Lu4Hf3O12 crystallizes in the triclinic P-1 space group. There are four inequivalent Lu sites. In the first Lu site, Lu(1) is bonded to one O(1), one O(11), one O(12), one O(5), one O(7), one O(8), and one O(9) atom to form distorted LuO7 pentagonal bipyramids that share  a cornercorner with one Hf(1)O6 octahedra and  an edgeedge with one Hf(1)O6 octahedra. The corner-sharing octahedral tilt angles are 52°. In the second Lu site, Lu(2) is bonded in a 7-coordinate geometry to one O(1), one O(10), one O(12), one O(3), one O(4), and two equivalent O(6) atoms. In the third Lu site, Lu(3) is bonded in a 7-coordinate geometry to one O(5), one O(8), one O(9), two equivalent O(2), and two equivalent O(7) atoms. In the fourth Lu site, Lu(4) is bonded in a 7-coordinate geometry to one O(10), one O(11), one O(12), one O(3), one O(6), one O(8), and one O(9) atom. There are three inequivalent Hf sites. In the first Hf site, Hf(1) is bonded to one O(1), one O(11), one O(2), one O(3), one O(4), and one O(8) atom to form HfO6 octahedra that share  a cornercorner with one Lu(1)O7 pentagonal bipyramid and  an edgeedge with one Lu(1)O7 pentagonal bipyramid. In the second Hf site, Hf(2) is bonded in a 7-coordinate geometry to one O(1), one O(2), one O(3), one O(7), one O(9), and two equivalent O(5) atoms. In the third Hf site, Hf(3) is bonded in a 6-coordinate geometry to one O(11), one O(12), one O(4), one O(6), and two equivalent O(10) atoms. There are twelve inequivalent O sites. In the first O site, O(1) is bonded in a distorted rectangular see-saw-like geometry to one Lu(1), one Lu(2), one Hf(1), and one Hf(2) atom. In the second O site, O(2) is bonded to two equivalent Lu(3), one Hf(1), and one Hf(2) atom to form distorted OLu2Hf2 tetrahedra that share  a cornercorner with one O(8)Lu3Hf tetrahedra, corners with two equivalent O(5)Lu2Hf2 tetrahedra, corners with three equivalent O(7)Lu3Hf tetrahedra, corners with three equivalent O(9)Lu3Hf tetrahedra,  an edgeedge with one O(2)Lu2Hf2 tetrahedra,  an edgeedge with one O(3)Lu2Hf2 tetrahedra,  an edgeedge with one O(5)Lu2Hf2 tetrahedra,  an edgeedge with one O(7)Lu3Hf tetrahedra, and  an edgeedge with one O(8)Lu3Hf tetrahedra. In the third O site, O(3) is bonded to one Lu(2), one Lu(4), one Hf(1), and one Hf(2) atom to form distorted OLu2Hf2 tetrahedra that share  a cornercorner with one O(6)Lu3Hf tetrahedra,  a cornercorner with one O(7)Lu3Hf tetrahedra, corners with two equivalent O(10)Lu2Hf2 tetrahedra, corners with two equivalent O(5)Lu2Hf2 tetrahedra, corners with two equivalent O(12)Lu3Hf tetrahedra, corners with two equivalent O(8)Lu3Hf tetrahedra,  an edgeedge with one O(2)Lu2Hf2 tetrahedra,  an edgeedge with one O(6)Lu3Hf tetrahedra, and  an edgeedge with one O(9)Lu3Hf tetrahedra. In the fourth O site, O(4) is bonded in a 3-coordinate geometry to one Lu(2), one Hf(1), and one Hf(3) atom. In the fifth O site, O(5) is bonded to one Lu(1), one Lu(3), and two equivalent Hf(2) atoms to form OLu2Hf2 tetrahedra that share  a cornercorner with one O(12)Lu3Hf tetrahedra, corners with two equivalent O(2)Lu2Hf2 tetrahedra, corners with two equivalent O(3)Lu2Hf2 tetrahedra, corners with two equivalent O(8)Lu3Hf tetrahedra, corners with two equivalent O(9)Lu3Hf tetrahedra, corners with three equivalent O(7)Lu3Hf tetrahedra,  an edgeedge with one O(2)Lu2Hf2 tetrahedra,  an edgeedge with one O(5)Lu2Hf2 tetrahedra,  an edgeedge with one O(7)Lu3Hf tetrahedra, and  an edgeedge with one O(9)Lu3Hf tetrahedra. In the sixth O site, O(6) is bonded to one Lu(4), two equivalent Lu(2), and one Hf(3) atom to form distorted OLu3Hf tetrahedra that share  a cornercorner with one O(3)Lu2Hf2 tetrahedra,  a cornercorner with one O(8)Lu3Hf tetrahedra,  a cornercorner with one O(9)Lu3Hf tetrahedra, corners with two equivalent O(12)Lu3Hf tetrahedra, corners with three equivalent O(10)Lu2Hf2 tetrahedra,  an edgeedge with one O(10)Lu2Hf2 tetrahedra,  an edgeedge with one O(3)Lu2Hf2 tetrahedra,  an edgeedge with one O(12)Lu3Hf tetrahedra, and  an edgeedge with one O(6)Lu3Hf tetrahedra. In the seventh O site, O(7) is bonded to one Lu(1), two equivalent Lu(3), and one Hf(2) atom to form OLu3Hf tetrahedra that share  a cornercorner with one O(3)Lu2Hf2 tetrahedra,  a cornercorner with one O(12)Lu3Hf tetrahedra,  a cornercorner with one O(8)Lu3Hf tetrahedra, corners with two equivalent O(9)Lu3Hf tetrahedra, corners with three equivalent O(2)Lu2Hf2 tetrahedra, corners with three equivalent O(5)Lu2Hf2 tetrahedra,  an edgeedge with one O(2)Lu2Hf2 tetrahedra,  an edgeedge with one O(5)Lu2Hf2 tetrahedra,  an edgeedge with one O(7)Lu3Hf tetrahedra,  an edgeedge with one O(8)Lu3Hf tetrahedra, and  an edgeedge with one O(9)Lu3Hf tetrahedra. In the eighth O site, O(8) is bonded to one Lu(1), one Lu(3), one Lu(4), and one Hf(1) atom to form distorted OLu3Hf tetrahedra that share  a cornercorner with one O(10)Lu2Hf2 tetrahedra,  a cornercorner with one O(2)Lu2Hf2 tetrahedra,  a cornercorner with one O(6)Lu3Hf tetrahedra,  a cornercorner with one O(7)Lu3Hf tetrahedra,  a cornercorner with one O(9)Lu3Hf tetrahedra, corners with two equivalent O(3)Lu2Hf2 tetrahedra, corners with two equivalent O(5)Lu2Hf2 tetrahedra, corners with two equivalent O(12)Lu3Hf tetrahedra,  an edgeedge with one O(2)Lu2Hf2 tetrahedra,  an edgeedge with one O(7)Lu3Hf tetrahedra, and  an edgeedge with one O(9)Lu3Hf tetrahedra. In the ninth O site, O(9) is bonded to one Lu(1), one Lu(3), one Lu(4), and one Hf(2) atom to form distorted OLu3Hf tetrahedra that share  a cornercorner with one O(10)Lu2Hf2 tetrahedra,  a cornercorner with one O(6)Lu3Hf tetrahedra,  a cornercorner with one O(8)Lu3Hf tetrahedra, corners with two equivalent O(5)Lu2Hf2 tetrahedra, corners with two equivalent O(7)Lu3Hf tetrahedra, corners with three equivalent O(2)Lu2Hf2 tetrahedra,  an edgeedge with one O(3)Lu2Hf2 tetrahedra,  an edgeedge with one O(5)Lu2Hf2 tetrahedra,  an edgeedge with one O(12)Lu3Hf tetrahedra,  an edgeedge with one O(7)Lu3Hf tetrahedra, and  an edgeedge with one O(8)Lu3Hf tetrahedra. In the tenth O site, O(10) is bonded to one Lu(2), one Lu(4), and two equivalent Hf(3) atoms to form OLu2Hf2 tetrahedra that share  a cornercorner with one O(8)Lu3Hf tetrahedra,  a cornercorner with one O(9)Lu3Hf tetrahedra, corners with two equivalent O(3)Lu2Hf2 tetrahedra, corners with two equivalent O(12)Lu3Hf tetrahedra, corners with three equivalent O(6)Lu3Hf tetrahedra,  an edgeedge with one O(10)Lu2Hf2 tetrahedra,  an edgeedge with one O(12)Lu3Hf tetrahedra, and  an edgeedge with one O(6)Lu3Hf tetrahedra. In the eleventh O site, O(11) is bonded in a 4-coordinate geometry to one Lu(1), one Lu(4), one Hf(1), and one Hf(3) atom. In the twelfth O site, O(12) is bonded to one Lu(1), one Lu(2), one Lu(4), and one Hf(3) atom to form OLu3Hf tetrahedra that share  a cornercorner with one O(5)Lu2Hf2 tetrahedra,  a cornercorner with one O(7)Lu3Hf tetrahedra, corners with two equivalent O(10)Lu2Hf2 tetrahedra, corners with two equivalent O(3)Lu2Hf2 tetrahedra, corners with two equivalent O(6)Lu3Hf tetrahedra, corners with two equivalent O(8)Lu3Hf tetrahedra,  an edgeedge with one O(10)Lu2Hf2 tetrahedra,  an edgeedge with one O(6)Lu3Hf tetrahedra, and  an edgeedge with one O(9)Lu3Hf tetrahedra.
Lu4Hf3O12 crystallizes in the triclinic P-1 space group. There are four inequivalent Lu sites. In the first Lu site, Lu(1) is bonded to one O(1), one O(11), one O(12), one O(5), one O(7), one O(8), and one O(9) atom to form distorted LuO7 pentagonal bipyramids that share  a cornercorner with one Hf(1)O6 octahedra and  an edgeedge with one Hf(1)O6 octahedra. The corner-sharing octahedral tilt angles are 52°. The Lu(1)-O(1) bond length is 2.25 Å. The Lu(1)-O(11) bond length is 2.25 Å. The Lu(1)-O(12) bond length is 2.21 Å. The Lu(1)-O(5) bond length is 2.33 Å. The Lu(1)-O(7) bond length is 2.12 Å. The Lu(1)-O(8) bond length is 2.45 Å. The Lu(1)-O(9) bond length is 2.24 Å. In the second Lu site, Lu(2) is bonded in a 7-coordinate geometry to one O(1), one O(10), one O(12), one O(3), one O(4), and two equivalent O(6) atoms. The Lu(2)-O(1) bond length is 2.60 Å. The Lu(2)-O(10) bond length is 2.32 Å. The Lu(2)-O(12) bond length is 2.13 Å. The Lu(2)-O(3) bond length is 2.17 Å. The Lu(2)-O(4) bond length is 2.22 Å. There is one shorter (2.22 Å) and one longer (2.25 Å) Lu(2)-O(6) bond length. In the third Lu site, Lu(3) is bonded in a 7-coordinate geometry to one O(5), one O(8), one O(9), two equivalent O(2), and two equivalent O(7) atoms. The Lu(3)-O(5) bond length is 2.15 Å. The Lu(3)-O(8) bond length is 2.20 Å. The Lu(3)-O(9) bond length is 2.32 Å. There is one shorter (2.21 Å) and one longer (2.58 Å) Lu(3)-O(2) bond length. There is one shorter (2.20 Å) and one longer (2.21 Å) Lu(3)-O(7) bond length. In the fourth Lu site, Lu(4) is bonded in a 7-coordinate geometry to one O(10), one O(11), one O(12), one O(3), one O(6), one O(8), and one O(9) atom. The Lu(4)-O(10) bond length is 2.16 Å. The Lu(4)-O(11) bond length is 2.56 Å. The Lu(4)-O(12) bond length is 2.22 Å. The Lu(4)-O(3) bond length is 2.19 Å. The Lu(4)-O(6) bond length is 2.33 Å. The Lu(4)-O(8) bond length is 2.18 Å. The Lu(4)-O(9) bond length is 2.22 Å. There are three inequivalent Hf sites. In the first Hf site, Hf(1) is bonded to one O(1), one O(11), one O(2), one O(3), one O(4), and one O(8) atom to form HfO6 octahedra that share  a cornercorner with one Lu(1)O7 pentagonal bipyramid and  an edgeedge with one Lu(1)O7 pentagonal bipyramid. The Hf(1)-O(1) bond length is 2.07 Å. The Hf(1)-O(11) bond length is 2.10 Å. The Hf(1)-O(2) bond length is 2.09 Å. The Hf(1)-O(3) bond length is 2.05 Å. The Hf(1)-O(4) bond length is 2.06 Å. The Hf(1)-O(8) bond length is 2.06 Å. In the second Hf site, Hf(2) is bonded in a 7-coordinate geometry to one O(1), one O(2), one O(3), one O(7), one O(9), and two equivalent O(5) atoms. The Hf(2)-O(1) bond length is 2.11 Å. The Hf(2)-O(2) bond length is 2.09 Å. The Hf(2)-O(3) bond length is 2.49 Å. The Hf(2)-O(7) bond length is 2.17 Å. The Hf(2)-O(9) bond length is 2.05 Å. There is one shorter (2.12 Å) and one longer (2.18 Å) Hf(2)-O(5) bond length. In the third Hf site, Hf(3) is bonded in a 6-coordinate geometry to one O(11), one O(12), one O(4), one O(6), and two equivalent O(10) atoms. The Hf(3)-O(11) bond length is 2.08 Å. The Hf(3)-O(12) bond length is 2.15 Å. The Hf(3)-O(4) bond length is 2.09 Å. The Hf(3)-O(6) bond length is 2.04 Å. There is one shorter (2.08 Å) and one longer (2.17 Å) Hf(3)-O(10) bond length. There are twelve inequivalent O sites. In the first O site, O(1) is bonded in a distorted rectangular see-saw-like geometry to one Lu(1), one Lu(2), one Hf(1), and one Hf(2) atom. In the second O site, O(2) is bonded to two equivalent Lu(3), one Hf(1), and one Hf(2) atom to form distorted OLu2Hf2 tetrahedra that share  a cornercorner with one O(8)Lu3Hf tetrahedra, corners with two equivalent O(5)Lu2Hf2 tetrahedra, corners with three equivalent O(7)Lu3Hf tetrahedra, corners with three equivalent O(9)Lu3Hf tetrahedra,  an edgeedge with one O(2)Lu2Hf2 tetrahedra,  an edgeedge with one O(3)Lu2Hf2 tetrahedra,  an edgeedge with one O(5)Lu2Hf2 tetrahedra,  an edgeedge with one O(7)Lu3Hf tetrahedra, and  an edgeedge with one O(8)Lu3Hf tetrahedra. In the third O site, O(3) is bonded to one Lu(2), one Lu(4), one Hf(1), and one Hf(2) atom to form distorted OLu2Hf2 tetrahedra that share  a cornercorner with one O(6)Lu3Hf tetrahedra,  a cornercorner with one O(7)Lu3Hf tetrahedra, corners with two equivalent O(10)Lu2Hf2 tetrahedra, corners with two equivalent O(5)Lu2Hf2 tetrahedra, corners with two equivalent O(12)Lu3Hf tetrahedra, corners with two equivalent O(8)Lu3Hf tetrahedra,  an edgeedge with one O(2)Lu2Hf2 tetrahedra,  an edgeedge with one O(6)Lu3Hf tetrahedra, and  an edgeedge with one O(9)Lu3Hf tetrahedra. In the fourth O site, O(4) is bonded in a 3-coordinate geometry to one Lu(2), one Hf(1), and one Hf(3) atom. In the fifth O site, O(5) is bonded to one Lu(1), one Lu(3), and two equivalent Hf(2) atoms to form OLu2Hf2 tetrahedra that share  a cornercorner with one O(12)Lu3Hf tetrahedra, corners with two equivalent O(2)Lu2Hf2 tetrahedra, corners with two equivalent O(3)Lu2Hf2 tetrahedra, corners with two equivalent O(8)Lu3Hf tetrahedra, corners with two equivalent O(9)Lu3Hf tetrahedra, corners with three equivalent O(7)Lu3Hf tetrahedra,  an edgeedge with one O(2)Lu2Hf2 tetrahedra,  an edgeedge with one O(5)Lu2Hf2 tetrahedra,  an edgeedge with one O(7)Lu3Hf tetrahedra, and  an edgeedge with one O(9)Lu3Hf tetrahedra. In the sixth O site, O(6) is bonded to one Lu(4), two equivalent Lu(2), and one Hf(3) atom to form distorted OLu3Hf tetrahedra that share  a cornercorner with one O(3)Lu2Hf2 tetrahedra,  a cornercorner with one O(8)Lu3Hf tetrahedra,  a cornercorner with one O(9)Lu3Hf tetrahedra, corners with two equivalent O(12)Lu3Hf tetrahedra, corners with three equivalent O(10)Lu2Hf2 tetrahedra,  an edgeedge with one O(10)Lu2Hf2 tetrahedra,  an edgeedge with one O(3)Lu2Hf2 tetrahedra,  an edgeedge with one O(12)Lu3Hf tetrahedra, and  an edgeedge with one O(6)Lu3Hf tetrahedra. In the seventh O site, O(7) is bonded to one Lu(1), two equivalent Lu(3), and one Hf(2) atom to form OLu3Hf tetrahedra that share  a cornercorner with one O(3)Lu2Hf2 tetrahedra,  a cornercorner with one O(12)Lu3Hf tetrahedra,  a cornercorner with one O(8)Lu3Hf tetrahedra, corners with two equivalent O(9)Lu3Hf tetrahedra, corners with three equivalent O(2)Lu2Hf2 tetrahedra, corners with three equivalent O(5)Lu2Hf2 tetrahedra,  an edgeedge with one O(2)Lu2Hf2 tetrahedra,  an edgeedge with one O(5)Lu2Hf2 tetrahedra,  an edgeedge with one O(7)Lu3Hf tetrahedra,  an edgeedge with one O(8)Lu3Hf tetrahedra, and  an edgeedge with one O(9)Lu3Hf tetrahedra. In the eighth O site, O(8) is bonded to one Lu(1), one Lu(3), one Lu(4), and one Hf(1) atom to form distorted OLu3Hf tetrahedra that share  a cornercorner with one O(10)Lu2Hf2 tetrahedra,  a cornercorner with one O(2)Lu2Hf2 tetrahedra,  a cornercorner with one O(6)Lu3Hf tetrahedra,  a cornercorner with one O(7)Lu3Hf tetrahedra,  a cornercorner with one O(9)Lu3Hf tetrahedra, corners with two equivalent O(3)Lu2Hf2 tetrahedra, corners with two equivalent O(5)Lu2Hf2 tetrahedra, corners with two equivalent O(12)Lu3Hf tetrahedra,  an edgeedge with one O(2)Lu2Hf2 tetrahedra,  an edgeedge with one O(7)Lu3Hf tetrahedra, and  an edgeedge with one O(9)Lu3Hf tetrahedra. In the ninth O site, O(9) is bonded to one Lu(1), one Lu(3), one Lu(4), and one Hf(2) atom to form distorted OLu3Hf tetrahedra that share  a cornercorner with one O(10)Lu2Hf2 tetrahedra,  a cornercorner with one O(6)Lu3Hf tetrahedra,  a cornercorner with one O(8)Lu3Hf tetrahedra, corners with two equivalent O(5)Lu2Hf2 tetrahedra, corners with two equivalent O(7)Lu3Hf tetrahedra, corners with three equivalent O(2)Lu2Hf2 tetrahedra,  an edgeedge with one O(3)Lu2Hf2 tetrahedra,  an edgeedge with one O(5)Lu2Hf2 tetrahedra,  an edgeedge with one O(12)Lu3Hf tetrahedra,  an edgeedge with one O(7)Lu3Hf tetrahedra, and  an edgeedge with one O(8)Lu3Hf tetrahedra. In the tenth O site, O(10) is bonded to one Lu(2), one Lu(4), and two equivalent Hf(3) atoms to form OLu2Hf2 tetrahedra that share  a cornercorner with one O(8)Lu3Hf tetrahedra,  a cornercorner with one O(9)Lu3Hf tetrahedra, corners with two equivalent O(3)Lu2Hf2 tetrahedra, corners with two equivalent O(12)Lu3Hf tetrahedra, corners with three equivalent O(6)Lu3Hf tetrahedra,  an edgeedge with one O(10)Lu2Hf2 tetrahedra,  an edgeedge with one O(12)Lu3Hf tetrahedra, and  an edgeedge with one O(6)Lu3Hf tetrahedra. In the eleventh O site, O(11) is bonded in a 4-coordinate geometry to one Lu(1), one Lu(4), one Hf(1), and one Hf(3) atom. In the twelfth O site, O(12) is bonded to one Lu(1), one Lu(2), one Lu(4), and one Hf(3) atom to form OLu3Hf tetrahedra that share  a cornercorner with one O(5)Lu2Hf2 tetrahedra,  a cornercorner with one O(7)Lu3Hf tetrahedra, corners with two equivalent O(10)Lu2Hf2 tetrahedra, corners with two equivalent O(3)Lu2Hf2 tetrahedra, corners with two equivalent O(6)Lu3Hf tetrahedra, corners with two equivalent O(8)Lu3Hf tetrahedra,  an edgeedge with one O(10)Lu2Hf2 tetrahedra,  an edgeedge with one O(6)Lu3Hf tetrahedra, and  an edgeedge with one O(9)Lu3Hf tetrahedra.