Interpenetrating lattices

Figure 29.1 Crystal structures of ZnS. (a) Zinc blende, consisting of two, interpenetrating, cep lattices of Zn and S atoms displaced with respect to each other so that the atoms of each achieve 4-coordination (Zn-S = 235 pm) by occupying tetrahedral sites of the other lattice. The face-centred cube, characteristic of the cep lattice, can be seen — in this case composed of S atoms, but an extended diagram would reveal the same arrangement of Zn atoms. Note that if all the atoms of this structure were C, the structure would be that of diamond (p. 275). (b) Wurtzite. As with zinc blende, tetrahedral coordination of both Zn and S is achieved (Zn-S = 236 pm) but this time the interpenetrating lattices are hexagonal, rather than cubic, close-packed.

Fig. 1. The crystal structure of a hydroquinone clathrate according to Palin and Powell. 8 The balls inside the transparent spheres represent argon atoms encaged in the cavities formed by the two interpenetrating lattices, (photograph kindly supplied by Dr. Powell).

The order-disorder transition of a binary alloy (e.g. CuZn) provides another instructive example. The body-centred lattice of this material may be described as two interpenetrating lattices, A and B. In the disordered high-temperature phase each of the sub-lattices is equally populated by Zn and Cu atoms, in that each lattice point is equally likely to be occupied by either a Zn or a Cu atom. At zero temperature each of the sub-lattices is entirely occupied by either Zn or Cu atoms. In terms of fractional occupation numbers for A sites, an appropriate order parameter may be defined as... 

The points are not all equivalent, that is, in identical environments. There are in fact three distinct kinds. (Select a representative example of each kind and label them a, / , and y.) The array shown is three interpenetrating lattices, of a points, / points, and y points. All three lattices are identical but displaced from each other by the shortest horizontal and/or shortest vertical distance shown. 

Silver(I) oxide, [CAS 20667-12-3]. AgjO. is made by action of oxygen under pressure on silver at 300°C, or by precipitation of a silver salt with carbonate-free alkali metal hydroxide it is covalent, each silver atom (in solid AgjO) having two collinear bonds and each oxygen atom four tetrahedral ones two such interpenetrating lattices constitute the structure. Silver(I) oxide is die normal oxide of silver. Silver(II) oxide, AgO, is formed when ozone reacts with silver, and thus was once considered to be a peroxide, Silvcr(III) oxide, Ag203, has been obtained in impure state by anodic oxidation of silver. 

It turns out that this is not true generally, but a model built on this assumption does a fairly good job of explaining a rather small but important class of compounds that are called ionic solids. The most well known example of such a compound is sodium chloride, which consists of two interpenetrating lattices of Na+ and CE ions arranged in such as way that every ion of one type is surrounded (in three dimensional space) by six ions of opposite charge. 

It is of interest to note in connection with the question of the nature of the hydrogen ion in solution that the crystalline hydrate of perchloric acid, HC104-H20, has been shown by X-ray diffraction methods to have the same fundamental structure as ammonium perchlorate. Since the latter consists of interpenetrating lattices of NHt and ClOr ions, it is probable that the former is built up of H3O+ and CIO4 ions. 

Fig. 9.39. Interpenetrating lattices used to consider grain boundary structure (courtesy of D. Pawaskar). Open and filled circles correspond to the two host lattices and filled squares correspond to those atoms that are common to both lattices (i.e. the lattice of coincident sites).

In addition, as was mentioned above, it is possible that there is a shift of origin associated with some translation vector t. The matrix Q imposes a linear transformation which, since it describes a rotation, may be characterized by three parameters. We now have two interpenetrating lattices as indicated in the simplified two-dimensional setting in fig. 9.39. As yet, there is no grain boundary. The remainder of our work in identifying the macroscopic degrees of freedom is to select the boundary plane itself. This plane is characterized by a unit vector n... 

If we wish to locate comparatively large particles in these holes, then all the particles of the original lattice must move apart the structure expands to enlarge the holes to sufficient size. We can view the sodium chloride structure as an fee arrangement of chloride ions that has expanded sufficiently to permit the sodium ions in the octahedral holes. As a result, neither of these interpenetrating lattices is closely packed in the sense of having all the particles in contact as they are in metals, but both have the symmetry of the close-packed fee lattices. Each sodium ion is in contact with six chloride ions, and each chloride ion is in contact with six sodium ions 6-6 coordination. Figure 27.8 shows the NaCl... 

The strategy is to assemble a lattice of defects having a particular space group symmetry and fill the interstices with nematic material. The director is then allowed to relax everywhere (except of course at the defects) and the free energy calculated. From this process one finds that there is also an interpenetrating lattice of double-twist tubes and that the whole structure is stable between the helical and isotropic phases. Figure 7.7 shows both the double-twist lattice and the defect lattices for the proposed structures scO and bcc(9 . Other structures—bccO and bccO +—have also been worked out [42]. 

We use the formulas from the preceding sections to study the short-range ordered states in some typical disordered alloys. Our Monte Carlo calculations start from a code written for the simple cubic lattice by Loren P. Meissner to illustrate the use of array intrinsic functions in fortran 90 and 95. It treats the 2 interpenetrating cubic lattices in bcc or the 4 interpenetrating lattices in fee as blocks. We use the Ising Hamiltonian,... 

Finally one can consider systems, which certainly are somewhat remote from experiments, where the only source of disorder is due to the occurrence of knotted loops. Such are the interpenetrating lattice networks. They can be analyzed and investigated in a variety of ways in order to shed some light on the question of entanglements and conserved topology. 

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