BaMg15 crystallizes in the hexagonal P-6m2 space group. Ba(1) is bonded to six equivalent Mg(2) and six Mg(4,4) atoms to form BaMg12 cuboctahedra that share corners with six equivalent Ba(1)Mg12 cuboctahedra; corners with twelve equivalent Mg(3)Mg12 cuboctahedra; edges with twelve Mg(4,4)BaMg11 cuboctahedra; faces with two equivalent Mg(1)Mg12 cuboctahedra; faces with six Mg(4,4)BaMg11 cuboctahedra; and faces with six equivalent Mg(5)Mg12 cuboctahedra. There are six inequivalent Mg sites. In the first Mg site, Mg(1) is bonded to six equivalent Mg(3) and six Mg(4,4) atoms to form MgMg12 cuboctahedra that share corners with six equivalent Mg(1)Mg12 cuboctahedra; edges with six equivalent Mg(3)Mg12 cuboctahedra; edges with twelve Mg(4,4)BaMg11 cuboctahedra; faces with two equivalent Ba(1)Mg12 cuboctahedra; faces with six Mg(4,4)BaMg11 cuboctahedra; faces with six equivalent Mg(3)Mg12 cuboctahedra; and faces with six equivalent Mg(5)Mg12 cuboctahedra. In the second Mg site, Mg(2) is bonded in a 12-coordinate geometry to two equivalent Ba(1); two equivalent Mg(5); four equivalent Mg(2); and four Mg(4,4) atoms. In the third Mg site, Mg(3) is bonded to two equivalent Mg(1); two equivalent Mg(5); four equivalent Mg(3); and four Mg(4,4) atoms to form MgMg12 cuboctahedra that share corners with four equivalent Ba(1)Mg12 cuboctahedra; corners with six equivalent Mg(3)Mg12 cuboctahedra; edges with two equivalent Mg(1)Mg12 cuboctahedra; edges with four equivalent Mg(3)Mg12 cuboctahedra; edges with four equivalent Mg(5)Mg12 cuboctahedra; edges with eight Mg(4,4)BaMg11 cuboctahedra; faces with two equivalent Mg(1)Mg12 cuboctahedra; faces with two equivalent Mg(5)Mg12 cuboctahedra; faces with four equivalent Mg(3)Mg12 cuboctahedra; and faces with ten Mg(4,4)BaMg11 cuboctahedra. In the fourth Mg site, Mg(4) is bonded to one Ba(1), one Mg(1), two equivalent Mg(2), two equivalent Mg(3), two equivalent Mg(5), and four equivalent Mg(4) atoms to form distorted MgBaMg11 cuboctahedra that share corners with four equivalent Mg(5)Mg12 cuboctahedra; corners with fourteen Mg(4,4)BaMg11 cuboctahedra; edges with two equivalent Ba(1)Mg12 cuboctahedra; edges with two equivalent Mg(1)Mg12 cuboctahedra; edges with two equivalent Mg(5)Mg12 cuboctahedra; edges with four equivalent Mg(4)BaMg11 cuboctahedra; edges with four equivalent Mg(3)Mg12 cuboctahedra;  a faceface with one Ba(1)Mg12 cuboctahedra;  a faceface with one Mg(1)Mg12 cuboctahedra; faces with two equivalent Mg(5)Mg12 cuboctahedra; faces with five equivalent Mg(3)Mg12 cuboctahedra; and faces with six Mg(4,4)BaMg11 cuboctahedra. In the fifth Mg site, Mg(4) is bonded to one Ba(1); one Mg(1); two equivalent Mg(2); two equivalent Mg(3); two equivalent Mg(5); and four Mg(4,4) atoms to form distorted MgBaMg11 cuboctahedra that share corners with four equivalent Mg(5)Mg12 cuboctahedra; corners with fourteen Mg(4,4)BaMg11 cuboctahedra; edges with two equivalent Ba(1)Mg12 cuboctahedra; edges with two equivalent Mg(1)Mg12 cuboctahedra; edges with two equivalent Mg(5)Mg12 cuboctahedra; edges with four Mg(4,4)BaMg11 cuboctahedra; edges with four equivalent Mg(3)Mg12 cuboctahedra;  a faceface with one Ba(1)Mg12 cuboctahedra;  a faceface with one Mg(1)Mg12 cuboctahedra; faces with two equivalent Mg(5)Mg12 cuboctahedra; faces with five equivalent Mg(3)Mg12 cuboctahedra; and faces with six Mg(4,4)BaMg11 cuboctahedra. In the sixth Mg site, Mg(5) is bonded to three equivalent Mg(2); three equivalent Mg(3); and six Mg(4,4) atoms to form MgMg12 cuboctahedra that share corners with six equivalent Mg(5)Mg12 cuboctahedra; corners with twelve Mg(4,4)BaMg11 cuboctahedra; edges with six Mg(4,4)BaMg11 cuboctahedra; edges with six equivalent Mg(3)Mg12 cuboctahedra; faces with two equivalent Mg(5)Mg12 cuboctahedra; faces with three equivalent Ba(1)Mg12 cuboctahedra; faces with three equivalent Mg(1)Mg12 cuboctahedra; faces with three equivalent Mg(3)Mg12 cuboctahedra; and faces with six Mg(4,4)BaMg11 cuboctahedra.
BaMg15 crystallizes in the hexagonal P-6m2 space group. Ba(1) is bonded to six equivalent Mg(2) and six Mg(4,4) atoms to form BaMg12 cuboctahedra that share corners with six equivalent Ba(1)Mg12 cuboctahedra; corners with twelve equivalent Mg(3)Mg12 cuboctahedra; edges with twelve Mg(4,4)BaMg11 cuboctahedra; faces with two equivalent Mg(1)Mg12 cuboctahedra; faces with six Mg(4,4)BaMg11 cuboctahedra; and faces with six equivalent Mg(5)Mg12 cuboctahedra. All Ba(1)-Mg(2) bond lengths are 3.33 Å. All Ba(1)-Mg(4,4) bond lengths are 3.40 Å. There are six inequivalent Mg sites. In the first Mg site, Mg(1) is bonded to six equivalent Mg(3) and six Mg(4,4) atoms to form MgMg12 cuboctahedra that share corners with six equivalent Mg(1)Mg12 cuboctahedra; edges with six equivalent Mg(3)Mg12 cuboctahedra; edges with twelve Mg(4,4)BaMg11 cuboctahedra; faces with two equivalent Ba(1)Mg12 cuboctahedra; faces with six Mg(4,4)BaMg11 cuboctahedra; faces with six equivalent Mg(3)Mg12 cuboctahedra; and faces with six equivalent Mg(5)Mg12 cuboctahedra. All Mg(1)-Mg(3) bond lengths are 3.33 Å. All Mg(1)-Mg(4,4) bond lengths are 3.17 Å. In the second Mg site, Mg(2) is bonded in a 12-coordinate geometry to two equivalent Ba(1); two equivalent Mg(5); four equivalent Mg(2); and four Mg(4,4) atoms. Both Mg(2)-Mg(5) bond lengths are 3.26 Å. There are two shorter (3.18 Å) and two longer (3.48 Å) Mg(2)-Mg(2) bond lengths. All Mg(2)-Mg(4,4) bond lengths are 3.33 Å. In the third Mg site, Mg(3) is bonded to two equivalent Mg(1); two equivalent Mg(5); four equivalent Mg(3); and four Mg(4,4) atoms to form MgMg12 cuboctahedra that share corners with four equivalent Ba(1)Mg12 cuboctahedra; corners with six equivalent Mg(3)Mg12 cuboctahedra; edges with two equivalent Mg(1)Mg12 cuboctahedra; edges with four equivalent Mg(3)Mg12 cuboctahedra; edges with four equivalent Mg(5)Mg12 cuboctahedra; edges with eight Mg(4,4)BaMg11 cuboctahedra; faces with two equivalent Mg(1)Mg12 cuboctahedra; faces with two equivalent Mg(5)Mg12 cuboctahedra; faces with four equivalent Mg(3)Mg12 cuboctahedra; and faces with ten Mg(4,4)BaMg11 cuboctahedra. Both Mg(3)-Mg(5) bond lengths are 3.28 Å. There are two shorter (3.29 Å) and two longer (3.38 Å) Mg(3)-Mg(3) bond lengths. All Mg(3)-Mg(4,4) bond lengths are 3.13 Å. In the fourth Mg site, Mg(4) is bonded to one Ba(1), one Mg(1), two equivalent Mg(2), two equivalent Mg(3), two equivalent Mg(5), and four equivalent Mg(4) atoms to form distorted MgBaMg11 cuboctahedra that share corners with four equivalent Mg(5)Mg12 cuboctahedra; corners with fourteen Mg(4,4)BaMg11 cuboctahedra; edges with two equivalent Ba(1)Mg12 cuboctahedra; edges with two equivalent Mg(1)Mg12 cuboctahedra; edges with two equivalent Mg(5)Mg12 cuboctahedra; edges with four equivalent Mg(4)BaMg11 cuboctahedra; edges with four equivalent Mg(3)Mg12 cuboctahedra;  a faceface with one Ba(1)Mg12 cuboctahedra;  a faceface with one Mg(1)Mg12 cuboctahedra; faces with two equivalent Mg(5)Mg12 cuboctahedra; faces with five equivalent Mg(3)Mg12 cuboctahedra; and faces with six Mg(4,4)BaMg11 cuboctahedra. Both Mg(4)-Mg(5) bond lengths are 3.34 Å. There are two shorter (3.24 Å) and two longer (3.43 Å) Mg(4)-Mg(4) bond lengths. In the fifth Mg site, Mg(4) is bonded to one Ba(1); one Mg(1); two equivalent Mg(2); two equivalent Mg(3); two equivalent Mg(5); and four Mg(4,4) atoms to form distorted MgBaMg11 cuboctahedra that share corners with four equivalent Mg(5)Mg12 cuboctahedra; corners with fourteen Mg(4,4)BaMg11 cuboctahedra; edges with two equivalent Ba(1)Mg12 cuboctahedra; edges with two equivalent Mg(1)Mg12 cuboctahedra; edges with two equivalent Mg(5)Mg12 cuboctahedra; edges with four Mg(4,4)BaMg11 cuboctahedra; edges with four equivalent Mg(3)Mg12 cuboctahedra;  a faceface with one Ba(1)Mg12 cuboctahedra;  a faceface with one Mg(1)Mg12 cuboctahedra; faces with two equivalent Mg(5)Mg12 cuboctahedra; faces with five equivalent Mg(3)Mg12 cuboctahedra; and faces with six Mg(4,4)BaMg11 cuboctahedra. Both Mg(4)-Mg(5) bond lengths are 3.34 Å. There is one shorter (3.24 Å) and one longer (3.43 Å) Mg(4)-Mg(4) bond length. In the sixth Mg site, Mg(5) is bonded to three equivalent Mg(2); three equivalent Mg(3); and six Mg(4,4) atoms to form MgMg12 cuboctahedra that share corners with six equivalent Mg(5)Mg12 cuboctahedra; corners with twelve Mg(4,4)BaMg11 cuboctahedra; edges with six Mg(4,4)BaMg11 cuboctahedra; edges with six equivalent Mg(3)Mg12 cuboctahedra; faces with two equivalent Mg(5)Mg12 cuboctahedra; faces with three equivalent Ba(1)Mg12 cuboctahedra; faces with three equivalent Mg(1)Mg12 cuboctahedra; faces with three equivalent Mg(3)Mg12 cuboctahedra; and faces with six Mg(4,4)BaMg11 cuboctahedra.