Table 1 Predicted chemical orderings for the Al_xCrFeCoNi system as a function of x, as inferred from the described Landau-type linear-response theory for cases with both bcc and fcc as the underlying crystal lattice

Lattice	Composition	T_ord (K)	k_ord (2π/a)	Δ Al	Δ Cr	Δ Fe	Δ Co	Δ Ni	Ordered structure
fcc	CrFeCoNi	380	(0,0,1)		0.722	−0.049	−0.689	0.016	L1₂
	Al_0.5CrFeCoNi	562	(0,\(\frac{1}{2}\),1)	0.803	0.145	−0.167	−0.401	−0.381	D0₂₂
	AlCrFeCoNi	744	(0,\(\frac{1}{2}\),1)	0.871	−0.042	−0.195	−0.321	−0.314	D0₂₂
	Al_1.5CrFeCoNi	987	(0,\(\frac{1}{2}\),1)	0.882	−0.091	−0.204	−0.301	−0.286	D0₂₂
	Al₂CrFeCoNi	1082	(0,0,1)	0.871	−0.040	−0.194	−0.308	−0.328	L1₂
bcc	CrFeCoNi	488	(0,0,0)		0.684	−0.257	0.220	−0.647	Phase Seg.
	Al_0.5CrFeCoNi	1982	(0,0,1)	0.777	0.173	−0.117	−0.470	−0.363	B2
	AlCrFeCoNi	3006	(0,0,1)	0.831	0.073	−0.164	−0.404	−0.336	B2
	Al_1.5CrFeCoNi	3781	(0,0,1)	0.852	0.021	−0.184	−0.374	−0.315	B2
	Al₂CrFeCoNi	4386	(0,0,1)	0.863	−0.012	−0.195	−0.357	−0.298	B2

Within the concentration wave formalism, a chemical ordering is described by the temperature at which ordering occurs (T_ord), the wave-vector describing ordering (k_ord), and its chemical polarisation (Δc_α). On the fcc lattice, we find a tendency towards L1₂ ordering competing with more complex interactions, which we interpret as a precursor to the experimentally observed D8_b phase. On the bcc lattice, with increasing Al concentration, we find exceptionally strong B2 ordering tendencies, indicating this atomically ordered phase will form directly from the melt, consistent with experimental data.

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