Basic crystallography, Lecture notes of Physics

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DST-SERC School on Texture and Microstructure

BASIC CRYSTALLOGRAPHY

Rajesh Prasad

Department of Applied Mechanics

Indian Institute of Technology

New Delhi 110016

[email protected]

Contents

Crystal, Lattice and MotifUnit cells, Lattice Parameters and ProjectionsMiller Indices & Miller-Bravais Indices

Directions and Planes

Classification of Lattices:

7 crystal systems14 Bravais lattices

Reciprocal lattice

Lattice?

A 3D translationallyperiodic arrangementof points in space iscalled a lattice.

A 3Dtranslationallyperiodicarrangementof

atoms

Crystal

Lattice A 3Dtranslationallyperiodicarrangement of points

Crystal=lattice+basis

Lattice:

the underlying periodicity ofthe crystal,

Basis:

atom or group of atomsassociated with each lattice points

Lattice:

how to repeat

Motif:

what to repeat

A 3D translationally periodicarrangement of points

Each lattice point in a lattice

has identical neighbourhood of

other lattice points.

Lattice

Air,WaterandEarthby M.C.Esher

Everyperiodicpattern(and hencea crystal)has auniquelatticeassociatedwith it

Contents

Crystal, Lattice and Motif^ Unit cells, Lattice Parameters and Projections

Miller Indices & Miller-Bravais Indices

Directions and Planes

Classification of Lattices:

7 crystal systems14 Bravais lattices

Reciprocal lattice

Translational Periodicity

One can select a small volume ofthe crystal which by periodicrepetition generates the entirecrystal (without overlaps or gaps)

UnitCell

Unit cell description : 1

Size and shape of the unit cell:

1. A corner as origin

2. Three edge vectors {

a

b

c

from the origin definea CRSYTALLOGRAPHIC

COORDINATESYSTEM

3. The three

lengths

a

b

c

and

the three

interaxial angles α

are called the

LATTICE PARAMETERS

β^ α

γ

a

b

c

Unit cell description : 4

7 crystal Systems

Crystal System

Conventional Unit Cell

1. Cubic

a

b

c

α

β

γ

2. Tetragonal

a=b

c,

α

β

γ

3. Orthorhombic

a

b

c

α

β

γ

4. Hexagonal

a

b

c

α

β

γ

5. Rhombohedral

a

b

c

α

β

γ≠

OR Trigonal

6. Monoclinic

a

b

c

α

β

°≠γ

7. Triclinic

a

b

c,

α≠β≠γ

Unit cell description : 5

x

y

Projection/plan view of unit cells ½ ½

0

Example 1: Cubic close-packed (CCP) crystal

e.g. Cu, Ni, Au, Ag, Pt, Pb etc.

1 2

1 2 1 2

1 2

x

y

z

Plan description : 1

The six lattice parameters

a

b

c

The cell of the lattice

lattice crystal

• Motif