Hexagonal Close-Packed Materials

For both quantities, our predictions are very close (although not equal to) the experimental values. 

It is not hard to understand why many metals favor an fee crystal structure there is no packing of hard spheres in space that creates a higher density than the fee structure. (A mathematical proof of this fact, known as the Kepler conjecture, has only been discovered in the past few years.) There is, however, one other packing that has exactly the same density as the fee packing, namely the hexagonal close-packed (hep) structure. As our third example of applying DFT to a periodic crystal structure, we will now consider the hep metals. 

The supercell for an hep metal is a little more complicated than for the simple cubic or fee examples with which we have already dealt. The supercell can be defined using the following cell vectors  

The definition of the hep supercell given above is useful to introduce one more concept that is commonly used in defining atomic coordinates in periodic geometries. As our definition stands, the vectors defining the shape of the 

Because we can always choose each atom so it lies within the supercell, 0 j), 1 for all i and j. These coefficients are called the fractional coordinates of the atoms in the supercell. The fractional coordinates are often written in terms of a vector for each distinct atom. In the hep structure defined above, for example, the two atoms lie at fractional coordinates (0,0,0) and (5,5, 5). Notice that with this definition the only place that the lattice parameters appear in the definition of the supercell is in the lattice vectors. The definition of a supercell with a set of lattice vectors and a set of fractional coordinates is by far the most convenient way to describe an arbitrary supercell, and it is the notation we will use throughout the remainder of this book. Most, if not all, popular DFT packages allow or require you to define supercells using this notation. 

---

Figure Bl.21.1 shows a number of other clean umeconstnicted low-Miller-index surfaces. Most surfaces studied in surface science have low Miller indices, like (111), (110) and (100). These planes correspond to relatively close-packed surfaces that are atomically rather smooth. With fee materials, the (111) surface is the densest and smoothest, followed by the (100) surface the (110) surface is somewhat more open , in the sense that an additional atom with the same or smaller diameter can bond directly to an atom in the second substrate layer. For the hexagonal close-packed (licp) materials, the (0001) surface is very similar to the fee (111) surface the difference only occurs deeper into the surface, namely in the fashion of stacking of the hexagonal close-packed monolayers onto each other (ABABAB.. . versus ABCABC.. ., in the convenient layerstacking notation). The hep (1010) surface resembles the fee (110) surface to some extent, in that it also...

The hexagonal-close-packed (hep) metals generally exhibit mechanical properties intermediate between those of the fee and bcc metals. For example Zn encounters a ductile-to-brittle transition whereas Zr and pure Ti do not. The latter and their alloys with a hep structure remain reasonably ductile at low temperatures and have been used for many applications where weight reduction and reduced heat leakage through the material have been important. However, small impurities of O, N, H, and C can have a detrimental effect on the low temperature ductihty properties of Ti and its alloys. 

The chemical bonding and the possible existence of non-nuclear maxima (NNM) in the EDDs of simple metals has recently been much debated [13,27-31]. The question of NNM in simple metals is a diverse topic, and the research on the topic has basically addressed three issues. First, what are the topological features of simple metals This question is interesting from a purely mathematical point of view because the number and types of critical points in the EDD have to satisfy the constraints of the crystal symmetry [32], In the case of the hexagonal-close-packed (hep) structure, a critical point network has not yet been theoretically established [28]. The second topic of interest is that if NNM exist in metals what do they mean, and are they important for the physical properties of the material The third and most heavily debated issue is about numerical methods used in the experimental determination of EDDs from Bragg X-ray diffraction data. It is in this respect that the presence of NNM in metals has been intimately tied to the reliability of MEM densities. 

Mono- or single-crystal materials are undoubtedly the most straightforward to handle conceptually, however, and we start our consideration of electrochemistry by examining some simple substances to show how the surface structure follows immediately from the bulk structure we will need this information in chapter 2, since modern single-crystal studies have shed considerable light on the mechanism of many prototypical electrochemical reactions. The great majority of electrode materials are either elemental metals or metal alloys, most of which have a face-centred or body-centred cubic structure, or one based on a hexagonal close-packed array of atoms. 

In the wurtzite form of ZnS the sulfur atoms are arranged in hexagonal close packing, with the metal atoms in one-half of the tetrahedral positions. There are two layers of tetrahedra in the repeat distance, c, and these point in the same direction. This gives the materials a unique axis, the c axis, and these compounds show piezoelectricity. 

Hexagonal close-packed (hep) materials have a sixfold symmetry axis normal to the basal plane. Using a three-axis system to define Miller indices for this structure is unsatisfactory, as is demonstrated in Fig. 4.9. The two planes highlighted in Fig. 4.9 are equivalent by symmetry, and yet their Miller indices do not show this relationship. This is unfortunate since one of the reasons Miller indices are useful is that equivalent planes have similar... 

Related behaviour is exhibited by the dealkylated analogue, calix[4]arene (7.63). This material can also be sublimed to give a guest-free host but in this case the empty host forms an unusual trimer motif which is almost spherical and packs in a hexagonal close packed array with just a small interstitial void... 

---