Faceted crystallographic shape

Wang, Z. L., Mohamed, M. B., Link, S. and El-Sayed, M. A. (1999). Crystallographic facets and shapes of gold nanorods of different aspect ratios. Surf. Sci. 440 L809-L814. 

Transmission electron microscopy ( ) analysis reveals that these materials crystallize as hexagonal planar particles with marked anisotropic shape,8,37 as shown in Figure 6. When appropriate preparation methods are used, plate-like crystals are obtained with small thickness of about 20-30 nm and an aspect ratio D/h=5-10. Selected area diffraction (SAD) patterns of incident beams perpendicular and parallel to the large hexagonal facet show that they correspond to the crystallographic planes perpendicular to the c axis. The anisotropic shape of the... 

The means to determine the minimum-energy shape for a crystal of fixed volume was developed by Wulff (38), who showed that the equilibrium shape can be determined if the surface tension, y, at all crystallographic orientations is known. As illustrated in Fig. 2, on a polar y plot of the surface tension as a function of orientation, the inner envelope of the planes drawn perpendicular to and at the ends of the radius vectors gives the equilibrium shape of a crystal of constant volume. Faceting in the equilibrium crystal shape is due to cusps in the polar y plot. 

Fig. 10 Four common types of crystal habit in polyethylene and long alkanes (a) rhombic lozenge bounded by 110 facets (b) lozenge truncated by curved 100 faces (Toda s type B) (c) leaf-shaped crystal bounded solely by curved 100 faces (step propagation rate v equals h = Gno/sin(

For an isotropic material, the equilibrium shape is a sphere. For an anisotropic material, the equilibrium shape will depart from sphericity so that crystallographic orientations with relatively low values of y will be exposed. Based on Eq. 1, it has been demonstrated that for a crystal in equilibrium, the ratio of the energies of two orientations (y, and is equal to the ratio of the perpendicular distances from the center of the crystal to the facet and l ) [27] ... 

Fig. 6.9c) and a low (Fig.6.9d) aspect ratios, gold nanocrystals with hexagonal profiles (Fig. 6.9e) and cubic profiles (Fig. 6.9f), etc. can also provide ordered two-dimensional structures [40]. These results clearly show that the shape and the crystallographic facets of nanocrystals have a strong influence to the nature of the self-assembly patterns [40,42]. 

FIGURE 36.20 Different diamond shapes produced by diamonds faceting on three different crystallographic planes while keeping the same symmetry. 

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