Simple molecular lattices

We must exercise some caution when using ficticious lattices. A simple molecular lattice for Si()2 inevitably has the topology of a two-dimensional system, as is apparent in Fig. 11-4, and calculation of some properties can give results that arc qualitatively incorrect. To construct a three-dimensional molecular lattice for this coordination, we require at least Si204, which would then have the topology of... 

The simple molecular lattice for SiOi- The large circles represent silicon atoms the small, oxygen atoms. There is one Si02 per primitive cell. The coordination of each atom is the same as in real Si02, but the connectivity and symmetry have been simplified. 

It will help here to mention the structure of polar counterparts to SiC)2. These are constructed exactly as aluminum phosphide is constructed from silicon in the simple tetrahedral solid. The process is illustrated for the simple molecular lattice in I ig. 11-5, which shows how the structure for SiOj can be transformed to the structure for aluminum phosphate. Transferring an additional proton leads to magnesium sulphate. Indeed, the counterpart of each AB tetrahedral semiconductor, ABO4, is po.ssible in principle. The structures may be obtained from Wyck-hoff (1963), and if the structure has twofold-fourfold coordination, it can be analyzed by the methods outlined here. 

The lattice vibrations for the simple tetrahedral lattice wore studied in Section 9-A. The state of the distortion of the lattice was specified by giving the displacement (5r, of each atom. We then made a transformation to normal coordinates u., each corresponding to a normal mode frequency w(k), and these were plotted as a function of k in Fig. 9-2. There were three curves for each atom in the primitive cell. We see immediately that there will be difficulties in complex structures in quartz there arc 27 sets of modes, and oven in the simple molecular lattice there are 9. This complexity suggests that one should proceed by computer. One such approach was taken by Bell, Bird, and Dean (1968). They took a large cluster... 

We choose here instead an analytic formulation based upon the simple molecular lattice, which will immediately be generalized. This will bring the most important concepts to light and provide an interpretation of the spectrum. It can also provide the starting point for a quantitative application of the cluster -Belhc-latticc method use of the Belhc lattice should improve accuracy and reduce the computation required in comparison to the direct cluster technique. 

We initially consider the simple molecular lattice and only the modes with k = 0, those modes at F in the Brillouin Zone. These arc, in fact, just the modes that arc coupled to infrared light, (Light is of sufficiently long wavelength to be regarded as having k = 0.)... 

In the simple molecular lattice, at k = 0, the problems of finding the acoustical modes and the optical modes arc independent. Even at k 0 they would be independent problems if the ratio of the silicon to the oxygen mass were sufficiently large. In the end we shall see that the lowest optical mode frequencies overlap the acoustical mode frequencies, so to separate the problems is not entirely valid. However, for the present, let us separate the problems. 

These are the same for every oxygen if the environment of each oxygen is the same. Notice that there is one of each type of local mode for every oxygen present, and that fact does not depend upon the assumption of a simple molecular lattice. 

The simple molecular lattice for AIPO4. To make this, beginning with the simple molecular lattice of SiOj, from Fig. 11-4, we remove a proton from alternate silicon nuclei (open circles), making them aluminum, and place the proton in the other silicon nuclei (shaded circles), making them phosphorus. The primitive cell has become twice as large but the structure is the same as that of SiOj and isoelectronic with it. 

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