Bravais-Miller indices

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.

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).

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... 

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...

MILLER-BRAVAIS INDICES FOR HEXAGONAL COORDINATE SYSTEMS... 

Miller-Bravais indices (hkil) are used with ... 

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

FIGURE 2.44 Introducing Miller-Bravais indices by special geometrical system reference, after Chiriac-Putz-Chiriac (2005). 

Miller-Bravais indices Pearson classification point groups reducing cell method Schoenflies notation space groups stereographic projection symmetry perturbations Weiss zone law Wulff map... 

Fig. B.2. Coordinate system used for Miller-Bravais indices...

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