Chapter 1Chapter 1

Crystal StructuresCrystal Structures

Two Categories of Solid State MaterialsTwo Categories of Solid State Materials

Crystalline: quartz, diamond…..

Amorphous: glass, polymer…..

Ice Ice crystalscrystals

crylstalscrylstals

Lattice Points, Lattice and Unit CellLattice Points, Lattice and Unit Cell

How to define lattice points, lattice and unit cell?

LATTICELATTICE LATTICE = An infinite array of points in space, in

which each point has identical surroundings to all others.

CRYSTAL STRUCTURE = The periodic arrangement of atoms in the crystal.

It can be described by associating with each lattice point a group of atoms called the MOTIF (BASIS)

Notes for lattice pointsNotes for lattice points

Don't mix up atoms with lattice points Lattice points are infinitesimal points in space Atoms are physical objects Lattice Points do not necessarily lie at the centre of

atoms

An example of 2D latticeAn example of 2D lattice

An example of 3D latticeAn example of 3D lattice

Unit cellUnit cell

• • A repeat unit (or motif) of the regular A repeat unit (or motif) of the regular arrangements of a crystal arrangements of a crystal

•• •• is defined as the smallest repeating unit whichis defined as the smallest repeating unit which shows the full symmetry of the crystal structure shows the full symmetry of the crystal structure

More than one ways More than one ways

How to assign a unit cellHow to assign a unit cell

A cubic unit cellA cubic unit cell

3 cubic unit 3 cubic unit cellscells

Crystal systemCrystal system is governed by unit cell shape and symmetry

The Interconversion of Trigonal Lattices

t1t2

t1t2

γ=120°

兩正三角柱合併體

The seven crystal systemsThe seven crystal systems

SymmetrySymmetry

Space group = point group + translation

Definition of symmetry elementsDefinition of symmetry elements

------------------------------------------------------------- Elements of symmetry

------------------------------------------------

Symbol Description Symmetry operations

---------------------------------------------------------------------E Identity No change

Plane of symmetry Reflection through the plane

i Center of symmetry Inversion through the center

Cn Axis of symmetry Rotation about the axis by (360/n)o

Sn Rotation-reflection Rotation about the axis by (360/n)o

axis of symmetry followed by reflection through the

plane perpendicular to the axis

---------------------------------------------------------------------

Center of symmetry, Center of symmetry, ii

Rotation operation, CRotation operation, Cnn

Plane reflection , Plane reflection ,

Matrix representation of symmetry operatorsMatrix representation of symmetry operators

Symmetry operationSymmetry operation

Symmetry elementsSymmetry elements

space group space group = point group + translation= point group + translation

Symmetry elements

Screw axes 21(//a), 21(//b), 41(//c)

42(//c), 31(//c) etc

Glide planes c-glide (┴ b), n-glide,

d-glide etc

2211 screw axis // b-axis screw axis // b-axis

Glide planeGlide plane

Where are Where are glide planes?glide planes?

Examples for 2D symmetryExamples for 2D symmetry

http://www.clarku.edu/~djoyce/wallpaper/seventeen.html

Examples of 2D symmetryExamples of 2D symmetry

General positions of Group 14 General positions of Group 14 (P 2(P 2

11/c) [unique axis b]/c) [unique axis b]

1 x,y,z identity

2 -x,y+1/2,-z+1/2 Screw axis

3 -x,-y,-z i

4 x,-y+1/2,z+1/2 Glide plane

Multiplicity, Wyckoff Letter, Site SymmetryMultiplicity, Wyckoff Letter, Site Symmetry

4e 1 (x,y,z) (-x, ½ +y,½ -z) (-x,-y,-z) (x,½ -y, ½ +z)

2d ī (½, 0, ½) (½, ½, 0)

2c ī (0, 0, ½) (0, ½, 0)

2b ī (½, 0, 0) (½, ½, ½)

2a ī (0, 0, 0) (0, ½, ½)

General positions of Group 15 General positions of Group 15 (C 2/c) [unique axis b](C 2/c) [unique axis b]

1 x,y,z identity

2 -x,y,-z+1/2 2-fold rotation

3 -x,-y,-z inversion

4 x,-y,z+1/2 c-glide

5 x+1/2,y+1/2,z identity + c-center

6 -x+1/2,y+1/2,-z+1/2 2 + c-center

7 -x+1/2,-y+1/2,-z i + c-center

8 x+1/2,-y+1/2,z+1/2 c-glide + c-center

P21/c in international table AP21/c in international table A

P21/c in international table BP21/c in international table B

CCnn and and

Relation between cubic and tetragonal unit Relation between cubic and tetragonal unit

cellcell

LatticeLattice : : the manner of repetition of atoms, ions or the manner of repetition of atoms, ions or

molecules in a crystal by an array of pointsmolecules in a crystal by an array of points

Types of latticeTypes of lattice

Primitive lattice (P) - the lattice point only at corner

Face centred lattice (F) - contains additional lattice points in the center of each face

Side centred lattice (C) - contains extra lattice points on only one pair of opposite faces

Body centred lattice (I) - contains lattice points at the corner of a cubic unit cell and body

center

Examples of F, C, and I latticesExamples of F, C, and I lattices

14 Possible Bravais lattices 14 Possible Bravais lattices : : combination of four types of lattice and seven crystcombination of four types of lattice and seven cryst

al systemsal systems

How to index crystal planes?How to index crystal planes?

Lattice planes and Miller indicesLattice planes and Miller indices

Lattice planesLattice planes

Miller indicesMiller indices

Assignment of Miller indices to a set of Assignment of Miller indices to a set of planesplanes

1. Identify that plane which is adjacent to the one 1. Identify that plane which is adjacent to the one that passes through the origin.that passes through the origin.

2. Find the intersection of this plane on the three 2. Find the intersection of this plane on the three axes of the cell and write these intersections as axes of the cell and write these intersections as fractions of the cell edges. fractions of the cell edges.

3. Take reciprocals of these fractions.3. Take reciprocals of these fractions.

Example: fig. 10 (b) of previous pageExample: fig. 10 (b) of previous page

cut the x axis at a/2, the y axis at band the z axis at cut the x axis at a/2, the y axis at band the z axis at c/3;c/3; the reciprocals are therefore, 1/2, 1, 1/3; the reciprocals are therefore, 1/2, 1, 1/3; Miller index is ( 2 1 3 ) #Miller index is ( 2 1 3 ) #

Examples of Miller indicesExamples of Miller indices

Miller Index and other indicesMiller Index and other indices

(1 1 1), (2 1 0){1 0 0} : (1 0 0), (0 1 0), (0 0 1) ….

[2 1 0], [-3 2 3]<1 0 0> : [1 0 0], [0 1 0], [0 0 1]

考古題考古題Assign the Miller indices for the crystal faces

Descriptions of crystal structuresDescriptions of crystal structures

The close packing approach

The space-filling polyhedron approach

Materials can be described as close Materials can be described as close packedpacked

Metal- ccp, hcp and bccAlloy- CuAu (ccp), Cu(ccp), Au(ccp)Ionic structures - NaClCovalent network structures (diamond)Molecular structures

Close packed layerClose packed layer

A A NON-CLOSE-PACKEDNON-CLOSE-PACKED structure structure

Close packedClose packed

Two Two cpcp layers layers

P = sphere, O = octahedral hole, T+ / T- = tetrahedral holesP = sphere, O = octahedral hole, T+ / T- = tetrahedral holes

Three close packed layers in Three close packed layers in ccpccp sequence sequence

ccpccp

ABCABCABCABC.... repeat gives .... repeat gives Cubic Close-PackingCubic Close-Packing ( (CCPCCP))

Unit cell showing the full symmetry of the arrangement is Face-Centered Cubic

Cubic: a = b =c, = = = 90° 4 atoms in the unit cell: (0, 0, 0) (0, 1/2,

1/2) (1/2, 0,

1/2) (1/2,

1/2, 0)

hcp hcp

ABABABABABAB.... repeat gives .... repeat gives Hexagonal Close-PackingHexagonal Close-Packing

((HCPHCP))

Unit cell showing the full symmetry of the arrangement is Hexagonal

Hexagonal: a = b, c = 1.63a, = = 90°, = 120° 2 atoms in the unit cell: (0, 0, 0) (2/3,

1/3, 1/2)

Coordination number inCoordination number in hcp hcp and and ccpccp structuresstructures

hcphcp

Face centred cubic unit cell of a Face centred cubic unit cell of a ccpccp arrangement of arrangement of spheresspheres

Hexagonal unit cell of a Hexagonal unit cell of a hcphcp arrangement of arrangement of spheresspheres

Unit cell dimensions for a face centred unit Unit cell dimensions for a face centred unit cellcell

Density of metalDensity of metal

Tetrahedral sitesTetrahedral sites

Covalent network Covalent network structures of structures of

silicatessilicates

CC6060 and and

AlAl22BrBr66

The space-filling approachThe space-filling approachCorners and edges sharingCorners and edges sharing

Example of edge-sharingExample of edge-sharing

Example of edge-sharingExample of edge-sharing

Example of corner-sharingExample of corner-sharing

Corner-Corner-sharing of sharing of silicatessilicates