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Structure elucidation from XRD

X-rays are electromagnetic waves with a wavelength in the range of interatomicdistances (0.1-10 )

topics

Highlight on instrumentation Crystallography X-ray diffraction from polycrystalline

samples (powders); XRD data interpretation

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10 20 30 402

Atomic distribution in

the unit cell

Peak relative intensities

Unit cell Symmetry and size

Peak positions

a

cb

Peak shapes

Particle size and defects

Background

Diffuse scattering, sample holder,

matrix, amorphous phases, etc...

X-ray Powder Diffraction

Instrumentation highlight

Details should be given in Instrumentation lecture !

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What is X-ray Diffraction?

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Our powder diffractometers typically use the Bragg-Brentano geometry.

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The incident angle, , is defined between the X-ray source and the sample. The diffracted angle, 2, is defined between the incident beam and the

detector angle. The incident angle is always of the detector angle 2 . In a :2 instrument (e.g. Rigaku RU300, our XRD), the tube is fixed, the

sample rotates at /min and the detector rotates at 2 /min. In a : instrument (e.g. PANalytical XPert Pro, Bruker), the sample is fixed

and the tube rotates at a rate - /min and the detector rotates at a rate of /min.

X-ray tube

Detector

Braggs law is a simplistic model to understand what conditions are required for diffraction.

For parallel planes of atoms, with a space dhkl between the planes, constructive interference only occurs when Braggs law is satisfied.

In our diffractometers, the X-ray wavelength is fixed. Consequently, a family of planes produces a diffraction peak only at a specific angle . Additionally, the plane normal must be parallel to the diffraction vector

Plane normal: the direction perpendicular to a plane of atoms Diffraction vector: the vector that bisects the angle between the incident and diffracted beam

The space between diffracting planes of atoms determines peak positions. The peak intensity is determined by what atoms are in the diffracting plane.

sin2 hkld= d

hkld

hkl

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constructive interference results in diffraction line, but destructive interference does not

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monochromatic/single wavelength with highest intensity is used as the X-ray source

- 2 Scan

The The -- 22 scan maintains these angles with the scan maintains these angles with the sample, detector and Xsample, detector and X--ray sourceray source

Normal to surface

Only planes of atoms that share this normal will be seen in the - 2 Scan

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Solid matter

crystalline

The atoms and molecules are arranged in a random way similar to the disorder we find in a liquid do not form crystallites. Glasses are amorphous materials.Small particles with no long-range order (100 )

The atoms are arranged in a regular pattern, and there is as smallest volume element that by repetition in three dimensions describes the crystal (a unit cell).Long range order (>103 molecules)

polycrystallineSolids which contain many small, randomly oriented and joined crystallites/grains

2 (o)10 20 30 40 50 60 70 80 90

Inte

nsity

(a.u

)

2 h3 h

4 h

5 h

A [101]

AA A A

Am

Bnon-crystalline (amorphous)

single crystalSolid which contains single crystallite

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Fig. 10.2: Crystalline solids are characterized by highly ordered arrays of atoms, ions or molecules and give distinct X-ray diffraction patterns.

Amorphous solids have no long-range ordering in their structures.

What type of attractive forces are most important in the quartz structure?

Glass (SiO2), has the same chemical composition as quartz but exhibits no extensive, repeating organization.

Quartz (SiO2), a crystalline solid and the most abundant mineral in the earths crust)

Crystalline Versus Amorphous

Quartz Obsidian

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Crystalline materials are characterized by the orderly periodic arrangements of atoms.

The unit cell is the basic repeating unit that defines a crystal. Parallel planes of atoms intersecting the unit cell are used to define

directions and distances in the crystal. These crystallographic planes are identified by Miller indices.

The (200) planes of atoms in NaCl

The (220) planes of atoms in NaCl

The atoms in a crystal are a periodic array of coherent scatterers and thus can diffract light.

Diffraction occurs when each object in a periodic array scattersradiation coherently, producing concerted constructive interference at specific angles.

The electrons in an atom coherently scatter light. The electrons interact with the oscillating electric field of the light wave.

Atoms in a crystal form a periodic array of coherent scatterers. The wavelength of X rays are similar to the distance between atoms. Diffraction from different planes of atoms produces a diffraction pattern,

which contains information about the atomic arrangement within the crystal

X Rays are also reflected, scattered incoherently, absorbed, refracted, and transmitted when they interact with matter.

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X-Ray Powder Diffraction (XRPD) uses information about the position, intensity, width, and shape of diffraction peaks

in a pattern from a polycrystalline sample.

The x-axis, 2theta, corresponds to the angular position of the detector that rotates around the sample.

This is an example of an XRD spectrum for quartz.

Peaks result from constructive interference for the reflected X-rays

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Only crystallites having reflecting planes (h,k, l) parallel to the specimen surface will contribute to the reflected intensities.

If we have a truly random sample, each possible reflection from a given set of h, k, l planes an equal number of crystallites contributing to it.

We only have to rock the sample through the glancing angle THETA in order to produce all possible reflections.

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Crystallography: reviews

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There are seven crystal systems also knows as BravaisStructures.

In Chapter 11 we will cover only the cubic systems, sc, fcc, and bcc.

Sides:

a = b = c

Angles

= = = 90

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h, k, l are Miller indices a, b, c are unit cell distances , , are angles between the lattice directions

Complexity of calculations is dependent on the symmetry of the crystal system.

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XRD for polycrystalline samples

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Powder diffraction data consists of a record of photon intensity versus detector angle 2.

Diffraction data can be reduced to a list of peak positions and intensities Each dhkl corresponds to a family of atomic planes {hkl} individual planes cannot be resolved- this is a limitation of powder diffraction

versus single crystal diffraction

{202}

{113}

{006}

{110}

{104}

{012}

hkl

1.41.9680

100.02.0903

1.92.1701

36.12.3852

85.82.5583

49.83.4935

Relative Intensity (%)

dhkl ()

328.000025.7200

380.000025.6800

456.000025.6400

732.000025.6000

1216.000025.5600

1720.000025.5200

2104.000025.4800

1892.000025.4400

1488.000025.4000

1088.000025.3600

752.000025.3200

576.000025.2800

460.000025.2400

372.000025.2000

Intensity [cts]

Position[2]

Raw Data Reduced dI list

A single crystal specimen in a Bragg-Brentano diffractometer would produce only one family of peaks in

the diffraction pattern.

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At 20.6 2, Braggs law fulfilled for the (100) planes, producing a diffraction peak.

The (110) planes would diffract at 29.3 2; however, they are not properly aligned to produce a diffraction peak (the perpendicular to those planes does not bisect the incident and diffracted beams). Only background is observed.

The (200) planes are parallel to the (100) planes. Therefore, they also diffract for this crystal. Since d200 is d100, they appear at 42 2.

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A polycrystalline sample should contain thousands of crystallites. Therefore, all possible diffraction peaks should

be observed.

2 2 2

For every set of planes, there will be a small percentage of crystallites that are properly oriented to diffract (the plane perpendicular bisects the incident and diffracted beams).

Basic assumptions of powder diffraction are that for every set of planes there is an equal number of crystallites that will diffract and that there is a statistically relevant number of crystallites, not just one or two.

About 95% of all solid materials can be described as crystallineWhen X-rays interact with a crystalline substance (phase) gets a diffraction patternIn 1919 A.W.Hull gave a paper titled, A New Method of Chemical Analysis. Here he pointed out that .every crystalline substance gives a pattern; the same substance always gives the same pattern; and in a mixture of substances each produces its pattern independently of the others. The X-ray diffraction pattern of a pure substance a fingerprint finger print identification

The powder diffraction method is thus ideally suited for characterization and identification of

crystalline/polycrystalline phases

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Today about 50,000 inorganic and 25,000 organic single component, crystalline phases, diffraction patterns have been collected and stored on magnetic or optical media as standardsThe main use of powder diffraction is to identify components in a sample by a search/match procedureThe areas under the peak are related to the amount of each phase present in the sampleFor single-phase materials the crystal structure can be obtained directly using X-Ray diffraction (XRD)XRD can be used :

for phase identification (With the help of a database of known structures)

to determine crystal size, strain and preferred orientation of polycrystalline materials

the related technique of X-ray reflection enables accurate determination of film thickness

XRD applications and interpretation

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You can use XRD to determine Crystalline phase and sample identification Phase Composition of a Sample

Quantitative Phase Analysis: determine the relative amounts of phases in a mixture by referencing the relative peak intensities

Unit cell lattice parameters and Bravais lattice symmetry Index peak positions Lattice parameters can vary as a function of, and therefore give you information

about, alloying, doping, solid solutions, strains, etc. Residual Strain (macrostrain) Crystal Structure

By Rietveld refinement of the entire diffraction pattern Epitaxy/Texture/Orientation Crystallite Size and Microstrain

Indicated by peak broadening Other defects (stacking faults, etc.) can be measured by analysis of peak shapes

and peak width in-situ studies (evaluate all properties above as a function of time,

temperature, and gas environment)

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Phase Identification The diffraction pattern for every phase is as unique as your

fingerprint Phases with the same chemical composition can have drastically

different diffraction patterns. Use the position and relative intensity of a series of peaks to match

experimental data to the reference patterns in the database

Databases such as the Powder Diffraction File (PDF) contain d-I lists for thousands of crystalline phases.

The PDF contains over 200,000 diffraction patterns. Modern computer programs can help you determine what phases are

present in your sample by quickly comparing your diffraction data to all of the patterns in the database database XRD pattern matching

The PDF card for an entry contains a lot of useful information, including literature references.

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XRD patterns of furnace materials and reference patterns of identified phases

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23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 412 (deg.)

Inte

nsity

(a.u

.)

00-043-1002> Cerianite- - CeO2

Crystallite Size and Microstrain

Crystallites smaller than ~120nm create broadening of diffraction peaks this peak broadening can be used to quantify the average crystallite size of

nanoparticles using the Scherrer equation must know the contribution of peak width from the instrument by using a

calibration curve microstrain may also create peak broadening

analyzing the peak widths over a long range of 2theta using a Williamson-Hull plot can let you separate microstrain and crystallite size

cosKD =

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cosKD =

Different crystallite sizes

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Quantitative Phase Analysis

With high quality data, you can determine how much of each phase is present

must meet the constant volume assumption (see later slides)

The ratio of peak intensities varies linearly as a function of weight fractions for any two phases in a mixture

need to know the constant of proportionality RIR method is fast and gives semi-

quantitative results Whole pattern fitting/Rietveld refinement is

a more accurate but more complicated analysis

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

X(phase a)/X(phase b)

I(pha

sea)

/I(ph

ase

b)..

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Unit Cell Lattice Parameter Refinement

By accurately measuring peak positions over a long range of 2theta, you can determine the unit cell lattice parameters of the phases in your sample alloying, substitutional doping, temperature and pressure, etc

can create changes in lattice parameters that you may want to quantify

use many peaks over a long range of 2theta so that you can identify and correct for systematic errors such as specimen displacement and zero shift

measure peak positions with a peak search algorithm or profile fitting

profile fitting is more accurate but more time consuming then numerically refine the lattice parameters

determination and refinement of lattice parameters (indexing)

)(4

sin 22222

2 lkha

++= )(sin 2222 lkhC ++=

dividing the above equation with the first reflection angle gives the ratio of hkl

relationship between Miller indices and diffraction angles

)()(

sinsin

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21

21

222

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2

lkhlkh

++++

=

the ratio of hkl define the possible Miller indices: ratio = h2 + k2 + l2if there are some possible hkl, the highest number comes firstAfter indexing, one of the peaks can be used to calculate the cell parameters.As the error in measuring the diffraction angles is a systematic error, the last reflection data will be used

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exercise 1: indexing XRD data (cubic)

43.830

60.093

67.21370.634

56.331

63.705

33.60227.302

10010.027919.213

48.266

38.995

Miller indicesratiosin22 1. define the Miller indices

2. calculate the lattice parameters

lattice type and systematic absences on cubic system

destructive interferences occurring between the diffracted waves intensity cancels outeg:

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topics

Evaluation of the intensities of X-rays diffracted from polycrystalline samples;

Evaluating crystallinity

taking the sum total of relative intensities of ten individual characteristic peaks1 then taking the ratio over the corresponding relative intensities of standard materials E.g.:Comparing crystallinity of flyash-based zeolite-A using XRD and IR spectroscopy

1CURRENT SCIENCE, VOL. 89, NO. 12, 25 DECEMBER 2005

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% crystallinity = (AD4R)/(ATO4)

% crystallinity = (IR sample)/(IR standard)

72.8% 85%

98.6%

98.6%

98.6%

100%

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72.8%83.2%

92%

93.3%

96.2%

100%

Crystallinity = (A ratio of 560 over 464 cm-1 bands of sample/reference) x 100%

Preferred Orientation (texture)

Preferred orientation of crystallites can create a systematic variation in diffraction peak intensities can qualitatively analyze using a 1D diffraction pattern a pole figure maps the intensity of a single peak as a function of

tilt and rotation of the sample this can be used to quantify the texture

(111)

(311)(200)

(220)

(222)(400)

40 50 60 70 80 90 100Two-Theta (deg)

x103

2.0

4.0

6.0

8.0

10.0

Inte

nsity

(Cou

nts)

00-004-0784> Gold - Au

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For example MoO3 crystallizes in thin plates (like sheets of paper) these crystals will pack with the flat surfaces in a parallel orientation.

Comparing the intensity between a randomly oriented diffraction pattern and a preferred oriented diffraction pattern can look entirely different.

Quantitative analysis depend on intensity ratios which are greatly distorted by preferred orientation.

Careful sample preparation is most important to deal with preferred orientation samples

The following illustrations show the Mo O3 spectra's collected by using transmission , Debye-Scherrercapillary and reflection mode.

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