Diffraction evidence suggests that the postrecovered form of silicon has body-centered cubic symmetry with sixteen atoms in the cubic unit cell and a lattice parameter of 6.64Å. The space group of the structure is Ia - . The structure is then fully specified by a single lattice parameter, a, and a single positional parameter x which has been experimentally reported as being 0.1003 0.0008 in silicon. A projection of the BC8 structure is shown in Figure 4.1.
Figure 4.1: Projection of the structure of BC8 on (001). Elevations (in a/10) are given by the numbers inside the circles. A and B label the two distinct bonds.
The structure is based on tetrahedrally co-ordinated atoms, but with a rather more efficient packing than in diamond. This efficiency leads to a higher density structure (hence favoured at high pressures) at the expense of small distortions from the diamond bondlength but with an appreciable change in the tetrahedral angle relative to the diamond structure. The distortions can be described by the existence of two bondlengths (R and R ) and two bond angles ( and ). It is possible to choose special values of x to satisfy either R =R or but in practice the observed values (R =2.30Å, R =2.39Å, =117.9 and =99.2) seem to be a compromise between the two. All atoms are equivalent, and the structure contains only evenfold rings, the smallest being sixfold, as with diamond. Although BC8 exhibits a density some 10% higher than diamond, the B bondlengths are slightly larger. The relations connecting bondlengths and the internal parameter x are as follows:
Where and are the bondlengths of the A and B bonds, while is the length of the nearest non-bonded neighbour and is the lattice parameter (see Figure 4.1). The two bond angles also depend on x via the following relations:
This allows a perfect tetrahedral angle at . This is never realised in atomic systems because it would require the A bond to be only 62% as long as the B bond. These expressions are useful because they show that the bond angles become less ideal with increasing x. Other values of x having interesting structural consequences are where the structure becomes a threefold coordinated layer structure with graphitic layers of type B bonds, and which is a transformation which leaves the structure unchanged. Thus increasing x can be regarded as a `magic internal strain' similar to the so-called magic strains used to generate the BCT5 structure proposed as a high pressure phase of silicon.
The postrecovered form of germanium, ST12, is more complex. A projection of the ST12 structure is shown in Figure 4.2.
Figure 4.2: Projection of the structure of ST12 on (001). Elevations (in fractional co-ordinates) are given by the numbers inside the circles. A, B and C label the three distinct bonds.
Like BC8 it is based on tetrahedrally co-ordinated atoms packed in such a way as to increase the density to 10% above diamond. Crystallographic solution of this phase cites it as having a simple tetragonal unit cell with 12 atoms. The structure would appear to have unusual optical properties since it has left- and right-handed forms, although there has not yet been any experiment done to examine this, presumably because of the difficulty in obtaining suitable single crystals. The space group is or its enantiomorph. The fully relaxed structure can be defined by two lattice parameters and four atomic positional parameters.
In ST12 there are two distinct atomic environments, which leads to some rather complicated topological substructures. Four of the twelve atoms are in environment a, and the remaining eight in b. The b type can be viewed as forming spiral chains along the unique axis, while the a atoms bridge different spirals. All the spirals rotate the same way, giving the structure a well-defined helicity as noted above. Although the atoms are still fourfold co-ordinated, there are now five and seven membered rings, and the variation in bond-angles (ranging from to ) is greater than in BC8 but the bond lengths are clustered more closely around the value found in diamond. In Chapter 4 it is shown that the stability of ST12 against BC8 rests in the compromise between equalizing bond lengths at the expense of increasing bond angle distortions. ST12 occurs when the bond angles are relatively easier to distort.