The structure described by Mg14HfAlO16 is Molybdenum Carbide MAX Phase-derived structured and crystallizes in the tetragonal P4/mmm space group. There are five inequivalent Mg sites. In the first Mg site, Mg(1) is bonded to two equivalent O(1), two equivalent O(2), and two equivalent O(5) atoms to form MgO6 octahedra that share corners with two equivalent Mg(1)O6 octahedra; corners with two equivalent Hf(1)O6 octahedra; corners with two equivalent Al(1)O6 octahedra; edges with four equivalent Mg(2)O6 octahedra; edges with four Mg(3,3)O6 octahedra; and edges with four equivalent Mg(4)O6 octahedra. The corner-sharing octahedra are not tilted. In the second Mg site, Mg(2) is bonded to one O(4), one O(5), two equivalent O(1), and two equivalent O(3) atoms to form MgO6 octahedra that share corners with two equivalent Mg(3)O6 octahedra; corners with four equivalent Mg(2)O6 octahedra; edges with two equivalent Mg(1)O6 octahedra; edges with two equivalent Mg(2)O6 octahedra; edges with two Mg(3,3)O6 octahedra; edges with two equivalent Al(1)O6 octahedra; and edges with four equivalent Mg(4)O6 octahedra. The corner-sharing octahedral tilt angles range from 0-10°. In the third Mg site, Mg(3) is bonded to one O(5), one O(6), two equivalent O(2), and two equivalent O(3) atoms to form MgO6 octahedra that share corners with two equivalent Mg(2)O6 octahedra; corners with four Mg(3,3)O6 octahedra; edges with two equivalent Mg(1)O6 octahedra; edges with two equivalent Mg(2)O6 octahedra; edges with two Mg(3,3)O6 octahedra; edges with two equivalent Hf(1)O6 octahedra; and edges with four equivalent Mg(4)O6 octahedra. The corner-sharing octahedral tilt angles range from 0-5°. In the fourth Mg site, Mg(3) is bonded to one O(5), one O(6), two equivalent O(2), and two equivalent O(3) atoms to form MgO6 octahedra that share corners with two equivalent Mg(2)O6 octahedra; corners with four Mg(3,3)O6 octahedra; edges with two equivalent Mg(1)O6 octahedra; edges with two equivalent Mg(2)O6 octahedra; edges with two Mg(3,3)O6 octahedra; edges with two equivalent Hf(1)O6 octahedra; and edges with four equivalent Mg(4)O6 octahedra. The corner-sharing octahedral tilt angles range from 0-5°. In the fifth Mg site, Mg(4) is bonded to two equivalent O(1), two equivalent O(2), and two equivalent O(3) atoms to form MgO6 octahedra that share corners with six equivalent Mg(4)O6 octahedra,  an edgeedge with one Hf(1)O6 octahedra,  an edgeedge with one Al(1)O6 octahedra, edges with two equivalent Mg(1)O6 octahedra, edges with four equivalent Mg(2)O6 octahedra, and edges with four equivalent Mg(3)O6 octahedra. The corner-sharing octahedral tilt angles range from 1-10°. Hf(1) is bonded to two equivalent O(6) and four O(2,2) atoms to form HfO6 octahedra that share corners with two equivalent Hf(1)O6 octahedra; corners with four equivalent Mg(1)O6 octahedra; edges with four equivalent Mg(4)O6 octahedra; and edges with eight Mg(3,3)O6 octahedra. The corner-sharing octahedra are not tilted. Al(1) is bonded to two equivalent O(4) and four equivalent O(1) atoms to form AlO6 octahedra that share corners with two equivalent Al(1)O6 octahedra, corners with four equivalent Mg(1)O6 octahedra, edges with four equivalent Mg(4)O6 octahedra, and edges with eight equivalent Mg(2)O6 octahedra. The corner-sharing octahedra are not tilted. There are seven inequivalent O sites. In the first O site, O(1) is bonded to one Mg(1), two equivalent Mg(2), two equivalent Mg(4), and one Al(1) atom to form OMg5Al octahedra that share corners with two equivalent O(2)HfMg5 octahedra; corners with four equivalent O(1)Mg5Al octahedra; edges with two O(2,2)HfMg5 octahedra; edges with two equivalent O(4)Mg4Al2 octahedra; edges with two equivalent O(1)Mg5Al octahedra; edges with two equivalent O(5)Mg6 octahedra; and edges with four equivalent O(3)Mg6 octahedra. The corner-sharing octahedral tilt angles range from 0-10°. In the second O site, O(2) is bonded to one Mg(1), two equivalent Mg(3), two equivalent Mg(4), and one Hf(1) atom to form OHfMg5 octahedra that share corners with two equivalent O(1)Mg5Al octahedra; corners with four equivalent O(2)HfMg5 octahedra; edges with two equivalent O(6)Hf2Mg4 octahedra; edges with two O(2,2)HfMg5 octahedra; edges with two equivalent O(1)Mg5Al octahedra; edges with two equivalent O(5)Mg6 octahedra; and edges with four equivalent O(3)Mg6 octahedra. The corner-sharing octahedral tilt angles range from 0-9°. In the third O site, O(3) is bonded to two equivalent Mg(2), two equivalent Mg(3), and two equivalent Mg(4) atoms to form OMg6 octahedra that share corners with six equivalent O(3)Mg6 octahedra,  an edgeedge with one O(6)Hf2Mg4 octahedra,  an edgeedge with one O(4)Mg4Al2 octahedra, edges with two equivalent O(5)Mg6 octahedra, edges with four equivalent O(2)HfMg5 octahedra, and edges with four equivalent O(1)Mg5Al octahedra. The corner-sharing octahedral tilt angles range from 0-2°. In the fourth O site, O(4) is bonded to four equivalent Mg(2) and two equivalent Al(1) atoms to form OMg4Al2 octahedra that share corners with two equivalent O(4)Mg4Al2 octahedra, corners with four equivalent O(5)Mg6 octahedra, edges with four equivalent O(3)Mg6 octahedra, and edges with eight equivalent O(1)Mg5Al octahedra. The corner-sharing octahedra are not tilted. In the fifth O site, O(5) is bonded to two equivalent Mg(1); two equivalent Mg(2); and two Mg(3,3) atoms to form OMg6 octahedra that share corners with two equivalent O(6)Hf2Mg4 octahedra, corners with two equivalent O(4)Mg4Al2 octahedra, corners with two equivalent O(5)Mg6 octahedra, edges with four equivalent O(2)HfMg5 octahedra, edges with four equivalent O(1)Mg5Al octahedra, and edges with four equivalent O(3)Mg6 octahedra. The corner-sharing octahedra are not tilted. In the sixth O site, O(6) is bonded to four Mg(3,3) and two equivalent Hf(1) atoms to form OHf2Mg4 octahedra that share corners with two equivalent O(6)Hf2Mg4 octahedra; corners with four equivalent O(5)Mg6 octahedra; edges with four equivalent O(3)Mg6 octahedra; and edges with eight O(2,2)HfMg5 octahedra. The corner-sharing octahedra are not tilted. In the seventh O site, O(2) is bonded to one Mg(1), two equivalent Mg(3), two equivalent Mg(4), and one Hf(1) atom to form OHfMg5 octahedra that share corners with two equivalent O(1)Mg5Al octahedra; corners with four O(2,2)HfMg5 octahedra; edges with two equivalent O(6)Hf2Mg4 octahedra; edges with two equivalent O(2)HfMg5 octahedra; edges with two equivalent O(1)Mg5Al octahedra; edges with two equivalent O(5)Mg6 octahedra; and edges with four equivalent O(3)Mg6 octahedra. The corner-sharing octahedral tilt angles range from 0-9°. is represented by the CIF card [CIF]
data_HfMg14AlO16
_symmetry_space_group_name_H-M   'P 1'
_cell_length_a   8.651
_cell_length_b   8.651
_cell_length_c   4.299
_cell_angle_alpha   90.000
_cell_angle_beta   90.000
_cell_angle_gamma   90.000
_symmetry_Int_Tables_number   1
_cell_volume   321.736
_cell_formula_units_Z   1
loop_
 _symmetry_equiv_pos_site_id
 _symmetry_equiv_pos_as_xyz
  1  'x, y, z'
loop_
 _atom_site_type_symbol
 _atom_site_label
 _atom_site_symmetry_multiplicity
 _atom_site_fract_x
 _atom_site_fract_y
 _atom_site_fract_z
 _atom_site_occupancy
  Hf  Hf0  1  0.500  0.500  0.000  1.0
  Mg  Mg1  1  0.500  0.000  0.000  1.0
  Mg  Mg2  1  0.000  0.500  0.000  1.0
  Mg  Mg3  1  0.249  0.000  0.500  1.0
  Mg  Mg4  1  0.751  0.000  0.500  1.0
  Mg  Mg5  1  0.244  0.500  0.500  1.0
  Mg  Mg6  1  0.756  0.500  0.500  1.0
  Mg  Mg7  1  0.000  0.249  0.500  1.0
  Mg  Mg8  1  0.500  0.244  0.500  1.0
  Mg  Mg9  1  0.000  0.751  0.500  1.0
  Mg  Mg10  1  0.500  0.756  0.500  1.0
  Mg  Mg11  1  0.249  0.249  0.000  1.0
  Mg  Mg12  1  0.751  0.249  0.000  1.0
  Mg  Mg13  1  0.249  0.751  0.000  1.0
  Mg  Mg14  1  0.751  0.751  0.000  1.0
  Al  Al15  1  0.000  0.000  0.000  1.0
  O  O16  1  0.000  0.228  0.000  1.0
  O  O17  1  0.500  0.233  0.000  1.0
  O  O18  1  0.000  0.772  0.000  1.0
  O  O19  1  0.500  0.767  0.000  1.0
  O  O20  1  0.249  0.249  0.500  1.0
  O  O21  1  0.751  0.249  0.500  1.0
  O  O22  1  0.249  0.751  0.500  1.0
  O  O23  1  0.751  0.751  0.500  1.0
  O  O24  1  0.000  0.000  0.500  1.0
  O  O25  1  0.500  0.000  0.500  1.0
  O  O26  1  0.000  0.500  0.500  1.0
  O  O27  1  0.500  0.500  0.500  1.0
  O  O28  1  0.228  0.000  0.000  1.0
  O  O29  1  0.772  0.000  0.000  1.0
  O  O30  1  0.233  0.500  0.000  1.0
  O  O31  1  0.767  0.500  0.000  1.0
[/CIF]
.
The structure described by Mg14HfAlO16 is Molybdenum Carbide MAX Phase-derived structured and crystallizes in the tetragonal P4/mmm space group. There are five inequivalent Mg sites. In the first Mg site, Mg(1) is bonded to two equivalent O(1), two equivalent O(2), and two equivalent O(5) atoms to form MgO6 octahedra that share corners with two equivalent Mg(1)O6 octahedra; corners with two equivalent Hf(1)O6 octahedra; corners with two equivalent Al(1)O6 octahedra; edges with four equivalent Mg(2)O6 octahedra; edges with four Mg(3,3)O6 octahedra; and edges with four equivalent Mg(4)O6 octahedra. The corner-sharing octahedra are not tilted. In the second Mg site, Mg(2) is bonded to one O(4), one O(5), two equivalent O(1), and two equivalent O(3) atoms to form MgO6 octahedra that share corners with two equivalent Mg(3)O6 octahedra; corners with four equivalent Mg(2)O6 octahedra; edges with two equivalent Mg(1)O6 octahedra; edges with two equivalent Mg(2)O6 octahedra; edges with two Mg(3,3)O6 octahedra; edges with two equivalent Al(1)O6 octahedra; and edges with four equivalent Mg(4)O6 octahedra. The corner-sharing octahedral tilt angles range from 0-10°. In the third Mg site, Mg(3) is bonded to one O(5), one O(6), two equivalent O(2), and two equivalent O(3) atoms to form MgO6 octahedra that share corners with two equivalent Mg(2)O6 octahedra; corners with four Mg(3,3)O6 octahedra; edges with two equivalent Mg(1)O6 octahedra; edges with two equivalent Mg(2)O6 octahedra; edges with two Mg(3,3)O6 octahedra; edges with two equivalent Hf(1)O6 octahedra; and edges with four equivalent Mg(4)O6 octahedra. The corner-sharing octahedral tilt angles range from 0-5°. In the fourth Mg site, Mg(3) is bonded to one O(5), one O(6), two equivalent O(2), and two equivalent O(3) atoms to form MgO6 octahedra that share corners with two equivalent Mg(2)O6 octahedra; corners with four Mg(3,3)O6 octahedra; edges with two equivalent Mg(1)O6 octahedra; edges with two equivalent Mg(2)O6 octahedra; edges with two Mg(3,3)O6 octahedra; edges with two equivalent Hf(1)O6 octahedra; and edges with four equivalent Mg(4)O6 octahedra. The corner-sharing octahedral tilt angles range from 0-5°. In the fifth Mg site, Mg(4) is bonded to two equivalent O(1), two equivalent O(2), and two equivalent O(3) atoms to form MgO6 octahedra that share corners with six equivalent Mg(4)O6 octahedra,  an edgeedge with one Hf(1)O6 octahedra,  an edgeedge with one Al(1)O6 octahedra, edges with two equivalent Mg(1)O6 octahedra, edges with four equivalent Mg(2)O6 octahedra, and edges with four equivalent Mg(3)O6 octahedra. The corner-sharing octahedral tilt angles range from 1-10°. Hf(1) is bonded to two equivalent O(6) and four O(2,2) atoms to form HfO6 octahedra that share corners with two equivalent Hf(1)O6 octahedra; corners with four equivalent Mg(1)O6 octahedra; edges with four equivalent Mg(4)O6 octahedra; and edges with eight Mg(3,3)O6 octahedra. The corner-sharing octahedra are not tilted. Al(1) is bonded to two equivalent O(4) and four equivalent O(1) atoms to form AlO6 octahedra that share corners with two equivalent Al(1)O6 octahedra, corners with four equivalent Mg(1)O6 octahedra, edges with four equivalent Mg(4)O6 octahedra, and edges with eight equivalent Mg(2)O6 octahedra. The corner-sharing octahedra are not tilted. There are seven inequivalent O sites. In the first O site, O(1) is bonded to one Mg(1), two equivalent Mg(2), two equivalent Mg(4), and one Al(1) atom to form OMg5Al octahedra that share corners with two equivalent O(2)HfMg5 octahedra; corners with four equivalent O(1)Mg5Al octahedra; edges with two O(2,2)HfMg5 octahedra; edges with two equivalent O(4)Mg4Al2 octahedra; edges with two equivalent O(1)Mg5Al octahedra; edges with two equivalent O(5)Mg6 octahedra; and edges with four equivalent O(3)Mg6 octahedra. The corner-sharing octahedral tilt angles range from 0-10°. In the second O site, O(2) is bonded to one Mg(1), two equivalent Mg(3), two equivalent Mg(4), and one Hf(1) atom to form OHfMg5 octahedra that share corners with two equivalent O(1)Mg5Al octahedra; corners with four equivalent O(2)HfMg5 octahedra; edges with two equivalent O(6)Hf2Mg4 octahedra; edges with two O(2,2)HfMg5 octahedra; edges with two equivalent O(1)Mg5Al octahedra; edges with two equivalent O(5)Mg6 octahedra; and edges with four equivalent O(3)Mg6 octahedra. The corner-sharing octahedral tilt angles range from 0-9°. In the third O site, O(3) is bonded to two equivalent Mg(2), two equivalent Mg(3), and two equivalent Mg(4) atoms to form OMg6 octahedra that share corners with six equivalent O(3)Mg6 octahedra,  an edgeedge with one O(6)Hf2Mg4 octahedra,  an edgeedge with one O(4)Mg4Al2 octahedra, edges with two equivalent O(5)Mg6 octahedra, edges with four equivalent O(2)HfMg5 octahedra, and edges with four equivalent O(1)Mg5Al octahedra. The corner-sharing octahedral tilt angles range from 0-2°. In the fourth O site, O(4) is bonded to four equivalent Mg(2) and two equivalent Al(1) atoms to form OMg4Al2 octahedra that share corners with two equivalent O(4)Mg4Al2 octahedra, corners with four equivalent O(5)Mg6 octahedra, edges with four equivalent O(3)Mg6 octahedra, and edges with eight equivalent O(1)Mg5Al octahedra. The corner-sharing octahedra are not tilted. In the fifth O site, O(5) is bonded to two equivalent Mg(1); two equivalent Mg(2); and two Mg(3,3) atoms to form OMg6 octahedra that share corners with two equivalent O(6)Hf2Mg4 octahedra, corners with two equivalent O(4)Mg4Al2 octahedra, corners with two equivalent O(5)Mg6 octahedra, edges with four equivalent O(2)HfMg5 octahedra, edges with four equivalent O(1)Mg5Al octahedra, and edges with four equivalent O(3)Mg6 octahedra. The corner-sharing octahedra are not tilted. In the sixth O site, O(6) is bonded to four Mg(3,3) and two equivalent Hf(1) atoms to form OHf2Mg4 octahedra that share corners with two equivalent O(6)Hf2Mg4 octahedra; corners with four equivalent O(5)Mg6 octahedra; edges with four equivalent O(3)Mg6 octahedra; and edges with eight O(2,2)HfMg5 octahedra. The corner-sharing octahedra are not tilted. In the seventh O site, O(2) is bonded to one Mg(1), two equivalent Mg(3), two equivalent Mg(4), and one Hf(1) atom to form OHfMg5 octahedra that share corners with two equivalent O(1)Mg5Al octahedra; corners with four O(2,2)HfMg5 octahedra; edges with two equivalent O(6)Hf2Mg4 octahedra; edges with two equivalent O(2)HfMg5 octahedra; edges with two equivalent O(1)Mg5Al octahedra; edges with two equivalent O(5)Mg6 octahedra; and edges with four equivalent O(3)Mg6 octahedra. The corner-sharing octahedral tilt angles range from 0-9°. is represented by the CIF card [CIF]
data_HfMg14AlO16
_symmetry_space_group_name_H-M   'P 1'
_cell_length_a   8.651
_cell_length_b   8.651
_cell_length_c   4.299
_cell_angle_alpha   90.000
_cell_angle_beta   90.000
_cell_angle_gamma   90.000
_symmetry_Int_Tables_number   1
_cell_volume   321.736
_cell_formula_units_Z   1
loop_
 _symmetry_equiv_pos_site_id
 _symmetry_equiv_pos_as_xyz
  1  'x, y, z'
loop_
 _atom_site_type_symbol
 _atom_site_label
 _atom_site_symmetry_multiplicity
 _atom_site_fract_x
 _atom_site_fract_y
 _atom_site_fract_z
 _atom_site_occupancy
  Hf  Hf0  1  0.500  0.500  0.000  1.0
  Mg  Mg1  1  0.500  0.000  0.000  1.0
  Mg  Mg2  1  0.000  0.500  0.000  1.0
  Mg  Mg3  1  0.249  0.000  0.500  1.0
  Mg  Mg4  1  0.751  0.000  0.500  1.0
  Mg  Mg5  1  0.244  0.500  0.500  1.0
  Mg  Mg6  1  0.756  0.500  0.500  1.0
  Mg  Mg7  1  0.000  0.249  0.500  1.0
  Mg  Mg8  1  0.500  0.244  0.500  1.0
  Mg  Mg9  1  0.000  0.751  0.500  1.0
  Mg  Mg10  1  0.500  0.756  0.500  1.0
  Mg  Mg11  1  0.249  0.249  0.000  1.0
  Mg  Mg12  1  0.751  0.249  0.000  1.0
  Mg  Mg13  1  0.249  0.751  0.000  1.0
  Mg  Mg14  1  0.751  0.751  0.000  1.0
  Al  Al15  1  0.000  0.000  0.000  1.0
  O  O16  1  0.000  0.228  0.000  1.0
  O  O17  1  0.500  0.233  0.000  1.0
  O  O18  1  0.000  0.772  0.000  1.0
  O  O19  1  0.500  0.767  0.000  1.0
  O  O20  1  0.249  0.249  0.500  1.0
  O  O21  1  0.751  0.249  0.500  1.0
  O  O22  1  0.249  0.751  0.500  1.0
  O  O23  1  0.751  0.751  0.500  1.0
  O  O24  1  0.000  0.000  0.500  1.0
  O  O25  1  0.500  0.000  0.500  1.0
  O  O26  1  0.000  0.500  0.500  1.0
  O  O27  1  0.500  0.500  0.500  1.0
  O  O28  1  0.228  0.000  0.000  1.0
  O  O29  1  0.772  0.000  0.000  1.0
  O  O30  1  0.233  0.500  0.000  1.0
  O  O31  1  0.767  0.500  0.000  1.0
[/CIF]
.