An Introduction to the Theory of Crystalline Elemental Solids and ...

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and many other materials which are not obviously crystalline in appearance possess
a long range crystalline structure. Indeed metals and semiconductors and many other
insulators aside from just diamond and salt adopt crystalline structures.
The key property of crystalline solids is their inherent symmetry which arises because of
the regular arrangement of their nuclei. It is interesting to note, however, that the actual
definition of the term crystal provided by the International Union of Crystallography
(IUC) is somewhat broader than a discussion based on periodicity of the atoms would
imply. Specifically, IUC define a crystal as “any solid having an essentially discrete
diffraction diagram”. This definition, which is clearly somewhat ill-defined, has arisen
mainly to account for the observations, from the 1980s, of quasiperiodic crystals,
so-called “quasicrystals” [1]. Thus periodic crystals, which we will focus on here, are just
a subset. How the nuclei are arranged leads to the crystal structure which is the unique
arrangement of atoms in a crystal composed of a unit cell: a set of atoms arranged in a
particular way which is periodically repeated in three dimensions on a lattice. The unit
cell is given in terms of its lattice parameters, the length of the unit cell edges and the
angles between them.
Crystallographers are adept at categorizing, counting, and systematizing crystal types
according to their symmetry and crystal system. We will not venture into the world
of crystallography except to say that for a general crystal structure in which objects
of arbitrary symmetry are translated on a so-called Bravais lattice that there turn out
to be 230 different symmetry groups that a lattice can have, known as the 230 space
groups. When the object to be translated (the “basis”) is completely symmetric, such
as a single atom is, there turn out to be fourteen space groups (the fourteen Bravais
lattices) falling into seven crystal structures (cubic, tetragonal, orthorombic, monoclinic,
triclinic, trigonal, and hexagonal). Fortunately, for present purposes, we do not need to
worry about all 230 space groups or even all fourteen Bravais lattices since elemental
solids under standard conditions adopt a very small number of structures. Indeed more
than 70% [2] of all elemental solids adopt one of the following four crystal structures:
• Body-Centered Cubic (bcc): This is a simple cubic structure with atoms at the
corner of a cube along with an additional atom at the center of the cube (Fig. 1(a)).
The alkali metals and many early transition metals adopt this structure;
• Face-Centered Cubic (fcc): Again this is a simple cubic structure but now with an
additional atom at the center of each square face (Fig. 1(b)). Most late transition
and noble metals adopt this structure as well as the inert gas solids and some of the
alkaline earth elements;
• Hexagonal Close Packed (hcp): This consists of two interpenetrating simple hexagonal
lattices as shown in Fig. 1(c). Several transition metals adopt this structure;
• Diamond: This consists of two interpenetrating face-centered cubic lattices
(Fig. 1(d)). The group IV elements (C, Si, Ge, and Sn) often crystalize in this
structure.
All the elements we discuss in the following will be in one of these four crystal structures.
2.2 Cohesive energies