¯ ψ ϵ For small ϵ   the solid free-energy density can be well ap- proximated by only considering the contribution of the 〈 111 〉 and 〈 200 〉 RLVs. Accordingly  the crystal density field is expanded in the form where we have used the fact that all density waves have the same amplitude in the crystal ( | A$_{i}$ | = A$_{s}$ and | B$_{i}$ | = B$_{s}$ ) and the magnitude of the principal RLVs are unity in our dimensionless units so ( q = 1 $_{/}$√$_{3). The parameters}$ A$_{s}$ and B$_{s}$ are solved by substituting Eq. (27) into Eqs. (19) and (20) and by minimizing the resulting free-energy F with respect to A$_{s}$ and B$_{s}$ . This minimization yields the solid free energy density where A$_{s}$ and B$_{s}$ are themselves functions of ¯ ψ . The co- existence densities ¯ ψ$_{s}$ and ¯ ψ$_{l}$ are computed numerically using the standard common tangent construction  which consists of equating the chemical potentials f ′ $_{s}$( ¯ ψ$_{s}$ ) = f ′ $_{l}$( ¯ ψ$_{l}$ ) = µ$_{E}$ and grand potentials f$_{s}$ ( ¯ ψ$_{s}$ ) − µ$_{E}$ ¯ ψ$_{s}$ = f$_{l}$ ( ¯ ψ$_{l}$ ) − µ$_{E}$ ¯ ψ$_{l}$ of the two phases. It is also necessary to compute the solid free-energy curve for bcc since the latter can have a lower free-energy than fcc for some re- gions of the phase diagram. The bcc free-energy density was obtained by expanding the crystal density field us- ing a one-mode approximation  which only involves 〈 110 〉 RL Vs as in Ref. [14]  and substituting this expansion into the t wo-mode model defined by Eqs. (19) and (20). An example of the phase diagram for R$_{1}$ = 0 . 05 is sho wn in Fig. 1  where we also show for completeness the hexagonal and stripe phases. As desired  we obtain a large ϵ range of fcc-liquid coexistence. For small ϵ   how- ever  bcc becomes favored over fcc. A common tangent construction using fcc and bcc free-energy curves shows that the density range of bcc-fcc coexistence is extremely narrow for small values of ϵ and cannot be resolved on the scale of Fig. 1. As will be explained later in sec- tion III C  the range of ϵ where bcc is favored depends on the value of R$_{1}$ . In the limit R$_{1}$ ≫ 1  the two-mode model reduces to the standard one-mode model after a simple rescaling of parameters  which can be easily seen by comparing Eqs. (20) and (25). Hence  increasing R$_{1}$ reduces the contribution of the second mode. Conversely  reducing R$_{1}$ increases the contribution of this mode and tends to favor the fcc structure  which extends to smaller ϵ for smaller R$_{1}$ . In the extreme case where R$_{1}$ = 0  the region of fcc-liquid coexistence extends all the way to vanishingly small ϵ as shown in Fig. 2. Student Version of MATLAB The phase diagram of the two-mode PFC model is ob- tained by computing the free-energy density as a func- tion of the mean density ¯ ψ in solid and liquid  denoted by f$_{s}$ ( ¯ ψ )  and f$_{l}$ ( ¯ ψ )  respectively  and then using a stan- dard common tangent construction to obtain equilibrium values of ¯ ψ in solid ( ¯ ψ$_{s}$ ) and liquid ( ¯ ψ$_{l}$ ). Since the density is constant in the liquid  f$_{l}$ is ob- tained directly from Eq. (20) FIG. 1: Phase diagram of the two-mode PFC model for R$_{1}$ = 0 . 05 computed using two-mode and one-mode expansions of the crystal density field for fcc and bcc  respectively. Tuesday  January 5  2010 FIG. 2: Phase diagram of the two-mode PFC model showing only the fcc-solidus and liquidus for the case R$_{1}$ = 0 where fcc-liquid coexistence extends to vanishingly small ϵ . B. Phase diagram ψ ( ⃗ r ) ≈ ¯ ψ + ∑ ⃗ K$_{i}$ = 〈 111 〉$^{A$_{i}$}$ e i ⃗ K$_{i}$ · ⃗ r + ∑ ⃗ K$_{j}$ $^{′}$= 〈 200 〉$^{B$_{j}$}$ e i ⃗ K$_{j}$ $^{′}$· ⃗ r ≈ ¯ ψ + 8 A$_{s}$ cos qx cos qy cos qz +2 B$_{s}$ (cos 2 qx + cos 2 qy + cos 2 qz )   (27) f$_{s}$ ( ¯ ψ$_{s}$ ) = 4 ( − ϵ + 3 ¯ ψ 2 $_{s}$) A 2 s + 3 ( − ϵ + 3 ¯ ψ 2 s + R$_{1}$ 9 ) B 2 s +72 ¯ ψ$_{s}$A 2 $_{s}$B$_{s}$ + 144 A 2 $_{s}$B 2 s + 54 A 4 s + 45 2 B 4 s − ϵ 2 ¯ ψ 2 s + R$_{1}$ 2 ¯ ψ 2 s + 8 9 ¯ ψ 2 s + 1 4 ¯ ψ 4 $_{s}$  (28) f$_{l}$ ( ¯ ψ$_{l}$ ) = − ( ϵ − 16 9 − R$_{1}$ ) ¯ ψ 2 l 2 + ¯ ψ 4 l 4 . (26) − 0.3 − 0.25 − 0.2 − 0.15 − 0.1 − 0.05  0 0  0.05 0.1 0.15 ¯ ψ ϵ fcc liquid 5