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simplicity, we used the value R = 0 that yields a reason- able fit of this amplitude for pure Ni. The other input pa- rameters computed by Hoyt [27] using the EAM potential of Foiles, Baskes and Daw [28] (FBD) are given in Table I. The density wave amplitudes are calculated using the re- lation u$_{i}$ = exp( − K 2 i / 4 a ), which assumes that the crystal density field is a sum of Gaussians centered around each fcc lattice site. The value of a is obtained from the ex- pressions for the root-mean-square displacement of atoms in the solid $^{√}$< | ⃗ r | 2 > = 3 / (2 a ) derived from this den- sity field. For the value $^{√}$< | ⃗ r | 2 > ≈ 0 . 298 ˚ A from MD simulations, we obtain u$_{111}$ = exp( − K 2 $_{111}$/ 4 a ) = 0 . 6639 and u$_{200}$ = exp( − K 2 $_{111}$/ 4 a ) = 0 . 5791. It should be noted that, with the present fitting pro- 

cedure, the two-mode PFC model only reproduces the correct shape of the main peak of the liquid structure factor. The second peak is spurious and is only used to increase the amplitude of the second mode to some desired value. Since the second mode is critical to ob- tain solid-liquid coexistence, the lack of realism of the structure factor outside of the first peak is a limitation of the present two-mode model. The liquid structure factor could in principle be made more realistic by shift- ing the second peak to larger wavector and reducing its amplitude, which would couple the principal 〈 111 〉 RLVs to other sets such as 〈 222 〉 and 〈 311 〉 . However, larger k -modes require a finer mesh and are computationally more costly to resolve. Whether such a fit would offer specific advantages remains to be investigated. 

In this section, we derive analytical expressions for the elastic constants of the two-mode PFC model. We com- 

## V. ELASTIC CONSTANTS 

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pare the results to MD computations of elastic constants at the melting point for parameters of fcc Ni. For com- pleteness, we also carry out the same comparison for the standard one-mode PFC model for parameters of bcc Fe. Following the same approach as in Ref. [2], we obtain the elastic constants by deforming the lattice from its ideal structure and computing the corresponding change of free-energy density. We consider three different defor- mations 

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