of the two-mode crystal density field We compute the change of free-energy density where f$_{s}$ is the free-energy density of the unperturbed solid and F$_{i}$ is the free-energy integrated over the per- turbed unit cell of volume V$_{i}$ with where dV = dxdydz , a is the lattice spacing, and V = a 3 is the unperturbed unit cell volume. The elastic constants computed with the parameters of Table I are compared to the predictions of MD sim- ulations in Table II. The MD simulations for fcc Fe and bcc Ni were carried out using the EAM potentials from Mendelev, Han, Srolovitz, Ackland, Sun and Asta MH(SA) 2 [29], and Foiles, Baskes and Daw [28], respec- tively. The same EAM potentials were used to compute the input parameters for the PFC model and the elastic constants. The input parameters for Fe are the same as in Ref. [26]. The input parameters for Ni were computed by Hoyt [27]. The elastic constants for both Fe and Ni were computed by Foiles [30]. Their values at the melt- ing point are smaller than at zero temperature as shown by Foiles for a different Ni EAM potential [31]. and Using Eqs. (65)-(68) with the two-mode crystal den- sity field ψ ( ⃗ r ) defined by Eq. (27) and the free-energy density f ( ψ ( ⃗ r )) defined by Eq. (20), we obtain the di- mensionless elastic constants where we have defined For the one-mode model, we use again Eqs. (65)-(68) with the one-mode bcc crystal density field where q = 1 $_{/}$√$_{2 and the free-energy density}$ f ( ψ ( ⃗ r )) de- fined by Eq. (25). We obtain We can set R$_{1}$ ≈ 0 in the above expression since R$_{1}$ ≪ 1 for typical model parameters where the fcc lattice is favored. With only the principle 〈 111 〉 RLVs ( B$_{s}$ = 0), all three elastic constants are equal ˜ C$_{11}$ = ˜ C$_{12}$ = ˜ C$_{44}$ = 2 α/ 9, which gives a vanishing tetragonal shear modulus ˜ C ′ = ( ˜ C$_{11}$ − ˜ C$_{22}$ ) / 2. The inclusion of the 〈 200 〉 RLVs, however, raises the value of ˜ C$_{11}$ , which becomes while leaving the values of ˜ C$_{12}$ and ˜ C$_{44}$ unchanged, thereby making ˜ C ′ finite as desired. Finally, convert- ing back to dimensional units using the relation C$_{ij}$ = ( λ $^{2}$q16 0 /g ) ˜ C$_{ij}$ , we obtain where α$_{bcc}$ = 24 A 2 $_{s}$. This yields the dimensionless elas- tic constants ˜ C$_{11}$ = 2 ˜ C$_{12}$ = 2 ˜ C$_{44}$ = 8 A 2 $_{s}$. Finally, converting back to dimensional units using the relation C$_{ij}$ = ( λ $^{2}$q8 $_{0}$/g ) ˜ C$_{ij}$ , we obtain The elastic constants computed with the input parame- ters of Table III for bcc Fe are compared to the predic- tions of MD simulations in Table II. C$_{11}$ = − 4 9 n$_{0}$k$_{B}$T C ′′ ( K$_{111}$ ) q 2 $_{0}$u 2 111 ( 1 + 4 u 2 200 u 2 111 ) , (75) C$_{11}$ = 2 C$_{12}$ = 2 C$_{44}$ = − n$_{0}$k$_{B}$T C ′′ ( K$_{110}$ ) q 2 $_{0}$u 2 $_{110}$. (81) ˜ C$_{11}$ = 2 9 α$^{(}$ 1 + 4 B 2 s A 2 s ) , Δ f$_{3}$ = ˜ C$_{44}$ 2 ξ 2 = α$_{bcc}$ 12 ξ $^{2}$, (80) Δ f$_{2}$ = ( ˜ C$_{11}$ − ˜ C$_{12}$ ) ξ 2 = α$_{bcc}$ 6 ξ $^{2}$, (79) Δ f$_{1}$ = $^{(}$3 2 ˜ C$_{11}$ + 3 ˜ C$_{12}$$^{)}$ ξ 2 = α$_{bcc}$ξ $^{2}$, (78) ψ ( ⃗ r ) ≈ ¯ ψ +4 A$_{s}$ (cos qx cos qy +cos qx cos qz +cos qy cos qz ) . (77) δ = 8 9 R$_{1}$B 2 $_{s}$, (74) β = 8 27 (284 − 315 Q 2 1 + 81 Q 4 1 + 81 R$_{1}$ ) B 2 $_{s}$, (73) α = 16(1 − 2 Q 2 1 + Q 4 1 + R$_{1}$ ) A 2 $_{s}$, (72) Δ f$_{3}$ = ˜ C$_{44}$ 2 ξ 2 = ( α/ 9 + δ ) ξ $^{2}$, (71) Δ f$_{2}$ = ( ˜ C$_{11}$ − ˜ C$_{12}$ ) ξ 2 = 2 3 βξ $^{2}$, (70) Δ f$_{1}$ = $^{(}$3 2 ˜ C$_{11}$ + 3 ˜ C$_{12}$$^{)}$ ξ 2 = ( α + β ) ξ $^{2}$, (69) B. bcc elastic constants for the one-mode model A. fcc elastic constants for the two-mode model F$_{1}$ V$_{1}$ = 1 V (1 + ξ ) 3 ∫ a (1+ ξ ) 0 ∫ a (1+ ξ ) 0 ∫ a (1+ ξ ) 0 f ( ψ$_{1}$ ( ⃗ r )) dV, F$_{2}$ V$_{2}$ = 1 V (1 − ξ $^{2}$) ∫ a 0 ∫ a (1 − ξ ) 0 ∫ a (1+ ξ ) 0 f ( ψ$_{2}$ ( ⃗ r )) dV, F$_{3}$ V$_{3}$ = 1 V ∫ a 0 ∫ a 0 ∫ a − ξy − ξy f ( ψ$_{3}$ ( ⃗ r )) dV, C$_{12}$ = C$_{44}$ = − 4 9 n$_{0}$k$_{B}$T C ′′ ( K$_{111}$ ) q 2 $_{0}$u 2 $_{111}$. (76) Δ f$_{i}$ = F$_{i}$ V$_{i}$ − f$_{s}$, (68) TABLE II: Comparison of elastic constants at the melting point predicted by the two-mode PFC model for fcc Ni and the one-mode PFC model for bcc Fe and MD simulations [30]. Quantity PFC fcc MD fcc PFC bcc MD bcc C$_{11}$ (GPa) 106.6 155.4 90.0 128.0 C$_{12}$ (GPa) 31.4 124.7 45.0 103.4 C$_{44}$ (GPa) 31.4 66.0 45.0 63.9 Bulk Modulus (GPa) 56.5 134.9 60.0 111.6 ( C$_{11}$ + 2 C$_{12}$ ) / 3 13