Next Step for Bravais Lattice
Recall:
Mirror (reflection): Rotation:
3
6
Class16/1
Recall:
Examples of 3-fold rotations
in cubic crystal systems
Rocksalt
(e.g. MgO)
Crystal
Structure
Zincblende
(Sphalerite)
(e.g. GaP)
Crystal
Structure
Spinel
(e.g. MgAl2O4)
Crystal
Structure
Fluorite
(e.g. c-ZrO2)
Crystal
Structure
Class16/2
{111}’s
shown in
red
From
Crystal
Maker
Definition of Point and Space Groups
Class16/3
-- There are 10 of them in 2-D
-- Describes symmetry of properties such as thermal
expansion, elastic modulus, refractive index, conductivity,...
Point group symmetry plus
translational microscopic
symmetry (screw axes and
glide planes)
→ Movements that cause no change (i.e., remain invariant).
Symmetry Operations
•Symmetry operations
are essentially operations
on a 3-D shape, crystal,
lattice or whatever, which
leave it unchanged in
appearance.
Class16/4
Point Groups generated by Rotation
•Rotation axes of 5, 7, etc. can not exist in a repeating structure such as a crystal.
•Symbols for rotation are closed shapes indicating the fold of the rotation:
[100] stereographic
projection of cubic crystal
Used to visualize
crystallographic features:
--slip planes and directions
--crystal planes in electron
microscopy
--grain texture/orientation
relationships in x-ray diffraction
(XRD) and electron backscatter
diffraction (EBSD)→pole figuresClass16/5
•The large circles are called stereograms.
Stereographic projections are used to
represent 3-D info (+next slide):
Recall cubic rotation elements
Intro to Stereograms &
Stereographic ProjectionsRecall:
Class16/6
[001] stereographic
projection of cubic crystal
7
The 10 2-D Point Groups based on
rotation axes and mirror planes
One 1-fold
axis
(oblique)
One 2-fold
axis
(oblique)
One mirror
plane
(rectangular)
One 3-fold axis
& 1 mirror
plane (the
triad generates
the other 2)
(hexagonal)
Operation
(2-D system)
Point
Group
Point
GroupExamples ExamplesSymbolOperation
(2-D system)Symbol
One 3-fold
axis
(hexagonal)
One 2-fold
axis & two
mirror planes
(rectangular)
One 4-fold
axis
(square)
One 4-fold axis
& 2 mirror
planes at 45º to
each other (the
tetrad
generates the
other 2)
(square)
One 6-fold
axis
(hexagonal)
One 6-fold
axis, 2 mirror
planes at 30º to
each other
(other 4 by
hexad)
(hexagonal)
→4 in 2-D