Symmetry close-packing

Figure 20. Gibbs free energy of formation of clusters as a function of size N at different supersaturations A/u. k and j are calculated from il/i and iff a = lAi and the number of first neighbors of the cluster configuration of lowest AG value (fivefold symmetry close-packed structure) according to Stoyanov. ...

A LEED pattern is obtained for the (111) surface of an element that crystallizes in the face-centered close-packed system. Show what the pattern should look like in symmetry appearance. Consider only first-order nearest-neighbor diffractions. 

In the CHS model only nearest neighbors interact, and the interactions between amphiphiles in the simplest version of the model are neglected. In the case of the oil-water symmetry only two parameters characterize the interactions b is the strength of the water-water (oil-oil) interaction, and c describes the interaction between water (oil) and an amphiphile. The interaction between amphiphiles and ordinary molecules is proportional to a scalar product between the orientation of the amphiphile and the distance between the particles. In Ref. 15 the CHS model is generalized, and M orientations of amphiphiles uniformly distributed over the sphere are considered, with M oo. Every lattice site is occupied either by an oil, water, or surfactant particle in an orientation ujf, there are thus 2 + M microscopic states at every lattice site. The microscopic density of the state i is p.(r) = 1(0) if the site r is (is not) occupied by the state i. We denote the sum and the difference of microscopic oil and water densities by and 2 respectively and the density of surfactant at a point r and an orientation by p (r) = p r,U(). The microscopic densities assume the values = 1,0, = 1,0 and 2 = ill 0- In close-packing case the total density of surfactant ps(r) is related to by p = Ylf Pi = 1 - i i. The Hamiltonian of this model has the following form [15]... 

FIGURE 5.26 A fragment of the structure formed as described in Fig. 5.25 shows the hexagonal symmetry of the arrangement—and the origin of its name, "hexagonal close-packed."... 

In cubic close-packing each molecule is surrounded by twelve others, whose interaction with the central molecule can be represented by a potential function of cubic point-group symmetry in case that the twelve molecules are spherically symmetrical or oriented at random. The energy change produced by this potential function,/say, is... 

This completes our discussion of the beisis and factors developed by past investigators to describe and conceptulize the structure of solids. You will note that we have not yet fully described the s)unmetry factor of solids. The reason for this is that we use symmetry factors to characterize solid structure without resorting to the theoretical basis of structure determination. That is, we have a standard method for categorizing solid structures. We say that salt, NaCl, is cubic. That is, the Na" ion and the Cl ion are alternately arranged in a close-packed cubic structure. The next section now investigates these structure protocols. 

The term crystal structure in essence covers all of the descriptive information, such as the crystal system, the space lattice, the symmetry class, the space group and the lattice parameters pertaining to the crystal under reference. Most metals are found to have relatively simple crystal structures body centered cubic (bcc), face centered cubic (fee) and hexagonal close packed (eph) structures. The majority of the metals exhibit one of these three crystal structures at room temperature. However, some metals do exhibit more complex crystal structures. 

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. 

---