BaMg14Hf crystallizes in the hexagonal P-6m2 space group. Ba(1) is bonded to six Mg(1,1) and six Mg(3,3) atoms to form BaMg12 cuboctahedra that share corners with six equivalent Ba(1)Mg12 cuboctahedra; corners with twelve equivalent Mg(2)Hf2Mg10 cuboctahedra; edges with twelve Mg(3,3)BaHfMg10 cuboctahedra; faces with two equivalent Hf(1)Mg12 cuboctahedra; faces with six Mg(3,3)BaHfMg10 cuboctahedra; and faces with six equivalent Mg(4)Mg12 cuboctahedra. There are six inequivalent Mg sites. In the first Mg site, Mg(1) is bonded in a 12-coordinate geometry to two equivalent Ba(1); two equivalent Mg(4); four Mg(1,1); and four Mg(3,3) atoms. In the second Mg site, Mg(2) is bonded to two equivalent Mg(4); four equivalent Mg(2); four Mg(3,3); and two equivalent Hf(1) atoms to form MgHf2Mg10 cuboctahedra that share corners with four equivalent Ba(1)Mg12 cuboctahedra; corners with six equivalent Mg(2)Hf2Mg10 cuboctahedra; edges with two equivalent Hf(1)Mg12 cuboctahedra; edges with four equivalent Mg(2)Hf2Mg10 cuboctahedra; edges with four equivalent Mg(4)Mg12 cuboctahedra; edges with eight Mg(3,3)BaHfMg10 cuboctahedra; faces with two equivalent Mg(4)Mg12 cuboctahedra; faces with two equivalent Hf(1)Mg12 cuboctahedra; faces with four equivalent Mg(2)Hf2Mg10 cuboctahedra; and faces with ten Mg(3,3)BaHfMg10 cuboctahedra. In the third Mg site, Mg(1) is bonded in a 12-coordinate geometry to two equivalent Ba(1); two equivalent Mg(4); four equivalent Mg(1); and four Mg(3,3) atoms. In the fourth Mg site, Mg(3) is bonded to one Ba(1); two equivalent Mg(1); two equivalent Mg(2); two equivalent Mg(4); four Mg(3,3); and one Hf(1) atom to form distorted MgBaHfMg10 cuboctahedra that share corners with four equivalent Mg(4)Mg12 cuboctahedra; corners with fourteen Mg(3,3)BaHfMg10 cuboctahedra; edges with two equivalent Ba(1)Mg12 cuboctahedra; edges with two equivalent Mg(4)Mg12 cuboctahedra; edges with two equivalent Hf(1)Mg12 cuboctahedra; edges with four Mg(3,3)BaHfMg10 cuboctahedra; edges with four equivalent Mg(2)Hf2Mg10 cuboctahedra;  a faceface with one Ba(1)Mg12 cuboctahedra;  a faceface with one Hf(1)Mg12 cuboctahedra; faces with two equivalent Mg(4)Mg12 cuboctahedra; faces with five equivalent Mg(2)Hf2Mg10 cuboctahedra; and faces with six Mg(3,3)BaHfMg10 cuboctahedra. In the fifth Mg site, Mg(3) is bonded to one Ba(1); two Mg(1,1); two equivalent Mg(2); two equivalent Mg(4); four equivalent Mg(3); and one Hf(1) atom to form distorted MgBaHfMg10 cuboctahedra that share corners with four equivalent Mg(4)Mg12 cuboctahedra; corners with fourteen Mg(3,3)BaHfMg10 cuboctahedra; edges with two equivalent Ba(1)Mg12 cuboctahedra; edges with two equivalent Mg(4)Mg12 cuboctahedra; edges with two equivalent Hf(1)Mg12 cuboctahedra; edges with four equivalent Mg(3)BaHfMg10 cuboctahedra; edges with four equivalent Mg(2)Hf2Mg10 cuboctahedra;  a faceface with one Ba(1)Mg12 cuboctahedra;  a faceface with one Hf(1)Mg12 cuboctahedra; faces with two equivalent Mg(4)Mg12 cuboctahedra; faces with five equivalent Mg(2)Hf2Mg10 cuboctahedra; and faces with six Mg(3,3)BaHfMg10 cuboctahedra. In the sixth Mg site, Mg(4) is bonded to three Mg(1,1); three equivalent Mg(2); and six Mg(3,3) atoms to form MgMg12 cuboctahedra that share corners with six equivalent Mg(4)Mg12 cuboctahedra; corners with twelve Mg(3,3)BaHfMg10 cuboctahedra; edges with six Mg(3,3)BaHfMg10 cuboctahedra; edges with six equivalent Mg(2)Hf2Mg10 cuboctahedra; faces with two equivalent Mg(4)Mg12 cuboctahedra; faces with three equivalent Ba(1)Mg12 cuboctahedra; faces with three equivalent Mg(2)Hf2Mg10 cuboctahedra; faces with three equivalent Hf(1)Mg12 cuboctahedra; and faces with six Mg(3,3)BaHfMg10 cuboctahedra. Hf(1) is bonded to six equivalent Mg(2) and six Mg(3,3) atoms to form HfMg12 cuboctahedra that share corners with six equivalent Hf(1)Mg12 cuboctahedra; edges with six equivalent Mg(2)Hf2Mg10 cuboctahedra; edges with twelve Mg(3,3)BaHfMg10 cuboctahedra; faces with two equivalent Ba(1)Mg12 cuboctahedra; faces with six Mg(3,3)BaHfMg10 cuboctahedra; faces with six equivalent Mg(2)Hf2Mg10 cuboctahedra; and faces with six equivalent Mg(4)Mg12 cuboctahedra.
BaMg14Hf crystallizes in the hexagonal P-6m2 space group. Ba(1) is bonded to six Mg(1,1) and six Mg(3,3) atoms to form BaMg12 cuboctahedra that share corners with six equivalent Ba(1)Mg12 cuboctahedra; corners with twelve equivalent Mg(2)Hf2Mg10 cuboctahedra; edges with twelve Mg(3,3)BaHfMg10 cuboctahedra; faces with two equivalent Hf(1)Mg12 cuboctahedra; faces with six Mg(3,3)BaHfMg10 cuboctahedra; and faces with six equivalent Mg(4)Mg12 cuboctahedra. All Ba(1)-Mg(1,1) bond lengths are 3.33 Å. All Ba(1)-Mg(3,3) bond lengths are 3.39 Å. There are six inequivalent Mg sites. In the first Mg site, Mg(1) is bonded in a 12-coordinate geometry to two equivalent Ba(1); two equivalent Mg(4); four Mg(1,1); and four Mg(3,3) atoms. Both Mg(1)-Mg(4) bond lengths are 3.24 Å. There are two shorter (3.18 Å) and two longer (3.47 Å) Mg(1)-Mg(1,1) bond lengths. All Mg(1)-Mg(3,3) bond lengths are 3.32 Å. In the second Mg site, Mg(2) is bonded to two equivalent Mg(4); four equivalent Mg(2); four Mg(3,3); and two equivalent Hf(1) atoms to form MgHf2Mg10 cuboctahedra that share corners with four equivalent Ba(1)Mg12 cuboctahedra; corners with six equivalent Mg(2)Hf2Mg10 cuboctahedra; edges with two equivalent Hf(1)Mg12 cuboctahedra; edges with four equivalent Mg(2)Hf2Mg10 cuboctahedra; edges with four equivalent Mg(4)Mg12 cuboctahedra; edges with eight Mg(3,3)BaHfMg10 cuboctahedra; faces with two equivalent Mg(4)Mg12 cuboctahedra; faces with two equivalent Hf(1)Mg12 cuboctahedra; faces with four equivalent Mg(2)Hf2Mg10 cuboctahedra; and faces with ten Mg(3,3)BaHfMg10 cuboctahedra. Both Mg(2)-Mg(4) bond lengths are 3.25 Å. There are two shorter (3.31 Å) and two longer (3.34 Å) Mg(2)-Mg(2) bond lengths. All Mg(2)-Mg(3,3) bond lengths are 3.08 Å. Both Mg(2)-Hf(1) bond lengths are 3.33 Å. In the third Mg site, Mg(1) is bonded in a 12-coordinate geometry to two equivalent Ba(1); two equivalent Mg(4); four equivalent Mg(1); and four Mg(3,3) atoms. Both Mg(1)-Mg(4) bond lengths are 3.24 Å. All Mg(1)-Mg(3,3) bond lengths are 3.32 Å. In the fourth Mg site, Mg(3) is bonded to one Ba(1); two equivalent Mg(1); two equivalent Mg(2); two equivalent Mg(4); four Mg(3,3); and one Hf(1) atom to form distorted MgBaHfMg10 cuboctahedra that share corners with four equivalent Mg(4)Mg12 cuboctahedra; corners with fourteen Mg(3,3)BaHfMg10 cuboctahedra; edges with two equivalent Ba(1)Mg12 cuboctahedra; edges with two equivalent Mg(4)Mg12 cuboctahedra; edges with two equivalent Hf(1)Mg12 cuboctahedra; edges with four Mg(3,3)BaHfMg10 cuboctahedra; edges with four equivalent Mg(2)Hf2Mg10 cuboctahedra;  a faceface with one Ba(1)Mg12 cuboctahedra;  a faceface with one Hf(1)Mg12 cuboctahedra; faces with two equivalent Mg(4)Mg12 cuboctahedra; faces with five equivalent Mg(2)Hf2Mg10 cuboctahedra; and faces with six Mg(3,3)BaHfMg10 cuboctahedra. Both Mg(3)-Mg(4) bond lengths are 3.33 Å. There are two shorter (3.25 Å) and two longer (3.40 Å) Mg(3)-Mg(3,3) bond lengths. The Mg(3)-Hf(1) bond length is 3.12 Å. In the fifth Mg site, Mg(3) is bonded to one Ba(1); two Mg(1,1); two equivalent Mg(2); two equivalent Mg(4); four equivalent Mg(3); and one Hf(1) atom to form distorted MgBaHfMg10 cuboctahedra that share corners with four equivalent Mg(4)Mg12 cuboctahedra; corners with fourteen Mg(3,3)BaHfMg10 cuboctahedra; edges with two equivalent Ba(1)Mg12 cuboctahedra; edges with two equivalent Mg(4)Mg12 cuboctahedra; edges with two equivalent Hf(1)Mg12 cuboctahedra; edges with four equivalent Mg(3)BaHfMg10 cuboctahedra; edges with four equivalent Mg(2)Hf2Mg10 cuboctahedra;  a faceface with one Ba(1)Mg12 cuboctahedra;  a faceface with one Hf(1)Mg12 cuboctahedra; faces with two equivalent Mg(4)Mg12 cuboctahedra; faces with five equivalent Mg(2)Hf2Mg10 cuboctahedra; and faces with six Mg(3,3)BaHfMg10 cuboctahedra. Both Mg(3)-Mg(4) bond lengths are 3.33 Å. The Mg(3)-Hf(1) bond length is 3.12 Å. In the sixth Mg site, Mg(4) is bonded to three Mg(1,1); three equivalent Mg(2); and six Mg(3,3) atoms to form MgMg12 cuboctahedra that share corners with six equivalent Mg(4)Mg12 cuboctahedra; corners with twelve Mg(3,3)BaHfMg10 cuboctahedra; edges with six Mg(3,3)BaHfMg10 cuboctahedra; edges with six equivalent Mg(2)Hf2Mg10 cuboctahedra; faces with two equivalent Mg(4)Mg12 cuboctahedra; faces with three equivalent Ba(1)Mg12 cuboctahedra; faces with three equivalent Mg(2)Hf2Mg10 cuboctahedra; faces with three equivalent Hf(1)Mg12 cuboctahedra; and faces with six Mg(3,3)BaHfMg10 cuboctahedra. Hf(1) is bonded to six equivalent Mg(2) and six Mg(3,3) atoms to form HfMg12 cuboctahedra that share corners with six equivalent Hf(1)Mg12 cuboctahedra; edges with six equivalent Mg(2)Hf2Mg10 cuboctahedra; edges with twelve Mg(3,3)BaHfMg10 cuboctahedra; faces with two equivalent Ba(1)Mg12 cuboctahedra; faces with six Mg(3,3)BaHfMg10 cuboctahedra; faces with six equivalent Mg(2)Hf2Mg10 cuboctahedra; and faces with six equivalent Mg(4)Mg12 cuboctahedra.