Shortcourse+on+symmetry+and+ crystallography+ Part3...
Transcript of Shortcourse+on+symmetry+and+ crystallography+ Part3...
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 be all the symmetries of the crystal. Reminder: A symmetry is a combina/on of an orthogonal transforma/on and a transla/on: Defini&on: TranslaKon subgroup (normal subgroup): Point group (in general no subgroup):
{A1, b1}, {A2, b2}, {A3, b3}, . . .
{A, b} : x �→ T (x) = Ax + b
T = {b1, b2, b3, . . .}P = {A1, A2, A3, . . .}
M.C. Escher (1898-‐1972), mathemaKcally inspired Dutch graphic arKst
Point symmetries of Klings In other words: The point symmetries are the orthogonal part of the symmetries that remain a9er the transla/on is disregarded.
The 17 wallpaper groups
No. Space Group Hermann-‐Mauguin (PG) Schönflies (PG) La@ce
1 p1 1 C1 Oblique
2 p2 2 C2 Oblique
3 pm m D1 Rectangular
4 pg m D1 Rectangular
5 cm m D1 Rectangular
6 p2mm 2mm D2 Rectangular
7 p2mg 2mm D2 Rectangular
8 p2gg 2mm D2 Rectangular
9 c2mm 2mm D2 Rectangular
10 p4 4 C4 Square
11 p4mm 4mm D4 Square
12 p4gm 4mm D4 Square
13 p3 3 C3 Hexagonal
14 p3m1 3m D3 Hexagonal
15 p31m 3m D3 Hexagonal
16 p6 6 C6 Hexagonal
17 p6mm 6mm D6 Hexagonal
4 laXce systems 5 Bravais laXces 10 Point groups
Guide to recognizing wallpaper groups
1
2
3
3
3
4
3
3
4
Tetris Klings (Eric J.)
(4) (5) (6)
(1) (2) (3)
Exercise: Determine wallpaper groups
(1) (2)
(5) (6) (7) (8)
(3) (4)
h\p://en.wikipedia.org/wiki/Wallpaper_group
Cell structure of the wallpaper groups NotaKon: 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 reflecKons: (i) Mirror at the line. (ii) Shib along the line.
Oblique, Ci Oblique, C2 Rectangular, D1
Rectangular, D1 Rectangular, D1 Rectangular, D2
Rectangular, D2 Rectangular, D2 Rectangular, D2
Square, C4 Square, D4 Square, D4
Hexagonal, C3 Hexagonal, D3 Hexagonal, D3
Hexagonal, C6 Hexagonal, D6
Annotated example from the ITC (part 1)
Space group (H-‐M short)
Number, follows point groups
Point group (H-‐M)
Space group (H-‐M long) Symmetry of the diffrac/on paFern (includes inversion)
Bravais laIce
Cell structure One low symmetry orbit “,” means inversion Point symmetry at the origin
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
Classifica/on of symmetry orbits These are the lines/points shown in the cell structure
Ex/nc/on condi/ons for
diffrac/on (see later)
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
• InternaKonal 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
IdenKfy the space group of a crystal, given the posiKons of the atoms in a unit cell. Input: (i) LaXce parameters and angles or basis vectors of the laXce (ii) Number and posiKons 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-‐Kn): 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
excepKons (e.g. high pressure, low temperature etc.) • Reports are errors and
correcKons for symmetry, laXce/atomic parameters
File formats
• CIF: Crystallographic InformaKon 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 transiKons and phase diagrams
[Porter, Easterling: Phase Transforma/ons in Metals and Alloys]
Tangent construcKon
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