In nearly free electron (NFE) theory, the effects of the atoms are included via a weak ‘pseudopotential.’ The interatomic forces arise from the response of the electron gas to this perturbation. To examine the appropriate form for an interatomic potential, we consider a simple weak, local pseudopotential V0(r). The total potential actually seen at ri due to atoms at rj will be as follows:

V(ri) = X Vo(rj) + W(r)

j

where W(r) describes how the electrons interact with one another. Given the dielectric constant,

we can estimate W in reciprocal space using linear response theory:

W (q) = Vo(q)/e(q)

where e(q) is the dielectric function. In Thomas Fermi theory,

e(q) = 1 + (4kF=P00q2)

where a0 is the Bohr radius. A more accurate approach due to Lindhard:

4kF

%a0q2

where x = q/2kF accounts for the reduced screening at high q, and p0 is the mean electron density.

From this screened interaction, it is possible to obtain volume-dependent real space potentials.6

The contributions to the total energy are as follows:

• The free electron gas (including exchange and correlation)

• The perturbation to the free electron band structure

• Electrostatic energy (ion-ion, electron-electron, ion-electron)

• Core corrections (from treating the atoms as pseudopotentials)

In this model, interatomic pair potential terms arise only from the band structure and the electrostatic energy (the difference between the Ewald sum and a jellium model) and give a minor contribution to the total cohesive energy. However, these terms are totally responsible for the crystal structure.

A key concept emerging from representing the Lindhard screening in real space is the idea of a ‘Friedel oscillation’ in the long-range potential:

cos 2kFr (2kFr )3

This arises from the singularity in the Lindhard function at q = 2kF: physically, periodic lattice per­turbations at twice the Fermi vector have the largest perturbative effect on the energy. The effect of Friedel oscillations is to favor structures where the atoms are arranged with this preferred wavelength. It gives rise to numerous effects.

• Kohn anomalies in the phonon spectrum are par­ticular phonons with anomalously low frequency. The wavevector of these phonons is such as to match the Friedel oscillation.

• Soft phonon instabilities are an extreme case of the Kohn anomaly. They arise when the lowering of energy is so large that the phonon excitation has negative energy. In this case, the phonon ‘freezes in,’ and the material undergoes a phase transfor­mation to a lower symmetry phase.

• Quasicrystals are an example where the atoms arrange themselves to fit the Friedel oscillation. This gives well-defined Bragg Peaks for scattering in reciprocal space, and includes those at 2kF but no periodic repetition in real space.

• Charge density waves refer to the buildup of charge at the periodicity of the Friedel oscillation.

• ‘Brillouin Zone-Fermi surface interaction’ is yet another name for essentially the same phenome­non, a tendency for free materials from structures which respect the preferred 2kF periodicity for the ions — which puts 2kF at the surface.

• ‘Fermi surface nesting’ is yet another example of the phenomenon. It occurs for complicated crystal structures and/or many electron metals. Here, structures that have two planes of Fermi surface separated by 2kF are favored, and the wavevector q is said to be ‘nested’ between the two.

• Hume-Rothery phases are alloys that have ideal composition to allow atoms to exploit the Friedel oscillation.

NFE pseudopotentials enabled the successful predic­tion of the crystal structures of the sp3 elements. It is tempting to use this model for ‘empirical’ potential simulation, using the effective pseudopotential core radius and the electron density as fitting parameters; indeed such linear-response pair potentials do an excellent job of describing the crystal structures of sp elements.

For MD, however, there are difficulties: the elec­tron density cannot be assumed constant across a free surface and the elastic constants (which depend on the bulk term) do not correspond to long-wavelength phonons (which do not depend on the bulk term). Since most MD calculations of interest in radiation damage involve defects (voids, surfaces), phonons, and long-range elastic strains, NFE pseudopotentials have not seen much use in this area. They may be appropriate for future work on liquid metals (sodium, potassium, NaK alloys).

The key results from NFE theory are the following:

• The cohesive energy of a NFE system comes pri­marily from a volume-dependent free electron gas and depends only mildly on the interatomic pair potential.

• The pair potential is density dependent: structures at the same density must be compared to deter­mine the minimum energy structure.

• The pair potential has a long-ranged, oscillatory tail.

• These potentials work well for understanding crystal structure stability, but not for simulating defects where there is a big change in electron density.

• The reference state is a free electron gas: descrip­tion of free atoms is totally inadequate.