Set of supplementary notes.

34 CHAPTER 3. FROM ATOMS TO SOLIDS
Figure 3.1: Two dipoles represent model atoms that are arranged along a line, with the positive
charges (+e) fixed at the positions 0, R, and the negative charges (-e) at the points x 1 , R + x 2 .
If the atoms move together so that the electron charge distributions begin to overlap, repulsive
forces come into play. While there is of course a contribution from the direct electrostatic
repulsion of the electrons, more important is the Pauli exclusion principle that prevents two
electrons having their quantum numbers equal. The effect of Pauli exclusion can be seen by an
extreme example, of overlapping two Hydrogen atoms entirely, with the electrons for simplicity
assumed to be in the same spin state. In this case, while two separated atoms may be both in
the 1S ground state, the combined molecule must have a configuration 1s2s, and thus is higher
by the promotion energy.
Calculations of the repulsive interaction are complex but the answer is clearly short-ranged.
They are often modelled empirically by an exponential form e −R/Ro , or a power law with a
large power. A commonly used empirical form to fit experimental data on inert gases is the
Lennard-Jones potential
with A and B atomic constants obtained from gas-phase data.
U(R) = − A R 6 + B R 12 (3.3)
With the exception of He, the rare gases from close-packed (face-centered cubic) solids with
a small cohesive energy, and low melting temperatures. Helium is special because zero-point
motion of these light atoms is substantial enough that they do not solidify at zero pressure
down to the absolute zero of temperature. The quantum fluids 3 He and 4 He have a number of
extraordinary properties, including superfluidity.
Ionic Crystals
Given the stability of the electronic configurations of a rare gas, atoms that are close to a filled
shell will have a tendency to lose or gain electrons to fill the shell.
• The energy for the reaction M− > M + + e − in the gas phase is called the ionization
energy I.
• The energy for the reaction X + e − − > X − in the gas phase is called the electron affinity
A.
• The cohesion of an ionic molecule can overcome the energy cost I + A by the electrostatic
attraction, e 2 /R