Chapter 3

Crystal Geometry and

Structure Determination

Courtesy: H Bhadhesia

Crystals: long range periodicity, AnisotropicAmorphous: Homogeneous, isotropic

A 3D translationaly periodic arrangement of atoms in space is called a crystal.

Crystal ?

2D crystal

Translational Periodicity

One can select a small volume of the crystal which by periodic repetition generates the entire crystal (without overlaps or gaps)

Unit cell description : 1

Space filling

Crystal

Unit Cell

Building block ofcrystal

2D crystal

Unit cell is the imaginary, it doesn't really exist: We use them to understand the crystallography

It should be space filling, no gaps, no overlaps

We tend to choose unit cells with angles closeto 90° and shortest unit cell edge length

The most common shape of a unit cell is a parallelopiped.

Unit cell description : 23D UNIT CELL:

The description of a unit cell requires:

1. Its Size and shape (lattice parameters)

2. Its atomic content

(fractional coordinates)

Unit cell description : 3

Lets just think about size and shapefirst!!

Size and shape of the unit cell:

1. A corner as origin

2. Three edge vectors {a, b, c} from the origin define

a CRSYTALLOGRAPHIC COORDINATE

SYSTEM

3. The three lengths a, b, c and the three interaxial angles , , are called the LATTICE PARAMETERS

a

b

c

Unit cell description : 4

Crystal

Unit cell

Characterize the size and shape of a unit cell

Lattice

Lattice?A 3D translationally periodic arrangement of points in space is called a lattice.

A 3D translationally periodic arrangement of points

Each lattice point in a lattice has identical neighbourhood

of other lattice points.

Lattice

Primitivecell

Primitivecell

Non-primitive cell

A unit cell of a lattice is NOT unique.

If the lattice points are only at the corners, the unit cell is primitive otherwise non-primitive

UNIT CELLS OF A LATTICE

Primitivecell

UNIT CELLS OF A LATTICE

Can we select a triangular unit cell? Since it can give a very smallrepeat unit

Unit cell is a small volume of the crystal which by periodic repetition generates the entire crystal (without overlaps or gaps)

In 2D there are only 5 possible ways of arranging points which are regular in space

A 3D space lattice can be generated by repeatedtranslation of three vectors a, b and c

It turns out there are 14 distinguishable ways of arrangingpoints in 3 dimensional space such that each arrangementconforms to the definition of a space lattice

These 14 space lattices are known as Bravais lattices, named after their originator

Think about 2D crystal which is making big news??Carbon nanotube: Graphene sheet

A layer of C atoms in hexagonal arrangement

Cylindrical crystal

In general we mostly deal with 3 dimensional crystals

Classification of lattice

The Seven Crystal SystemAnd

The Fourteen Bravais Lattices

Crystal System Bravais Lattices

1. Cubic P I F

2. Tetragonal P I

3. Orthorhombic P I F C

4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. Triclinic PP: Primitive; I: body-centred; F: Face-centred; C: End-centred

*The notations comes from Germans

7 Crystal Systems and 14 Bravais Lattices

20/87

Cubic Crystals

a=b=c; ===90

The three cubic Bravais lattices

Crystal system Bravais lattices

1. Cubic P I F

Simple cubicPrimitive cubicCubic P

Body-centred cubicCubic I

Face-centred cubicCubic F

Orthorhombic CEnd-centred orthorhombicBase-centred orthorhombic

Unit cell parameters for different crystal systems

Courtesy: H Bhadhesia

24/87

Trinclinc Crystal

Courtesy: H Bhadhesia

Courtesy: H Bhadhesia

Why half the boxes are empty?

E.g. Why cubic C is absent?

Crystal System Bravais Lattices

1. Cubic P I F

2. Tetragonal P I

3. Orthorhombic P I FC

4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. Triclinic P

?

End-centred cubic not in the Bravais list ?

End-centred cubic = Simple Tetragonal

2

a2

a

14 Bravais lattices divided into seven crystal systems

Crystal system Bravais lattices

1. Cubic P I F C

2. Tetragonal P I

3. Orthorhombic P I F C

4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. Triclinic P

Similarly, answer why face centred tetragonalis not in the list?

Face-centred tetragonal = Body-centred Tetragonal

What is the basis for classification of lattices

into 7 crystal systems

and 14 Bravais lattices?

Lattices are classified on the basis of their

symmetry

Crystal class is defined by certain minimum symmetry (defining

symmetry)

If an object is brought into self-coincidence after some operation it said to possess symmetry with respect to that operation.

Symmetry?

Lattices also have translational symmetry

Translational symmetry

In fact this is the defining symmetry of a lattice

If an object come into self-coincidence through smallest non-zero rotation angle of then it is said to have an n-fold rotation axis where

0360n

=180

=90

Rotation Axis

n=2 2-fold rotation axis

n=4 4-fold rotation axis

Rotational Symmetries

Z180 120 90 72 60

2 3 4 5 6

45

8

Angles:

Fold:

Graphic symbols

Symmetry of lattices

Lattices have

Rotational symmetry

Reflection symmetry

Translational symmetry