iv. powder diffractometry – phase...
TRANSCRIPT
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