Set of supplementary notes.

Chapter 3
From atoms to solids
What holds a solid together Cohesion is ultimately produced by the interactions between the
nuclei and the electrons, which give rise to an effective interaction potential between atoms.
We distinguish between a number of distinct mechanisms (or types of bonds), however, which
can contribute to this.
3.1 The binding of crystals
Inert gases
The inert gases have filled electron shells and large ionisation energies. Consequently, the
electronic configuration in the solid is close to that of separated atoms. Since the atoms are
neutral, the interaction between them is weak, and the leading attractive force at large distances
comes from the van der Waals interaction, which gives an attractive potential proportional to
1/R 6 .
This form can be loosely derived by thinking of an atom as an oscillator, with the electron
cloud fluctuating around the nucleus as if on a spring.
The centre of the motion lies on top of the atom, but if the cloud is displaced, there will be
a small dipole induced, say p 1 . Such displacements happen as a result of zero-point motion of
the electron cloud in the potential of the nucleus.
A distance R away from the atom there is now an induced electric field ∝ p 1 /R 3 . A second
atom placed at this point will then have a dipole induced by the electric field of the first:
p 2 ∝ αp 1 /R 3 , where α is the atomic polarizability.
The second dipole induces an electric field at the first, which is now
The energy of the system is then changed by an amount
E 1 ∝ p 2 /R 3 ∝ αp 1 /R 6 . (3.1)
∆U = 〈−p 1 · E 1 〉 ∝ −α 〈 p 2 1〉
/R 6 . (3.2)
Notice that it depends on the expectation value of the square of the dipole moment < p 2 1 >,
which is non-zero, and not the square of the expectation value < p 1 > 2 , which would be zero.
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