Set of supplementary notes.

3.1. THE BINDING OF CRYSTALS 35
• In a solid, the electrostatic interaction energy for a diatomic crystal 1 is
U electrostatic = 1 ∑ ∑
U ij (3.4)
2
where U ij = ±q 2 /R ij is the sum of all Coulomb forces between ions. If the system is on
a regular lattice of lattice constant R, then we write the sum
i
j
U electrostatic = − 1 α M q 2
2 R
where α M is a dimensionless constant that depends only on the crystal structure.
(3.5)
• The evaluation of α M is tricky, because the sum converges slowly. Three common crystal
structures are NaCl (α M = 1.7476), CsCl (1.7627), and cubic ZnS or Zincblende
(1.6381).
• To the attractive Madelung term must be added the repulsive short range force, and we
now have the added caveat that ions have different sizes, explaining why NaCl has the
rocksalt structure, despite the better electrostatic energy of the CsCl structure.
Covalent crystals
The covalent bond is the electron pair or single bond of chemistry.
Model Hydrogen. Two overlapping atomic orbitals on identical neighbouring atoms will
hybridise. Because the Hamiltonian must be symmetric about a point centered between the
ions then the eigenstates must have either even or odd parity about this center. If we have a
simple system of two one electron atoms - model hydrogen - which can be approximated by a
basis of atomic states φ(r − R) (assumed real) centered on the nucleus R, then two states of
even and odd parity are
ψ ± (r) = φ(r − R a ) ± φ(r − R b ) (3.6)
ψ + has a substantial probability density between the atoms, where ψ − has a node. Consequently,
for an attractive potential E + < E − , and the lower (bonding) state will be filled
with two electrons of opposite spin. The antibonding state ψ − is separated by an energy gap
E g = E − − E + and will be unfilled. The cohesive energy is then approximately equal to the
2
gap E g
Covalent semiconductors. If we have only s-electrons, we clearly make molecules first,
and then a weakly bound molecular solid, as in H 2 . Using p, d, orbitals, we may however make
directed bonds, with the classic case being the sp 3 hybrid orbitals of C, Si, and Ge. These are
constructed by hybrid orbitals s + p x + p y + p z + 3 other equivalent combinations, to make new
orbitals that point in the four tetrahedral directions: (111), (¯1¯11), (¯11¯1), (1¯1¯1). These directed
orbitals make bonds with neighbours in these tetrahedral directions, with each atom donating
one electron. The open tetrahedral network is the familiar diamond structure of C, Si and Ge.
1 Beware the factor of 1/2, which avoids double counting the interaction energy. The energy of a single ion i
due to interaction with all the other ions is U i = ∑ j≠i U ij; the total energy is 1 ∑
2 i U i
2 Actually twice (two electrons) half the gap, if we assume that E ± = E atom ± 1 2 E g