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- 208016 Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh AN INTRODUCTORY E-BOOK Part of http://home.iitk.ac.in/~anandh/E-book.htm A Learner’s Guide Space Group diagrams and tables http://img.chem.ucl.ac.uk/sgp/mainmenu.htm Magnetic Space groups http://mpg.web.cmu.edu/

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

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)

Point Group Symmetry Present 2D 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

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

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

Getting the 32 point groups

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

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

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