Space Group = {Essential Symmetry Operations}  {Bravais Lattice}

1. 0 b a Crystal Class: Orthorhombic Lattice Type: Primitive Hand-Outs: 15 Symmetry in Crystalline Systems (3) 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 m a-axis = ( m100 | 0 ) ( m100 | 0 )( m010 | 0 ) = ( 2001 | 0 ) m b-axis = ( m010 | 0 ) ( 2001 | 0 )( m001 | 0 ) = ( | 0 ) m c-axis = ( m001 | 0 )

2. 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 D2hsymmetry (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 C2hsymmetry (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 D4hsymmetry (order = 16). Fm3m: all face-centered, cubic lattice. The point symmetry at each lattice point in a structure has Ohsymmetry. Hand-Outs: 15 Symmetry in Crystalline Systems (3) Space Groups: Notation Symmorphic space groups (73): {h essential symmetry operations} is a group.

3. 0 b a Hand-Outs: 15 Symmetry in Crystalline Systems (3) 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. Nonsymmorphic space groups (157): {h essential symmetry operations} is a not a group. Pnma Crystal Class: Orthorhombic Lattice Type: Primitive ( m100 | b/2 + c/2 )( m010 | 0 ) = ( 2001 | b/2 + c/2 ) n a-axis = ( m100 | b/2 + c/2 ) = ( 2001 | c/2 ) intersecting (0, 1/4) m b-axis = ( m010 | 0 ) a c-axis = ( m001 | a/2 )

4. Pnma: primitive, orthorhombic lattice. There is a n glide plane perpendicular to thea direction (the translation is b/2 + c/2), a regular mirror plane m perpendicular to theb direction, and an a glide plane perpendicular to thec direction (the translation is a/2). There are 8 essential symmetry operations (not a group). P21/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 (not a group). I41/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 (not 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 (not a group). Hand-Outs: 16 Symmetry in Crystalline Systems (3) Space Groups: Notation Nonsymmorphic space groups (157): {h essential symmetry operations} is a not a group.

5. Hand-Outs: 17 Symmetry in Crystalline Systems (3) Space Groups: Symmorphic vs. Nonsymmorphic Space Groups Consider the space groups P2and P21, and let theb axis be the C2 axis. P2: the essential symmetry operations = {( 1 0 ), ( 2 0 )}; P21:the essential symmetry operations = {( 1 |0 ), ( 2 b/2 )}. The multiplication tables for each set is:

6. Hand-Outs: 17 Symmetry in Crystalline Systems (3) Space Groups: Symmorphic vs. Nonsymmorphic Space Groups Consider the space groups P2and P21, and let theb axis be the C2 axis. P2: the essential symmetry operations = {( 1 0 ), ( 2 0 )}; P21:the essential symmetry operations = {( 1 |0 ), ( 2 b/2 )}. The multiplication tables for each set is: 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

7. Hand-Outs: 18 Symmetry in Crystalline Systems (3) Space Groups: International Tables (Symmorphic Group) Point Group of the Space Group Generating Operations Sites in Unit Cells Symmetry Operations

8. Hand-Outs: 19 Symmetry in Crystalline Systems (3) Space Groups: International Tables (Nonsymmorphic Group) NOTE: No sites have the full point symmetry of the space group (4/mmm).

9. Hand-Outs: 20 Symmetry in Crystalline Systems (3) Space Groups: Group-Subgroup Relationships A group G is a subgroup of G0 if all members of G are contained in G0. G is a proper subgroup if G0 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; i.e., the set of essential symmetry operations has fewer members. TiO2 (down the c-axis) P42/mnm (P 42/m 21/n 2/m) CaCl2 (HCP Cl) Pnnm (P 21/n 21/n 2/m)

10. Hand-Outs: 20 Symmetry in Crystalline Systems (3) 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)

11. Hand-Outs: 20 Symmetry in Crystalline Systems (3) Space Groups: Group-Subgroup Relationships Klassengleiche: preserves the point group of the space group, but loses some translations. TYPE IIb: conventional unit cell becomes larger (lose translations as periodicity changes) SrGa2 c High press. (P6/mmm) Low press. (P63/mmc) c TYPE IIc: two space groups are isomorphous Rutile Structure: TiO2, CrO2, RuO2 – P42/mnm Trirutile Structure: Ta2FeO6 (M3O6, P42/mnm; c goes to 3c)

12. Hand-Outs: 21 Symmetry in Crystalline Systems (4) Reciprocal Space: Reciprocal Lattice General, single-valued function, f (r), with total symmetry of Bravais lattice: Plane waves: ei = cos  + i sin  (r) = K  r, K: units of 1/distance K Tn = 2N {Km} = Reciprocal Lattice: Km = m1a1* + m2a2* + m3a3* (m1, m2, m3 integers) Therefore, for r = ua1 + va2 + wa3, the general periodic function of the lattice is