content_image/403358.jpg content_image/403354.jpg content_image/403355.jpg
For the case where the amplitudes are spatially uniform, we obtain the free-energy density 

<img src='content_image/403358.jpg'>

In general, the above free-energy density is a multi- variate function of the fourteen amplitudes A 0 i and B 0 i of different density waves, and cannot be represented graph- ically in a simple way. However, the free-energy barrier between solid and liquid can be made explicit by assum- ing that all the 〈 111 〉 and all the 〈 200 〉 density waves have the same amplitude (i.e., A 0 i = A and B 0 i = B , respec- tively). In this isotropic approximation (see [14] for the bcc analog), the free-energy density becomes 

This expression can also be obtained by evaluating di- rectly the difference between the solid and liquid free- energy densities, Δ f AE = ϵ − $^{2}$( f$_{s}$ − f$_{l}$ ), with f$_{l}$ and f$_{s}$ given by Eqs. (26) and (28), respectively, and the substi- tutions ¯ ψ$_{s}$ = ¯ ψ$_{l}$ = ψ$_{c}$ϵ 1 / $^{2}$, A$_{s}$ = Aϵ 1 / 2 and B$_{s}$ = Bϵ 1 / $^{2}$. For the parameters R = 0 and ψ$_{c}$ calculated from the 

<img src='content_image/403354.jpg'>

<img src='content_image/403355.jpg'>

common tangent construction, we plot in Fig. 8 the free- energy landscape as a function of amplitudes A and B . The free-energy landscape exhibits two minima that cor- respond to the stable liquid and solid phases. The above amplitude equation calculation shows that the two-mode PFC model describes well solid-liquid coexistence with a well-defined free-energy barrier between solid and liquid. 

We have only treated here the case where the ampli- tudes are spatially uniform to characterize the bulk free- energy landscape. A more general free-energy functional that includes gradient terms would be necessary to treat the case where the amplitudes are spatially varying. Such a functional could then be used to compute the excess free-energy of the solid-liquid interface and its anisotropy, as done previously for bcc [14]. Those computations will be presented elsewhere. 

So far, we have only examined the possibility of fcc- liquid coexistence. However, the phase-diagram of Fig. 1 shows that bcc can have a lower free-energy than fcc for small enough ϵ if R$_{1}$ is finite. We now use the amplitude equations to understand the relative stability of fcc and bcc. As a first step, it is useful to re-examine the scaling of the mean density that is controlled by the parameter ψ$_{c}$ . We computed previously the equilibrium solid and liquid densities using the common tangent construction, from which we obtained the scalings ( ¯ ψ$_{s}$ + ¯ ψ$_{l}$ ) / 2 = ψ$_{c}$ϵ 1 / $^{2}$, which defines ψ$_{c}$ , and ψ$_{s}$ − ¯ ψ$_{l}$ ∼ ϵ 3 / $^{2}$, which shows that the size of the density difference between solid and liq- 

## C. Relative stability of fcc and bcc 

