2 the two-mode and one-mode structures (fcc and bcc, re- spectively). In this section, we also use a standard com- mon tangent construction to compute the phase-diagram in the plane of density and ϵ for an illustrative choice of R$_{1}$ = 0 . 05. The phase diagram exhibits regions of bcc-liquid and fcc-liquid coexistence for small and large epsilon, respectively. The size of the fcc-liquid coexis- tence region depends generally on R$_{1}$ . For r$_{1}$ = 0 where Eq. (4) reduces to the free-energy density of Lifshitz and Petrich [18], the analog phase-diagram only exhibits fcc- liquid coexistence, so that a finite r$_{1}$ is necessary for the phase diagram to exhibit both bcc-liquid and fcc-liquid coexistence. We demonstrate the feasibility of the ap- proach with some simulations of polycrystalline growth and (111) twin growth. A numerical computation of the (111) twin boundary energy for parameters of Ni is given in an appendix. The ability to model twin growth is im- portant for solidification modeling since twins can dra- matically alter both eutectic [19, 20] and dendritic [21] microstructures. coefficient c$_{3}$ < 0. Since c$_{2}$ and c$_{4}$ are both positive, this cubic term is responsible for the existence of a free-energy barrier between the two minima of f corresponding to liquid ( u = 0) and solid ( u$_{s}$ > 0). In this paper, we use a “two-mode” phase-field-crystal model to model face-centered-cubic (fcc) structures, which has the free-energy density This model truncates the sum (1) to two sets of RLVs with magnitude | ⃗ K$_{i}$ | = q$_{0}$ and | ⃗ K ′ i | = q$_{1}$ , respectively, where the first set corresponds in general to the princi- pal RLVs and the second to some other set with larger wavevector magnitude; all other RLVs have much smaller amplitude. This construct provides more flexibility to form triad interactions by combining RLVs from those two sets, and hence to describe other crystal structures. We demonstrate this here for fcc ordering, which is ob- tained by choosing the sets { ⃗ K$_{i}$ } and { ⃗ K ′ i } to correspond to 〈 111 〉 (principal set) and 〈 200 〉 RLVs, respectively, with q$_{1}$/q$_{0}$ = $^{√}$4 / 3. While there is in principle freedom in the choice of the second set { ⃗ K ′ i } for a given structure, we have chosen this set such that q$_{1}$ > q$_{0}$ is as small as possible, as desired for computational efficiency. In section III, we carry out an amplitude expansion of the bulk free-energy density in the small ϵ limit. This expansion exploits the property that, with the scaling R$_{1}$ = ϵR , the amplitudes of the 〈 111 〉 and 〈 200 〉 den- sity waves scale as, A$_{s}$ ∼ Aϵ 1 / 2 and B$_{s}$ ∼ Bϵ 1 / $^{2}$, re- spectively, while the density difference between solid and liquid scales ∼ ϵ 3 / $^{2}$. Therefore, this density difference can be neglected in the small ϵ limit and the bulk free- energy density can be expressed solely in terms of those amplitudes. As required for solid-liquid coexistence, the free-energy density has minima in the ( A , B ) plane cor- responding to liquid ( A = B = 0) and fcc solid (finite A and B values that depend on R ). By comparing this form to the free-energy density for a single amplitude of bcc density waves (corresponding to 〈 110 〉 RLVs), we identify different regions of relative fcc and bcc stability, which explains the phase diagram computed in section II. The form (4) reduces in the limit r$_{1}$ = 0 to the free-energy density introduced by Lifshitz and Petrich [18] as a generalization of the Swift-Hohenberg model to describe two-dimensional quasiperiodic patterns ob- served in Faraday wave experiments, which result from the superposition of two frequencies. Although formu- lated primarily to describe those patterns, this model was also shown to describe other patterns, including regular square crystal lattices in two dimensions with the choice q$_{1}$/q$_{0}$ = √$_{2, which couples}$ 〈 10 〉 and 〈 11 〉 RLVs. In section IV, we discuss how to determine the two- mode PFC model parameters to relate them quantita- tively to different materials. We follow essentially the same approach developed by two of the authors for the standard PFC one-mode model for bcc ordering [14]. For bcc, the parameters were completely determined by fit- ting three parameters: (i) the peak value of the liquid structure factor, S ( q$_{0}$ ), where q$_{0}$ = | ⃗ K$_{110}$ | , (ii) the sec- ond derivative of the fourier transform of the direct cor- relation function at this peak, C ′′ ( q$_{0}$ ), and (iii) the solid density wave amplitude u$_{110}$ . For fcc, all the parame- ters except R$_{1}$ are determined by the same fit, where q$_{0}$ = | ⃗ K$_{111}$ | . (The shape of the structure factor at q$_{1}$ = | ⃗ K$_{200}$ | is not realistically modeled given the lim- ited number of model parameters.) R$_{1}$ then determines the ratio u$_{200}$/u$_{111}$ of the 〈 111 〉 and 〈 200 〉 solid ampli- tudes, which can be varied to alter the relative stability of fcc and bcc. The present introduction of the parameter r$_{1}$ in the form (4) provides the additional flexibility to change the relative stability of different crystal structures. This is because in the limit r$_{1}$ ≫ q 4 $_{0}$, this form reduces formally to the original Swift-Hohenberg form (3) after a simple rescaling of the parameters. Hence, as r$_{1}$ is increased the contribution of the second q$_{1}$ -mode becomes less signif- icant in comparison to the first q$_{0}$ -mode. Consequently, as r$_{1}$ is increased from zero, the crystal structure favored by the two-mode interaction becomes metastable with re- spect to the one-mode structure. This added capability to model the coexistence of two different crystal struc- tures, in addition to the coexistence of each structure with a liquid, should prove useful to model a wide range of phase transformations with a PFC approach. In the next section, we scale the parameters of the model to write the free-energy functional in a dimension- less form with only three parameters: ϵ , which is the standard PFC model parameter analogous to tempera- ture that controls the size of the solid-liquid coexistence regions as a function of density, Q$_{1}$ ≡ q$_{1}$/q$_{0}$ , whose value is generally determined by the choice of crystal struc- ture, and R$_{1}$ ≡ r$_{1}$/q 4 0 controls the relative stability of In section V, we derive analytical expressions for the three independent elastic constants of a cubic material, C$_{11}$ , C$_{12}$ , and C$_{44}$ , for both the standard one-mode PFC f = φ 2 $^{[}$a + λ ( ∇ 2 + q 2 $_{0}$)$^{2}$(( ∇ 2 + q 2 $_{1}$) 2 + r$_{1}$ ) $^{]}$+ g φ 4 4 . (4)