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in 2D

4/24/2013 L. Viciu| AC II | Symmetry in 2D 1 Outlook

• Symmetry: definitions, unit cell choice • Symmetry operations in 2D • Symmetry combinations • Point groups • Plane (space) groups • Finding the plane : examples

4/24/2013 L. Viciu| AC II | Symmetry in 2D 2 Symmetry  Symmetry is the preservation of form and configuration across a point, a line, or a plane.

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

 Inorganic crystals usually have the shape which reflects their internal symmetry

4/24/2013 L. Viciu| AC II | Symmetry in 2D 3 = 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

4/24/2013 L. Viciu| AC II | Symmetry in 2D 4 Unit cell convention By convention the unit cell is chosen so that it is as small as possible while reflecting the full symmetry of the lattice

(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)4/24/2013 L. Viciu| AC II | Symmetry in 2D 5 A. West: Solid state chemistry and its applications Some Definitions

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

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

point: point that maps onto itself

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

4/24/2013 L. Viciu| AC II | Symmetry in 2D 6 From molecular to space groups Complete consideration of all symmetry elements and yields to the space groups benzene graphene graphite

D6h or 6/mmm p6mm P63/mmc Point group Plane group = point = point group symmetry + group symmetry + in plane translation in 3D translation 4/24/2013 L. Viciu| AC II | Symmetry in 2D 7 • Symmetry operations in 2D*: 1. translation 2. 3. reflections 4. glide reflections

• Symmetry operations in 3D: the same as in 2D + inversion center, rotoinversions and screw axes

* Besides identity 4/24/2013 L. Viciu| AC II | Symmetry in 2D 8 1. Translation (“move”)

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 of figures in the plane.

4/24/2013 L. Viciu| AC II | Symmetry in 2D 9 2. Rotations A 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.

4/24/2013 L. Viciu| AC II | Symmetry in 2D 10 Symbols for symmetry axes

Drawn symbol

One fold rotation axis --- (monad)

two fold rotation axis (diad) Axes to the plane

Axes to the plane three fold rotation axis (triad) four fold rotation axis (Tetrad) six fold rotation axis (Hexad)

CRYSTALS MOLECULES

4/24/2013 L. Viciu| AC II | Symmetry in 2D 11 3. Reflections

A 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

• Symbol: m • Representation: a solid line

4/24/2013 L. Viciu| AC II | Symmetry in 2D 12 4. Glide Reflections reflects the asymmetric unit across a mirror and then translates it parallel to the mirror

There are no invariant points under a glide reflection.

A changes the chirality of figures in the asymmetric unit.

• Symbol: g • Representation: a dashed line 4/24/2013 L. Viciu| AC II | Symmetry in 2D 13 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 or plane crystallographic group is a mathematical classification of a two-dimensional repetitive pattern, based on the in the pattern

4/24/2013 L. Viciu| AC II | Symmetry in 2D 14 Examples of plane symmetry in architecture

4/24/2013 L. Viciu| AC II | Symmetry in 2D 15 Crystallographic plane point groups = 10 1. 1 (one fold axis) 6. m (mirror line)

2. 2 (two fold axis) 7. 2 mm (two mirror lines and a 2-fold axis)*

3. 3 (three fold axis) 3 m (one 3-fold axis and 8. three mirror lines)

4. 4 (four fold axis) 9. 4 mm (4-fold axis and four mirror lines)*

6 mm (6-fold axis 10. and 6 mirror lines)* 5. 6 (six fold axis) 4/24/2013 L. Viciu| AC II | Symmetry in 2D 16 * Second “m” in the symbol refers to the second type of mirror line Non-periodic 2D patterns • 5-fold , 7-fold, etc. axes are not compatible with translation  non-periodic two dimensional patterns Ex: Starfish

Wikipedia.org 5m (five fold axis + mirror) A Penrose tiling Group of atoms or viruses can form “quasicrystals” (quasicristals = ordered structural forms that are non-periodic)

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

4/24/2013 L. Viciu| AC II | Symmetry in 2D 17 http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/press.html

4/24/2013 L. Viciu| AC II | Symmetry in 2D 18 Combining symmetry operations

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

 Five different cell lattice types:

1. oblique(parallelogram) (a ≠ b, ≠ 90°) 2. Rectangular (a  b, 90ᵒ) 3. (a = b, 90ᵒ) 4. Centered rectangular or diamond (a  b, 90ᵒ) 5. Rhombic or hexagonal (a = b, 120ᵒ) When point group symmetries are combined with the possible lattice cells  17 plane groups.

4/24/2013 L. Viciu| AC II | Symmetry in 2D 19 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)

Ex: 2-fold rotation followed by translation (=180) 1 The second rotation will be on T in the middle at B B T 180 A x

Steps: 2 3 1. 2-fold rotation through A moves the motif from 1 to 2 2. translation by T moves the motif from 2 to 3 is the motif Or 1. 2-fold rotation through B moves the motif from 1 to 3 4/24/2013 L. Viciu| AC II | Symmetry in 2D 20 2-fold axis combined with translation 1 8 4 1 4 T2 T1 T1 T1+T2 Pair of motifs: 6 7

T2 2 9 3 2 3

•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

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 same4/24/2013 pair -wise relationship L. Viciu| AC II | Symmetry in 2D 21 6-fold axis combined with translation

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

4/24/2013 L. Viciu| AC II | Symmetry in 2D 22 Combination of the rotation axes with a plane lattice = translation Two fold axis Three fold axis

Four non-equivalent 2-fold axes  to the plane Three non-equivalent 3-fold axes  to the plane (0 0; ½ ½ , ½ 0, 0 ½ ) 2 1 1 2 00, /3 /3, /3 /3) Four fold axis Six fold axis

Two non-equivalent 4-fold axes  to the plane; One non-equivalent 6-fold axes  to the plane; One non-equivalent 2-fold axis  to the plane; One non-equivalent 3-fold axis  to the plane; 2 1 1 2 (00, ½ ½) and ( ½ 0, 0 ½ ) (0 0) ; ( /3 /3 , /3 /3) and ( ½ 0) 4/24/2013 L. Viciu| AC II | Symmetry in 2D Martin Buerger: An introduction to fundamental geometric features of crystals 23 2. Combining a reflection with translation A reflection combined with a translation  to it is another reflection at ½ of that perpendicular translation 1. A rectangular cell 11 2 Translation3 1 2 2 1 3 *  2

*the mirror 2 is situated at ½ distance of the translation - 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)

4/24/2013 L. Viciu| AC II | Symmetry in 2D 24 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

1 2 21 2 2 2 - Pair of motifs 1 1 1 12

A glide is the result of a reflection and a translation

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

T

4/24/2013 L. Viciu| AC II | Symmetry in 2D 25 3. Combining a glide with a translation 1. A rectangular cell 1gliding2 Translation3

g g 1 2 3 gliding by g 2 2 1 The glide g2 is situated at half of the translation which is perpendicular to it - Motif

g1 g2 Reflecting 1 by a 3 2 mirror in the 1 center of the 3’ edges gives 3’; T() Gliding 3’ half of Tparallel gives 3

4/24/2013 L. Viciu| AC II | Symmetry in 2D 26 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.

g1 g2 2 1

4/24/2013 L. Viciu| AC II | Symmetry in 2D 27 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

1 Guide to the eye  1’ 2 1 2

Two reflections: One rotation: 1  1’ by reflection on 1 1  2 by two times  rotation 1’  2 by reflection on 2 11 1'  2 2

rotation by 2 4/24/2013 L. Viciu| AC II | Symmetry in 2D 28 5. Combining a rotation with a reflection

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

 3 2 1 3

 1 1 2

2 1rotationby 2 reflectionby1 3

reflection by 2

4/24/2013 L. Viciu| AC II | Symmetry in 2D 29 Combining symmetry operations 1. Oblique (parallelogram) (a ≠ b, ≠ 90°)

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

Examples of motifs having point group 1: (The motif itself should have no symmetry) and

Examples of motifs having point group 2: and (The motif itself should have a 2-fold axis) 4/24/2013 L. Viciu| AC II | Symmetry in 2D 30 Plane group symbol rules/meaning

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

4/24/2013 m  y L. Viciu| AC II | Symmetry in 2D m  x m  z 31 2. Rectangular (a  b, 90ᵒ)

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

pmg2 pgg2 pmm2

Possible motifs: 4/24/2013 m 2mm 32 L. Viciu| AC II | Symmetry in 2D 2. Examples of Rectangular plane groups with glide lines

motif: motif:

pmg2

pgg2

pmg2 4/24/2013 L. Viciu| AC II | Symmetry in 2D pgg2 33 3. Square (a = b, 90ᵒ)

Plane groups: p4, p4mm and p4gm

Possible motifs:

4/24/2013 L. Viciu| AC II | Symmetry in 2D 34 4 4mm 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

4/24/2013 L. Viciu| AC II | Symmetry in 2D 35 4. Centered rectangular (a  b, 90ᵒ)

The dash lined cell is known as diamond or rhombus cell

Plane groups: cm and cmm2

Possible motifs: cmm2

m 2mm

4/24/2013 L. Viciu| AC II | Symmetry in 2D 36 Diamond vs. centered rectangular

The centered rectangular lattice has now 2 atoms per The diamond lattice has a mirror unit cell through it such that always a = b a a=b but the angle is general

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

4/24/2013 L. Viciu| AC II | Symmetry in 2D 37 5. Rhombic or hexagonal (a = b, 120ᵒ)

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

Possible motifs:

4/24/2013 6 L. Viciu| AC II6mm | Symmetry in 2D 3 3m 38 How the motifs are oriented in p3m plane group

p3m1 p31m The mirrors are  to the translation The translation is along (the translation comes in the middle of the the mirror planes mirrors)

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 4/23/2013 L. Viciu| AC II | Symmetry in 2D 39 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

1 2

a) the inclination of translation b) the location of glide (between the relative to the mirrors mirrors at the half distance)

4/23/2013 L. Viciu| AC II | Symmetry in 2D 40 When we have translations which are inclined to the mirrors like in p31m plane group, a glide is always interleaved between the two mirrors. The glide is parallel to the mirrors at half distance between them.

a) The inclination of the translation b) The location of the glides (between relative to the mirrors the mirrors at the half distance)

4/23/2013 L. Viciu| AC II | Symmetry in 2D 41 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.

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;

4/23/2013 L. Viciu| AC II | Symmetry in 2D 42 Symmetry Elements of the 2D Space Groups

Unit cell edge glide line 4/23/2013 mirror line 2, 3, 4, 6 – fold axes 43 L. Viciu| AC II | Symmetry in 2D The equivalence of atom positions results from translation y

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

The atom at the lattice point has the coordinates: (x, y) The 2 – fold axes place the atoms at the opposite direction

It is possible to say also 1-x 1-y x y But is more esthetic to give the 1-y 1-x positions x y and x y 4/23/2013 L. Viciu| AC II | Symmetry in 2D 44 1. Highest 2. Has reflection? order rotation? Yes No 6-fold p6mm p6 3. Has mirrors at 45°? 4-fold p4 Yes: p4mm No: p4gm 3. Has rot. centre off mirrors? 3-fold p3 Yes: p31m No: p3m1 3. Has perpendicular reflections? Has glide reflection? Yes No 2-fold Has rot. centre off mirrors? pmg2 Yes: pgg2 No: p2 Yes: cmm2 No: pmm2 Has glide axis off mirrors? Has glide reflection? none Yes: cm No: pm Yes: pg No: p1

4/23/2013 L. Viciu| AC II | Symmetry in 2D 45 Fundamental Steps in Plane Groups Identification

1. Locate the motif present in the pattern. This can be a molecule, molecules, atom, group of atoms, a shape or group of . 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.

4/23/2013 L. Viciu| AC II | Symmetry in 2D 46 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.

4/23/2013 L. Viciu| AC II | Symmetry in 2D 47 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.

4/23/2013 L. Viciu| AC II | Symmetry in 2D 48 Finding the plane group

4/23/2013 L. Viciu| AC II | Symmetry in 2D 49 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

4/23/2013 L. Viciu| AC II | Symmetry in 2D 50 Finding the plane group

The unit cell is square. Symmetry elements: -2-fold axis -Two mirror lines ( to each other) - Two glide lines  Plane group: cmm2

4/23/2013 L. Viciu| AC II | Symmetry in 2D 51 Finding the plane group

4/23/2013 L. Viciu| AC II | Symmetry in 2D 52 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

4/23/2013 L. Viciu| AC II | Symmetry in 2D 53 Finding the plane group

4/23/2013 L. Viciu| AC II | Symmetry in 2D 54 Finding the plane group

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

4/23/2013 L. Viciu| AC II | Symmetry in 2D 55 Finding the plane group

4/23/2013 L. Viciu| AC II | Symmetry in 2D 56 Christopher Hammond: The basics of and diffraction (third edition) Finding the plane group

4/23/2013 L. Viciu| AC II | Symmetry in 2D 57 Christopher Hammond: The basics of crystallography and diffraction (third edition) Finding the plane group

4/23/2013 L. Viciu| AC II | Symmetry in 2D 58 p4gm

4/23/2013 L. Viciu| AC II | Symmetry in 2D 59 Finding the plane group

4/23/2013 L. Viciu| AC II | Symmetry in 2D 60 Finding the plane group

4/23/2013 L. Viciu| AC II | Symmetry in 2D 61