Chapter 3 Crystal Geometry and Structure Determination

Crystals: long range periodicity, Anisotropic Amorphous: Homogeneous, isotropic Courtesy: H Bhadhesia

Crystal ? A 3D translationaly periodic arrangement of atoms in space is called a crystal. 2D crystal

Translational Periodicity Unit cell description : 1 Translational Periodicity Crystal One can select a small volume of the crystal which by periodic repetition generates the entire crystal (without overlaps or gaps) Unit Cell Space filling Building block of crystal

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 close to 90° and shortest unit cell edge length

The most common shape of a unit cell is a parallelopiped. 3D UNIT CELL: Unit cell description : 2 The most common shape of a unit cell is a parallelopiped.

The description of a unit cell requires: Unit cell description : 3 The description of a unit cell requires: 1. Its Size and shape (lattice parameters) 2. Its atomic content (fractional coordinates)

Lets just think about size and shape first!!

Size and shape of the unit cell: Unit cell description : 4 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 a b c    3. The three lengths a, b, c and the three interaxial angles , ,  are called the LATTICE PARAMETERS

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.

Lattice A 3D translationally periodic arrangement of points Each lattice point in a lattice has identical neighbourhood of other lattice points.

A unit cell of a lattice is NOT unique. UNIT CELLS OF A LATTICE If the lattice points are only at the corners, the unit cell is primitive otherwise non-primitive Non-primitive cell Primitive cell A unit cell of a lattice is NOT unique. Primitive cell

Can we select a triangular unit cell? Since it can give a very small UNIT CELLS OF A LATTICE Can we select a triangular unit cell? Since it can give a very small repeat unit Unit cell is a small volume of the crystal which by periodic repetition generates the entire crystal (without overlaps or gaps) Primitive cell

In 2D there are only 5 possible ways of arranging points which are regular in space A 3D space lattice can be generated by repeated translation of three vectors a, b and c It turns out there are 14 distinguishable ways of arranging points in 3 dimensional space such that each arrangement conforms 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 System And The Fourteen Bravais Lattices

7 Crystal Systems and 14 Bravais Lattices Crystal System Bravais Lattices Cubic P I F Tetragonal P I Orthorhombic P I F C Hexagonal P Trigonal P Monoclinic P C Triclinic P P: Primitive; I: body-centred; F: Face-centred; C: End-centred *The notations comes from Germans

Cubic Crystals a=b=c; ===90

The three cubic Bravais lattices Crystal system Bravais lattices Cubic P I F Simple cubic Primitive cubic Cubic P Body-centred cubic Cubic I Face-centred cubic Cubic F

Orthorhombic C End-centred orthorhombic Base-centred orthorhombic

Unit cell parameters for different crystal systems Courtesy: H Bhadhesia

Trinclinc Crystal

Courtesy: H Bhadhesia

Courtesy: H Bhadhesia

Why half the boxes are empty? Crystal System Bravais Lattices Cubic P I F Tetragonal P I Orthorhombic P I F C Hexagonal P Trigonal P Monoclinic P C Triclinic P ? E.g. Why cubic C is absent?

End-centred cubic not in the Bravais list ? End-centred cubic = Simple Tetragonal

14 Bravais lattices divided into seven crystal systems Crystal system Bravais lattices Cubic P I F C Tetragonal P I Orthorhombic P I F C Hexagonal P Trigonal P Monoclinic P C Triclinic P

Similarly, answer why face centred tetragonal is 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)

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

Translational symmetry Lattices also have translational symmetry In fact this is the defining symmetry of a lattice

Rotation Axis 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 =180 n=2 2-fold rotation axis =90 n=4 4-fold rotation axis

Rotational Symmetries Z Angles: 180 120 90 72 60 45 Fold: 2 3 4 5 6 8 Graphic symbols

Symmetry of lattices Lattices have Translational symmetry Rotational symmetry Reflection symmetry