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.

Hand-Outs: 15

Symmetry in Crystalline Systems (3) Space Groups: Notation

Pmmm

Crystal Class: OrthorhombicLattice Type: Primitive

m a-axis = ( m100 | 0 )

m b-axis = ( m010 | 0 )

m c-axis = ( m001 | 0 )

( m100 | 0 )( m010 | 0 ) = ( 2001 | 0 )

a

b0

Symmorphic space groups (73): {h essential symmetry operations} is a group.

( 2001 | 0 )( m001 | 0 ) = ( | 0 )1

Hand-Outs: 15

Symmetry in Crystalline Systems (3) Space Groups: Notation

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 D2h 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 C2h 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 D4h symmetry (order = 16).

Fm3m: all face-centered, cubic lattice. The point symmetry at each lattice point in a structure has Oh symmetry.

Symmorphic space groups (73): {h essential symmetry operations} is a group.

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.

Hand-Outs: 15

Symmetry in Crystalline Systems (3) Space Groups: Notation

Pnma

Crystal Class: OrthorhombicLattice Type: Primitive

n a-axis = ( m100 | b/2 + c/2 )

m b-axis = ( m010 | 0 )

a c-axis = ( m001 | a/2 )

( m100 | b/2 + c/2 )( m010 | 0 ) = ( 2001 | b/2 + c/2 )

a

b0

Nonsymmorphic space groups (157): {h essential symmetry operations} is a not a group.

= ( 2001 | c/2 ) intersecting (0, 1/4)

Nonsymmorphic space groups (157): {h essential symmetry operations} is a not a group.

Hand-Outs: 16

Symmetry in Crystalline Systems (3) Space Groups: Notation

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 an a glide plane perpendicular to the c 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).

Consider the space groups P2 and P21, and let the b 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:

P2 ( 1 0 ) ( 2 0 ) P21 ( 1 0 )( 2 b/2 )

( 1 0 ) ( 1 0 )

( 2 0 ) ( 2 b/2 )

Hand-Outs: 17

Symmetry in Crystalline Systems (3) Space Groups: Symmorphic vs. Nonsymmorphic Space Groups

Consider the space groups P2 and P21, and let the b 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:

P2 ( 1 0 ) ( 2 0 ) P21 ( 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: 17

Symmetry in Crystalline Systems (3) Space Groups: Symmorphic vs. Nonsymmorphic Space Groups

Point Group of the Space Group

Symmetry Operations

Hand-Outs: 18

GeneratingOperations

Sites inUnit Cells

Symmetry in Crystalline Systems (3) Space Groups: International Tables (Symmorphic Group)

NOTE: No sites have the full pointsymmetry of the space group (4/mmm).

Hand-Outs: 19

Symmetry in Crystalline Systems (3) Space Groups: International Tables (Nonsymmorphic Group)

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)

A group is a subgroup of 0 if all members of are contained in 0. is a proper subgroup if 0 contains members that are not in . is a maximal subgroup if there is no other subgroup such that is a proper subgroup of .

Translationengleiche: retains all translations, but the order of the point group is reduced; i.e., the set of essential symmetry operations has fewer members.

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)

Hand-Outs: 20

Symmetry in Crystalline Systems (3) Space Groups: Group-Subgroup Relationships

TYPE IIb: conventional unit cell becomes larger (lose translations as periodicity changes)

SrGa2

High press.(P6/mmm)

Low press.(P63/mmc)

TYPE IIc: two space groups are isomorphous

cc

Rutile Structure: TiO2, CrO2, RuO2 – P42/mnm

Trirutile Structure: Ta2FeO6 (M3O6, P42/mnm; c goes to 3c)

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.

General, single-valued function, f (r), with total symmetry of Bravais lattice:

( )( ) : ( ) ( ) inf f f Ae rr r T r Plane waves: ei = cos + i sin

(r) = K r, K: units of 1/distance

( )( ) ( )n ni iinf Ae Ae e f K r T K TK rr T r

Therefore, 1nie K T K Tn = 2N

{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

wmvmumii AeAef m 3212)( rKr

Hand-Outs: 21

Symmetry in Crystalline Systems (4) Reciprocal Space: Reciprocal Lattice