Introduction to point and space group symmetry

Humboldt-Universität zu Berlin, Institut für Physik, AG TEM Newtonstrasse 15, D-12489 Berlin E-mail: holm.kirmse@physik.hu-berlin.de Web: http://crysta.physik.hu-berlin.de/ag_tem/

Holm Kirmse

Workshop on Electron Crystallography, Nelson Mandela Metropolitan University, South Africa, October 14-16, 2013

Embedding Crystallography

Mathematics Lattice, Symmetry operations, Group theory

2

Physics Transparency

Conductivity

Birefringence

Piezoelectricity

Pyroelectricity

Chemistry Bonding

Composition

Structure

Cleavability

Polarity

CRYSTALLO-GRAPHY

Topics of Crystallography

3

Geometric Crystallography

Structure analysis X ray and electron

diffraction

Crystal growth

Crystal chemistry Crystal physics Crystal defects

Geometrical Crystallography, e.g., Au

4

Crystal system Coordinate system

Symmetry element

Projection

view [100] Indices Crystal classes (cubooktahedral)

Crystal shape

(111)

facet

of

hedron

octa-

(100)

Road map Introduction

• What is a crystal?

• Definition of the 7 crystal systems

• Indexing planes and directions

• Bravais lattices

• Stereographic projection

• Symmetry operations of point groups

• The 32 point groups

• From point groups to layer groups

• Symmetry operations of layer groups

• The 17 layer groups

• Transition to third dimension: space groups

• Symmetry operations of space groups

• Example for determination of a space group

• Notations 5

What is a crystal?

• Characteristics of an ideal crystal:

– Flat regular surfaces/facets

– Characteristic symmetry

– Chemically homogeneous object

– Anisotropic properties

– Periodic arrangement of constituents like atoms or molecules along the three spatial directions

6

Rock crystal, SiO2

Definition of a crystal

– A crystal is a homogeneous anisotropic solid with periodic arrangement of constituents like atoms or molecules in all three spatial dimensions.

– Since 1991: A crystal is defined as a solid exhibiting discrete diffraction spots. This definition includes real crystals with defects, incommensurate crystals and quasicrystals.

7

Coordinate system

• 3-dimensional lattices are described by three not necessarily orthogonal directions x, y, z having lattice parameters a, b, c.

• The angle between the three directions are denoted as a, b, g.

• As a convention a right-handed coordinate system is used.

9

a b

c

a b

g x

z

y

The 7 crystal systems

Crystal system Lattice parameters

Triclinic a ≠ b ≠ c a ≠ b ≠ g ≠ 90°

Monoclinic a ≠ b ≠ c a = g = 90° b ≠ 90°

Orthorhombic a ≠ b ≠ c a = b = g = 90°

Tetragonal a = b ≠ c a = b = g = 90°

Trigonal a = b = c a = b = g ≠ 90°

a = b ≠ c a = b = 90° g = 120°

Hexagonal a = b ≠ c a = b = 90° g = 120°

Cubic a = b = c a = b = g = 90°

10

Indexing of crystallographic directions

• Two-dimensional lattice

12

Crystallographic direction: straight line crossing two arbitrary points of the lattice

a

b

[010] [120] [110] [210]

Iuvw = 0·a + 1·b + 0·c Description analogous to vectors

Indexing of crystallographic directions

• Three-dimensional lattice

13

Triple product: (a × b) · c > 0, i.e. non-coplanar

Point in space: Iuvw = u·a + v·b + w·c ; u,v,w Z

Infinite periodic arrangement of points within the 3-d space

c

a

b

Indexing of crystallographic planes

• Derivation of MILLER indices:

14

O

a

A

c

b

C

B

p:n:mc

OC:b

OB:a

OA

000

m : n : p = 2 : 4 : 3

3

1:4

1:2

1

p

1:n

1:m

1

h : k : l = 6 : 3 : 4 (6 3 4) plane

WEISS indices:

MILLER indices

The 14 Bravais lattices (A. Bravais, 1850)

20

Crystal system

Cente-ring

Symbol Lattice

Triclinic Prim. aP

Monocl.

Prim. mP

Face cent.

mA

Orthorh.

Prim. oP

Body cent.

oI

Basal-pl. cent.

oC

Face cent.

oF

Crystal system

Cente-ring

Symbol Lattice

Tetrag.

Prim. tP

Body cent.

tI

Trigonal Rhom-bohedr.

hR

Trig. + Hexag.

Prim. hP

Cubic

Prim. cP

Body cent.

cI

Face cent.

cF

From inner structure to morphology: description of crystals

• Correspondence between morphology and structure

• Any crystal face (morphology) is oriented parallel to a lattice plane (structure).

• The symmetry of the outer shape of a crystal is higher or equal to the symmetry of its inner structure.

22

Example: Galenite (PbS):

Crystal projections: stereographic projection

23

• Projection method:

Pole sphere

Observation spot

Face normal

Pole of face

North pole of pole sphere

Projection plane Projected pole

South pole of pole sphere

Crystal projections

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Stereographic projection Projection onto pole sphere

Observation spot

Crystal projections: stereographic projection

25

• Examples for stereograms:

Hexahedron Octahedron

Rhombic dodecahedron Polyhedron:

Stereo- gram

111

010

001

010001

111111

111 101

011

110 011

110

101011

011

100

100

111

111111

111

101

110

110 101

( )

( )

( )

( )

( )

( ) ( )

( ) ( )

( )

( ) ( )

( )

( )

( )

( ) ( ) ( )

( )

( )

( ) ( )

( )

( ) ( )

( )

Symmetry of crystals

Point symmetry operations:

27

Symmetry operation with at least one point of the object remains at its original position. The corresponding symmetry element is called point symmetry element.

Trivial point symmetry operations : ROTATION, INVERSION, REFLECTION

Combined point symmetry operations : ROTATION INVERSION, REFLECTION ROTATION

Point symmetry elements in 3-d space: Rotation axis, inversion centre, mirror plane, rotation inversion axis, reflection rotation plane

Point symmetry operations

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• Rotation:

4-fold rotation Rotation angle: 90° Symbol: 4 Graphic symbol:

Stereogram: Shape: Impact on structure motive:

Tetragonal pyramid

Point symmetry operations

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• Rotation Number of positions

Angle Symbol Stereogram Shape

1 360° 1 - Pedion

2 180° 2 Sphenoid

3 120° 3 Trigonal pyramid

4 90° 4 Tetragonal pyramid

6 60° 6 Hexagonal pyramid

Point symmetry operations

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• Inversion

Stereogram: Shape: Impact on structure motiv:

Symbol: 1 Graphic symbol:

Pinakoid

Inversion centre

Point symmetry operations

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• Reflection

Stereogram: Shape:

Doma

Impact on structure motiv:

Mirror plane

Symbol: m (mirror) = 2 + 1 Graphic symbol:

Point symmetry operations

32

• Rotation inversion

4-fold rotation inversion Symmetry operation: 90° + Inversion Symbol: 4 Graphic symbol:

Stereogram: Shape: Impact on structure motiv:

Tetragonal disphenoid

Point symmetry operations

33

• Rotation inversion Number of positions

Angle Symbol Stereogram Shape

1 360° 1 Pinakoid

2 180° 2 Doma

3 120° 3 Rhombo-hedron

4 90° 4 Tetragonal disphenoid

6 60° 6 Hexagonal dipyramid

The 32 point symmetry groups

35

X X Xm Xm X2 m

tric

linic

1

mo

no

clin

ic

ort

ho

rh.

2

trig

on

al

3

tetr

ago

nal

4

hex

ago

nal

6

cub

ic

23

X m

X m

1

2

3

4

6

3m

232

2m

1

m

2

m

4

m

6

6m

3

3m

2

2m1 m

2m1 212 2mmmm

1

2mm

m3

mm4

mm6

3m

23m2

2mmm2

m

23

m24

m26

m34

222

32

422

622

432

m

2

m

2

m

2

m26mm

3

m

2

m

2

m

4

m

2

m

2

m

6

m

23m

4

Notation of point symmetry groups

• Notation following Hermann-Maugin

36

HA: main axis, NA: minor axis, ZA: intermediate axis

Crystal system 1st position 2nd position 3rd position

Triclinic x - -

Monoclinic y - -

Orthorhombic x y z

Trigonal z (HA) x (NA) -

Tetragonal z (HA) x (NA) xy (ZA)

Hexagonal z (HA) x (NA) xy (ZA)

Cubic [100] [111] [110]

n: n-fold rotation axis, n: n-fold rotation inversion axis, m: mirror plane, : n-fold rotation axis with mirror plane

m

n

Example for the determination of a point symmetry group

49

Crystal system: tetragonal

Crystal class: ditetragonal-pyramidal

Symbol following Hermann-Maugin:

Symmetry elements

Stereographic projection

Ditetragonal pyramid

mm4

8 (+1) facets

Morphology / shape of crystal

The two-dimensional lattice

• Infinite periodic arrangement of points (i.e. atoms, ions, or molecules) at a plane

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a

b

|a| = a |b| = b |a x b| > 0 (i.e. non-collinear)

Expression for a single point: Iuv = u·a + v·b; u,v Z

Two-dimensional patterns

Topkapi palace

Europe

Asia

Black sea

Sea of Marmara

Istanbul

Symmetry operation: Translation

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• Shift of a motive (asymmetric unit: atoms, ions, molecules) by translation vector t

t

No longer pure point symmetry only

Space filling

New symmetry operations

Combination of translation and reflection

55

• Translation t • Reflection m

New symmetry operation: Glide reflection g

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Simultaneous application of translation and reflection:

Step 1: Translation by t = ½ a0

Step 2: Reflection

Symmetry elements of layer groups (wallpaper groups)

Reflection m Rotation

Glide reflection g

Motive Angle Multiplicity Symbol

360° 1

180° 2

120° 3

90° 4

60° 6

Symbol

Glide component: t/2

t

Lattice types

Primitive: p Centered: c

The 17 layer groups (wallpaper groups) oblique

rectangular

quadratic

hexagonal

p 1 p 211

p 1m1 p 1g1

c 1m1 p 2mm

p 2mg p 2gg c 2mm

p 4

p 4gm

p 4mm

p 3 p 3m1

p 31m p 6 p 6mm

The 17 layer groups (wallpaper groups) p 1 p 211

p 1m1 p 1g1

c 1m1 p 2mm

p 2mg p 2gg c 2mm

p 4

p 4gm

p 4mm

p 3 p 3m1

p 31m p 6 p 6mm

oblique

rectangular

quadratic

hexagonal

Quadratic System: p 4gm

p 4gm

Topkapi palace

The three-dimensional lattice

• Infinite periodic arrangement of points within the 3-d space

63

c

a

b

Space groups

65

• Space groups for describing symmetry relations in 3-d space.

• Space groups include all symmetry operations of a 3-dimensional, infinitely extended, and perfect crystal structure.

• The notation of the space group is done after Hermann-Mauguin.

• Number of Space groups in the 3-d space: 230

• But! By definition there is an infinite number of space groups!

• There are 73 types of space groups comprising the identical (point) symmetry elements as the point symmetry group but extended by the translation operation: These are the symmorphic space groups.

Symmetry operation: Glide reflection

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m a, b c n e

Symmetry operation: Glide reflection

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n d

(a+b)/4, (a+c)/4, (b+c)/4, (a+b+c)/4

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

Symmetry operation: Screw rotation

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• 2-fold rotation • 2-fold screw rotation

= ½ ao

• Translation period of a screw axis np: = p/n

2 21

Symmetry operation: Screw rotation

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= 1/4 = 2/4 = 3/4

41 43

42

• 41 and 43 are enantiomorphous screw axis

• 41 dextrorotatory, 43 laevorotatory, 42 without rotary sense like 2

• 4-fold screw rotation:

Symbols for orientation of rotation axis along the viewing direction

Symbols of rotation axes and screw rotation axes

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Symbols for inclined rotation axis

Example for a space group

• Marcasite (FeS2)

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½ +

½ +

½ -

½ -

½

+

-

+

-

Projection onto basal plane

Example for a space group

• Marcasite (FeS2)

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½ +

½ +

½ -

½ -

½

+

-

+

-

Space group:

¼, ¾ ¼, ¾

¼, ¾ ¼, ¾ ¼, ¾

n

2

m

2

n

2P 11

Notation of space groups

• Notation following Hermann-Maugin

76

HA: main axis, NA: minor axis, ZA: intermediate axis

Crystal system 1st position 2nd position 3rd position 4th position

Triclinic Lattice type x - -

Monoclinic Lattice type y - -

Orthorhombic Lattice type x y z

Trigonal Lattice type z (HA) x (NA) -

Tetragonal Lattice type z (HA) x (NA) xy (ZA)

Hexagonal Lattice type z (HA) x (NA) xy (ZA)

Cubic Lattice type [100] [111] [110]

n: n-fold rotation axis, n: n-fold rotation inversion axis, np: n-fold screw axis, m: mirror plane, : n-fold rotation axis with mirror plane

m

n

Introduction

What is a crystal?

Definition of the 7 crystal systems

Indexing planes and directions

Bravais lattices

Stereographic projection

Symmetry operations of point groups

The 32 point groups

From point groups to layer groups

Symmetry operations of layer groups

The 17 layer groups

Transition to third dimension: space groups

Symmetry operations of space groups

Example for determination of a space group

Notations

Road map

77