Page 88 - Electrical Properties of Materials

70 Bonds

we counted each pair twice, or we may say that it is the cohesive energy per
NaCl unit.
The infinite summations look a bit awkward, but fortunately there are math-
ematicians who are fond of problems of this sort; they have somehow managed
to sum up all these series, not only for the cubical structure of NaCl, but for the
more complicated structures of some other ionic crystals as well. Their labour
brought forth the formula
M is called the Madelung con- e 2
stant. For a simple cubic structure Electrostatic energy = –M . (5.14)
4π 0 a
its value is 1.748.
Taking a = 0.28 nm and putting the constants into eqn (5.14) we get for the
cohesive energy,
E = 8.94 eV, (5.15)
which is about ten per cent above the experimentally observed value. There are
other types of energies involved as well (as, e.g. the energy due to the slight
deformation of the atoms) but, as the numerical results show, they must be of
lesser significance. We have thus confirmed our starting point that NaCl may
be regarded as an ionic bond.

5.3.2 Metallic bonds

Having studied the construction of atoms, we are now in a somewhat better
position to talk about metals. Conceptually, the simplest metal is a monovalent
alkali metal, where each atom contributes one valence electron to the com-
mon pool of electrons. So we are, in fact, back to our very first model, when
we regarded a conductor as made up of lattice ions and charged billiard balls
bouncing around.
We may now ask the question: how is a piece of metal kept together? ‘By
electrostatic forces’, is the simplest, though not quite accurate, answer. Thus,
the metallic bond is similar to the ionic bond in the sense that the main role is
played by electrostatic forces, but there is a difference as far as the positions
of the charges are concerned. In metals the carriers of the negative charge are
highly mobile; thus we may expect a bond of somewhat different properties.
Since electrons whizz around and visit every little part of the metal, the electro-
static forces are ubiquitous and come from all directions. So we may regard the
electrons as a glue that holds the lattice together. It is quite natural, then, that a
small deformation does not cause fracture. Whether we compress or try to pull
apart a piece of metal the cohesive forces are still there and acting vigorously.
This is why metals are so outstandingly ductile and malleable.

5.3.3 The covalent bond

So far we have discussed two bonds, which depend on the fact that unlike
charges attract—a familiar, old but nevertheless true, idea. But why should
atoms like carbon or silicon hang together? It is possible to purify silicon, so
that its resistivity is several ohm metres—there can be no question of a lot
of free electrons swarming around, nor is there an ionic bond. Carbon in its
diamond form is the hardest material known. Not only must it form strong
bonds, but they must also be exceptionally precise and directional to achieve
this hardness.