Powder X-ray Diffraction

Brendan J. Kennedy
School of Chemistry
The University of Sydney

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.

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

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
β α
a γ
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
●

Name

Bravis Lattice Conditions

Triclinic

1 (P)Monoclinic

2 (P, C)

a≠b≠c
α≠β≠γ
a≠b≠c
α = β = 90° ≠ γa≠b≠c
α = β = γ = 90°a=b≠cα = β = γ = 90°
a=b=cα = β = γ = 90°zOrthorhombic 4 (P,F,I,A)

β α
γ

c

Tetragonal

2 (P, I)

Cubic

3 (P, F,I)

Trigonal

1 (P)

a=b=c
α = β = γ < 120° ≠ 90°

1 (P)

a=b≠c
α = β = 90°
γ = 120°

a

x

b

y

Lattice parameters:
a, b, c; α, β, γHexagonal

P

C

I

F

PC Lattice

BCC Latticeα-Po is primitiveCubic
Identical atoms at
corners but nothingat the and body orface centers.
Lattice type P

α-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….

Young’s Double Slit Experiment• Thomas Young first
demonstrated interference in lightwaves from two sources in 1801
• Light is incident on a screen witha narrow slit, So• The light waves emerging fromthis slit arrive at a second screenthat contains two narrow, parallelslits, S1 and S2• The narrow slits, S1 and S2 act assources of waves• The waves emerging from theslits originate from the same
wave front and therefore arealways in phaseResulting Interference Pattern• The light from the two slits form a visible pattern ona screen• The pattern consists of a series of bright and darkparallel bands called fringes• Constructive interference occurs where a brightfringe occurs• Destructive interference results in a dark fringeInterference Patterns• Constructive interference occurs
at the center point• The two waves travel the same
distance– Therefore, they arrive in phase• The upper wave travels one
wavelength farther than the lower
wave– Therefore, they arrive in phase• A bright fringe occurs

Interference Patterns• The upper wave travels
one-half of a wavelength
farther than the lower
wave• The trough of the bottom
wave overlaps the crest of
the upper wave (180°phase shift)• This is destructive
interference– A dark fringe occurs

Interference Equations• The path difference, δ, is found from the tan triangle• δ = r2 – r1 = d sin θ••

For a bright fringe, produced by constructive interference, the path
difference must be either zero or some integral multiple of of the
wavelengthδ = d sin θbright = m λ– m = 0, ±1, ±2, …
– m is called the order number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

Bragg’s Law
Incident radiation“Reflected” radiation

1
2

P

θ
S

d

θ
R

Q

Transmitted radiation

Beam “2” travels the extra distance SQR

nλ = SQ + QR
= d hkl sin θ + d hkl sin θ= 2d hkl sin θ
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}Calcium oxalate solvates: COM & COD
CaOx Monohydrate(symptomatic)

CaOx Dihydrate(protective)
polyD
H O H
*

C C N

*

n

C

25 µm

O

C

O-

P21/c
(a = 6.290 Å, b = 14.580 Å, c = 10.116 Å, β = 109.46o)

25 µm

I4/m
(a = b = 12.371 Å, c =7.357 Å, α = β = γ = 90o)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 not observed

(100)

(200)

(400)

(300)

Lattice Spacing1h2 + k 2 + l 2
=
2
d hkl
a2

For cubic system with a = 4.0 A

ad100 = 4.0

d110 = 2.828

Single Crystal vs PowderPowder – A Polycrystalline Mass
All orientations ofcrystallites possible
Single crystal reciprocal lattice- smeared into spherical shells• By varying the angle θ, 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)

2θ

NaCl - 0.809 Å

(222)

(220)

(311)

2750
2500
2250
2000
1750
1500
1250
1000
750
500
250
0

(111)

Intensity

Sample15

20

Two Theta (degrees)

25

30

My Powder X-ray Diffraction Beamline

Powder Diffraction Beamline

Verticalcollimating
mirror

Double crystal
monochromator
Si(111) flat pairSi(311) flat/bent pair

Vertical focusing
mirror002

002

Intensity (arbitrary units)

Preferred orientation of lead pencil

10

20

30

006

004

Lead of pencil
Experimental data
(λ=1.0Å)

40

50

60

006

10

20

30

006

004
006

Graphite
Simulation data

004

002

002

004

Intensity (arbitrary units)2θ (degree)40

2θ (degree)

Kim BL02B2 Spring-8

50

60

Information Contained in a
Diffraction Pattern
15000

Ca0.8Sr0.1Ce0.1MnO3

10000

Intensity

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

5000
0

20

3000
Intensity

Peak Intensities
Unit Cell Contents
Point Symmetry
Quantitative Phase Fractions

4000

60

Ca0.8Sr0.1Ce0.1MnO3

80

100

120

Ca0.1Sr0.8Ce0.1MnO3

2000
1000
0
29.8

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

40

30.0 30.2
Two Theta

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

NaCl Fm3m
FCC

(511)

(422)

(400)

(113)

(111)

100

600

(222)

Intensity

(211)

200

(200)

Intensity

300

(331)

1200

400

(420)

(220)

500

(200)

(110)

α−Fe - Im3m
BCC

0

0
20

30

20

40

30

40

Two Theta (degrees)

Two Theta (degrees)

Monoclinic
a = 5.45 b = 5.25 c = 5.00 Å
β= 100o

Pm3m

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

55

Two Theta

Influence of centeringcubic
a = 5.45 Å
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 θ (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 ∆λ ∆θ=+λ tan θdSampleSource +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

Sourceλ = 2d sin θ

MRPD 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

1/2

2• The weighted ProfileR• The expected Profile
R

R wp

i

i

2
iobs

 N-P R exp = 2 ∑ w i y iobs 
w (y
∑
χ =
2

i

1/2

− y icalc )

2

iobs

N-P

• The Goodness of fit

icalc

iobs

 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 (2θ) degrees

40

The background
8000

Intensity• Fluorescent radiationfrom the sample• Diffraction from thecontinuous spectrum• Diffuse scattering– Incoherent– Temperature diffuse– Short range order• Other materials– Specimen holder– air etc

6000

4000

2000

10

20

30

40

Angle (2θ) 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

NeutronBragg-Brentano
Synchrotron

H2 = Utan2θ+ Vtanθ + W

FWHM (degrees)

0.4• 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%
χ2 25440000

Intensity• The fit to a single
phase sample looks
good

30000
20000
10000
0

10

• BUT…...

20

25

30

35

40

45

Angle (2θ) degrees

Rp 2.42%
Rwp 3.85%
χ2 254

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

15

35

40

Angle (2θ) 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%
χ2182.9

10000

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

0
35

40

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

45

The finished Product
SrRuO3 828 K λ= 0.700 Å
RP 1.79%
Rwp 2.22%
χ2 84.5

60000
50000

Intensity (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

Crystallography gives average structure
800000Intensity

600000400000200000020402θ (degrees)

60

80• The Rietveld method
has served us well forover 45 years – but it
only uses part of theinformation of thediffraction
experiments, namely
the intensity of the
Bragg peaks

Crystallography challenged: materials withdisorderRietveld approach assumption: crystals are perfectly periodic…
…but this is not always the case!

Intensity

100000

10000

1000
10

20

30

40

50

60

70

2θ (degrees)

Just using the Bragg reflections means that we “ignore” theinformation in the background, but what information is this?

Crystallography challenged: nano-crystals
From E. Bozin BNL

Rietveld methodin powders

Bragg peak info ONLYDiffuse intensity contains information on
From E. Bozin BNL
short-range order

Bragg intensity → long range order

Diffuse intensity →short range order

G (r ) =2∞

Q[ S (Q ) − 1] sin(Qr ) dQ∫
π0

From E. Bozin BNL

Total Scattering - PDF

4.26Å

Pair distribution function (PDF)
gives the probability of finding
an atom at a distance “r” from agiven atom.

2.84Å

1.42Å

2.46Å
3.76Å
4.92Å
5.11Å

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.

X-rays and Neutrons, Same
Same but Different
Neutron

S-XRD

4000

25000026 28 30 32 34 36 38

3000

511

300000

331

2
0.9
1
0

Intensity

Intensity (Counts)

5000

350000

422

511

422

331

400

6000

400

400000

7000

y=2
y=0

200000

20

22

24

26

1500002000

100000
1000

50000
0

0
20

40

60

80

100

2θ (degrees)

120

140

160

20

30

40

50

60

70

80

2θ (Degrees)

Anion disorder clearly evident in neutron profiles but absent in XRD patterns
No evidence for cation disorder in the XRD patterns
J. Solid State Chem 2011 184 2108

Sr0.65Pr0.35MnO3 – structuralparameters
58.0

56.5

Lattice Parameter (Å)

56.0
3.92

I4/mcm

ce

3.88

Pm3m

ao

3.84

1.95

Mn-O1
1.94
1.93
1.92
1.91

ae

3.80

Average Mn-O bond length
in tetragonal region
Fit of Mn-O bond lengthfrom the cubic phase

1.96

Bond Distance (Å)

57.0

Cell Volume (Å3)

57.5

1.97

Mn-O2

1.90
3.76

0
0

100

200

300

400

500

600

Temperature (K)

700

800

900

1000

200

400

600

800

1000

Temperature (K)Single structural phase transition observed. Apparently conventional thermal
expansion of the cell volume ( fitted as Vo=V1+V2Θcoth(Θ/T) )Clear change in the anisotropy of the unit cell near 250 K.
Large tetragonal distortion of the MnO6 octahedra. Consequence of the JT active
Mn3+ cations – evidence for orbital ordering and suggests coupling between thelattice and the orbitals

Oxygen Conduction in Y1-xTaxO1.5+x

Representation of the fluorite
structure of Y1-xTaxO1.5+x

Scattering amplitude distribution (a) on the (002) plane of theorthorhombic Cmmm fluorite-related Y0.7Ta0.3O1.8, and on the (110)planes of the cubic fluorite-type Y0.785Ta0.215O1.715 (b) at 299 K and (c)at 808 K.. Lines with arrows indicate the diffusion paths along theAcetylene Absorption in Framework
SolidsCrystal structures with the adsorption ofacetylene.
a) Views from the side of the nanochannels.
Pillar-molecules (pyrazine) and adsorbed
acetylene molecules are shown by CPK
model. Otherwise are connected by lines. b)
Views from the nanochannel direction by the
CPK model. Adsorbed acetylene molecules
are omitted in the lower central pore in the