Short-course on symmetry and crystallography
Part 3: Wallpaper groups (2D) Space groups (3D)
Michael Engel Ann Arbor, June 2011 Symmetry group of a crystal
Defini on: A space group (3D) or wallpaper group (2D) of a crystal consists of all symmetries that leave the crystal invariant.
Let A1 ,b1 , A2,b2 , A3,b3 ,... be all the symmetries of the { } { } { } crystal.
Reminder: A symmetry is a combina on of an orthogonal transforma on and a transla on: A, b : x (x)=Ax + b { } → T
Defini on: Transla on subgroup (normal subgroup): T = b ,b ,b ,... { 1 2 3 } Point group (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 a er the transla on is disregarded.
M.C. Escher (1898-1972), mathema cally inspired Dutch graphic ar st The 17 No. Space Group Hermann-Mauguin (PG) Schönflies (PG) La ce 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 la ce 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 la ces 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)
h p://en.wikipedia.org/wiki/Wallpaper_group Cell structure of the wallpaper groups Nota on:
Example:
The 17 wallpaper groups can be found at Wikipedia: h p://en.wikipedia.org/wiki/ Wallpaper_group Example: p4m (No. 11)
Note: Dashed lines are glide reflec ons: (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 la ce
Number, follows Space group (H-M long) Symmetry of the diffrac on point groups pa ern (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) transla on Note: overbar means minus Group elements that generate the symmetry group
Ex nc on condi ons for diffrac on (see later)
Classifica on 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 transi ons) h p://en.wikipedia.org/wiki/Space_group Resources
• Interna onal Tables of Crystallography A, pages 112-725. The absolute source! • Hypertext book of Crystallographic Space Group Diagrams and Tables: h p://img.chem.ucl.ac.uk/sgp/mainmenu.htm • Three-dimensional space groups: h p://www.uwgb.edu/DutchS/SYMMETRY/3dSpaceGrps/ 3dspgrp.htm Findsym (by Harold T. Stokes, BYU) h p://stokes.byu.edu/findsym.html
Iden fy the space group of a crystal, given the posi ons of the atoms in a unit cell.
Input: (i) La ce parameters and angles or basis vectors of the la ce (ii) Number and posi ons 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 excep ons (e.g. high pressure, low temperature etc.) • Reports are errors and correc ons for symmetry, la ce/atomic parameters File formats
• CIF: Crystallographic Informa on File Interna onal Union of Crystallography Hall SR, Allen FH, Brown ID (1991). "The Crystallographic Informa on 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 crystal structure database (ICSD)
• Homepage: h p://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: h p://icsd.ornl.gov/index.php Cambride Structural Database (CSD)
• Homepage: h p://www.ccdc.cam.ac.uk/products/csd/
• All crystal structures with do contain C—H bonds
• Ca. 500000 entries.
• Demo/teaching access: h p://webcsd.ccdc.cam.ac.uk/teaching_database_demo.php Metals Crystallographic Data File (CRYST-MET)
• Homepage: h p://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: h p://nanocrystallography.research.pdx.edu/search.py/ search?database=wcd
Side remarks:
Phase transi ons and phase diagrams
[Porter, Easterling: Phase Transforma ons in Metals and Alloys] Tangent construc on
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