right) are also inserted with specimen thickness of 30 nm, and lens focuses of—30 nm (a) and —10 nm (b).

I/15,000 1.6.000. which in lowest terms arc 3 2 5. These smallest integers are the Miller indices of the family lo which this plane belongs, and the family is thus designated (325). The family (2011 is parallel to the Y axis but intersects the X and Z axes. (The hexagonal system has four Bravais-Miller indices lor each plane-family. I... 

Example 16.1-1 Find the Bravais lattices, crystal systems, and crystallographic point groups that are consistent with a C3z axis normal to a planar hexagonal net. 

The Miller-Bravais index system for identifying planes and directions in hexagonal crystals is similar to the Miller index system except that it uses four axes rather than three. The advantage of the four-index system is that the symmetry is more apparent. Three of the axes, ai, a2, and a3, he in the hexagonal (basal) plane at 120° to one another and the fourth or c-axis is perpendicular to then, as shown in Figure 3.1. 

There is also the three-digit system for directions in hexagonal crystals. For planar indices, it uses intercepts on the ai, a2, and c axes. The indices (HKL) are related to the Miller-Bravais indices (hkil) by... 

Bravais showed in 1850 that all three-dimensional lattices can be classified into 14 distinct types, namely the fourteen Bravais lattices, the unit cells of which are displayed in Fig. 9.2.3. Primitive lattices are given the symbol P. The symbol C denotes a C face centered lattice which has additional lattice points at the centers of a pair of opposite faces defined by the a and b axes likewise the symbol A or B describes a lattice centered at the corresponding A or B face. When the lattice has all faces centered, the symbol F is used. The symbol I is applicable when an additional lattice point is located at the center of the unit cell. The symbol R is used for a rhombohedral lattice, which is based on a rhombohedral unit cell (with a = b = c and a = ft = y 90°) in the older literature. Nowadays the rhombohedral lattice is generally referred to as a hexagonal unit cell that has additional lattice points at (2/3,1 /3, /s) and (V3,2/3,2/3) in the conventional obverse setting, or ( /3,2/3, ) and (2/3, /3,2/3) in the alternative reverse setting. In Fig. 9.2.3 both the primitive rhombohedral (.R) and obverse triple hexagonal (HR) unit cells are shown for the rhombohedral lattice. 

For a crystal having a hexagonal symmetry, a set of four numbers, [uvtw], named the Miller-Bravais coordinate system (see Figure 1.5), is used to describe the crystallographic directions, where the first three numbers, that is, u, v, t, are projections along the axes at, a2, and a3, describing the basal plane of the hexagonal structure, and w is the projection in the z-direction [2,3],... 

Figure 1.3 Five two-dimensional Bravais lattices. Clockwise from upper left square, rectangular, oblique, hexagonal, and (center) centered rectangular.

In direct analogy with two dimensions, we can define a primitive unit cell that when repeated by translations in space, generates a 3D space lattice. There are only 14 unique ways of connecting lattice points in three dimensions, which define unit cells (Bravais, 1850). These are the 14 three-dimensional Bravais lattices. The unit cells of the Bravais lattices may be described by six parameters three translation vectors (a, b, c) and three interaxial angle (a, (3, y). These six parameters differentiate the seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

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