1. GEOMETRY OF CRYSTALS Space Lattices Crystal Structures
Symmetry, Point Groups and Space Groups
Acknowledgments: Prof. Rajesh Prasad for a lot of things

2. Crystal = Lattice + Motif
The language of crystallography is one succinctness
Crystal = Lattice + Motif
Motif or basis: an atom or a group of atoms associated with each lattice point

3. An array of points such that every point has identical surroundings
Space Lattice
An array of points such that every point has identical surroundings
In Euclidean space  infinite array
We can have 1D, 2D or 3D arrays (lattices) or
Translationally periodic arrangement of points in space is called a lattice

4. A 2D lattice

5. Crystal = Lattice + Motif
Translationally periodic arrangement of points
Translationally periodic arrangement of motifs
Crystal = Lattice + Motif
Lattice  the underlying periodicity of the crystal
Basis  atom or group of atoms associated with each lattice points
Lattice  how to repeat
Motif  what to repeat

6. Lattice
Motif + 

7. = Crystal                              
Courtesy Dr. Rajesh Prasad

8. Cells A cell is a finite representation of the infinite lattice
Instead of drawing the whole structure I can draw a representative part and specify the repetition pattern
A cell is a finite representation of the infinite lattice
A cell is a parallelogram (2D) or a parallelopiped (3D) with lattice points at their corners.
If the lattice points are only at the corners, the cell is primitive.
If there are lattice points in the cell other than the corners, the cell is nonprimitive.

9. Nonprimitive cell Primitive cell Primitive cell
Courtesy Dr. Rajesh Prasad

10. Nonprimitive cell Primitive cell Primitive cell Double Triple
Symmetry of the Lattice or the crystal is not altered by our choice of unit cell!!

11. Centred square lattice = Simple/primitive square lattice
Nonprimitive cell
Primitive cell b a
4- fold axes
Shortest lattice translation vector  ½ [11]

12. Centred rectangular lattice
Maintains the symmetry of the lattice  the usual choice
Nonprimitive cell
Primitive cell
Lower symmetry than the lattice  usually not chosen
2- fold axes

13. Centred rectangular lattice Simple rectangular Crystal
Primitive cell
Not a cell
MOTIF
Shortest lattice translation vector  [10]
Courtesy Dr. Rajesh Prasad

14. Cells- 3D
In order to define translations in 3-d space, we need 3 non-coplanar vectors
Conventionally, the fundamental translation vector is taken from one lattice point to the next in the chosen direction
With the help of these three vectors, it is possible to construct a parallelopiped called a CELL

15. Different kinds of CELLS
Unit cell
A unit cell is a spatial arrangement of atoms which is tiled in three-dimensional space to describe the crystal.
Primitive unit cell
For each crystal structure there is a conventional unit cell, usually chosen to make the resulting lattice as symmetric as possible. However, the conventional unit cell is not always the smallest possible choice. A primitive unit cell of a particular crystal structure is the smallest possible unit cell one can construct such that, when tiled, it completely fills space.
Wigner-Seitz cell
A Wigner-Seitz cell is a particular kind of primitive cell
which has the same symmetry as the lattice.

16. SYMMETRY SYMMETRY OPERATOR
If an object is brought into self-coincidence after some operation it said to possess symmetry with respect to that operation.
SYMMETRY OPERATOR
Given a general point a symmetry operator leaves a finite set of points in space
A symmetry operator closes space onto itself

17. Symmetry operators Translation Rotation Symmetries Type I
Takes object to same form → Proper
Type I
Rotation
Symmetries
Roto- reflection
Mirror
Type II
Inversion
Takes object to enantiomorphic form → improper
Roto- inversion
Classification based on the dimension invariant entity of the symmetry operator
Operator
Dimension
Inversion 0D
Rotation 1D
Mirror 2D

18. Symmetry operators Symmetries Rotation Mirror Macroscopic Inversion
Influence the external shape of the crystal
Macroscopic
Inversion
Symmetries
Screw Axes
Microscopic
Glide Reflection
Do not Influence the external shape of the crystal

19. Ones with built in translation
Minimum set of symmetry operators required
R  Rotation
G  Glide reflection
R  Roto-inversion
S  Screw axis
Ones acting at a point
Ones with built in translation

20. =180 n=2 2-fold rotation axis =120 n=3 3-fold rotation axis
If an object come into self-coincidence through smallest non-zero rotation angle of  then it is said to have an n-fold rotation axis where
=180
n=2
2-fold rotation axis
=120
n=3
3-fold rotation axis

21. =90 n=4 4-fold rotation axis =60 n=6 6-fold rotation axis
The rotations compatible with translational symmetry are  (1, 2, 3, 4, 6)

22. Symmetries acting at a point
Point group symmetry of Lattices →
7 crystal systems
Symmetries acting at a point
R  R
32 point groups
Along with symmetries having a translation
G + S
230 space groups
Space group symmetry of Lattices →
14 Bravais lattices
R + R → rotations compatible with translational symmetry (1, 2, 3, 4, 6)

23. Crystal = Lattice (Where to repeat) + Motif (What to repeat)
Previously
Crystal = Lattice (Where to repeat) Motif (What to repeat) a = a +

24. Space group (how to repeat) + Asymmetric unit (Motif’: what to repeat)
Crystal =
Space group (how to repeat)
+ Asymmetric unit (Motif’: what to repeat)
Now a =
Glide reflection operator a +
Usually asymmetric units are regions of space within the unit cell- which contain atoms

25. a mmm Three mirror planes mirror glide reflection
Progressive lowering of symmetry in an 1D lattice  illustration using the frieze groups
Consider a 1D lattice with lattice parameter ‘a’
Asymmetric Unit
Unit cell a
mmm
Three mirror planes
The intersection points of the mirror planes give rise to redundant inversion centres
mirror
glide reflection

26. Decoration of the lattice with a motif  may reduce the symmetry of the crystal1
mmm
Decoration with a “sufficiently” symmetric motif does not reduce the symmetry of the lattice 2 mm
Loss of 1 mirror plane

27. 3 mg 4 ii 2 inversion centres
Presence of 1 mirror plane and 1 glide reflection plane, with a redundant inversion centre the translational symmetry has been reduced to ‘2a’ 4 ii
2 inversion centres

28. 5 m
1 mirror plane 6 g
1 glide reflection translational symmetry of ‘2a’ 7
No symmetry except translation

29. Redundant mirrors which need not be drawn
Effect of the decoration  a 2D example
Two kinds of decoration are shown  (i) for an isolated object, (ii) an object which can be an unit cell.
4mm
Redundant mirrors which need not be drawn
Redundant inversion centre
Can be a unit cell for a 2D crystal
4mm
Decoration retaining the symmetry

30. mm m m

31. 4
No symmetry
If this is an unit cell of a crystal → then the crystal would still have translational symmetry

32. Lattices have the highest symmetry  Crystals based on the lattice can have lower symmetry

33. Amorphous arrangement No unit cell
Unit cell of Triclinic crystal

34. Object with bilateral symmetry
Positioning a object with respect to the symmetry elements
mmm
Three mirror planes
The intersection points of the mirror planes give rise to redundant inversion centres
Left handed object
Right handed object
Object with bilateral symmetry

35. Positioning a object with respect to the symmetry elements
General site  8 identiti-points
On mirror plane (m)  4 identiti-points
On mirror plane (m)  4 identiti-points
Site symmetry 4mm  1 identiti-point
Note: this is for a point group and not for a lattice  the black lines are not unit cells

36. Positioning of a motif w. r
Positioning of a motif w.r.t to the symmetry elements of a lattice  Wyckoff positions
A 2D lattice with symmetry elements

37. Number of Identi-points
Multi-plicity
Wyckoff letter
Site symmetry
Coordinates 8 g
Area 1
(x,y)
(-x,-y)
(-y,x)
(y,-x)
(-x,y)
(x,-y)
(y,x)
((-y,-x) 4 f
Lines
..m
(x,x)
(-x,-x)
(x,-x)
(-x,x) e
.m.
(x,½)
(-x, ½)
(½,x)
(½,-x) d
(x,0)
(-x,0)
(0,x)
(0,-x) 2 c
Points
2mm.
(½,0)
(0,½) b
4mm
(½,½) a
(0,0) f g e b d a c
Any site of lower symmetry should exclude site(s) of higher symmetry [e.g. (x,x) in site f cannot take values (0,0) or (½, ½)]
Number of Identi-points

38. g e b d a c d e f f Exclude these points Exclude these points

39. Bravais Space Lattices  some other view points
Conventionally, the finite representation of space lattices is done using unit cells which show maximum possible symmetries with the smallest size.
Considering
Maximum Symmetry, and
Minimum Size
Bravais concluded that there are only 14 possible Space Lattices (or Unit Cells to represent them).
These belong to 7 Crystal systems
Or  the technical definition 
There are 14 Bravais Lattices which are the space group symmetries of lattices

40. Bravais Lattice
A lattice is a set of points constructed by translating a single point in discrete steps by a set of basis vectors. In three dimensions, there are 14 unique Bravais lattices (distinct from one another in that they have different space groups) in three dimensions. All crystalline materials recognized till now fit in one of these arrangements. or
In geometry and crystallography, a Bravais lattice is an infinite set of points generated by a set of discrete translation operations. A Bravais lattice looks exactly the same no matter from which point one views it.

41. Arrangement of lattice points in the unit cell. & No
Arrangement of lattice points in the unit cell & No. of Lattice points / cell
Position of lattice points
Effective number of Lattice points / cell 1 P
8 Corners
= 8 x (1/8) = 1 2 I
8 Corners body centre
= 1 (for corners) + 1 (BC) 3 F
8 Corners +
6 face centres
= 1 (for corners) + 6 x (1/2) = 4 4 A/ B/ C
8 corners + 2 centres of opposite faces
= 1 (for corners) + 2x(1/2) = 2

42. 14 Bravais lattices divided into seven crystal systems
Crystal system Bravais lattices
Cubic P I F
Tetragonal P I
Orthorhombic P I F C
Hexagonal P
Trigonal P
Monoclinic P C
Triclinic P
Courtesy Dr. Rajesh Prasad

43. 14 Bravais lattices divided into seven crystal systems
Crystal system Bravais lattices
Cubic P I F C
Tetragonal P I
Orthorhombic P I F C
Hexagonal P
Trigonal P
Monoclinic P C
Triclinic P
Courtesy Dr. Rajesh Prasad

44. Cubic F  Tetragonal I
The symmetry of the unit cell is lower than that of the crystal

45. 14 Bravais lattices divided into seven crystal systems
Crystal system Bravais lattices
Cubic P I F C
Tetragonal P I F
Orthorhombic P I F C
Hexagonal P
Trigonal P
Monoclinic P C
Triclinic P x
Courtesy Dr. Rajesh Prasad

46. The following 4 things are different
Lattice
Motif
Symmetry of the
Crystal
Unit Cell
Eumorphic crystal (equilibrium shape and growth shape of the crystal)
The shape of the crystal corresponds to the point group symmetry of the crystal

47. FCT = BCT

48. Crystal system
The crystal system is the point group of the lattice (the set of rotation and reflection symmetries which leave a lattice point fixed), not including the positions of the atoms in the unit cell.
There are seven unique crystal systems.

49. Polar coordinates (, )
Concept of symmetry and choice of axes
(a,b)
The centre of symmetry of the object does not coincide with the origin
Polar coordinates (, )
The type of coordinate system chosen is not
according to the symmetry of the object
Mirror
Our choice of coordinate axis does not alter the symmetry of the object (or the lattice)
Centre of Inversion

50. THE 7 CRYSTAL SYSTEMS

51. N is the number of point groups for a crystal system

52. (Truncated Octahedron)
Cubic Crystals
a = b= c  =  =  = 90º
Simple Cubic (P)
Body Centred Cubic (I) – BCC
Face Centred Cubic (F) - FCC
Vapor grown NiO crystal
[2]
Tetrakaidecahedron
(Truncated Octahedron)
Pyrite Cube
[1]
[1]
Fluorite
Octahedron
Garnet Dodecahedron
[1]
[1]
[2] L.E. Muir, Interfacial Phenomenon in Metals, Addison-Wesley Publ. co.

53. Tetragonal Crystals a = b  c  =  =  = 90º Simple Tetragonal
Body Centred Tetragonal
Zircon
[1]
[1]
[1]
[1]

54. Orthorhombic Crystals
a  b  c  =  =  = 90º
Simple Orthorhombic
Body Centred Orthorhombic
Face Centred Orthorhombic
End Centred Orthorhombic
[1]
Topaz
[1]
[1]

55. Hexagonal Crystals a = b  c  =  = 90º  = 120º Simple Hexagonal
[1]
Corundum
[1]

56. 5. Rhombohedral Crystals
a = b = c  =  =   90º
Rhombohedral (simple)
[1]
[1]
Tourmaline
[1]

57. Monoclinic Crystals a  b  c  =  = 90º   Simple Monoclinic
End Centred (base centered) Monoclinic (A/C)
[1]
Kunzite
[1]

58. 7. Triclinic Crystals a  b  c      Simple Triclinic Amazonite
[1]
Amazonite
[1]

59. Polar coordinates (, )
Concept of symmetry and choice of axes
(a,b)
The centre of symmetry of the object does not coincide with the origin
Polar coordinates (, )
The type of coordinate system chosen is not
according to the symmetry of the object
Mirror
Our choice of coordinate axis does not alter the symmetry of the object (or the lattice)
Centre of Inversion

60. Alternate choice of unit cells for Orthorhombic lattices
Alternate choice of unit cell for “C”(C-centred orthorhombic) case.
The new (orange) unit cell is a rhombic prism with (a = b  c,  =  = 90o,   90o,   120o)
Both the cells have the same symmetry  (2/m 2/m 2/m)
In some sense this is the true Ortho-”rhombic” cell

61. z = 0 & z = 1
Conventional
Alternate choice
(“ortho-rhombic”) P
C 2ce the size I F
2ce the size
I 1/2 the size C
P 1/2 the size
z = ½
Note: All spheres represent lattice points. They are coloured differently but are the same
 A consistent alternate set of axis can be chosen for Orthorhombic lattices

62. 2d produces this additional point not part of the original lattice
Intuitively one might feel that the orthogonal cell has a higher symmetry  is there some reason for this?
2d produces this additional point not part of the original lattice
This is in addition to our liking for 90!
The 2x and 2y axes move lattice points out the plane of the sheet in a semi-circle to other points of the lattice (without introducing any new points)
The 2d axis introduces new points which are not lattice points of the original lattice
The motion of the lattice points under the effect of the artificially introduced 2-folds is shown as dashed lines (---)

63. Progressive lowering of symmetry amongst the 7 crystal systems
Cubic48
Increasing symmetry
Hexagonal24
Tetragonal16
Trigonal12
Orthorhombic8
Monoclinic4
Triclinic2
Arrow marks lead from supergroups to subgroups
Superscript to the crystal system is the order of the lattice point group

64. Progressive relaxation of the constraints on the lattice parameters amongst the 7 crystal systems
Cubic (p = 2, c = 1, t = 1) a = b = c  =  =  = 90º
Increasing number t
Tetragonal (p = 3, c = 1 , t = 2) a = b  c  =  =  = 90º
Hexagonal (p = 4, c = 2 , t = 2) a = b  c  =  = 90º,  = 120º
Trigonal (p = 2, c = 0 , t = 2) a = b = c  =  =   90º
Orthorhombic1 (p = 4, c = 1 , t = 3) a  b  c  =  =  = 90º
Orthorhombic2 (p = 4, c = 1 , t = 3) a = b  c  =  = 90º,   90º
p = number of independent parameters = (p  e)
c = number of constraints (positive  “=“)
t = terseness = (p  c) (is a measure of the ‘expenditure’ on the parameters
Monoclinic (p = 5, c = 1 , t = 4) a  b  c  =  = 90º,   90º
Triclinic (p = 6, c = 0 , t = 6) a  b  c       90º
Orthorhombic1 and Orthorhombic2 refer to the two types of cells

65. Minimum symmetry requirement for the 7 crystal systems

66. REGULAR SOLIDS IN VARIOUS DIMENSIONSnD
No.
SYMBOL 1
1/ 2 
{p} 3 5
{p, q} 4 6
{p, q, r}
{p, q, r, s}
POINT
LINE SEGMENT
TRIANGLE {3}
SQUARE {4}
PENTAGON {5}
HEXAGON {6}
TETRAHEDRON {3, 3}
OCTAHEDRON {3, 4}
DODECAHEDRON {5, 3}
CUBE {4, 3}
ICOSAHEDRON {3, 5}
SIMPLEX {3, 3, 3}
16-CELL {3, 3, 4}
120-CELL {5, 3, 3}
DRP
24-CELL {3, 4, 3}
HYPERCUBE {4, 3, 3}
600-CELL {3, 3, 5}
CRN
REGULAR SIMPLEX {3, 3, 3, 3}
CROSS POLYTOPE {3, 3, 3, 4}
MEASURE POLYTOPE {4, 3, 3, 3}