Lecture 5

Crystal Geometry

and

Structure Determination

Recap Bravais lattices: characteristic symmetry of each crystal

system

Mystery of missing entries in the Bravais list

Todays lecture

Making crystals

Identification of crystal planes and directions

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

FCC, BCC crystals can be obtained by decorating FCC/BCClattices with motif.

The structures obtained may be simpler or complicateddepending on what the motif is??

Crystal structure description requires the specificationof following things-

1. Number of atoms and their kind

2. Internuclear spacing

3. Orientation in space

Note in our discussion we will assume atoms to be hardsphere though this may not be very true

Position of motif can be specified by coordinates

Mn occurs in number of allotropic forms

Alpha Mn at RT, BCC space lattice, 29 atoms in motif

Beta Mn, SC space lattice, 20 atoms motif

Polymeric materials can have thousands of atoms as motif

Some facts

Lets take some examples of simple systems.

Some real important crystals

8Create the crystal structure of brass

Cubic P

Each of these points are lattice points

1/2 1/2

1/21/2

Crystal Structure

MotifCoordinates of Cu and Zn atomsStructure of brassCourtesy: H Bhadhesia

10

Courtesy: H Bhadhesia

lattice + motif = structure

primitive cubic lattice

motif = Cu at 0,0,0

Zn at 1/2, 1/2, 1/2

Courtesy: H Bhadhesia

+Face Centred Cubic (FCC) Lattice Two Ion Motif

=

NaCl Crystal

Na+ Ion at (, 0, 0)Cl Ion at (0, 0, 0)

This was one of the first structure that was analysed by Braggs in X-ray diffraction

13

Create some complicated crystal structure:

Structure of diamond

Face-centred cubicCubic F

1/4

1/4

1/4

1/4

1/43/4

3/4

3/4

3/4

Lattice: face-centred cubic

Motif: C at 0,0,0 C at 1/4,1/4,1/4 Courtesy: H Bhadhesia

Two Carbon atom Motif(0,0,0) & (, , )

+Face Centred Cubic (FCC) Lattice Two Carbon atom Motif(0,0,0) & (, , )

=

Diamond Cubic Crystal

Tetrahedral bonding of C

(sp3 hybridized)

Four fold rotation axis of cube is lost, but 3 fold of cube remains intact. Completely covalently bonded crystal

Structure of Diamond

All the C atoms are tetrahedrally bonded by covalent bond

Courtesy: H Bhadhesia

How many C atoms per unit cell??

You know about total no. of lattice points in cubic F

4

How many C atoms you are putting per lattice point?

2

So total no. of C atoms per unit cell would be

8

1/4

1/43/4

3/4

Structure of ZnS

Courtesy: H Bhadhesia

1/4

1/43/4

3/4

Lattice: face-centred cubic

Motif: Zn at 0,0,0 S at 1/4,1/4,1/4

Structure of ZnS

Courtesy: H Bhadhesia

Courtesy: H Bhadhesia

Miller Indices of directions and planes

William Hallowes Miller(1801 1880)

University of Cambridge

In crystallography planes and directions are representedby Miller indices

Miller indices of direction is very easy to understand compared to that of plane

Miller Indices for Direction: Vectorial direction

This concept can be equally used for both lattice and crystal

Miller indices can be used to specify directions and planes

Concept of family of directions and planes

Some facts