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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
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 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
Rotation-Inversion axis
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
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
2/m
2/m
2/m
2/m
Alreadydiscussed
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
2/m
2/m
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
Xm
2 orm even
X2 + centre, Xm +centre Include m (odd order)
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
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
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
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-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
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
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 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
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)
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)
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 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)
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
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
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
Temperature
183 Krhombohedral
(R3m)
278 KOrthorhombic
(Amm2)
393 K
Tetragonal(P4mm).
Cubic(Pmm)