In both dense phases, as the volume is reduced the distortions of the
tetrahedra increase. In the BC8 structure these distortions are
uniquely described by the variation in the *x* parameter as described in
Section 4.3. For the BC8 structures of Si, Ge and C the
structural response to applied pressure is shown is represented in Figure
4.11 where the calculated variation in the free structural
parameter, *x*, is shown. It is clear that both BC8-Si and BC8-Ge behave
similarly under pressure as the equilibrium value of *x* and its slope are
nearly equal in the two materials. For Si, the value of *x* which fully
relaxes the structure is found to be *x*=0.1001 at the equilibrium
lattice constant of 6.54Å. The experimental value is reported to be
*x*=0.1003+/-0.0008. For fully relaxed BC8-Ge, *x*=0.1013. In
BC8-carbon, the equilibrium value of *x*=0.0935 is approximately 6%
smaller than it is for Si or Ge.

Although plots of *x* give a complete representation of the data from
the calculations, which are done at constant volume, it is more
informative to examine the variation of bondlengths with pressure, and
this is plotted in Fig. 4.16. Notice that under pressure *x*
increases, which tends to compensate for decreasing in , but
enhances the reduction in . The effect on is small
but gives rise to a small
additional decrease in with pressure, in addition to the
reduction in .

**Figure 4.16:** Variations in bondlengths and near neighbour distances with
volume change for BC8 silicon. The graph for germanium is similar. Note
that is most sensitive to isotropic compression. The *A* bond
changes by less than 0.1Å over the range shown. This insensitivity
results in a bondlength crossover at a lattice parameter of 6.25Å.

The effect of this compensation against decreasing is that whilst at ambient pressure the A bonds are shorter, as the pressure increases the B bond contracts until at all bonds are the same length. There is no special symmetry associated with this coincidence, and the B bond length continues to contract. A qualitative explanation of this comes from the empirical model in Chapter 4.

Consideration of the differentials of expressions (4.1)
to (4.5) makes it clear that decreases
with increasing *x* (pressure) while increases. Thus
pressure serves to increase the distortion of the tetrahedra: the price
which must be paid for maintaining bondlengths. It is thus clear that
the observation of increasing *x* with pressure means that the
bond-stretching forces have a greater bearing on the structure than the
bond-bending ones.

In ST12 the structure is only fully defined by four internal parameters. For this reason it is essential to relax the structure under the Hellmann-Feynman forces, and thus study of ST12 is ideal for the plane-wave basis set method. The relationships between these internal parameters and the bondlengths and bond angles are complex, and for clarity only the latter quantities are considered. The evolution of the internal structure under pressure is described by Figure 4.17 which depict the change of the three different bondlengths under pressure.

**Figure 4.17:** Variation of the three different bondlengths on ST12
germanium with volume. The graph for silicon is similar.

Following the notation of Kasper and Richards[22] for
ST12-Ge, the four atomic positional parameters are shown for axial
ratios of 1.2, 1.25 and 1.3 in Figure 4.18. It is
evident from the figure that the *z*-parameter for the 8(*b*) site has
the strongest dependence on isotropic compression of the unit cell and
that the positional parameters are not greatly affected by changes in
axial ratio over the range considered here. The ST12-Ge positional
parameters near the equilibrium volume and *c*/*a* ratio may be found
from Table 4.1. The calculated value of *x* for the
4(a) sites is 0.0882 compared to 0.0912 found experimentally. The
calculated *x*, *y* and *z* values for the 8(b) sites are *x*=0.1693,
*y*=0.3771 and *z*=0.2454. The corresponding experimental values at
ambient pressure are *x*=0.1730, *y*=0.3784 and *z*=0.2486. The
agreement is encouraging and suggests that future experiments using
recent advances in angle-dispersive powder diffraction[60]
to determine the pressure dependence of this structure would be of
value.

**Figure 4.18:** The relaxed ST12 structural
parameters for germanium against volume. The circles, squares and
triangles represent axial ratios of 1.25, 1.30 and 1.35
respectively.

As in BC8, the ST12 internal parameters vary with pressure to maintain
the bondlengths at the expense of further distorting the bond angles.
For a constant *c*/*a* ratio the bondlengths are reduced more slowly as a
function of volume than the unit cell parameters. In practice the *c*/*a*
ratio also changes under pressure in such a way as to reduce the
changes in bondlengths. The fully relaxed unit cell dimensions and
atomic positions in for silicon, germanium and
carbon in the BC8 and ST12 structures are summarised in Tables
4.3 and 4.4.

**Table 4.3:** The full set of positions are given for the BC8 structure in
terms of the one free structural parameter *x*. Also shown is the
relaxed lattice parameter *a* in Å and the corresponding *x* for
carbon, silicon and germanium.

**Table 4.4:** The full set of positions are given for the ST12 structure in
terms of the 4 free positional parameters *x*, *y*, *z* and x which
correspond to the (8b)-x, (8b)-y, (8b)-z and (4a)-x sites respectively.
Also shown is the relaxed lattice parameters *a* and *c* in Å and the
corresponding positional parameters for carbon, silicon and
germanium.

Thu Oct 31 19:32:00 GMT 1996