− ¯ ψ ∗ ϵ 10 − 2 10 − 1 10 − 1 10 0 R=0 R=3 R=5 7 FIG. 5: Simulation of the growth of a twin crystal in a super- cooled liquid for R = 0, ϵ = 0 . 00823, and ¯ ψ = − 0 . 06. Saturday, December 19, 2009 FIG. 6: (Color Online) Plots of ¯ ψ ∗ ≡ 1 $_{2}$( ¯ ψ$_{s}$ + ¯ ψ$_{l}$ ) versus ϵ for different values of R, with ¯ ψ$_{s}$ and ¯ ψ$_{l}$ calculated from the common tangent construction. Fits to the numerical results of the form ¯ ψ ∗ = ψ$_{c}$ϵ 1 / 2 yield for R = 0, ψ$_{c}$ = − 0 . 6901, for R = 3, ψ$_{c}$ = − 0 . 6578, and for R = 5, ψ$_{c}$ = − 0 . 6396. The solid black line has a slope of exactly 1 / 2 on this log-log plot showing that the mean density ∼ ϵ 1 / $^{2}$. III. AMPLITUDE EQUATIONS A. Scalings B. Free-energy functional 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 / $^{2}$. 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. The above scalings suggest that we can expand the crystal density field in powers of ϵ 1 / 2 as and expand accordingly the average densities and 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 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 ¯ ψ$_{l}$ = ψ$_{l}$$_{0}$ϵ 1 / 2 + ψ$_{l}$$_{1}$ϵ + ψ$_{l}$$_{2}$ϵ 3 / 2 + · · · , (30) ψ ( ⃗ r ) = ψ$_{0}$ ( ⃗ r ) ϵ 1 / 2 + ψ$_{1}$ ( ⃗ r ) ϵ + ψ$_{2}$ ( ⃗ r ) ϵ 3 / 2 + · · · , (29) ¯ ψ$_{s}$ = ψ$_{s}$$_{0}$ϵ 1 / 2 + ψ$_{s}$$_{1}$ϵ + ψ$_{s}$$_{2}$ϵ 3 / 2 + · · · , (31) ψ$_{l}$$_{0}$ = ψ$_{s}$$_{0}$ ≡ ψ$_{c}$, (32)