Post on 31-Dec-2015
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
continuousradiation
characteristicradiation
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.
minmax
hcheV
A10240.1 4
min VeV
hcSWL
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
04/19/23
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 MN
K
KL
Energy K state(shell)
L state
M state
N state
ground state
K
K
L
L
K1
K2
I II III
M
subshells
EK>EL>EM
EK>EK
K excitation
L excitation M
Element
K
(weightedaverage), Å
K1
very strong,Å
K2
strong, ÅK
weak, ÅK
Absorptionedge, Å
Excitationpotential
(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
K1
K
K
I
(Å)
<0.001Å
K2
K and K2 will causeExtra 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
xx eIeII
00
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.”
Targetmaterial
Ag Mo Cu Ni Co Fe Cr
Filtermaterial
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
xx eIeII
00
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
2B
I
http://www.youtube.com/watch?v=1FwM1oF5e6o to~1:17 diffraction & interference
Constructive and Destructive Interference of Waves
Constructive Interference Destructive InterferenceIn 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_7XBng8whttp://micro.magnet.fsu.edu/primer/java/interference/waveinteractions/index.html
Bragg’s Law and X-ray DiffractionHow waves reveal the atomic structure of crystals
n = 2dsin()
Atomicplane
d=3 Å
=3Å=30o
n-integer
X-ray1
X-ray2
2-diffraction angle
Diffraction occurs only when Bragg’s Law is satisfiedCondition 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 1X-ray 2Constructive interferenceoccurs only when
n= AB + BC
AB=BC
n= 2AB
Sin=AB/d
AB=dsin
n =2dsin
=2dhklsinhkl
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
ab
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
cTetragonal
a=bc===90o
ba
cOrthorhombic
abc===90o
a
Rhombohedral
a=b=c==90o
Hexagonala=bc=90o =120o
c
ab
Monoclinicabc==90o
ba
cTriclinicabc90o
c
Seven crystal Systems
a
Plane Spacings for Seven Crystal Systems
1hkl
hkl
hkl
hkl
hkl
hkl
hkl
Miller Indices - hkl
(010)
Miller indices-the reciprocals of thefractional intercepts which the planemakes with crystallographic axes
Axial length 4Å 8Å 3ÅIntercept lengths 1Å 4Å 3ÅFractional intercepts ¼ ½ 1Miller indices 4 2 1
h k l
4Å 8Å 3Å 8Å /4 1 /3 0 1 0 h k l
a b ca 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
ab
c
ab
c(111)
[110]
a direction [uvw]a set of equivalentdirections <uvw><100>:[100],[010],[001][100],[010] and [001]
a plane (hkl)a set of equivalentplanes {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
=2dhklsinhklBragg’s Law: (Cu K)=1.5418Å
BaTiO3 at T>130oC
dhkl
20o 40o 60o
(hkl)
XRD PatternSignificance 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 PositionDetermine d-spacings and lattice parameters
Fix (Cu k)=1.54Å dhkl = 1.54Å/2sinhkl
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, 2220=65.9o, 220=32.95o,
d220 =1.4156Å, a0=4.0039Å
2
Peak Intensity
X-ray intensity: Ihkl lFhkll2
Fhkl - Structure Factor
Fhkl = fjexp[2i(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 jth 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 Structuresa = 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
aa
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
aa
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
= 2dhklsinhkl
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
12
31’
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
12
31’
2’3’
/2 /2
/4 /4
http://emalwww.engin.umich.edu/education_materials/microscopy.html
d001
d002
=2dhklsinhkld001sin001=d002sin002 since d001=2d002
If sin002=2sin001 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
yA
B
C
D
For a certain set of plane, (hkl)F = f () exp[2i(hu+kv+lw)] = f () exp[2i(hu+kv+lw)] = f (){exp[2i(0)] + exp[2i(h/2 + l/2)] + exp[2i(h/2 + k/2)] + exp[2i(k/2 + l/2)]} = f (){1 + ei(h+k) + ei(k+l) + ei(l+h)}Since e2ni = 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 lk,h,0
evenalloroddalllk,h,4 fF
jjj lwkvhui
jj efF
2
A(0,0,0), B(½,0,½),
C(½,½,0) & D(0,½,½)
2i
Ihkl lFhkll2
XRD Patterns of
Simple Cubic and
FCC
Diffraction angle 2 (degree)
I Simple Cubic
FCC
2
h2 + k2 + l2simple cubic
(anycombination)
FCC(either all odd
or all even)
BCC(h + k + l) is
even1 100 - -2 110 - 1103 111 111 -4 200 200 2005 210 - -6 211 - 2117 - - -8 220 220 2209 300, 221 - -10 310 - 31011 311 311 -12 222 222 222
Diffractions Possibly Present for Cubic Structures
Peak Width - Full Width at Half Maximum(FWHM)
1. Particle orgrain size
2. Residualstrain
Determine
Effect of Particle (Grain) Size
(331) Peak of cold-rolled andannealed 70Cu-30Zn brass
2
I
K1K2
As rolled
200oC
250oC
300oC
450oC
As rolled 300oC
450oC
Grain size
t
B = 0.9
t cos Peakbroadening
As grain size decreaseshardness increases andpeak 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 d1constant
Peak moves, no shape changes
Peak broadens
XRD patterns from other states of matter
Constructive interferenceStructural periodicity
DiffractionSharp maxima
Crystal
Liquid or amorphous solid
Lack of periodicity One or twoShort range order broad maxima
Monatomic gas
Atoms are arranged Scattering Iperfectly 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
FilmX-ray
crystal
crystal Film
Transmission Laue
[001]
http://www.youtube.com/watch?v=UfDW0-kghmI at~1:20-3:00http://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.
22
Polycrystallinesample
Powdersample
crystallite
Diffraction of X-rays by Polycrystals
http://www.youtube.com/watch?v=lwV5WCBh9a0 at~1:20-1:56
d1
d3
d2
d1d2
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-raytube
X-raydetector
Sampleholder
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 otherdata 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 (10-4Å), 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?v=CpJZfeJ4poE phased contrast x-ray imaging
https://www.youtube.com/watch?v=6POi6h4dfVsDetermining 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=bc (a and c)
c. Orthorhombic abc (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 orientationPreferred orientation
Grain
https://www.youtube.com/watch?v=UfDW0-kghmI at~1:20
Preferred Orientation (Texture)
I
(110)
Random orientation
Preferred Orientation
Preferred Orientation (Texture)
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 cubicI
2
I I
20 30 40 50 60 70
PbTiO3 (PT)simple tetragonal
(110)
(111)
TexturePbTiO3 (001) MgO (001)highly c-axisoriented
Random orientation
Preferred orientation
By rotating the specimen about three major axes as shown, these spatial variations in diffraction intensity can be measured.
Preferred Orientation (Texture)
4-Circle GoniometerFor 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)
K1
K2
K1
K2
(330) peak splitting is due toPresence of <111> domains
Rhombohedral phase
No (330) peak splitting
Tetragonal phase
Specimen Preparation
Double sided tape
Glass slide
Powders: 0.1m < 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