1. 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

2. X-ray and X-ray Diffraction
at~0:40-3:10
X-ray and X-ray Diffraction
X-ray was first discovered by W. C. Roentgen in Diffraction of X-ray was discovered by W.H. Bragg and W.L. Bragg in 1912
Bragg’s law: n=2dsin
X-rays were discovered in 1895 by the German physicist Roentgen and were so named because their nature was unknown at the time. Unlike ordinary light, these rays were invisible, but they traveled in straight lines and affected photographic film in the same way as light. On the other hand, they were much more penetrating than light and could easily pass through the human body, wood, quite thick pieces of metal and other “opaque” objects. X-rays are a form of electromagnetic radiation (like light); they are of higher energy, however, and can penetrate the body to form an image on film.
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 ( 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
h=4.136x10-15 eV sec.

4. A Modern Automated X-ray Diffractometer
X-ray Tube
Detector
at~6:00 how to operate XRD equipment
Sample stage
Cost: $560K to 1.6M

5. Production of X-rays - +
to~1:10 Production of X-rays
Production of X-rays
Cross section of sealed-off filament X-ray tube W
filament + -
target
X-rays
Vacuum
How X-ray works
The target is always water-cooled to prevent its heating since most of the kinetic energy of the electrons is converted into heat in the target, less than 1 percent being transformed into x-rays. The filament is heated by a filament current of ~3 amp and emits electrons which are rapidly drawn to the target by the high negative voltage between the cathode and anode. A small metal cup maintained at the same negative voltage as the filament repels the electrons and tends to focus them into a narrow region of the target. X-rays are emitted from the focal spot in all directions and escape from the tube through windows in the tube housing. Since the windows must be vacuum tight and yet highly transparent to x-rays, they are usually made of beryllium.
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.
How does X-ray tube work

6. 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 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
Bremsstrahlung
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.
Planck constant h=6.63x10-34 J.s and speed of light c=3.0x108 m/s
Electron charge e=1.6x10-19 C.
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
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 L K
K2
K1 K I II
III
K L M N L
subshells
K state
(shell)
Energy K K
Since the energy of the emitted x-ray photon is related to the difference in energy between the sharply defined levels of the atom, it is referred as a characteristic x-ray.
L state
K excitation L
M state
EK>EL>EM
EK>EK M
L excitation
N state
4/23/2017
ground state

10. Characteristic x-ray SpectraZ

11. Characteristic X-ray LinesK
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
<0.001Å
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
 is dependent on the density,  and thus / is a constant of the materials.
Mass absorption is also wavelength dependent. x x

13. Effect of , / (Z) and t on Intensity of Diffracted X-ray
incident beam
crystal
diffracted beam
film

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.

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.
                                                                                                         
Bragg’s Law:
n= 2dsin()
 ~ d
2B
atomic plane
In XRD, X-rays with ~0.5-2Å, is incident on a specimen and is diffracted by crystalline phase in specimen according to Bragg’s Law. The intensity of the diffracted X-ray is measured as a function of the diffraction angle and the specimen orientation.
XRD patterns will provide a lot of information about materials structure and properties.
X-ray of  B I d
to~1:17 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
For demonstration of constructive and destructive interference of waves.
Constructive Interference Destructive Interference
In Phase Out Phase

20. http://www.youtube.com/watch?v=UfDW0-kghmI at~3:00-6:00
Bragg’s Law and X-ray Diffraction How waves reveal the atomic structure of crystals
nl = 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 l
=3Å
=30o
Atomic
plane
d=3 Å
2-diffraction angle
When Lambda=0.15nm, path difference between X-rays 1 and 2 equals to nxLambda.
Braggs were awarded the Nobel Prize in physics in 1915 for their work in determining
Crystal structures beginning with NaCl, ZnS and diamond.
Bragg’s law can also be used to explain the interference pattern any beam, e.g., electrons, neutrons, ions and protons with a wavelength similar to the distance between the atomic or molecular structures of interest.
at~3:00-6:00

21. Deriving Bragg’s Law - nl = 2dsin
Constructive interference
occurs only when
nl = AB + BC
X-ray 2
X-ray 1
AB=BC
nl = 2AB
Sin=AB/d
AB=dsin
nl =2dsin
What is hkl, basics of crystallography
l=2dhklsinhkl
n – integer, called the order of diffraction

22. Basics of Crystallography
crystal packing in lattices
to~2:25
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.
The size and shape of the unit cell can be described by the three vectors, a, b and c.
also called the crystallographic axes of the cell. How to define atomic planes?
Lattice structures

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

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

25. Miller Indices - hkl Miller indices-the reciprocals of the
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
h k l
4Å 8Å 3Å
 8Å 
/ /3
h k l
Miller indices example crystallography for everyone

26. Planes and Spacings - a

27. Indexing of Planes and Directions
Directions in crystals
Indexing of Planes and Directions
(111) c c
[111]
(110) b b
[110] a a
[] square bracket, <> angular bracket, () parenthesis, {} brace
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.

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

29. XRD Pattern Significance of Peak Shape in XRD
Peak position
Peak width
Peak intensity I 2
XRD peak analysis

30. Peak Position Determine d-spacings and lattice parameters
Fix l (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
Because the value of sin changes very slows with  in the neighborhood of 90o. For this reason, a very accurate value of sin can be obtained from a measurement of  which is itself not particular precise, provided that  is near 90o or diffraction angle 2 is near 180o.
Note: Most accurate d-spacings are those calculated
from high-angle peaks.

31. Peak Intensity X-ray intensity: Ihkl  lFhkll2 Fhkl - Structure Factor
Determine crystal structure and atomic arrangement in a unit cell
X-ray intensity: Ihkl  lFhkll2
Fhkl - Structure Factor N
Fhkl =  fjexp[2i(huj+kvj+lwj)]
j=1
fj – atomic scattering factor
fj  Z, sin/
In order to describe the intensity of the diffraction peaks of various crystal plane of different unit cells, the Structure Factor is introduced.
Table of fj values, as a function of (sin/), for the elements and some ionic states of the elements can be found from references.
The summation is over the contents of the unit cell, i.e. the structure factor describes the collective scattering effect of all atoms in a unit cell. F depends on atomic arrangement in a unit cell and orientation of specimen. To calculate F, one has to know the number of atoms present in the unit cell and their locations.
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 Structures a = 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 atoms
[100] x
8 x 1/8 =1 8 x 1/8 + 1 = x 1/8 + 6 x 1/2 = 4
Location: 0,0,0 0,0,0, ½, ½, ½, ,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
d002 =
½ a
[010]
[010]
axis 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 diffractionz
Structure factor and intensity of diffraction
(001)
(002)
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.
FCC
d001
d002  1 2 3 1’ 2’ 3’

35. Structure factor and intensity of diffraction1 2 3 1’ 2’ 3’
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. l=2dhklsinhkl d002 d001 d001sin001=d002sin002 since d001=2d002
l=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 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 FCCz
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
Ihkl  lFhkll2
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
A(0,0,0), B(½,0,½),
C(½,½,0) & D(0,½,½)
e2ni = con(2n) + isin(2n) = 1

39. XRD Patterns of Simple Cubic and FCC2
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
The smaller FWHM, the better the peak location and phase identification.

42. Effect of Particle (Grain) Size
As rolled
300oC
As rolled t
Grain
size
200oC I
K1 B
K2
(FWHM)
250oC
Grain
size
450oC
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
Measure residual strains or stresses in samples.
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
Bragg’s Law
Basics of Crystallography
XRD Pattern
Powder Diffraction
Applications of XRD 45

46. Diffraction of X-rays by Crystals-Laue Method
Back-reflection Laue
crystal
Film
X-ray
[001]
Transmission Laue
crystal
Film
at~1:20-3:00

47. Powder Diffraction (most widely used)
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 d1 d2 2
Powder
sample d1
crystallite d2 d3
Polycrystalline
sample
at~1:20-1:56

48. Detection of Diffracted X-ray by A Diffractometer
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 
The focusing circle – X-ray source, sample and detector on same circle, detector moving through 2 theta and sample moving through theta. 2
at~1:44-1: and 15:44-16:16

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
Intensities are expressed as percentages of I1, the intensity of the strongest
line on the pattern. There are about 50,000 JCPDS cards of inorganic materials
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
The Pacific Northwest National Laboratory is developing a range of technologies to broaden the field of explosives detection. Phased contrast X-ray imaging, which uses silicon gratings to detect distortions in the X-ray wave front, may be applicable to mail or luggage scanning for explosives; it can also be used in detecting other contraband, small-parts inspection, or materials characterization. - Uploaded on Feb 9, 2010
phased contrast x-ray imaging
Determining strain pole figures from diffraction experiments

52. Phase Identification -Effect of Symmetry on XRD Pattern
a b c
Phase Identification -Effect of Symmetry on XRD Pattern
Cubic
a=b=c, (a)
b. Tetragonal
a=b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
at~1:20

55. Preferred Orientation (Texture)
(110)
Random orientation

56. Preferred Orientation (Texture)
Simple cubic
Random orientation
Texture
PbTiO3 (001)  MgO (001)
highly c-axis
oriented 2 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 600°C.
Figure 2. X-ray diffraction  scan patterns from (a) PbTiO3 (101) and (b) MgO (202) reflections.

57. Preferred Orientation (Texture)
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 
A pole figure is a graphical representation of the orientation of objects in space. For example, pole figures in the form of stereographic projections are used to represent the orientation distribution of crystallographic lattice planes in crystallography and texture analysis in materials science.
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-
Collecting diffraction patterns every few minutes. High temperature reactions can be mapped over hundreds of different temperatures in an overnight run. BaCuO2 start to form at ~950C. T 2

60. 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
(330) peak splitting is due to
Presence of <111> domains
Rhombohedral phase
No (330) peak splitting
Tetragonal phase
K1
K2
E=10kV/cm
K1
O-experimental data and --- fit data
K2

61. 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.
at~2:00-5:10

62. a b Do review problems for XRD Next Lecture
Transmission Electron Microscopy a b 62