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
http://www.youtube.com/watch?v=vYztZlLJ3ds
at~0:40-3:10
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.
Properties and Generation of X-ray
 X-rays are
electromagnetic
radiation with very
short wavelength (
10-8 -10-12 m)
 The energy of the xray can be calculated
with the equation
E = h = hc/
 e.g. the x-ray photon
with wavelength 1Å
has energy 12.5 keV
http://www.youtube.com/watch?v=lwV5WCBh9a0
to~1:08
A Modern Automated
X-ray Diffractometer
Detector
X-ray Tube
Sample stage
Cost: $560K to 1.6M
http://www.youtube.com/watch?v=Bc0eOjWkxpU
rays
to~1:10 Production of X-
Production of X-rays
Cross section of sealed-off filament X-ray tube
+
target
X-rays
W filament
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.
https://www.youtube.com/watch?v=3_bZCA7tlFQ How does X-ray tube work
http://www.youtube.com/watch?v=Bc0eOjWkxpU
at~1:06-3:10
X-ray Spectrum
 A spectrum of x-ray is
produced as a result of the
I
Mo
interaction between the
k
incoming electrons and the
characteristic
nucleus or inner shell
radiation
continuous
electrons of the target
radiation
element.
k
 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
http://www.youtube.com/watch?v=3fe6rHnhkuY
http://www.youtube.com/watch?v=n9FkLBaktEY
Bremsstrahlung
characteristic X-ray
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.
eV  h max 
 min
hc
min
hc 1.240 104 
 SW L 

A
eV
V
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
Excitation of K, L, M and N shells and
Formation of K to M Characteristic X-rays
 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).
M
K
EK>EL>EM
EK>EK
3/23/2016
K2
K1
K
 One of the outer electron falls
into the K-shell vacancy,
Energy
emitting the excess energy as
a x-ray photon.
 Characteristic x-ray energy:
Ex-ray=Efinal-Einitial
L
K excitation
K
L
M
N
I
II
III
L
subshells
K state
(shell)
K
K
L state
L
M state
L excitation
M
N state
ground state
Characteristic x-ray Spectra
Element
Z
K
K1
(weighted very strong,
average), Å
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 Lines
K
I
K1
<0.001Å
K
K and K2 will cause
Extra peaks in XRD
pattern, but can be
eliminated by adding
filters.
----- is the mass
absorption coefficient
K2
of Zr.
=2dsin
 (Å)
Spectrum of Mo at 35kV
Absorption of x-ray
• All x-rays are absorbed to some extent in
passing through matter due to electron
ejection or scattering.
• The absorption follows the equation
I  I 0 e  x  I 0 e




 x

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;
I0
II
,

(/) is the mass absorption coefficient (cm2/gm).
xx
Effect of , / (Z) and t on
Intensity of Diffracted X-ray
incident beam
crystal
diffracted beam
film
http://www.matter.org.uk/diffraction/x-ray/x_ray_diffraction.htm
Absorption of x-ray
• 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 xrays possess
sufficient energy to
eject an electron from
one of the shells.
Absorption edges
/

Absorption coefficients of Pb,
showing K and L absorption edges.
Filtering of X-ray
• The absorption behavior of x-ray by matter
can be used as a means for producing quasimonochromatic x-ray which is essential for
XRD experiments.
• The rule: “Choose for the filter an element
whose K absorption edge is just to the shortwavelength side of the K line of the target
material.”
Target
material
Filter
material
Ag
Mo
Cu
Ni
Co
Fe
Cr
Pd
Nb,
Zr
Ni
Co
Fe
Mn
V
Filtering of X-ray
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.
 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.
I  I 0 e  x  I 0 e




 x

K absorption
edge of Ni
Cu K 
1.5405Å

No filter
1.4881Å

Ni filter
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.
What Is Diffraction?
A wave interacts with
A single particle
The particle scatters the
incident beam uniformly
in all directions.
A crystalline material
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.
Bragg’s Law: n= 2dsin()
~d
2B
atomic plane
X-ray of  B
d
http://www.youtube.com/watch?v=1FwM1oF5e6o
I
to~1:17 diffraction & interference
Constructive and Destructive
Interference of Waves
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
Constructive Interference
In Phase
Destructive Interference
Out Phase
http://www.youtube.com/watch?v=kSc_7XBng8w
http://micro.magnet.fsu.edu/primer/java/interference/waveinteractions/index.html
http://www.eserc.stonybrook.edu/ProjectJava/Bragg/index.html
Bragg’s Law and X-ray Diffraction
How waves reveal the atomic structure of crystals
n = 2dsin()
n-integer
Diffraction occurs only when Bragg’s Law is satisfied
Condition for constructive interference (X-rays 1 & 2) from planes with
spacing d
X-ray1
X-ray2
=3Å

=30o
d=3 Å
Atomic
plane
2-diffraction angle
http://www.youtube.com/watch?v=UfDW0-kghmI
at~3:00-6:00
Deriving Bragg’s Law - n = 2dsin
Constructive interference X-ray 2 X-ray 1
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
http://www.youtube.com/watch?v=Mm-jqk1TeRY crystal packing in lattices
to~2:25
Basics of Crystallography
smallest building block
c
Single crystal
CsCl
z [001]
d3

b
a 
Unit cell (Å)
y [010]
x [100] crystallographic
axes
d1
Lattice
d2
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.
http://www.youtube.com/watch?v=Rm-i1c7zr6Q&list=TLyPTUJ62VYE4wC1snHSChDl0NGo9IK-Nl
Lattice structures
Seven crystal Systems
System
Axial lengths
and angles
Cubic
Tetragonal
Unit cell
a
a=b=c
===90o
Rhombohedral
a=b=c
==90o
a
a=bc
===90o
c
c
b
c
a
Monoclinic
abc
==90o c
a
Orthorhombic
abc
===90o
Hexagonal
a=bc
=90o
=120o
a
Triclinic
abc
90o
b
a
c
b
a
Plane Spacings for Seven
Crystal Systems
h
1kl
hkl
hkl
hkl
hkl
hkl
hkl
Miller Indices - hkl
Miller indices form a notation system in crystallography for planes in crystal lattices.
Miller indices-the reciprocals of the
fractional intercepts which the plane
makes with crystallographic axes
(010)
a
Axial length
4Å
Intercept lengths
1Å
Fractional intercepts ¼
Miller indices
4
h
b
c
8Å 3Å
4Å 3Å
½ 1
2 1
k
l
https://www.youtube.com/watch?v=enVpDwFCl68
a
b
4Å 8Å
 8Å
/4 1
0
1
h
k
c
3Å

/3
0
l
Miller indices example crystallography for everyone
Planes and Spacings
-a
http://www.matter.org.uk/diffraction/geometry/planes_in_crystals.htm
http://www.youtube.com/watch?v=9Rjp9i0H7GQ
Directions in crystals
Indexing of Planes and Directions
c
(111)
[111]
b
c
(110)
b
[110]
a
a
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.
X-ray Diffraction Pattern
(hkl)
BaTiO3 at T>130oC
Simple Cubic
I
20o
Bragg’s Law:
40o 2
dhkl
=2dhklsinhkl
60o
(Cu K)=1.5418Å
XRD Pattern
Significance of Peak Shape in XRD
1.Peak position
2.Peak width
3.Peak intensity
I
2
http://www.youtube.com/watch?v=MU2jpHg2vX8
XRD peak analysis
Peak Position
Determine d-spacings and lattice parameters
Fix  (Cu k)=1.54Å
dhkl = 1.54Å/2sinhkl
For a simple cubic (a=b=c=a0)
2
a0 = dhkl (h2+k2+l2)½
e.g., for BaTiO3, 2220=65.9o, 220=32.95o,
d220 =1.4156Å, a0=4.0039Å
Note: Most accurate d-spacings are those calculated
from high-angle peaks.
Peak Intensity
Determine crystal structure and atomic arrangement
in a unit cell
2
X-ray intensity: Ihkl  lFhkll
Fhkl - Structure Factor
N
Fhkl =  fjexp[2i(huj+kvj+lwj)]
j=1
fj –
atomic scattering factor
fj  Z, sin/
Low Z elements may be difficult to detect by XRD
N – number of atoms in the unit cell,
uj,vj,wj - fractional coordinates of the jth atom
in the unit cell
Cubic Structures
a=b=c=a
Body-centered Cubic
BCC
Simple Cubic
[001]
z axis
Face-centered Cubic
FCC
a
a
y
1 atom
[100]
x
[010]
a
2 atoms
8 x 1/8 =1
Location:
0,0,0
8 x 1/8 + 1 = 2
0,0,0, ½, ½, ½,
8 unit cells
4 atoms
8 x 1/8 + 6 x 1/2 = 4
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
Structures of Some Common Metals
 = 2dhklsinhkl
[001] axis
(001) plane
d010
Mo
Cu
a
d001
(002)
d002 = ½ a
[100]
a
a
[010]
BCC
FCC
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
[010]
axis
z
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.
(001)
(002)
FCC
3
d001
2
1
1’
2’
3’

d002
Structure factor and
intensity of diffraction
3
d001
2
1
1’
2’
3’
/4 d002
/4
/2
/2
 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.
http://emalwww.engin.umich.edu/education_materials/microscopy.html
=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
2
1
2’
2
002
/2
3
d001
3
1’
1’
/4
2’
001
3’
002
001
When =001 no diffraction occurs, while 
increases to 002, diffraction occurs.
d002
3’
Structure factor and intensity
of diffraction for FCC
z
 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)
D
B
y
A
x
C
Structure factor and intensity of
i 2 hu  kv  lw 
diffraction for FCC F     f j   e 2i
j
Ihkl  lFhkll2
j
j
For a certain set of plane, (hkl)
A(0,0,0), B(½,0,½),
F =  f () exp[2i(hu+kv+lw)]
C(½,½,0) & D(0,½,½)
= 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
4 f h, k, l all odd or all even
F 
h, k, l mixed
0
j
I
XRD
Patterns of
Simple
Cubic and
FCC
Simple Cubic
2
FCC
Diffraction angle 2 (degree)
Diffractions Possibly Present for
Cubic Structures
h2 + k2 + l2
1
2
3
4
5
6
7
8
9
10
11
12
simple cubic
(any
combination)
100
110
111
200
210
211
220
300, 221
310
311
222
FCC
(either all odd
or all even)
111
200
220
311
222
BCC
(h + k + l) is
even
110
200
211
220
310
222
Peak Width - Full Width at Half Maximum
(FWHM)
Determine
1. Particle or
grain size
2. Residual
strain
Effect of Particle (Grain) Size
As rolled
I
300oC
Grain
size
450oC
Grain
size
200oC
250oC
t
As rolled
K1 B
K2
(FWHM)
300oC
0.9
B = t cos
450oC
2
(331) Peak of cold-rolled and
annealed 70Cu-30Zn brass
Peak
broadening
As grain size decreases
hardness increases and
peak become broader
Effect of Lattice Strain
on Diffraction Peak
Position and Width
No Strain
Uniform Strain
(d1-do)/do
Peak moves, no shape changes
Non-uniform Strain
d1constant
Peak broadens
XRD patterns from
other states of matter
Crystal
Constructive interference
Diffraction
Sharp maxima
Structural periodicity
Liquid or amorphous solid
Lack of periodicity
Short range order
One or two
broad maxima
Monatomic gas
Atoms are arranged
perfectly at random
Scattering I
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
crystal
X-ray
Film
[001]
Transmission Laue
crystal
Film
http://www.youtube.com/watch?v=UfDW0-kghmI at~1:20-3:00
http://www.youtube.com/watch?v=2JwpHmT6ntU
Diffraction of X-rays by Polycrystals
Powder Diffraction (most widely used)
A powder sample is in fact an assemblage of small
crystallites, oriented at random in space.
d3
d2
d1
d2
2
d3
d1
Powder
sample
crystallite
2
Polycrystalline
sample
http://www.youtube.com/watch?v=lwV5WCBh9a0
at~1:20-1:56
Detection of Diffracted X-ray
X-ray
detector
by A Diffractometer
 x-ray detectors (e.g. Geiger
X-ray
tube
counters) is used instead of
the film to record both the
position and intensity of the
x-ray peaks
 The sample holder and the xray detector are mechanically
linked
 If the sample holder turns ,
the detector turns 2, so that
2
the detector is always ready to
detect the Bragg diffracted
x-ray
http://www.youtube.com/watch?v=lwV5WCBh9a0
at~1:44-1:56
Sample
holder

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
Quality of data
1.file number 2.three strongest lines
3.lowest-angle line 4.chemical formula and name 5.data on diffraction method used 6.crystallographic data 7.optical and other
data 8.data on specimen 9.data on diffraction pattern.
Other Applications of XRD
XRD is a nondestructive technique
• To identify crystalline phases
• To determine structural properties:
Lattice parameters (10-4Å), 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
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
a
b
c
Phase Identification
-Effect of Symmetry
on XRD Pattern
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
Grain
diffraction cone will not be a
complete cone
Random orientation
Preferred orientation
https://www.youtube.com/watch?v=UfDW0-kghmI
at~1:20
Preferred Orientation (Texture)
Preferred Orientation
I
(110)
Random orientation
Preferred Orientation (Texture)
Simple cubic
I
Random orientation
Texture
20
PbTiO3 (001)  MgO (001)
highly c-axis
I
oriented
PbTiO3 (PT)
simple tetragonal
Preferred
orientation
30
40
2
50
60
70
I
(110)
(111)

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.
Preferred Orientation (Texture)
https://www.youtube.com/watch?v=R9o39StS5ik
Goniometer Rotations for X-Ray Crystallography
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://en.wikipedia.org/wiki/Pole_figure
Pole figures displaying crystallographic texture of -TiAl in
an 2-gamma alloy, as measured by high energy X-rays.[
In Situ XRD Studies
• Temperature
• Electric Field
• Pressure
High Temperature XRD Patterns of
Decomposition of YBa2Cu3O7-
I
T
2
In Situ X-ray Diffraction Study of an Electric
Field Induced Phase Transition
(330)
Single Crystal Ferroelectric
92%Pb(Zn1/3Nb2/3)O3 -8%PbTiO3
E=6kV/cm
K1
K2
E=10kV/cm
K1
K2
(330) peak splitting is due to
Presence of <111> domains
Rhombohedral phase
No (330) peak splitting
Tetragonal phase
Specimen Preparation
Powders:
0.1m < particle size <40 m
Peak broadening
less diffraction occurring
Double sided tape
Glass slide
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
Do review problems for XRD
Next Lecture
a
Transmission Electron
Microscopy
b