X-ray Diffraction
Tohoku University
Masaki Takata

courtesy ofProf. Brendan J. Kennedy
School of Chemistry
The University of Sydney
brendan.kennedy@sydney.edu.auTypes of Forces
1) Strong forces
2) Weak forces
3) Electromagnetic forces
4) Gravity

Types of Matter
1) Atoms
2) Molecules
3) Crystals
4) Particles, Solids, Surfaces, Liquids, Glasses,
Gases

Electromagnetic Radiation at a
Synchrotron
Energy transmitted in the form of waves or particles
γ-rays

X-rays

UV

Vis

IR

μ-waveshigher energy  lower energy
For particles:
de Broglie wavelength = Planck’s const./momentum [λ = h/p]
For an electron accelerated through 100 Volts, λ = 1.2 Å,Why is wavelength important?
To probe a sample you need radiation with a wavelength of similar, or
smaller, magnitude to the size of the “object” under investigation.

samplesampleVisible light

X-raysTo investigate atomic/molecular structure, we use X-rays, electrons,
and neutrons, since they can have wavelengths about the sizes ofatoms.

X-rays• Hard X-ray wavelengths ~ 0.1 Å to 60 Å
• Soft X-ray wavelengths ~ 60 Å to 120 Å• Interact weakly, ie. penetrating – therefore, can see
inside of a structure, but most of the beam passes
through, unperturbed.
• 10 x more photons are absorbed than scattered.
• X-rays scatter off of valence electrons (~ 1% off nuclei).

Interaction of X-rays and Matter

Scattering.
coherent, incoherent
elastic (Thomson), inelastic (Compton)
Absorption.
atoms: can then be emitted as fluorescence, photoelectrons, Auger
electrons
molecules: can emit fluorescence, phosphorescence, transfer heat,(stimulated emission)
Diffraction. The bending of waves due to obstructions and small apertures, aswith crystals.
Refraction. The bending of a wave as it passes from one medium to another
Reflection. Radiation bouncing back from one medium to the original medium,
where the wavelength << size of the object.

Photoelectric EffectIonization occurs when a light has sufficient energy to eject
an electron. The kinetic energy of the emitted electron isdiagnostic of the orbital from which it was ejected. Allows
Photoelectron spectroscopy.
e-

e-

e-

hυ
Ehυ = electron kinetic energy + electron binding energy
Valence spectroscopy: information on bondingCore spectroscopy: qualitative and quantitative analysis,“chemical shift”

X-ray Scattering
Coherent scattering occurs when the Xray particle collides with an atom anddeviates without a loss in energy.

Coherent

Incoherent scattering is where theincident X-ray loses some of its energy to
the scattering electron. As total momentum
is preserved, the wavelength of thescattered photon is increased.

IncoherentCompton Scattering
Compton scattering occurs when the incident x-ray photon is deflectedfrom its original path by an interaction with an electron. The electron isejected from its orbital position and the x-ray photon loses energy
because of the interaction but continues to travel through the material
along an altered path. Energy and momentum are conserved in thisprocess. The energy shift depends on the angle of scattering and not on
the nature of the scattering medium.

 h 
scattered  initial   (1  cos  ) me c 
where h is Plancks constant; me the mass of the electron; c the speed oflight; and θ the angle the photon is deflected by.
Since the scattered x-ray photon has less energy, it has a longer
wavelength and less penetrating than the incident photon.

Thompson Scattering•

An electron will oscillate in phase with an x-ray beam according to thefollowing equation:

2

2 
2

I0 e
1  cos (2 )I 2 
2
2
r  me c 
where I0 is the intensity of the incident beam; e the charge on theelectron; me the mass of the electron; c the speed of light; andr the distance from the scattering electron to the detector.
Clearly (by the second term) the scattered energy from a single electron
is quite low. Third term, involving the cosine function, is called thepolarisation factor because it indicates that the incoming non-polarised xray is polarised by the scattering process, resulting in a directionalvariation in the scattered intensity.

Absorption
When X-rays pass through a material, the intensity isattenuated (decreased) through absorption by the elementsin the material.
Attenuation is due to two effects:
Compton scattering (σ) of the beam, which diverts
photons in directions different from the primary beam:
Photoelectric absorption (τ), which produces
fluorescence or Auger electrons.

Absorption
These two effects may be combined into a single bulk mass
absorption coefficient μ). This coefficient is different for eachelement and wavelength and is defined as a the sum ofphotoelectric and Compton scattering:  

Whereτ = photoelectric absorption coefficient andσ is the Compton Scattering coefficient
In many materials photoelectric absorption accounts forabout 95 percent of the absorption and Compton scattering
can be ignored. Further, Compton scattering is not important
at wavelengths less than 1 Å.

Fluoresence
If an X-ray photon of sufficiently
high energy strikes an innershell electron the electron canbe ejected it from the atom.
Higher energy electrons cascadeto fill vacancy, giving off
characteristic fluorescent Xrays.
Higher energy electrons cascadeto fill vacancy, giving off
characteristic fluorescent Xrays.

Secondary Fluorescence
The absorption of an incident X-ray to produce another X-ray produces aneffect termed secondary fluorescence. Secondary fluorescence produces anexcess intensity. Some elements undergo self-absorption: K-line X-rays may be
absorbed by other atoms of same element to produce L-line X-rays. For a given
element of interest, any X-rays of energy higher than Ec produced from otherelements can cause fluorescence. The energy at which the mass absorption
coefficient increases abruptly is the absorption edge and corresponds to Ec
exactly.
For example, consider a material composed of 1/3 Fe, 1/3 Ni, and 1/3 Co. One
would expect about 33.3% of the total X-rays observed to be produced by eachelement, but instead we see 40% Fe, 28% Ni, and 32% Co. The Fe in thematerial is fluoresced by Ni-Kα producing Fe-Kα X-rays and absorbing Ni-K α.
Plot of the X-ray absorption curves for Fe (black), Co(red), and Ni (blue), showing the Kα absorption edges.
The locations of the Kα lines for these elements is also
shown. Note than Ni-Kα x-rays have sufficient energy tofluoresce Fe-Kα x-rays.

Scattering and Diffraction Techniques
3-Dimensional
2-Dimensiional

X-ray crystallography
Phasing methods
Grazing incidence diffraction

Low-angle scatter
Dynamics
Equilibrium: B factors, diffuse scatter
Non-equilibrium: Laue method

Other Synchrotron Techniques
Scattering:- Raman (inelastic, due to vibrations)- visible Rayleigh (elastic, information on particle sizeand dynamics)
Absorption and related techniques:
EXAFS (measure fluorescence in an excitation spectrum)
UV, vis, IR (electronic, vibrations)- absorption (CD, linear, time-resolved)-fluorescence-phosphorescence (luminescence)Infra-red Spectroscopy
Displacement of atoms during vibration lead to distortion of electricalcharge distribution of the molecule.

r
F

re

F

atoms (mass) are connectedwith bonding electrons. re is
the equilibrium distance and F:
force to restore equilibrium.

1
i 
2

ki

i

F(x) = -kx where X is displacement from equilibrium ki is the force
constant and μi is reduce mass of a particular motion.
Molecular vibration lead to oscillation of electric charge governed byvibration frequencies of the system.
Oscillating molecular dipole can interact directly with oscillating electric vector ofelectromagnetic radiation of the same frequencyh=hEnergy is quantized and vibrations are in the range 1011 to 1013 Hz=> 30 - 3,000 cm-1

Circular Dichroism•

CD measures the difference between the absorption of left and righthanded circularly-polarized light by a chiral chromophore.

•

This difference is measured as a function of wavelength, and is always
very small (<<1/10000 of total). Therefore CD spectroscopy requires
a high flux source that provides stable linear polarization over andextended wavelength range (VUV-Near IR)X-ray Scattering by an Atom•

Scattering by an atom is essentially thesum of the scattering of the electron“cloud” around the nucleus.

•

Scattering from each electron follows
the Thompson equation. Because of
the distance between electrons
scattering within the atom and the factthat the x-ray wavelength is of the sameorder as the atomic dimensions, there
will be path differences between the
scattered waves. These differences will
always be less than one wavelength, so
the interference will always be partiallydestructive.

•This phenomenon is called the atomic scattering factor, described by the quantity
f0. This function is normalized in units of the amount of scattering occurring from asingle electron in the Thompson equation. At zero degrees, f0 will be equal to thenumber of electrons surrounding the atom or ion. At higher scattering angles, thefactor will be less.

Intensity• The intensity of the diffracted beam can be given by:2
I  SLP Fhkl
• S is a scale factor that is dependent on the intensityof the source and the detector efficiency.
• LP is a combination of the Lorentz and polarisation
factors which for a non-polarised X-ray beam is given
by:
2

LP 

1  cos 2
2sin   cosThe Structure Factor Fhkl
• For a reflection from a plane hkl. The structure factor
Fhkl is:Fhkl 

unit  cell



i 1

2

sin  hkl 
f i  exp  Bi
 exp[2ihkl  xi ]

2




• xI = (xiyizi) are the fractional coordinates for atom i in
the cell• hkl are the Miller indices for the reflection• Bi is the Debye-Waller displacement (thermal) factor
for atom (in Å2)• fi. Is the scattering factor.

X-ray Diffraction• In X-ray diffraction fi is given by:

f i2  ( f 0  f ' ) 2  (f " ) 2• f’ and f” are the real and anomalous dispersion
terms.
• The first term f0 depends on the diffraction angle 2.

Anomalous X-ray Scattering22

f i  ( f 0  f ' )  (f " )2• Anomalous scattering or anomalous dispersion occurs
when the incident X-ray energy is sufficient to cause
photoelectric x-ray production in a target atom. The
process is called fluorescence. This phenomenon is
responsible for “absorption edge” observed when certain
elements interacting with particular wavelength x-rays. In
this process a characteristic X-ray photon is produced in
the target; subsequent interaction produces coherent X rays which are slightly out of phase with other coherently
scattered X-rays. The net result is a reduction of thescattered intensity from the element.

12
10
8
6
4
2
0

f' (e.u.)

-5

f"
f'

-10

m

-15
150

2

cm /g

Just as the absorption
coefficient shows large
jumps at “edges” thescattering power of
elements shows rapidchanges near edges.
• Minimum absorption is
obtained near 13.31 eV(0.93 Å)• Maximum ∆f’ is ca 8electrons by measuring at
13.3 and 13.5 eV.

f" (e.u.)

Bi Wavelength Dependence

100

m

50
0
10

15

20

Energy (keV)25

Dynamics
Equilibrium

Non-equilibrium• B factors• Lauecrystallography• Diffuse scatter• Rapid Mixing andSmall angle
scattering

The temperature (B) factor• Atoms are not located at fixed points - undergo
vibrations about their mean positions.
• The amplitude of these vibrations increase as thetemperature increases.
• Due to such motion, the scattering factor falls off
exponentially.The greatest reduction in intensity is athigh angles (low d-values).

Debye-Waller temperature factor B = 8π2<u>2
Where <u>2 is mean-square amplitude of atomic vibration. This isdirectional and can be anisotropic

The temperature (B) factor
Increasing motion of
an atom induces anangular dependent
reduction in intensity.
The greatest
reduction is at highangles (low d-values).

Debye-Waller temperature factor B = 8<u>2

Effect of Temperature Ag2O
Intensity

Ag2O 0.5 Å X-rays

T = 100 K

T = 290 K
25

30

35

40

45

50

55

60

Angle (2) Degrees
4.746

Lattice Parameter (Å)

4.744

The latticeshrinks due tounusual motionof the Ag atoms!

4.742

4.740

4.738

4.736
0

50

100

150

200

Temperature (K)

250

300

Diffuse Scatter• Diffuse Scatter is the scatter that is notin the Bragg reflections. Every crystal
has it.
• Diffuse scatter can arise from:– Thermal displacement– Occupational disorder– Displacive Disorder– Short Range Order– Stacking Faults

Diffuse Scatter

Powder X-ray Diffraction

Uses of Powder Diffraction
Qualitative Analysis
Identification of single-phasematerials
Identification of multiple
phases in microcrystalline
mixtures
Recognition of amorphousmaterials in partially crystalline
mixtures
Quantitative Analysis
Lattice Parameter
Determination
Phase Fraction Analysis
Peak Shape Analysis
Crystallite Size Distribution
Microstrain Analysis
Extended DefectConcentration

Structure Refinement
Rietveld Method
Structure Solution
Reciprocal Space Methods
Real Space Methods
Thermal expansion and Phase
Transitions

Three Unique Features ofSynchrotron Radiation•Intensity•Enables Rapid Data
Collection
Kinetics
Unstable Compounds
Environmental Cells–Enables Focussing
Small Samples
Small areas/volumes•Energy Range•Enables Spectroscopy–Elemental Identification–Bonding Studies–Speciation•Enables Optimal Conditions–Environmental Cells–Selected Elements•Low Divergence
Enables High Resolution•Micro Beams•Small Volumes•Complex Materials

What is special about a crystal?
Solid phases are often crystalline, but need not be e.g. glass an “amorphous material”
CrystalGlass• Fractures into shards• Takes on any shape,depending on preparation• Properties do not varywith orientation.

Si• Cleaves along preferred
directions• Grows with welldeveloped crystal faces• Properties depend onorientation in which they
are measured.

OxygenCrystal Structure• CRYSTAL: Contains a periodical array of atoms/ions.
This can be represented by a simple lattice of points.
• A group of atoms is associated with each lattice points.
• LATTICE: An infinite array of points in space, in which
each point has identical surroundings to all others.
• CRYSTAL STRUCTURE: The periodic arrangement of
atoms in the crystal.

The Unit Cell
c
 
a 
b
The unit cell is a basic parallelopiped shaped block from which thewhole volume of the crystal may be built by repetition in 3 dimensions.
Any point in the unit cell may be specified with respect to the origin byparameters x, y, z measured parallel to the unit cell axes and expressed
as fractions.

Example of 2D symmetry ina wallpaper pattern
To show symmetry:
1. Pick a point
2. Find all equivalent points(http://www.clarku.edu/~djoyce/wallpaper/)Example of 2D symmetry in awallpaper pattern
To show symmetry:
1. Pick a point●

2. Find all equivalent points●

These points form a 2D lattice●

Connecting 4 lattice points toform a parallelogram gives apossible unit cell●

Example of 2D symmetry in awallpaper pattern
Connecting 4 lattice points toform a parallelogram gives apossible unit cell●●

Unit cell – the basic unit that

repeats in every direction

Different unit cells can bechosen●But some unit cells are
preferable for higher symmetry

●

z

a
x

b

Bravis Lattice Conditions

Triclinic

1 (P)Monoclinic

2 (P, C)

abc

abc
 =  = 90  abc =  =  = 90
a=bc =  =  = 90
a=b=c =  =  = 90

Orthorhombic 4 (P,F,I,A)

 


c

Name

y

Tetragonal

2 (P, I)

Cubic

3 (P, F,I)

Trigonal

1 (P)

a=b=c
 =  =  < 120  90

1 (P)

a=bc
 =  = 90
 = 120

Lattice parameters:
a, b, c; 
Hexagonal

P

C

I

F

PCC Lattice-Po is primitiveCentered Cubic
Identical atoms at
corners but nothingat the and body orface centers.
Lattice type P

BCC Lattice

-Iron is BodyCentered Cubic
Identical atoms at
corners and body
center (nothing at
face centers)
Lattice type I
Also Nb, Ta, Ba, Mo...

FCC Lattice

Sodium Chloride(NaCl) Na is muchsmaller than Cl
Face Centered
Cubic
Rocksalt structure
Lattice type F
Also NaF, KBr,
MgO….

Diffraction of X-ray Waves• Diffraction: When light passes sharpedges or goes through narrow slits therays are deflected and produce fringes oflight and dark bands.
Diffraction grating and helium-neon laser

LightInterference fringes

Constructive

DestructiveBragg’s Law
Incident radiation“Reflected” radiation

1
2

P


S

d


R

Q

Transmitted radiation

Beam “2” travels the extra distance SQRn  SQ  QR
 d hkl sin   d hkl sin  2d hkl sin 
But not all planes result in diffraction !!!

Lattice Planes• It is possible to describe certain directions andplanes with respect to the crystal lattice using a
set of integers referred to as Miller Indicies

Crystallographic Directions And
Planes
Lattice Directions
Individual directions: [uvw]Symmetry-related directions: <uvw>
Miller Indices:
1. Find the intercepts on the axes in terms of the lattice
constant a, b, c
2. Take the reciprocals of these numbers, reduce to thethree integers having the same ratio(hkl)
Set of symmetry-related planes: {hkl}Examples of Miller Indices
a

(100)

(200)

y
x
z

(110)

Families of Planes• Miller indices describe the orientation of a family of planes– the spacing between adjacent planes in a family is referred to as a“d-spacing”• different families of planes– d-spacing between (400) planes is 1/4 that of the (100)spacing.
– The (300) plane does not contain atoms and so is notobserved

(100)

(200)

(400)

(300)

Lattice Spacing1h k l

2
d hkl
a2
2

2

2

For cubic system with a = 4.0 A

ad100 = 4.0

d110 = 2.828

Single Crystal vs Powder

McCuskerPowder – A Polycrystalline Mass
All orientations ofcrystallites possible
Single crystal reciprocal lattice- smeared into spherical shells• By varying the angle , the Bragg's Law conditions are
satisfied by different d-spacings in polycrystalline
materials.
• Plotting the angular positions and intensities of theresultant diffracted peaks produces a pattern which ischaracteristic of the sample.

Powder Diffraction
MonochromatorθSourceθ
λ=2dsinθ
Detector(200)

Sample2750
2500

NaCl - 0.809 Å
(220)

2250
2000
1500

(222)

1250
1000
500

(311)

750

(111)

Intensity

1750

250
0
15

20

Two Theta (degrees)

25

30

2θ

Information Contained in a
Diffraction Pattern
a0
C

15000

3
O
n
M
1
.
0
e
C
1
.
0
r
S
8
.

10000

Intensity

Peak Positions
Crystal System
Space Group Symmetry
Unit Cell Dimensions
Qualitative Phase Identification

5000
0

20

80

100

120

3
O
n
M
1
.
0
e
C
8
.
0
r
S
1
.
0
a
C

3
O
n
M
1
.
0
e
C
1
.
0
r
S
8
.

Intensity

3000
2000
1000
0
29.8

30.0

30.2

Two Theta

Peak Shapes & Widths
Crystallite Size (2-200 nm)
Non-uniform microstrain
Extended Defects (stacking faults,
etc.)

60

a0
C

Peak Intensities
Unit Cell Contents
Point Symmetry
Quantitative Phase Fractions

4000

40

30.4

29.2 29.4 29.6 29.8 30.0 30.2
Two Theta

Changes in symmetry and microstrain
upon chemical substitution can beestablished by examination of thepatterns

Centering and Absences••

The positions of the atoms in a unit cell determine the intensities of
the reflections
Consider diffraction from (100) planes in

If the pathlength between
rays 1 and 2 differs by λ, thepath length between rays 1
and 3 will differ by λ/2 anddestructive interference in(b) will lead to no diffracted
intensity

Centering and Absences• We can extend these types of calculation
to include other modes of lattice centering.
They all lead to systematic absences
Bravais lattice
Simple (Primitive)

Reflections that must
be absentnone

Base (C) centeredh and k mixed

Body (I) centered(h+k+l) odd

Face (F) centeredh, k and l mixed

BCC

(511)

(422)

(400)

(113)

(111)

100

600

(222)

Intensity

(211)

200

(200)

Intensity

300

(331)

1200

400

(420)

(220)

500

NaCl Fm3m
FCC

(200)

(110)

Fe - Im3m

0

0
20

30

20

40

30

40

Two Theta (degrees)

Two Theta (degrees)

Monoclinic
a = 5.45 b = 5.25 c = 5.00 Åo= 100Pm3m

Orthorhombic
a = 5.45 b = 5.25 c = 5.00 Å

Im3m

Tetragonal
a = 5.45 c = 5.00 Å

Fm3m
10

15

20

25

30

35

40

45

50

cubic
a = 5.45 Å
55

Two Theta

Influence of centering

10

15

20

25

30

35

40

45

50

55

60

65

70

Two Theta

Influence of symmetry75

Multiplicity•

For high symmetry materials the Bragg angles and d-spacings fordifferent reflections may be equivalent to one anotherFor example (100), (010), (001) etc are equivalent in a cubic
material•

In a powder, all planes with the same d-spacing contribute to thescattered intensity at a given Bragg angle•

The number of planes that are symmetry equivalent is referred to as
the multiplicity and its appears as a multiplicative term in powder
diffraction intensity calculations•

The multiplicity of a reflection depends upon the symmetry of thecrystalMultiplicity of {100} for cubic is 6, but for tetragonal it would
only be 4 as (100) and (001) are not equivalentDiffraction Patterns• Spacing of peaks depends on size of unit celland the space group.
• The bigger the unit cell and/or the lower thesymmetry the more diffraction peaks areobserved.
• Intensity of peaks depends on (amongst otherthings) the arrangement of the atoms in the unitcell.
• For two materials that had identical unit cells, thepeak positions would be IDENTICAL, however
their intensities would be DIFFERENT.

Need for High Q
50000

Bi2InNbO7 RT
0.79 Å X-rays

orderedmodel

disordered
model

There are many more reflections at higher Q.
Therefore, most of the structural information is
at higher Q

40000

Intensity

30000

20000

56

58

60

62

64

56

58

60

62

64

10000

0

20

30

40

50

60

70

80

Two  (Degrees)

Refinement of structure gave
unusual displacement parameters
for the Bi cations, indicative of
cation disorder. The patterns
could only be adequately fitted byincluding 6-fold disorder of the Bi.
This involves a displacement alongthe (1 -1 0) direction

Atom
Bi
In/Nb
O(1)
O(2)
Bi

Site

x

y

z

Model 1. Ordered Bi. Rp 4.08 Rwp 6.07%
16d
0
0.25
0.75

16c

0

0

0

Biso

2.74(6)3.00(8)48f
0.350(3)
0.125
0.125
7.6(7)
8b
0.375
0.375
0.375
7.6(7)
Model 2. Disordered Bi Rp 3.09 Rwp 3.93%96h
00.2249(1) 0.7751(1) 0.96(7)

In/Nb

16c

0

0

0

0.61(3)

O(1)
O(2)

48f
8b

0.322(1)
0.375

0.125
0.375

0.125
0.375

1.7(2)
1.7(2)

Need for High Resolution
Differentiating Braggs Law gives theresolution as:d  d tan Sample  2d sin 

Source +Instrument

Resolution
In Powder Diffraction it typically refers
to the width of the peaks.
In Single Crystal Diffraction it typicallyrefers to the minimum d-space studied.

Rutile
HRPD 1.49 Å

Intensity

SourceMRPD 1.66 Å

20

40

60

80

100

120

Two Theta (Degrees)

Both definitions are relevant.

140

Peak Overlap• Powder Diffraction
patterns are a one
dimensional
representation of athree dimensional
structure.
• Often peaks due toindividual Braggreflections overlap

The Solution - Rietveld
k p2

y icalc  y iback    G ikp I 2k
p k  k1p

•
•
•
•
•

yic the net intensity calculated at point i in the pattern,yiback is the background intensity,
Gik is a normalised peak profile function,
Ik is the intensity of the kth Bragg reflection,
k1 ... k2 are the reflections contributing intensity topoint i,• the superscript p corresponds to the possible phasespresent in the sample.

The Answers• The Profile R

 y iobs  y icalc
Rp 
 y iobs
w y y    w y2• The weighted ProfileR• The expected Profile
R

R wp

i

iobs
i

2
iobs

w y

χ 
i

1/2

 y icalc 

2

iobs

N-P

• The Goodness of fit

icalc

 N-P R exp  
2  w i y iobs 2

1/2

 R wp  R exp 

2

Phase Analysis
Anatase + Rutile

12000

Intensity (a.u.)

10000

8000

Anatase
6000

4000

Rutile

2000

0
20

40

60

80

Two Theta (degrees)• Where a mixture of different phases is present,
the resultant diffraction pattern is formed byaddition of the individual patterns.
• The intensity of the peaks is proportional to theamount of the phase present.

Quantitative Phase Analysis• Bragg scattering is proportional to N/V where N is thenumber of unit cells and V the unit cell volume.
There for the weight of a phase in the beam is:( SZMV ) P
WP  (SMPV )ii

S - the scale factor
Z the number of formula unites per unit cell
M the molecular weight of the formula unit
I is the index running over all phases• Hence SZVM is proportional to the weight of thediffracting sample

An Example• Synchrotron X-ray Diffraction pattern for
SrRuO370000

SrRuO3 Observed

60000
50000

Intensity

40000
30000
20000
10000
0
10

20

30

Angle (2) degrees

40

The background– Incoherent– Temperature diffuse• Other materials– Specium holder– air etc

8000

6000

Intensity• Fluorescentradiation from thesample• Diffraction from thecontinuous
spectrum• Diffuse scattering

4000

2000

10

20

30

40

Angle (2) degrees• Background can beeither fitted or
estimated.
• Here the capillary isa feature.

Peak Shapes• Different
Diffractometers havedifferent peak shapes.
• The most widely
function is a pseudoVoigt (mixed Gaussianand Lorentzian).

Intensity

gamma0
gamma05
gamma1

Angle



1/2
1
C1/2
C
2
G ik  γ 0 1  C 0 X ik  (1  γ) 1 1/2 exp - C1X ik2
Hk π
Hk π

H2 = Utan2+ Vtan + W

NeutronBragg-Brentano
Synchrotron

0.4

FWHM (degrees)• The width of peaks is
usually not constant.



0.2

0.0
0

20

40

60

80

Two Theta (degrees)

100

120

140

Peak Asymmetry
asym
perfect

Intensity

9.60

9.65

9.70

9.75

9.80

9.85

Angle

asym
perfect
Intensity• Beam Divergence canresults in asymmetricpeaks at low angles.
• Results from not
integrating over theentire Debye cone.

0

35.50

35.55

35.60

35.65

Angle

35.70

35.75

35.80

The Simple Structural Model
60000
50000

Rp 2.42%
Rwp 3.85%

40000

Intensity• The fit to a single
phase sample looks
good

2

 254

30000
20000
10000
0

• BUT…...

15

20

25

30

35

40

45

Angle (2) degrees

Rp 2.42%
Rwp 3.85%
2

 254

Intensity• The detail of the fit
is not satisfactory the model is missing
something!

10

35

40

Angle (2) degrees

45

A Common Problem• If the structural model is wrong then the mostcommon response of Rietveld programs is to:– broaden the peaks,– Increase the displacement parameters,• The former is most noticeable at high angles
where intensity is lowest.
• Due to absorption of the X-rays powder X-ray
diffraction often yields poor displacement
parameters

An Alternate ModelRp 2.25%
Rwp 3.27%

10000

2

 182.9

Intensity• The high angle
splitting is wellmodeled by atetragonal model - but
this overestimatessome intensities.

0
35

40

Angle (2) degrees• The Truth lies somewhere in the middle

45

The finished Product
60000

SrRuO3 828 K = 0.700 Å
RP 1.79%
Rwp 2.22%

50000

2

 84.5Intensity (au)

40000
30000
20000
10000
0

10

20

30

40

50

60

70

80

Two Theta (degrees)• The sample contains a mixture of bothphases!

90

Strengths and Limitations of
Powder X-ray Diffraction
Strengths
•
•
•
•
•
•

Non-destructive – small
amount of sample
Relatively rapid
Identification of compounds /phases – not just elements
Quantification of concentrationof phases – (sometimes)
Classically for powders, but
solids possible too
Gives information regarding
crystallinity, strain, crystallite
size, and orientation

Limitations••••

Bulk technique – generally –unless a microfocus source isused
Not a “stand-alone” technique– often need chemical data
Complicated appearance
multiphase materials –identification /quantification
can be difficult

Experiment Design Issues
What Wavelength?
• Absorption is your enemy!
• Short Wavelengths are best! BUT….
• Consider required resolution. And…
• Avoid Absorption Edges.
What Size Capillary?
• Small capillaries reduce absorption AND (witharea detectors) improve resolution.
• BUT reduce amount of material.

0.000004

c

41.3

41.4

41.0

41.2

40.9

41.1

40.8

41.0

40.7

40.9
3.88

b

3.84

e
c
i
t
t
a
3.86
L

2
e

0.0000005

0.000001

BaBi4Ti4O15

0.000000

3.84
42.3

0.0000000
0.0000015

0.000004

PbBi4Ti4O15

42.1

r
e
t
e
m
a
r
a
P
3.89
e
c
i
3.88
t
t
a
L

41.6

0.000003

42.0

41.5

3.86

b

b

3.87

a

a

3.88

Y Axis Title

41.9

41.4

BaBi4Ti4O15

42.2

2
e
0.0000005

0.000001

3.87

3.85

0.0000010

0.000002

3.86
500

600

700

800

900

1000

300

400

Temperature (K)

500

600

700

800

900

0.0000000
300

400

500

600

700

800

900

1000

300

400

Temperature (K)

16000
3000

Intensity (au)

12000

︶

u
a

︵

8000y

t
i
s
n
e
4000t
n
I

500

600

700

800

900

1000

Temperature (K)

K
5
7
7

K
0
0
0
1

CaBi4Ti4O15

0.000000

1000

Temperature (K)

SrBi4Ti4O15

8000
6000

u
a

2000

4000
y

t
i
s
n
e
I

t
2000
n

1000

0

0

16000

8000

︵

8000y

t
i
s
n
e
t
4000n
I

6000

u
a

2000

4000
y

︵

︶

u
a

Intensity (au)Intensity (au)

12000

t
i
s
n
e

t
2000
n
I

1000

0

0

16000

8000

︵

t
i
s
n
e
4000t
n
I

4000

2000

6000

4000
y

t
i
s
n
e
n
I

t
2000

1000

0
5.4

5.6

5.8

6.0

6.2

6.4

6.6

6.8

Two Theta (degreees)

7.0

7.2

17.0
7.4

17.2

17.4

17.6

17.8

18.0

18.2

18.4

Two Theta (degrees)

18.6

18.8

19.0

0
5.4

︶

u
a

2000

︵

︶

u
a

8000y

K
0
5
3

12000

Intensity (au)

3000

6000

K
0
5
3

Intensity (au)

K
0
0
7

K
0
5
7

3000

8000

︶

400

︵

300

︶

3.84

Intensity (au)

Lattice Parameter

PbBi4Ti4O15

c

c

3.82
41.9

41.7

0.0000010
0.000002

3.85

3.83

41.8

0.0000015

3.87

b

3.85

SrBi4Ti4O15

0.000003

r
e
t
e
m
a
r
a
P

a

3.86

CaBi4Ti4O15

41.5

41.3

a

Lattice Parameter

SrBi4Ti4O15c

CaBi4Ti4O15

41.1

Y Axis Title

41.2

5.6

5.8

6.0

6.2

6.4

6.6

6.8

Two Theta (degreees)

7.0

7.2

17.6

17.8

18.0

18.2

Two Theta (degrees)

18.4

Neutron Diffraction

XRD 0.85A PF

BaBi4Ti4O15

BaBi4Ti4O15

SrBi4Ti4O15

Intensity (au)Intensity (au)SrBi4Ti4O15

PbBi4Ti4O15

PbBi4Ti4O15

CaBi4Ti4O15

20

40

60

80

100

Two Theta (degrees)

120

140

CaBi4Ti4O15

32

34

36

38

40

10

20

30

Two Theta (degrees)

40

30.5

31.0

Powder Diffraction Software
http://www.ccp14.ac.uk/• Structure Solution from Powder Diffraction– Some pathways for Structure Solution
from Powder Diffraction• Search-Match phase Identification• Powder Diffraction Data Conversion /Interconversion• Powder Diffraction Data Visualisation• Peak finding and Peak Profiling• Powder Indexing (finding a unit cell frompowder data)• Unit Cell Refinement• Le Bail Whole Profile Fitting• Pawley Whole Profile Fitting• Rietveld Refinement• Restrained Rietveld Structure Refinement ofOrganics• View Fourier Maps from Rietveld Refined Data
• Structure Conversion and Transformation• Structure Drawing and Visualisation• Modelling of Amorphous Materials• Incommensurate/Modulated Structure solution
and refinement• Anharmonic Thermal Refinement• Bond Length and Bond Angle Calculation• Bond Valence Calculations• Pole Figure and Texture Analysis Software• Polyhedral Volume Calculations• High Q Powder Diffraction Analysis for PDF/Local
Order Determination• TOF (Time of Flight) Neutron Diffraction• Variable Count Time (VCT) Data Collection for
Powder Diffraction• Integration of Image Plate Data

Challenge!

DATA

SR