1. I. Structural Aspects Space GroupsFranzen, pp. 55-77 Combine Translational and Rotational symmetry operations  230 Space Groups Both types must be compatible. Rotation-Translation symmetry operationsSeitz Notation ( R |  ):( R |  ) x = R x +  R = 3  3 rotation matrix;  = vector A space group G is the set of operations, {( R |  )}, that (i) is closed under multiplication: ( R 2 |  2 )( R 1 |  1 ) = ( R 2 R 1 | R 2  1 +  2 )  G ; (ii) contains an identity operation: ( 1 | 0 ) x = x (iii) contains the inverse of every operation: ( R |  )  1 = ( R  1 |  R  1  ) Hand-Outs: 27

2. I. Structural Aspects Space Groups: Allowed Symmetry OperationsFranzen, pp. 55-77 ( R |  ):( R |  ) x = R x +  (1) Pure (lattice) translations: ( 1 | t ), t  {Bravais lattice vectors} (2) Pure rotations: ( n | 0 ), ( | 0 ) Hand-Outs: 27

3. I. Structural Aspects Space Groups: Allowed Symmetry OperationsFranzen, pp. 55-77 ( R |  ):( R |  ) x = R x +  (1) Pure (lattice) translations: ( 1 | t ), t  {Bravais lattice vectors} (2) Pure rotations: ( n | 0 ), ( | 0 ) (3) Screw (rotation) axes: ( n t | jt/n ) – rotation by 2  /n followed by displacement jt/n along the axis direction. 2 1, 3 1, 3 2, 4 1, 4 2, 4 3, 6 1, 6 2, 6 3, 6 4, 6 5 screw axes allowed. 2 1 = ( 2 | c/2 ) c C2C2 c/2 Hand-Outs: 27-28

4. I. Structural Aspects Space Groups: Allowed Symmetry OperationsFranzen, pp. 55-77 ( R |  ):( R |  ) x = R x +  (1) Pure (lattice) translations: ( 1 | t ), t  {Bravais lattice vectors} (2) Pure rotations: ( n | 0 ), ( | 0 ) (3) Screw (rotation) axes: ( n t | jt/n ) – rotation by 2  /n followed by displacement jt/n along the axis direction. 2 1, 3 1, 3 2, 4 1, 4 2, 4 3, 6 1, 6 2, 6 3, 6 4, 6 5 screw axes allowed. 2 1 = ( 2 | c/2 ) c C2C2 c/2 6 3 = ( 6 | 3c/6 )4 1 = ( 4 | c/4 ) Hand-Outs: 27-28

5. I. Structural Aspects Space Groups: Allowed Symmetry OperationsFranzen, pp. 55-77 ( R |  ):( R |  ) x = R x +  (4) Glide (reflection) planes: ( m t' | t/2 ), t'  t/2 – reflection in a plane followed by displacement in directions parallel to the plane. Axial glides: (“m” replaced by “a,” “b,” or “c”). t = a, b, c (unit cell vectors). a glide: a/2, and the reflection plane can be either the ab- or the ac-plane. b a a glide: ( m 010 | a/2 ) Hand-Outs: 27-28

6. I. Structural Aspects Space Groups: Allowed Symmetry OperationsFranzen, pp. 55-77 ( R |  ):( R |  ) x = R x +  (4) Glide (reflection) planes: ( m t' | t/2 ), t'  t/2 – reflection in a plane followed by displacement in directions parallel to the plane. Axial glides: (“m” replaced by “a,” “b,” or “c”). t = a, b, c (unit cell vectors). a glide: a/2, and the reflection plane can be either the ab- or the ac-plane. Diagonal glides: (“m” replaced by “n”). t = a + b, a + c, or b + c (diagonal vectors). Diamond glides: (“m” replaced by “d”). Only possible for centered lattices. b a a glide: ( m 010 | a/2 ) b a n glide: ( m 010 | (a + c)/2 ) Hand-Outs: 27-28

7. I. Structural Aspects Space Groups: Notation Space Group = {essential symmetry operations}  {Bravais lattice} N = # of translation operations in the Bravais lattice (N is a very large number) h = # of rotation-translations – isomorphous with one of the 32 crystallographic point groups (h  48) The space group has hN symmetry operations. Symmorphic space groups (73): {h essential symmetry operations} is a group. Pmmm: primitive, orthorhombic lattice. There are mirror planes perpendicular to each crystallographic axis and the point symmetry at each lattice point in a structure has D 2h symmetry (order = 8). C2/m: base-centered, monoclinic lattice. Lattice centering occurs in the ab-planes. There is a mirror plane perpendicular to the twofold rotation axis through each lattice point. The point symmetry at each lattice point in a structure has C 2h symmetry (order = 4). I4/mmm: body-centered, tetragonal lattice. There are mirror planes perpendicular to each crystallographic axis and to the face diagonals. The point symmetry at each lattice point in a structure has D 4h symmetry (order = 16). Fm3m: all face-centered, cubic lattice. The point symmetry at each lattice point in a structure has O h symmetry. Hand-Outs: 29

8. I. Structural Aspects Space Groups: Notation Nonsymmorphic space groups (157): {h essential symmetry operations} is a not a group. Pnma: primitive, orthorhombic lattice. There is a n glide plane perpendicular to the a direction (the translation is b/2 + c/2), a regular mirror plane m perpendicular to the b direction, and a a glide plane perpendicular to the c direction (the translation is a/2). There are 8 essential symmetry operations, but these do not form a group. P2 1 /c: primitive, monoclinic lattice. The twofold rotation axis is actually a twofold screw axis, i.e., 180º rotation followed by translation by b/2. There is also a glide reflection perpendicular to this screw axis, i.e., reflection through a plane perpendicular to b followed by translation by c/2. There are 4 essential symmetry operations, but these do not form a group. I4 1 /amd: body-centered, tetragonal lattice. The fourfold rotation axis is actually a screw axis, i.e., 90º rotation followed by translation by c/4. There is a glide reflection perpendicular to this screw axis, i.e., reflection through a plane perpendicular to c followed by translation by a/2. There are mirror planes perpendicular to the a and b directions. And, there are diamond glide planes perpendicular to (a+b) and (a−b) directions. There are 16 essential symmetry operations, but these do not form a group. Fd3m: all face-centered, cubic lattice. There are diamond glide reflections perpendicular to the crystallographic a, b, and c axes. There are 48 essential symmetry operations, but these do not form a group. Hand-Outs: 30

9. I. Structural Aspects Space Groups: Symmorphic vs. Nonsymmorphic Space Groups Consider the space groups P2 and P2 1, and let the b axis be the C 2 axis. P2: the essential symmetry operations = {( 1  0 ), ( 2  0 )}; P2 1 : the essential symmetry operations = {( 1 | 0 ), ( 2  b/2 )}. The multiplication tables for each set is: P2P2 ( 1  0 )( 2  0 ) P21P21 ( 1  0 )( 2  b/2 ) ( 1  0 ) ( 2  0 )( 2  b/2 ) Hand-Outs: 31

10. I. Structural Aspects Space Groups: Symmorphic vs. Nonsymmorphic Space Groups Consider the space groups P2 and P2 1, and let the b axis be the C 2 axis. P2: the essential symmetry operations = {( 1  0 ), ( 2  0 )}; P2 1 : the essential symmetry operations = {( 1 | 0 ), ( 2  b/2 )}. The multiplication tables for each set is: P2P2 ( 1  0 )( 2  0 ) P21P21 ( 1  0 )( 2  b/2 ) ( 1  0 ) ( 1 | 0 )( 2 | 0 ) ( 1  0 ) ( 1 | 0 )( 2 | b/2 ) ( 2  0 ) ( 2 | 0 )( 1 | 0 ) ( 2  b/2 ) ( 2 | b/2 )( 1 | b ) Point Group of the Space Group: set all translations/displacements to 0; one of the 32 crystallographic point groups Order of this Point Group = # of general equivalent positions in one unit cell International Tables of Crystallography Hand-Outs: 31

11. I. Structural Aspects Space Groups: International Tables (Symmorphic Group) Hand-Outs: 32

12. I. Structural Aspects Space Groups: International Tables (Symmorphic Group) Point Group of the Space Group Symmetry Operations Hand-Outs: 32

13. I. Structural Aspects Space Groups: International Tables (Symmorphic Group) Generating Operations Sites in Unit Cells Have full point symmetry of the space group Hand-Outs: 32

14. I. Structural Aspects Space Groups: International Tables (Nonsymmorphic Group) NOTE: No sites have the full point symmetry of the space group (4/mmm). Hand-Outs: 33

15. I. Structural Aspects Space Groups: Group-Subgroup Relationships TiO 2 (down the c-axis) P4 2 /mnm (P 4 2 /m 2 1 /n 2/m) CaCl 2 (HCP Cl) Pnnm (P 2 1 /n 2 1 /n 2/m ) A group G is a subgroup of G 0 if all members of G are contained in G 0. G is a proper subgroup if G 0 contains members that are not in G. G is a maximal subgroup if there is no other subgroup H such that G is a proper subgroup of H. Translationengleiche: retains all translations, but the order of the point group is reduced. Hand-Outs: 34

16. I. Structural Aspects Space Groups: Group-Subgroup Relationships Klassengleiche: preserves the point group of the space group, but loses some translations. TYPE IIa: conventional unit cells are identical (lose lattice centering) CuZn High temp. (Im3m) Low temp. (Pm3m) Hand-Outs: 34

17. I. Structural Aspects Space Groups: Group-Subgroup Relationships TYPE IIb: conventional unit cell becomes larger (lose translations as periodicity changes) SrGa 2 High press. (P6/mmm) Low press. (P6 3 /mmc) TYPE IIc: two space groups are isomorphous c c Rutile Structure: MO 2 – P4 2 /mnm Trirutile Structure: Ta 2 FeO 6 (M 3 O 6, P4 2 /mnm; c goes to 3c) Hand-Outs: 34