Chapter 1 Crystal Structures

Two Categories of Solid State Materials Crystalline: quartz, diamond….. Amorphous: glass, polymer…..

Ice crystals

crylstals

Lattice Points, Lattice and Unit Cell How to define lattice points, lattice and unit cell?

LATTICE 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 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 lattice

An example of 3D lattice

Unit cell A repeat unit (or motif) of the regular arrangements of a crystal is defined as the smallest repeating unit which shows the full symmetry of the crystal structure Unit cell A repeat unit (or motif) of the regular arrangements of a crystal is defined as the smallest repeating unit which shows the full symmetry of the crystal structure

More than one ways

How to assign a unit cell

A cubic unit cell

3 cubic unit cells

Crystal system Crystal system is governed by unit cell shape and symmetry

The Interconversion of Trigonal Lattices t1t1 t2t2 t1t1 t2t2 γ=120° 兩正三角 柱合併體

The seven crystal systems

Symmetry Space group Space group = point group + translation point group Space group point group

Definition 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 C n Axis of symmetry Rotation about the axis by (360/n) o S n 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, i

Rotation operation, C n

Plane reflection, 

Matrix representation of symmetry operators

Symmetry operation

Symmetry elements

space group = point group + translation Symmetry elements Screw axes2 1 (//a), 2 1 (//b), 4 1 (//c) 4 2 (//c), 3 1 (//c) etc Glide planes c-glide ( ┴ b), n-glide, d-glide etc

2 1 screw axis // b-axis

Glide plane

Where are glide planes?

Examples for 2D symmetry

Examples of 2D symmetry

General positions of Group 14 (P 2 1 /c) [unique axis b] 1x,y,z identity 2-x,y+1/2,-z+1/2Screw axis 3-x,-y,-zi 4x,-y+1/2,z+1/2Glide plane

Multiplicity, Wyckoff Letter, Site Symmetry 4e1(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 (C 2/c) [unique axis b] 1x,y,zidentity 2-x,y,-z+1/22-fold rotation 3-x,-y,-zinversion 4x,-y,z+1/2c-glide 5x+1/2,y+1/2,zidentity + c-center 6-x+1/2,y+1/2,-z+1/22 + c-center 7-x+1/2,-y+1/2,-zi + c-center 8x+1/2,-y+1/2,z+1/2c-glide + c-center

P21/c in international table A

P21/c in international table B

C n and 

Relation between cubic and tetragonal unit cell

Lattice : the manner of repetition of atoms, ions or molecules in a crystal by an array of points

Types 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 lattices

14 Possible Bravais lattices : combination of four types of lattice and seven crystal systems

How to index crystal planes?

Lattice planes and Miller indices

Lattice planes

Miller indices

Assignment of Miller indices to a set of planes 1. Identify that plane which is adjacent to the one that passes through the origin. 2. Find the intersection of this plane on the three axes of the cell and write these intersections as fractions of the cell edges. 3. Take reciprocals of these fractions. Example: fig. 10 (b) of previous page cut the x axis at a/2, the y axis at band the z axis at c/3; the reciprocals are therefore, 1/2, 1, 1/3; Miller index is ( ) #

Examples of Miller indices

Miller 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], [0 1 0], [0 0 1]

考古題 Assign the Miller indices for the crystal faces

Descriptions of crystal structures The close packing approach The space-filling polyhedron approach

Materials can be described as close packed Metal- ccp, hcp and bcc Alloy- CuAu (ccp), Cu(ccp), Au(ccp) Ionic structures - NaCl Covalent network structures (diamond) Molecular structures

Close packed layer

A NON-CLOSE-PACKED structure

Close packed

Two cp layers

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

Three close packed layers in ccp sequence

ccp

ABCABC.... repeat gives Cubic Close-Packing (CCP) 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

ABABAB.... repeat gives Hexagonal Close-Packing (HCP) 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 in hcp and ccp structures

hcp

Face centred cubic unit cell of a ccp arrangement of spheres

Hexagonal unit cell of a hcp arrangement of spheres

Unit cell dimensions for a face centred unit cell

Density of metal

Tetrahedral sites

Covalent network structures of silicates

C 60 and Al 2 Br 6

The space-filling approach Corners and edges sharing

Example of edge-sharing

Example of corner-sharing

Corner- sharing of silicates