Environmental Engineering Reference
In-Depth Information
3.4.1
Properties of a Metal: Electrons in an Empty Box (II)
We come back to our description of metals, rst to remark that forming the empty
box, that is, the binding of the metal atoms to make the container for the electrons,
can be imagined to start with the fundamental van der Waals short-range interatomic
attraction [34], but it is made stronger by an electron delocalization effect when a
valence electron is weakly bound and can easily leave an atom.
Once the empty box is formed of real atoms, we can discuss how the electrons are
bound inside it. To address the
first question, we can apply Equation 2.35 to the
binding of a group of metal atoms, if we replace the nuclear binding energy
U
o
! U
coh
, by the
cohesive energy U
coh
per atom for the metal. For gold
U
coh
is
listed as 3.81 eV per atom, while for the rare gas atoms from Ne to Rn the cohesive
energy per atom ranges from0.02 to 0.2 eV/atom [35]. The value 3.81 eV/atomgreatly
exceeds the values for rare gases, which are in the range expected for the van der
Waals attraction. The rare gases form liquids or weakly bound solids only at very low
temperatures, but of course metals are strongly bound with highmelting points. The
cohesive energy is the energy needed to remove one neutral atom from the metal,
which does not involve ionization of any atom.
Since the only difference between the rare gas like Radon and a monovalent metal
like gold is one valence electron on the outside of a
filled rare gas electron shell, this
one valence electron greatly strengthens the binding.
The surface energy correction term (2.35) again means only that outside the
metal there are no more atoms to bind a surface atom. This term describes why a
liquid metal like Hg or molten solder will minimize its surface area to form a
spherical drop. The collection of atoms in a metal becomes more strongly bound if
the outer valence electrons delocalize to go into extended states similar to
Equation 2.16a, which have lower kinetic energies than the bound atomic valence
states.
The outer valence electron localized states are similar in concept to the hydrogenic
state (Section 3.1). The valence electron for gold has principal quantum number
n¼
6, and is called a 6s state, to indicate a spherical state of zero angular
momentum. These states oscillate rapidly varying with radius (following Equa-
tion 3.23, we state that the number of radial nodes is
in l
1, thus 5 for the 6s
state), and the sharp d
y
/d
x
variations lead to large kinetic energy. This part of the
binding energy of ametal comes from the reduction in electron kinetic energy related
to the delocalization of its wave functions. The smooth electron states extended away
fromthe atomic cores, de
nitely have lower d
y
/d
x¼p
/
h
than localized atomic states,
which have rapidly varying wavefunctions (large d
y
/d
x
), as one can see by looking at
Equations 2.4 and 2.9. This clearly implies a reduction of kinetic energy
p
2
/2
m
, and
this contributes to the strength of the
metallic bond
. Amore detailed estimate coming
to the same conclusion is given by Kittel [36].
That the starting point for an electron in the metallic empty box is a free particle,
rather than an atomic state, is suggested by the fact that the atomic density exceeds the
Mott critical value, mentioned in Chapter 2 following Equation 2.2. Mott predicted
