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In this section, we analyze in more detail the proper- ties of the model by expanding the free-energy in terms of the amplitudes of density waves. In the one-mode bcc case analyzed in Ref. [14], a similar expansion exploited the fact that the amplitude of 〈 110 〉 density waves scales as ϵ 1 / 2 in the small ϵ limit. In the present case, the ex- pansion is rendered more difficult by the presence of two different sets of density waves with amplitudes A$_{s}$ and B$_{s}$ corresponding to 〈 111 〉 and 〈 200 〉 RLVs, respectively. Therefore, it is not a priori obvious how A$_{s}$ and B$_{s}$ should scale in the small ϵ limit. If R$_{1}$ is kept constant, the bcc structure turns out to always be favored in the small ϵ limit as apparent in the phase diagram of Fig. 1. Con- sequently, a small ϵ amplitude expansion that captures the fcc structure cannot be carried out at fixed R$_{1}$ . How- ever, if R$_{1}$ is decreased proportionally to ϵ by imposing the additional scaling R$_{1}$ = ϵR , both A$_{s}$ and B$_{s}$ scale as ϵ 1 / $^{2}$, thereby making a rigorous expansion possible. This expansion may seem artificial since the phase diagram of Fig. 1 is computed at fixed R$_{1}$ . However, as we show below, the results of this expansion can be used to un- derstand the small ϵ structure of the phase diagram, in particular the relative stability of fcc and bcc. 

To demonstrate the feasibility of this expansion, we first analyze fcc-liquid coexistence for small ϵ with the scaling R$_{1}$ = ϵR . The equilibrium densities are calcu- lated using the common tangent construction described in the previous section. To make the dependence of the coexistence densities on ϵ explicit, we make a log-log plot of the mean coexistence density ¯ ψ ∗ ≡ 1 $_{2}$( ¯ ψ$_{l}$ + ¯ ψ$_{s}$ ) versus ϵ 

## III. AMPLITUDE EQUATIONS 

## A. Scalings 

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for three different values of R . The results in Fig. 6 show that the mean coexistence density scales as ϵ 1 / $^{2}$. Next in Fig. 7, we show a log-log plot of the density difference be- tween solid and liquid versus ϵ for the same three values of R . The results show that ¯ ψ$_{s}$ − ¯ ψ$_{l}$ ∼ ϵ 3 / $^{2}$. Together, these two log-log plots show that, in the small ϵ limit, the two-mode PFC model exhibits a weak first-order freezing transition where the size of the solid-liquid coexistence region is at the order of ϵ 3 / 2 that is much smaller than the mean value of the density ∼ ϵ 1 / $^{2}$. 

The above scalings suggest that we can expand the crystal density field in powers of ϵ 1 / 2 as 

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and expand accordingly the average densities 

and 

in the liquid and solid, respectively. The numerically determined scaling relations ( ¯ ψ$_{l}$ + ¯ ψ$_{s}$ ) / 2 ∼ ϵ 1 / 2 and ¯ ψ$_{s}$ − ¯ ψ$_{l}$ ∼ ϵ 3 / 2 then imply that 

## B. Free-energy functional 

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