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and 

Next, to carry out the amplitude expansion, we start from the equilibrium equation δF/δψ = µ$_{E}$ , where µ$_{E}$ is the equilibrium value of the chemical potential. With F defined by Eqs. (19) and (20), we obtain 

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We substitute the small ϵ expansion of the density field (29) into Eq. (34) and collect terms with the same power of ϵ . We find at the order ϵ 1 / 2 

which has the solution 

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where the summations are over 〈 111 〉 and 〈 200 〉 RLVs, respectively, and | ⃗ K$_{i}$ | = 1 , | ⃗ $^{$_{K$_{j}$}$′}$| = $^{√}$4 / 3, in our scaled units. At order ϵ , we obtain 

which has the solution 

and collecting the terms at order ϵ 3 / 2 yields 

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Since ( ∇ 2 + 1) $^{2}$( ∇ 2 + Q 2 $_{1}$)$^{2}$ψ$_{2}$ gives a vanishing contri- bution for all density waves associated with sets { K$_{i}$ } and { K ′ j } , all remaining terms ∼ e i ⃗ K$_{i}$ · ⃗ r and e i ⃗ K$_{j}$ $^{′}$· ⃗ r must balance each other in order for a solution of Eq. (39) to exist. F or example, the condition that the coefficients of e i ⃗ K$_{111}$ · ⃗ r balance each other yields 

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and requiring that the coefficients of e i ⃗ K$_{200}$ · ⃗ r balance each other yields in turn 

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The above solvability condition must be satisfied inde- pendently for each reciprocal lattice vector. This yields a set of fourteen coupled amplitude equations that are straightforward to obtain. From those amplitude equa- tions, it is useful to express the free-energy of the system measured from its constant value in the liquid, defined as the difference Δ F $^{AE}$, as a functional of the density wave amplitudes A 0 i and B 0 $_{i}$. This quantity can be ex- pressed solely in terms of the amplitudes of density waves owing to the property that the size of the coexistence region ( ∼ ϵ 3 / $^{2}$) is much smaller than the mean density ( ∼ ϵ 1 / $^{2}$) in the small ϵ limit. Since the amplitudes are not conserved order parameters, the equilibrium state simply corresponds to a minimum of this free-energy. Hence the amplitude equations must be equivalent to 

and 

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