III Crystal Symmetry

III Crystal Symmetry3-3 Point group and space group

A. Point groupSymbols of the 32 three dimensional point groups

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1, 2, 3, 4, 6, 23 + diad axis, mirror, inversion 32 point groups

General symbol Triclinic Monoclinic

1st setting Tetragonal Trigonal Hexagonal Cubic

X 1 2 4 3 6 23

m

X + centre (include

odd order) (2/m) m3

Monoclinic2nd setting

Orthorho-mbic

X2 2 12 222 422 32 622 432

Xm m 1m mm2 (2mm) 4mm 3m 6mm 3m

2 or meven 2m m2

X2 + centreXm + centreInclude m odd order

mmm mm m mm m3m

1. Rotation axis X

2. Rotation-Inversion axis

x

X

3. X + centre (inversion): Include for odd order4. Rotation axis + diad axis (axes) normal to it: X2

x

x

x

X

5. Rotation axis + mirror plane (planes) parallel to it: Xm

mirror

6. 2 orm even: only for even rotation symmetry (refer to 4 and 5 except the rotation axis is )

7. X2 + centre; Xm +centre: the same result (Include m for odd order)

Rotation axis with mirror plane normal to it: X/m

x

X

mirror

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

x

xmirror

mirrorX/m

System and point group

Position in point group symbol Stereographic representationPrimary Secondary Tertiary

Triclinic1,

Only one symbol which denotes all directions in the crystal.

Monoclinic2, m, 2/m

The symbol gives the nature of the unique diad axis (rotation and/or inversion).1st setting: z-axis unique2nd setting: y-axis unique

1st setting

2nd setting



Orthorhombic222, 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

Tetragonal4, , 4/m, 422, 4mm, 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 [10] axis



Trigonal and Hexagonal3, , 32, 3m, m, 6, , 6/m, 622, 6mm, 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)

Cubic23, m3, 432,3m, 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 SystemSymmetry Direction

Primary Secondary Tertiary

Triclinic None

Monoclinic [010]

Orthorhombic [100] [010] [001]

Tetragonal [001] [100]/[010] [110]

Hexagonal [001] [100] [120]

Trigonal [111] [010] [10]

Cubic [100]/[010]/[001] [111] [110]

TriclinicRotation axis X 1

Rotation-Inversion axis

X + centreInclude (odd order) 1

1

For odd order includes already!

Monoclinic1st setting

X 2

XX + centreInclude (odd order)

2𝑚

= m

2mirrormirror

2𝑚 = m2

Monoclinic2st setting

X2 12

Xm

2 orm (even)

X2 + centre, Xm +centre Include m (odd order)

1m

2/m

2/m

2

1

Orthorhombic

Rotation axis X


X + centreInclude (odd order)

X2

Xm

2 orm even


222

2mm (2D) = mm2

mmm 2/m2/m2/m

mm2

2/m

2/m

2/2mIncluded inmonoclinic

system

2 orm

RL

LR

mirror

mirror

2

2mm

Question: mmm 2/m2/m2/mterminologynew old

Using mmm symmetry operation, one canget the same point group as !

Tetragonal

Rotation axis X 4



4𝑚

X2

Xm

2 orm even


4

422

4mm

4 2𝑚4/mmm

4/m 2/m 2/m

Trigonal

Rotation axis X 3



3

X2

Xm

2 orm even


32

3m

3

Hexagonal

Rotation axis X 6



6𝑚

X2

Xm

2 orm even


6

622

6mm

6𝑚26/mmm

6/m 2/m 2/m

Cubic

Rotation axis X 23



m3 2/m

X2 (+2 [110] direction)

Xm (+m [110] direction)

2 orm even


432

4 3𝑚

m3m 4/m 2/m

23

R

+ mirror [110] direction

R

R

R

R

RR

R

RR

RR

L

L

LL

L

L

L

L

L

L

LL

4 3𝑚

Examples of point group operation

#1 Point group 222

(1) At a general position [x y z], the symmetry is 1, Multiplicity = 4

x

y

The multiplicity tells us how many atoms are generated by symmetry if we place a single atom at that position.

(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

(1)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

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

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

Original vector is

i.e.

When symmetry operation transform the original axes to the new axes

The same vector also transform with the axes and remain the same coordination (x, y, z)

i.e.

],,[],,[ 321 zyxpppP

zzyyxxP ˆˆˆ

)ˆ,ˆ,ˆ( zyx )ˆ,ˆ,ˆ( zyx

],,[],,[ 321 zyxpppP

zzyyxxP ˆˆˆ

Finding the new coordination for a point aftersymmetry operation:

xˆ

yˆ

x

yP

P ],[ yx

],[ yx],[ vu

system ˆ ˆ in yx system ˆˆ in yx

yvxuyyxxP ˆˆˆˆ

What is the coordination of (symmetryoperation of ) in term of old coordination?

P

P

zwyvxuzzyyxxP ˆˆˆˆˆˆ

We want to know the coordinates of thevector in the same coordination system

Consider orthogonal coordination system, i.eall base vectors are perpendicular to each other

zwxyvxxuxxPx ˆˆˆˆˆˆˆ

zwyyvyxuyyPy ˆˆˆˆˆˆˆ

zwzyvzxuzzPz ˆˆˆˆˆˆˆ

wvu

zzyzxzzyyyxyzxyxxx

zyx

ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ

wvu

zzyzxzzyyyxyzxyxxx

zyx

ˆˆcosˆˆcosˆˆcosˆˆcosˆˆcosˆˆcosˆˆcosˆˆcosˆˆcos

zyx

zzyzxzzyyyxyzxyxxx

wvu

ˆˆcosˆˆcosˆˆcosˆˆcosˆˆcosˆˆcosˆˆcosˆˆcosˆˆcos

Similarly,

The angular relations between the axes may be specified by drawing up a table of direction cosines.

New Axes

old Axes

zyx

zˆyˆxˆxxa ˆˆcos11 yxa ˆˆcos12 zxa ˆˆcos13

xya ˆˆcos21 yya ˆˆcos22 zya ˆˆcos23

xza ˆˆcos31 yza ˆˆcos32 zza ˆˆcos33

For example: #1 Point group 4

The direction cosines for the first operation isNew Axes

Old Axes

zz ˆˆ xy ˆˆ yx ˆˆ

zyx 011 a

121 a

031 a

012 a

022 a

032 a

013 a

023 a

133 a

4

After symmetry operation, the new position is [x y z] in new axes.

We can express it in old axes by

zx

y

zyx

wvu

100001010

B. Space group

14 lattices + 32 point groups 230 Space groups

Crystal Class Bravais Lattices Point Groups

Triclinic P 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, , 32, 3m, 2/m

Hexagonal P 6, , 6/m, 622, 6mm, m2,6/m 2/m 2/m

Tetragonal P, I 4, , 4/m, 422, 4mm, 2m,4/m 2/m 2/m

Isometric P, F, I 23, 2/m, 432, 3m, 4/m2/m

Table for all space groupsLook at the notes!

http://www.uwgb.edu/dutchs/SYMMETRY/3dSpaceGrps/3dspgrp.htm

Good web site to read about space group

http://img.chem.ucl.ac.uk/sgp/mainmenu.htm



http://img.chem.ucl.ac.uk/sgp/mainmenu.htm


The first character: P: primitiveA, B, C: A, B, C-base centeredF: Face centeredI: Body centeredR: Romohedral

Symmetry elements in space group(1)Point group(2)Translation symmetry + point group

Translational symmetry operations

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 mGlide plane 1/2 along line a, b, or c

Glide plane 1/2 normal to plane a, b, or c

Double glide plane

1/2 along line &1/2 normal to plane

e

Diagonal glide plane


n

Diamond glide plane


d

Projection plane

Symmetry planes normal to the plane of projection

1/8

Symmetry plane 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 d3/8

Symmetry planes parallel to plane of projection

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)

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 the axial glide plane is normal to projection plane, the graphic symbol change to

c glide

z c

ybx

a

glide plane axis⊥

If b glide plane is axis ⊥ z

y

x

glide plane symbol

b,

underneath the glide plane

c glide: along z axis

along [111] on rhombohedral axis

or

ii. Diagonal glide (n)

, , or (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)

, (tetragonal, cubic system)

Glide plane

Glide direction

Symmetry Element Graphical Symbol Translation Symbol

Identity None None 12-fold page⊥ None 22-fold in page None 2

2 sub 1 page⊥ 1/2 21

2 sub 1 in page 1/2 21

3-fold None 33 sub 1 1/3 31

3 sub 2 2/3 32


4 sub 2 1/2 42

4 sub 3 3/4 43


6 sub 2 1/3 62

6 sub 3 1/2 63

Symbols of symmetry axes

Symmetry Element Graphical Symbol Translation Symbol

6 sub 4 2/3 64

6 sub 5 5/6 65

Inversion None 13 bar None 34 bar None 46 bar None 6 = 3/m

2-fold and inversion None 2/m

2 sub 1 and inversion None 21/m





•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 followed by either a 1 or a (-1).

How to quickly identify the crystal class of a spacegroup?

P1 C11 No. 1 P1 1 Triclinic

Origin on 1 Number of positions

Wyckoff notation

Point symmetry

Coordinates of equivalent positions

Condition limiting possible reflections

1 a 1 x, y, z No conditions

ExamplesSpace group P1

http://img.chem.ucl.ac.uk/sgp/large/001az1.htm


Space group P P1ത Ci1 No. 2 P1ത 1ത Triclinic

Origin on 1ത Number of positions

Wyckoff notation

Point symmetry



2 i 1 x, y, z；xത, yത, zത General: No conditions

1 h 1ത 12, 12, 12 Special: No conditions

1 g 1ത 0, 12, 12 1 f 1ത 12, 0, 12 1 e 1ത 12, 12, 0 1 d 1ത 12, 0, 0 1 c 1ത 0, 12, 0 1 b 1ത 0, 0, 12 1 a 1ത 0, 0, 0

Space group P112P112 C21 No. 3 P112 2 Monoclinic

Ist setting Origin on 2; unique axis c Number of positions

Wyckoff notation

Point symmetry



2 e 1 x, y, z; xത, yത, z General:

൝hklhk000lൡ

No conditions 1 d 2 12, 12, z Special:

No conditions 1 c 2 12, 0, z 1 b 2 0, 12, z 1 a 2 0, 0, z


Origin on 2; unique axis b 2nd setting Number of positions

Wyckoff notation

Point symmetry



2 e 1 x, y, z; xത, y, zത General:

൝hklh0l0k0ൡ

No conditions

1 d 2 12, y, 12 Special: No conditions

1 c 2 12, y, 0 1 b 2 0, y, 12 1 a 2 0, y, 0

http://img.chem.ucl.ac.uk/sgp/large/003ay1.htm

http://img.chem.ucl.ac.uk/sgp/large/003ay1.htm

P21 C22 No. 4 P1121 2 Monoclinic


Wyckoff notation

Point symmetry



2 a 1 x, y, z; xത, yത, 12 +z General: hkl: No conditions hk0: No conditions 00l: l=2n

Space group P1121

Explanation: #1 Consider the diffraction condition from plane (h k 0) Two atoms at x, y, z; xത, yത, 12 +z The diffraction amplitude F can be expressed as F= fi ∗e−2πiሾh k lሿ∗ሾx y zሿ

i

= fi ∗e−2πiሾh k 0ሿ∗ሾx y zሿi

= fi ∗e−2πiሾh k 0ሿ∗ሾx y zሿ+ fi ∗e−2πiሾh k 0ሿ∗ሾ xത yഥ 1/2 +z ሿ = fi ∗e−2πi(hx+ky) + fi ∗e−2πiሺ−hx−kyሻ = fi ∗൫e−2πi(hx+ky) + e2πi(hx+ky)൯ = fi ∗൫2cos൫2πiሺhx + kyሻ൯൯ = 2fi

Therefore, no conditions can limit the (h, k, 0) diffraction


#2 For the planes (00l) Two atoms at x, y, z; xത, yത, 12+z The diffraction amplitude F can be expressed as F= fi ∗e−2πiሾh k lሿ∗ሾxi yi ziሿi

= fi ∗e−2πiሾ0 0 lሿ∗ሾxi yi ziሿi

= fi ∗e−2πiሾ0 0 lሿ∗ሾx y zሿ+ fi ∗e−2πiሾ0 0 lሿ∗ሾ xത yഥ 1/2 +z ሿ = fi ∗e−2πilz + fi ∗e−2πi൬l2+lz൰ = fi ∗e−2πilz ∗൫1+ e−πil൯ = fi ∗൫1+ e−πil൯ If l=2n, then F=2fi If l=2n+1, then F=0

Therefore, the condition l=2n limit the (0, 0 ,l) diffraction.


Origin on 21; unique axis b 2nd setting Number of positions

Wyckoff notation

Point symmetry



2 a 1 x, y, z; xത, 12+y, zത General: hkl: No conditions h0l: No conditions 0k0: k=2n

Space group B112B2 C23

No. 5 B112 2 Monoclinic


Wyckoff notation

Point symmetry



4 c 1 x, y, z; xത, yത, z General: hkl: h+l=2n hk0: h=2n 00l: l=2n

2 b 2 0, 12, z Special: as above only 2 a 2 0, 0, z

+(

(x,y,z)

(,,z)

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

x

y

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

http://it.iucr.org/

and all the space groups

http://it.iucr.org/Ab/contents/

Here you can access the “International Tables for Crystallography”

http://it.iucr.org/

http://it.iucr.org/



1. Generating a Crystal Structure from its Crystallographic Description

What can we do with the space group informationcontained in the International Tables?

2. Determining a Crystal Structure from Symmetry & Composition

Example: Generating a Crystal Structure http://chemistry.osu.edu/~woodward/ch754/sym_itc.htm

Description of crystal structure of Sr2AlTaO6

Space Group = Fmm; a = 7.80 ÅAtomic Positions

Atom x y zSr 0.25 0.25 0.25Al 0.0 0.0 0.0Ta 0.5 0.5 0.5O 0.25 0.0 0.0

http://chemistry.osu.edu/~woodward/ch754/sym_itc.htm

http://chemistry.osu.edu/~woodward/ch754/sym_itc.htm

From the space group tables http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wp-list?gnum=225

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

24 d mmm 0 ¼ ¼, 0 ¾ ¼, ¼ 0 ¼,¼ 0 ¾, ¼ ¼ 0, ¾ ¼ 0

8 c 3m ¼ ¼ ¼ , ¼ ¼ ¾ 4 b mm ½ ½ ½4 a mm 000

http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wp-list?gnum=225


Sr 8c; Al 4a; Ta 4b; O 24e 40 atoms in the unit cellstoichiometry Sr8Al4Ta4O24 Sr2AlTaO6

Another way of thinking: F lattice has for lattice sites per unit cell, each site have to put a molecule Sr2AlTaO6 (10 atoms) 4x10 = 40 atoms per unit cell

F: face centered (000) (½ ½ 0) (½ 0 ½) (0 ½ ½)

8c: ¼ ¼ ¼ (¼¼¼) (¾¾¼) (¾¼¾) (¼¾¾) ¼ ¼ ¾ (¼¼¾) (¾¾¾) (¾¼¼) (¼¾¼)

¾ + ½ = 5/4 =¼

Sr

Al4a: 0 0 0 (000) (½ ½ 0) (½ 0 ½) (0 ½ ½)

(000) (½½0) (½0½) (0½½)

Ta4b: ½ ½ ½ (½½½) (00½) (0½0) (½00)

O

24e: ¼ 0 0 (¼00) (¾½0) (¾0½) (¼½½)

(000) (½½0) (½0½) (0½½)

(000) (½½0) (½0½) (0½½)

¾ 0 0 (¾00) (¼½0) (¼0½) (¾½½)

x00

-x00

0 ¼ 0 (0¼0) (½¾0) (½¼½) (½¾½)0x00-x0 0 ¾ 0 (0¾0) (½¼0) (½¾½) (0¼½)

0 0 ¼ (00¼) (½½¼) (½0¾) (0½¾)00x00-x 0 0 ¾ (00¾) (½½¾) (½0¼) (0½0¼)

Bond distances:Al ion is octahedrally coordinated by six OAl-O distanced = 7.80 Å = 1.95 Å

Ta ion is octahedrally coordinated by six OTa-O distanced = 7.80 Å = 1.95 Å

Sr ion is surrounded by 12 OSr-O distance: d = 2.76 Å

Determining a Crystal Structure fromSymmetry & Composition

Example:Consider the following information:Stoichiometry = SrTiO3

Space Group = Pmma = 3.90 ÅDensity = 5.1 g/cm3

First step:calculate the number of formula units per unit cell :Formula Weight SrTiO3 = 87.62 + 47.87 + 3 (16.00) = 183.49 g/mol (M)

Unit Cell Volume = (3.9010-8 cm)3 = 5.93 10-23 cm3 (V)

(5.1 g/cm3)(5.93 10-23 cm3) : weight in aunit cell

(183.49 g/mole) / (6.022 1023/mol) : weightof one molecule of SrTiO3

number of molecules per unit cell : 1 SrTiO3.

(5.1 g/cm3)(5.93 10-23 cm3)/(183.49 g/mole/6.022 1023/mol) = 0.99

6 e 4mm x00, -x00, 0x0,0-x0,00x, 00-x

3 d 4/mmm ½ 0 0, 0 ½ 0, 0 0 ½3 c 4/mmm 0 ½ ½ , ½ 0 ½ , ½ ½ 0 1 b mm ½ ½ ½1 a mm 000

From the space group tables (only part of it)





Sr: 1a or 1b; Ti: 1a or 1b Sr 1a Ti 1b or vice verseO: 3c or 3d

Evaluation of 3c or 3d: Calculate the Ti-O bond distances:d (O @ 3c) = 2.76 Å (0 ½ ½) D (O @ 3d) = 1.95 Å (½ 0 0, Better)

Atom x y zSr 0.5 0.5 0.5Ti 0 0 0O 0.5 0 0

The usage of space group for crystal structure identificationSpace group P 4/m 2/m

Reference to note chapter 3-2 page 26

Another example from the note

6 e 4mm x00, -x00, 0x0,0-x0,00x, 00-x

3 d 4/mmm ½ 0 0, 0 ½ 0, 0 0 ½3 c 4/mmm 0 ½ ½ , ½ 0 ½ , ½ ½ 0 1 b mm ½ ½ ½1 a mm 000

From the space group tables (only part of it)





#1 Simple cubic

Number of positions

Wyckoff notation

Point symmetry


1 A m3m 0, 0, 0

Example:

CsCl Vital StatisticsFormula CsCl

Crystal System CubicLattice Type PrimitiveSpace Group Pm3m, No. 221

Cell Parameters a = 4.123 Å, Z=1

Atomic Positions Cl: 0, 0, 0 Cs: 0.5, 0.5, 0.5(can interchange if desired)

Density 3.99

#2 CsCl structureatoms Number of

positions Wyckoff notation

Point symmetry


Cl 1 a m3m 0, 0, 0 Cs 1 b m3m 12, 12, 12

atoms Number of positions

Wyckoff notation

Point symmetry


Ba 1 a m3m 0, 0, 0 Ti 1 b m3m 12, 12, 12 O 3 c 4/mmm 0, 12, 12; 12, 0, 12; 12, 12, 0

#3 BaTiO3 structure

Temperature183 K

rhombohedral (R3m)

278 KOrthorhombic

(Amm2)

393 KTetragonal(P4mm).

Cubic(Pmm)

III Crystal Symmetry

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