10 0 0.05 0.1 0.15 0.2 − 0.02 0 0.02 0.04 0.06 A f bcc R=0 R=2 R=5 Δ f AE$_{bcc}$ 0 2 4 ! 0.70  ! 0.69  ! 0.68  ! 0.67  ! 0.66  ! 0.65  ! 0.64  R ! c Stable Fcc, Unstable Bcc Stable Fcc, Metastable Bcc Metastable Fcc, Stable Bcc ψ$_{c}$ FIG. 10: Plots of free-energy density of bcc relative to the liquid, Δ f AE bcc , as a function of the amplitude A of 〈 110 〉 den- sity waves for different values of R . When the value of Δ f AE bcc corresponding to the solid bcc free-energy minimum is nega- tive, bcc is favored over fcc. As R decreases, bcc first becomes metastable with respect to fcc and then unstable as the local solid free-energy minimum disappears. FIG. 9: (Color Online) ψ$_{c}$ as a function of R showing different ranges of bcc or fcc metastability, stability, or instability. uid can be neglected in the small ϵ limit. We can also compute ψ$_{c}$ more directly from Eq. (45) by requiring Student Version of MATLAB from the one between fcc and liquid, Δ f $^{AE}$, defined by Eq. (45). By definition, Δ f AE = 0 for solid fcc in equi- librium with the liquid. Therefore, to assess the relative stability of bcc and fcc, we can plot Δ f AE bcc defined by Eq. (49) versus A and check if the value correspond- ing to the solid bcc minimum is above (below) zero in which case fcc (bcc) has a lower free-energy than bcc (fcc). Such plots shown in Fig. 10 show that bcc be- comes metastable with respect to fcc and then unstable (with the disappearance of the local solid bcc free-energy minimum) as R is decreased. A detailed study as a func- tion of R shows that bcc first becomes metastable for R < R$_{c}$ where R$_{c}$ = 2 . 68 and then unstable as R is de- creased below a second threshold value ( ≈ 1 . 43), giving rise to the three different stability regimes as a function of R shown in Fig. 9. Translated in terms of the phase diagram constructed at fixed R$_{1}$ , this implies that bcc becomes favored over fcc when ϵ < ϵ$_{c}$ where and with all the above relations evaluated at the equilibrium values of A and B in the solid. Eq. (46) stems from the requirement that the solid amplitudes must correspond to a free-energy minimum, which fixes those amplitudes uniquely as functions of ψ$_{c}$ . Eq. (47), in turn, is the re- quirement that the free-energies of solid and liquid must be equal in equilibrium, which fixes ψ$_{c}$ uniquely for a given R . A plot of ψ$_{c}$ versus R obtained in this way using Eqs. (46) and (47) is shown in Fig. 9. The relationship between ψ$_{c}$ and R , denoted by ψ$_{c}$ ( R ), can now be used to assess the relative stability of bcc and fcc. To obtain an analogous expression to Eq. (45) for bcc, we substitute into the two-mode free-energy func- tional defined by Eqs. (19) and (20), the one-mode ex- pansion of the bcc crystal density field in terms of prin- cipal set of 〈 110 〉 density waves For R$_{1}$ = 0 . 05, the above expression predicts ϵ$_{c}$ ≈ 0 . 019 that is in good quantitative agreement with the phase diagram of Fig. 1. As R$_{1}$ increases, ϵ$_{c}$ increases and the switch from stable bcc-liquid to fcc-liquid coexistence moves to higher values of ϵ in the phase diagram. In this section, we derive expressions to relate the two- mode PFC model parameters to material parameters by where q = 1 $_{/}$√$_{2. We obtain}$ where A now denotes the amplitude of 〈 110 〉 density waves and we have used the subscript “bcc” to distin- guish this free-energy difference between bcc and liquid IV. PARAMETER DETERMINATION ∂ Δ f AE ∂A = ∂ Δ f AE ∂B = 0 , (46) Δ f AE = 0 , (47) ϵ$_{c}$ = R$_{1}$ R$_{c}$ . (50) ψ ( ⃗ r ) ≈ ϵ 1 / $^{2}$ψ$_{c}$ + 4 ϵ 1 / $^{2}$A (cos qx cos qy + cos qx cos qz + cos qy cos qz ) , (48) Δ f AE bcc = 6(3 ψ 2 c − 1) A 2 + 48 ψ$_{c}$A 3 + 135 A $^{4}$, (49)