CHEMICAL BONDING AND THE PROPERTIES OF MINERALS - Chemical Fundamentals of Geology and Environmental Geoscience - page 118

The upper limit for octahedral co-ordination is met at
a radius ratio of 0.732, at which the cation is sufficiently
large to touch eight equidistant neighbours at the same
time (Table 7.1). The requirement of maximum co-ord-
ination leads to a new structure in which the anion nuclei
lie at the corners of a cube, with the cation enjoying eight-
fold co-ordination at the centre ( caesium chloride structure ,
see Figure 7.3b). The mineral fluorite CaF 2 (Figure. 7.3d)
is an example of an AB 2 compound with the cation in
eight-fold co-ordination. Each fluoride F - ion lies at the
centre of a tetrahedral group of Ca 2+ ions.
If the radius ratio exceeds 1.0, the co-ordination
number rises to 12. Eight-fold and larger sites cannot
exist as interstitial sites in a close-packed assemblage
of anions, and the presence of large ions like K + requires
the host crystal to have a more open structure, as for
example in feldspars.
Analysing ionic crystal structures as if they were 3D
assemblages of hard spheres thus explains why the
structure of halite, for example, differs from that of
rutile, and also provides a basis for understanding the
preferences of major elements for specific sites in sili-
cate minerals (Table 8.2). One must recognize, how-
ever, that the ionic model is an idealization, and real
crystal structures are complicated by other factors. In
many minerals the bonding involves a degree of elec-
tron-sharing (covalency - see the following section)
which undermines the assumption that bonding is
non-directional, and where transition metals are con-
cerned (Chapter 9) the presence of d-orbitals intro-
duces still more complications. Predictions of crystal
structure based on the radius ratio must therefore be
seen only as general guidelines.

far considered. These are the characteristics of covalent
bonding, which operates through the sharing of unpaired
electrons between neighbouring atoms.
An unpaired electron is one that occupies an orbital
on its own. When such singly occupied orbitals in adj-
acent atoms overlap each other, they coalesce to form a
molecular orbital , allowing the electrons to pass freely
between one atom and the other. In this shared state
the electrons have a total energy lower than that in
either atom individually (Box 7.3), and consequently
the atoms have greater stability attached to each other
than they had as separate atoms. The greater the degree
of overlap, the stronger the attractive force becomes;
however, as with ionic bonding (Box 7.1), the tendency
for the atoms to continue approaching each other is
restricted by a short-range repulsion between the core
electrons of the two atoms, and ultimately between
their nuclei. The equilibrium covalent bond length
between two atoms can be divided up into the covalent
radii of the individual atoms, but the values differ
numerically from the corresponding ionic radii.

σ - and π -bonds

The unpaired electron in a hydrogen atom is found in
the 1 s orbital, so the coupling in a hydrogen molecule
H 2 is due to 1 s overlap. The molecular orbital so
formed (Box 7.3) is designated 1s σ , and its occupation
by two electrons can be symbolized by the electron
configuration 1s σ 2 . A σ-bond is one with cylindrical
symmetry about the line joining the two nuclei
(Figure 7.4a). σ -bonds can also form when two p-orbitals
overlap end-on, or when a p-orbital in one atom over-
laps with an s-orbital in another, as in the water mole-
cule (Figure 7.4b). The participation of p-orbitals, with
their elongated shape (Figure 5.5), gives a σ -bond a
specific direction, whose orientation in relation to
other bonds determines the shape of a multi-atom mol-
ecule like H 2 O. Note that the two σ -bonds in the water
molecule involve separate p-orbitals of the oxygen
atom, each of which contributes an unpaired electron;
it is not possible for the hydrogen atoms to be attached
to opposite lobes of the same p-orbital.
Geometric constraints prevent the formation of more
than one σ -bond between the same pair of atoms. There
is, however, another way in which p-orbitals can link
atoms together. Figure 7.4c shows two atoms that are
already joined by a 2p σ bond as described above; the σ
molecular orbital is shown by the heavy stipple.

The covalent model of bonding

Many substances exhibit chemical bonding between
atoms having the same, or very similar, electronegativ-
ity. Among them are some of the hardest, most strongly
bonded materials known, including diamond, silicon
carbide and tungsten carbide. In materials like these,
ionic bonding cannot operate and a different bonding
mechanism must be at work. Although capable of form-
ing extended crystalline structures like diamond, this
bonding is also responsible for small discrete molecules
like O 2 , CH 4 , CO 2 and H 2 O, the shape of which often
indicates a directional type of bond completely foreign
to the electrostatically bonded, close-packed materials so