Crystal Structures & Crystal GeometryCrystal Structures & Crystal Geometry

The Platonic SolidsThe Platonic Solids

http://home.teleport.com/~tpgettys/platonic.shtmlTetrahedronOctahedron

Icosahedron

20 faces and 12 verticesCube

Dodecahedron

Christian Huygens, 1690

Snow flakesSnow flakes

Bentley captured more than 5000 snowflakes during his lifetime, not

finding any two alike.

http://www.its.caltech.edu/~atomic/snowcrystals/http://www.its.caltech.edu/~atomic/snowcrystals/

Caltech work

What I expect you to know for this courseWhat I expect you to know for this course

Space LatticeSpace Lattice

Atomic arrangements in crystalline solids can be described with respect to a network of lines in three dimensions.

The intersections of the lines are called “lattice sites” (or lattice points). Each lattice site has the same environment in the same direction.

A particular arrangement of atoms in a crystal structure can be described by specifying the atom positions in a

repeating “unit cell”.

Lattice: regular patternLattice: regular pattern

BravaisBravais

14 BravaisLattices14 BravaisLattices

Why not pentagons???Why not pentagons???

Principal Metal Crystal StructuresPrincipal Metal Crystal Structures

There are three principle crystal structures for metals:–(a) Body-centered cubic (BCC)–(b) Face-centered cubic (FCC) –(c) Hexagonal close-packed (HCP)

Reallyimportant for

Thiscourse!!

Body-centered cubic (BCC)

BCCBCC

Face-centered cubic (FCC)

Geometry of the BCC StructureGeometry of the BCC Structure

FCCFCC

Geometry of the FCC StructureGeometry of the FCC Structure

Hexagonal close-packed (HCP)Hexagonal close-packed (HCP)

HCPHCP

Geometry of the HCP StructureGeometry of the HCP Structure

RelationshipsRelationships

Simple cube

a = 2 r

BCC FCC

Directions in a crystalDirections in a crystal

Directions in a crystalDirections in a crystal

Negative: use overbar

ExampleExample

FractionFraction

For cubic systemFor cubic system

<100> edge direction<110> face diagonal<111> body diagonal

Directions FCCDirections FCC

Directions BCCDirections BCC

Interstial spacesInterstial spaces

It depends on the r/R ratios, leading to different coordination numbers

4

Interstial spacesInterstial spaces

6

Interstial spaces -8Interstial spaces -8

Interstial spaces -12Interstial spaces -12

ExamplesExamples

NaCl (FCC, octahedral bonding)

Examples – SiO2Examples – SiO2

Some Materials have Different Crystal Structures at Different TemperaturesSome Materials have Different Crystal Structures at Different Temperatures

Many elements and compounds exist in more than one crystalline form under different conditions of temperature and pressure.

This phenomenon is called polymorphism or allotropy.

Last Century workLast Century work

The Nobel Prize in Physics 1914"for his discovery of the diffraction of

X-rays by crystals"

PaulingPauling

The Nobel Prize in Chemistry 1954"for his research into the nature of

the chemicalbond and its application to the

elucidation of thestructure of complex substances"

Penrose tilingPenrose tiling

The C60 or 'Buckball' belongs to a verysmall set of known molecules withicosahedral symmetry

The C60 or 'Buckball' belongs to a verysmall set of known molecules withicosahedral symmetry

Curl, Kroto, Smalleyreceived the Nobel

prize in Chemistry 1996.