X-ray Diffraction (XRD)

1
X-ray Diffraction (XRD)

• What is X-ray Diffraction
• Properties and generation of X-ray
• Braggs Law
• Basics of Crystallography
• XRD Pattern
• Powder Diffraction
• Applications of XRD

http//www.matter.org.uk/diffraction/x-ray/default
.htm
2
X-ray and X-ray Diffraction
http//www.youtube.com/watch?vvYztZlLJ3ds
at040-310
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 Braggs law
n?2dsin?



Photograph of the hand of an old man using X-ray.
3
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

4
A Modern Automated X-ray Diffractometer
http//www.youtube.com/watch?vlwV5WCBh9a0
to108
X-ray Tube
Detector
Sample stage
Cost 560K to 1.6M
5
Production of X-rays
http//www.youtube.com/watch?vBc0eOjWkxpU
to110 Production of X-rays
Cross section of sealed-off filament X-ray tube
W
filament
-

target
X-rays
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?v3_bZCA7tlFQ How
does X-ray tube work
6
X-ray Spectrum
http//www.youtube.com/watch?vBc0eOjWkxpU at106
-310

• 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.

I
Mo
k?
characteristic radiation
continuous radiation
k?
?
SWL - short-wavelength limit
http//www.youtube.com/watch?v3fe6rHnhkuY
Bremsstrahlung
http//www.youtube.com/watch?vn9FkLBaktEY
characteristic X-ray
7
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.

V applied voltage
8
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
9
Excitation of K, L, M and N shells and Formation
of K? to M? Characteristic X-rays
M?

• 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-rayEfinal-Einitial

L?
K?
K?2
K?1
K?
I II III
K L M N
L
subshells
K state (shell)
Energy
K?
K?
L state
K excitation
L?
M state
EKgtELgtEM EK?gtEK?
M?
L excitation
N state
ground state
10
Characteristic x-ray Spectra
Z
11
Characteristic X-ray Lines
K?
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.
I
K?1
lt0.001Å
K?2
K?
?2dsin?
? (Å)
Spectrum of Mo at 35kV
12
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
• 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).

I0
I
?, ?
I
x
x
13
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_d
iffraction.htm
14
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 x-rays possess
  sufficient energy to eject an electron from one
  of the shells.

Absorption edges
?/?
?
Absorption coefficients of Pb, showing K and L
absorption edges.
15
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.

16
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.

K absorption edge of Ni
Cu K ? 1.5405Å
1.4881Å
?
?
No filter 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.
17
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
18
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.
n? 2dsin(?)
Braggs Law
? d
2?B
atomic plane
?B
X-ray of ?
I
d
http//www.youtube.com/watch?v1FwM1oF5e6o
to117 diffraction interference
19
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 Destructive
Interference In Phase Out Phase
http//www.youtube.com/watch?vkSc_7XBng8w http//
micro.magnet.fsu.edu/primer/java/interference/wave
interactions/index.html
20
Braggs Law and X-ray DiffractionHow waves
reveal the atomic structure of crystals
http//www.eserc.stonybrook.edu/ProjectJava/Bragg/
index.html

• nl 2dsin(?)

n-integer
Diffraction occurs only when Braggs Law is
satisfied Condition for constructive interference
(X-rays 1 2) from planes with spacing d
X-ray1
X-ray2
l
?3Å
?30o
Atomic plane
d3 Å
2?-diffraction angle
http//www.youtube.com/watch?vUfDW0-kghmI
at300-600
21
Deriving Braggs Law - nl 2dsin?
X-ray 1
Constructive interference occurs only when
nl AB BC
X-ray 2
ABBC nl 2AB Sin?AB/d ABdsin? nl 2dsin?
l2dhklsin?hkl
n integer, called the order of diffraction
22
Basics of Crystallography
http//www.youtube.com/watch?vMm-jqk1TeRY
crystal packing in lattices
to225
smallest building block
c
Single crystal
d3
CsCl

?
?
b
?
a
Unit cell (Å)
z 001
d1
y 010
Lattice
d2
x 100
crystallographic axes
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?vRm-i1c7zr6QlistTL
yPTUJ62VYE4wC1snHSChDl0NGo9IK-Nl
Lattice structures
23
Seven crystal Systems
System Axial lengths Unit cell and
angles
Rhombohedral
a
abc ????90o
Cubic
abc ???90o
a
Hexagonal

• ab?c
• ?90o
• ?120o

c
Tetragonal
c
ab?c ???90o
a
Monoclinic
a
a?b?c ??90o??
c
b
Orthorhombic
c
a
a?b?c ???90o
Triclinic
c
a?b?c ??????90o
a
a
b
b
24
Plane Spacings for Seven Crystal Systems
1
hkl
hkl
hkl
hkl
hkl
hkl
hkl
25
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 b c
a b c
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
https//www.youtube.com/watch?venVpDwFCl68
Miller indices example crystallography for
everyone
26
Planes and Spacings
-
a
http//www.matter.org.uk/diffraction/geometry/plan
es_in_crystals.htm
27
Indexing of Planes and Directions
http//www.youtube.com/watch?v9Rjp9i0H7GQ
Directions in crystals
(111)
c
c
111
(110)
b
b
110
a
a
a direction uvw a set of equivalent directions
ltuvwgt lt100gt100,010,001 100,010 and
001
a plane (hkl) a set of equivalent planes
hkl 110(101),(011),(110) (101),(101),(101),et
c.
28
X-ray Diffraction Pattern
BaTiO3 at Tgt130oC
(hkl)
Simple Cubic
I
2?
20o
40o
60o
dhkl
l2dhklsin?hkl
Braggs Law
l(Cu K?)1.5418Å
29
XRD PatternSignificance of Peak Shape in XRD

1. Peak position
2. Peak width
3. Peak intensity

I
2?
http//www.youtube.com/watch?vMU2jpHg2vX8 XRD
peak analysis
30
Peak PositionDetermine d-spacings and lattice
parameters
Fix l (Cu k?)1.54Å dhkl 1.54Å/2sin?hkl
For a simple cubic (abca0)
2?
a0 dhkl (h2k2l2)½
e.g., for BaTiO3, 2?22065.9o, ?22032.95o,
d220 1.4156Å, a04.0039Å
Note Most accurate d-spacings are those
calculated from high-angle peaks.
31
Peak Intensity
Determine crystal structure and atomic
arrangement in a unit cell
X-ray intensity Ihkl ? lFhkll2
Fhkl - Structure Factor
N
Fhkl ? fjexp2?i(hujkvjlwj)
j1
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
32
Cubic Structuresa b c a
Simple Cubic Body-centered Cubic
Face-centered Cubic BCC FCC
001 z axis
a
a
010 y
a
1 atom 2 atoms 4 atoms
100 x
8 x 1/8 1 8 x 1/8 1 2 8 x 1/8
6 x 1/2 4
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
33
Structures of Some Common Metals
001 axis
l 2dhklsin?hkl
(001) plane
d010
Mo
Cu
a
d001
(010) plane
(002)
a
½ a
d002
010 axis
010
a
BCC FCC
100
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.
34
Structure factor and intensity of diffraction
z
(001)
(002)

• Sometimes, even though the Braggs 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.

FCC
35
Structure factor and intensity of diffraction

• 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.

?/4
?/4
?/2
?/2
36
http//emalwww.engin.umich.edu/education_materials
/microscopy.html
l2dhklsin?hkl
d001sin?001d002sin?002 since d0012d002 If
sin?0022sin?001 i.e., ?002gt?001 Braggs law
holds and (002) diffraction peak appears
1
1
1
1
2
2
2
2
?001
?002
d002
?/2
3
3
3
3
?/4
d001
?002
?001
When ??001 no diffraction occurs, while ?
increases to ?002, diffraction occurs.
37
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
C
x
38
Structure factor and intensity of diffraction for
FCC
2?i

Ihkl ? lFhkll2

• For a certain set of plane, (hkl)
• F ? f (?) exp2?i(hukvlw)
• f (?) ? exp2?i(hukvlw)
• f (?)exp2?i(0) exp2?i(h/2 l/2)
• exp2?i(h/2 k/2) exp2?i(k/2
  l/2)
• f (?)1 e?i(hk) e?i(kl) e?i(lh)
• Since e2n?i 1 and e(2n1)?i -1,
• ?if h, k l are all odd or all even, then (hk),
  (kl), and (lh) are all even and F 4f
  otherwise, F 0

A(0,0,0), B(½,0,½), C(½,½,0) D(0,½,½)
39
XRD Patterns of Simple Cubic and FCC
I
Simple Cubic
2?
FCC
Diffraction angle 2? (degree)
40
Diffractions Possibly Present for Cubic Structures
41
Peak Width - Full Width at Half Maximum
(FWHM)
Determine

• Particle or
• grain size
• 2. Residual
• strain

42
Effect of Particle (Grain) Size
As rolled
300oC
As rolled
t
Grain size
200oC
I
K?1
B
K?2
(FWHM)
250oC
450oC
Grain size
300oC
0.9?
Peak broadening
B
t cos?
450oC
As grain size decreases hardness increases
and peak become broader
2?
(331) Peak of cold-rolled and annealed 70Cu-30Zn
brass
43
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
44
XRD patterns from other states of matter
Crystal
Constructive interference Structural periodicity
Diffraction Sharp maxima
2?
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 ?
45
X-ray Diffraction (XRD)

• What is X-ray Diffraction
• Properties and generation of X-ray
• Braggs Law
• Basics of Crystallography
• XRD Pattern
• Powder Diffraction
• Applications of XRD

http//www.matter.org.uk/diffraction/x-ray/laue_me
thod.htm
46
Diffraction of X-rays by Crystals-Laue Method
Back-reflection Laue
crystal
Film
X-ray
001
Transmission Laue
Film
crystal
http//www.youtube.com/watch?vUfDW0-kghmI
at120-300
http//www.youtube.com/watch?v2JwpHmT6ntU
47
Powder Diffraction (most widely used)
Diffraction of X-rays by Polycrystals

• A powder sample is in fact an assemblage of small
  crystallites, oriented at random in space.

d3
d1
d2
Powder sample
d1
crystallite
d2
d3
Polycrystalline sample
http//www.youtube.com/watch?vlwV5WCBh9a0
at120-156
48
Detection of Diffracted X-ray by A Diffractometer
X-ray detector
Sample holder

• 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
?
2?
http//www.youtube.com/watch?vlwV5WCBh9a0
at144-156 and 1544-1616
49
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).

50
JCPDS Card
Quality of data
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.
51
Other Applications of XRD

• 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

XRD is a nondestructive technique
https//www.youtube.com/watch?vCpJZfeJ4poE
phased contrast x-ray imaging
https//www.youtube.com/watch?v6POi6h4dfVs Determ
ining strain pole figures from diffraction
experiments
52
Phase Identification -Effect of Symmetry on
XRD Pattern
a b c

• Cubic
• abc, (a)
• b. Tetragonal
• ab?c (a and c)
• c. Orthorhombic
• a?b?c (a, b and c)

2?

• Number of reflection
• Peak position
• Peak splitting

53
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 ?
54
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

Grain
Random orientation Preferred orientation
https//www.youtube.com/watch?vUfDW0-kghmI
at120
55
Preferred Orientation (Texture)
Preferred Orientation
I
(110)
Random orientation
56
Preferred Orientation (Texture)
Simple cubic
I
Random orientation
Texture
20 30 40 50
60 70
2?
PbTiO3 (001) ?? MgO (001) highly c-axis oriented
I
I
(110)
PbTiO3 (PT) simple tetragonal
(111)
Preferred orientation
?
Figure 1. X-ray diffraction ?-2? scan profile of
a PbTiO3 thin film grown on MgO (001) at 600C.
Figure 2. X-ray diffraction ? scan patterns from
(a) PbTiO3 (101) and (b) MgO (202) reflections.
57
Preferred Orientation (Texture)
https//www.youtube.com/watch?vR9o39StS5ik
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.
58
In Situ XRD Studies

• Temperature
• Electric Field
• Pressure

59
High Temperature XRD Patterns of Decomposition of
YBa2Cu3O7-?
I
T
2?
60
In Situ X-ray Diffraction Study of an Electric
Field Induced Phase Transition
(330)
Single Crystal Ferroelectric 92Pb(Zn1/3Nb2/3)O3
-8PbTiO3
E6kV/cm
(330) peak splitting is due to Presence of lt111gt
domains Rhombohedral phase No (330) peak
splitting Tetragonal phase
K?1
K?2
E10kV/cm
K?1
K?2
61
Specimen Preparation
Powders 0.1?m lt particle size lt40 ?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?vlwV5WCBh9a0
at200-510
62
Do review problems for XRD
Next Lecture Transmission Electron Microscopy
a
b