The embedded atom and FS potentials fall into a general class of potentials of the form:

U, =£ F

i

with a many-body cohesive part and a two-body repulsion. Both f(rj) and V(r, y) are short ranged, so MD with these potentials is at worst only as costly as a simple pair potential (computer time is propor­tional to number of particles).

These models are sometimes referred to as glue potentials,16 the many-body F term being thought of as describing how strongly an atom is held by the electron ‘glue’ provided by its environment.

The pragmatic approach to fitting in all glue schemes is to regard the pair potential as repulsive at short-range with long-range Friedel oscillations. Compared with most pair potential approaches, this is unusual in that the repulsive term is longer ranged than the cohesive one.

1.10.7.1 Crystal Structure

According to the tight-binding theory on which the FS potentials are based, the relative stability of bcc and fcc is determined by moments above the second, which in turn relate to three center and higher hops. These third and higher moments effects are explicitly absent in second-moment models, and so by implica­tion, the correct physics of phase stability is not contained in them. There is no such clear result in the derivation of the EAM; however, since the forms are so similar, the same problem is implicit.

In glue models, energy is lowered by atoms having as many neighbors as possible; thus, fcc, hcp, and bcc crystal structures (and their alloy analogs) are nor­mally stable (see Table 1); bcc is normally stable in potentials when the attractive region is broad enough
to include 14 neighbors, fcc/hcp are stable for nar­rower attractive regions in which only the eight near­est bcc-neighbors contribute significantly to the bonding. Indeed, without second neighbor interac­tions, bcc is mechanically unstable to Bain-type shear distortion. The fcc-hcp energy difference is related to the stacking fault energy: it is common to see MD simulations with too small an hcp-fcc energy differ­ence producing unphysically many stacking faults and over widely separated partial dislocations.

Phase transitions are observed in some potentials. As free energy calculations are complicated and time consuming,17 it is impractical to use them directly in fitting — one would require the differen­tial of the free energy with respect to the potential parameters, and this could only be obtained numer­ically. Consequently, most potentials are only fitted to reproduce the zero temperature crystal structure, and high-temperature phase stability is unknown for the majority of published potentials. One counter­example is in metals such as Ti and Zr, where the bcc structure is mechanically unstable with respect to hcp, but becomes dynamically stabilized at high temperatures. Here, the transition temperature is directly related to a single analytic quantity: the energy difference between the phases. Although about half of this difference comes from electronic entropy,18 which suggests a temperature-dependent potential, phase transition calculations have been explicitly included in some recent fits.19 The case of iron is also anomalous, as the phase transition is related to changes in the magnetic structure.