1. 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

2. 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)

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

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

5. 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

6. 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

7. 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

8. 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

9. 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]

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

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

12. X2 + centre, Xm +centre Include X m (odd order) 2/m1
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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. (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

20. #2 Point group 4
At a general position [x y z], the symmetry
is 1. Multiplicity = 4

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

22. 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

23. (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

24. 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′

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. Symmetry planes normal to the plane of projection
Projection plane

35. 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!

36. 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

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

38. 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

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

40. 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

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

42. 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

43. 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

44. 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

45. 2 21 3 31 32 4 41 42 43

46. 6 61 62 63 64 65

47. 62

48. 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.

49. 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).

50. Examples
Space group P1

52. Space group P 1

54. Space group P112

56. Space group P121

58. Space group P1121

60. Condition limiting possible reflections

62. Space group P1211

63. Space group B112
+( )

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

65. 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

66. 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

67. 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

68. 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)

69. 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)

70. 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

71. 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

72. 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

73.  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

74. 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

75. 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

76. 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

77. #1 Simple cubic

78. 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

79. #3 BaTiO3 structure Temperature 183 K 278 K 393 K rhombohedral (R3m)
Orthorhombic
(Amm2)
Tetragonal
(P4mm).
Cubic
(Pm 3 m)