Modern Crystallographic Methods€¦ · 2. Methods of diffraction: Powder diffraction and structure elucidation Modern Crystallographic Methods 2.1 Different diffractometers 2.2 Detectors

$Page 1: Modern Crystallographic Methods€¦ · 2. Methods of diffraction: Powder diffraction and structure elucidation Modern Crystallographic Methods 2.1 Different diffractometers 2.2 Detectors$
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)


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

http://ruby.chemie.uni-freiburg.de/Vorlesung/methoden_0.html

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



7



In this lecture:

STM

Scanning Tunnel

Microscopy

AFM

Atomic Force

Microscopy

http://www.uni-tuebingen.de/Teilchenoptik/html/fprakt/tem.html 8



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


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


The symmetry is determined by the arrangement of the atoms.

Structure

and data

Diffractogram

with indices



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

15

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

19

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

20

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


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



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

27

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


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



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.

35


Focusing

in powder

diffractometry

Seemann-Bohlin

diffractometer

For flat samples:

violation of

focusing condition

and

broadening of the

reflexesAssembly

36


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


Design

Bragg-Brentano

Source and detector are

moving to keep focus

Use of films not possible

Good resolution and high

intensity

38


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



• Matrix

• Atmosphere








• …

46


Diffractograms of various matrices

Influence of the atmosphere

47



• Matrix

• Atmosphere








• …

48


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


Angle deviation due to sample height displacement

is one of the most common sources of measurement

problems!

50


Deviations in the diffraction angle, caused by

systematic errors

51


Errors of intensity: Texture and overshooting

no

texture

strong

texture

Sample has not

been illuminated

correctly

52



• Matrix

• Atmosphere








• …

53


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



• Matrix

• Atmosphere








• …

55


Requirements on the crystallites:

Crystallites should have a size between

1-10 µm.

at 1 µm size: 38000 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.



• Matrix

• Atmosphere








• …

Chapter 2.5

57


Bragg equation

…followed by indexing

Zero point correction

Locating the peak

Background subtraction

Smoothing

Data collection

58

Removal of Ka1


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


• Gauß

• Lorentz (Cauchy)

• Pearson VII

• Pseudo-Voigt

• Definition der Linienbreite

2ln4;22exp 0

2

2

00

C

CCG ki

kk

qq

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

qq

GLpV 1

WVU kkk qq tantan22

65


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


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

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Modern Crystallographic Methods€¦ · 2. Methods of diffraction: Powder diffraction and structure elucidation Modern Crystallographic Methods 2.1 Different diffractometers 2.2 Detectors