Lecture presentation on x-ray diffraction.ppt

X-ray Diffraction (XRD)
• What is X-ray Diffraction
Properties and generation of X-ray
• Bragg’s Law
• Basics of Crystallography
• XRD Pattern
• Powder Diffraction
• Applications of XRD
http://www.matter.org.uk/diffraction/x-ray/default.htm

X-ray and X-ray Diffraction
X-ray was first discovered by W. C. Roentgen in
1895. Diffraction of X-ray was discovered by
W.H. Bragg and W.L. Bragg in 1912
Bragg’s law: n=2dsin
Photograph of the hand of
an old man using X-ray.
http://www.youtube.com/watch?v=vYztZlLJ3ds at~0:40-3:10

Properties and Generation of X-ray
 X-rays are
electromagnetic
radiation with very
short wavelength (
10-8
-10-12
m)
 The energy of the x-
ray can be calculated
with the equation
E = h = hc/
 e.g. the x-ray photon
with wavelength 1Å
has energy 12.5 keV

A Modern Automated
X-ray Diffractometer
Cost: $560K to 1.6M
X-ray Tube
Sample stage
Detector
http://www.youtube.com/watch?v=lwV5WCBh9a0 to~1:08

Production of X-rays
Cross section of sealed-off filament X-ray tube
target X-rays
W
Vacuum
X-rays are produced whenever high-speed electrons collide with a
metal target.
A source of electrons – hot W filament, a high accelerating
voltage (30-50kV) between the cathode (W) and the anode,
which is a water-cooled block of Cu or Mo containing desired
target metal.
http://www.youtube.com/watch?v=Bc0eOjWkxpU to~1:10 Production of X-rays
https://www.youtube.com/watch?v=3_bZCA7tlFQ How does X-ray tube work
filament
+ -

X-ray Spectrum
 A spectrum of x-ray is
produced as a result of the
interaction between the
incoming electrons and the
nucleus or inner shell
electrons of the target
element.
 Two components of the
spectrum can be identified,
namely, the continuous
spectrum caused by
bremsstrahlung (German
word: braking radiation)
and the characteristic
spectrum.
SWL - short-wavelength limit
continuous
radiation
characteristic
radiation
k
k
I

Mo
http://www.youtube.com/watch?v=n9FkLBaktEY characteristic X-ray
http://www.youtube.com/watch?v=Bc0eOjWkxpU at~1:06-3:10
http://www.youtube.com/watch?v=3fe6rHnhkuY Bremsstrahlung

Short-wavelength Limit
• The short-wavelength limit (SWL or SWL)
corresponds to those x-ray photons
generated when an incoming electron
yield all its energy in one impact.
min
max


hc
h
eV 


A
10
240
.
1 4
min
V
eV
hc
SWL




 

V – applied voltage

Characteristic x-ray Spectra
 Sharp peaks in the
spectrum can be seen if the
accelerating voltage is high
(e.g. 25 kV for molybdenum
target).
 These peaks fall into sets
which are given the names,
K, L, M…. lines with
increasing wavelength.
Mo

08/21/24
 If an incoming electron has
sufficient kinetic energy for
knocking out an electron of the
K shell (the inner-most shell),
it may excite the atom to an
high-energy state (K state).
 One of the outer electron falls
into the K-shell vacancy,
emitting the excess energy as
a x-ray photon.
 Characteristic x-ray energy:
Ex-ray=Efinal-Einitial
Excitation of K, L, M and N shells and
Formation of K to M Characteristic X-rays
K
L
M
N
K
K
L
Energy K state
(shell)
L state
M state
N state
ground state
K
K
L
L
K1
K2
I
II
III
M
subshells
EK>EL>EM
EK>EK
K excitation
L excitation M

Element
K
(weighted
average), Å
K1
very strong,
Å
K2
strong, Å
K
weak, Å
K
Absorption
edge, Å
Excitation
potential
(kV)
Ag 0.56084 0.55941 0.56380 0.49707 0.4859 25.52
Mo 0.710730 0.709300 0.713590 0.632288 0.6198 20.00
Cu 1.541838 1.540562 1.544390 1.392218 1.3806 8.98
Ni 1.65919 1.65791 1.66175 1.50014 1.4881 8.33
Co 1.790260 1.788965 1.792850 1.62079 1.6082 7.71
Fe 1.937355 1.936042 1.939980 1.75661 1.7435 7.11
Cr 2.29100 2.28970 2.293606 2.08487 2.0702 5.99
Characteristic x-ray Spectra
Z

Characteristic X-ray Lines
Spectrum of Mo at 35kV
K1
K
K
I
 (Å)
<0.001Å
K2
K and K2 will cause
Extra peaks in XRD
pattern, but can be
eliminated by adding
filters.
----- is the mass
absorption coefficient
of Zr.
=2dsin

• All x-rays are absorbed to some extent in
passing through matter due to electron
ejection or scattering.
• The absorption follows the equation
where I is the transmitted intensity;
I0 is the incident intensity
x is the thickness of the matter;
 is the linear absorption coefficient
 (element dependent);
 is the density of the matter;
(/) is the mass absorption coefficient (cm2
/gm).
Absorption of x-ray
x
x
e
I
e
I
I



 










 0
0
I0 I
,

x
I
x

Effect of , / (Z) and t on
Intensity of Diffracted X-ray
incident beam
diffracted beam
film
crystal
http://www.matter.org.uk/diffraction/x-ray/x_ray_diffraction.htm

• The mass absorption
coefficient is also
wavelength
dependent.
• Discontinuities or
“Absorption edges”
can be seen on the
absorption coefficient
vs. wavelength plot.
• These absorption
edges mark the point
on the wavelength
scale where the x-
rays possess
sufficient energy to
eject an electron from
one of the shells.
Absorption of x-ray
Absorption coefficients of Pb,
showing K and L absorption edges.

/
Absorption edges

Filtering of X-ray
• The absorption behavior of x-ray by matter
can be used as a means for producing quasi-
monochromatic x-ray which is essential for
XRD experiments.
• The rule: “Choose for the filter an element
whose K absorption edge is just to the short-
wavelength side of the K line of the target
material.”
Target
material
Ag Mo Cu Ni Co Fe Cr
Filter
material
Pd Nb,
Zr
Ni Co Fe Mn V

 A common example is
the use of nickel to cut
down the K peak in the
copper x-ray spectrum.
 The thickness of the filter
to achieve the desired
intensity ratio of the
peaks can be calculated
with the absorption
equation shown in the
last section.
Filtering of X-ray
x
x
e
I
e
I
I



 










 0
0
Comparison of the spectra of Cu
radiation (a) before and (b) after
passage through a Ni filter. The
dashed line is the mass absorption
coefficient of Ni.
No filter Ni filter
K absorption
edge of Ni
1.4881Å
Choose for the filter an element whose K absorption edge is just to the
short-wavelength side of the K line of the target material.


Cu K 
1.5405Å

What Is Diffraction?
A wave interacts with
A single particle
A crystalline material
The particle scatters the
incident beam uniformly
in all directions.
The scattered beam may
add together in a few
directions and reinforce
each other to give
diffracted beams.
http://www.matter.org.uk/diffraction/introduction/what_is_diffraction.htm

What is X-ray Diffraction?
The atomic planes of a crystal cause an incident beam of x-rays (if
wavelength is approximately the magnitude of the interatomic
distance) to interfere with one another as they leave the crystal.
The phenomenon is called x-ray diffraction.
atomic plane
X-ray of 
d
n= 2dsin()
Bragg’s Law:
B
 ~ d
2B
I
http://www.youtube.com/watch?v=1FwM1oF5e6o to~1:17 diffraction & interference

Constructive and Destructive
Interference of Waves
Constructive Interference Destructive Interference
In Phase Out Phase
Constructive interference occurs only when the path
difference of the scattered wave from consecutive layers
of atoms is a multiple of the wavelength of the x-ray.
/2
http://www.youtube.com/watch?v=kSc_7XBng8w
http://micro.magnet.fsu.edu/primer/java/interference/waveinteractions/index.html

Bragg’s Law and X-ray Diffraction
How waves reveal the atomic structure of crystals
n = 2dsin()
Atomic
plane
d=3 Å
=3Å
=30o
n-integer
X-ray1
X-ray2

2-diffraction angle
Diffraction occurs only when Bragg’s Law is satisfied
Condition for constructive interference (X-rays 1 & 2) from planes with
spacing d
http://www.youtube.com/watch?v=UfDW0-kghmI at~3:00-6:00
http://www.eserc.stonybrook.edu/ProjectJava/Bragg/index.html

Deriving Bragg’s Law - n = 2dsin
X-ray 1
X-ray 2
Constructive interference
occurs only when
n= AB + BC
AB=BC
n= 2AB
Sin=AB/d
AB=dsin
n =2dsin
=2dhklsinhkl
n – integer, called the order of diffraction

Basics of Crystallography
A crystal consists of a periodic arrangement of the unit cell
into a lattice. The unit cell can contain a single atom or
atoms in a fixed arrangement.
Crystals consist of planes of atoms that are spaced a
distance d apart, but can be resolved into many atomic
planes, each with a different d-spacing.
a,b and c (length) and ,  and  (angles between a,b and c)
are lattice constants or parameters which can be
determined by XRD.
smallest building block
Unit cell (Å)
Lattice
CsCl
d1
d2
d3
a
b
c



z [001]
y [010]
x [100] crystallographic
axes
Single crystal
http://www.youtube.com/watch?v=Rm-i1c7zr6Q&list=TLyPTUJ62VYE4wC1snHSChDl0NGo9IK-Nl
http://www.youtube.com/watch?v=Mm-jqk1TeRY crystal packing in lattices
to~2:25
Lattice structures

System Axial lengths Unit cell
and angles
Cubic a=b=c
===90o
a
a
c
Tetragonal
a=bc
===90o
b
a
c
Orthorhombic
abc
===90o
a
Rhombohedral
a=b=c
==90o
Hexagonal
a=bc
=90o
=120o
c
a
b
Monoclinic
abc
==90o

b
a
c
Triclinic
abc
90o
c
Seven crystal Systems
a

Plane Spacings for Seven
Crystal Systems
1
hkl
hkl
hkl
hkl
hkl
hkl
hkl

Miller Indices - hkl
(010)
Miller indices-the reciprocals of the
fractional intercepts which the plane
makes with crystallographic axes
Axial length 4Å 8Å 3Å
Intercept lengths 1Å 4Å 3Å
Fractional intercepts ¼ ½ 1
Miller indices 4 2 1
h k l
4Å 8Å 3Å
 8Å 
/4 1 /3
0 1 0
h k l
a b c
a b c
https://www.youtube.com/watch?v=enVpDwFCl68 Miller indices example crystallography for everyone
Miller indices form a notation system in crystallography for planes in crystal lattices.

Planes and Spacings
a
-
http://www.matter.org.uk/diffraction/geometry/planes_in_crystals.htm

Indexing of Planes and Directions
a
b
c
a
b
c
(111)
[110]
a direction [uvw]
a set of equivalent
directions <uvw>
<100>:[100],[010],[001]
[100],[010] and [001]
a plane (hkl)
a set of equivalent
planes {hkl}
{110}:(101),(011),(110)
(101),(101),(101),etc.
(110)
[111]
http://www.youtube.com/watch?v=9Rjp9i0H7GQ Directions in crystals

X-ray Diffraction Pattern
2
I Simple Cubic
=2dhklsinhkl
Bragg’s Law: (Cu K)=1.5418Å
BaTiO3 at T>130o
C
dhkl
20o 40o
60o
(hkl)

XRD Pattern
Significance of Peak Shape in XRD
1.Peak position
2.Peak width
3.Peak intensity
http://www.youtube.com/watch?v=MU2jpHg2vX8 XRD peak analysis
I
2

Peak Position
Determine d-spacings and lattice parameters
Fix  (Cu k)=1.54Å dhkl = 1.54Å/2sinhkl
Note: Most accurate d-spacings are those calculated
from high-angle peaks.
For a simple cubic (a=b=c=a0)
a0 = dhkl (h2
+k2
+l2
)½
e.g., for BaTiO3, 2220=65.9o
, 220=32.95o
,
d220 =1.4156Å, a0=4.0039Å
2

Peak Intensity
X-ray intensity: Ihkl  lFhkll2
Fhkl - Structure Factor
Fhkl =  fjexp[2i(huj+kvj+lwj)]
j=1
N
fj – atomic scattering factor
fj  Z, sin/
N – number of atoms in the unit cell,
uj,vj,wj - fractional coordinates of the j
th atom
in the unit cell
Low Z elements may be difficult to detect by XRD
Determine crystal structure and atomic arrangement
in a unit cell

Cubic Structures
a = b = c = a
Simple Cubic Body-centered Cubic Face-centered Cubic
BCC FCC
8 x 1/8 =1 8 x 1/8 + 1 = 2 8 x 1/8 + 6 x 1/2 = 4
1 atom 2 atoms 4 atoms
a
a
a
[001]
z axis
[100]
x
[010]
y
Location: 0,0,0 0,0,0, ½, ½, ½, 0,0,0, ½, ½, 0,
½, 0, ½, 0, ½, ½,
- corner atom, shared with 8 unit cells
- atom at face-center, shared with 2 unit cells
8 unit cells

Structures of Some Common Metals
BCC FCC
a
a
a
[001] axis
[100]
[010]
(001) plane
(002)
h,k,l – integers, Miller indices, (hkl) planes
(001) plane intercept [001] axis with a length of a, l = 1
(002) plane intercept [001] axis with a length of ½ a, l = 2
(010) plane intercept [010] axis with a length of a, k = 1, etc.
(010)
plane
½ a [010]
axis
 = 2dhklsinhkl
Mo Cu
d002 =
d001
d010

Structure factor and
intensity of diffraction
• Sometimes, even though
the Bragg’s condition is
satisfied, a strong
diffraction peak is not
observed at the expected
angle.
• Consider the diffraction
peak of (001) plane of a
FCC crystal.
• Owing to the existence of
the (002) plane in
between, complications
occur.
d001
d002

1
2
3
1’
2’
3’
z
(001)
(002)
FCC

 ray 1 and ray 3 have path
difference of 
 but ray 1 and ray 2 have
path difference of /2. So
do ray 2 and ray 3.
 It turns out that it is in
fact a destructive
condition, i.e. having an
intensity of 0.
 the diffraction peak of a
(001) plane in a FCC
crystal can never be
observed.
Structure factor and
intensity of diffraction
d001
d002
1
2
3
1’
2’
3’
/2 /2
/4 /4

http://emalwww.engin.umich.edu/education_materials/microscopy.html
d001
d002
=2dhklsinhkl
d001sin001=d002sin002 since d001=2d002
If sin002=2sin001 i.e., 002>001
Bragg’s law holds and (002) diffraction peak appears
1
3
2
1’
2’
/4
001
3’
1
2
3
1’
2’
3’
002
/2
001
002
When =001 no diffraction occurs, while 
increases to 002, diffraction occurs.

 e.g., Aluminium (FCC),
all atoms are the same
in the unit cell
 four atoms at positions,
(uvw):
A(0,0,0), B(½,0,½),
C(½,½,0) & D(0,½,½)
Structure factor and intensity
of diffraction for FCC
z
x
y
A
B
C
D

For a certain set of plane, (hkl)
F =  f () exp[2i(hu+kv+lw)]
= f ()  exp[2i(hu+kv+lw)]
= f (){exp[2i(0)] + exp[2i(h/2 + l/2)]
+ exp[2i(h/2 + k/2)] + exp[2i(k/2 + l/2)]}
= f (){1 + ei(h+k)
+ ei(k+l)
+ ei(l+h)
}
Since e2ni
= 1 and e(2n+1)i
= -1,
if h, k & l are all odd or all even, then (h+k),
(k+l), and (l+h) are all even and F = 4f;
otherwise, F = 0
Structure factor and intensity of
diffraction for FCC




mixed
l
k,
h,
0
even
all
or
odd
all
l
k,
h,
4 f
F
     
j
j
j lw
kv
hu
i
j
j e
f
F



 




2
A(0,0,0), B(½,0,½),
C(½,½,0) & D(0,½,½)
2i
Ihkl  lFhkll2

XRD
Patterns of
Simple
Cubic and
FCC
Diffraction angle 2 (degree)
I Simple Cubic
FCC
2

h2
+ k2
+ l2
simple cubic
(any
combination)
FCC
(either all odd
or all even)
BCC
(h + k + l) is
even
1 100 - -
2 110 - 110
3 111 111 -
4 200 200 200
5 210 - -
6 211 - 211
7 - - -
8 220 220 220
9 300, 221 - -
10 310 - 310
11 311 311 -
12 222 222 222
Diffractions Possibly Present for
Cubic Structures

Peak Width - Full Width at Half Maximum
(FWHM)
1. Particle or
grain size
2. Residual
strain
Determine

Effect of Particle (Grain) Size
(331) Peak of cold-rolled and
annealed 70Cu-30Zn brass
2
I
K1
K2
As rolled
200o
C
250o
C
300o
C
450o
C
As rolled 300o
C
450o
C
Grain
size
t
B =
0.9
t cos
Peak
broadening
As grain size decreases
hardness increases and
peak become broader
Grain
size
B
(FWHM)

Effect of Lattice Strain
on Diffraction Peak
Position and Width
No Strain
Uniform Strain
(d1-do)/do
Non-uniform Strain
d1constant
Peak moves, no shape changes
Peak broadens

XRD patterns from
other states of matter
Constructive interference
Structural periodicity
Diffraction
Sharp maxima
Crystal
Liquid or amorphous solid
Lack of periodicity One or two
Short range order broad maxima
Monatomic gas
Atoms are arranged Scattering I
perfectly at random decreases with 
2

X-ray Diffraction (XRD)
• What is X-ray Diffraction
Properties and generation of X-ray
• Bragg’s Law
• Basics of Crystallography
• XRD Pattern
• Powder Diffraction
• Applications of XRD
http://www.matter.org.uk/diffraction/x-ray/laue_method.htm

Diffraction of X-rays by Crystals-Laue Method
Back-reflection Laue
Film
X-ray
crystal
crystal Film
Transmission Laue
[001]
http://www.youtube.com/watch?v=UfDW0-kghmI at~1:20-3:00
http://www.youtube.com/watch?v=2JwpHmT6ntU

Powder Diffraction (most widely used)
A powder sample is in fact an assemblage of small
crystallites, oriented at random in space.
2
2
Polycrystalline
sample
Powder
sample
crystallite
Diffraction of X-rays by Polycrystals
http://www.youtube.com/watch?v=lwV5WCBh9a0 at~1:20-1:56
d1
d3
d2
d1
d2
d3

Detection of Diffracted X-ray
by A Diffractometer
 x-ray detectors (e.g. Geiger
counters) is used instead of
the film to record both the
position and intensity of the
x-ray peaks
 The sample holder and the x-
ray detector are mechanically
linked
 If the sample holder turns ,
the detector turns 2, so that
the detector is always ready to
detect the Bragg diffracted
x-ray
X-ray
tube
X-ray
detector
Sample
holder
2

http://www.youtube.com/watch?v=lwV5WCBh9a0 at~1:44-1:56 and 15:44-16:16

Phase Identification
One of the most important uses of XRD
• Obtain XRD pattern
• Measure d-spacings
• Obtain integrated intensities
• Compare data with known standards in
the JCPDS file, which are for random
orientations (there are more than 50,000
JCPDS cards of inorganic materials).

JCPDS Card
1.file number 2.three strongest lines
3.lowest-angle line 4.chemical formula and name 5.data on dif-
fraction method used 6.crystallographic data 7.optical and other
data 8.data on specimen 9.data on diffraction pattern.
Quality of data

Other Applications of XRD
• To identify crystalline phases
• To determine structural properties:
Lattice parameters (10-4
Å), strain, grain size, expitaxy,
phase composition, preferred orientation
order-disorder transformation, thermal expansion
• To measure thickness of thin films and multilayers
• To determine atomic arrangement
• To image and characterize defects
Detection limits: ~3% in a two phase mixture; can be
~0.1% with synchrotron radiation.
Lateral resolution: normally none
XRD is a nondestructive technique
https://www.youtube.com/watch?v=CpJZfeJ4poE phased contrast x-ray imaging
https://www.youtube.com/watch?v=6POi6h4dfVs
Determining strain pole figures from diffraction experiments

Phase Identification
-Effect of Symmetry
on XRD Pattern
a b c
2
a. Cubic
a=b=c, (a)
b. Tetragonal
a=bc (a and c)
c. Orthorhombic
abc (a, b and c)
•Number of reflection
•Peak position
•Peak splitting

Finding mass fraction of
components in mixtures
 The intensity of
diffraction peaks depends
on the amount of the
substance
 By comparing the peak
intensities of various
components in a mixture,
the relative amount of
each components in the
mixture can be worked out
ZnO + M23C6 + 

Preferred Orientation (Texture)
 In common polycrystalline
materials, the grains may not be
oriented randomly. (We are not
talking about the grain shape, but
the orientation of the unit cell of
each grain, )
 This kind of ‘texture’ arises from all
sorts of treatments, e.g. casting,
cold working, annealing, etc.
 If the crystallites (or grains) are not
oriented randomly, the diffraction
cone will not be a complete cone
Random orientation
Preferred orientation
Grain
https://www.youtube.com/watch?v=UfDW0-kghmI at~1:20

I
(110)
Random orientation
Preferred Orientation

Figure 1. X-ray diffraction -2 scan
profile of a PbTiO3 thin film grown
on MgO (001) at 600°C.
Figure 2. X-ray diffraction  scan
patterns from (a) PbTiO3 (101) and
(b) MgO (202) reflections.
Simple cubic
I
2

I I
20 30 40 50 60 70
PbTiO3 (PT)
simple tetragonal
(110)
(111)
Texture
PbTiO3 (001)  MgO (001)
highly c-axis
oriented
Random orientation
Preferred
orientation

By rotating the specimen about
three major axes as shown, these
spatial variations in diffraction
intensity can be measured.

4-Circle Goniometer
For pole-figure measurement
https://www.youtube.com/watch?v=R9o39StS5ik Goniometer Rotations for X-Ray
Crystallography
Pole figures displaying crystallographic texture of -TiAl in
an 2-gamma alloy, as measured by high energy X-rays.[
https://en.wikipedia.org/wiki/Pole_figure

In Situ XRD Studies
• Temperature
• Electric Field
• Pressure

High Temperature XRD Patterns of
Decomposition of YBa2Cu3O7-
T
2
I

In Situ X-ray Diffraction Study of an Electric
Field Induced Phase Transition
Single Crystal Ferroelectric
92%Pb(Zn1/3Nb2/3)O3 -8%PbTiO3
E=6kV/cm
E=10kV/cm
(330)
K1
K2
K1
K2
(330) peak splitting is due to
Presence of <111> domains
Rhombohedral phase
No (330) peak splitting
Tetragonal phase

Specimen Preparation
Double sided tape
Glass slide
Powders: 0.1m < particle size <40 m
Peak broadening less diffraction occurring
Bulks: smooth surface
after polishing, specimens should be
thermal annealed to eliminate any
surface deformation induced during
polishing.
http://www.youtube.com/watch?v=lwV5WCBh9a0 at~2:00-5:10

a b
Next Lecture
Transmission Electron Microscopy
Do review problems for XRD

Lecture presentation on x-ray diffraction.ppt

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