1. Anandh Subramaniam & Kantesh Balani
POINT GROUPS &
SPACE GROUPS
A Detailed Exploration
MATERIALS SCIENCE
&
ENGINEERING
Anandh Subramaniam & Kantesh Balani
Materials Science and Engineering (MSE)
Indian Institute of Technology, Kanpur
URL: home.iitk.ac.in/~anandh
AN INTRODUCTORY E-BOOK
Part of
A Learner’s Guide
Space Group diagrams and tables
Magnetic Space groups

2. Click here to read about basics of symmetry
Point Groups and Space Groups: a detailed look
Click here to read about basics of symmetry
We have already considered an overview of point groups and space groups.
Here we have a more detailed look at various related aspects.
The number of lattices, crystal systems, point groups and space groups in 1D, 2D & 3D is listed in the table below. We will get to the details soon.
Number of 1D 2D 3D
Lattices 1 5 14
Crystal Systems 2 4 7
Point Groups 10 32
Magnetic 90
Space Groups 17
230
1651

3. 1D m1 m2 There is only one type of lattice (the simple lattice).
There are only two point groups in 1D 1, m
This ‘m’ can be thought of as ‘m’ or a 2-fold or a ‘i’  as in 1D all these are equivalent.
There are two crystal systems having 1, m symmetry.
Two mirror points (extended for better visibility- planes become points in 1D !!) m m1 m2
1 (with only t)

4. Point Group Symmetry Present2D
There are 5 plane lattices  parallelogram, simple rectangle, centred rectangle, square, 120 rhombus.
4 crystal systems  parallelogram, rectangle, (120) rhombus, square.
10 point groups  1, 2, m, 2mm, 4, 4mm, 3, 3m, 6, 6mm.
17 space groups (next slide).
Point Group Symmetry Present
Lattice Type
Unit Cell Shape
Single
Combinations
1, 2
Parallelogram m
2mm
Rectangle
Centred Rectangle 4
4mm
Square
3, 6
3m, 6mm
120 Rhombus

5. Highest Point Group Symmetry
System
(& Unit Cell Shape)
Lattice
Highest Point Group Symmetry
Space Group symbols
Space group number
Full
Short
Parallelogram
Primitive (p) 1 p1
" 2
p211 p2
Rectangle m
p1m1 pm 3
p1g1 pg 4
Centred (c)
c1m1 cm 5
2mm
p2mm
pmm 6
p2mg
pmg 7
p2gg
pgg 8
c2mm
cmm 9
Square p4 10
4mm
p4mm
p4m 11
p4gm
p4g 12
120 Rhombus p3 13 3m
p3m1 14
p31m 15 p6 16
6mm
p6mm
p6m 17

6. Highest symmetry class is in blue
The 32 Point Groups
Highest symmetry class is in blue
The possible combinations of crystallographic symmetry operators

7. Getting the 32 point groups

8. Order of the point group of the lattice*
Crystal system
Number of point groups
Order of the point group of the lattice*
Sum of the orders of various point groups
Cubic 5 48
132
Hexagonal 7 24 84
Tetragonal 16 56
Trigonal 12 33
Orthorhombic 3 8
Monoclinic 4
Triclinic 2
Total 32
332
* The order for the highest symmetry point group for each crystal system is given. E.g. for cubic (4/m 3 2/m) point group has a order 48 → if we start with a general point then a total of 48 points is obtained

9. Laue groups
A centrosymmetric property imparts a (pseudo) centre of symmetry to a crystal . The crystal will seem to have a centre of symmetry with respect to that property even if is actually absent + = 2 i
2/m

10. Asymmetric unit for various point groups
Let us consider the aysmmetric unit for each of the point groups using sterographic projections.