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

III Crystal Symmetry3-3 Point group and space groupA.Point group

Symbols of the 32 three dimensional point groups

Rotation axis X

Rotation-Inversion axis

x

X

X + centre (inversion): Include for odd order

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

2 orm even: only for even rotation symmetry

Ratation axis with mirror plane normal to it X/m

x

x

x

x

Rotation axis with mirror plane (planes) parallel to it Xm

mirror

mirror

Rotation axis with diad axis (axes) normal to it X2

x

x

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

X

X

Rotation-inversion axis with mirror plane (planes) parallel to it m

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

mirror

X

X

x

xmirror

mirror

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 System

Symmetry Direction

Primary Secondary Tertiary

Triclinic None

Monoclinic [010]

Orthorhombic [100] [010] [001]

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

Hexagonal/Trigonal [001] [100]/[010] [120]/[10]

Cubic[100]/[010]/

[001] [111] [110]

Triclinic

Rotation axis X 1


X + centreInclude (odd order) 1

1

For odd order includes already!

Monoclinic1st setting

X 2

X

X + centreInclude (odd order)

2𝑚

= m

2

mirrormirror2𝑚 = 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



X2

Xm

2 orm even


222

2mm (2D) = mm2

mmm 2/m2/m2/m

2/m

2/m

2/m

2/m

Alreadydiscussed

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

2/m

2/m

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

Xm

2 orm even


432

4 3𝑚m3m

4/m 2/m

2/m

2/m

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

(3) At a special position [000], the symmetry is . Multiplicity = 1

P

4 h 1 xyz, -x-yz, y-x-z, -yx-z

2 g m 0 ½ z, ½ 0 -z

2 f m ½ ½ z, ½ ½ -z

2 e m 0 0 z, 0 0 -z

1 d ½ ½ ½

1 c ½ ½ 0

1 b 0 0 ½

1 a 000

Transformation of vector components

Original vector is [, , ]

P=𝑥 x+𝑦 y+𝑧 zi.e.

When symmetry operation transform the original axesto the new axes

New vector after transformation of axes becomes i.e.

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

Old axes

New axes

a11 = a12 = a13 =

a21 = a22 = a23 =

a31 = a32 = a33 =

Then

i.e.

In a dummy notation

Similarly

i.e.

Moreover, by repeating the argument for the reverse transformation and we have

Similarly,

i.e. “old” in terms of “new”

For example: #1 Point group 4

The direction cosines for the first operation is Old axes

New axes

= - a11 = a12 = a13 =

= a21 = a22 = a23 =

= a31 = a32 = a33 =

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

We can express it in old axes by

𝑝i=a ji∗𝑝 ′ j=𝑝 ′ j∗a ji

i.e. [

¿ [𝑦 𝑥 𝑧 ]

[𝑝1

𝑝2

𝑝3]=[ 0 1 0

− 1 0 00 0 1 ][ x

yz ]=[ y

xz ]or

14 plane 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

B. Space group

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 m

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

Glide plane1/2 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

y

bxa

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)

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


None 4/m


None 42/m


None 6/m


None 63/m

i. All possible screw operationsscrew axis --- translation τ plus rotation

screw Rn along c axis= counterclockwise rotation o + translation

41 42 434

212 31 323

61 62 63 64 656

62

T

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.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).

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


positions

Condition

limiting

possible

reflections

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


positions

Condition

limiting

possible

reflections

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



Space group P121P121 C21 No. 3 P121 2

Monoclinic

Origin on 2; unique axis b 2nd setting

Number

of

positions

Wyckoff

notation

Point

symmetry


positions

Condition

limiting

possible

reflections

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


Number

of

positions

Wyckoff

notation

Point

symmetry


positions

Condition

limiting

possible

reflections

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 zሿi

= 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

Condition limiting possible reflections

#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.

Space group P1211P21 C22 No. 4 P1211 2

Monoclinic

Origin on 21; unique axis b 2nd setting

Number

of

positions

Wyckoff

notation

Point

symmetry


positions

Condition

limiting

possible

reflections

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


Number

of

positions

Wyckoff

notation

Point

symmetry


positions

Condition

limiting

possible

reflections

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)

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 z

Sr 0.25 0.25 0.25

Al 0.0 0.0 0.0

Ta 0.5 0.5 0.5

O 0.25 0.0 0.0



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 ½

4 a mm 000



Sr 8c; Al 4a; Ta 4b; O 24e

40 atoms in the unit cellstoichiometry Sr8Al4Ta4O24 Sr2AlTaO6

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

Al

4a: 0 0 0 (000) (½ ½ 0) (½ 0 ½) (0 ½ ½)

(000) (½½0) (½0½) (0½½)

Ta

4b: 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)0x0

0-x0 0 ¾ 0 (0¾0) (½¼0) (½¾½) (0¼½)

0 0 ¼ (00¼) (½½¼) (½0¾) (0½¾)00x

00-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 ½

3 c 4/mmm 0 ½ ½ , ½ 0 ½ , ½ ½ 0

1 b mm ½ ½ ½

1 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 z

Sr 0.5 0.5 0.5

Ti 0 0 0

O 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

Coordinates of equivalent positions

1 A m3m 0, 0, 0

CsCl Vital StatisticsFormula CsCl

Crystal System CubicLattice Type PrimitiveSpace Group Pm3m, No. 221

Cell Parameters a = 4.123 Å, Z=1

Atomic PositionsCl: 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


positions

Cl 1 a m3m 0, 0, 0

Cs 1 b m3m 12, 12, 12

atoms Number of

positions

Wyckoff

notation

Point

symmetry


positions

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

Temperature

183 Krhombohedral

(R3m)

278 KOrthorhombic

(Amm2)

393 K

Tetragonal(P4mm).

Cubic(Pmm)

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

Documents

x2 x x slide

xm x x x x rotation

xmm mirror x x slide

hexagonal rotation axis

trigonal rotation axis

cubic rotation axis

triclinic rotation axis

tetragonal rotation