X-Ray Diffraction

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X-ray Diffraction

Merve AyvazChemical Engineering DepartmentBoğaziçi University

ChE 592

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Introduction• Atomic radii of atoms are smaller than 1/1000 of the

wavelengths present in the visible light.• A suitable wavelength to observe individual atoms is x-

rays.

For Catalysis• It is used to identify crystalline phases inside catalysts

by means of lattice structural parameters, kinetics of bulk transformation and to obtain an indication of particle size.

Niemantsverdriet J.W.

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Introduction• The scattered monochromic X-rays that are in phase give

constructive interference.• Diffraction of x-rays by crystal planes allows one to drive

lattice spacing by using

Bragg relation: n λ=2d sinθ• d is charactesitic for a given compound

n: order of reflection; λ: wavelength of x-rays; d: distance between two lattice planes; θ: angle between the incoming x-rays and the normal to

the reflecting lattice plane

Jenkins R.,Snyder R.L.

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Nature and Properties of X-rays• The x-rays have wavelength from 0.1 Å to 100 Å, which are

located between gamma radiation and ultraviolet rays.• The wavelengths most commonly used in crystallography,

range between 0.5 Å to 2.5 Å. If d < λ/2, then sin θ > 1, which is impossible. n λ=2d sinθ

• They are of the same order of magnitude as the shortest interatomic distances observed both organic and inorganic materials.


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Production of X-rays

• X-ray tube

▫Sealed Tube▫Rotating Anode Tube

•Synchrotron

Percharsky V.K., Zavalij P.Y.

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X-ray Tube• Known as a laboratory or conventional x-ray source,• Electromagnetic waves are generated from impacts

of high-energy electrons with a metal target.• Brightness is limited with the thermal properties of

target material,

• Simple,• Most commonly used,• Must be continuously cooled,• Low efficiency,


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Sealed Tube• Consist of a stationary anode coupled

with cathode, placed in a metal/glass or metal/ceramic container sealed under vacuum

• Electrons are emitted by the cathode, accelerate through the anode (30 to 60 kV),

• Typical current is between 10 to 50 mA,

• The x-rays are generated by the impacts of high energy electrons,

• The exit the tube from Be window,


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Synchrotron• Advance source of x-ray radiation,• High energy electrons are confined in a

storage ring,• They move in a circular orbit, electrons

accelerate towards the center of the ring, thus emitting electromagnetic radiation.

• Extremely bright, (limited by the flux of electrons)

• Thermal losses are minimized,• No target to cool,


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Collimation and Monochromatization• The radiation comes out from the x-ray source is

polychromatic radiation. (Kα, Kα1, Kα2, Kβ)


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Monochromatization• Reducing the intensity of white radiation,• Eliminating the undesirable characteristic

wavelengths from x-ray spectrum to Kα1, Kα2,


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Adsorption of x-rays and B-filter• Simplest method performed by means of

filtering.• When x-ray penetrate into a matter, they are

partially transmitted and partially adsorbed.

It=I0 exp(-µx)µ:linear absorption coefficient of the material

Examples of filter elements: Sc,Ti,Cr,Mn,Fe,Co,Ni,Cu


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Angular Divergence and Collimation• The simplest collimation can be achieved by

placing a slit between the x-ray source and the sample.

• Angular divergence of thus collimated beam is established by the dimensions of the source, the size and the placement of the slit.


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Detection of X-rays• Role is to measure intensity,• Sensitive to x-rays,• The oldest detector of x-ray is photographic film,• In modern detectors the signal, usually electric

current, easily digitized and transferred to a computer for further processing and analysis,


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Gas Proportional Counter-Point Detector

• The x-ray photons enter to the window and is absorbed by the gas, it ionizes Xe atoms producing positively charged ions and electrons.

• The resulting electric current is measured • The number of current pulses is proportional to

the photons absorbed.• The second window is usually added to enable

exit of non-absorbed photons.


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Other types of detectors

•Scintillation detector•Position sensitive detector•Area detectors


Theory of X-Ray Diffraction

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Diffraction• X-rays scattered from different parts of the atom

(nucleus and electrons) combine to give the effect of a point source.

• The radiation scattered by the atom depends on the number of electrons associated to atom and their distribution.

Nuffield E.W.

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Intensity of the DiffractionThe intensities of the diffracted waves;

• Depend on the kind and arrangement of atoms in the crystal structure

• Proportional to the squares of amplitudes of the composite reflected waves

Nuffield E.W.

Derivation of Braggs Law

• Beams are parallel to each other• The second beam must travel the extra distance AB + BC • nλ = AB +BC • AB = BC• nλ = 2AB

AB = d sinΘnλ = 2 d sinΘ

Glenn A. Richard, Mineral Physics Institute, SUNY Stony Brook

Bragg Relation

Diffraction Pattern• A diffraction pattern records the X-ray intensity as a

function of 2-theta angle.

• Tube voltage • Current • --A starting 2-theta angle. • --A step-size (typically 0.005 degrees). • --A count time per step (typically 0.05-1 second).• --An ending 2-theta angle.

Purudue University, Geoscience Department

Lattice Structure

• 0A = 0B = 0C = a =

the length of the edge of

the unit cell

•Bragg Equation

Satisfied for all planes

crystal can be arranged in sheets in a number of ways

University of Saskatchewan, Physics Department

Miller Indices h,k,l plane

• h, k, l are required to describe the order of the diffracted waves

• Denote the orientation of the reflecting sheets with respect to the unit cell

• The path difference in units of wavelength between identical reflecting sheets

dhkl

University of Saskatchewan, Physics Department

dhkl= a/(h2 + k2 + l2)½

Determination of the unit cell structure for a cubic latticeFor a cubic lattice, primitive, fcc, bcc reflections are obsrevedSee h,k,l in the increasing order of sum h2 + k2 + l2

P 100 110 111 200 210 211 220 221 300 310 311F 111 200 220 311 222 400 331 420 422 333 511B 110 200 211 220 310 222 321 400 411 330 420

sin2 θ1/ sin2 θ2 = (h12 + k1

2 + l12)/(h2

2 + k22 + l2

2)

P 1:2:3:4:5:6:8F 1:1 1/3 : 2 2/3: 3 2/3: 4: 5 1/3: 6 1/3 B 1:2:3:4:5:6:7

All odd or even

Sum is even

Atkins,Physical Chemistry

Determination of the unit cell structure for a cubic lattice• sin2 θ1/ sin2 θ2 = (h1

2 + k12 + l1

2)/(h22 + k2

2 + l22)

• Powder diffraction angles (θ) are red from the pattern are: 17.660, 25.400, 31.700, 37.350, 42.710, 47.980, 59.080

• sin2 θ gives

0.0920, 0.1840, 0.2761, 0.3681, 0.4601, 0.5519, 0.7360

• These number give the ratio 1:2:3:4:5:6:8

• So the cell is primitive

P 1:2:3:4:5:6:8F 1:1 1/3 : 2 2/3: 3 2/3: 4: 5 1/3: 6 1/3 B 1:2:3:4:5:6:7

Atkins,Physical Chemistry

Determination of unit cell edge length for a cubic cell

•dhkl= a/(h2 + k2 + l2)½

•λ = 2 d sin θ

•(4 sin2 θ)/λ2 = (h2 + k2 + l2)/a2

(4 sin2 θ)/λ2 = (h2 + k2 + l2) *(1/a2)Augustana Collage, Chemistry and Physics

Department

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Principles of Determination of Crystallite Size• Rays A, D and M all make exact Bragg angle with diffracting

planes so A’, D’ and M’ differ in phase by an integer number of wavelengths (M-M’ has a path length mλ greater than A-A’)

• Consider rays coming in at angle θ1 (B and L). If θ1 is selected so that the path length B-B’ differs from that of L-L’ by (m+1)λ, there will be a plane in the middle of the crystal that scatters with path length difference (m+1)λ/2 and destructively interferes with x-rays on the B-B’ path. θ1 represent the highest angle you can go to before you get complete destructive interference.

• Consider rays coming in at angle θ2 (C and N). If θ2 is selected so that the path length C-C’ differs from that of N-N’ by (m-1)λ, then there will be a plane in the middle of the crystal that scatters with path length difference (m-1)λ/2 and destructively interferes with Xrays on the C-C’ path. θ2 represent the lowest angle you can go to before you get complete destructive interference.

Georgia Tech., Chemistry and Biochemisrty Department

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Principles of Determination of Crystallite Size• Considering the path length differences between x-rays scattered from the front

and back planes of the crystal• – 2tsinθ1 = (m+1)λ

• – 2tsinθ2 = (m-1)λ• If we subtract them – t(sinθ1 – sinθ2)= λ – 2tcos((θ1+θ2)/2)sin((θ1- θ2)/2) = λ Use small angle approximation and (θ1 + θ2)/2 = θB

– 2t[(θ1- θ2)/2]cosθB = λ , t = λ/(BcosθB)

• More rigorous treatment gives t = 0.9λ/(BcosθB) for spherical crystals of diameter t.

• – This is the Scherrer equation

Georgia Tech., Chemistry and Biochemisrty Department

Scherrer Equation

• Peak width (B) is inversely proportional to crystallite thickness (t)• As the crystallites in a powder get smaller the diffraction peaks in a powder

pattern get wider

B = (2θ High) – (2θ Low)• B is the difference in angles at half max• The constant of proportionality, K(the Scherrer constant) depends on the shape of the crystal, and the size distribution

▫ the most common values for K are: 0.94 -0.89 K actually varies from 0.62 to 2.08

cos

2t

KB

MIT, Center for Materials Science and Engineering

References

• Glenn A. Richard, Mineral Physics Institute, SUNY Stony Brook • University of Saskatchewan, Physics Department• Atkins,Physical Chemistry• Augustana Collage, Chemistry and Physics Department• Purudue University, Geoscience Department• MIT, Center for Materials Science and Engineering • Jenkins R.,Snyder R.L., Introduction to X-ray Powder

Diffractometry, John Wiley & Sons, Inc. ,1996• Niemantsverdriet J.W., Spectroscopy in Catalysis, Wiley-VCH,2000• Nuffield E.W., X-ray Diffraction Methods, John Wiley & Sons, Inc.,

1966• Percharsky V.K., Zavalij P.Y., Fundamentals of Powder Diffraction

and Structural Characterization of Materials, Springer, 2005

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Thanks to you

X-Ray Diffraction

Documents

xray source

xray spectrum

adsorption of xrays

electromagnetic radiation

gamma radiation

polychromatic radiation

diffraction of

lattice planes