iii crystal symmetry
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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
System and point group
Position in point group symbol Stereographic representationPrimary Secondary Tertiary
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
System and point group
Position in point group symbol Stereographic representationPrimary Secondary Tertiary
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
Rotation-Inversion axis
X + centreInclude (odd order)
X2
Xm
2 orm even
X2 + centre, Xm +centre Include m (odd order)
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
Rotation-Inversion axis
X + centreInclude (odd order)
4𝑚
X2
Xm
2 orm even
X2 + centre, Xm +centre Include m (odd order)
4
422
4mm
4 2𝑚4/mmm
4/m 2/m 2/m
Trigonal
Rotation axis X 3
Rotation-Inversion axis
X + centreInclude (odd order)
3
X2
Xm
2 orm even
X2 + centre, Xm +centre Include m (odd order)
32
3m
3
Hexagonal
Rotation axis X 6
Rotation-Inversion axis
X + centreInclude (odd order)
6𝑚
X2
Xm
2 orm even
X2 + centre, Xm +centre Include m (odd order)
6
622
6mm
6𝑚26/mmm
6/m 2/m 2/m
Cubic
Rotation axis X 23
Rotation-Inversion axis
X + centreInclude (odd order)
m3 2/m
X2 (+2 [110] direction)
Xm (+m [110] direction)
2 orm even
X2 + centre, Xm +centre Include m (odd order)
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
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
1/2 along line &1/2 normal to plane
n
Diamond glide plane
1/4 along line &1/4 normal to 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-fold None 44 sub 1 1/4 41
4 sub 2 1/2 42
4 sub 3 3/4 43
6-fold None 66 sub 1 1/6 61
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
4-fold and inversion None 4/m
4 sub 2 and inversion None 42/m
6-fold and inversion None 6/m
6 sub 3 and inversion None 63/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
Space group P P1ത Ci1 No. 2 P1ത 1ത Triclinic
Origin on 1ത Number of positions
Wyckoff notation
Point symmetry
Coordinates of equivalent 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
Coordinates of equivalent 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
Coordinates of equivalent 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
P21 C22 No. 4 P1121 2 Monoclinic
Ist setting Origin on 21; unique axis c Number of positions
Wyckoff notation
Point symmetry
Coordinates of equivalent 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 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
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
Coordinates of equivalent 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
Ist setting Origin on 2; unique axis c Number of positions
Wyckoff notation
Point symmetry
Coordinates of equivalent 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)
http://it.iucr.org/
and all the space groups
http://it.iucr.org/Ab/contents/
Here you can access the “International Tables for Crystallography”
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
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
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)
http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wp-list?gnum=221
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)
http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wp-list?gnum=221
#1 Simple cubic
Number of positions
Wyckoff notation
Point symmetry
Coordinates of equivalent positions
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
Coordinates of equivalent positions
Cl 1 a m3m 0, 0, 0 Cs 1 b m3m 12, 12, 12
atoms Number of positions
Wyckoff notation
Point symmetry
Coordinates of equivalent 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
Temperature183 K
rhombohedral (R3m)
278 KOrthorhombic
(Amm2)
393 KTetragonal(P4mm).
Cubic(Pmm)
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