IV. Powder Diffractometry – Phase Analysis

Powder Diffraction

λ ϑ= 2dhkl

sin

There are many different ways to fulfill Bragg’s equation

• Utilize polychromatic radiation in combination with a perfect single crystal

� Laue technique

• Systematic variation of the orientation of a single crystal relative to a

monochromatic incoming beam

� Rotating crystal, Weißenberg-, Buerger Precession Technique

• Diffraction of monochromatic X-rays on polycrystalline material (i.e. a largevariety of crystals with random orientation

Powder Technique

Powder

Very large amount of very small crystals with random spatial orientation

(easy to produce)

Powder Diffraction

History

Independent development by

Peter W. Debye und Paul Scherrer (1916) and Albert Wallace Hull (1917)

Peter W. Debye, 1884-1966Nobel prize for chemistry,1936

Paul Scherrer,1890-1969 Albert Wallace Hull,1880-1966

Powder Diffraction

Basic Principle

(Source: http://de.wikipedia.org/wiki/Debye-Scherrer-Verfahren)

Powder Diffraction

monochromatic X-rays Crystal powder

(Laue Cones) With half opening angle 2ϑ

Back Scattering Regime Transmission Regime

Powder Diffraction

Basic Principle

Debye-Scherrer Camera

Source: W. Kleber, Einführung in die Kristallographie, Verlag Technik Berlin, 1990

Powder Diffraction

Debye-Scherrer Method

Principle

• Not the full Laue cone will be imaged on the film, but

just a small segment only

• If only a few powder particles contribute the Laue cones

appear as single dots, where each dot represents a

different particle

• Randomization can be enlarged by continuous rotation

of the powder during illumination

Sample Preparation

• Sample diameter has to be much smaller than diameter

of sample chamber

• Stick like samples (e.g. wires)

• Use of capillaries as sample containers

• Preparation on glass-wires

Powder Diffraction

Powder Diffraction

Debye-Scherrer Method

For historical reasons:

Traditional Debye-Scherrer Cameras with film can be

bought in two standard sizes, allowing a direct readout of

mm → degrees

� 180 mm Film (57.3 mm Diameter):

� Routine investigations

� “Small” illumination times (a few hours)

� 360 mm Film (114.59 mm Diameter):

� “High precision” investigations

� Lattice parameter determination

� Investigation of substances exhibiting many Bragg

reflections or mixtures of substances

Advantages

• Precise rotation of the sample is not necessary

• Careful adjustment of the sample is not necessary (except for centralizing the sample)

• Fast data acquisition through multi-detection of all Bragg reflections

• Quantitative evaluation of intensities through use of position sensitive detectors

(1D curved wire detectors, 2D-curved area detectors)

Drawbacks

• Indexing of net-planes is already difficult for unit cells with reduced symmetry

• Determination of crystal structure for medium organic molecules is basically impossible

• Small wire shaped samples (i.e. no large area samples) can be investigated only

Advantages/Drawbacks of the Debye-Scherrer Technique

Powder Diffraction

Identification and Indexing of Cubic Crystals

Orthorhombic Lattice:

Cubic Lattice: (b1 = b2 = b3 = 2π/a) �

Since

We can write

2

3

22

2

22

1

22blbkbh ++=G

( )222

2

22 4

lkha

++=π

G

λ

ϑπsin4=G

( )222

2

22 lkh

a4sin ++

λ=ϑ

22

22

22

21

21

21

22

12

lkh

lkh

sin

sin

++

++=

ϑ

ϑ

This quadratic equation is starting point basis of indexing of cubic crystals.

For two different lines (ϑ1,ϑ2) on the film we can state:

Powder Diffraction

� Correct indexing of the Debye-Scherrer rings, how?

� It is advantageous to start with indexing of the smallest ring as (100), (110)

or (111) and then continue indexing of the outer rings

� Example: Indexing of powder rings of copper: a = 361.2(4) pm

Nr ϑ sin2ϑ sin2ϑn /sin2ϑ h2 + k2 + l2 h k l a /pm

1 21.7 0.1367 - 3 111 361.1

2 25.3 0.1826 1.336 4 200 360.8

3 37.2 0.3655 2.673 8 220 360.7

4 45.1 0.5017 3.670 11 311 361.0

5 47.6 0.5453 3.989 12 222 361.6

6 58.6 0.7285 5.329 16 400 361.3

7 68.3 0.8633 6.315 19 331 361.7

8 72.5 0.9096 6.653 20 420 361.5

22

22

22

21

21

21

22

12

lkh

lkh

sin

sin

++

++=

ϑ

ϑ

Powder Diffraction

Identification and Indexing of Cubic Crystals

Phase Analysis and Identification of Unknown Substances

• Each powder diagram is characteristic for a particular substance

• ICDD (International Centre for Diffraction Data)

� Before 1978: JCPDS (Joint Committee on Powder Diffraction Standards)

� Collecting X-ray data of all substances

� Powder Diffraction Files (previously: ASTM - Database)

• PDF-2 Database

� Involves dhkl, Intensities, Lattice parameters and angles, crystal symmetry

� Data of 265000 Substances (2013)

� 85 000 measured X-ray data, 46 000 calculated X-ray data

� Sorted in nonorganic, organic and metal-organic substances

� Subsorting with respect to main phases, minerals, metals, and alloys

• Database is helpful for

� Analysis of unknown substances

� Identification of powder mixtures

� Quantitative Analysis

� Detection and identification of sample impurities

Powder Diffraction

Zr O2

111

200

220

311

400

331

420222

PDF 03-065-0461 (Zr 0)2

Powder diagram of Zr2O and corresponding PDF card

2ϑ (Degrees)

Powder Diffraction

Phase Analysis and Identification of Unknown Substances

Refinement of Structural Analysis: The Rietveld-Method

Hugo M. Rietveld

(1932-2016)

Question: How can more complex and less symmetrical crystal structures be

indexing and structurally analyzed?

• Refinement of crystal structure through iterative comparison of

experimental and calculated powder diagrams

• First introduced by Hugo M. Rietveld (from the Netherlands)

� 1967 initially developed for neutron powder diagrams

� H.M. Rietveld, Acta Crystallographica 22, 151 (1967)

� H.M. Rietveld, J. Appl. Cryst. 2, 65 (1969)

� 1977 applied for the first time for X-ray powder diagrams

• Evaluation of complex powder diagrams

Powder Diffraction

Refinement of Structural Analysis: The Rietveld-Method

Procedure:

1. Initial model of unit cell including atomic arrangement inside

2. Calculation of corresponding powder diagram

3. Comparison with experimental powder diagram

� Peak position, peak profile

� Intensity

4. Subsequent Refinement (minimization of root mean square

deviation)

� Crystal structure

� Instrumental parameters

Powder Diffraction

Sufficiently good

agreement

between

experiment and

simulation?

Determination of Particle Size

How does the particle size influence the widths of the Bragg reflections?

)aQ(sin

)aQN(sin

)aQ(sin

)aQN(sin

)aQ(sin

)aQN(sinG

32

12

332

12

22

12

222

12

12

12

112

122

vr

vr

vr

vr

vr

vr

⋅⋅=

0

10

20

30In

ten

sity

Qa

2 /Nπ

N2

0−2π−3π 2π 3π−π π

Laue Function: N = 5 N = 10

0

5000

10000

Inte

nsi

ty

Qa

0−2π−3π 2π 3π−π π

Laue Function: N = 100

1 1

• D: thickness of the crystal along Q, i.e. perpendicular to the net-planes

• ∆Q is independent of the type of reflection (hkl)

∆Q = 2π/aN= 2π/D

Powder Diffraction

In angular space we can express the equation ∆Q = 2π/aN= 2π/D as

Θ

λ=Θ∆

cosD

K)2(

Scherrer formula (1918)

∆(2Θ): angular width of Bragg reflection (on Film/Detector)

K: Formfactor of crystallite (K = 1 für Laue Function)

λ: X-ray wavelength

D: (mean) size of crystallites perpendicular to net-planes

Θ: Bragg angle of the reflection

Home work: Derivation of Scherrer formula (starting from Laue function)

Typical Values:

D = 1 µm .. 10 µm � Sharp reflections

D < 1 µm � Peak broadening can be observed

D > 10 µm � Only a few crystallites are illuminated by x-ray beam

� Debye-Scherrer rings show up as single dots

D > 10 µmD = 1 µm .. 10 µm

Powder Diffraction

Determination of Particle Size

Comment:

Alternative technique for the determination of the size/shape of very small particles (D < 1 µm) in crystalline or even amorphous substances � Small Angle X-Ray Scattering (Lecture 10)

Example: Pt-Ru-catalytic particles before and after usage in a Fuel Cell (Brennstoffzelle)

Widths are in the range of ∆(2Θ) = 1° (λ = 0.154 nm, K = 1)

� D ~ 10 nm (order of magnitude)

Powder Diffraction

Determination of Particle Size

Formation of Alloys – Vegard’s Law

“Binary“ alloy AxB1-x with corresponding lattice parameters aA und aB

Vegard‘s Law:

• Linear relationship between molar concentration x und lattice parameter

• Holds at least for small (x << 1) or large (x ≈ 1) concentrations x (in atomic percent)

a(x) = x· aA + (1-x) · aB

Example: Ruthenium concentration in Pt-Ru catalytic particles

Powder Diffraction

Distinction between “Crystalline” and “Amorphous“

Powder Diffraction

Diffraction lines for polycrystalline sample,

7 Peaks in diffraction diagram

Diffraction lines for textured sample,

3 Peaks in diffraction diagram, contrary

intensity ratios

Diffraction lines for single crystalline sample,

No peaks in diffraction diagram

Background for amorphous sample,

No peaks in diffraction diagram

Transmission BackscatteringTrace of measurement

for a diffractometer

2Theta10.0 20.0 30.0 40.0 50.0 60.00.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

400.0

Ab

so

lute

In

ten

sity

Incirkus zeolith X (Range 1)

Amorphous parts (broad feature) of a crystalline sample

Powder Diffraction

Distinction between “Crystalline” and “Amorphous“

Bragg-Brentano Focusing

Solution: Bragg-Brentano Focusing

All beams are diffracted by 2θ with

respect to incoming beam

Source/Entrance Slit

Sample

Detector/Exit Slit

Focusing of diffracted beam

onto detector/exit slit

Powder Diffraction

are located on

the same circle

(Rowland circle)

Entrance Slit

Exit SlitDetector

Sample

Rowland CircleRadius R

R

Drawback of Debye-Scherrer Camera

no area samples

Bragg-Brentano Focusing

Exact focusing for:

• Source/ Entrance Slit is a mathematical

point

• Sample is curved and always touching

Rowland circle (Radius R)

“Reality”:

• Angular resolution at detector is limited

by finite source/ entrance slit size

• Finite size of detector slit (if no film is

used)

• Often plane samples are used

• Finite penetration depth of X-rays into

sample

Powder Diffraction

Entrance Slit

Exit SlitDetector

Sample

Rowland CircleRadius R

R

Seemann-Bohlin Technique (Seemann (1919), Bohlin (1920))

Advantage:

� Area samples

� High angular resolution (line width is solely a function of the entrance slit size

� Debye-Scherrer cameras can be easily modified for Seemann-Bohlin geometry

Drawbacks:

� Only backscattering geometry� Requirement of precise sample adjustment (out-of-plane)

Probe

Eintritts-

spalt

Powder Diffraction

Bragg-Brentano Focusing - Examples

Monochromatizing by using Filters

• If we need not resolve the Kα1-Kα2 Doublet, a (strong) suppression of the Kβ Line is at least

required

• Use of thin filter foils is sufficient for this goal

Which filter has to be chosen?

Anode material Z

Filter material Z -1

Powder Diffraction

Monochromatizing by using a Crystal Monochromator

Filter foils: Kα1−α2 Doublet cannot be resolved

• We make use of small intrinsic line widths of the Bragg

reflections of single crystals

• Pure spectrum is achieved (only Kα1 can reach the sample)

• We need focusing geometry owing to subsequent focusing

Bragg Brentano geometry of sample

Bent Crystal Monochromators

Powder Diffraction

Pulverdiffraktometrie

What requirements have to be fulfilled?

• Focusing of entrance slit S onto exit slit F

� Entrance slit, crystal and exit slit are placed on a Rowland circle with radius

� Surface of crystal is curved with radius R

• We have to fulfill the Bragg condition everywhere on the monochromator crystal

� Net-planes of the crystal are curved with curvature radius 2R

Monochromatizing by using a Crystal Monochromator

Net-planes

D

d=∆θ

θλ

λcot

D

d=

∆

� Source size / Entrance slit size: d

� Outer dimension of crystal: << R

� Distance of crystal to source: D >> d

� From every place on the crystal we see a finite source size leading to a finite angular

divergence ∆θ of the incoming beam of

If we use the differential form of Bragg’s law

we obtain for the wavelength resolution (energy resolution)

Numerical Values: θ = 14° (Si 111, Cu Kα), d = 0.05 mm, D = 100 mm � ∆λ/λ = 2 ·10-3

Energy Resolution for Johann/Johansson Geometry

θθλ

λ∆=

∆cot

Powder Diffraction

Machining of a Johansson Curvature

Guinier Camera

Advantages:

• No spectral impurities, Bremsstrahlung is missing

• Weak Bragg reflections can be detected with large illumination times

• Sharp diffracted lines � Good angular resolution

• Investigation of complex mixtures of substances is possible (no overlapping due to

presence of Kα1 and Kα2

• It is possible to simultaneously measure two, three, four, .. samples

• With one of them being a calibration sample

Powder Diffraction

Guinier Camera

Drawbacks:

• Precise adjustment of both monochromator and sample is necessary

• Loss of intensity due to crystal monochromator (at least one order of magnitude)

• Loss of intensity will be (partly) compensated by using area samples which is enabled by

focusing

Powder Diffraction

Modern Powder Diffractometers

Pulverdiffraktometrie

• Use of a single channel detector (alternative: curved line detector)

• Use of a secondary monochromator

� Polycrystalline, bent graphite crystal

� Suppression of Kβ as well as background

• Separation of Kα1−α2 doublet by deconvolution procedure (� Rachinger correction)

Rachinger Correction

William Albert RachingerA Correction for the α1α2 Doublet in the Measurement of Widths of X-ray Diffraction Lines, Journal of Scientific Instruments 25, 254–255 (1948).

• Fixed sample

• Source and detector are rotated with a

fixed ratio of 1:1

• Theta-Theta-Diffractometer

1:1 Scan 1:2 Scan

• Fixed source

• Sample and detector are rotated with a

fixed ratio of 1:2

• Theta-2Theta-Diffraktometer

Powder Diffraction

Modern Powder Diffractometers

Scan-Modi

Determination of Texture in Polycrystalline Materials

Pulverdiffraktometrie

“Texture“

Statistical Ensemble of crystal orientations in a “Multi-Crystal“

natural (a,b) and experimentally formed Hematite (Brazil)

“Grey Texture”Random distribution

of orientation

“Sharp Texture”Predominant crystal

orientation

� Preferential orientations are usually caused by the production process:

• Freezing (e.g. wire drawing)

• Nucleation

• Growth of grains

• Recrystallisation

• Phase transitions

• Sintering

• Plastic deformation

• Shape anisotropy

� How do these preferential orientations influence the diffraction diagram?

• No Debye-Scherrer cones (rings) anymore

• No sharp Bragg reflections yet

Powder Diffraction

Determination of Texture in Polycrystalline Materials

Powder Diffraction

Determination of Texture in Polycrystalline Materials

Diffraction lines for polycrystalline sample,

7 Peaks in diffraction diagram

Diffraction lines for textured sample,

3 Peaks in diffraction diagram, contrary

intensity ratios

Diffraction lines for single crystalline sample,

No peaks in diffraction diagram

Background for amorphous sample,

No peaks in diffraction diagram

Transmission BackscatteringTrace of measurement

for a diffractometer

Powder Diffraction

Experimental Characterization of “Texture“

• Three-dimensional orientation can be

described by two angles

• Polar and azimuthal angle

• Measurement of so-called pole figures

• The (hkl) pole figure describes the

distribution of the net-plane normal {hkl} of

all grains in angle preserving stereographic

projection

Measurement of Pole figure

• We first choose a particular Bragg

reflection (hkl)

• We keep the point detector fixed at twice

the Bragg angle

• For a set of polar angles (psi = 0° .. 90°) we perform azimuthal scans (phi = 0° .. 360°)

• Experimentally the psi-range is often limited to e.g. psi = 0° .. 60°

• Use of point detector makes pole figures very time consuming (several hours for one)

• Alternative: Use of an area detector which simultaneously measures (i) a set of Bragg

reflections and (ii) a limited range of polar angles psi

� Reduction of data acquisition time (we need to scan the azimuth angle only)

� A set of different Bragg reflections are simultaneously investigated

Powder Diffraction

Experimental Characterization of “Texture“

Application of Powder Techniques - Summary

Crystal Structure Analysis

• Measurement of many Bragg reflections

• Refinement of crystal structure by applying the Rietveld Method

• Determination of lattice parameters and thermal expansion

Identification of (unknown) Substances

• Chemical analysis of crystalline substances

• quantitative und qualitative analysis of mixtures

• Isomorphisms, Polymorphisms, Determination of phase diagrams

• In situ high/low temperature and high pressure investigations

• Solid state chemical reactions

Real Structure Analysis

• Identification of crystalline and amorphous phases

• Texture analysis

• Micro structure (grain sizes, strain, stacking faults) from line broadening

Industrial Applications (enabled through large sample discharges)

Powder Diffraction