I. Structural Aspects Space GroupsFranzen, pp. 55-77 Combine Translational and Rotational symmetry...

I. Structural Aspects Space Groups Franzen, pp. 55-77

Combine Translational and Rotational symmetry operations 230 Space GroupsBoth types must be compatible.

Rotation-Translation symmetry operations Seitz 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: ( R2 | 2 )( R1 | 1 ) = ( R2R1 | R21 + 2 ) G;

(ii) contains an identity operation: ( 1 | 0 ) x = x

(iii) contains the inverse of every operation: ( R | )1 = ( R1 | R1 )

Hand-Outs: 27

I. Structural Aspects Space Groups: Allowed Symmetry Operations Franzen, pp. 55-77

( R | ): ( R | ) x = R x +

(1) Pure (lattice) translations: ( 1 | t ), t {Bravais lattice vectors}

(2) Pure rotations: ( n | 0 ), ( | 0 )n

Hand-Outs: 27


( R | ): ( R | ) x = R x +


(2) Pure rotations: ( n | 0 ), ( | 0 )

(3) Screw (rotation) axes: ( nt | jt/n ) – rotation by 2/n followed by displacement jt/n along the axis direction.

21, 31, 32, 41, 42, 43, 61, 62, 63, 64, 65 screw axes allowed.

n

21 = ( 2 | c/2 )

c

C2

c/2

Hand-Outs: 27-28


( R | ): ( R | ) x = R x +


(2) Pure rotations: ( n | 0 ), ( | 0 )

(3) Screw (rotation) axes: ( nt | jt/n ) – rotation by 2/n followed by displacement jt/n along the axis direction.

21, 31, 32, 41, 42, 43, 61, 62, 63, 64, 65 screw axes allowed.

n

21 = ( 2 | c/2 )

c

C2

c/2

63 = ( 6 | 3c/6 ) 41 = ( 4 | c/4 )

Hand-Outs: 27-28


( R | ): ( R | ) x = R x +

(4) Glide (reflection) planes: ( mt' | 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.

ba

a glide: ( m010 | a/2 )

Hand-Outs: 27-28


( R | ): ( R | ) x = R x +

(4) Glide (reflection) planes: ( mt' | 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.

ba

a glide: ( m010 | a/2 )

ba

n glide: ( m010 | (a + c)/2 )

Hand-Outs: 27-28

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 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.

Hand-Outs: 29

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.

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, but these do not form 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, 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

I. Structural Aspects 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 ) ( 2 b/2 )

Hand-Outs: 31

I. Structural Aspects 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: 31

I. Structural Aspects Space Groups: International Tables (Symmorphic Group)

Hand-Outs: 32


Point Group of the Space Group

Symmetry Operations

Hand-Outs: 32


GeneratingOperations

Sites inUnit Cells

Have full point symmetry of the space group

Hand-Outs: 32

I. Structural Aspects Space Groups: International Tables (Nonsymmorphic Group)

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

Hand-Outs: 33

I. Structural Aspects Space Groups: Group-Subgroup Relationships

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 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.

Hand-Outs: 34


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


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: MO2 – P42/mnm

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

Hand-Outs: 34

I. Structural Aspects Space GroupsFranzen, pp. 55-77 Combine Translational and Rotational symmetry...

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Transcript of I. Structural Aspects Space GroupsFranzen, pp. 55-77 Combine Translational and Rotational symmetry...