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What is x ray diffraction properties and generation of x ray bragg s law basics of crystallography

Attention Please!

From: LAM, Mandi [mailto:[email protected]] Sent: Thursday, October 10, 2013 3:46 PMTo: Email ListSubject: Reminder: Completion of Teaching and Learning Questionnaire (TLQ) - AP 5301

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  • What is X-ray Diffraction

    Properties and generation of X-ray

  • Bragg’s Law

  • Basics of Crystallography

  • XRD Pattern

  • Powder Diffraction

  • Applications of XRD


Lecture 5 x ray diffraction xrd

Lecture-5 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/Default.htm


X ray and x ray diffraction

http://www.youtube.com/watch?v=vYztZlLJ3ds ~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.

http://www.youtube.com/watch?v=IRBKN4h7u80


Properties and generation of x ray

Properties and Generation of X-ray

  • X-rays are electromagneticradiation 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

http://www.youtube.com/watch?v=lwV5WCBh9a0 to~1:08

A Modern Automated X-ray Diffractometer

X-ray Tube

Detector

Sample stage

Cost: $560K to 1.6M


Production of x rays

http://www.youtube.com/watch?v=Bc0eOjWkxpU 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

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=UjyHK7jy1VwX-ray tube


X ray spectrum

http://www.youtube.com/watch?v=Bc0eOjWkxpUat~1:10-

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 spectrumcaused 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?v=3fe6rHnhkuY Bremsstrahlung

http://www.youtube.com/watch?v=n9FkLBaktEY characteristic X-ray


Short wavelength limit

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


Characteristic x ray spectra

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


Characteristic x ray spectra1

Characteristic x-ray Spectra

Z


Characteristic x ray lines

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

<0.001Å

K2

K

=2dsin

 (Å)

Spectrum of Mo at 35kV


Absorption of x ray

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

    whereI 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


Effect of z and t on intensity of diffracted x ray

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 ray1

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 coefficients of Pb, showing K and L absorption edges.


Filtering of x ray

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.”


Filtering of x ray1

Filtering of X-ray

K absorption edge of Ni

  • 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.

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.


What is diffraction

What Is Diffraction?

A wave interacts with

A single particle

The particle scatters the incident beam uniformly in alldirections.

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

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

B

X-ray of 

I

d

http://www.youtube.com/watch?v=1FwM1oF5e6o to~1:17 diffraction & interference


Constructive and destructive interference of waves

Constructive and Destructive Interference of Waves

Constructiveinterference 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 PhaseOut Phase

http://www.youtube.com/watch?v=kSc_7XBng8w


Bragg s law and x ray diffraction how waves reveal the atomic structure of crystals

http://www.youtube.com/watch?v=hQUsnMzTdpU

Bragg’s Law and X-ray DiffractionHow waves reveal the atomic structure of crystals

n-integer

Diffraction occurs only when Bragg’s Law is satisfied

Condition for constructive interference (X-rays 1 & 2) from planes with

spacing d

nl = 2dsin()

X-ray1

X-ray2

l

=3Å

=30o

Atomic

plane

d=3Å

2-diffraction angle

http://www.youtube.com/watch?v=UfDW0-kghmI at~3:00-5.50


Deriving bragg s law n l 2dsin

Deriving Bragg’s Law - nl = 2dsin

Constructive interference

occurs only when

nl = AB + BC

X-ray 1

X-ray 2

AB=BC

nl = 2AB

Sin=AB/d

AB=dsin

nl =2dsin

l=2dhklsinhkl

n – integer, called the order of diffraction


Basics of crystallography

http://www.youtube.com/watch?v=wJ1s-Ztxuzg crystal lattice

Basics of Crystallography

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?v=Rm-i1c7zr6Q&list=TLyPTUJ62VYE4wC1snHSChDl0NGo9IK-Nl


Seven crystal systems

Seven crystal Systems

SystemAxial lengths Unit cell

and angles

Rhombohedral

a=b=c

==90o

a

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

a

c

abc

===90o

Triclinic

abc

90o

c

a

a

b

b


Plane spacings for seven crystal systems

Plane Spacings for Seven Crystal Systems

1

hkl

hkl

hkl

hkl

hkl

hkl

hkl


Miller indices hkl

Miller Indices - hkl

Miller indices-the reciprocals of the

fractional intercepts which the plane

makes with crystallographic axes

(010)

a b c

a b c

Axial length4Å 8Å 3Å

Intercept lengths1Å 4Å 3Å

Fractional intercepts¼ ½ 1

Miller indices 4 2 1

h k l

4Å 8Å 3Å

 8Å 

/4 1 /3

0 1 0

h k l

http://www.youtube.com/watch?v=C9h1gLQmUto Miller indices of a given plane


Planes and spacings

Planes and Spacings

-

a

http://www.matter.org.uk/diffraction/geometry/planes_in_crystals.htm


Indexing of planes and directions

Indexing of Planes and Directions

(111)

c

c

[111]

(110)

b

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

X-ray Diffraction Pattern

BaTiO3 at T>130oC

(hkl)

Simple Cubic

I

40o

2

60o

20o

dhkl

Bragg’s Law:

l=2dhklsinhkl

l(Cu K)=1.5418Å


Xrd pattern significance of peak shape in xrd

XRD PatternSignificance of Peak Shape in XRD

  • Peak position

  • Peak width

  • Peak intensity

http://www.youtube.com/watch?v=nstYtUFELVQ


Peak position determine d spacings and lattice parameters

Peak PositionDetermined-spacings and lattice parameters

Fix l (Cu k)=1.54Å dhkl = 1.54Å/2sinhkl

For a simple cubic (a=b=c=a0)

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

Peak Intensity

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/

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 jthatom

in the unit cell


Scattering of x ray by an atom

Scattering of x-ray by an atom

  • x-ray also interact with the electrons in an atom through scattering, which may be understood as the redistribution of the x-ray energy spatially.

  • The Atomic Scattering Factor, f is defined to describe this distribution of intensity with respect to the scattered angle, .


Atomic scattering factor f

Atomic Scattering Factor - f

  • f is element-dependent and also dependent on the bonding state of the atoms.

  • This parameter influence directly the diffraction intensity.

  • Table of f values, as a function of (sin/), for the elements and some ionic states of the elements can be found from references.

I  f

Direction of incident beam

atom


Cubic structures a b c a

Cubic Structuresa = b = c = a

Simple Cubic Body-centered Cubic Face-centered Cubic

BCCFCC

[001]

z axis

a

a

[010]

y

a

1 atom2 atoms 4 atoms

[100]

x

8 x 1/8 =18 x 1/8 + 1 = 2 8 x 1/8 + 6 x 1/2 = 4

Location: 0,0,00,0,0,½, 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

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]

axis

[010]

a

BCCFCC

[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.


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.

1

1’

2

2’

3

3’

d001

d002

z

Structure factor and intensity of diffraction

(001)

(002)

FCC


Structure factor and intensity of diffraction1

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.

1

1’

2

2’

3

3’

d001

d002

Structure factor and intensity of diffraction

/4

/4

/2

/2


Structure factor and intensity of diffraction for fcc

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

D

B

y

A

C

x


Structure factor and intensity of diffraction for fcc1

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

Structure factor and intensity of diffraction for FCC

2i

Ihkl lFhkll2

A(0,0,0),B(½,0,½),

C(½,½,0) & D(0,½,½)


Xrd patterns of simple cubic and fcc

I

Simple Cubic

XRD Patterns of Simple Cubic and FCC

2

FCC

Diffraction angle 2 (degree)


Diffractions possibly present for cubic structures

Diffractions Possibly Present for Cubic Structures


Peak width full width at half maximum

Peak Width - Full Width at Half Maximum

(FWHM)

Determine

  • Particle or

    grain size

    2.Residual

    strain


Effect of particle grain size

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


Effect of lattice strain on diffraction peak position and width

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

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 orderbroad maxima

Monatomic gas

Atoms are arranged Scattering I

perfectly at random decreases with 


Diffraction of x rays by crystals laue method

Diffraction of X-rays by Crystals Laue Method

http://www.youtube.com/watch?v=UfDW0-kghmI at~1:20-3:00

Back-reflection Laue

crystal

Film

X-ray

[001]

Transmission Laue

Film

crystal

http://www.youtube.com/watch?v=2JwpHmT6ntU


Powder diffraction most widely used

Diffraction of X-rays by Polycrystals

2

2

Powder Diffraction (most widely used)

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?v=lwV5WCBh9a0 at~1:08-1:46


Detection of diffracted x ray by a diffractometer

X-ray

detector

Sample

holder

Detection of Diffracted X-ray by A Diffractometer

X-ray

tube

  • 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

2

http://www.youtube.com/watch?v=lwV5WCBh9a0 at~1:44-1:56 and15:44-16:16


Phase identification

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

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 other

data 8.data on specimen 9.data on diffraction pattern.


Other applications of xrd

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


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


Finding mass fraction of components in mixtures

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

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


Preferred orientation texture1

Preferred Orientation (Texture)

I

Simple cubic

Random orientation

Texture

20 30 40 50 60 70

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.


Preferred orientation texture2

Preferred Orientation (Texture)

By rotating the specimen about the three major axes as shown, these spatial variations in diffraction intensity can be measured.

4-Circle Goniometer

For pole-figure measurement


In situ xrd studies

In Situ XRD Studies

  • Temperature

  • Electric Field

  • Pressure


High temperature xrd patterns of decomposition of yba 2 cu 3 o 7

High Temperature XRD Patterns of Decomposition of YBa2Cu3O7-

I

T

2


In situ x ray diffraction study of an electric field induced phase transition

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

K2


Specimen preparation

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


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