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[56] 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[56]) 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'[57] similar to the
so-called magic strains used to generate the BCT5 structure proposed as
a high pressure phase of silicon[58].
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[22] 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.
