# Symbols of the 32 three dimensional point groups

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Symbols of the 32 three dimensional point groups
III Crystal Symmetry 3-3 Point group and space group Point group Symbols of the 32 three dimensional point groups Rotation axis X x X Rotation-Inversion axis X

X + centre (inversion): Include X for odd order
X 2 or X m even: only for even rotation symmetry X2 + centre; Xm +centre: the same result (Include X m for odd order)

Ratation axis with mirror plane normal to it X/m
Rotation axis with mirror plane (planes) parallel to it Xm x mirror x

Rotation axis with diad axis (axes) normal to it X2
Rotation-inversion axis with diad axis (axes) normal to it X 2 X X

Rotation-inversion axis with mirror plane (planes) parallel to it X m
Rotation axis with mirror plane (planes) normal to it and mirror plane (planes) parallel to it X/mm x mirror x mirror

Only one symbol which denotes all directions in the crystal.
System and point group Position in point group symbol Stereographic representation Primary Secondary Tertiary Triclinic 1, 1 Only one symbol which denotes all directions in the crystal. Monoclinic 2, m, 2/m The symbol gives the nature of the unique diad axis (rotation and/or inversion). 1st setting: z-axis unique 2nd setting: y-axis unique 1st setting 2nd setting

System and point group Position in point group symbol Stereographic representation Primary Secondary Tertiary Orthorhombic 222, mm2, mmm Diad (rotation and/or inversion) along x-axis Diad (rotation and/or inversion) along y-axis Diad (rotation and/or inversion) along z-axis Tetragonal 4, 4 , 4/m, 422, 4mm, 4 2m, 4/mmm Tetrad (rotation and/or inversion) along z-axis Diad (rotation and/or inversion) along x- and y-axes Diad (rotation and/or inversion) along [110] and [1 1 0] axis

System and point group Position in point group symbol Stereographic representation Primary Secondary Tertiary Trigonal and Hexagonal 3, 3 , 32, 3m, 3 m, 6, 6 , 6/m, 622, 6mm, 6 m2, 6/mmm Triad or hexad (rotation and/or inversion) along z-axis Diad (rotation and/or inversion) along x-, y- and u-axes Diad (rotation and/or inversion) normal to x-, y-, u-axes in the plane (0001) Cubic 23, m3, 432, 4 3m, m3m Diads or tetrad (rotation and/or inversion) along <100> axes Triads (rotation and/or inversion) along <111> axes Diads (rotation and/or inversion) along <110> axes

Crystal System Symmetry Direction Primary Secondary Tertiary Triclinic None Monoclinic [010] Orthorhombic [100] [001] Tetragonal [100]/[010] [110] Hexagonal/ Trigonal [120]/[1 1 0] Cubic [100]/[010]/ [001] [111]

Triclinic Rotation axis X 1 Rotation-Inversion axis X 1 X + centre Include X (odd order) 1 For odd order includes X already!

Monoclinic 1st setting X 2 X 2 = m X + centre Include X (odd order) 2 𝑚 2 mirror mirror 2 𝑚 2 = m 2

X2 + centre, Xm +centre Include X m (odd order) 2/m
1 Monoclinic 2st setting 2 X2 12 ≡2 Xm 1m ≡m X 2 or X m even 2/m X2 + centre, Xm +centre Include X m (odd order) 2/m

Rotation-Inversion axis X
Orthorhombic 2/m Rotation axis X Already discussed Rotation-Inversion axis X 2/m X + centre Include X (odd order) 2/m X2 222 Xm 2mm (2D) = mm2 X 2 or X m even 2/m X2 + centre, Xm +centre Include X m (odd order) mmm ≡ 2/m2/m2/m

Tetragonal Rotation axis X 4 4 Rotation-Inversion axis X X + centre Include X (odd order) 4 𝑚 X2 422 Xm 4mm X 2 or X m even 4 2𝑚 X2 + centre, Xm +centre Include X m (odd order) 4/mmm ≡ 4/m 2/m 2/m

Rotation-Inversion axis X
Trigonal Rotation axis X 3 Rotation-Inversion axis X 2/m X + centre Include X (odd order) 3 X2 32 Xm 3m X 2 or X m even 2/m X2 + centre, Xm +centre Include X m (odd order) 3 𝑚≡ 3 2 𝑚

Hexagonal Rotation axis X 6 6 Rotation-Inversion axis X X + centre Include X (odd order) 6 𝑚 X2 622 Xm 6mm X 2 or X m even 6 𝑚2 X2 + centre, Xm +centre Include X m (odd order) 6/mmm ≡ 6/m 2/m 2/m

Rotation-Inversion axis X m3≡ 2/m 3 X + centre Include X (odd order)
Cubic Rotation axis X 23 Rotation-Inversion axis X 2/m m3≡ 2/m 3 X + centre Include X (odd order) X2 432 Xm 2/m X 2 or X m even 4 3𝑚 X2 + centre, Xm +centre Include X m (odd order) m3m ≡ 4/m 3 2/m

Examples of point group operation
At a general position [x y z], the symmetry is 1, Multiplicity = 4 The multiplicity tells us how many atoms are generated by symmetry if we place a single atom at that position. y x

(2)At a special position [100], the symmetry is
2. Multiplicity = 2 At a special position [010], the symmetry is 2. Multiplicity = 2 At a special position [001], the symmetry is 2. Multiplicity = 2

#2 Point group 4 At a general position [x y z], the symmetry is 1. Multiplicity = 4

(2) At a special position [001], the symmetry is
4. Multiplicity = 1 #3 Point group 4

At a general position [x y z], the symmetry
is 1. Multiplicity = 4 (2) At a special position [001], the symmetry is 𝑚. Multiplicity = 2

(3) At a special position [000], the symmetry is 4 . Multiplicity = 1
xyz, -x-yz, y-x-z, -yx-z 2 g m 0 ½ z, ½ 0 -z f ½ ½ z, ½ ½ -z e 0 0 z, 0 0 -z d ½ ½ ½ c ½ ½ 0 b 0 0 ½ a 000

Transformation of vector components
Original vector is P =[ 𝑝 1 , 𝑝 2 , 𝑝 3 ]= 𝑥, 𝑦, 𝑧 P =𝑥 x +𝑦 y +𝑧 z i.e. When symmetry operation transform the original axes x , y , z to the new axes x′ , y′ , z′ New vector after transformation of axes becomes P ′ =[ 𝑝 ′ 1, 𝑝′ 2, 𝑝′ 3 ]= 𝑢, 𝑣, 𝑤 i.e. P′ =𝑢 x ′ +𝑣 y′ +𝑤 z′

The angular relations between the axes may be specified by drawing up a table of direction cosines.
Old axes x y z New axes x′ a11 = cos x′ x a12 = cos x′ y a13 = cos x′ z y′ a21 = cos y′ x a22 = cos y′ y a23 = cos y′ z z′ a31 = cos z′ x a32 = cos z′ y a33 = cos z′ z

Then 𝑢=𝑥∗cos x′ x +𝑦∗cos x′ y + 𝑧∗cos x′ z i.e. 𝑝′ 1 = a 11 ∗ 𝑝 1 + a 12 ∗ 𝑝 2 + a 13 ∗ 𝑝 3 In a dummy notation 𝑝′ 1 = a 1j ∗ 𝑝 j Similarly 𝑝′ 2 = a 2j ∗ 𝑝 j 𝑝′ 3 = a 3j ∗ 𝑝 j i.e. 𝑝′ i = a ij ∗ 𝑝 j

Moreover, by repeating the argument for the reverse transformation and we have
𝑥=𝑢∗cos x ′ x +𝑣∗cos y ′ x + 𝑤∗cos z ′ x 𝑝 1 = a j1 ∗ 𝑝′ j Similarly, 𝑝 2 = a j2 ∗ 𝑝′ j 𝑝 3 = a j3 ∗ 𝑝′ j i.e. “old” in terms of “new” 𝑝 i = a ji ∗ 𝑝′ j

For example: #1 Point group 4
The direction cosines for the first operation is Old axes x y z New axes x′ = - y a11 = 0 a12 =−1 a13 = 0 y′ = x a21 = 1 a22 = 0 a23 = 0 z′ = z a31 = 0 a32 = 0 a33 =1

After symmetry operation, the new position is [x y z] in new axes.
We can express it in old axes by 𝑝 i = a ji ∗ 𝑝′ j = 𝑝′ j ∗a ji i.e. [ 𝑝 1 𝑝 2 𝑝 3 ]=[ 𝑥 𝑦 𝑧 ] 0 − =[ 𝑦 𝑥 𝑧 ] 𝑝 1 𝑝 2 𝑝 3 = − x y z = y x z or

14 plane lattices + 32 point groups
 230 Space groups Crystal Class Bravais Lattices Point Groups Triclinic P 1, 1 Monoclinic P, C 2, m, 2/m Orthorhombic P, C, F, I 222, mm2, 2/m 2/m 2/m Trigonal P, R 3, 3 , 32, 3m, 3 2/m Hexagonal 6, 6 , 6/m, 622, 6mm, 6 m2, 6/m 2/m 2/m Tetragonal P, I 4, 4 , 4/m, 422, 4mm, 4 2m, 4/m 2/m 2/m Isometric P, F, I 23, 2/m 3 , 432, 4 3m, 4/m 3 2/m

Table for all space groups Look at the notes!
B. Space group Table for all space groups Look at the notes! Good web site to read about space group

Symmetry elements in space group
Point group Translation symmetry + point group Translational symmetry operations The first character: P: primitive A, B, C: A, B, C-base centered F: Face centered I: Body centered R: Romohedral

Glide plane also exists for 3D space group with more possibility
Symmetry planes normal to the plane of projection Symmetry plane Graphical symbol Translation Symbol Reflection plane None   m Glide plane 1/2 along line   a, b, or c 1/2 normal to plane Double glide plane 1/2 along line & 1/2 normal to plane   e Diagonal glide plane   n Diamond glide plane 1/4 along line & 1/4 normal to plane   d

Symmetry planes normal to the plane of projection
Projection plane

Symmetry planes parallel to plane of projection
Graphical symbol Translation Symbol Reflection plane None   m Glide plane 1/2 along arrow   a, b, or c Double glide plane 1/2 along either arrow   e Diagonal glide plane 1/2 along the arrow   n Diamond glide plane 1/8 or 3/8 along the arrows   d 3/8 1/8 The presence of a d-glide plane automatically implies a centered lattice!

Glide planes ---- translation plus reflection across the glide plane * axial glide plane (glide plane along axis) ---- translation by half lattice repeat plus reflection ---- three types of axial glide plane i. a glide, b glide, c glide (a, b, c) 1 2 along line in plane ≡ along line parallel to projection plane

e.g. b glide , b --- graphic symbol for the axial glide plane along y axis c.f. mirror (m) graphic symbol for mirror

If b glide plane is ⊥ 𝑧 axis
If the axial glide plane is normal to projection plane, the graphic symbol change to c glide glide plane⊥ 𝑧 axis If b glide plane is ⊥ 𝑧 axis glide plane symbol

b , underneath the glide plane c glide: 𝑐 2 along z axis or 𝑎+𝑏+𝑐 2 along [111] on rhombohedral axis

ii. Diagonal glide (n) 𝑎+𝑏 2 , 𝑏+𝑐 2 , 𝑎+𝑐 2 or 𝑎+𝑏+𝑐 2 (tetragonal, cubic system) If glide plane is perpendicular to the drawing plane (xy plane), the graphic symbol is If glide plane is parallel to the drawing plane, the graphic symbol is

iii. Diamond glide (d) 𝑎+𝑏 4 , 𝑎+𝑏+𝑐 4 (tetragonal, cubic system)

Symbols of symmetry axes
Symmetry Element Graphical Symbol Translation Symbol Identity None   1 2-fold ⊥ page   2 2-fold in page 2 sub 1 ⊥ page 1/2   21 2 sub 1 in page 3-fold   3 3 sub 1 1/3   31 3 sub 2 2/3   32 4-fold   4 4 sub 1 1/4   41 4 sub 2   42 4 sub 3 3/4   43 6-fold   6 6 sub 1 1/6   61 6 sub 2   62 6 sub 3   63

Symmetry Element Graphical Symbol Translation Symbol 6 sub 4 2/3   64 6 sub 5 5/6   65 Inversion None   1 3 bar   3 4 bar   4 6 bar   6 = 3/m 2-fold and inversion   2/m 2 sub 1 and inversion   21/m 4-fold and inversion   4/m 4 sub 2 and inversion   42/m 6-fold and inversion   6/m 6 sub 3 and inversion   63/m

i. All possible screw operations
screw axis --- translation τ plus rotation screw Rn along c axis = counterclockwise rotation 360/R o + translation n/R c

2 21 3 31 32 4 41 42 43

6 61 62 63 64 65

62

Symmorphic space group is defined as a space group that may be specified entirely by symmetry operation acting at a common point (the operations need not involve τ) as well as the unit cell translation. (73 space groups) Nonsymmorphic space group is defined as a space group involving at least a translation τ.

Cubic – The secondary symmetry symbol will always be either 3 or –3 (i
Cubic – The secondary symmetry symbol will always be either 3 or –3 (i.e. Ia3, Pm3m, Fd3m) Tetragonal – The primary symmetry symbol will always be either 4, (-4), 41, 42 or 43 (i.e. P41212, I4/m, P4/mcc) Hexagonal – The primary symmetry symbol will always be a 6, (-6), 61, 62, 63, 64 or 65 (i.e. P6mm, P63/mcm) Trigonal – The primary symmetry symbol will always be a 3, (-3) 31 or 32 (i.e P31m, R3, R3c, P312) Orthorhombic – All three symbols following the lattice descriptor will be either mirror planes, glide planes, 2-fold rotation or screw axes (i.e. Pnma, Cmc21, Pnc2) Monoclinic – The lattice descriptor will be followed by either a single mirror plane, glide plane, 2-fold rotation or screw axis or an axis/plane symbol (i.e. Cc, P2, P21/n) Triclinic – The lattice descriptor will be followed by either a 1 or a (-1).

Examples Space group P1

Space group P 1

Space group P112

Space group P121

Space group P1121

Condition limiting possible reflections

Space group P1211

Space group B112 +( )

( 𝑥 , 𝑦 ,z) (1/2+ 𝑥 , 𝑦 ,1/2+z) (x,y,z) y x (1/2+x,y,1/2+z)

What can we do with the space group information
contained in the International Tables? 1. Generating a Crystal Structure from its Crystallographic Description 2. Determining a Crystal Structure from Symmetry & Composition

Example: Generating a Crystal Structure
Description of crystal structure of Sr2AlTaO6 Space Group = Fm 3 m; a = 7.80 Å Atomic Positions Atom x y z Sr 0.25 Al 0.0 Ta 0.5 O

From the space group tables
32 f 3m xxx, -x-xx, -xx-x, x-x-x, xx-x, -x-x-x, x-xx, -xxx 24 e 4mm x00, -x00, 0x0, 0-x0,00x, 00-x d mmm 0 ¼ ¼, 0 ¾ ¼, ¼ 0 ¼, ¼ 0 ¾, ¼ ¼ 0, ¾ ¼ 0 8 c 4 3m ¼ ¼ ¼ , ¼ ¼ ¾ 4 b m 3 m ½ ½ ½ a 000

Sr 8c; Al 4a; Ta 4b; O 24e 40 atoms in the unit cell stoichiometry Sr8Al4Ta4O24  Sr2AlTaO6 F: face centered  (000) (½ ½ 0) (½ 0 ½) (0 ½ ½) (000) (½½0) (½0½) (0½½) Sr 8c: ¼ ¼ ¼  (¼¼¼) (¾¾¼) (¾¼¾) (¼¾¾) ¼ ¼ ¾  (¼¼¾) (¾¾¾) (¾¼¼) (¼¾¼) Al ¾ + ½ = 5/4 =¼ 4a:  (000) (½ ½ 0) (½ 0 ½) (0 ½ ½)

Ta (000) (½½0) (½0½) (0½½) 4b: ½ ½ ½  (½½½) (00½) (0½0) (½00) (000) (½½0) (½0½) (0½½) O x00 24e: ¼ 0 0  (¼00) (¾½0) (¾0½) (¼½½) -x00 ¾ 0 0  (¾00) (¼½0) (¼0½) (¾½½) 0x0 0 ¼ 0  (0¼0) (½¾0) (½¼½) (½¾½) 0-x0 0 ¾ 0  (0¾0) (½¼0) (½¾½) (0¼½) 00x 0 0 ¼  (00¼) (½½¼) (½0¾) (0½¾) 00-x 0 0 ¾  (00¾) (½½¾) (½0¼) (0½0¼)

Bond distances: Al ion is octahedrally coordinated by six O Al-O distance d = 7.80 Å  − − − = 1.95 Å Ta ion is octahedrally coordinated by six O Ta-O distance d = 7.80 Å  − − − = 1.95 Å Sr ion is surrounded by 12 O Sr-O distance: d = 2.76 Å

Determining a Crystal Structure from
Symmetry & Composition Example: Consider the following information: Stoichiometry = SrTiO3 Space Group = Pm 3 m a = 3.90 Å Density = 5.1 g/cm3

First step: calculate the number of formula units per unit cell : Formula Weight SrTiO3 =  (16.00) = g/mol (M) Unit Cell Volume = (3.9010-8 cm)3 = 5.93  cm3 (V) (5.1 g/cm3)(5.93  cm3) : weight in a unit cell ( g/mole) / (6.022 1023/mol) : weight of one molecule of SrTiO3

 number of molecules per unit cell : 1 SrTiO3.
 (5.1 g/cm3)(5.93  cm3)/ ( g/mole/6.022 1023/mol) = 0.99  number of molecules per unit cell : 1 SrTiO3. From the space group tables (only part of it) 6 e 4mm x00, -x00, 0x0, 0-x0,00x, 00-x 3 d 4/mmm ½ 0 0, 0 ½ 0, 0 0 ½ c 0 ½ ½ , ½ 0 ½ , ½ ½ 0 1 b m 3 m ½ ½ ½ a 000

Calculate the Ti-O bond distances:
Sr: 1a or 1b; Ti: 1a or 1b  Sr 1a Ti 1b or vice verse O: 3c or 3d Evaluation of 3c or 3d: Calculate the Ti-O bond distances: d 3c) = 2.76 Å (0 ½ ½) D 3d) = 1.95 Å (½ 0 0, Better) Atom x y z Sr 0.5 Ti O

Another example from the note
The usage of space group for crystal structure identification Space group P 4/m 3 2/m Reference to note chapter 3-2 page 26

From the space group tables (only part of it)
6 e 4mm x00, -x00, 0x0, 0-x0,00x, 00-x 3 d 4/mmm ½ 0 0, 0 ½ 0, 0 0 ½ c 0 ½ ½ , ½ 0 ½ , ½ ½ 0 1 b m 3 m ½ ½ ½ a 000

#1 Simple cubic

Cl: 0, 0, 0 Cs: 0.5, 0.5, 0.5 (can interchange if desired)
#2 CsCl structure CsCl Vital Statistics Formula CsCl Crystal System Cubic Lattice Type Primitive Space Group Pm3m, No. 221 Cell Parameters a = Å, Z=1 Atomic Positions Cl: 0, 0, 0   Cs: 0.5, 0.5, 0.5 (can interchange if desired) Density 3.99

#3 BaTiO3 structure Temperature 183 K 278 K 393 K rhombohedral (R3m)
Orthorhombic (Amm2) Tetragonal (P4mm). Cubic (Pm 3 m)