MgLa8Hf4O20 crystallizes in the monoclinic Pm space group. Mg(1) is bonded in a 4-coordinate geometry to one O(16), one O(2), and two equivalent O(17) atoms. There are eight inequivalent La sites. In the first La site, La(1) is bonded in a 7-coordinate geometry to one O(20), one O(4), one O(6), two equivalent O(19), and two equivalent O(3) atoms. In the second La site, La(2) is bonded in a 7-coordinate geometry to one O(20), one O(6), one O(8), two equivalent O(1), and two equivalent O(7) atoms. In the third La site, La(3) is bonded in a 7-coordinate geometry to one O(11), one O(3), one O(5), two equivalent O(10), and two equivalent O(4) atoms. In the fourth La site, La(4) is bonded to one O(11), one O(5), one O(7), two equivalent O(12), and two equivalent O(8) atoms to form distorted LaO7 hexagonal pyramids that share  a cornercorner with one Hf(2)O5 trigonal bipyramid, corners with two equivalent Hf(3)O5 trigonal bipyramids, edges with two equivalent La(4)O7 hexagonal pyramids, and edges with two equivalent Hf(1)O5 trigonal bipyramids. In the fifth La site, La(5) is bonded to one O(10), one O(14), one O(16), two equivalent O(13), and two equivalent O(9) atoms to form distorted LaO7 hexagonal pyramids that share  a cornercorner with one Hf(3)O5 trigonal bipyramid, corners with two equivalent Hf(2)O5 trigonal bipyramids, edges with two equivalent La(5)O7 hexagonal pyramids, and edges with two equivalent Hf(4)O5 trigonal bipyramids. In the sixth La site, La(6) is bonded in a 7-coordinate geometry to one O(10), one O(16), one O(18), two equivalent O(11), and two equivalent O(17) atoms. In the seventh La site, La(7) is bonded in a body-centered cubic geometry to one O(1), one O(13), one O(15), one O(19), two equivalent O(14), and two equivalent O(20) atoms. In the eighth La site, La(8) is bonded to one O(1), one O(15), one O(17), two equivalent O(18), and two equivalent O(2) atoms to form distorted LaO7 hexagonal pyramids that share  a cornercorner with one Hf(4)O5 trigonal bipyramid, corners with two equivalent Hf(1)O5 trigonal bipyramids, edges with two equivalent La(8)O7 hexagonal pyramids, and edges with two equivalent Hf(3)O5 trigonal bipyramids. There are four inequivalent Hf sites. In the first Hf site, Hf(1) is bonded to one O(2), one O(4), one O(8), and two equivalent O(5) atoms to form distorted HfO5 trigonal bipyramids that share corners with two equivalent La(8)O7 hexagonal pyramids, corners with two equivalent Hf(1)O5 trigonal bipyramids, and edges with two equivalent La(4)O7 hexagonal pyramids. In the second Hf site, Hf(2) is bonded to one O(3), one O(7), one O(9), and two equivalent O(6) atoms to form distorted HfO5 trigonal bipyramids that share  a cornercorner with one La(4)O7 hexagonal pyramid, corners with two equivalent La(5)O7 hexagonal pyramids, and corners with two equivalent Hf(2)O5 trigonal bipyramids. In the third Hf site, Hf(3) is bonded to one O(12), one O(14), one O(18), and two equivalent O(15) atoms to form distorted HfO5 trigonal bipyramids that share  a cornercorner with one La(5)O7 hexagonal pyramid, corners with two equivalent La(4)O7 hexagonal pyramids, corners with two equivalent Hf(3)O5 trigonal bipyramids, and edges with two equivalent La(8)O7 hexagonal pyramids. In the fourth Hf site, Hf(4) is bonded to one O(13), one O(17), one O(19), and two equivalent O(16) atoms to form distorted HfO5 trigonal bipyramids that share  a cornercorner with one La(8)O7 hexagonal pyramid, corners with two equivalent Hf(4)O5 trigonal bipyramids, and edges with two equivalent La(5)O7 hexagonal pyramids. There are twenty inequivalent O sites. In the first O site, O(1) is bonded to one La(7), one La(8), and two equivalent La(2) atoms to form OLa4 tetrahedra that share  a cornercorner with one O(13)La3Hf tetrahedra,  a cornercorner with one O(19)La3Hf tetrahedra, corners with two equivalent O(6)La2Hf2 tetrahedra, corners with two equivalent O(14)La3Hf tetrahedra, corners with two equivalent O(18)La3Hf tetrahedra, corners with two equivalent O(7)La3Hf tetrahedra, corners with two equivalent O(8)La3Hf tetrahedra, corners with two equivalent O(1)La4 tetrahedra, corners with two equivalent O(2)La2HfMg trigonal pyramids,  an edgeedge with one O(7)La3Hf tetrahedra, and edges with two equivalent O(20)La4 tetrahedra. In the second O site, O(2) is bonded to one Mg(1), two equivalent La(8), and one Hf(1) atom to form distorted OLa2HfMg trigonal pyramids that share  a cornercorner with one O(4)La3Hf tetrahedra,  a cornercorner with one O(8)La3Hf tetrahedra, corners with two equivalent O(5)La2Hf2 tetrahedra, corners with two equivalent O(18)La3Hf tetrahedra, corners with two equivalent O(1)La4 tetrahedra,  a cornercorner with one O(16)La2Hf2Mg trigonal bipyramid, corners with two equivalent O(2)La2HfMg trigonal pyramids, and  an edgeedge with one O(18)La3Hf tetrahedra. In the third O site, O(3) is bonded to one La(3), two equivalent La(1), and one Hf(2) atom to form OLa3Hf tetrahedra that share  a cornercorner with one O(5)La2Hf2 tetrahedra,  a cornercorner with one O(7)La3Hf tetrahedra,  a cornercorner with one O(11)La4 tetrahedra, corners with two equivalent O(19)La3Hf tetrahedra, corners with two equivalent O(3)La3Hf tetrahedra, corners with two equivalent O(10)La4 tetrahedra, corners with two equivalent O(20)La4 tetrahedra,  an edgeedge with one O(19)La3Hf tetrahedra, edges with two equivalent O(6)La2Hf2 tetrahedra, and edges with two equivalent O(4)La3Hf tetrahedra. In the fourth O site, O(4) is bonded to one La(1), two equivalent La(3), and one Hf(1) atom to form OLa3Hf tetrahedra that share  a cornercorner with one O(6)La2Hf2 tetrahedra,  a cornercorner with one O(8)La3Hf tetrahedra,  a cornercorner with one O(20)La4 tetrahedra, corners with two equivalent O(19)La3Hf tetrahedra, corners with two equivalent O(4)La3Hf tetrahedra, corners with two equivalent O(10)La4 tetrahedra, corners with two equivalent O(11)La4 tetrahedra,  a cornercorner with one O(2)La2HfMg trigonal pyramid,  an edgeedge with one O(10)La4 tetrahedra, edges with two equivalent O(5)La2Hf2 tetrahedra, and edges with two equivalent O(3)La3Hf tetrahedra. In the fifth O site, O(5) is bonded to one La(3), one La(4), and two equivalent Hf(1) atoms to form distorted OLa2Hf2 tetrahedra that share  a cornercorner with one O(3)La3Hf tetrahedra,  a cornercorner with one O(7)La3Hf tetrahedra, corners with two equivalent O(5)La2Hf2 tetrahedra, corners with two equivalent O(10)La4 tetrahedra, corners with two equivalent O(2)La2HfMg trigonal pyramids,  an edgeedge with one O(11)La4 tetrahedra, edges with two equivalent O(4)La3Hf tetrahedra, and edges with two equivalent O(8)La3Hf tetrahedra. In the sixth O site, O(6) is bonded to one La(1), one La(2), and two equivalent Hf(2) atoms to form distorted OLa2Hf2 tetrahedra that share  a cornercorner with one O(4)La3Hf tetrahedra,  a cornercorner with one O(8)La3Hf tetrahedra, corners with two equivalent O(6)La2Hf2 tetrahedra, corners with two equivalent O(19)La3Hf tetrahedra, corners with two equivalent O(1)La4 tetrahedra,  an edgeedge with one O(20)La4 tetrahedra, edges with two equivalent O(3)La3Hf tetrahedra, and edges with two equivalent O(7)La3Hf tetrahedra. In the seventh O site, O(7) is bonded to one La(4), two equivalent La(2), and one Hf(2) atom to form distorted OLa3Hf tetrahedra that share  a cornercorner with one O(5)La2Hf2 tetrahedra,  a cornercorner with one O(3)La3Hf tetrahedra,  a cornercorner with one O(11)La4 tetrahedra, corners with two equivalent O(7)La3Hf tetrahedra, corners with two equivalent O(1)La4 tetrahedra, corners with two equivalent O(20)La4 tetrahedra,  an edgeedge with one O(1)La4 tetrahedra, edges with two equivalent O(6)La2Hf2 tetrahedra, and edges with two equivalent O(8)La3Hf tetrahedra. In the eighth O site, O(8) is bonded to one La(2), two equivalent La(4), and one Hf(1) atom to form OLa3Hf tetrahedra that share  a cornercorner with one O(6)La2Hf2 tetrahedra,  a cornercorner with one O(4)La3Hf tetrahedra,  a cornercorner with one O(20)La4 tetrahedra, corners with two equivalent O(8)La3Hf tetrahedra, corners with two equivalent O(1)La4 tetrahedra, corners with two equivalent O(11)La4 tetrahedra,  a cornercorner with one O(2)La2HfMg trigonal pyramid, edges with two equivalent O(5)La2Hf2 tetrahedra, and edges with two equivalent O(7)La3Hf tetrahedra. In the ninth O site, O(9) is bonded in a distorted trigonal planar geometry to two equivalent La(5) and one Hf(2) atom. In the tenth O site, O(10) is bonded to one La(5), one La(6), and two equivalent La(3) atoms to form OLa4 tetrahedra that share  a cornercorner with one O(14)La3Hf tetrahedra,  a cornercorner with one O(18)La3Hf tetrahedra, corners with two equivalent O(5)La2Hf2 tetrahedra, corners with two equivalent O(13)La3Hf tetrahedra, corners with two equivalent O(3)La3Hf tetrahedra, corners with two equivalent O(4)La3Hf tetrahedra, corners with two equivalent O(10)La4 tetrahedra,  an edgeedge with one O(4)La3Hf tetrahedra, edges with two equivalent O(11)La4 tetrahedra, and  an edgeedge with one O(16)La2Hf2Mg trigonal bipyramid. In the eleventh O site, O(11) is bonded to one La(3), one La(4), and two equivalent La(6) atoms to form OLa4 tetrahedra that share  a cornercorner with one O(3)La3Hf tetrahedra,  a cornercorner with one O(7)La3Hf tetrahedra, corners with two equivalent O(18)La3Hf tetrahedra, corners with two equivalent O(4)La3Hf tetrahedra, corners with two equivalent O(8)La3Hf tetrahedra, corners with two equivalent O(11)La4 tetrahedra, corners with two equivalent O(16)La2Hf2Mg trigonal bipyramids,  an edgeedge with one O(5)La2Hf2 tetrahedra, and edges with two equivalent O(10)La4 tetrahedra. In the twelfth O site, O(12) is bonded in a distorted trigonal planar geometry to two equivalent La(4) and one Hf(3) atom. In the thirteenth O site, O(13) is bonded to one La(7), two equivalent La(5), and one Hf(4) atom to form OLa3Hf tetrahedra that share  a cornercorner with one O(1)La4 tetrahedra, corners with two equivalent O(13)La3Hf tetrahedra, corners with two equivalent O(10)La4 tetrahedra, corners with two equivalent O(20)La4 tetrahedra,  an edgeedge with one O(19)La3Hf tetrahedra, edges with two equivalent O(14)La3Hf tetrahedra, and edges with two equivalent O(16)La2Hf2Mg trigonal bipyramids. In the fourteenth O site, O(14) is bonded to one La(5), two equivalent La(7), and one Hf(3) atom to form distorted OLa3Hf tetrahedra that share  a cornercorner with one O(18)La3Hf tetrahedra,  a cornercorner with one O(10)La4 tetrahedra, corners with two equivalent O(14)La3Hf tetrahedra, corners with two equivalent O(19)La3Hf tetrahedra, corners with two equivalent O(1)La4 tetrahedra, corners with two equivalent O(20)La4 tetrahedra,  a cornercorner with one O(16)La2Hf2Mg trigonal bipyramid,  an edgeedge with one O(20)La4 tetrahedra, and edges with two equivalent O(13)La3Hf tetrahedra. In the fifteenth O site, O(15) is bonded in a 4-coordinate geometry to one La(7), one La(8), and two equivalent Hf(3) atoms. In the sixteenth O site, O(16) is bonded to one Mg(1), one La(5), one La(6), and two equivalent Hf(4) atoms to form distorted OLa2Hf2Mg trigonal bipyramids that share  a cornercorner with one O(14)La3Hf tetrahedra,  a cornercorner with one O(18)La3Hf tetrahedra, corners with two equivalent O(19)La3Hf tetrahedra, corners with two equivalent O(11)La4 tetrahedra, corners with two equivalent O(16)La2Hf2Mg trigonal bipyramids,  a cornercorner with one O(2)La2HfMg trigonal pyramid,  an edgeedge with one O(10)La4 tetrahedra, and edges with two equivalent O(13)La3Hf tetrahedra. In the seventeenth O site, O(17) is bonded in a 6-coordinate geometry to two equivalent Mg(1), one La(8), two equivalent La(6), and one Hf(4) atom. In the eighteenth O site, O(18) is bonded to one La(6), two equivalent La(8), and one Hf(3) atom to form OLa3Hf tetrahedra that share  a cornercorner with one O(14)La3Hf tetrahedra,  a cornercorner with one O(10)La4 tetrahedra, corners with two equivalent O(18)La3Hf tetrahedra, corners with two equivalent O(1)La4 tetrahedra, corners with two equivalent O(11)La4 tetrahedra,  a cornercorner with one O(16)La2Hf2Mg trigonal bipyramid, corners with two equivalent O(2)La2HfMg trigonal pyramids, and  an edgeedge with one O(2)La2HfMg trigonal pyramid. In the nineteenth O site, O(19) is bonded to one La(7), two equivalent La(1), and one Hf(4) atom to form OLa3Hf tetrahedra that share  a cornercorner with one O(1)La4 tetrahedra, corners with two equivalent O(6)La2Hf2 tetrahedra, corners with two equivalent O(14)La3Hf tetrahedra, corners with two equivalent O(19)La3Hf tetrahedra, corners with two equivalent O(3)La3Hf tetrahedra, corners with two equivalent O(4)La3Hf tetrahedra, corners with two equivalent O(16)La2Hf2Mg trigonal bipyramids,  an edgeedge with one O(13)La3Hf tetrahedra,  an edgeedge with one O(3)La3Hf tetrahedra, and edges with two equivalent O(20)La4 tetrahedra. In the twentieth O site, O(20) is bonded to one La(1), one La(2), and two equivalent La(7) atoms to form OLa4 tetrahedra that share  a cornercorner with one O(4)La3Hf tetrahedra,  a cornercorner with one O(8)La3Hf tetrahedra, corners with two equivalent O(13)La3Hf tetrahedra, corners with two equivalent O(14)La3Hf tetrahedra, corners with two equivalent O(3)La3Hf tetrahedra, corners with two equivalent O(7)La3Hf tetrahedra, corners with two equivalent O(20)La4 tetrahedra,  an edgeedge with one O(6)La2Hf2 tetrahedra,  an edgeedge with one O(14)La3Hf tetrahedra, edges with two equivalent O(19)La3Hf tetrahedra, and edges with two equivalent O(1)La4 tetrahedra.
MgLa8Hf4O20 crystallizes in the monoclinic Pm space group. Mg(1) is bonded in a 4-coordinate geometry to one O(16), one O(2), and two equivalent O(17) atoms. The Mg(1)-O(16) bond length is 2.42 Å. The Mg(1)-O(2) bond length is 2.02 Å. Both Mg(1)-O(17) bond lengths are 2.28 Å. There are eight inequivalent La sites. In the first La site, La(1) is bonded in a 7-coordinate geometry to one O(20), one O(4), one O(6), two equivalent O(19), and two equivalent O(3) atoms. The La(1)-O(20) bond length is 2.51 Å. The La(1)-O(4) bond length is 2.67 Å. The La(1)-O(6) bond length is 2.68 Å. Both La(1)-O(19) bond lengths are 2.49 Å. Both La(1)-O(3) bond lengths are 2.46 Å. In the second La site, La(2) is bonded in a 7-coordinate geometry to one O(20), one O(6), one O(8), two equivalent O(1), and two equivalent O(7) atoms. The La(2)-O(20) bond length is 2.40 Å. The La(2)-O(6) bond length is 2.55 Å. The La(2)-O(8) bond length is 2.58 Å. Both La(2)-O(1) bond lengths are 2.47 Å. Both La(2)-O(7) bond lengths are 2.54 Å. In the third La site, La(3) is bonded in a 7-coordinate geometry to one O(11), one O(3), one O(5), two equivalent O(10), and two equivalent O(4) atoms. The La(3)-O(11) bond length is 2.62 Å. The La(3)-O(3) bond length is 2.48 Å. The La(3)-O(5) bond length is 2.61 Å. Both La(3)-O(10) bond lengths are 2.52 Å. Both La(3)-O(4) bond lengths are 2.42 Å. In the fourth La site, La(4) is bonded to one O(11), one O(5), one O(7), two equivalent O(12), and two equivalent O(8) atoms to form distorted LaO7 hexagonal pyramids that share  a cornercorner with one Hf(2)O5 trigonal bipyramid, corners with two equivalent Hf(3)O5 trigonal bipyramids, edges with two equivalent La(4)O7 hexagonal pyramids, and edges with two equivalent Hf(1)O5 trigonal bipyramids. The La(4)-O(11) bond length is 2.66 Å. The La(4)-O(5) bond length is 2.62 Å. The La(4)-O(7) bond length is 2.44 Å. Both La(4)-O(12) bond lengths are 2.52 Å. Both La(4)-O(8) bond lengths are 2.42 Å. In the fifth La site, La(5) is bonded to one O(10), one O(14), one O(16), two equivalent O(13), and two equivalent O(9) atoms to form distorted LaO7 hexagonal pyramids that share  a cornercorner with one Hf(3)O5 trigonal bipyramid, corners with two equivalent Hf(2)O5 trigonal bipyramids, edges with two equivalent La(5)O7 hexagonal pyramids, and edges with two equivalent Hf(4)O5 trigonal bipyramids. The La(5)-O(10) bond length is 2.49 Å. The La(5)-O(14) bond length is 2.51 Å. The La(5)-O(16) bond length is 2.68 Å. Both La(5)-O(13) bond lengths are 2.39 Å. Both La(5)-O(9) bond lengths are 2.51 Å. In the sixth La site, La(6) is bonded in a 7-coordinate geometry to one O(10), one O(16), one O(18), two equivalent O(11), and two equivalent O(17) atoms. The La(6)-O(10) bond length is 2.38 Å. The La(6)-O(16) bond length is 2.58 Å. The La(6)-O(18) bond length is 2.57 Å. Both La(6)-O(11) bond lengths are 2.38 Å. Both La(6)-O(17) bond lengths are 2.98 Å. In the seventh La site, La(7) is bonded in a body-centered cubic geometry to one O(1), one O(13), one O(15), one O(19), two equivalent O(14), and two equivalent O(20) atoms. The La(7)-O(1) bond length is 2.57 Å. The La(7)-O(13) bond length is 2.59 Å. The La(7)-O(15) bond length is 2.71 Å. The La(7)-O(19) bond length is 2.52 Å. Both La(7)-O(14) bond lengths are 2.60 Å. Both La(7)-O(20) bond lengths are 2.44 Å. In the eighth La site, La(8) is bonded to one O(1), one O(15), one O(17), two equivalent O(18), and two equivalent O(2) atoms to form distorted LaO7 hexagonal pyramids that share  a cornercorner with one Hf(4)O5 trigonal bipyramid, corners with two equivalent Hf(1)O5 trigonal bipyramids, edges with two equivalent La(8)O7 hexagonal pyramids, and edges with two equivalent Hf(3)O5 trigonal bipyramids. The La(8)-O(1) bond length is 2.32 Å. The La(8)-O(15) bond length is 2.47 Å. The La(8)-O(17) bond length is 2.50 Å. Both La(8)-O(18) bond lengths are 2.44 Å. Both La(8)-O(2) bond lengths are 2.57 Å. There are four inequivalent Hf sites. In the first Hf site, Hf(1) is bonded to one O(2), one O(4), one O(8), and two equivalent O(5) atoms to form distorted HfO5 trigonal bipyramids that share corners with two equivalent La(8)O7 hexagonal pyramids, corners with two equivalent Hf(1)O5 trigonal bipyramids, and edges with two equivalent La(4)O7 hexagonal pyramids. The Hf(1)-O(2) bond length is 2.04 Å. The Hf(1)-O(4) bond length is 2.04 Å. The Hf(1)-O(8) bond length is 2.04 Å. Both Hf(1)-O(5) bond lengths are 2.09 Å. In the second Hf site, Hf(2) is bonded to one O(3), one O(7), one O(9), and two equivalent O(6) atoms to form distorted HfO5 trigonal bipyramids that share  a cornercorner with one La(4)O7 hexagonal pyramid, corners with two equivalent La(5)O7 hexagonal pyramids, and corners with two equivalent Hf(2)O5 trigonal bipyramids. The Hf(2)-O(3) bond length is 2.03 Å. The Hf(2)-O(7) bond length is 2.06 Å. The Hf(2)-O(9) bond length is 1.94 Å. Both Hf(2)-O(6) bond lengths are 2.07 Å. In the third Hf site, Hf(3) is bonded to one O(12), one O(14), one O(18), and two equivalent O(15) atoms to form distorted HfO5 trigonal bipyramids that share  a cornercorner with one La(5)O7 hexagonal pyramid, corners with two equivalent La(4)O7 hexagonal pyramids, corners with two equivalent Hf(3)O5 trigonal bipyramids, and edges with two equivalent La(8)O7 hexagonal pyramids. The Hf(3)-O(12) bond length is 1.95 Å. The Hf(3)-O(14) bond length is 2.03 Å. The Hf(3)-O(18) bond length is 2.07 Å. Both Hf(3)-O(15) bond lengths are 2.08 Å. In the fourth Hf site, Hf(4) is bonded to one O(13), one O(17), one O(19), and two equivalent O(16) atoms to form distorted HfO5 trigonal bipyramids that share  a cornercorner with one La(8)O7 hexagonal pyramid, corners with two equivalent Hf(4)O5 trigonal bipyramids, and edges with two equivalent La(5)O7 hexagonal pyramids. The Hf(4)-O(13) bond length is 2.00 Å. The Hf(4)-O(17) bond length is 2.14 Å. The Hf(4)-O(19) bond length is 1.98 Å. Both Hf(4)-O(16) bond lengths are 2.15 Å. There are twenty inequivalent O sites. In the first O site, O(1) is bonded to one La(7), one La(8), and two equivalent La(2) atoms to form OLa4 tetrahedra that share  a cornercorner with one O(13)La3Hf tetrahedra,  a cornercorner with one O(19)La3Hf tetrahedra, corners with two equivalent O(6)La2Hf2 tetrahedra, corners with two equivalent O(14)La3Hf tetrahedra, corners with two equivalent O(18)La3Hf tetrahedra, corners with two equivalent O(7)La3Hf tetrahedra, corners with two equivalent O(8)La3Hf tetrahedra, corners with two equivalent O(1)La4 tetrahedra, corners with two equivalent O(2)La2HfMg trigonal pyramids,  an edgeedge with one O(7)La3Hf tetrahedra, and edges with two equivalent O(20)La4 tetrahedra. In the second O site, O(2) is bonded to one Mg(1), two equivalent La(8), and one Hf(1) atom to form distorted OLa2HfMg trigonal pyramids that share  a cornercorner with one O(4)La3Hf tetrahedra,  a cornercorner with one O(8)La3Hf tetrahedra, corners with two equivalent O(5)La2Hf2 tetrahedra, corners with two equivalent O(18)La3Hf tetrahedra, corners with two equivalent O(1)La4 tetrahedra,  a cornercorner with one O(16)La2Hf2Mg trigonal bipyramid, corners with two equivalent O(2)La2HfMg trigonal pyramids, and  an edgeedge with one O(18)La3Hf tetrahedra. In the third O site, O(3) is bonded to one La(3), two equivalent La(1), and one Hf(2) atom to form OLa3Hf tetrahedra that share  a cornercorner with one O(5)La2Hf2 tetrahedra,  a cornercorner with one O(7)La3Hf tetrahedra,  a cornercorner with one O(11)La4 tetrahedra, corners with two equivalent O(19)La3Hf tetrahedra, corners with two equivalent O(3)La3Hf tetrahedra, corners with two equivalent O(10)La4 tetrahedra, corners with two equivalent O(20)La4 tetrahedra,  an edgeedge with one O(19)La3Hf tetrahedra, edges with two equivalent O(6)La2Hf2 tetrahedra, and edges with two equivalent O(4)La3Hf tetrahedra. In the fourth O site, O(4) is bonded to one La(1), two equivalent La(3), and one Hf(1) atom to form OLa3Hf tetrahedra that share  a cornercorner with one O(6)La2Hf2 tetrahedra,  a cornercorner with one O(8)La3Hf tetrahedra,  a cornercorner with one O(20)La4 tetrahedra, corners with two equivalent O(19)La3Hf tetrahedra, corners with two equivalent O(4)La3Hf tetrahedra, corners with two equivalent O(10)La4 tetrahedra, corners with two equivalent O(11)La4 tetrahedra,  a cornercorner with one O(2)La2HfMg trigonal pyramid,  an edgeedge with one O(10)La4 tetrahedra, edges with two equivalent O(5)La2Hf2 tetrahedra, and edges with two equivalent O(3)La3Hf tetrahedra. In the fifth O site, O(5) is bonded to one La(3), one La(4), and two equivalent Hf(1) atoms to form distorted OLa2Hf2 tetrahedra that share  a cornercorner with one O(3)La3Hf tetrahedra,  a cornercorner with one O(7)La3Hf tetrahedra, corners with two equivalent O(5)La2Hf2 tetrahedra, corners with two equivalent O(10)La4 tetrahedra, corners with two equivalent O(2)La2HfMg trigonal pyramids,  an edgeedge with one O(11)La4 tetrahedra, edges with two equivalent O(4)La3Hf tetrahedra, and edges with two equivalent O(8)La3Hf tetrahedra. In the sixth O site, O(6) is bonded to one La(1), one La(2), and two equivalent Hf(2) atoms to form distorted OLa2Hf2 tetrahedra that share  a cornercorner with one O(4)La3Hf tetrahedra,  a cornercorner with one O(8)La3Hf tetrahedra, corners with two equivalent O(6)La2Hf2 tetrahedra, corners with two equivalent O(19)La3Hf tetrahedra, corners with two equivalent O(1)La4 tetrahedra,  an edgeedge with one O(20)La4 tetrahedra, edges with two equivalent O(3)La3Hf tetrahedra, and edges with two equivalent O(7)La3Hf tetrahedra. In the seventh O site, O(7) is bonded to one La(4), two equivalent La(2), and one Hf(2) atom to form distorted OLa3Hf tetrahedra that share  a cornercorner with one O(5)La2Hf2 tetrahedra,  a cornercorner with one O(3)La3Hf tetrahedra,  a cornercorner with one O(11)La4 tetrahedra, corners with two equivalent O(7)La3Hf tetrahedra, corners with two equivalent O(1)La4 tetrahedra, corners with two equivalent O(20)La4 tetrahedra,  an edgeedge with one O(1)La4 tetrahedra, edges with two equivalent O(6)La2Hf2 tetrahedra, and edges with two equivalent O(8)La3Hf tetrahedra. In the eighth O site, O(8) is bonded to one La(2), two equivalent La(4), and one Hf(1) atom to form OLa3Hf tetrahedra that share  a cornercorner with one O(6)La2Hf2 tetrahedra,  a cornercorner with one O(4)La3Hf tetrahedra,  a cornercorner with one O(20)La4 tetrahedra, corners with two equivalent O(8)La3Hf tetrahedra, corners with two equivalent O(1)La4 tetrahedra, corners with two equivalent O(11)La4 tetrahedra,  a cornercorner with one O(2)La2HfMg trigonal pyramid, edges with two equivalent O(5)La2Hf2 tetrahedra, and edges with two equivalent O(7)La3Hf tetrahedra. In the ninth O site, O(9) is bonded in a distorted trigonal planar geometry to two equivalent La(5) and one Hf(2) atom. In the tenth O site, O(10) is bonded to one La(5), one La(6), and two equivalent La(3) atoms to form OLa4 tetrahedra that share  a cornercorner with one O(14)La3Hf tetrahedra,  a cornercorner with one O(18)La3Hf tetrahedra, corners with two equivalent O(5)La2Hf2 tetrahedra, corners with two equivalent O(13)La3Hf tetrahedra, corners with two equivalent O(3)La3Hf tetrahedra, corners with two equivalent O(4)La3Hf tetrahedra, corners with two equivalent O(10)La4 tetrahedra,  an edgeedge with one O(4)La3Hf tetrahedra, edges with two equivalent O(11)La4 tetrahedra, and  an edgeedge with one O(16)La2Hf2Mg trigonal bipyramid. In the eleventh O site, O(11) is bonded to one La(3), one La(4), and two equivalent La(6) atoms to form OLa4 tetrahedra that share  a cornercorner with one O(3)La3Hf tetrahedra,  a cornercorner with one O(7)La3Hf tetrahedra, corners with two equivalent O(18)La3Hf tetrahedra, corners with two equivalent O(4)La3Hf tetrahedra, corners with two equivalent O(8)La3Hf tetrahedra, corners with two equivalent O(11)La4 tetrahedra, corners with two equivalent O(16)La2Hf2Mg trigonal bipyramids,  an edgeedge with one O(5)La2Hf2 tetrahedra, and edges with two equivalent O(10)La4 tetrahedra. In the twelfth O site, O(12) is bonded in a distorted trigonal planar geometry to two equivalent La(4) and one Hf(3) atom. In the thirteenth O site, O(13) is bonded to one La(7), two equivalent La(5), and one Hf(4) atom to form OLa3Hf tetrahedra that share  a cornercorner with one O(1)La4 tetrahedra, corners with two equivalent O(13)La3Hf tetrahedra, corners with two equivalent O(10)La4 tetrahedra, corners with two equivalent O(20)La4 tetrahedra,  an edgeedge with one O(19)La3Hf tetrahedra, edges with two equivalent O(14)La3Hf tetrahedra, and edges with two equivalent O(16)La2Hf2Mg trigonal bipyramids. In the fourteenth O site, O(14) is bonded to one La(5), two equivalent La(7), and one Hf(3) atom to form distorted OLa3Hf tetrahedra that share  a cornercorner with one O(18)La3Hf tetrahedra,  a cornercorner with one O(10)La4 tetrahedra, corners with two equivalent O(14)La3Hf tetrahedra, corners with two equivalent O(19)La3Hf tetrahedra, corners with two equivalent O(1)La4 tetrahedra, corners with two equivalent O(20)La4 tetrahedra,  a cornercorner with one O(16)La2Hf2Mg trigonal bipyramid,  an edgeedge with one O(20)La4 tetrahedra, and edges with two equivalent O(13)La3Hf tetrahedra. In the fifteenth O site, O(15) is bonded in a 4-coordinate geometry to one La(7), one La(8), and two equivalent Hf(3) atoms. In the sixteenth O site, O(16) is bonded to one Mg(1), one La(5), one La(6), and two equivalent Hf(4) atoms to form distorted OLa2Hf2Mg trigonal bipyramids that share  a cornercorner with one O(14)La3Hf tetrahedra,  a cornercorner with one O(18)La3Hf tetrahedra, corners with two equivalent O(19)La3Hf tetrahedra, corners with two equivalent O(11)La4 tetrahedra, corners with two equivalent O(16)La2Hf2Mg trigonal bipyramids,  a cornercorner with one O(2)La2HfMg trigonal pyramid,  an edgeedge with one O(10)La4 tetrahedra, and edges with two equivalent O(13)La3Hf tetrahedra. In the seventeenth O site, O(17) is bonded in a 6-coordinate geometry to two equivalent Mg(1), one La(8), two equivalent La(6), and one Hf(4) atom. In the eighteenth O site, O(18) is bonded to one La(6), two equivalent La(8), and one Hf(3) atom to form OLa3Hf tetrahedra that share  a cornercorner with one O(14)La3Hf tetrahedra,  a cornercorner with one O(10)La4 tetrahedra, corners with two equivalent O(18)La3Hf tetrahedra, corners with two equivalent O(1)La4 tetrahedra, corners with two equivalent O(11)La4 tetrahedra,  a cornercorner with one O(16)La2Hf2Mg trigonal bipyramid, corners with two equivalent O(2)La2HfMg trigonal pyramids, and  an edgeedge with one O(2)La2HfMg trigonal pyramid. In the nineteenth O site, O(19) is bonded to one La(7), two equivalent La(1), and one Hf(4) atom to form OLa3Hf tetrahedra that share  a cornercorner with one O(1)La4 tetrahedra, corners with two equivalent O(6)La2Hf2 tetrahedra, corners with two equivalent O(14)La3Hf tetrahedra, corners with two equivalent O(19)La3Hf tetrahedra, corners with two equivalent O(3)La3Hf tetrahedra, corners with two equivalent O(4)La3Hf tetrahedra, corners with two equivalent O(16)La2Hf2Mg trigonal bipyramids,  an edgeedge with one O(13)La3Hf tetrahedra,  an edgeedge with one O(3)La3Hf tetrahedra, and edges with two equivalent O(20)La4 tetrahedra. In the twentieth O site, O(20) is bonded to one La(1), one La(2), and two equivalent La(7) atoms to form OLa4 tetrahedra that share  a cornercorner with one O(4)La3Hf tetrahedra,  a cornercorner with one O(8)La3Hf tetrahedra, corners with two equivalent O(13)La3Hf tetrahedra, corners with two equivalent O(14)La3Hf tetrahedra, corners with two equivalent O(3)La3Hf tetrahedra, corners with two equivalent O(7)La3Hf tetrahedra, corners with two equivalent O(20)La4 tetrahedra,  an edgeedge with one O(6)La2Hf2 tetrahedra,  an edgeedge with one O(14)La3Hf tetrahedra, edges with two equivalent O(19)La3Hf tetrahedra, and edges with two equivalent O(1)La4 tetrahedra.