Page 301 - Instant notes

H5
STRONG SOLID PHASE INTERACTIONS

Key Notes
Molecular orbitals in large, extended solids form energy bands,
rather than discrete energy levels, and an electron may hold any
of a continuous range of energies. Two complementary theories
form the basis of band theory: the tightly bound electron model
(also known as the tight binding approximation) and the free
electron model.
The tightly bound electron model extends the LCAO approach to
the overlapping of very large numbers of molecular orbitals. This
results in large numbers of molecular orbitals with the fully
bonding orbital at the lowest energy, and the fully antibonding
orbital at the highest. The energy difference between the upper
and lower orbitals is known as the band width, and the number of
molecular states per unit binding energy is referred to as the
density of states. In real solids bands are formed in three
dimensions using s, p, d, and f orbitals and both the structure of
the bands and the density of states diagrams may be extremely
complex.
The free electron model reduces the problem of the energy levels
to that of a particle in a box The maximum energy of an electron,
mass m e , in this model is given by:

where ρ is the density of the electrons. Bands in the free electron
model are open-ended, and do not have an upper energy limit.
More accurate approaches to band theory account for both the
highly delocalized nature of the electrons, and the regular array of
nuclear potentials.
As with molecular orbitals, the energy levels in a band are
progressively filled from the lowest energy upwards. The Fermi
level, E f , is defined as the energy of the orbital for which the
probability of electron occupation is ½. The distribution of
electrons is described by Fermi-Dirac statistics. The probability
f(E) of an electron occupying an energy E at a temperature, T is
given by: