The Miller-Bravais Indices

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... 

The Miller-Bravais indices following a hardness value indicate the crystallographic face on which the measurement was made. 

Determination of the Miller-Bravais Indices for a Plane within a Hexagonal Unit Cell... 

Determine the Miller-Bravais indices for the plane shown in the hexagonal unit cell. 

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... 

Fig. 2.3 Evidence of monolayer graphene from TEM [72]. (a) and (b) High-resolution TEM images of solution-cast monolayer (a) and bilayer (b) graphene (scale bar 500 nm) (c) electron diffraction pattern of the sheet in (a), with the peaks labeled by Miller-Bravais indices (d) and (e) electron diffraction patterns taken from the positions of the black (d) and white spots (e), respectively, of the sheet shown in (b), using the same labels as in (c). fhe graphene is clearly one layer thick in (d) and two layers thick in (e) (f)-(h) Diffracted intensity taken along the 1210 to 2110 axis for the patterns shown in (c)-(e), respectively (i) Histogram of the ratios of the intensity of the 1100 and 2110 diffraction peaks for all the diffraction patterns collected. A ratio > 1 is a signature of graphene.

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). 

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).

Examples of direction indices with the Miller-Bravais system. 

Auguste Bravais (1811-1863) first proposed the Miller-Bravais system for indices. Also, as a result of his analyses of the external forms of crystals, he proposed the 14 possible space lattices in 1848. His Etudes Cristallographiques, published in 1866, after his death, treated the geometry of molecular polyhedra. 

For hexagonal crystal planes, a slightly different indexing nomenclature is used relative to cubic crystals. To index a plane in the hexagonal system, four axes are used, called Miller-Bravais indices. In addition to both a and b axes, another axis... 

Since i is determined by h and k, it is sometimes replaced by a dot and the plane symbol writ n hk /). Sometimes even the dot is omitted. However, this usage defeats the purpose for which Miller-Bravais indices were devised, namely, to give similar indices to similar planes. For example, the side planes of the hexagonal prism in Fig. 2-11(b) are all similar and symmetrically located, and their relationship is clearly shown in their full Miller-Bravais symbols (lOTO), (OlTO), (TlOO), (To 10), (OTlO), (iTOO). On the other hand, the abbreviated symbols of these planes, (10-0), (01 -0), (Tl 0), (TO-O), (OT-0), (lT-0) do not immediately suggest this relationship. 

Figure 2.15 Miller-Bravais indices in hexagonal lattices. The three sets of identical planes marked have different Miller indices but similar Miller-Bravais indices...

In hexagonal crystals, where the base plane exhibits a 120° symmetry, the Miller-Bravais system is used to specify planes and directions, using a coordinate system with four axes Three of these, with angles of 120° between them, lie in the base plane and are equivalent, the fourth is perpendicular to the base plane [hkil] (figure B.2). In this way, the symmetry of the crystal is reflected in the indices. The first three indices obey the additional constraint... 

Fig. 8 TEM image of a single GO sheet on a lacey carbon support a double fold is visible in the top right corner, (b) Selected-area diffraction pattern of the centre of the region shown in (a), the diffraction spots are labelled with Miller-Bravais indices. (Reproduced with permission.)... 

The designation of planes is more straightforward and proceeds as before. Plane A intersects the a-axis at -1/2 and axes b and c at 1. If it is perpendicular to the basal plane, it would be designated as (2110). If it also intersected the z-axis at one unit, it would be (2111). Plane B intercepts the b-axis at 1, the a-axis at —1, and the c-axis not at all. It would thus be denoted (ilOO). Again note that the sum of the first three Miller-Bravais indices is always zero in this notation. 

Here, uppercase U, V, and W indices are associated with the three-index scheme (instead of u, V, and w as previously), whereas lowercase u, v, f, and w correlate with the Miller-Bravais four-index system. For example, using these equations, the [010] direction becomes [1210]. Several directions have been drawn in the hexagonal unit cell of Figure 3.9. 

For crystals having hexagonal symmetry, it is desirable that equivalent planes have the same indices as with directions, this is accomplished by the Miller-Bravais system shown in Figure 3.8. This eonvention leads to the four-index (hkil) scheme, which is favored in most instances because it more clearly identifies the orientation of a plane in a hexagonal crystal. There is some redimdaney in that i is determined by the sum of h and k through... 

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... 

The rules for determining Miller-Bravais planar indices are similar to those for Miller indices with three axes. 

Index the lattice planes drawn in the figure below using Miller - Bravais (hkil) and Miller indices (hid). The lattice is hexagonal with the c-axis is normal to the plane of the page and hence the index l is 0 in all cases. 

FIGURE 5.6 Indices of planes (using Miller-Bravais notation) and directions (using three-index Miller notation) in the hexagonal unit cell. 

Mostly, the Miller indices are integers and small numbers, rarely larger than 3 and even rarely greater than 5 (as based on empirical observations). These characteristics have substance in the Bravais observation that only the planes (reticular layers) with large reticular density will appear as natural faces, so these faces being simple planes in the unit cell with small Miller indices. 

Generalizing the previous procedure, the transition from a Miller index ot pqr directions to associated [uvtw] Miller-Bravais one it will be made by the indices transformations ... 

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