Short-course on and

Part 3: groups (2D) Space groups (3D)

Michael Engel Ann Arbor, June 2011 Symmetry of a

Definion: A (3D) or (2D) of a crystal consists of all that leave the crystal .

Let A1 ,b1 , A2,b2 , A3,b3 ,... be all the symmetries of the { } { } { } crystal.

Reminder: A symmetry is a combinaon of an orthogonal transformaon and a translaon: A, b : x (x)=Ax + b { } ￿→ T

Definion: Translaon subgroup (normal subgroup): T = b ,b ,b ,... { 1 2 3 } (in general no subgroup): P = A ,A ,A ,... { 1 2 3 } Point symmetries of lings

In other words: The point symmetries are the orthogonal part of the symmetries that remain aer the translaon is disregarded.

M.C. Escher (1898-1972), mathemacally inspired Dutch graphic arst The 17 No. Space Group Hermann-Mauguin (PG) Schönflies (PG) Lace 1 p1 1 C1 Oblique wallpaper 2 p2 2 C2 Oblique groups 3 pm m D1 Rectangular 4 pg m D1 Rectangular

5 cm m D1 Rectangular 6 p2mm 2mm D Rectangular 4 lace 2 7 p2mg 2mm D Rectangular systems 2 8 p2gg 2mm D Rectangular 2 9 c2mm 2mm D Rectangular 5 Bravais 2 10 p4 4 C Square laces 4 11 p4mm 4mm D Square 4 12 p4gm 4mm D Square 10 Point 4 13 p3 3 C Hexagonal groups 3 14 p3m1 3m D3 Hexagonal

15 p31m 3m D3 Hexagonal

16 p6 6 C6 Hexagonal

17 p6mm 6mm D6 Hexagonal Guide to recognizing wallpaper groups

2 1

3

3

3 3

4

3 4 Tetris lings (Eric J.)

(1) (2) (3)

(4) (5) (6) Exercise: Determine wallpaper groups

(1) (2) (3) (4)

(5) (6) (7) (8)

hp://en.wikipedia.org/wiki/Wallpaper_group Cell structure of the wallpaper groups Notaon:

Example:

The 17 wallpaper groups can be found at Wikipedia: hp://en.wikipedia.org/wiki/ Wallpaper_group Example: p4m (No. 11)

Note: Dashed lines are glide reflecons: (i) Mirror at the line. (ii) Shi along the line. Oblique, Ci Oblique, C2 Rectangular, D1

Rectangular, D Rectangular, D 1 1 Rectangular, D2

Rectangular, D2 Rectangular, D2 Rectangular, D2 Square, C4 Square, D4 Square, D4

Hexagonal, C 3 Hexagonal, D3 Hexagonal, D3

Hexagonal, C6 Hexagonal, D6

Annotated example from the ITC (part 1)

Space group (H-M short) Point group (H-M) Bravais lace

Number, follows Space group (H-M long) Symmetry of the diffracon point groups paern (includes inversion)

Cell structure One low symmetry orbit Point symmetry at the origin “,” means inversion

Fundamental domain of points that are (i) non-equivalent under symmetry and (ii) are mapped by symmetry to fill all space. Annotated example from the ITC (part 2)

All symmetry operators (i) Orthogonal part (ii) translaon Note: overbar means minus Group elements that generate the

Exncon condions for diffracon (see later)

Classificaon of symmetry orbits These are the lines/points shown in the cell structure Annotated example from the ITC (part 3)

Maximal subgroups and supergroups allow to study symmetry breaking (second order phase transions) hp://en.wikipedia.org/wiki/Space_group Resources

• Internaonal Tables of Crystallography A, pages 112-725. The absolute source! • Hypertext book of Crystallographic Space Group Diagrams and Tables: hp://img.chem.ucl.ac.uk/sgp/mainmenu.htm • Three-dimensional space groups: hp://www.uwgb.edu/DutchS/SYMMETRY/3dSpaceGrps/ 3dspgrp.htm Findsym (by Harold T. Stokes, BYU) hp://stokes.byu.edu/findsym.html

Idenfy the space group of a crystal, given the posions of the atoms in a .

Input: (i) Lace parameters and angles or basis vectors of the lace (ii) Number and posions of the atoms (iii) Tolerance Examples

• Michael’s phase (chiral: P4132 (No. 213)

• Kevin’s phase (α-O2): C2/m (No. 12)

• Ryan’s phi35 phase (β-n): I41/amd (No. 141)

• Ryan’s phi40 phase (columnar): cmm (No. 9 (2D))

• Ryan’s phi40_2 phase (double gyroid, BC8): Ia-3 (No. 206) Crystal structures

• As of 2008 ca. 700,000 crystal structures have been published • Ca. 50,000 crystal structures are currently discovered every year • Nevertheless, most “elemental” ones are known with some excepons (e.g. high pressure, low temperature etc.) • Reports are errors and correcons for symmetry, lace/atomic parameters File formats

• CIF: Crystallographic Informaon File Internaonal Union of Crystallography Hall SR, Allen FH, Brown ID (1991). "The Crystallographic Informaon File (CIF): a new standard archive file for crystallography”. Acta Crystallographica A47 (6): 655–685.

• PDB: Protein Data Bank format Biology/Biochemistry Brown ID, McMahon B (2002). "CIF: the computer language of crystallography”. Acta Crystallographica B 58 (Pt 3 Pt 1): 317–24. Inorganic database (ICSD)

• Homepage: hp://www.fiz-karlsruhe.de/icsd.html

• (Originally) All Structures that have no C—H bonds and are not metals or alloys.

• Ca. 100000 entries.

• Free (old) access: hp://icsd.ornl.gov/index.php Cambride Structural Database (CSD)

• Homepage: hp://www.ccdc.cam.ac.uk/products/csd/

• All crystal structures with do contain C—H bonds

• Ca. 500000 entries.

• Demo/teaching access: hp://webcsd.ccdc.cam.ac.uk/teaching_database_demo.php Metals Crystallographic Data File (CRYST-MET)

• Homepage: hp://www.tothcanada.com/

• Metals, alloys, and also semiconductors

• > 50000 entries

• Last update 2005 (?) Open databases

• Crystallography Open Database: www.crystallography.net

• Wiki Crystallography Database Search: hp://nanocrystallography.research.pdx.edu/search.py/ search?database=wcd

Side remarks:

Phase transions and phase diagrams

[Porter, Easterling: Phase Transformaons in Metals and Alloys] Tangent construcon

Equilibrium phases can be characterinzed by a mixture of phase α and phase β. Simple phase diagram (completely miscible) A system with a miscibility gap at low temperatures Complex phase diagram