Fig. 1. (a-c) HAADF-STEM images showing β′′ phase along (a) [$2\bar{1}\bar{1}0$]α, (b) [$01\bar{1}0$]α and (c) [0001]α, respectively, in samples that are solution treated and then irradiated for 1 h by electron beam. (d-f) SAED patterns of electron beam irradiated areas along (d) [$2\bar{1}\bar{1}0$]α, (e) [$01\bar{1}0$]α and (f) [0001]α, respectively.

Fig. 2. (a) [$2\bar{1}\bar{1}0$]α, (b) [$01\bar{1}0$]α and (c) [0001]α atomic-resolution HAADF-STEM images showing the structure of β′′. (d) Schematic diagram showing a β′′ unit cell. (e-g) Perspective views of β′′ along (e) [$2\bar{1}\bar{1}0$]β′′, (f) [$01\bar{1}0$]β′′ and (g) [0001]β′′, respectively. Yellow lines outline the arrangements of bright atomic columns in (a-c) and Sn-rich columns in (e-g). A and B represent the stacking sequence of the closely packed planes of α-Mg in (a).

Fig. 3. Simulated superimposed SAED patterns of β′′ phase and α-Mg matrix along (a) [$2\bar{1}\bar{1}0$]β′′/[$2\bar{1}\bar{1}0$]α, (b) [$01\bar{1}0$]β′′/[$01\bar{1}0$]α and (c) [0001]β′′/[0001]α zone axes. Hollow circle represents the diffraction spot of α-Mg and solid dot represents the diffraction spot of β′′. Symbol × represents the spot that does not meet the reflection conditions.

Fig. 4. Schematic diagrams showing the relaxed supercell structures of two states. (a) Initial state: two Sn atoms dissolving separately in α-Mg matrix. (b) Final state: two Sn atoms binding together to form a β′′ unit cell. The β′′ unit cell is outlined by blue rhomboid in (b).

Fig. 5. Convex hull of formation energy showing the stability of β′′, β′, and β phases.