Introduction to crystals symmetry

Symmetry in 3D

5/1/2013 L. Viciu| AC II | Symmetry in 3D 1 Outlook

• Symmetry elements in 3D: rotoinversion, screw axes • Combining symmetry elements with the lattice • Space group symbols/representation • Wyckoff positions • Crystallographic conventions • Symmetry in crystal systems

5/1/2013 L. Viciu| AC II | Symmetry in 3D 2 Definitions - a remainder

• Unit cell: The smallest volume that can generate the entire crystal structure only by means of translation in three dimensions.

• Lattice: A rule of translation.

5/1/2013 L. Viciu| AC II | Symmetry in 3D 3 Bravais Basis/ Crystal + lattice Motif structure

1. single atom: Au, Al, Cu, Pt 1. FCC 2. molecule: solidCH4 2. FCC + - + 3. ion pairs: Na /Cl 3. Rock salt

4. atom pairs: Carbon, Si, Ge 4. diamond

•A crystal system is described only in terms of the unit cell geometry, i.e. cubic, tetragonal, etc

•A crystal structure is described by both the geometry of, and atomic arrangements within, the unit cell, i.e. face centered cubic, body centered5/1/2013 cubic, etc. L. Viciu| AC II | Symmetry in 3D 4 The Bravais lattices

5/1/2013 L. Viciu| AC II | Symmetry in 3D 5 From 2D to 3D

• Bravais lattices may be seen as build up layers of the five plane lattices:

cubic and tetragonal stacking of square lattice layers orthorhombic P and I  stacking of rectangular layers orthorhombic C and F stacking of rectangular centered layers rhombohedral  stacking of hexagonal layers hexagonal  stacking of hexagonal layers monoclinic  stacking of oblique layers triclinic stacking of oblique layers

5/1/2013 L. Viciu| AC II | Symmetry in 3D 6 Combining symmetry elements

For two dimensions – 5 lattices – 10 point groups – 17 plane groups

For three dimensions – 14 Bravais lattices – 32 point groups – 230 space groups

5/1/2013 L. Viciu| AC II | Symmetry in 3D 7 • 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

* Besides identity 5/1/2013 L. Viciu| AC II | Symmetry in 3D 8 Symmetry elements in 3D

• In three dimensions we have additional symmetry operators: 1. inversion center = symmetry center 2. inversion axis = rotoinversion or improper rotation axis 3. screw axes

• In addition, the mirror line becomes a mirror plane, and the glides become glide planes.

5/1/2013 L. Viciu| AC II | Symmetry in 3D 9 1. The center of symmetry/inversion center

Symbol: 1 “one bar”; Graphical: 

• The symmetry center: the intersection point of a mirror plane and a 2-fold axis

Inversion center  (x,y,z)  (x, y, z)

• It is always present in one of the situations:

a rotation axis with even multiplicity and a  if an inversion axis with odd multiplicity reflection  to it is present is present

P2/m P3

5/1/2013 L. Viciu| AC II | Symmetry in 3D 10 • An inversion center requires a 2-fold axis and a mirror plane  to it. • Therefore, an even rotation with a reflection perpendicular to it will give an inversion center.

A A= inversion, 1 2 1

 i 1rotationby1802 reflection3

3 inversion

*The order of 1  has the coordinates (x y z) the operation is not important

2 has the coordinates ( x y z )

3 has the coordinates ( x y z ) 5/1/2013 L. Viciu| AC II | Symmetry in 3D 11 • The point groups that contain an inversion center are called Laue groups.

• If you take away the translational part of the space group symmetry and add an inversion center, you end up with the Laue group.

• The Laue groups are used in diffraction: describe the symmetry of the diffraction pattern.

5/1/2013 L. Viciu| AC II | Symmetry in 3D 12 5/1/2013 L. Viciu| AC II | Symmetry in 3D 13 Centrosymmetric structures

• If an inversion center is present then the structure is centrosymmetric • In a centrosymmetric structure, the unit cell is chosen so that the origin lays on the inversion center

82% of inorganic crystals are centrosymmetric because Inversion center leads to equal forces in opposing directions favoring stability

George M. Sheldrick. • In a non-centrosymmetric space group, the origin is chosen at a highest symmetry point. 5/1/2013 L. Viciu| AC II | Symmetry in 3D 14 32 point groups

11 centrosymmetric point group 20 non-centrosymmetric point groups: + piezoelectrics 432 class*

*432 class has a polar axis but no piezoelectricity

10 point groups 10 point groups with unique polar with no unique axis: polar axis Pyroelectric + ferroelectrics

• Polar direction = a crystal direction that is not related by symmetry to the opposed direction

5/1/2013 L. Viciu| AC II | Symmetry in 3D 15 Space groups and enantiomorphous molecules

Enantiomorphous molecules crystallise in space groups without inversion centre , i, and without reflection planes, m.

Altogether there are 11 Point groups possible for enantiomorphous molecules. (“Biological Point Groups”)

Enantiomorphous crystals of tartaric acid

(monoclinic structure, space group P21)

The most common chiral space groups are P212121, P21, P1 and C2221.

 15%5/1/2013 of all crystals are enantiomorphicL. Viciu| AC II | Symmetry and in 3D potentially optically active16 Klockmanns Lehrbuch der Mineralogie, 16. Auflage, Enke VerlagStuttgart 1978 2. Axes of inversion/rotoinversion

• Rotation + center of symmetry = inversion axis, n , pronounced n bar

• It is also called improper rotation axes (Schoenflies symbols, Sn) • Two objects related by an operation of inversion axis are enantiomorphous 2  m Symbol/graphic 1. inversion 2. 180ᵒ 1  axis atom

2  m

3 

3. 4. 4 

6  m

5/1/2013 L. Viciu| AC II | Symmetry in 3D 17 if 3  31, 4  41 Therefore 4 bar is a distinct operation

Spinoid (like a squashed Td)

5/1/2013 L. Viciu| AC II | Symmetry in 3D 18 3. Screw axes: nm Rotation + Translation in axial direction

The rotation is combined with a translation in a defined crystallographic direction by a defined step (half a unit cell for twofold screw axes, a third of a unit cell for threefold screw axes, etc

counter clockwise rotation with 360ᵒ/n +

Translation with m/n (m

c. Hammond, the basics of crystallography and diffraction Srew axis nm

5/1/2013 L. Viciu| AC II | Symmetry in 3D 19 3. Screw axes: nm Rotation + Translation in axial direction:

21: ½ displacement on c 1 2 31: /3 displacemnt on c; 32: /3 displacement on c; 2 41: ¼ displacemnt on c; 42 /4 = ½ displacement on c; 43: ¾ displacement on c 1 2 1 3 61: /6 displacement on c; 62: /6= /3 displacement on c; 63: /6= ½ displacement on c, etc

5/1/2013 NotL. shownViciu| AC II | Symmetry in 3D 20 (Fig. 140) Inorganic structural chemistry, U. Mueller Screw axis of order 3: 31; 32

31 32 rotation by a 3-fold axis and rotation by a 3-fold axis and then translation by 1/3 then translation by 2/3

T 

T is translation periodicity Red color is used only to  Is translation component = 1/3 show the objects obtained by translation periodicity

5/1/2013 L. Viciu| AC II | Symmetry in 3D 21 Screw axis of order 3: 31; 32

31 32 rotation by a 3-fold axis and rotation by a 3-fold axis and then translation by 1/3 then translation by 2/3

T 

3 gives a clockwise spiral 31 gives an anticlockwise spiral 2

5/1/2013 L. Viciu| AC II | Symmetry in 3D 22 41 4 43 2

T T   T 4 gives an  1 4 gives a anticlockwise spiral 3 clockwise spiral Color is used only to show the objects obtained by translation periodicity; otherwise the color represent the same 5/1/2013 L. Viciu| AC II | Symmetry motifin 3D 23 42

42 gives two motifs at every level and they are 180 away from each other

42 gives two spirals - one left handed (anticlockwise) and one right handed (clockwise) – in the same pattern

T 

5/1/2013 L. Viciu| AC II | Symmetry in 3D 24 Quartz symmetry

View along c axis

  - quartz (hexagonal) 573C  - quartz (trigonal) 5/1/2013 L. Viciu| AC II | Symmetry in 3D 25 P6221 and P6421 P3121 and P3221 Glide planes in Space Groups

• In 2D the gliding (translation) is in the direction of the dashes: g1 g2 2 3 1

• In 3D we can look at a glide from 4 different directions:  Down from the top (1)  Edge on and  to the gliding direction (2)  Edge on and  to the gliding direction (3)  Edge on and in between normal to and parallel to the gliding direction (4) (1)

(3) * * - gliding direction 5/1/2013 (2)L. Viciu| AC II | Symmetry in 3D 26 (4) (1) Looking down from top of a glide  use or or

a glide b glide n glide = diagonal glide

b b

   a a a

 = ½ of a axis  = ½ of b axis  = ½ (a+b) axis

Ex: n-glide b 

a

c glide is represented in plane as well and the translation is ½ of c axis 5/1/2013 L. Viciu| AC II | Symmetry in 3D 27 (2) Looking normal to the gliding direction  use a dashed line b Ex: b glide

a 

(3) Looking in the direction of gliding  use a dotted line b Ex: b glide

a 

(4) Looking not parallel not perpendicular to the direction of gliding  use a mixed line b Ex: b glide

a 

5/1/2013 L. Viciu| AC II | Symmetry in 3D 28 d-glide = diamond glide

• Found in centered cells only (not in primitive lattices) • It originates from the diamond structure where the diamond glide has been first observed. • The translation is ½ of the ½(a+b) axis

½ (a+b)

 a+b

½ (½ (a+b))

5/1/2013 L. Viciu| AC II | Symmetry in 3D 29 Graphical symbols for symmetry elements

Equivalent position diagrams: motif in a general position in the plane + motif at arbitrary height above the plane  Enantiomorph of motif in the plane  + Enantiomorph of motif at arbitrary height above the plane

Yellow box: Rotation and inversion axes Blue box: screw axes Green box: inversion center Red box: glide planes * = axis in the plane; # = glide in the screen plane; Cyan box: mirror planes & = perpendicular to the screen plane n - diagonal glide either of the two directions d- diamond glide in centered cells only 5/1/2013 L. Viciu| AC II | Symmetry in 3D one direction only 30 Massa, Werner. Crystal Structure Determination Direction of the 2-fold rotation axes when looked at by the side

Direction of the 2-fold screw axes when looked at by the side

Only 2-fold rotations and 2-fold screw axes are shown graphically!

5/1/2013 L. Viciu| AC II | Symmetry in 3D 31 Combining symmetry operations in 3D 1. Combining rotations Two crystallographic rotations (1-,2-, 3-, 4- and 6-fold) at an intersection point will give a 3rd rotation which must be crystallographic (1-,2-, 3-, 4- and 6-fold).

C B B  A = C A 2 1rotationbyA 2 rotationbyB 3 3 1 Rotation by C

Euller constrictions in 3D: there are 11 combinations for the crystallographic rotations 11 axial combinations: 1, 2, 3, 4, 6, 222, 322, 422, 622 (dihedral 23, 432 (in Schoenfliess notation 23 is the Td and 432 is the Oh) 32 5/1/2013 L. Viciu| AC II | Symmetry in 3D 3. A rotation and reflection in 3D A rotation, , followed by a reflection with a mirror parallel to it and at an angle half of the rotation angle, /2, gives another reflection.

Ex: 2-fold rotation + reflection

A 1A=2

1 2mm

*The two mirrors will contain the rotation 2

If the mirror is perpendicular to the rotation, it does not create another reflection! In this case the resulted point group is labeled 2/m 5/1/2013 L. Viciu| AC II | Symmetry in 3D 33 A rotation followed by a reflection with a mirror perpendicular to it gives an inversion.

A 2 1 1  has the coordinates (x y z)

2 has the coordinates ( ) x yz A= inversion, 1 3 has the coordinates ( x y z )

3

A 2 1 1rotation2 reflection3 2 m 3 Inversion (inversion, mirror, 2-fold  identity) 5/1/2013 L. Viciu| AC II | Symmetry in 3D 34 Crystallographic Conventions: unit cell and unique axes

• Unit cell: Right handed system a, b, c, α, β, γ • Unique axis: the direction with highest rotation symmetry The monoclinic angle is β with β≥ 90°. This makes b the unique axis in the monoclinic system In the tetragonal, trigonal and hexagonal systems, c is the unique axis.

5/1/2013 L. Viciu| AC II | Symmetry in 3D 35 Symmetry directions • How the symmetry elements are oriented with respect to the axes of the unit cell, a, b and c ( the places in the Hermann-Mauguin symbol for point group)

Crystal system Order of direction Triclinic - Monoclinic b Orthorhombic a, b, c Tetragonal c, a Trigonal c, a Hexagonal c, a cubic c

Ex: monoclinic

5/1/2013 P2/m L. Viciu| AC II | SymmetryP2 in 3D 36 Crystal System Symmetry Direction Primary Secondary Tertiary Triclinic None Monoclinic [010] b Orthorhombic [100] a [010]  b [001] c Tetragonal [001]  c [100]/[010]  a/b [110] Hexagonal/Trigonal [001]  c [100]/[010] a/b [120] Cubic [100]/[010]/[001]  a/b/c [111] [110]

[100] – Axis  or plane  to the x-axis . [010] – Axis  or plane  to the y-axis. [001] – Axis  or plane  to the z-axis. [110] – Axis  or plane  to the line running at 45° to the x and y axes (face diagonal). [120] – Axis  or plane  to the line running  on the a or on the b axis in the hexagonal cell. [111] – Axis parallel or plane perpendicular to the body diagonal. 5/1/2013 L. Viciu| AC II | Symmetry in 3D 37 Space group symbol rules/meaning

1. First letter: P, A, B, C, F, I or R  translation symmetry + type of centering ex: P 4mmm; C mm2; F mmm 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: P 21/c: 21 axis in the z direction  If highest multiplicity axis is 2-fold the sequence is x-y-z ex: Cmm2: 2-fold axis on z; or Cm2m the2-fold axis on y  The highest symmetry axis is mentioned first ex: I 4/mcm: 4-fold axis on z and two 2-fold axes on a and b 3. An inversion center is mentioned only if it is the only symmetry element ex: P1 4. A reflection plane  to a symmetry axis is designated by a fraction bar ”/” ex: P 2/mmm: mirror palne  on 2-fold axis 5/1/2013 L. Viciu| AC II | Symmetry in 3D 38  exception m  to odd rotation axis: 3/m  6 the inversion axis used instead Triclinic system

No symmetry elements or only inversion center present: P1 and P1

b b

a a

Note: the projection of objects in an oblique lattice should be made parallel with the translation otherwise the number of the obtained points will be different from the number of lattice points. 5/1/2013 L. Viciu| AC II | Symmetry in 3D 39 We consider two unit cells with one lattice point inside the cell (primitive) stacked on top of each other

Projecting  Projecting 

c c b b a a

The projection of the atoms into the The projection of the atoms into the basal plane makes the lattice points basal plane gives one point in be different5/1/2013 from a primitive cell. L. Viciu| AC II |agreement Symmetry in 3D with a primitive cell. 40 We consider two unit cells with one lattice point inside the cell (primitive) stacked on top of each other

Projecting  Projecting 

c c b b a a

The projection of the atoms into the The projection of the atoms into the basal plane makes the lattice points basal plane gives one point in be different5/1/2013 from a primitive cell. L. Viciu| AC II |agreement Symmetry in 3D with a primitive cell. 41 Monoclinic System In a monoclinic system there are possible two settings 1. c is  to the (ab) plane and the angle , between a and b, the general angle 2. b is  to the (ca) plane and the angle , between c and a, is general angle b c

90 a 90 b 90 90  

c a 2nd setting 1st setting the unique axis (either b or c) the 2-fold axis parallel to it and/or the mirror plane  to it; 5/1/2013 L. Viciu| AC II | Symmetry in 3D 42 Monoclinic space group P 2

1st setting: Unique axis c 2nd setting: Unique axis b

b c o o o a b o

ap a cp c

c is the direction of 2-fold axis (c  (ab)) b is the direction of 2-fold axis (b(ac)) and The direction of the 2-fold axis when looking on the side (an arrow is the symbol for a 2-fold axis along on the page)

This is the symbol for the 2-fold axis  to the plane

5/1/2013 L. Viciu| AC II | Symmetry in 3D 43 P2 – unique axis c

Atoms at + z (the atom in the cell gives the atom outside by rotation with a 2- fold axis)

5/1/2013 L. Viciu| AC II | Symmetry in 3D 44 Space group Pm: unique axis c (mirror  c) Above view

Side view-along the c axis (parallel to Mirror plane the mirror plane) viewed directly from above (the drawn plane is the mirror plane)

Side view :  to the mirror plane

- ,‘ + Two atoms superimposed in projection: one at +z and one to –z The right side of the symbol says that the atom is at +z 5/1/2013 The left side of the symbolL. Viciu| ACsays II | Symmetry that an in 3D enantiomorph sits at -z 45 Monoclinic System Either a two fold axis (2) or a mirror plane (m) or both (2/m) • 2implies that the 2-fold axis is parallel to the unique axis (b or c depending on the setting) • m  implies a mirror plane  to the unique axis

It can also have glide planes or screw axis (i.e. Cc, P2, P21/n

2 (P 2)  2-fold axis  to b 2/m (P 2/m)  [unique axis c] 2/m = reflection plane  to the 2-fold axis

b axis  to the b axis  to the 2-fold axis  to c 2-fold axis  to c plane* plane* ( to the plane)* ( to the plane)*

*Please note the a, b, and c axes labeling: The reflection plane is  to the paper/figure in the left the b axis is  to the plane Inversion center because we have a 2-fold 5/1/2013 L. Viciu| AC II | Symmetry in 3D 46 in the right, the b axis is  to the plane and a mirror plane Unconventional Lattices

Monoclinic A (B or C ), can always be transformed into monoclinic P (red cell), with c half the unit cell volume.

a b

Monoclinic I, can always be transformed into monoclinic B (A or C) -dashed cell, with the same unit cell volume

That is why the list of the 14 Bravais lattices does not include monoclinic B nor monoclinic I nor several other unconventional lattices. 5/1/2013 L. Viciu| AC II | Symmetry in 3D 47 What information is included in the space group C2/m?

C- side centered cell 2/m – 2-fold axis along b and mirror plane  to it (or  on b axis)

Ex: Na0.5CoO2 – monoclinic structure in space group C2/m with a = 4.9043(2), b = 2.8275(1), c = 5.7097(3) and  = 106.052(3)

5/1/2013 L. Viciu| AC II | Symmetry in 3D 48 Orthorhombic system

In an orthorhombic symmetry no direction is special than any other.  three 2-fold axes parallel to the cell edge three mirror planes parallel to the faces  The symmetry elements are along all three directions, a, b and c

(i.e. Pnma, Cmc21, Pnc2)

The magnitude of the translations is: b  a  c m m 2 (P m m 2) 5/1/2013 L. Viciu| AC II | Symmetry in 3D 49 Space group symbol for orthorhombic

Symbol for the lattice type axis a axis b axis c plane  a plane  b plane  c

•If there is a 2-fold and a 2-fold screw symmetry along the same axis, the 2-fold is preferred in the symbol •If there are more mirror planes perpendicular to the same axis, the preferred order is m>a>b>c>n>d

5/1/2013 L. Viciu| AC II | Symmetry in 3D 50 What information is included in the space group

symbol Pna21?

P primitive lattice

21  orthorhombic system with screw axis 21 along (parallel) c n  diagonal glide plane  on a (moves the motif ½ of the diagonal of b and c) a  axial glide plane  on b (moves the motif ½ a -  to a)

Ex: LiB3O5

5/1/2013 L. Viciu| AC II | Symmetry in 3D 51 Tetragonal symbol 4/mmm

• 1st  the 4-fold symmetry along c direction • 2nd  symmetry element along a (which must apply to b as well) • 3rd  symmetry element along the a,b diagonal

The primary symmetry symbol will always be either 4, (-4), 41, 42 or 43 (i.e.P41212, I4/m, P4/mcc)

5/1/2013 4/m m mL. Viciu| (P AC4/ II m| Symmetry m m in) 3D 52 What information is included in the space group symbol I41cd?

I body centered lattice

41  tetragonal system with screw axis 41 along (parallel) c c  glide plane  on a and b because a = b (moves the motif ½ c -  to c) d  diamond glide plane  on [110] direction (the diagonal of xy plane)

5/1/2013 L. Viciu| AC II | Symmetry in 3D 53 What information is included in the space group symbol P4mm?

P primitive lattice 4 tetragonal system with 4-fold axis  to c m  mirror plane  on a and b m  mirror plane  on the ab diagonal [110]

Ex: PuS2

5/1/2013 L. Viciu| AC II | Symmetry in 3D 54 Trigonal and hexagonal systems

• 1st  the symmetry element along c direction: - 3-fold axes (rotation and screw axis) in trigonal i.e P31m, R3, R3c, P312)

- 6-fold axes in hexagonal (i.e. P6mm, P63/mcm) • 2nd  symmetry element with respect to a/b (a = b) • 3rd  symmetry element with respect to a, b diagonal (s) Trigonal Hexagonal

3 (P 3) 3m1(P3m1) 5/1/2013 L. Viciu| AC II | Symmetry in 3D 6 m m (P 6 m m) 55 For trigonal lattice there are two possibilities: 1. The third translation  to a1 and a2 2. The third translation inclined so that the projection falls either on (1/3, 2/3) or (2/3, 1/3). The two situations are actually the same

1/3, 2/3)

2/3, 1/3)

Rhombohedral lattice

Hexagonal/trigonal lattice (1/3, 2/3, 2/3)

(2/3, 1/3, 1/3)

The “double body centered cell” is then described by a primitive cell taking as one base the point at 2/3, 1/3, 1/3 and the points at the corner of the triangle of which the 5/1/2013 L. Viciu| AC II | Symmetry in 3D 56 point is related The rhombohedral R cell expressed in hexagonal axes

three hexagonal cells Rhombohedral R cell (red) one hexagonal cell with additional nodes inscribed in hexagonal

1 2 2 3 3 3

2 1 1 3 3 3

The hexagonal cell is three times larger than the rhombohedral and has additional 2 1 1 1 2 2 57 nodes at ( 3 , 3 , 3) and ( 3 , 3 , 3) L. Viciu| AC II | Symmetry in 3D The relation between rhombohedral R Rhombohedral R cell (brown) inscribed cell (brown) and hexagonal cell (black) in an F-centered cubic cell (black)

Blue - the rhombohedron projection on ab plane of the hexagonal cell In the rhombohedral (R) lattice, the first character (3 or 3 bar) denotes the unique space diagonal of the cell and the next defines the directions that are perpendicular to the 3-fold axis. 5/1/2013 L. Viciu| AC II | Symmetry in 3D 58 Cubic System

Isometric system = the translations are identical in all 3 directions (a=b=c)

4/ m32/ m  m3m the direction of the 4-fold axis is always along the edge of the unit cell • 1st  symmetry element along c direction (either a 2-fold axis or a 4-fold axis parallel to, and/or a plane  to, c direction) • 2nd  3-fold axis or 3-fold inversion axis along the body diagonals • 3rd  symmetry element along the face diagonals (either 2-fold axis parallel to, or mirror plane  to, the face diagonals)

The secondary symmetry symbol will always be either 3 or –3 (i.e. Ia3, Pm3m, Fd3m) 4/m  m in cubic symmetry because the four 3-fold axes and the nine mirror planes automatically generates the three 4-fold axes, six 2-fold axes and a center of symmetry 5/1/2013 L. Viciu| AC II | Symmetry in 3D 59 Cubic symmetry: axes

2-fold rotation axes 4 fold rotation axes (passing through 3-fold rotation axes (passing through pairs of opposite (passing through cube diagonal edge centers) face centres, body diagonals) TOTAL = 6 parallel to cell axes) TOTAL = 3 TOTAL = 4 5/1/2013 L. Viciu| AC II | Symmetry in 3D 60 Cubic symmetry: planes

3 equivalent planes 6 equivalent in a cube planes in a cube

5/1/2013 L. Viciu| AC II | Symmetry in 3D 61 What information is included in the space group Fm3m? - F: face centered cubic - m: mirror ( 4/m)  on c axis - 3: 3-fold rotoinversion on the body diagonal - m: mirrors  on the face diagonal

Ex: NaCl

5/1/2013 L. Viciu| AC II | Symmetry in 3D 62 Essential Symmetry

Essential symmetry is that which defines the crystal system (i.e. is unique to that shape).

System Essential Symmetry Symmetry axes Cubic four 3-fold axes along the body diagonals Tetragonal one 4-fold axis parallel to c, in the centre of ab Orthorhombic three mirrors or three 2-fold perpendicular to each other axes Hexagonal one 6-fold axis down c Trigonal (R) one 3-fold axis down the long diagonal Monoclinic one 2-fold axis down the “unique” axis Triclinic no symmetry

5/1/2013 L. Viciu| AC II | Symmetry in 3D 63 Herman Mauguin symbol

Long notation Short notation •C 1 2/m 1 •C2/m •P 2 2 2 1 1 •P21212 •P 2/m 2/n 21/a •Pmna •I 4 /a 2/m 2/d 1 •I41/amd •P 6 /m 2/m 2/c 3 •P63/mmc •F 4/m (-3) 2/m •Fm(-3)m

Rotation/screw Only mirror/glide axes + mirror/glide planes (in their absence  planes to them rotation/screw axes)*

*In monoclinic, tetragonal and hexagonal both rotation/screw axis and the mirror/glide plane for the primary direction only is retained in the short H.M. symbol!

5/1/2013 L. Viciu| AC II | Symmetry in 3D 64 Occurrence of crystal systems

Inorganic materials Organic materials

1. Cubic 1. Monoclinic 2. Orthorhombic 2. Orthorhombic 3. Monoclinic 3. Tetragonal 4. triclinic

5/1/2013 L. Viciu| AC II | Symmetry in 3D 65 Reduction in symmetry

Cubic Tetragonal Three 4-fold axes One 4-fold axis

Four 3-fold axes No 3-fold axes Six 2-fold axes Two 2-fold axes Nine mirrors Five mirrors

5/1/2013 L. Viciu| AC II | Symmetry in 3D 66 BaTiO3 (1) At temp. >120ᵒC : cubic perovskite structure (a=4.018Å) (2) At temp.< 120ᵒC : tetragonal structure (a=3.997Å, c=4.031 Å)

Views on the [100] direction = a axis

(1) (2)

c

 cubic (1)  tetragonal (2) distortion goes with an off-centre displacement of Ti4+

and the dipoles are pointing along c axis  tetragonal BaTiO3 is ferroelectric 5/1/2013 L. Viciu| AC II | Symmetry in 3D 67 Changing the symmetry it changes the properties

BaTiO3

120ᵒC 5ᵒC -90ᵒC cooling cubic tetragonal orthorhombic rhombohedral

KNbO3 435ᵒC 225ᵒC -10ᵒC cooling cubic tetragonal orthorhombic hexagonal crystal structure phase I: cubic, space group –Pm3m (paraelectric phase) phase II: tetragonal, space group – P4mm (ferroelectrical phase) phase III: orthorhombic, space group – Bmm2 (ferroelectric phase) phase5/1/2013 IV: rhombohedral, spaceL. Viciu| group AC II | Symmetry – R3m in 3D (ferroelectric phase) 68 General and special positions: Wyckoff positions

 General position = no symmetry elements lying on it the number of points in general position = the number of symmetry operations  Special position = symmetry element(s) lying on the position point

Ex: space group Pm: monoclinic with two mirror planes  on b axis

o b Multiplicity Wyckoff Site Coordinates Letter Symmetry 2 c 1 (1) x,y,z (2) x,-y,z 1 b m x, ½ , z a ½ 1 a m x,0,z

Number of Alphabetic order created from bottom up (no positions by physical meaning symmetry 5/1/2013 L. Viciu| AC II | Symmetry in 3D 69

Generating crystal structure from crystallographic description

SrTiO3 Space group: Pm 3 m ; a = 3.90 Å Wyckoff Atom Symmetry x y z sites Ti 1a m3m 0 0 0 Sr 1b 0.5 0.5 0.5 O 3d 4/m m m 0.5 0 0

5/1/2013 L. Viciu| AC II | Symmetry in 3D 70

Generating crystal structure from crystallographic description

SrTiO3 Space group: Pm 3 m ; a = 3.90 Å Wyckoff Atom Symmetry x y z sites Ti 1a m3m 0 0 0 Sr 1b 0.5 0.5 0.5 O 3d 4/m m m 0.5 0 0 Z=1

O: 0.5 0 0; 0 0.5 0; and 0 0 0.5

5/1/2013 L. Viciu| AC II | Symmetry in 3D 71 Example of crystal with P4mm: PuS2

Lattice parameters: a = b = 3.943Å; c = 7.962Å

Multiplicity Atom Site Coordinates And Symmetry Wyckoff letter x y z 1a Pu1 4mm 0 0 0 1b Pu2 4mm ½ ½ 0.464 1a S1 4mm 0 0 0.367 1b S2 4mm ½ ½ 0.097 2c S3 2mm ½ 0 0.732

Coordinates 2c 2mm ( ½ , 0, z) and (0, ½ z)

5/1/2013 L. Viciu| AC II | Symmetry in 3D 72 Wyckoff positions of space group P4mm

5/1/2013 L. Viciu| AC II | Symmetry in 3D 73 Crystal structure of PuS2

S2 ( ½ , 0, 0.732)

0.097, ½

S1 (0, 0, 0.367) S2 ( ½ , ½ 0.097)

0.732 5/1/2013 L. Viciu| AC II | Symmetry in0, 3D 0.367,1 74 What information is included in the space group C2/m?

C- side centered cell 2/m – 2-fold axis along b and mirror plane  to it (or  on b axis)

Ex: Na0.5CoO2 – monoclinic structure in space group C2/m with a = 4.9043(2), b = 2.8275(1), c = 5.7097(3) and  = 106.052(3)

Atom Wyckoff x y z Occ position Co 2a 0 0 0 1 Na 4i 0.806(2) 0 0.491(2) 0.25 O 4i 0.3871(3) 0 0.1740(4) 1

Coordinates (0,0,0) + (1/2,1/2,0) +

5/1/2013 L. Viciu|4i AC II | Symmetry in 3Dm (x,0,z) and (-x, 0, -z) 75 Wyckoff positions of space group C2/m [unique axis b]

5/1/2013 L. Viciu| AC II | Symmetry in 3D 76 Generating crystal structure of NaCl

NaCl: unit cell, a = 5.652Å; space group Fm3m

Multiplicity Atom Site Coordinates And Symmetry Wyckoff letter x y z 4a Cl m-3m 0 0 0 4b Na m-3m ½ ½ ½

5/1/2013 L. Viciu| AC II | Symmetry in 3D 77 Generating crystal structure of NaCl

NaCl: unit cell, a = 5.652Å; space group Fm3m

Multiplicity Atom Site Coordinates And Symmetry Wyckoff letter x y z 4a Cl m-3m 0 0 0 4b Na m-3m ½ ½ ½

5/1/2013 L. Viciu| AC II | Symmetry in 3D 78