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Symmetry in 2D

1 4/23/2013 L. Viciu| AC II | Symmetry in 2D

Outlook

• Symmetry: definitions, unit cell choice

• Symmetry operations in 2D

• Symmetry combinations

• Plane Point groups

• Plane (space) groups

• Finding the plane group: examples

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Symmetry

The techniques that are used to "take a shape and match it exactly to another” are called transformations

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Inorganic crystals usually have the shape which reflects their internal symmetry

Symmetry is the preservation of form and configuration across a point, a line, or a plane.

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Lattice = an array of points repeating periodically in space (2D or 3D). Motif/Basis = the repeating unit of a pattern (ex. an atom, a group of atoms, a molecule etc.) Unit cell = The smallest repetitive volume of the crystal, which when stacked together with replication reproduces the whole crystal

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Unit cell convention

(b) to (e) correct unit cell: choice of origin is arbitrary but the cells should be identical; (f) incorrect unit cell: not permissible to isolate unit cells from each other (1 and 2 are not identical)

A. West: Solid state chemistry and its applications

By convention the unit cell is chosen so that it is as small as possible while reflecting the full symmetry of the lattice

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Some Definitions

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• Symmetry element: An imaginary geometric entity (line, point, plane) about which a symmetry operation takes place

• Symmetry Operation: a permutation of atoms such that an object (molecule or crystal) is transformed into a state indistinguishable from the starting state

• Invariant point: point that maps onto itself

• Asymmetric unit: The minimum unit from which the structure can be generated by symmetry operations

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D6h or 6/mmm

Plane group = point group symmetry + in plane translation

Space group = point group symmetry + in 3D translation

benzene graphene graphite

From molecular point group to space groups Complete consideration of all symmetry elements and translation yields to the space groups

p6mm P63/mmc

Point group

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• Symmetry operations in 2D*: 1. translation

2. rotations

3. reflections

4. glide reflections

• Symmetry operations in 3D: the same as in 2D

+

inversion center, rotoinversions and screw axes

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* Besides identity

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1. Translation (“move”)

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Translation moves all the points in the asymmetric unit the same distance in the same direction.

There are no invariant points (points that map onto themselves) under a translation. Translation has no effect on the chirality of figures in the plane.

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2. Rotations

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A rotation turns all the points in the asymmetric unit around one axis, the center of rotation.

The center of rotation is the only invariant point. A rotation does not change the chirality of figures.

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Symbols for symmetry axes

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Drawn symbol ---

One fold rotation axis

two fold rotation axis

three fold rotation axis

four fold rotation axis

six fold rotation axis

Axes parallel to the plane

Axes perpendicular to the plane

CRYSTALS MOLECULES

(diad)

(monad)

(triad)

(Tetrad)

(Hexad)

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3. Reflections

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• Symbol: m

• Representation: a solid line

A reflection flips all points in the asymmetric unit over a line called mirror.

The points along the mirror line are all invariant points A reflection changes the chirality of any figures in the asymmetric unit

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4. Glide Reflections

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There are no invariant points under a glide reflection. A glide plane changes the chirality of figures in the asymmetric unit.

Glide reflection reflects the asymmetric unit across a mirror and then translates it parallel to the mirror

• Symbol: g

• Representation: a dashed line 4/23/2013 L. Viciu| AC II | Symmetry in 2D

Point group symmetry

• Point group = the collection of symmetry elements of an isolated shape

• Point group symmetry does not consider translation!

• The symmetry operations must leave every point in the lattice identical therefore the lattice symmetry is also described as the lattice point symmetry

• Plane symmetry group or plane crystallographic group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern

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Examples of plane symmetry in architecture

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2 (two fold axis) 2 mm (two mirror lines and a 2-fold axis)*

4 (four fold axis) 4 mm (4-fold axis and four mirror lines)*

6 mm (6-fold axis and 6 mirror lines)*

6 (six fold axis)

3 m (one 3-fold axis and three mirror lines)

3 (three fold axis)

1 (one fold axis) m (mirror line)

Crystallographic plane point groups = 10 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

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• 5-fold , 7-fold, etc. axes are not compatible with translation non-periodic two dimensional patterns

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Non-periodic 2D patterns

Starfish

5m (five fold axis + mirror)

Group of atoms or viruses can form “quasicrystals” (quasicristals = ordered structural forms that are non-periodic)

Ex:

A Penrose tiling Wikipedia.org

Electron diffraction of a Al-Mn quasicrystal showing 5-fold symmetry by Dan Shechtman

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http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/press.html

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Combining symmetry operations

Five different cell lattice types: 1. oblique(parallelogram) (a ≠ b, ≠ 90°) 2. Rectangular (a b, 90ᵒ) 3. Square (a = b, 90ᵒ) 4. Centered rectangular or diamond (a b, 90ᵒ) 5. Rhombic or hexagonal (a = b, 120ᵒ)

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Ten different plane point groups : 1, 2, 3, 4, 6, m, 2 mm, 3 m, 4 mm, 6 mm

When point group symmetries are combined with the possible lattice cells 17 plane groups.

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1. Combining rotation with translation

1. The rotations will always be to the plane (the space in 2D)

2. An -fold rotation followed by translation to it gives another rotation of the same angle (same order), in the same sense

3. The new rotation will be located at a distance x = T/2 x cotg /2 along perpendicular bisector of T (T=cell edge translation)

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Ex: 2-fold rotation followed by translation (=180)

Steps: 1. 2-fold rotation through A moves the motif from 1 to 2 2. translation by T moves the motif from 2 to 3 Or 1. 2-fold rotation through B moves the motif from 1 to 3

The second rotation will be on T in the middle at B

180 T

x

1

2 3

B A

is the motif

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•2-fold rotation at 1 combined with translation T 1 gives the rotation 6 (rotation 6 is translated to 7 by T2) •2-fold rotation at 1 combined with translation T 2 gives the rotation 8 (rotation 8 is translated to 9 by T1) •2-fold rotation at 1 combined with translation T 1+T2 gives the rotation in the middle of the 8 (same 9)

1

2

4

3

T2

T1

T2

T1

6 7

8

9

T1+T2

The blue, red, green and yellow marked are independent 2-fold axes: they relate different objects pair-wise in the pattern no any pair of the blue and one of the red, green or yellow 2-fold axis describe the same pair-wise relationship

1

2

4

3

Pair of motifs:

2-fold axis combined with translation

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6-fold axis contains 2/6, 2/3, 2/2 rotations

6-fold axis combined with translation

All the operations of a 3-fold axis combined with translation and of a 2-fold axis combined with translation will be included for a p6 plane group

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Combination of the rotation axes with a plane lattice = translation

Two fold axis Three fold axis

Four fold axis Six fold axis

Martin Buerger: An introduction to fundamental geometric features of crystals

Four non-equivalent 2-fold axes to the plane (0 0; ½ ½ , ½ 0, 0 ½ )

Three non-equivalent 3-fold axes to the plane 00, 2/3 1/3, 1/3 2/3)

Two non-equivalent 4-fold axes to the plane; One non-equivalent 2-fold axis to the plane; (00, ½ ½) and ( ½ 0, 0 ½ )

One non-equivalent 6-fold axes to the plane; One non-equivalent 3-fold axis to the plane; (0 0) ; (2/3 1/3 , 1/3 2/3) and ( ½ 0)

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2. Combining a reflection with translation

A reflection combined with a translation to it is another reflection at ½ of that perpendicular translation

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*the mirror 2 is situated at ½ distance of the translation

1 2 3

2 1

1. A rectangular cell

- Pair of motifs

The mirror 2 is independent from 1 because the position of the objects (1 and 2) relative to the mirror in the center (2)of the cell is distinct from the position of the same objects relative to the first mirror (1)

321 1 nTranslatio

*2

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1

T

T

T *T(T+T)=glide plane The glide will be at the half distance of T

2. A centered rectangular cell

1 and 2 are equivalent because we must have a motif in the center

A glide line results in here

A glide is the result of a reflection and a translation

1 11 12

2 1

2 21

- Pair of motifs

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3. Combining a glide with a translation

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1

g2 g1

2 3

The glide g2 is situated at half of the translation which is perpendicular to it

Reflecting 1 by a mirror in the center of the edges gives 3’; Gliding 3’ half of Tparallel gives 3

1

g2 g1

2 3

3’

T()

1. A rectangular cell

321 nTranslatiogliding

gliding by g2

- Motif

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2. A centered rectangular cell

Combining a glide plane with a translation in a centered rectangular lattice gives a mirror plane situated at ½ of T/2.

1 2

g2 g1

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4. Combining two reflections

• The operation of applying two reflections in which the mirror planes (1 and 2) are making an angle with each other is the same with the rotation by an 2 angle

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Guide to the eye

Two reflections: 1 1’ by reflection on 1

1’ 2 by reflection on 2

1 2

1

1’ 2

One rotation: 1 2 by two times rotation

2'11 21

rotation by 2

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5. Combining a rotation with a reflection

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A rotation by followed by a reflection 1 will result in another reflection which will be situated at an angle /2 relative to the first reflection

321 1 byreflectionbyrotation

reflection by 2

1

3

2 1

2 3 1

2

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Combining symmetry operations

1. Oblique (parallelogram) (a ≠ b, ≠ 90°)

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Plane groups p1 and p2 p stands for the fact that we have only one lattice point per cell primitive lattice

p1 p2

Examples of motifs having point group 1:

and

Examples of motifs having point group 2: and (The motif itself should have a 2-fold axis)

(The motif itself should have no symmetry)

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Plane group symbol rules/meaning

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1. First letter: p or c translation symmetry + type of centering

2. The orientation of the symmetry elements: to coordinate system x, y and z.

The highest multiplicity axis or if only one symmetry axis present they are on z

Ex: p4mm: 4-fold axis in the z direction; p3m1: 3-fold axis in the z direction

The highest symmetry axis is mentioned first and the rest are omitted

ex: p4mmm: 4-fold axis on z and two 2-fold axes are omitted

If highest multiplicity axis is 2-fold the sequence is x-y-z

ex: pmm2; pgm2; cmm2: 2-fold axis on z

3. The addition of 1 is often used as a place holder to ensure the mirror or glide line

is correctly placed

ex: p3m1 and p31m

m y m x m z

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2. Rectangular (a b, 90ᵒ)

Plane groups: pm, pg, pmg2, pmm2 and pgg2

Possible motifs: m 2mm

pmg2 pgg2 pmm2

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2. Examples of Rectangular plane groups with glide lines

pmg2

pgg2

motif:

pmg2 pgg2

motif:

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3. Square (a = b, 90ᵒ)

Plane groups: p4, p4mm and p4gm

Possible motifs:

4 4mm 4/23/2013 L. Viciu| AC II | Symmetry in 2D

Questions to recognize a square plane group

1. Is there a 4-fold axis?

It should be otherwise it cannot be a square lattice

2. Is there a mirror line in there?

If No, then is a p4 plane group

If “Yes”,

3. Is the mirror line passing through a 4-fold axis?

If “Yes” then the plane group is p4mm

If “No” then the group is a p4mg

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4. Centered rectangular (a b, 90ᵒ)

Plane groups: cm and cmm2

m 2mm

Possible motifs:

The dash lined cell is known as diamond or rhombus cell

cmm2

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a a=b

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The diamond lattice has a mirror through it such that always a = b but the angle is general

The centered rectangular lattice has now 2 atoms per unit cell

The centered rectangular lattice has 2-fold redundancy (two diamond unit cells) but it has the big advantage of an orthogonal coordinate system. Therefore it is the standard cell

Diamond vs. centered rectangular

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5. Rhombic or hexagonal (a = b, 120ᵒ)

Plane groups: p3, p31m, p3m1, p6 and p6mm

6 6mm 3 3m

Possible motifs:

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How the motifs are oriented in p3m plane group

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p3m1 p31m The mirrors are to the translation (the translation comes in the middle of the mirrors)

The translation is along the mirror planes

On the second place in the plane group symbol comes what is to the cell edge and on the third place comes what is to the cell edge

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When we have translations which are inclined to the mirrors like in p3m1 plane group, a glide is always interleaved between the two mirrors

The glide is parallel to the mirrors at half distance between them

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1 2

a) the inclination of translation relative to the mirrors

b) the location of glide (between the mirrors at the half distance)

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a) The inclination of the translation relative to the mirrors

b) The location of the glides (between the mirrors at the half distance)

When we have translations which are inclined to the mirrors like in p3m1 plane group, a glide is always interleaved between the two mirrors.

The glide is parallel to the mirrors at half distance between them.

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The p6mm plane group has the symmetry elements of both p3m1 and p31m groups because both of these groups are present simultaneously in p6mm plane group.

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p3m1 +p31m

When we add the symmetry elements we should make sure that all the symmetry elements are left invariant (we don’t create additional translations or consequently more axes and planes;

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Symmetry Elements of the 2D Space Groups

glide line 2, 3, 4, 6 – fold axes

Unit cell edge mirror line 4/23/2013

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The atom at the lattice point has the coordinates: (x, y)

y

x The atom will be then moved by translation to every lattice point

y

x

y

x

The 2 – fold axes place the atoms at the opposite direction

1-x

1-y

y

x

It is possible to say also 1-x 1-y

But is more esthetic to give the

positions x y and yx

The equivalence of atom positions results from translation

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1. Highest order

rotation?

2. Has reflection?

Yes No

6-fold p6mm p6

4-fold 3. Has mirrors at 45°?

p4 Yes: p4mm No: p4gm

3-fold 3. Has rot. centre off mirrors?

p3 Yes: p31m No: p3m1

2-fold

3. Has perpendicular reflections? Has glide reflection?

Yes No

Has rot. centre off mirrors? pmg2 Yes: pgg2 No: p2

Yes: cmm2 No: pmm2

none Has glide axis off mirrors? Has glide reflection?

Yes: cm No: pm Yes: pg No: p1

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1. Locate the motif present in the pattern. This can be a molecule, molecules, atom,

group of atoms, a shape or group of shapes. The motif can usually be discovered

by noting the periodicity of the pattern.

2. Identify any symmetry elements in the motif.

3. Locate a single lattice point for each occurrence of the motif. It is a good idea to

locate the lattice points at a symmetry element location.

4. Connect the lattice points to form the unit cell.

5. Determine the plane group by comparing the symmetry elements present to the

17 plane patterns.

Fundamental Steps in Plane Groups Identification

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Finding the plane group

No symmetry besides translation: The lattice type is oblique, plane group p1. Each unit mesh (unit cell) contains 1 white bird and 1 blue bird.

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Finding the plane group

No symmetry besides translation: The lattice type is oblique, plane group p1. Each unit mesh (unit cell) contains 1 white bird and 1 blue bird.

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Finding the plane group

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Finding the plane group

1. Highest order rotation? A: 2 2. Has reflections? A: yes 3. Has rotation centers off mirrors? A: yes 4. Space group: A: cmm2

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The unit cell is square.

Symmetry elements:

-2-fold axis

-Two mirror lines ( to each

other)

- Two glide lines

Plane group: cmm2

Finding the plane group

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Finding the plane group

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Finding the plane group

1. Highest order rotation? A: 3 2. Has reflections? A: yes 3. Has rotation centers off mirrors? A: No 4. Space group: A: p3m1

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Finding the plane group

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Finding the plane group

1. Highest order rotation? A: 6 2. Has reflections? A: yes 3. Space group: A: p6mm

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56 Christopher Hammond: The basics of crystallography and diffraction (third edition)

Finding the plane group

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57 Christopher Hammond: The basics of crystallography and diffraction (third edition)

Finding the plane group

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Finding the plane group

p4gm

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Finding the plane group

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Finding the plane group