X-RAY DIFFRACTION
 X- Ray Sources

Part of

MATERIALS SCIENCE
& AALearner’s
Learner’sGuide
Guide
ENGINEERING
AN INTRODUCTORY E-BOOK

 Diffraction: Bragg’s Law
 Crystal Structure Determination

Anandh Subramaniam & Kantesh Balani
Materials Science and Engineering (MSE)
Indian Institute of Technology, Kanpur- 208016
Email: , URL: home.iitk.ac.in/~anandh
/>
Elements of X-Ray Diffraction
B.D. Cullity & S.R. Stock
Prentice Hall, Upper Saddle River (2001)
A very good book for
practical aspects

X-Ray Diffraction: A Practical Approach
C. Suryanarayana & M. Norton Grant
Plenum Press, New York (1998)

Recommended websites:

 /> />Caution Note: In any chapter, amongst the first few pages (say 5 pages) there will be some ‘big picture’
overview information. This may lead to ‘overloading’ and readers who find this ‘uncomfortable’ may skip
particular slides in the first reading and come back to them later.


What will you learn in this ‘sub-chapter’?
 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…
Other relevant topics

Laue_picture.ppt
line_broadening.ppt
other_signals_xray.ppt
reciprocal_lattice.ppt
structure_factor_calculations.ppt
Understanding_diffraction.ppt
XRD_lattice_parameter_calculation.ppt
XRD_powder_diffraction.ppt
XRD_sample_patterns.ppt


Some Basics

 For electromagnetic radiation to be diffracted* the spacing in the grating ( grating refers to
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.

** If the wavelength is of the order of the lattice spacing, then diffraction effects will be prominent.
 Three possibilities (regimes) exist based on the wavelength (λ) and the spacing between the scatteres (a).
 λ < a → transmission dominated.
 λ ~ a → diffraction dominated.
 λ > a → reflection dominated.
** 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)

Click here to know more about this


Generation of X-rays
 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. λX-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.

Beam of electrons

Target


X-rays

An accelerating (or decelerating) charge radiates electromagnetic radiation


Mo Target impacted by electrons accelerated by a 35 kV potential shows the emission
spectrum as in the figure below (schematic)
X-ray sources with different λ for
doing XRD studies
Target
Metal

λ Of Kα
radiation (Å)

Mo

0.71

Cu

1.54

Co

1.79

Fe


1.94

Cr

2.29

The high intensity nearly monochromatic Kα 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


X-ray sources with different λ for doing XRD studies
Elements

(KV)

λ Of Kα1
radiation
(Å)
 

λ Of Kα2
radiation (Å)
 

Kβ-Filter
λ Of Kβ
(mm)
radiation (Å)
 


Ag

25.52

0.55941

0.5638

0.49707

Mo

20

0.7093

0.71359

0.63229

Cu

8.98

1.540598

1.54439

1.39222


Ni

8.33

1.65791

1.66175

1.50014

Co

7.71

1.78897

1.79285

1.62079

Fe

7.11

1.93604

1.93998

1.75661


Cr

5.99

2.2897

2.29361

2.08487

Pd
0.0461
Zr
0.0678
Ni
0.017
Co
0.0158
Fe
0.0166
Mn
0.0168
V
0.169

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


• 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.

Incident X-rays

SPECIMEN

Fluorescent X-rays

Electrons

Scattered X-rays
Coherent
Coherent
From
Frombound
boundcharges
charges

Absorption (Heat)

Compton recoil

Photoelectrons

Incoherent (Compton modified)
From loosely bound charges

Transmitted beam


Click
Clickhere
heretotoknow
knowmore
more

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


Diffraction

Click
Clickhere
heretoto“Understand
“UnderstandDiffraction”
Diffraction”

 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 Xray 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 λ).
Sample related Crystalline array of scatterers* with spacing of the order of (~) λ.
Diffraction geometry related Fraunhofer diffraction geometry ( this is actually part of the Fraunhofer geometry)
&


Aspects related to the wave

Diffraction pattern
with sharp peaks

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

Coherent, monochromatic, parallel wave
Crystalline*,**
Fraunhofer geometry

[1] The intensity-θ plot looks like a ‘δ’ 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 broad peak).


Some comments and notes
 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 scatterers
(the object), and the screen (to capture the diffraction pattern) is placed far away from the
object.

** With a de Broglie wavelength

Click here to know more about Fraunhofer and Fresnel diffraction geometries


XRD → the first step
 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.

Schematics


Some points to recon with
 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 θ 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

In plane scattering is in phase
Scattering from across planes is in phase

A Laue diffraction pattern
with ‘sharp’ peaks.

 *Max von Laue’s postulate: If (i) crystals have a periodic arrangement of atoms and if (ii)
x-rays of waves (concepts which were not confirmed till then), then crystals should act like
a diffraction grating for x-rays. Both these postulates (i & ii) were proved by a single
experiment by Laue (published in 1912 which won him the noble prize in 1914).


Let us consider in-plane scattering
A B

X

Y

Atomic Planes


There is more to this
Click here to know more and get
introduced to Laue equations describing
diffraction

Extra path traveled by incoming waves →AY
Extra path traveled by scattered waves → XB

But this is still reinforced scattering
and NOT reflection

A B

X

These can be in phase if
→ θincident = θscattered

Y


BRAGG’s EQUATION

Warning: we are using ray diagrams in spite of
being in the realm of ‘physical optics’

Let us consider scattering across planes

Click
Clickhere

heretotovisualize
visualize
constructive
constructiveand
and
destructive
interference
destructive interference
See Note Ӂ later

 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 Sinθ + d Sinθ) = (2d.Sinθ).
 For constructive interference, this path difference should be an integral multiple of λ:

nλ = 2d Sinθ → the Bragg’s equation. (More about this sooner).
 The path difference between Ray-1 and Ray-3 is = 2× (2d.Sinθ) = 2× nλ = 2nλ. This implies that if Ray1 and Ray-2 constructively interfere Ray-1 and Ray-3 will also constructively interfere. (And so forth).


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

interference
of
destructive interference of
just
just‘of-Bragg
‘of-Braggrays’
rays’occur
occur

Interference of Ray-1 with Ray-2
Which remains same
thereafter (like in the
BB’ plane)

Note that they ‘almost’ constructively interfere!

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

Clearly A’BC’ > ABC


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
Clickhere
heretotoknow
knowmore

moreabout
aboutthe
theLaue
LauePicture
Picture

(More about the Bragg’s viewpoint soon).

 *Sir William Henry Bragg and William Lawrence Bragg (this won the father and son team the noble prize in
1915).
Since there are two Bragg’s involved, wherever we
refer to the law or the equation it has be Braggs’ (and
not Bragg’s as I have done in this chapter)

[1]
[1]

“The important thing in science is not so much to obtain new facts as to discover
new ways of thinking about them”. William Lawrence Bragg.
[1] />

Reflection versus Diffraction
 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).
Reflection

Diffraction
Occurs throughout the bulk


Occurs from surface

(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
Jump

Quantum

Planes are imaginary constructs

If nλ = 2d Sinθ is the Bragg’s equation, then what is the (famous) Bragg’s
law?

Braggs Law
 The diffracted beam appears to be specularly reflected from a set of crystal lattice planes.
 Angle of incidence = Angle of reflection.


Understanding the Bragg’s equation
 nλ = 2d Sinθ

The equation is written better with some descriptive subscripts:

n λCu Kα = 2 d hkl Sinθ hkl

If this equation is satisfied, then θ is θBragg

 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: Ӂ

Note: if hkl reflection (corresponding to n=1) occurs at θhkl then 2h 2k 2l reflection (n=2) will occur at a higher angle θ2h 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 θincident = θscattered); 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, Sinθ
decreases (and so does θ).
 The smallest interplanar spacing from which Bragg diffraction can be obtained is λ/2 →
maximum value of θ is 90°, Sinθ is 1 ⇒ from Bragg equation d = λ/2.


Order of the reflection (n)
 For Cu Kα radiation (λ = 1.54 Å) and d110= 2.22 Å

n

Sinθ = nλ/2d

θ

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

Relation between dnh nk nl and dhkl

d

Cubic crystal
hkl


d nh nk nl =

d nh nk nl

=

a
h2 + k 2 + l 2
a

(nh) 2 + (nk ) 2 + (nl ) 2

d
=
= hkl
n
n h2 + k 2 + l 2
a

e.g.

a
d 220 =
8
a
d110 =
2

d 220 1
=

d110 2
d 220

d110
=
2


In XRD nth order reflection from (h k l) is considered as 1st order reflection from (nh nk nl).

nλ = 2d hkl sin θ hkl

d nh nk nl

d hkl
sin θ
d hkl
n
λ = 2d nh nk nl sin θ nh nk nl

λ=2

1
=
n

d 200 1
=
d100 2
All these form the (200) set


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

d300 1
=
d100 3
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.


I have seen diagrams like in Fig.1 where rays seem to be scattered from nothing!
Funda Check What does this mean?
 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).
Wave impinging on a crystal (parallel wavefront)


(note there are no ‘rays’)

??

Fig.
2

Fig.
1
of
ction
e
r
i
D

e
w av

* 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.


More about the Bragg’s viewpoint


“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.


Funda Check 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?

 Diffraction is a process in which
‘linear information’ (the d-spacing of the planes)
is converted to ‘angular information’ (the angle of diffraction, θBragg).
 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
(2θ) is plotted instead of θ.


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


Funda Check

 What is θ (theta) in the Bragg’s equation?

 θ 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, θ in this context implies θBragg (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).


Funda Check

What do the terms ‘sharp’ and ‘diffuse’ mean with regard to peaks and scattering?

 Sharp peaks are obtained from crystalline materials (using parallel, monochromatic
radiation), while typically a broad peak is obtained from an amorphous material. The sharp

peak is referred to as a Bragg peak.
 Defects (like point defects, thermal vibration, partial ordering) in the crystal can give rise to
low intensity scattering between the Bragg peaks. This is termed as diffuse scattering.

d
re obtaine
a
s
k
a
e
p
t sharp
In contras

from crys

tals

Note the broad peak

SAD pattern (TEM ) from the
amorphous sample

XRD pattern showing the formation of amorphous
structure in the suction cast (Cu64Zr36)96Al4 alloy