Page 57 - Materials Chemistry, Second Edition

44 2 Solid-State Chemistry

lattice arrangement, and is determined from electron density maps and empirical
X-ray diffraction data. Some general trends for cationic radii are:
1. For a given species and charge, the radius increases as the coordination number
increases.
2. For a given charge, the radius decreases with increasing effective nuclear charge,
Z eff . [17]
3. For a given species, the radius decreases with increasing ionic charge.
4. For a given species and charge, the radius is larger for high-spin (weak field) ions
than for low-spin (strong field) ions.
Most inorganic chemistry texts list cut-off values for the r þ /r ratios corres-
ponding to the various geometries of interstitial sites (Table 2.4). For instance, the
halite or rocksalt structure exhibited by MX (M ¼ Grp I, Mg, Pb, Ag; X ¼ F, Cl,
Br, I) are predicted to have occupation of octahedral interstitial sites. Indeed, these

structures are described as a fcc array of the halide ion (except for v. small F ),
with the cation occupying all of the octahedral interstitial sites (i.e., 4 MX units
per unit cell).
However, it should also be pointed out that deviations in these predictions are found
for many crystals due to covalent bonding character. In fact, the bonding character for
compounds is rarely 100% covalent or ionic in nature, especially for inorganic species.
For instance, consider the zinc sulfide (ZnS) crystal structure. The ionic radius ratio
for this structure is 0.52, which indicates that the cations should occupy octahedral
interstitial sites. However, due to partial covalent bonding character, the anions are
closer together than would occur from purely electrostatic attraction. This results in an
“effective radius ratio” that is decreased, and a cation preference for tetrahedral sites
rather than octahedral. One crystal structure for this complex lattice (a-ZnS, Wurtzite
structure – also found for b-AgI, ZnO, a-CdS, CdSe, a-SiC, GaN, AlN, oBN, and
BeO) is shown in Figure 2.23. This is best described as a hcp lattice of sulfide ions,
with zinc ions occupying one-half of the available tetrahedral interstitial sites. As you
might expect, a hybrid of ionic/covalent bonding will greatly affect the physical
properties of the solid; for instance, the hardness of ZnS is significantly greater than
what would be expected for a purely ionic solid.
Interestingly, zinc sulfide (b-ZnS) may also crystallize in a cubic lattice, which
consists of a fcc array of S , with Zn occupying 1/2 of the available tetrahedral
2
sites. This structure is known as sphalerite or zincblende, and is shared with other
compounds such as a-AgI, b-BN, CuBr, and b-CdS. When the same atom occupies
both the fcc and tetrahedral interstitials of the sphalerite structure, it is described as
the diamond lattice, shared with elemental forms (allotropes) of silicon, germanium,
and tin, as well as alloys thereof. Important semiconductors such as GaAs, b-SiC,
and InSb also adopt the sphalerite crystal structure.
If the cation in the crystal lattice exhibits a cubic environment (coordination
number of 8), the fluorite structure is commonly observed (Figure 2.24). Lattices
of this variety consist of an fcc arrangement of cations, with all eight tetrahedral
interstitial sites (e.g., (1/4, 1/4, 1/4), etc.) occupied by the anionic species. Of course,
this will only be prevalent when the size of the anion is much smaller than the cation,