2.5 Metallic Bonding
31
2.5 Metallic Bonding
In a metal, the positively charged atomic cores are embedded in a more or less homogeneous sea of
electrons. The valence electrons of the atoms become the conduction electrons of the metal. These
are freely moveable and at T = 0 K there is no energy gap between filled and empty states. The
bonding is mediated by the energy reduction for the conduction electrons in the periodic potential of
the solid compared to free atoms. This will be clearer when the band structure is discussed (Chap. 6).
In transition metals the overlap of inner shells (d or f) can also contribute to the bonding.
2.6 Van-der-Waals Bonds
The van-der-Waals bond is a dipole bond that leads to bonding in the noble-gas crystals (at low
temperature). Ne, Ar, Kr and Xe crystallize in the densely packed fcc lattice (cf. Sect. 3.3.2.1). He
3
and He
4 represent an exception. They do not solidify at zero pressure at T = 0 K due to the large
zero-point energy. This quantum-mechanical effect is especially strong for oscillators with small mass.
When two neutral atoms come near to each other (distance of the nuclei R), an attractive dipole–
dipole interaction −AR
−6 arises (London interaction) the van-der-Waals interaction. The quantummechanical overlap of the (filled) shells leads to a strong repulsion +B R
−12 . Altogether, a binding
energy minimum results for the Lennard–Jones potential V LJ (see Fig. 2.16)
V LJ (R) = −
A
R 6 +
B
R 12 .
(2.12)
The energy minimum E min = −A
2
/(2B) is at R = (2B/A)
1/6 .
The origin of the attractive dipole–dipole interaction can be understood from a one-dimensional
(1D) model as follows: Two atoms are modeled by their fixed positively charged nuclei in a distance
R and their negatively charged electron shells that are polarizable, i.e. can be displaced along one
direction x. Additionally, we assume (two identical) 1D harmonic oscillators for the electron motion
at the positions 0 and R. Then, the Hamilton operator H 0 of the system without interaction (R is very
large)
H 0 =
1
2m
p
2
1 + C x
2
1 +
1
2m
p
2
2 + C x
2
2 .
(2.13)
Fig. 2.16 Lennard–Jones
potential (2.12) for A = 1
and two values of B
31
2.5 Metallic Bonding
In a metal, the positively charged atomic cores are embedded in a more or less homogeneous sea of
electrons. The valence electrons of the atoms become the conduction electrons of the metal. These
are freely moveable and at T = 0 K there is no energy gap between filled and empty states. The
bonding is mediated by the energy reduction for the conduction electrons in the periodic potential of
the solid compared to free atoms. This will be clearer when the band structure is discussed (Chap. 6).
In transition metals the overlap of inner shells (d or f) can also contribute to the bonding.
2.6 Van-der-Waals Bonds
The van-der-Waals bond is a dipole bond that leads to bonding in the noble-gas crystals (at low
temperature). Ne, Ar, Kr and Xe crystallize in the densely packed fcc lattice (cf. Sect. 3.3.2.1). He
3
and He
4 represent an exception. They do not solidify at zero pressure at T = 0 K due to the large
zero-point energy. This quantum-mechanical effect is especially strong for oscillators with small mass.
When two neutral atoms come near to each other (distance of the nuclei R), an attractive dipole–
dipole interaction −AR
−6 arises (London interaction) the van-der-Waals interaction. The quantummechanical overlap of the (filled) shells leads to a strong repulsion +B R
−12 . Altogether, a binding
energy minimum results for the Lennard–Jones potential V LJ (see Fig. 2.16)
V LJ (R) = −
A
R 6 +
B
R 12 .
(2.12)
The energy minimum E min = −A
2
/(2B) is at R = (2B/A)
1/6 .
The origin of the attractive dipole–dipole interaction can be understood from a one-dimensional
(1D) model as follows: Two atoms are modeled by their fixed positively charged nuclei in a distance
R and their negatively charged electron shells that are polarizable, i.e. can be displaced along one
direction x. Additionally, we assume (two identical) 1D harmonic oscillators for the electron motion
at the positions 0 and R. Then, the Hamilton operator H 0 of the system without interaction (R is very
large)
H 0 =
1
2m
p
2
1 + C x
2
1 +
1
2m
p
2
2 + C x
2
2 .
(2.13)
Fig. 2.16 Lennard–Jones
potential (2.12) for A = 1
and two values of B