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