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