modern crystallographic methods€¦ · 2. methods of diffraction: powder diffraction and structure...
TRANSCRIPT
1. Theory
1.0 Literature
1.1 The electromagnetic spectrum and ist application for structure
elucidation
1.2 Diffraction in the crystal
1.3 Symmetry: what are space groups?
2. Diffraction: Powderdiffraction and structure elucidation
2.1 Different diffractometers
2.2 Detectors
2.3 Monochromators
2.4 Sample preparation
2.5 Working with the data
2.6 Profile fitting
2.7 Factors for quality
2.8 Structure elucidation from powder data: Rietveld
2.9 Examples and applications
Modern Crystallographic Methods
(nach
Priv.-Doz. Dr. Tom Nilges)
Modern Crystallographic Methods
3. Diffraction: X-ray (single crystal), neutron and synchrotron
3.1 Single crystal structure elucidation with x-rays
3.2 Neutron diffraction
3.3 Synchrotron
4. Other methods
5.1 AFM (Atomic Force Microscopy) and STM (Scanning Tunnel Microscopy)
Lecture: Friday, 13:15 - 15:45 Uhr
Practical work: to be announced
Room: G180
Exam: End of semester
About: Lecture and excersises
Organization
1.0 Literature
• M. J. Buerger, Kristallographie, W. de Gruyter Verlag, 1. Aufl. 1977
• H. Krischner, B. Koppelhuber-Bitschnau, Röntgenstrukturanalyse und
Rietveldmethode, Vieweg Verlag, 5. Auflage. 1994
• W. Massa, Kristallstrukturbestimmung, Teubner Verlag, 2. Auflage 1996
• D. Haarer, H. W. Spiess, Strukturbestimmung amorpher und kristalliner
Festkörper, Steinkopf Verlag Darmstadt, 1. Auflage 1995
• Reviews in Mineralogy: Modern powder diffraction, Vol. 20, D. L. Bish, J. E. Post,
The Mineralogical Soc. of America, Washington.
• Crystallographic Computing 6: A window in modern crystallography, H. D. Flack,
L. Parkanyi, K. Simon, International Union of Crystallography, Oxford Science
Press 1993
• Server der Uni Freiburg, Prof. Dr. C. Röhr
http://ruby.chemie.uni-freiburg.de/Vorlesung/methoden_0.html
• R. Allmann, Röntgenpulverdiffraktometrie, 2. Aufl. 2002, Springer Verlag Berlin.
4
1.1 The electromagnetic spectrum and
5
Quelle: http://ruby.chemie.uni-freiburg.de/Vorlesung/Vorlagen/methoden_0_3.pdf
1.1 The electromagnetic spectrum and ist application for structure elucidation
Information about the structure are gained via spectroscopy, microscopy and diffraction
6
Quelle: http://ruby.chemie.uni-freiburg.de/Vorlesung/Vorlagen/methoden_0_3.pdf
1.1 The electromagnetic spectrum and ist application for structure elucidation
7
Quelle: http://ruby.chemie.uni-freiburg.de/Vorlesung/Vorlagen/methoden_0_3.pdf
1.1 The electromagnetic spectrum and ist application for structure elucidation
In this lecture:
STM
Scanning Tunnel
Microscopy
AFM
Atomic Force
Microscopy
http://www.uni-tuebingen.de/Teilchenoptik/html/fprakt/tem.html 8
Quelle: http://ruby.chemie.uni-freiburg.de/Vorlesung/Vorlagen/methoden_0_3.pdf
1.1 The electromagnetic spectrum and ist application for structure elucidation
In this lecture:
X-ray diffraction
Powder
Single crystal
Neutron diffraction
Electroc diffraction (TEM)
X-ray diffraction (single crystal)
Neutron
diffraction
X-ray
diffraction
9
1.2 Crystal diffraction
The long road to the structure of a compound ….
X-ray diffraction
10
1.2 Crystal diffraction
Observation: Type of atom and and the structure influences the diffractogram
Diffractogram
with indices
Structure and
data
Quelle: http://ruby.chemie.uni-freiburg.de/Vorlesung/Vorlagen/methoden_II_43.pdf 11
1.2 Crystal diffraction
The symmetry is determined by the arrangement of the atoms.
Structure
and data
Diffractogram
with indices
Quelle: http://ruby.chemie.uni-freiburg.de/Vorlesung/Vorlagen/methoden_II_43.pdf 12
1.2 Crystal diffraction
Conclusion: The structure (and symmetry) can be determined from diffraction patterns.
Cellparameters, symmetrie and atom coordinates are required
to uniquely describe the structure.
13
Lattice plane 1
Lattice plane 2
Lattice plane 3
X-ray diffracted
x-ray pos. interference
at n l
1.2 Crystal diffraction X-ray: Diffraction at electrons
1. Theorie
Bragg-Equation
n l = 2 d sin qBC = d sin
Length difference: D = BC + CD
2 * length difference: D = 2 BC = 2 d sin
Derivation:
14
1.2 Crystal diffraction Miller Indices
1. Theorie
Miller Indizes:
Point in the cell Direction in the crystal [ ]
“Miller indices are reciprocal, reduced to a common denominator
intersections of the planes with the axes.
Construction
of points
and directions
Lattice planes correspond to vectors in reciprocal space
Construction of
reciprocal space
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Simulation tool:
http://tiny.cc/n3p34y
1.2 Crystal diffraction Indexing
1. Theorie
2hkl
2
2
2
2
2
2
d
1
c
l
b
k
a
h
quadratic Bragg equation
2hkl
2
222
d
1
a
lkh
orthorhombic lattice
cubic lattice
2
22
d4sin
lq
)lkh(a4
sin 2222
22
lq
Angle of diffraction lattice parameter (h k l) indices
Relation between lattice parameters, (h,k,l) indices and angles of diffraction
16
Supplement
quadr.
Bragg eq.
1.2 Crystal diffraction Single crystal
1. Theorie
The result of diffraction on a single crystal
Single crystal
Detektor
Calcite
17
1.2 Crystal diffraction Single crystal
1. Theorie 18
Single crystalline:
Crystallinity decreases
Polycrystalline (powder)
The result of diffraction on a powder
Detector
1.2 Crystal diffraction Powder diffraction
1. Theorie
Polycrystalline material
Powder diffractogramScan along the diffraction cones
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1.3 Theory of symmetry Basics
Symmetry in different dimensions…
Line
1-D
Plane
2-D
Space
3-D
Quelle: http://ruby.chemie.uni-freiburg.de/Vorlesung/Vorlagen/methoden_II_21.pdf
Point
0-D
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1.3 Theory of symmetry Symmetry elements
Point symmetry: axis of rotation and rotoinversion
Quelle: http://ruby.chemie.uni-freiburg.de/Vorlesung/Vorlagen/methoden_II_21.pdf
• Axis of rotation is sticking up
• Red symbols show the center of rotation
• Rotation by 360/n degrees
Supplement
Symbols21
1.3 Theory of symmetry Symmetry elements
Screw axis
Translation by p/n parallel to the axis of rotation; rotation by 360°/n
Example when np = 21: Translation by ½; rotation by 360/2 = 180°
Suppl.
Symbols
22
Glide reflection
Translation parallel to mirror plane m by 1/2; mirroring along m
m
1.3 Theory of symmetry Symmetry elements
Quelle: http://ruby.chemie.uni-freiburg.de/Vorlesung/Vorlagen/methoden_II_21.pdf 23
8 basic symmetry elements
1, 2, 3, 4, 6, 1 = i, 2 = m, 3 = 3+i, 4, 6 = 3/m
Meaningful combination and application to a point
32 point groups
crystal systems
1.3 Theory of symmetry space groups
TranslationsCenter: (P), R, C, I, F
Glide reflection: n c, a, n
Screw axis: 21, 41, …
230 space groups 7 crystal systems
24
Types of Bravais lattices
• 7 crystal systems
• P, C (A, B), I, F - centered
1.3 Theory of symmetry 14 Bravais lattices
1. Theorie
Possible lattices with and without centering
2525
Why is there no base-centered
(cC) cubic?
When looking at the atoms in
pairs, this is just a primitive cubic
cell!
Base-centered cubic (cC)
=
Primitive cubic (cP)
1.3 Theory of symmetry 14 Bravais lattices
1. Theorie 2626
1.3 Theory of symmetry space groups
List of space groups
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1.3 Theory of symmetry Space groups
Quelle: http://ruby.chemie.uni-freiburg.de/Vorlesung/Vorlagen/e_in_fk_3_2.pdf 28
Quelle: http://ruby.chemie.uni-freiburg.de/Vorlesung/Vorlagen/e_in_fk_3_2.pdf 29
Quelle: http://ruby.chemie.uni-freiburg.de/Vorlesung/Vorlagen/e_in_fk_3_2.pdf
1.3 Theory of symmetry Space groups
Point of view:
30
Crystal system Order of
viewing axis:Example
1.3 Theory of symmetry Example
Crystal structure Structural data
X-ray diffractogram
31
1.3 Theory of symmetry Example
Quelle: http://ruby.chemie.uni-freiburg.de/Vorlesung/Vorlagen/methoden_II_25.pdf 32
1.3 Theory of symmetry Determination of the space group
Extinction means that several reflexes
of groups of reflexes are missing
from the diffractogram.
By indexing the missing reflexes it
is possible to determine the symmetry
element that is responsible.
complete table with all extinction rules
can be found in the annex
33
The space group can be determined from reflexes in the diffractogram!
Extinction Reflex Condition Element Remarks
2. Methods of diffraction: Powder diffraction and structure elucidation
Modern Crystallographic Methods
2.1 Different diffractometers
2.2 Detectors
2.3 Monochromators
2.4 Sample preparation, conditions and mistakes
2.5 Working with diffraction data
2.6 Profile fitting and profile functions
2.7 Quality factors
2.8 Structure elucidation from powder patterns: Rietveld analyses
2.9 Examples
34
2. Methods of diffraction: Powder diffraction and structure elucidation
General aspects:
Structure elucidation and refinement from powder data is not a trivial task!
Structure elucidation and refinement from single crystal data is easier
than from microcrystalline samples!
High demands on the diffractometer, the adjustment,
sample preparation and measurement parameters.
Structure solution from powders possible, but much more difficult than from
single crystal data. The amount of independent data for SC measurements is
much bigger.
For a Rietveld refinement, structural data must be available.
(initial model).
Many parameters in Rietveld refinement are directly interdependent.
Experience and practice are essential for a successful determination.
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2.1 Different diffractometers
Focusing
in powder
diffractometry
Seemann-Bohlin
diffractometer
For flat samples:
violation of
focusing condition
and
broadening of the
reflexesAssembly
36
2.1 Different diffractometers
Design
Guinier diffractometer
Detector on focal cylinder
z. B. Fa. Huber, München
Blanking of Cu Ka2 radiation
by quartz monochromator
also for focusing!
Very high resolution of less than
0.1 mm on film
Monochromatic radiation
Three to four simultaneous
measurements possible (on film)
37
2.1 Different diffractometers
Design
Bragg-Brentano
Source and detector are
moving to keep focus
Use of films not possible
Good resolution and high
intensity
38
2.1 Different diffractometers
STOE
Powder diffractometer
curved CCD on the
focal cylinder
Very fast measurements
Lower resolution than Guinier
Can be used to observe chemical reactions
39
2.2 Detectors
What reaches the detector?
For refinement we only want signals coming from the sample!
Device, measurement method and detector influence the measurement data.
1) SC: Scintillation counter
2) CCD: Spatially resolved detector
3) Kb-filter, no monochromator
4) Film
12
3 4
Sample +
background
SampleBackground
40
2.2 Detectors
Kinds of detectors:
Image plates: BaFBr doped with Eu2+; X-ray Eu2+ to Eu3+ and e-
(F-center); red laser causes e- to return to the ground state
and emit light
Proportional counter: Geiger-Müller counter
Scintillator: X-ray photon reaches NaI:Tl, produces a large number of
visible photons that can be detected
Spatial resolution detector: Proportional counter which detects the electrons time
delayed at the end of the wire
OEDProportionalitätszählrohr
Szintillationszähler
41
2.2 Detectors
Slit size influences the intensity
and resolution of the
diffractometer
Variation of the detector slit
from 0.025 to 0.4 °
0.025° Low intensity
High resolution
0.4° High intensity
Low resolution
For structural elucidation from powder samples
Aperture and slits
42
2.3 Monochromators
Wavelengths: Cu Ka1 1.54053 Å
Mo Ka 0.71073 Å
Ag Ka 0.56089 Å
Selection of a
defined
wavelength:
(cf Bragg!)
Filters
Filter
Monochromators
43
2.3 Monochromators
General function of an X-ray monochromator
Single crystal monochromator after Johansson
Cut and
curved
single crystal
Diffraction on
selected lattice planes
Focusing on
one point
44
2.4 Sample preparation, requirements and mistakes
Sources of error in a Rietveld refinement
• Matrix
• Atmosphere
• Deviation of angle
• Wrong intensities
• Instrument profile
• Cystallite size and distribution
• Unknown impurity phases
• Determination of peak position
• Background correctin
• …
45
2.4 Sample preparation, requirements and mistakes
Sources of error in a Rietveld refinement
• Matrix
• Atmosphere
• Deviation of angle
• Wrong intensities
• Instrument profile
• Cystallite size and distribution
• Unknown impurity phases
• Determination of peak position
• Background correctin
• …
46
2.4 Sample preparation, requirements and mistakes
Diffractograms of various matrices
Influence of the atmosphere
47
2.4 Sample preparation, requirements and mistakes
Sources of error in a Rietveld refinement
• Matrix
• Atmosphere
• Deviation of angle
• Wrong intensities
• Instrument profile
• Cystallite size and distribution
• Unknown impurity phases
• Determination of peak position
• Background correctin
• …
48
2.4 Sample preparation, requirements and mistakes
Systematic errors in diffraction angle and intensity
Angle of diffraction: - Misalignment of the diffractometer
- Mechanics of the diffractometer
- Zero shift
- Axial divergence of the x-ray beam
- Superposition of Ka1 und Ka2 resulting in
deformation of peaks
- Electronics of the detector
Sample:
- Level of the sample
- Transparency of the sample
Intensity: - Texture
- Overspill
49
2.4 Sample preparation, requirements and mistakes
Angle deviation due to sample height displacement
is one of the most common sources of measurement
problems!
50
2.4 Sample preparation, requirements and mistakes
Deviations in the diffraction angle, caused by
systematic errors
51
2.4 Sample preparation, requirements and mistakes
Errors of intensity: Texture and overshooting
no
texture
strong
texture
Sample has not
been illuminated
correctly
52
2.4 Sample preparation, requirements and mistakes
Sources of error in a Rietveld refinement
• Matrix
• Atmosphere
• Deviation of angle
• Wrong intensities
• Instrument profile
• Cystallite size and distribution
• Unknown impurity phases
• Determination of peak position
• Background correctin
• …
53
2.4 Sample preparation, requirements and mistakes
The instrumental profile is the
product of all instrument-related
effects.
Instrumental profiles are a constant
that is not influenced by sample
preparation.
Instrumental profiles should either
be known prior to refinement or
be determined during the
refinement.
The experimentally observed peak is
a result of specimen profile and instrument
profile.
54
2.4 Sample preparation, requirements and mistakes
Sources of error in a Rietveld refinement
• Matrix
• Atmosphere
• Deviation of angle
• Wrong intensities
• Instrument profile
• Cystallite size and distribution
• Unknown impurity phases
• Determination of peak position
• Background correctin
• …
55
2.4 Sample preparation, requirements and mistakes
Requirements on the crystallites:
Crystallites should have a size between
1-10 µm.
at 1 µm size: 38000 particles in diffraction condition
at 10 µm size: 760 particles in diffraction condition
at 40 µm size: 12 particles in diffraction condition
Rotating
the sample
Particle size distribution causes errors in the intensity of the peaks!
56
Crystallite size and size
distribution play a decisive
role in refinement.
If the crystallite size is too
small, the peaks broaden
and a poor resolution
results.
2.4 Sample preparation, requirements and mistakes
Sources of error in a Rietveld refinement
• Matrix
• Atmosphere
• Deviation of angle
• Wrong intensities
• Instrument profile
• Cystallite size and distribution
• Unknown impurity phases
• Determination of peak position
• Background correctin
• …
Chapter 2.5
57
2.5 Working with diffraction data
Bragg equation
…followed by indexing
Zero point correction
Locating the peak
Background subtraction
Smoothing
Data collection
58
Removal of Ka1
2.5 Working with diffraction data
Determination of peak location via 2nd derivative
Profile
1st derivative
2nd derivative
59
2.5 Working with diffraction data Smoothing
Can be used with noisy diffractograms.
Results can be wrong!
Proper choice of step
width and measurement
time are crucial!
60
2.5 Working with diffraction data Background subtraction
Wrong description can influence
the peak profile and intensities
61
2.5 Working with diffraction data Data correction
1. Zero-point correction
2. Identification of impurity phases
3. Analysis/exclusion of systematic errors via calibration with a standard
- external standard
- internernal standard
62
2.5 Working with diffraction data Rietveld
Rietveld-Refinement:
Developed by Hugo Rietveld (*1932) between 1967 and 1969, gained
importance with the emergence of powerful computers in the 1980s
1. Least-Squares-Refinement of the free parameters of a theoretical powder
diffraction pattern against all data points of an experimental powder
pattern
2. Free parameters
• structure (lattice constants, atom coordinates, etc.)
• background and profile parameters
3. Description of
• structure (multiple phases if applicable)
• sample: crystallinity, size of crystallites, stress, etc.
• equipment and measurement specific parameters
63
2.6 Profile functions
SiO2
triplet
Cu-Ka1 and a2
Profile parameters and their usage
64
2.6 Profile functions
• Gauß
• Lorentz (Cauchy)
• Pearson VII
• Pseudo-Voigt
• Definition der Linienbreite
2ln4;22exp 0
2
2
00
C
CCG ki
kk
4;
221
120
2
2
0
0
CC
CL
ki
k
k qq
5.0
122;22
1241
2
1
0
2
2
0
m
mC
CP
mm
ki
k
m
k
VII
GLpV 1
WVU kkk qq tantan22
65
2.6 Profile functions
Why are profile functions useful?
Depending on the measurement
conditions the peak intensity can vary
from one measurement to the other.
This can be corrected by
profile fitting.
66
2.6 Profile functions
Treatment of an asymmetric peak with
various profile functions
Flanks too narrow
Middle part too broad
Maximum shifted
Gauß
Lorentz
split-pearsonVII Best fit!
67
• Profile R-factor Rp
• weighted Rwp
• Bragg R-factor
• The expected Rf
• The goodness of fit
2.7 Quality factors
i
io
i
icio
py
yy
R
2
1
2
2
i
ioi
i
icioi
wpyw
yyw
R
i
ko
i
kcko
BI
II
R
2
1
2exp
i
ioi yw
PNR
2
exp
2
R
R
PN
yyw
GOFwpi
icioi
68
Profile-factors
quantify the goodness
of the profile fitting
at each data point
Quality factor on the basis
of individual peaks Ik
(if individual peaks can be
resolved)
N Number of data points
P Number of parameters
yi intensity at data point i
yi Contribution of structure + background
wi weight factor
Ik integrated intensity
Estimation of the
goodness of the
refinement
2.7 Quality factors Step width and measurement duration
Dependence of Rbragg
on the step width and
dwell time
Rule:
0.02 to 0.04° step width
sufficient
Dwell time has to be
considered individually
for each specific case!
69