CRYSTAL STRUCTURE & X-RAY DIFFRACTION

GETACHEW YIRGA

HARAMAYA UNIVERSITY

Classification of Matter

SolidsSolids are again classified in to two types Crystalline (orderly repeating pattern ) Non-Crystalline (Amorphous)So a crystal is characterized by regular arrangement of atoms or molecules

Why study crystal structures?

When we look around us many of the organic materials are noncrystalline But, many of the common inorganic materials are usually* crystalline: Metals: Cu, Zn, Fe, Cu-Zn alloys Semiconductors: Si, Ge, GaAs Ceramics: Alumina (Al2O3), Zirconia (Zr2O3), SiC, SrTiO3 Also, the usual form of crystalline materials (say a Cu wire or a piece of alumina) is polycrystalline and special care has to be taken to produce single crystals The crystal structure directly influences the properties of the material

Crystalline Solid

Single crystal

Amorphous Solid

Amorphous (Non-crystalline) Solid is composed of randomly orientated atoms , ions, or molecules that do not form defined patterns or lattice structures.

Noncrystalline

Amorphous materials have order only within a few atomic or molecular dimensions.

General Crystal properties? o Crystals have sharp melting points o They have long range positional order o Crystals are anisotropic (Properties change depending on the direction)

Atoms that are bound together, do so in a way that minimizes their energy. This most often leads to a periodic arrangement of the atoms in space. If the arrangement is purely periodic we say that it is crystalline.

Definition 1

Crystal = Lattice + MotifMotif or Basis: typically an atom or a group of atoms associated with each lattice point Lattice the underlying periodicity of the crystal Basis Entity associated with each lattice points Lattice how to repeat Motif what to repeat

LatticeTranslationally periodic arrangement of points

CrystalTranslationally periodic arrangement of motifs

Crystal = Lattice (Where to repeat) + Motif (What to repeat)Cryst al

Definition 1

aLattic e

=Note: all parts of the motif do not

a

+Moti fa 2 Motifs are associated

sit on the lattice point

with lattice points they need NOT sit physically at the lattice point

Space lattice ?A lattice is also called a Space Lattice

Translationally periodic arrangement of points in space is called a lattice Sp.latt. An array of points such that every point has identical surroundings A periodic array of dots (or lattice y points) with infinite repetition. In reality, we have to deal with finite sizes.

B b

C O a A

D x

E

A lattice can be described in terms of unit cell and lattice parameters (constants): (a, b, c) and (, , ). Arrays are arranged exactly in a periodic manner.

Lattice

+

Motif

b

a Continued

Crystal As before there are many ways of associating the motif with a lattice point (one of these is shown)

=

Note: Each motif is identically oriented (orientationally ordered)and is associated exactly at the same position with each lattice point (positionally ordered)

Translational Lattice Vectors 2DA space lattice is a set of points such that a translation from any point in the lattice by a vector; R=la+mb locates an exactly equivalent point, i.e. a point with the same environment as P . This is translational symmetry. The vectors a, b are known as lattice vectors and (l,m) is a pair of integers whose values depend on the lattice point.

For a three dimensional lattice R = la + mb +nc

Crystal structure !!

Crystal structure can be obtained by attaching atoms, groups of atoms or molecules which are called basis (motif) to the lattice sides of the lattice point.

Crystal lattice + basis = Crystal structure

Primitive cell

The unit cell formed by the primitives a,b and c is called primitive cell. A primitive cell will have only one lattice point. If there are two or more lattice points it is not considered as a primitive cell.

As most of the unit cells of various crystal lattice contains two or more lattice points, its not necessary that every unit cell is primitive.

Theunitcelland,consequently,theentirelattice,is

uniquelydeterminedbythesixlatticeconstants:a,b,c, Thesesixparametersarealsocalledasbasiclattice parameters.

Fundamental types of lattices

Crystal lattices can be carried or mapped into themselves by Lattice translations T, Rotation and reflection. SymmetryOperator=anoperationthatleaves thestructureunchanged. Wecanhaveamirrorreflectionaboutaplane throughalatticepoint. Theinversionoperationiscomposedofrotation bypifollowedbyreflectioninaplanenormalto thetherotationaxis

Mirror Symmetry

Inversion

2-D Lattices and Unit CellsIn 2-D we have five distinct lattice

3-D lattices, 7 crystal systems,14 Bravais lattices

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Table-1 Crystal systemCubic Tetragonal Orthorhombic Monoclinic Triclinic Trigonal Hexagonal Unit vector a= b=c a = b c abc abc abc a= b=c a= b c Angles = ==90 = ==90 = ==90 = =90 90 = =90 ==90 =120

Bravais lattices

In 1850, M. A. Bravais showed that identical points can be arranged spatially to produce 14 types of regular pattern. These 14 space lattices are known as Bravais lattices. The Bravais lattice are the distinct lattice types which when repeated can fill the whole space. The lattice can therefore be generated by three unit vectors, a1, a2 and a3 and a set of integers k, l and m so that each lattice point, identified by a vector r, can be obtained from: r = k a1 + l a2 + m a3

1. Name of crystal systemlattice parameters and relationship amongst them (preferred Unit Cell) Possible Bravais lattices

Diagram of preferred UC

Point groups belonging to the crystal system

1. Cubic Crystalsa = b= c = = = 90

Simple Cubic (P) - SC Body Centred Cubic (I) BCC Face Centred Cubic (F) - FCC

SC, BCC, FCC are lattices while HCP & DC are crystals!

Elements with Cubic structure SC: F, O, Po || BCC: Cr, Fe, Nb, K, W, V|| FCC: Al, Ar, Pb, Ni, Pd, Pt, Ge

Examples of elements with Cubic Crystal Structure

Pon=1 SC

Fen=2 BCC

Cun=4 FCC/CCP

n=8

DC

C (diamond)

2. Tetragonal Crystalsa=b c = = = 90 Simple Tetragonal Body Centred Tetragonal -BCT

Elements with Tetragonal structure In, Sn

3. Orthorhombic Crystalsa b c = = = 90 Simple Orthorhombic Body Centred Orthorhombic Face Centred Orthorhombic End Centred OrthorhombicOne convention

a