CYL729: Materials Characterization

CYL729: Materials CharacterizationCYL729: Materials Characterization

DiffractionDiffraction

MicroscopyMicroscopy

Thermal AnalysisThermal Analysis

A. RamananA. RamananDepartment of ChemistryDepartment of Chemistry

[email protected]@chemistry.iitd.ac.in

Reference Books

George M. Crankovic (Editor)

Electro-magnetic Spectrum

History of X-raysHistory of X-rays

1885-1895 Wm. Crookes sought unsuccessfully the cause of 1885-1895 Wm. Crookes sought unsuccessfully the cause of repeated fogging of photographic plates stored near his cathode repeated fogging of photographic plates stored near his cathode ray tubes.ray tubes.

X-rays discovered in 1895 by Roentgen, using ~40 keV electrons X-rays discovered in 1895 by Roentgen, using ~40 keV electrons (1st Nobel Prize in Physics 1901)(1st Nobel Prize in Physics 1901)

1909 Barkla and Sadler discovered characteristic X-rays, in 1909 Barkla and Sadler discovered characteristic X-rays, in studying fluorescence spectra (though Barkla incorrectly studying fluorescence spectra (though Barkla incorrectly understood origin) (Barkla got 1917 Nobel Prize) understood origin) (Barkla got 1917 Nobel Prize)

1909 Kaye excited pure element spectra by electron 1909 Kaye excited pure element spectra by electron bombardmentbombardment

History of X-rays - cont’dHistory of X-rays - cont’d

1912 von Laue, Friedrich and Knipping observe X-ray diffraction 1912 von Laue, Friedrich and Knipping observe X-ray diffraction (Nobel Prize to von Laue in 1914) (Nobel Prize to von Laue in 1914)

1912-13 Beatty demonstrated that electrons directly produced 1912-13 Beatty demonstrated that electrons directly produced two radiations: (a) independent radiation, Bremsstrahlung, and (b) two radiations: (a) independent radiation, Bremsstrahlung, and (b) characteristic radiation only when the electrons had high enough characteristic radiation only when the electrons had high enough energy to ionize inner electron shells.energy to ionize inner electron shells.

1913 WH + WL Bragg build X-ray spectrometer, using NaCl to 1913 WH + WL Bragg build X-ray spectrometer, using NaCl to resolve Pt X-rays. Braggs’ Law. (Nobel Prize 1915)resolve Pt X-rays. Braggs’ Law. (Nobel Prize 1915)

n = 2d sin

History of X-rays - cont’dHistory of X-rays - cont’d

1913 Moseley constructed an x-1913 Moseley constructed an x-ray spectrometer covering Zn to ray spectrometer covering Zn to Ca (later to Al), using an x-ray Ca (later to Al), using an x-ray tube with changeable targets, a tube with changeable targets, a potassium ferrocyanide crystal, potassium ferrocyanide crystal, slits and photographic platesslits and photographic plates

1914, figure at right is the first 1914, figure at right is the first electron probe analysis of a man-electron probe analysis of a man-made alloymade alloy

T. Mulvey Fig 1.5 (in Scott & Love, 1983). Note impurity lines in Co and Ni spectra

History of X-rays - cont’dHistory of X-rays - cont’dMoseley found that wavelength of characteristic X-Moseley found that wavelength of characteristic X-rays varied systematically (inversely) with atomic rays varied systematically (inversely) with atomic number number

Z

• Using wavelengths, Moseley developed the concept of atomic number and how elements were arranged in the periodic table.

• The next year, he was killed in Turkey in WWI. “In view of what he might still have accomplished (he was only 27 when he died), his death might well have been the most costly single death of the war to mankind generally,” says Isaac Asimov (Biographical Encyclopedia of Science &Technology).

1923 Manne Siegbahn published 1923 Manne Siegbahn published The Spectroscopy of X-raysThe Spectroscopy of X-rays in in which he shows that the Bragg equation must be revised to take which he shows that the Bragg equation must be revised to take refraction into account, and he lays out the “Siegbahn notation” refraction into account, and he lays out the “Siegbahn notation” for X-raysfor X-rays1931 Johann developed bent crystal spectrometer (higher 1931 Johann developed bent crystal spectrometer (higher efficiency)efficiency)

Historical Summary of X-raysHistorical Summary of X-rays1859 Kirchhoff and Bunsen showed patterns of lines given off by 1859 Kirchhoff and Bunsen showed patterns of lines given off by incandescent solid or liquid are characteristic of that substanceincandescent solid or liquid are characteristic of that substance 1904 Barkla showed each element could emit ≥1 characteristic 1904 Barkla showed each element could emit ≥1 characteristic groups (K,L,M) of X-rays when a specimen was bombarded with groups (K,L,M) of X-rays when a specimen was bombarded with beam of x-raysbeam of x-rays 1909 Kaye showed same happened with bombardment of cathode 1909 Kaye showed same happened with bombardment of cathode rays (electrons)rays (electrons)1913 Moseley found systematic variation of wavelength of 1913 Moseley found systematic variation of wavelength of characteristic X-rays of different elementscharacteristic X-rays of different elements1922 Mineral analysis using X-ray spectra (Hadding)1922 Mineral analysis using X-ray spectra (Hadding)

1923 Hf discovered by von Hevesy (gap in Moseley plot at1923 Hf discovered by von Hevesy (gap in Moseley plot at Z=72). Z=72). Proposed XRF (secondary X-ray fluorescence)Proposed XRF (secondary X-ray fluorescence)

Summary of X-ray PropertiesSummary of X-ray PropertiesX-rays are considered both particles and waves, X-rays are considered both particles and waves, i.e., consisting of small packets of electromagnetic i.e., consisting of small packets of electromagnetic waves, or photons.waves, or photons.

X-rays produced by accelerating HV electrons in X-rays produced by accelerating HV electrons in a vacuum and colliding them with a target.a vacuum and colliding them with a target.

The resulting spectrum contains (1) continuous The resulting spectrum contains (1) continuous background (Bremsstrahlung;“white X-rays”), (2) background (Bremsstrahlung;“white X-rays”), (2) occurrence of sharp lines (characteristic X-rays), occurrence of sharp lines (characteristic X-rays), and (3) a cutoff of continuum at a short and (3) a cutoff of continuum at a short wavelength.wavelength.

X-rays have no mass, no charge (vs. electrons)X-rays have no mass, no charge (vs. electrons)

X-ray CrystallographyX-ray Crystallography

DIFFRACTIONDIFFRACTION

What is a Unit Cell?What is a Unit Cell?

Unit cell can be chosen in different Unit cell can be chosen in different ways!ways!

Unit Cells?Unit Cells?

White and black birds by the artist, M. C. Escher.

A unit cell chosen such that it contains minimum volume but exhibit maximum symmetry

{R = n1 a1 + n2 a2 + n3 a3}

Translationalvector

Crystal Structure

Ideal Crystal: Contain periodical array of atoms/ionsRepresented by a simple lattice of pointsA group of atoms attached to each lattice points

Basis

LATTICE = An infinite array of points in space, in which each point has identical surroundings to all others.

CRYSTAL STRUCTURE = The periodic arrangement of atoms in the crystal.

It can be described by associating with each lattice point a group of atoms called the MOTIF (BASIS)

Lattice parameters: a, b, c;

7 Crystal Systems

Bravais Lattice: an infinite array of discrete points with an arrangement and orientation that appears exactly the same from whichever of the points the array is viewed.

Crystal SystemCrystal System Bravais Bravais LatticeLattice

Essential Essential Symmetry Symmetry

ConditionsConditions

CubicCubic P, F, IP, F, I 4 C4 C33 aa==bb==cc

====90900 0

TetragonalTetragonal P, IP, I 1 C1 C44 along [ along [cc-axis]-axis] aa==bb==cc

====909000

HexagonalHexagonal PP 1 C1 C66 along [ along [cc-axis]-axis] aa==bb==cc

====909000

RhombohedralRhombohedral RR 1 C1 C33 along body along body

diagonaldiagonalaa==bb==cc

= = = = 90 9000

OrthorhombicOrthorhombic P, F, I, CP, F, I, C 3 C3 C22 mutually mutually

perpedicular along perpedicular along the three axesthe three axes

a a b b cc

====909000

MonoclinicMonoclinic P, CP, C 1 C1 C22 along [ along [bb-axis]-axis] a a bb cc

===90=9000 & & 90 9000

TriclinicTriclinic PP CC22 or inversion centre or inversion centre a a b b cc

909000

14 Bravais lattices14 Bravais lattices

Unit cell symmetries - cubicUnit cell symmetries - cubic

3 C3 C4 4 - passes through pairs of opposite - passes through pairs of opposite face centers, parallel to cell axes face centers, parallel to cell axes

4 C4 C3 3 - passes through cubic - passes through cubic diagonalsdiagonals

A cube need not have C4 !!

Copper metal is Copper metal is face-centered face-centered cubiccubic

Identical atoms at corners and at Identical atoms at corners and at face centers face centers

Lattice type FLattice type F

also Ag, Au, Al, Ni...also Ag, Au, Al, Ni...

-Iron is body-centered cubic

Identical atoms at corners and body center (nothing at face centers)

Lattice type I

Also Nb, Ta, Ba, Mo...

periodic tableperiodic table

Hexagonal closedpacked (hcp)

face-centered cubic (fcc)

body-centered cubic (bcc)

Caesium Chloride (CsCl) isCaesium Chloride (CsCl) is primitive primitive cubiccubic

Different atoms at corners and body Different atoms at corners and body center. NOT body centered, therefore.center. NOT body centered, therefore.

Lattice type PLattice type P

Also CuZn, CsBr, LiAgAlso CuZn, CsBr, LiAg

Sodium Chloride (NaCl) - Na is much smaller than Cs

Face Centered Cubic

Rocksalt structure

Lattice type F

Also NaF, KBr, MgO….

Diamond Structure: two sets of FCC Lattices

Z = 8 C atoms per unit cell

one C4

Why not F tetragonal?

Tetragonal: P, I

Yellow and green colors represents same atoms but different depths.

ExampleExample

CaCCaC22 - has a rocksalt-like structure but with - has a rocksalt-like structure but with

non-spherical carbidesnon-spherical carbides

2-C

C

Carbide ions are aligned parallel to c

c > a,b tetragonal symmetry

Orthorhombic: P, I, F, C

C F

Side centering

Side centered unit cell

Notation:

A-centered if atom in bc plane

B-centered if atom in ac plane

C-centered if atom in ab plane

Trigonal: P : 3-fold rotation

Hexagonal

Monoclinic Triclinic

Unit cell contentsUnit cell contents Counting the number of atoms Counting the number of atoms withinwithin the unit cell the unit cell

Many atoms are shared between unit cells

Atoms Shared Between: Each atom counts:corner 8 cells 1/8face center 2 cells 1/2body center 1 cell 1edge center 4 cells 1/4

lattice type cell contentsP 1 [=8 x 1/8]I 2 [=(8 x 1/8) + (1 x 1)]F 4 [=(8 x 1/8) + (6 x 1/2)]C 2 [=(8 x 1/8) + (2 x 1/2)]

e.g. NaCl Na at corners: (8 1/8) = 1 Na at face centres (6 1/2) = 3Cl at edge centres (12 1/4) = 3 Cl at body centre = 1

Unit cell contents are 4(Na+Cl-)

(0,0,0)(0, ½, ½)(½, ½, 0)(½, 0, ½)

Fractional Coordinates

Cs (0,0,0)Cl (½, ½, ½)

Density Calculation

AC NV

nA

n: number of atoms/unit cell

A: atomic mass

VC: volume of the unit cell

NA: Avogadro’s number (6.023x1023 atoms/mole)

Calculate the density of copper.

RCu =0.128nm, Crystal structure: FCC, ACu= 63.5 g/mole

n = 4 atoms/cell, 333 216)22( RRaVC

3

2338/89.8

]10023.6)1028.1(216[

)5.63)(4(cmg

8.94 g/cm3 in the literature

Miller IndicesMiller Indices

describe which plane of atom isdescribe which plane of atom isinteracting with the x-raysinteracting with the x-rays

How to Identify Miller indices (hkl)?How to Identify Miller indices (hkl)?

direction: [hkl]family of directions: <hkl>planes: (hkl)family of planes: {hkl}

[001]

[010]

[001]

ab

c

to identify planes:Step 1 : Identify the intercepts on the x- , y- and z- axes.Step 2 : Specify the intercepts in fractional coordinatesStep 3 : Take the reciprocals of the fractional intercepts

Miller indices (hkl)Miller indices (hkl)

e.g.: cubic system:

to identify planes:Step 1 : Identify the intercepts on the x- , y- and z- axes (a/2, ∞, ∞)Step 2 : Specify the intercepts in fractional co-ordinates (a/2a, ∞, ∞) = (1/2,0,0) Step 3 : Take the reciprocals of the fractional intercepts (2, 0, 0)

(210)

(110) (111) (100)



Crystallographic Directions And Planes

Lattice DirectionsIndividual directions: [uvw][uvw]Symmetry-related directions: <uvw><uvw>

Miller Indices:1. Find the intercepts on the axes in terms of the lattice

constant a, b, c2. Take the reciprocals of these numbers, reduce to the

three integers having the same ratio(hkl)(hkl)

Set of symmetry-related planes: {hkl}{hkl}

(100) (111)

(200) (110)

In cubic system,

[hkl] direction perpendicular to (hkl) plane

Wilhelm Conrad Röntgen

Wilhelm Conrad Röntgen discovered 1895 the X-rays. 1901 he was honoured by the Noble prize for physics. In 1995 the German Post edited a stamp, dedicated to W.C. Röntgen.

The Principles of an X-ray Tube

Anode

focus

Fast electronsCathode

X-Ray

The Principle of Generation of X-ray

X-ray

Fast incident electron

nucleus

Atom of the anodematerial

electrons

Ejected electron(slowed

down and changed

direction)

The Principle of Generation the Characteristic Radiation

K

L

K

K

L

M

EmissionPhotoelectron

Electron

The Generating of X-rays

Bohr`s model


M

K

L

K K K K

energy levels (schematic) of the electrons

Intensity ratios KKK


Anode

Mo

Cu

Co

Fe

(kV)

20.0

9.0

7.7

7.1

K1 : 0,70926

K2 : 0,71354

K1 : 0,63225

Filter

K1 : 1,5405

K2 : 1,54434

K1 : 1,39217

K1 : 1,78890

K2 : 1,79279

K1 : 1,62073

K1 : 1,93597

K2 : 1,93991

K1 : 1,75654

Zr0,08mm

Mn0,011mm

Fe0,012mm

Ni0,015mm

Wavelength (Angström)


Emission Spectrum of aMolybdenum X-Ray Tube

Bremsstrahlung = continuous spectra

characteristic radiation = line spectra

Interaction between X-ray and Matter

d

wavelength Pr

intensity Io

incoherent scattering

Co (Compton-Scattering)

coherent scattering

Pr(Bragg´s-scattering)

absorbtionBeer´s law I = I0*e-µd

fluorescense

> Pr

photoelectrons

C. Gordon Darwin

C. Gordon Darwin, grandson of C. Robert Darwin developed 1912 dynamic theory of scattering of X-rays at crystal lattice

P. P. Ewald

P. P. Ewald 1916 published a simple and more elegant theory of X-ray diffraction by introducing the reciprocal lattice concept. Compare Bragg’s law (left), modified Bragg’s law (middle) and Ewald’s law (right).

sin2

n

d

2

1sin d

12

sin

Bragg’s Description

• The incident beam will be scattered at all scattering centres, which lay on lattice planes.

• The beam scattered at different lattice planes must be scattered coherent, to give an maximum in intensity.

• The angle between incident beam and the lattice planes is called .

• The angle between incident and scattered beam is 2 .

• The angle 2 of maximum intensity is called the Bragg angle.W.H. Bragg (father) and William

Lawrence.Bragg (son) developed a simple relation for scattering angles, now call Bragg’s law.

sin2 n

d

Bragg’s Law

• A powder sample results in cones with high intensity of scattered beam.

• Above conditions result in the Bragg equation

• or

sin2 dns

sin2

n

d

/hchE

35KeV ~ 0.1-1.4ACu K 1.54 A

Mo:

X-Ray Diffraction

Structure Determination

High Voltage

X-Ray DiffractionX-ray Tube

Lead Screen

X-ray Beam

Crystal

Photographic Plate

Projection Screen

Visible Light Laser 35mm slide

Optical Transforms

L

X

LightInterference fringes

Constructive

Destructive

Diffraction

Diffraction Conditions

Diffraction Conditions

Fraunhofer diffraction Bragg diffraction

For constructive interference, d sin = n

For constructive interference, 2(d sin ) = n

}d

d }}

d

d sin

}

}d

d sin d sin

2

222

2

1

a

lkh

dhkl

For cubic system

Lattice spacing

Bragg’s Law

For cubic system: But not all planes have the diffraction !!!

sin2

sinsin

hkl

hklhkl

d

dd

QTSQn

222 lkh

adhkl

(200)(211)

Powder diffraction

X-Ray

Powder X-ray Diffraction

Tube

Powder

Film

The Elementary Cell

a

b

c

a = b = c = = = 9

0

o

Relationship between d-value and the Lattice Constants

=2dsin Bragg´s law The wavelength is known

Theta is the half value of the peak position

d will be calculated

1/d2= (h2 + k2)/a2 + l2/c2 Equation for the determination of the d-value of a tetragonal elementary cell

h,k and l are the Miller indices of the peaks

a and c are lattice parameter of the elementary cell

if a and c are known it is possible to calculate the peak position

if the peak position is known it is possible to calculate the lattice parameter

D8 ADVANCE Bragg-Brentano Diffractometer

• A scintillation counter may be used as detector instead of film to yield exact intensity data.

• Using automated goniometers step by step scattered intensity may be measured and stored digitally.

• The digitised intensity may be very detailed discussed by programs.

• More powerful methods may be used to determine lots of information about the specimen.

The Bragg-Brentano Geometry

Tube

measurement circle

focusing-circle

2

Detector

Sample

The Bragg-Brentano Geometry

Divergence slit

Detector-

slitTube

Antiscatter-slit

Sample

Mono-chromat

or

Powder Diffraction Pattern

What is a Powder Diffraction Pattern?

A powder diffractogram is the result of a convolution of a) the diffraction capability of the sample (Fhkl) and b) a complex system function.

The observed intensity yoi at the data point i is the result of

yoi = of intensity of "neighbouring" Bragg peaks + background

The calculated intensity yci at the data point i is the result of

yci = structure model + sample model + diffractometer model + background model

Which Information does a Powder Pattern offer?

peak position - dimension of the elementary cell

peak intensity - content of the elementary cell

peak broadening - strain/crystallite size/nanostr.

Powder Pattern and Structure

• The d-spacings of lattice planes depend on the size of the elementary cell and determine the position of the peaks.

• The intensity of each peak is caused by the crystallographic structure, the position of the atoms within the elementary cell and their thermal vibration.

• The line width and shape of the peaks may be derived from conditions of measuring and properties - like particle size - of the sample material.

(110)

(200)(211)

Powder diffraction

X-Ray

)(75.0);(5.0sin

sin2

2

FCCBCCB

A

Example: layered silicatesmica

2*theta d

7.2 12.1

14.4 6.1

22 4.0

nd sin2

growth oriented along c-axis

(hkl)

(001)

(002)

(003)

C~12.2 A

(00l)

What we will see in XRD of simple cubic, BCC, FCC?

What we will see in XRD of simple cubic, BCC, FCC?

)(75.0);(5.0sin

sin2

2

FCCBCCB

A

222 lkh

adhkl

Observable diffraction peaks

222 lkh Ratio

Simple cubic

SC: 1,2,3,4,5,6,8,9,10,11,12..

BCC: 2,4,6,8,10, 12….

FCC: 3,4,8,11,12,16,24….

222 lkh

adhkl

nd sin2

Ex: An element, BCC or FCC, shows diffraction peaks at 2: 40, 58, 73, 86.8,100.4 and 114.7. Determine:(a) Crystal structure?(b) Lattice constant?(c) What is the element?

2theta theta (hkl)

40.0 20 0.117 1 (110)

58.0 29 0.235 2 (200)

73.0 36.5 0.3538 3 (211)

86.8 43.4 0.4721 4 (220)

100.4 50.2 0.5903 5 (310)

114.7 57.35 0.7090 6 (222)

2sin 222 lkh

a =3.18Å, BCC, W

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CYL729: Materials Characterization