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References

Space Groups for Solid State Scientists, G. Burns and A. M. Glazer(Little mathematical formalism; prose style)

International Tables for Crystallography (http://it.iucr.org/) Bilbao Crystallographic Server (www.cryst.ehu.es)

(Comprehensive resources for all space groups)

Physics 590“International Tables of Crystallography”

“Everything you wanted to know about beautiful flies, but were afraid to ask.”

Schönflies

Proposed Plan(1) What basic information is found on the space group pages…(2) Stoichiometry of the unit cell (Wyckoff sites)(3) Site (point) symmetry of atoms in solids(4) Solid-solid phase transitions (group-subgroup relationships)(5) Diffraction conditions – what to expect in a XRD powder pattern.

Gordie Miller (321 Spedding)

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BaFe2As2

Intensity(Arb. Units)

2θ (Cu Kα)

Space Group: I4/mmmLattice Constants: a = 3.9630 Å

c = 13.0462 ÅAsymmetric Unit:

Ba (2a): 0 0 0Fe (4d): ½ 0 ¼As (4e): 0 0 0.3544

(002) (011)

(004)

(013)

(112)

(015)

(200) (116)(213)

(215) (028)

(hkl) Indices

What can we learn from the International Tables?

h + k + l = even integer (2n)

a b

c

Typical Space Group Pages…

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Symbolism

Point Symmetry FeaturesStoichiometryStructure of Unit Cell

Subgroup/SupergroupRelationships

DiffractionExtinctionConditions

Symbolism

Space Group

CrystalSystem

Molecules Solids

4

NOTE: nS nIn Schönflies notation, what does the symbol S2 mean?

Why are the symbols S2 and not used?

In International notation, what does the symbol mean?2

Point Group of the Space Group

C2 rotation followed by sh C2 axis(x,y,z)

( )x, y,z ,( )x, y,z

2-fold (C2) rotation followed by inversion ( )

Symmetry Operation Schönflies Notation International Notation

Proper Rotation (by 2π/n) Cn (C2, C3, C4, …) “n” (2, 3, 4, …)

Identity E = C1 1

Improper Rotation Sn = h Cn (S3, S4, S5, …)

Inversion (x,y,z) (–x,–y,–z) i = S2

Mirror plane Principal Axis h = S1 /m (n/m is the designator: 4/m)

Mirror plane Principal Axis v , d (= S1) m =

1

1 (3, 4,5, )n n

2

(x,y,z)

( )x, y,z

,

( )x, y,z

S2 = h C2

2 1 2

x

y

x

y1

(z)

(z)

S2 = inversion

2 = reflection

2

2 2S

Symbolism: Crystal Systems

What rotational symmetries are consistent with a lattice (translational symmetry)?

C1 C2 (2π/2) C3 (2π/3) C4 (2π/4) C6 (2π/6)

Crystal System Minimum Symmetry Primitive Unit Cell Lattice Types

Triclinic None a b c;

Monoclinic One 2-fold axis (b-axis) a b c; = = 90, 90

Orthorhombic Three orthogonal 2-fold axes a b c; = = = 90

Tetragonal One 4-fold axis (c-axis) a = b c; = = = 90

Cubic Four 3-fold axes a = b = c; = = = 90

Trigonal One 3-fold axisa = b = c; = = a = b c; = = 90, = 120

Hexagonal One 6-fold axis (c-axis) a = b c; = = 90, = 120

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ba

= angle between b and c

= angle between a and c

= angle between a and bc

Symbolism: Bravais Lattices

cba

= angle between b and c

= angle between a and c

= angle between a and b

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Crystal System Minimum Symmetry Primitive Unit Cell Lattice Types

Triclinic None a b c; P

Monoclinic One 2-fold axis (b-axis) a b c; = = 90, 90 P C

Orthorhombic Three orthogonal 2-fold axes a b c; = = = 90 P C (A) I F

Tetragonal One 4-fold axis (c-axis) a = b c; = = = 90 P I

Cubic Four 3-fold axes a = b = c; = = = 90 P I F

Trigonal One 3-fold axisa = b = c; = = a = b c; = = 90, = 120

RP

Hexagonal One 6-fold axis (c-axis) a = b c; = = 90, = 120 P

7 Crystal Systems = 7 Primitive Lattices (Unit Cells): P

?

(rhombohedral)

“Centered Lattices”

I F C B A

Body- (All) Face- Base-

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Symbolism: Point Groups

Type Symbol Features

Uniaxial n Single rotation axis Cn

nh + mirror plane Cn axis

nv + n mirror planes || Cn axis

Low

Symmetry

1 Asymmetric (NO symmetry)

s Mirror plane only

i Inversion center only

Dihedral n Rotation axis Cn + n C2 axes Cn axis

nd + n mirror planes || Cn axis

nh + mirror plane Cn axis

Polyhedral T, Th , Td Tetrahedral; 4 C3 axes (cube body-diagonals)

O, Oh Octahedral; 4 C3 axes + 3 C4 axes (cube faces)

I, Ih Icosahedral; 6 C5 axes

Schönflies Notation

Symbolism: Crystallographic Point Groups

Crystal System

Schönflies Symbol

International SymbolOrder /Inversion?Full Abbrev. Directions

Triclinic 11 1 1 / No

(Holohedral) i2 / Yes

Monoclinic s1m1 or 11m m [010] or [001] 2 / No

2121 or 112 2 2 / No

(Holohedral) 2h12/m1 or 112/m 2/m 4 / Yes

Orthorhombic 2v2mm 2mm [100][010][001] 4 / No

2222 222 4 / No

(Holohedral) 2h2/m 2/m 2/m mmm 8 / Yes

Tetragonal 44 4 [001]{100}{110} 4 / No

44 / No

4h4/m 4/m 8 / Yes

2d 8 / No

4v4mm 4mm 8 / No

4422 422 8 / No

(Holohedral) 4h4/m 2/m 2/m 4/mmm 16 / Yes

Allowed Rotations = C1 C2 C3 C4 C6

b c

b ca

c ab

a+ba–b

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1 1

4 4

42m 42m

32 Point Groups

4 2m

m2m or mm2

Yes:Laue Groups(2)

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Questions for Friday…

For the following space group symbol

(a) What is the crystal class?

(b)What is the lattice type?

Which face(s) are centered?

(c) What is the point group of the space group using Schönflies notation?

(d)Does the point group contain the inversion operation?

Cmm2

Orthorhombic

Base (C)-centered

C2v

No

ab-faces

Symbolism: Crystallographic Point Groups (cont.)

Crystal System

Schönflies Symbol

International SymbolOrder / Inversion?Full Abbrev. Directions

Trigonal 33 3 [001]{100}{210} 3 / No

66 / Yes

3v3m or 3m1 3m or 3m1 6 / No

332 or 321 32 or 321 6 / No

(Holohedral) 3d 12 / Yes

Hexagonal 66 6 [001]{100}{210} 6 / No

3h6 / No

6h6/m 6/m 12 / Yes

3h 12 / No

6v6mm 6mm 12 / No

6622 622 12 / No

(Holohedral) 6h6/m 2/m 2/m 6/mmm 24 / Yes

Cubic T 23 23 {100}{111}{110} 12 / No

Th 24 / Yes

Td 24 / No

O 432 432 24 / No

(Holohedral) Oh 48 / Yes

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c ab

3 3

31m

32 / or 32 / 1m m 31m

6 6 c ab

6 2m 6 2m 62m

2 / 3m 3m abc

43m 43m

4 / 3 2 /m m 3m m

312

3 or 3 1m m

Schönflies International Coordinates Schönflies International Coordinates

(1) E 1 x, y, z (9) i –x, –y, –z

(2) C2 = C42 2 0,0,z –x, –y, z (10) σh m x,y,0 x, y, –z

(3) C4 4+ 0,0,z –y, x, z (11) S43 0,0,z; 0,0,0 y, –x, –z

(4) C43 4– 0,0,z y, –x, z (12) S4 0,0,z; 0,0,0 –y, x, –z

(5) C2 2 0,y,0 –x, y, –z (13) σv m x,0,z x, –y, z

(6) C2 2 x,0,0 x, –y, –z (14) σv m 0,y,z –x, y, z

(7) C2 2 x,x,0 y, x, –z (15) σd m x,,z –y, –x, z

(8) C2 2 x,,0 –y, –x, –z (16) σd m x,x,z y, x, z

x

y

Proper Rotations

33 Matrices:

4

0 1 0

1 0 0

0 0 1

C

(Determinant = +1)

Improper Rotations

4

0 1 0

1 0 0

0 0 1

S

(Determinant = 1)

Symbolism: Symmetry Operations (4h = 4/mmm = 4/m 2/m 2/m)

+

++

+

–

––

–

–

, ,

,

,

,

,

,,

–

–

–

+

+

+

+

11

4 / m m m

Symmorphic Space Groups

12

GeneralPosition

SpecialPositions

Space Group = {Essential Symmetry Operations} {Bravais Lattice} (230)

Point Group = {Symmetry operations intersecting in one point} (32)

Symmorphic Space Groups

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Space Group: I4/mmmLattice Constants: a = 3.9630 Å c = 13.0462 ÅAsymmetric Unit:

Ba (2a): 0 0 0Fe (4d): ½ 0 ¼As (4e): 0 0 0.3544

BaFe2As2

GeneralPosition

SpecialPositions“Ba2Fe4As4”

Z = 2

Ba (2a): 4/mmm (D4h)

Fe (4d): m2 (D2d)

As (4e): 4mm (C4v)

Space Groups (230)

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Symmorphic Space Groups (73): {Essential Symmetry Operations} is a group.

Nonsymmorphic Space Groups (157): {Essential Symmetry Operations} is a not a group.

Point Group of the Space Group

Space Group Operations: Screw Rotations and Glide Reflections

Screw Rotations: Rotation by 2/n (Cn) then Displacement j/n lattice vector || Cn axis (allowed integers j = 1,…, n–1)Symbol = nj

21, 31, 32, 41, 42, 43, 61, 62, 63, 64, 65

Glide Reflections: Reflection then Displacement 1/2 lattice vector || reflection

plane

Axial: a, b, c (lattice vectors = a, b, c)Diagonal: n (vectors = a+b, a+c, b+c)Diamond: d (vectors = (a+b+c)/2, (a+b)/2,

(b+c)/2, (b+c)/2)

I41/amd

P42/ncm

4/mmm

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Nonsymmorphic Space Groups (157)

Point Group of the Space Group

The Origin! Si: 0, 0, 0

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