Fig. 1 Schematic structure of the SAE model in a body-center-cubic (BCC) lattice of Fe-28Mn-18.5Al (wt%) quaternary alloy. a 3D views of the configurational transitions of the Al clusters via atomic rearrangements, including ten random arrangements. b Topological structure of Fe and Al atoms in the crystal lattice of A2, B2, D03,B32 and Fe13Al3 phases. The various colors characterize the different kinds of alloying elements for the given equiatomic Fe-Mn-Al BCC MEA alloy, which are kept consistently in the following related figures. In particular, the Fe, Mn and Al elements are in yellow, purple and green, respectively. Meanwhile, the free energies of 10 random arrangement configurations were obtained by first-principles calculations

Fig. 2 Predicted fundamental properties of the 5 selected stable random arrangements (Best-1, Best-2, Best-3, Best-4, Best-5) for the Fe-Mn-Al MEA, illustrating the effect of configurational transition on the a total energy E0, b electron work function Φ, c Fermi energy EF, d magnetic moment μMag, respectively. e A radar chart that displays the rank of the calculated data for Fe-Mn-Al MEA

Fig. 3 Bonding-charge-density isosurface of five selected configurations for the random solutions of the given BCC Fe-Mn-Al MEA in a 3D view, respectively. a Isosurfaces in red display the atomic sites absorbing electrons (Δρ > 0), while b the isosurfaces in blue identify the atomic sites contributing electrons (Δρ < 0). Those solid ellipses highlight the weak bond features caused by the chemical (solute atoms) and the mechanical (local lattice distortion) contributions. The red, green and blue dash ellipses highlight the different bonding features of the Fe, Al and Mn atoms, respectively. In particular, the Fe, Mn and Al elements are in yellow, purple and green, respectively. Meanwhile, the free energies of 5 stable random arrangement configurations were obtained by first-principles calculations

Fig. 4 a Influence of solute atoms and clusters on the bond structures of five selected configurations of the BCC Fe-Mn-Al MEA are characterized by the 3D view of counter plots of bonding charge density. The spaces in red are for Δρ > 0, while those in blue are for Δρ < 0. b The (100) and (010) view of contour plots of △ρ 0.0015 e-/Å3 intervals, generated by VESTA package. In particular, the Fe, Mn and Al elements are in yellow, purple and green, respectively

Fig. 5 Collinear spin alignments of selected stable random arrangement structures in a 3D view for BCC Fe-Mn-Al MEA. The arrows in yellow and in purple identify Fe and Mn atoms, respectively. The spin flipping caused by local lattice distortions is highlighted by the dashed ellipse in red

Fig. 6 a-e Total density of states (DOS) of the same alloying elements in various configurations generated by the SAE model, the corresponding configurations can be found in Fig. 3. The Fermi level defines the zero of energy

Fig. 7 Total density of states (DOS) and partial density of states (pDOS) of the same alloying elements in various configurations generated by the SAE model, the corresponding configurations can be found in Fig. 3. The Fermi level defines the zero of energy