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Theory and practice of X-ray diffraction

experiment

Power Point presentation from lecture 2 is

available at http//dl.dropbox.com/u/23622306/UIU

C/Lecture202.ppt

Power Point presentation from lecture 4 is

available at http//dl.dropbox.com/u/23622306/UIU

C/lecture204.ppt

Grain size and types of diffraction experiment

- Single crystal
- Bulk powder
- Nanopowder
- Amorphous/glass

Reciprocal space

- Direct space relates to atoms in the unit cell,

while reciprocal space relates to peaks in

diffraction experiment. - In diffraction experiment planes of atoms in the

crystal act as a selective mirror and reflect

(in the same way as in optics) incoming beam of

x-rays if specific geometric conditions are

satisfied. - Vectors in reciprocal space correspond to

families of planes in the crystal (the direction

of the rec. vector is normal to the family of

planes). - In a conventional crystal (as opposed to

incommensurately modulated crystal or

quasi-crystal) vectors in reciprocal space can

occupy only points on a 3-dimensional grid. - Grid coordinates of reciprocal vectors are known

as Miller indices hkl - Geometry of the reciprocal space grid is

determined by the unit cell of the crystal. And

can be directly measured. - D-spacing is equal to inverse of the reciprocal

vector length. - Coordinates of vectors in reciprocal space are

described in laboratory (instrument-related)

reference coordinate system.

y

Orientation matrix

- UB matrix relates the Miller indices of

reciprocal vector (hkl) with its Cartesian

coordinates in lab reference system (xyz) at zero

goniometer position. - xyzUB hkl
- Columns in the orientation matrix are coordinates

of the principal vectors in reciprocal space 1 0

0 , 0 1 0 and 0 0 1. - UB matrix is composed of two sub-matrices, U and

B - U describes the orientation of the crystal axes

with respect to the laboratory reference system,

while B stores information about the unit cell

parameters. - By inverting the above equation one can calculate

what are the Miller indices of a measured

reciprocal vector xyz - hklUB-1 xyz

z

hkl

X-ray

x

y

Diffraction condition and Bragg equation for a

single crystal

1q

hkl

diffracted X-ray

nl2dsinq

2q

incident X-ray

1q

2q

Diffraction geometry is analogous to reflection

in a selective mirror Diffraction is effective

only at selected incidence angles with respect to

the reflecting plane, determined by the Bragg

equation. When diffraction is effective

reflection angle equals to the incidence angle,

with the reciprocal vector bisecting the

two. Intensity of the reflected beam is not

equal to the intensity of the incident beam, it

is related to the mean electron density within

the direct lattice plane.

Explanation of Bragg derivation

nl2dsinq

Ewald construction

- In diffraction experiment we measure the

direction and intensity of the diffracted beams

as well as the orientation of the crystal

corresponding to each of the diffraction peaks. - Diffraction does not occur at any arbitrary

crystal orientation. In general to bring a given

reciprocal vector to a diffraction position one

may need to rotate the sample using goniometer or

change the incident wavelength (an exception is

polychromatic experiment in which a range of

incident wavelengths is available). - When the diffraction event occurs, the following

relation between the incident and diffracted beam

vectors, incident wavelength and the reciprocal

vector is satisfied - R is the goniometer rotation matrix bringing the

crystal from the zero-goniometer position to the

position at which maximum peak intensity is

observed. - In the Ewald construction the center of the

crystal and the center of reciprocal space do not

coincide.

1/l(S0Sd)R xyz

Rxyz

Sd

S0

Radius of the Evald sphere is 1/l

Experimental approach to single crystal

diffraction

- Effective diffraction vs. observed diffraction
- Cause effective diffraction and find (observe)

diffraction signal - Measure Sd and R for a number of diffraction

peaks - Calculate xyz using the Ewald equation
- Find orientation matrix/index the peaks (assign

Miller indices) - Measure/calculate peak intensities I(hkl)

Wide oscillation 13

Monochromatic SXD experiment

Goniometry in matrix representation

Atomic scattering factor

- The incoming X-rays are scattered by the

electrons of the atoms. As the wavelength of the

X-rays (1.5 to 0.5 A) is of the order of the atom

diameter, most of the scattering is in the

forward direction. For neutrons of the same

wavelength the scattering factor is not angle

dependent due to the fact that the atomic nucleus

is magnitudes smaller than the electron cloud. It

is also obvious that the X-ray scattering power

will depend on the number of electrons in the

particular atom. The X-ray scattering power of an

atom decreases with increasing scattering angle

and is higher for heavier atoms. A plot of

scattering factor f in units of electrons vs.

sin(theta)/lambda shows this behavior. Note that

for zero scattering angle the value of f equals

the number of electrons.

Electron density around an isolated atom ?(r) is

spherically symmetric, with max at nucleus

position, and falls off smoothly with distance.

Cromer and Mann 9-parameter equation

Anomalous dispersion

The scattering factor contains additional

(complex) contributions from anomalous dispersion

effects (essentially resonance absorption) which

become substantial in the vicinity of the X-ray

absorption edge of the scattering atom. These

anomalous contributions can be calculated as well

and their presence can be exploited in the MAD

phasing technique.

Excitation Scans We can observe the ?f by

measuring the absorption of the x-rays by the

atom. Often we us the fluorescence of the

absorbing atom as a measure of absorptivity. That

is, we measure an excitation spectrum.

How to get ?f ? The real, dispersive component

is calculated from ?f by the Kramers-Kronig

relationship. Very roughly, ?f is the negative

first derivative of ?f.

From Ramakrishnans study of GH5

Thermal vibrations and Debye-Waller factor

- There is an additional weakening of the

scattering power of the atoms by the so called

Temperature-, B- , or Debye-Waller factor. This

exponential factor is also angle dependent and

effects the high angle reflections substantially

(one of the reasons for cryo-cooling crystals is

to reduce the attenuation of the high angle

reflections due to this B-factor).

Structure factor

Lorenz correction

Accounts for the different speed with which the

reciprocal vectors move through the Ewald sphere

Polarization correction

Other intensity corrections

- Absorption correction
- Extinction correction
- Preferred orientation
- Illuminated volume
- Incident intensity

Symmetry of peak intensities. Friedel pairs and

Laue classes

Friedel's law, named after Georges Friedel, is a

property of Fourier transforms of real

functions.1 Given a real function , its Fourier

transform has

the following properties. where is the

complex conjugate of . Centrosymmetric points

are called Friedel's pairs. The Friedel pair

symmetry is broken by anomalous dispersion. The

effect is strongly pronounced for incident

energies close to the absorption edge.

Systematic absences and space group determination

A centered hkl k l 2n B centered h l

2n C centered h k 2n F centered k l 2n,

h l 2n, h k 2n I centered h k l

2n R (obverse) -h k l 3n R (reverse) h -

k l 3n

Glide reflecting in a 0kl b glide k

2n c glide l 2n n glide k l

2n d glide k l 4n Glide reflecting in

b h0l a glide h 2n c glide l

2n n glide h l 2n d glide h l

4n Glide reflecting in c hk0 b glide k

2n a glide h 2n n glide k h

2n d glide k h 4n

Screw 100 h00 21, 42 h 2n 41,

43 h 4n Screw 010 0k0 21, 42 k

2n 41, 43 k 4n Screw 001 00l 21,

42, 63 l 2n 31, 32, 62, 64 l

3n 41, 43 l 4n 61, 65 l 6n

Point detector experiment

- Scintillator detector converts x-ray to electric

signal - Center the sample on rotation axis and with the

beam - Search for peaks (usually 10-20) at different

2theta and sample orientations - Center the peaks that were found
- Determine orientation matrix
- Find more peaks to constrain and refine the

orientation matrix better - Calculate position for a list of peaks that need

to be measured based on orientation matrix - Position each peak individually, record a rocking

curve/peak profile - Integrate, scale and correct peak intensities
- Solve/refine the structure

Area detector experiment

- CCD detector detects visible light. Phosphor

screen in front of the detector converts x-ray to

visible light. - Center the sample on rotation axis and with the

beam - Collect diffraction images while rotating the

sample - Determine detector coordinates and sample

orientations for each diffraction peak - Reconstruct the reciprocal space in 3-d and

determine the orientation matrix (index) - Predict peak positions in recorded diffraction

images and retrieve peak intensities (structure

factor amplitudes) - Solve/refine the structure

Area detector and step scan approach

With wide rotation image we determine the

direction of the diffracted beam Sd but the

rotation angles (necessary to calculate R) at

which each peak occurs are unknown. Step scan

allows to determine R for each peak by finding a

step image at which the peak has maximum

intensity. With R and Sd we can calculate xyz.

Step scan images

General algorithm for analysis of single-crystal

XRD data

Image Initial peak list (Sd) Step scan analysis

(R) Orientation matrix (UB) Predicted peak

list Integrated intensities I(hkl) Scaled and

corrected intensities

Figures of merit

Demonstration of single-crystal data processing

with GSE_ADA

Powder diffraction

- Sample is composed of a very large (preferably

gt106) number of single crystal specimens with

random distribution of crystal orientations. - Diffraction for all reciprocal vectors occurs (is

effective) at any arbitrary orientation of the

sample. Sample rotation is not necessary. - Single crystal diffraction peaks (directional

beams) become cones of radiation. - Peaks corresponding to different reciprocal

vectors (different hkls) that have similar length

(d-spacing) overlap. - Diffraction signal carries only information about

the length of the reciprocal vectors, but not

their orientation.

Different experimental approaches to powder

diffraction

- Polychromatic EDX approach
- White beam
- Point energy-dispersive detector at fixed angle
- Non-scanning signal accumulation
- Can provide access to large d-spacing range w/o

much of angular access - Suffers from preferred orientation problems
- Peak usually quite broad
- Peak intensity interpretation difficult

(correction) - Monochromatic approach with point detector
- Mono beam
- Point detector at variable angle
- Scanning signal accumulation
- Suffers from preferred orientation problems
- Can provide patterns with very narrow peak

profiles - Monochromatic approach with area detector
- Mono beam
- Area detector at fixed or variable angle/position
- Non-scanning signal accumulation
- Does not suffer from preferred orientation

problems

High resolution powder diffraction with analyzer

crystal

Bragg-Brentano powder diffractometer

Integration of 2-dimensional diffraction pattern

- Calibration of detector geometry
- Detector coordinates of the point of intersection

of the incident beam and the detector surface - Sample-to-detector distance
- Detector non-orthogonality with respect to the

beam - Pixel size
- Goniometer geometry calibration
- Goniometer zeros
- Goniometer axis alignment with the detector

orientation - Instrumental function
- Calibrating with a diffraction standard
- Cake transform

Energy dispersive powder diffraction

Ge solid state semiconductor detector

Multiple wavelengths

Powder pattern fitting techniques

- Individual peak fitting
- Peak positions are not constrained they are

individually refined - Peak profiles are not related with each other
- Closely overlapping peaks are very hard to deal

with - LeBail (unit-cell constrained) refinement
- Individual peak positions are not refined they

are calculated from the unit cell parameters

which are refined - Individual peak profiles are not refined there

is a global function (e.g. Cagliotti function)

that connects all peak profiles. - Peak intensities are refined free (not

constrained by the structure model) - Gives much more reliable way of refining

positions and intensities of closely overlapping

peaks. - Rietveld (model-biased) refinement
- Individual peak positions are not refined they

are calculated from the unit cell parameters

which are refined - Individual peak profiles are not refined there

is a global function (e.g. Cagliotti function)

that connects all peak profiles. - Individual peak intensities are not refined free,

they are calculated from the structure model

which is refined.

Peak width analysis

- Instrumental function
- Focused beam divergence away from the sample
- Detector pixel size
- Phosphor point spread function
- Sample-related factors
- Scherrer formula
- Hall-Williamson plot
- Cagliotti function
- Deconvolution

size K l / FW(S) cos(q)

FW(S) cos(q) K l / size 4 strain sin(q)

FW(S)D FWHMD - FW(I)D

The dimensionless shape factor K has a typical

value of about 0.9, but varies with the actual

shape of the crystallite. Scherrer formula is

not applicable to grains larger than about 0.1

µm, which precludes those observed in most

metallographic and ceramographic microstructures.

Peak width analysis

13nm nano powder

FW(S) cos(q) K l / size 4 strain sin(q)

Bulk fine powder

SM and Quantitative analysis demonstration

Tools of the tradePowder diffraction

- Image integration
- Fit2d
- Powder3d
- Powder pattern analysis
- Jade
- GSE_Shell
- Indexing
- MDI Jade
- Treor (index permutation method)
- Ito (zone indexing method)
- Dicvol (dichotomy algorithm)
- Database and SM
- American Mineralogist Crystal Structure Database
- Project RRUFF
- Crystallography Open Database
- ICSD
- Peak and pattern fitting, refinement of structure
- GSAS

Tools of the tradeSingle-crystal diffraction

- Commercial
- Bruker Saint/SMART, APEX
- HKL Research Denzo/HKL2000
- Oxford (Agilent) Crysalis
- Academic
- XDS (Wofgang Kabsch) Linux
- GSE_ADA (Przemek Dera) Windows

Structure determination

Structure identification Logical

choices Search and match Indexing-based Structur

e refinement (requires the structure model to be

approximately known) Rietveld method Single-crys

tal refinement Structure solution (ab

initio) direct methods charge

flipping simulated annealing

High pressure apparatus

- Sample is immersed in hydrostatic liquid, which

freezes at some point during compression (usually

below 10 GPa). - Diffraction pressure calibrant is placed in the

sample chamber along with the sample. - Both incident and diffracted beam travel though

diamonds, Be disks, pressure medium, and sample.

(No Transcript)

High-pressure crystallography challenges

- Experimental challenges
- Very small sample (lt0.01mm)
- Angular access restricted (low completeness)
- Absorption limits the incident energy to gt15 keV
- Absorption and extinction affect intensity

measurement - High background (scattering, Compton, etc.)
- Multiple SXD diffraction signal
- Contamination by PXD signal
- Poor sample quality (strain, multi-grain

assemblages) - More challenging sample centering
- Beam size vs. sample size

Homework (detailed instructions will be sent by

Friday 9/16), due on Tuesday 9/27

- Download and install Rosetta. Feel free to use

any other program of your preference to complete

the tasks described below (in which case you can

skip 1). - Download the example unknown powder patterns

xx1.chi, xx2.chi, xx3.chi - Identify minerals present in each sample
- Download the example QA powder patterns of

quartz-albite mixture QA25.chi, QA50.chi. The

number in file name stands for the albite content

(as prepared). - Verify how accurate is the quantitative analysis

done with Rosetta.

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