X-RAY DIFFRACTIONX-RAY DIFFRACTION

X- Ray Sources

Diffraction: Bragg’s Law

Crystal Structure Determination

Elements of X-Ray DiffractionB.D. Cullity & S.R. Stock

Prentice Hall, Upper Saddle River (2001)

Recommended websites: http://www.matter.org.uk/diffraction/ http://www.ngsir.netfirms.com/englishhtm/Diffraction.htm

MATERIALS SCIENCEMATERIALS SCIENCE&&

ENGINEERING ENGINEERING

Anandh Subramaniam & Kantesh Balani

Materials Science and Engineering (MSE)

Indian Institute of Technology, Kanpur- 208016

Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh

AN INTRODUCTORY E-BOOKAN INTRODUCTORY E-BOOK

Part of

http://home.iitk.ac.in/~anandh/E-book.htmhttp://home.iitk.ac.in/~anandh/E-book.htm

A Learner’s GuideA Learner’s GuideA Learner’s GuideA Learner’s Guide

How to produce monochromatic X-rays? How does a crystal scatter these X-rays to give a diffraction pattern?

Bragg’s equation What determines the position of the XRD peaks? Answer) the lattice. What determines the intensity of the XRD peaks? Answer) the motif. How to analyze a powder pattern to get information about the lattice type?

(Cubic crystal types).

What other uses can XRD be put to apart from crystal structure determination? Grain size determination Strain in the material…

What will you learn in this ‘sub-chapter’?

For electromagnetic radiation to be diffracted* the spacing in the grating (~a series of obstacles or a series of scatterers) should be of the same order as the wavelength.

In crystals the typical interatomic spacing ~ 2-3 Å** so the suitable radiation for the diffraction study of crystals is X-rays.

Hence, X-rays are used for the investigation of crystal structures. Neutrons and Electrons are also used for diffraction studies from materials. Neutron diffraction is especially useful for studying the magnetic ordering in

materials.

Some Basics

** Lattice parameter of Cu (aCu) = 3.61 Å dhkl is equal to aCu or less than that (e.g. d111 = aCu/3 = 2.08 Å)

** If the wavelength is of the order of the lattice spacing, then diffraction effects will be prominent.Click here to know more about thisClick here to know more about this

Beam of electrons Target X-rays

An accelerating (or decelerating) charge radiates electromagnetic radiation

X-rays can be generated by decelerating electrons. Hence, X-rays are generated by bombarding a target (say Cu) with an electron beam. The resultant spectrum of X-rays generated (i.e. X-rays versus Intensity plot) is shown in

the next slide. The pattern shows intense peaks on a ‘broad’ background. The intense peaks can be ‘thought of’ as monochromatic radiation and be used for X-ray

diffraction studies.

Generation of X-rays

Mo Target impacted by electrons accelerated by a 35 kV potential shows the emission spectrum as in the figure below (schematic)

The high intensity nearly monochromatic K x-rays can be used as a radiation source for X-ray diffraction (XRD) studies a monochromator can be used to further decrease the spread of wavelengths in the X-ray

Inte

nsit

y

Wavelength ()0.2 0.6 1.0 1.4

White radiation

Characteristic radiation →due to energy transitionsin the atom

K

KIntense peak, nearly monochromatic

X-ray sources with different for doing XRD studies

Target Metal

Of K radiation (Å)

Mo 0.71

Cu 1.54

Co 1.79

Fe 1.94

Cr 2.29

Elements (KV) Of K1

radiation (Å)

 

Of K2

radiation (Å) 

Of Kβ

radiation (Å) 

Kβ-Filter

(mm)

Ag 25.52 0.55941 0.5638 0.49707 Pd0.0461

Mo 20 0.7093 0.71359 0.63229 Zr0.0678

Cu 8.98 1.540598 1.54439 1.39222 Ni0.017

Ni 8.33 1.65791 1.66175 1.50014 Co0.0158

Co 7.71 1.78897 1.79285 1.62079 Fe0.0166

Fe 7.11 1.93604 1.93998 1.75661 Mn0.0168

Cr 5.99 2.2897 2.29361 2.08487 V0.169

C.Gordon Darwin, Grandson of C. Robert Darwin developed the dynamic theory of scattering of x-rays (a tough theory!) in 1912

X-ray sources with different for doing XRD studies

Absorption (Heat)

Incident X-rays

SPECIMEN

Transmitted beam

Fluorescent X-raysElectrons

Compton recoil PhotoelectronsScattered X-rays

CoherentFrom bound charges

CoherentFrom bound charges

X-rays can also be refracted (refractive index slightly less than 1) and reflected (at very small angles)

• When X-rays hit a specimen, the interaction can result in various signals/emissions/effects.

• The coherently scattered X-rays are the ones important from a XRD perspective.

Incoherent (Compton modified)From loosely bound charges

Click here to know moreClick here to know more

Now we shall consider the important topic as to how X-rays interact with a crystalline array (of atoms, ions etc.) to give rise to the phenomenon known as X-ray diffraction (XRD).

Let us consider a special case of diffraction → a case where we get ‘sharp[1] diffraction peaks’.

Diffraction (with sharp peaks) (with XRD being a specific case) requires three important conditions to be satisfied:Radiation related Coherent, monochromatic, parallel waves& (with wavelength ). Sample related Crystalline array of scatterers* with spacing of the order of (~) .Diffraction geometry related Fraunhofer diffraction geometry (& this is actually part of the Fraunhofer geometry)

[1] The intensity- plot looks like a ‘’ function (in an ideal situation).* A quasicrystalline array will also lead to diffraction with sharp peaks (which we shall not consider in this text).

** Amorphous material will give broadened (diffuse) peak (additional factors related to the sample can also give a diffuse peak).

Diffraction Click here to “Understand Diffraction”Click here to “Understand Diffraction”

Coherent, monochromatic, parallel wave

Fraunhofer geometry

Diffraction pattern with sharp peaks

Crystalline*,**

Aspects related to the wave

Aspects related to the material

Aspects related to the diffraction set-up (diffraction geometry)

The waves could be: electromagnetic waves (light, X-rays…), matter waves** (electrons, neutrons…) or mechanical waves (sound, waves on water surface…).

Not all objects act like scatterers for all kinds of radiation. If wavelength is not of the order of the spacing of the scatterers, then the number

of peaks obtained may be highly restricted (i.e. we may even not even get a single diffraction peak!).

In short diffraction is coherent reinforced scattering (or reinforced scattering of coherent waves).

In a sense diffraction is nothing but a special case of constructive (& destructive) interference.To give an analogy the results of Young’s double slit experiment is interpreted as interference, while the result of multiple slits (large number) is categorized under diffraction.

Fraunhofer diffraction geometry implies that parallel waves are impinging on the scatteres (the object), and the screen (to capture the diffraction pattern) is placed far away from the object.

** With a de Broglie wavelength

Some comments and notes

Click here to know more about Fraunhofer and Fresnel diffraction geometriesClick here to know more about Fraunhofer and Fresnel diffraction geometries

Sets Electron cloud into oscillation

Sets nucleus into oscillation

Small effect neglected

A beam of X-rays directed at a crystal interacts with the electrons of the atoms in the crystal. The electrons oscillate under the influence of the incoming X-Rays and become secondary sources

of EM radiation. The secondary radiation is in all directions. The waves emitted by the electrons have the same frequency as the incoming X-rays coherent. The emission can undergo constructive or destructive interference.

XRD the first step

Schematics

Incoming X-rays

Secondaryemission

Oscillating charge re-radiates In phase with the incoming x-rays

We can get a better physical picture of diffraction by using Laue’s formalism (leading to the Laue’s equations).

However, a parallel approach to diffraction is via the method of Bragg, wherein diffraction can be visualized as ‘reflections’ from a set of planes.

As the approach of Bragg is easier to grasp we shall use that in this elementary text. We shall do some intriguing mental experiments to utilize the Bragg’s equation (Bragg’s model) with

caution.

Let us consider a coherent wave of X-rays impinging on a crystal with atomic planes at an angle to the rays.

Incident and scattered waves are in phase if the: i) in-plane scattering is in phase and ii) scattering from across the planes is in phase.

Incident and scattered waves are in phase if

Scattering from across planes is in phase

In plane scattering is in phase

Some points to recon with

Extra path traveled by incoming waves AY

A B

X Y

Atomic Planes

Extra path traveled by scattered waves XB

These can be in phase if incident = scattered

A B

X Y

But this is still reinforced scatteringand NOT reflection

Let us consider in-plane scattering

There is more to this Click here to know more and get

introduced to Laue equations describing diffraction

BRAGG’s EQUATION

A portion of the crystal is shown for clarity- actually, for destructive interference to occur many planes are required (and the interaction volume of x-rays is large as compared to that shown in the schematic).

The scattering planes have a spacing ‘d’. Ray-2 travels an extra path as compared to Ray-1 (= ABC). The path difference between

Ray-1 and Ray-2 = ABC = (d Sin + d Sin) = (2d.Sin). For constructive interference, this path difference should be an integral multiple of :

n = 2d Sin the Bragg’s equation. (More about this sooner).

The path difference between Ray-1 and Ray-3 is = 2(2d.Sin) = 2n = 2n. This implies that if Ray-1 and Ray-2 constructively interfere Ray-1 and Ray-3 will also constructively interfere. (And so forth).

Let us consider scattering across planes

Click here to visualize constructive and

destructive interference

Click here to visualize constructive and

destructive interference

See Note Ӂ later

The previous page explained how constructive interference occurs. How about the rays just of Bragg angle? Obviously the path difference would be just off as in the figure below. How come these rays ‘go missing’?

Click here to understand how destructive interference of just ‘of-Bragg rays’ occur

Click here to understand how destructive interference of just ‘of-Bragg rays’ occur

Interference of Ray-1 with Ray-2

Note that they ‘almost’ constructively interfere!

How to ‘see’ that path difference increases with angle?Funda Check

Which remains same thereafter (like in the

BB’ plane)

Clearly A’BC’ > ABC

Reflection versus Diffraction

Reflection Diffraction

Occurs from surfaceOccurs throughout the bulk

(though often the penetration of x-rays in only of the order of 10s of microns in a material)

Takes place at any angle Takes place only at Bragg angles

~100 % of the intensity may be reflected Small fraction of intensity is diffracted

Note: X-rays can ALSO be reflected at very small angles of incidence

Though diffraction (according to Bragg’s picture) has been visualized as a reflection from a set of planes with interplanar spacing ‘d’ diffraction should not be confused with reflection (specular reflection).

Planes are imaginary constructsPlanes are imaginary constructs

Laue versus Bragg

In Laue’s picture constructive and destructive interference at various points in space is computed using path differences (and hence phase differences) given a crystalline array of scatterers.

Bragg simplified this picture by considering this process as ‘reflections from atomic planes’.

Click here to know more about the Laue PictureClick here to know more about the Laue Picture

n = 2d SinThe equation is written better with some descriptive subscripts:

n is an integer and is the order of the reflection (i.e. how many wavelengths of the X-ray go on to make the path difference between planes).Note: if hkl reflection (corresponding to n=1) occurs at hkl then 2h 2k 2l reflection (n=2) will occur at a higher angle 2h 2k 2l.

Bragg’s equation is a negative statement If Bragg’s eq. is NOT satisfied NO ‘reflection’ can occur If Bragg’s eq. is satisfied ‘reflection’ MAY occur

(How?- we shall see this a little later).

The interplanar spacing appears in the Bragg’s equation, but not the interatomic spacing ‘a’ along the plane (which had forced incident = scattered); but we are not

free to move the atoms along the plane ‘randomly’ click here to know more. For large interplanar spacing the angle of reflection tends towards zero → as d increases,

Sin decreases (and so does ). The smallest interplanar spacing from which Bragg diffraction can be obtained is /2 → maximum value of is 90, Sin is 1 from Bragg equation d = /2.

Understanding the Bragg’s equation

2 SinCu K hkl hkln d If this equation is satisfied, then is Bragg

Note: Ӂ

For Cu K radiation ( = 1.54 Å) and d110= 2.22 Å

n Sin = n/2d

1 0.34 20.7º • First order reflection from (110) 110

2 0.69 43.92º• Second order reflection from (110) planes 110

• Also considered as first order reflection from (220) planes 220

2 2 2

Cubic crystalhkl

ad

h k l

8220

ad

2110

ad

2

1

110

220 d

d

Relation between dnh nk nl and dhkl

e.g.

2 2 2( ) ( ) ( )nhnk nl

ad

nh nk nl

2 2 2

hklnhnk nl

dad

nn h k l

Order of the reflection (n)

2 sinhkl hkln d In XRD nth order reflection from (h k l) is considered as 1st order reflection from (nh nk nl)

sin2n

dhkl

n n n n n n2 sinh k l h k ld

1nhnk nl

hkl

d

d n

300

100

1

3

d

d

200

100

1

2

d

d

Hence, (100) planes are a subset of (200) planes

Important point to note:In a simple cubic crystal, 100, 200, 300… are all allowed ‘reflections’. But, there are no atoms in the planes lying within the unit cell! Though, first order reflection from 200 planes is equivalent (mathematically) to the second order reflection from 100 planes; for visualization purposes of scattering, this is better thought of as the later process (i.e. second order reflection from (100) planes).

Note:Technically, in Miller indices we factor out the common factors. Hence, (220) 2(110) (110).In XRD we extend the usual concept of Miller indices to include planes, which do not pass through lattice points (e.g. every alternate plane belonging to the (002) set does not pass through lattice points) and we allow the common factors to remain in the indices.

All these form the (200) set

I have seen diagrams like in Fig.1 where rays seem to be scattered from nothing! What does this mean?Funda Check

Few points are to be noted in this context. The ray ‘picture’ is only valid in the realm of geometrical optics, where the wave nature of light is not considered (& also the discrete nature of matter is ignored; i.e. matter is treated like a continuum). In diffraction we are in the domain of physical optics.

The wave impinges on the entire volume of material including the plane of atoms (the effect of which can be quantified using the atomic scattering power* and the density of atoms in the plane). Due to the ‘incoming’ wave the atomic dipoles are set into oscillation, which further act like emitter of waves

In Bragg’s viewpoint, the atomic planes are to be kept in focus and the wave (not just a ray) impinges on the entire plane (some planes have atoms in contact and most have atoms, which are not in contact along the plane see Fig.2).

* To be considered later A plane in Bragg’s viewpoint can be characterized by two factors: (a) atomic density (atoms/unit area on the plane), (b) atomic scattering

factor of the atoms.

Fig.1

Fig.2

Wave impinging on a crystal (parallel wave-front)

(note there are no ‘rays’) ??

Direction of wave

“It is difficult to give an explanation of the nature of the semi-transparent layers or planes that is immediately convincing, as they are a concept rather than a physical reality. Crystal structures, with their regularly repeating patterns, may be referred to a 3D grid and the repeating unit of the grid, the unit cell, can be found. The grid may be divided up into sets of planes in various orientations and it is these planes which are considered in the derivation of Bragg’s law. In some cases, with simple crystal structures, the planes also correspond to layers of atoms, but this is not generally the case. See Section 1.5 for further information.

Some of the assumptions upon which Bragg’s law is based may seem to be rather dubious. For instance, it is known that diffraction occurs as a result of interaction between X-rays and atoms. Further, the atoms do not reflect X-rays but scatter or diffract them in all directions. Nevertheless, the highly simplified treatment that is used in deriving Bragg’s law gives exactly the same answers as are obtained by a rigorous mathematical treatment. We therefore happily use terms such as reflexion (often deliberately with this alternative, but incorrect, spelling!) and bear in mind that we are fortunate to have such a simple and picturesque, albeit inaccurate, way to describe what in reality is a very complicated process.” [1]

[1] Anthony R West, Solid State Chemistry and its Applications, Second Edition, John Wiley & Sons Ltd., Chichester, 2014.

More about the Bragg’s viewpoint

How is it that we are able to get information about lattice parameters of the order of Angstroms (atoms which are so closely spaced) using XRD?

Funda Check

Diffraction is a process in which ‘linear information’ (the d-spacing of the planes) is converted to ‘angular information’ (the angle of diffraction, Bragg).

If the detector is placed ‘far away’ from the sample (i.e. ‘R’ in the figure below is large) the distances along the arc of a circle (the detection circle) get amplified and hence we can make ‘easy’ measurements.

This also implies that in XRD we are concerned with angular resolution instead of linear resolution.

Later we will see that in powder diffraction this angle of deviation (2) is plotted instead of .

Forward and Back Diffraction

Here a guide for quick visualization of forward and backward scattering (diffraction) is presented

Funda Check What is (theta) in the Bragg’s equation?

is the angle between the incident x-rays and the set of parallel atomic planes (which have a spacing dhkl). Which is 10 in the above figure.

Usually, in this context implies Bragg (i.e. the angle at which Bragg’s equation is satisfied).

It is NOT the angle between the x-rays and the sample surface (note: specimens could be spherical or could have a rough surface).

We had mentioned that Bragg’s equation is a negative statement: i.e. just because Bragg’s equation is satisfied a ‘reflection’ may not be observed.

Let us consider the case of Cu K radiation ( = 1.54 Å) being diffracted from (100) planes of Mo (BCC, a = 3.15 Å = d100).

The missing ‘reflections’

100 1002 d Sin 100100

1.540.244

2 2(3.15)Sin

d

100 14.149

But this reflection is absent in BCC Mo

The missing reflection is due to the presence of additional atoms in the unit cell

(which are positions at lattice points) which we shall consider next

The wave scattered from the middle plane is out of phase with the ones scattered from top and bottom planes. I.e. if the green rays are in phase (path difference of ) then the red ray will be exactly out of phase with the green rays (path difference of /2).

However, the second order reflection from (100) planes (which is equivalent to the first order reflection from the (200) planes is observed

100100

2 1.540.48

2 3.15Sin

d

2 1100 200~ 29.26

nd ndorder order

This is because if the green rays have a path difference of 2 then the red ray will have path difference of → which will still lead to constructive interference!

Continuing with the case of BCC Mo…

Why does the 110 reflection not go missing? (Why is it present?)Funda Check

Let us look at the (110) planes in projection.

Note that (110)blue coloured planes existed before and after introducing an

atom at unit cell centre at (½, ½ ½)grey coloured. Thus lattice centering does

not lead to any waves being scattered out of phase.

Presence of additional atoms/ions/molecules in the UC at lattice points (as we may chose a non-primitive unit cell) or as a part of the motif can alter the intensities of some of the reflections

Some of the reflections may even go missing

Important points

Position of the ‘reflections’/‘peaks’ tells us about the lattice type.

The Intensities tells us about the motif.

Intensity of the Scattered Waves

Electron

Atom

Unit cell (uc)

Scattering by a crystal can be understood in three steps

A

B

C

Polarization factor

Atomic scattering factor (f)

Structure factor (F)

To understand the scattering from a crystal leading to the ‘intensity of reflections’ (and why some reflections go missing), three levels of scattering have to be considered: 1) scattering from electrons2) scattering from an atom3) scattering from a unit cell

Click here to know the details

Structure factor calculations&

Intensity in powder patterns

Structure factor calculations&

Intensity in powder patterns

Structure Factor (F): The resultant wave scattered by all atoms of the unit cell

The Structure Factor is independent of the shape and size of the unit cell; but is dependent on the position of the atoms/ions etc. within the cell

Click here to know more about

Bragg’s equation tells us about the position of the diffraction peaks (in terms of ) but tells us nothing about the intensities. The intensities of the peaks depend on many factors as considered here.

The concept of a Reciprocal lattice and the Ewald Sphere construction: Reciprocal lattice and Ewald sphere constructions are important tools towards

understanding diffraction. (especially diffraction in a Transmission Electron Microscope (TEM))

A lattice in which planes in the real lattice become points in the reciprocal lattice is a very useful one in understanding diffraction.

click here to go to a detailed description of these topics.

Reciprocal Lattice & Ewald Sphere constructionReciprocal Lattice & Ewald Sphere constructionClick here to know more about

Bravais Lattice Reflections which may be present Reflections necessarily absent

Simple all None

Body centred (h + k + l) even (h + k + l) odd

Face centred h, k and l unmixed h, k and l mixed

End centred (C centred) h and k unmixed h and k mixed

Bravais Lattice Allowed Reflections

SC All

BCC (h + k + l) even

FCC h, k and l unmixed

DCEither, h, k and l are all odd or

all are even & (h + k + l) divisible by 4

Selection / Extinction Rules

As we have noted before even if Bragg’s equation is satisfied, ‘reflections may go missing’ this is due to the presence of additional atoms in the unit cell.

The reflections present and the missing reflections due to additional atoms in the unit cell are listed in the table below. Click here to see the derivations

Structure factor calculationsClick here to see the derivations

Structure factor calculations

h2 + k2 + l2 SC FCC BCC DC

1 100

2 110 110

3 111 111 111

4 200 200 200

5 210

6 211 211

7

8 220 220 220 220

9 300, 221

10 310 310

11 311 311 311

12 222 222 222

13 320

14 321 321

15

16 400 400 400 400

17 410, 322

18 411, 330 411, 330

19 331 331 331

Allowed reflections in SC*, FCC*, BCC* & DC crystals

* lattice decorated with monoatomic/monoionic motif

Cannot be expressed as (h2+k2+l2)

Crystal structure determination

Monochromatic X-rays

Panchromatic X-rays

Monochromatic X-rays

Many s (orientations)Powder specimen

POWDER METHOD

Single LAUETECHNIQUE

Varied by rotation

ROTATINGCRYSTALMETHOD

λ fixed

θ variable

λ fixed

θ rotated

λ variable

θ fixed

As diffraction occurs only at specific Bragg angles, the chance that a reflection is observed when a crystal is irradiated with monochromatic X-rays at a particular angle is small (added to this the diffracted intensity is a small fraction of the beam used for irradiation).

The probability to get a diffracted beam (with sufficient intensity) is increased by either varying the wavelength () or having many orientations (rotating the crystal or having multiple crystallites in many orientations).

The three methods used to achieve high probability of diffraction are shown below.

Only the powder method (which is commonly used in materials science) will be considered in this text.

THE POWDER METHOD

2222 sin)( lkh

22

2222 sin

4)(

alkh

)(sin4

2222

22 lkha

2 2 2hkl Cubicad

h k l

2d Sin

222

222 sin4

lkh

a

Cubic crystal

In the powder method the specimen has crystallites (or grains) in many orientations (usually random).

Monochromatic* X-rays are irradiated on the specimen and the intensity of the diffracted beams is measured as a function of the diffracted angle.

In this elementary text we shall consider cubic crystals.

(1) (2)

(2) in (1)

* In reality this is true only to an extent

The ratio of (h2 + k2 + l2) derived from extinction rules (earlier page)

As we shall see soon the ratios of (h2 + k2 + l2) is proportional to Sin2 which can be used in the determination of the lattice type

SC 1 2 3 4 5 6 8 …

BCC 1 2 3 4 5 6 7 …

FCC 3 4 8 11 12 …

DC 3 8 11 16 …

Note that we have to consider the ratio of only two lines to distinguish FCC and DC. I.e. if the ratios are 3:4 then the lattice is FCC.

But, to distinguish between SC and BCC we have to go to 7 lines!

In the powder sample there are crystallites in different ‘random’ orientations (a polycrystalline sample too has grains in different orientations)

The coherent x-ray beam is diffracted by these crystallites at various angles to the incident direction All the diffracted beams (called ‘reflections’) from a single plane, but from different crystallites lie

on a cone. Depending on the angle there are forward and back reflection cones. A diffractometer can record the angle of these reflections along with the intensities of the reflection The X-ray source and diffractometer move in arcs of a circle- maintaining the Bragg ‘reflection’

geometry as in the figure (right)

POWDER METHOD

Different cones for different reflections

Also called Debye ring

Usually the source is fixed and the

detector and sample are rotated

How to visualize the occurrence of peaks at various angles

It is ‘somewhat difficult’ to actually visualize a random assembly of crystallites giving peaks at various angels in a XRD scan. The figures below are expected to give a ‘visual feel’ for the same. [Hypothetical crystal with a = 4Å is assumed with =1.54Å. Only planes of the type xx0 (like (100,110)are considered].

Random assemblage of crystallites in a material

The sample is not

rotating only the source

and detector move in

arcs of a circle

As the scan takes place at increasing angles, planes with suitable ‘d’,

which diffract are ‘picked out’ from favourably oriented crystallites

h2 hkl d Sin()

1 100 4.00 0.19 11.10

2 110 2.83 0.27 15.80

3 111 2.31 0.33 19.48

4 200 2.00 0.39 22.64

5 210 1.79 0.43 25.50

6 211 1.63 0.47 28.13

8 220 1.41 0.54 32.99

9 300 1.33 0.58 35.27

10 310 1.26 0.61 37.50

For convenience the source may be stationary (and the sample and detector may rotate– but the effect is

equivalent)

In the power diffraction method a 2 versus intensity (I) plot is obtained from the diffractometer (and associated instrumentation).

The ‘intensity’ is the area under the peak in such a plot (NOT the height of the peak). The information of importance obtained from such a pattern is the ‘relative intensities*’ and the absolute value of the intensities is of little importance (the longer we irradiate the sample the more will be

the intensity under the peak) (for now). I is really diffracted energy (as Intensity is Energy/area/time).

A table is prepared as in the next slide to tabulate the data and make calculations to find the crystal structure (restricting ourselves to cubic crystals for the present).

Determination of Crystal Structure from 2 versus Intensity Data in Powder Method

Powder diffraction pattern from Al

Radiation: Cu K, = 1.54 Å

Increasing

Increasing d

Intensity (I) has units of [Energy/area/time] → but here it is plotted as arbitrary units.

Usually in degrees ()

This is peak (sometimes called a line- a hangover from Debye Scherrer camera usage)

* Relative intensity: Intensity of any peak divided by the intensity of the ‘strongest’ peak.

n 2→ Intensity Sin Sin2 ratio

Determination of Crystal Structure (lattice type) from 2 versus Intensity Data

The following table is made from the 2 versus Intensity data (obtained from a XRD experiment on a powder sample (empty starting table of columns is shown below- completed table shown later).

Powder diffraction pattern from Al Radiation: Cu K, = 1.54 Å

Note: This is a schematic pattern In real patterns peaks or not idealized peaks broadened Increasing splitting of peaks with g

(1 & 2 peaks get resolved in the high angle peaks)

Peaks are all not of same intensity No brackets are used around the indexed numbers

(the peaks correspond to planes in the real space)

Note that there are no brackets around the indices!

These are Miller indices in reciprocal space (these are not planes they

correspond to panes in real space)

Powder diffraction pattern from Al

111

200

220

311

222

400

K1 & K2 peaks resolved in high angle peaks(in 222 and 400 peaks this can be seen)

Radiation: Cu K, = 1.54 Å

Note: Peaks or not idealized peaks broadened. Increasing splitting of peaks with g . Peaks are all not of same intensity. There is a ‘noisy’ background.

In low angle peaks K1 & K2 peaks merged

What is the maximum value of possible (experimentally)?

Funda Check How are real diffraction patterns different from the ‘ideal’ computed ones?

We have seen real and ideal diffraction patterns. In ideal patterns the peaks are ‘’ functions.

Real diffraction patterns are different from ideal ones in the following ways: Peaks are broadened Could be due to instrumental, residual ‘non-uniform’ strain (microstrain), grain size etc. broadening.

Peaks could be shifted from their ideal positions Could be due to uniform strain→ macrostrain.

Relative intensities of the peaks could be altered Could be due to texture in the sample.

Funda Check Ans: 90

At = 90 the ‘reflected ray’ is opposite in direction to the incident ray.

Beyond this angle, it is as if the source and detector positions are switched.

2max is 180.

Instrumental broadening

Crystal defects (‘bent’ planes)

Peak Broadening Small crystallite size

Note peak splitting has not been included here as this

comes from ‘symmetry lowering’ (i.e. crystal with

lower symmetry)

Including those coming from strain fields associated with these defects Click here to know moreClick here to know more

Funda Check What will determine how many peaks I will get?

1) smaller the wavelength of the X-rays, more will be the number of peaks possible.

From Bragg’s equation: [=2dSin], (Sin)max will correspond to dmin. (Sin)max=1.

Hence, dmin=/2. Hence, if is small then planes with smaller d spacing (i.e. those which

occur at higher 2 values) will also show up in a XRD patter (powder pattern). Given that

experimentally cannot be greater than 90.

2) Lattice type in SC we will get more peaks as compared to (say) FCC/DC. Other things being

equal.

3) Lower the symmetry of the crystal, more the number of peaks (e.g., in tetragonal crystal

the 100 peak will lie at a different 2 as compared to the 001 peak).

2dSin min

max

2dSin

min2

d

# 2 Sin Sin2 ratio Index d

1 38.52 19.26 0.33 0.11 3 111 2.34

2 44.76 22.38 0.38 0.14 4 200 2.03

3 65.14 32.57 0.54 0.29 8 220 1.43

4 78.26 39.13 0.63 0.40 11 311 1.22

5 82.47 41.235 0.66 0.43 12 222 1.17

6 99.11 49.555 0.76 0.58 16 400 1.01

7 112.03 56.015 0.83 0.69 19 331 0.93

8 116.60 58.3 0.85 0.72 20 420 0.91

9 137.47 68.735 0.93 0.87 24 422 0.83

10 163.78 81.89 0.99 0.98 27 333 0.78

Determination of Crystal Structure (lattice type) from 2 versus Intensity Data

From the ratios in column 6 we conclude that FCC

Let us assume that we have the 2 versus intensity plot from a diffractometer To know the lattice type we need only the position of the peaks (as tabulated below)

Solved exampleSolved example

2 d Sin 111 1111.54 2 2 0.33

3

ad Sin

o

4.04Aa Al

Using

We can get the lattice parameter which correspond to that for Al

1

Note: Error in d spacing decreases with → so we should use high angle lines for lattice parameter calculation

Click here to know moreClick here to know more

Note that Sin cannot be > 1

XRD_lattice_parameter_calculation.ppt

2222 sin)( lkhNote

2→ Sin Sin2 Ratiosof Sin2

Dividing Sin2 by

0.134/3 = 0.044667

Whole number ratios

1 21.5 0.366 0.134 1 3

2 25 0.422 0.178 1.33 3.99 4

3 37 0.60 0.362 2.70 8.10 8

4 45 0.707 0.500 3.73 11.19 11

5 47 0.731 0.535 4 11.98 12

6 58 0.848 0.719 5.37 16.10 16

7 68 0.927 0.859 6.41 19.23 19

FCC

Another exampleGiven the positions of the Bragg peaks we find the lattice type

Solved exampleSolved example

2

More SolvedExamples on XRDClick here

Comparison of diffraction patterns of SC, BCC & B2 structures

Click here

Aluminium = 1.54 Å = 3 Å = 0.1 Å

hkl d Sin() 2 Sin() 2 Sin() 2

111 2.34 0.33 19.26 38.52 0.64 39.87 79.74 0.02 1.22 2.45

200 2.03 0.38 22.38 44.76 0.74 47.64 95.28 0.02 1.41 2.82

220 1.43 0.54 32.57 65.14 1.05 - - 0.03 2.00 4.01

311 1.22 0.63 39.13 78.26 1.23 - - 0.04 2.35 4.70

222 1.17 0.66 41.24 82.47 1.28 - - 0.04 2.45 4.90

400 1.01 0.76 49.56 99.11 1.49 - - 0.05 2.84 5.68

331 0.93 0.83 56.02 112.03 1.61 - - 0.05 3.08 6.16

420 0.91 0.85 58.30 116.60 1.65 - - 0.05 3.15 6.30

422 0.83 0.93 68.74 137.47 1.81 - - 0.06 3.45 6.91

333 0.78 0.99 81.89 163.78 1.92 - - 0.06 3.68 7.35

Funda CheckWhat happens when we increase or decrease ?

We had pointed out that ~ a is preferred for diffraction. Let us see what happens if we ‘drastically’ increase or decrease .(This is only a thought experiment!!)

If we ~double → we get too few peaks

If we make small→ all the peaks get crowded to small

angles

With CuK = 1.54 Å

And the detector may not be able to resolve these peaks if they come too close!

Bravais lattice determination

Lattice parameter determination

Determination of solvus line in phase diagrams

Long range order

Applications of XRD

Crystallite size and Strain

Determine if the material is amorphous or crystalline

We have already seen these applications

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Next slide

And More….

Diffraction angle (2) →

Inte

nsit

y →

90 1800

Crystal

90 1800

Diffraction angle (2) →

Inte

nsit

y →

Liquid / Amorphous solid

90 1800

Diffraction angle (2) →

Inte

nsit

y →

Monoatomic gas

Schematic of difference between the diffraction patterns of various phases

Sharp peaks

Diffuse Peak

No peak

Schematics

Diffuse peak from Cu-Zr-Ni-Al-Si Metallic glass

(XRD patterns) courtesy: Dr. Kallol Mondal, MSE, IITK

Actual diffraction pattern from an amorphous solid

A amorphous solid which shows glass transition in a Differential Scanning Calorimetry (DSC) plot is also called a glass. In ‘general usage’ a glass may be considered equivalent to a amorphous solid (at least loosely in the structural sense).

Sharp peaks are missing. Broad diffuse peak survives → the peak corresponds to the average spacing between atoms which the diffraction experiment ‘picks out’

Amorphous solid

Funda CheckFunda Check What is the minimum spacing between planes possible in a crystal? How many diffraction peaks can we get from a powder pattern?

2 2 2

Cubic crystalhkl

adh k l

Let us consider a cubic crystal (without loss in generality)

As h,k, l increases, ‘d’ decreases we could have planes with infinitesimal spacing

10 1

ad a

112

ad

1310

ad

125

ad

34 525

a ad

With increasing indices the interplanar spacing decreases

The number of peaks we obtain in a powder diffraction pattern depends on the wavelength of x-ray we are using. Planes with ‘d’ < /2 are not captured in the diffraction pattern.

These peaks with small ‘d’ occur at high angles in diffraction pattern.

Q & A How to increase the number of peaks in a XRD pattern?

We have noted that (e.g. for DC crystal) the number of available peaks in the 2 regime could be insufficient for a given analysis.

The number of peaks can be increased in two ways:1) using Mo Kα instead of Cu Kα2) first obtain pattern with β filter and then remove the filter to get more lines.