content_image/380301.jpg content_image/380295.jpg content_image/380294.jpg content_image/380286.jpg content_image/380298.jpg content_image/380293.jpg content_image/380303.jpg content_image/380297.jpg content_image/380296.jpg content_image/380288.jpg content_image/380305.jpg content_image/380302.jpg content_image/380292.jpg content_image/380287.jpg content_image/380291.jpg content_image/380290.jpg content_image/380289.jpg
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 

<img src='content_image/380301.jpg'>

where dV = dxdydz , a is the lattice spacing, and V = a 3 is the unperturbed unit cell volume. 

## A. fcc elastic constants for the two-mode model 

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 

<img src='content_image/380295.jpg'>

<img src='content_image/380294.jpg'>

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 

<img src='content_image/380286.jpg'>

<img src='content_image/380298.jpg'>

<img src='content_image/380293.jpg'>

<img src='content_image/380303.jpg'>

<img src='content_image/380297.jpg'>

<img src='content_image/380296.jpg'>

<img src='content_image/380288.jpg'>

<img src='content_image/380305.jpg'>

and 

<img src='content_image/380302.jpg'>

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]. 

## B. bcc elastic constants for the one-mode model 

For the one-mode model, we use again Eqs. (65)-(68) with the one-mode bcc crystal density field 

<img src='content_image/380292.jpg'>

where q = 1 $_{/}$√$_{2 and the free-energy density}$ f ( ψ ( ⃗ r )) de- fined by Eq. (25). 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 

<img src='content_image/380287.jpg'>

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. 

<img src='content_image/380291.jpg'>

<img src='content_image/380290.jpg'>

<img src='content_image/380289.jpg'>

