X-RAY DIFFRACTION

CONTENTS

• Bragg`s Law

• X-ray Powder Diffraction

• X-ray Powder Diffractometer

X-ray diffraction• When a beam of X-ray radiation is incident upon a When a beam of X-ray radiation is incident upon a

substance, the electrons constituting the atoms of substance, the electrons constituting the atoms of the substance become as small oscillators.the substance become as small oscillators.

• These on oscillating at the same These on oscillating at the same frequency as that of incident X-ray frequency as that of incident X-ray radiation emit EM radiations in all radiation emit EM radiations in all directions at the same frequency.directions at the same frequency.

• Diffraction occurs as waves interact with a regular structure whose repeat distance is about the same as the wavelength .

• When certain geometric When certain geometric requirements are met, X-rays requirements are met, X-rays scattered from a crystalline solid can scattered from a crystalline solid can constructively interfere, producing a constructively interfere, producing a diffracted beam. diffracted beam.

Bragg Equation History• English physicists Sir W.H.

Bragg and his son Sir W.L. Bragg developed a relationship in 1913 to explain why the cleavage faces of crystals appear to reflect X-ray beams at certain angles of incidence (theta, θ).This observation is an example of X-ray wave interference.

Sir William Henry  Bragg (1862-1942),William Lawrence  Bragg (1890-1971)

o In 1915, the father and son were awarded the Nobel prize for physics "for their services in the analysis of crystal structure by means of X-rays".

Bragg’s law

• The conditions for diffraction are governed by Bragg’s law.

• When the path length in the crystal(2dsinθ) is a multiple of the wavelength, constructive interference occurs and diffracted intensity is obtained.

• Above figure shows a monochromatic beam of X-Above figure shows a monochromatic beam of X-rays incident on the surface of a crystal at an angle rays incident on the surface of a crystal at an angle θθ..

• P,Q & R represent the edges of a P,Q & R represent the edges of a family of planes distance ‘d’ apart.family of planes distance ‘d’ apart.

• Plane ‘P’ reflects AX in XD.Plane ‘P’ reflects AX in XD.

• Plane ‘Q’ reflects BY in YE at the same Plane ‘Q’ reflects BY in YE at the same angle angle θθ..

• Although the beam penetrates many Although the beam penetrates many more planes we need to consider only more planes we need to consider only the top two.the top two.

• Since ‘Q’ is lower than ‘P’, the beam path Since ‘Q’ is lower than ‘P’, the beam path ‘BYE’ is longer than ‘AXD’ by the amount ‘BYE’ is longer than ‘AXD’ by the amount GY+YH. This is called the path difference.GY+YH. This is called the path difference.

angle GXA=900

From ∆GXY, sinθ=GY/dHence GY=d sinθFrom ∆YXH, sinθ=YH/dHence YH=d sinθTherefore GY+YH=2d sinθ.

• Now the two reflected rays, XD & YE will Now the two reflected rays, XD & YE will constructively interfere when the path constructively interfere when the path difference is equal to the wavelength or difference is equal to the wavelength or a multiple of it.a multiple of it.

• Thus condition for X-ray diffraction isThus condition for X-ray diffraction is nλ=2d sinθ

where ‘n’ is an integer(1.2.3… etc) called where ‘n’ is an integer(1.2.3… etc) called the order of reflection. the order of reflection.

X-RAY DIFFRACTION METHODS

X-Ray Diffraction Method

Laue Rotating Crystal Powder

OrientationSingle Crystal

Polychromatic BeamFixed Angle

Lattice constantSingle Crystal

Monochromatic BeamVariable Angle

Lattice ParametersPolycrystal (powdered)Monochromatic Beam

Variable Angle

The Powder Method

The Powder MethodExperimental equipment consist of:Experimental equipment consist of:

• A X-ray sourceA X-ray source

• CollimatorCollimator

• Powdered CrystalsPowdered Crystals

• A cylindrical film or camera to record the A cylindrical film or camera to record the diffraction patterndiffraction pattern

• If a monochromatic x-ray beam is directed at a single crystal, then only one or two diffracted beams may result.

• A sample of some hundreds of crystals (i.e. a powdered sample) show that the diffracted beams form continuous cones. A circle of film is used to record the diffraction pattern as shown. Each cone intersects the film giving diffraction lines. The lines are seen as arcs on the film.

• If the sample consists of some tens of randomly orientated single crystals, the diffracted beams are seen to lie on the surface of several cones. The cones may emerge in all directions, forwards and backwards

• A particular family of planes in a crystal will only reflect an X-ray beam when the Bragg eqn is fulfilled.

• If a single crystal was placed in a X-ray beam, then it would be a mere chance that a particular family of plane was in the correct position to satisfy the Bragg eqn.

• Suppose we take the crystal and powder it. This does not destroy the crystal structure, it simply produces millions of very small crystals pointing in all possible directions.

• Thus if the powdered crystal is Thus if the powdered crystal is placed in a monochromatic X-ray placed in a monochromatic X-ray beam, then for any particular family beam, then for any particular family of planes, there will be at least a few of planes, there will be at least a few having those planes satisfying the having those planes satisfying the Bragg eqn.Bragg eqn.

Above figure shows a powdered Above figure shows a powdered crystal (A) irradiated with a crystal (A) irradiated with a

monochromatic X-ray beam.monochromatic X-ray beam.

• CD represents a particular family of planes in a CD represents a particular family of planes in a crystal satisfying the Bragg eqn, so part of the crystal satisfying the Bragg eqn, so part of the beam is reflected along AX.beam is reflected along AX.

• In the same way a crystal having the same In the same way a crystal having the same family of planes orient along EF it reflects part of family of planes orient along EF it reflects part of the incident beam along AY.the incident beam along AY.

• Now the 2 crystals have the planes oriented at Now the 2 crystals have the planes oriented at θθ to the beam and at right angles to the plane of to the beam and at right angles to the plane of the paper.the paper.

• The sample will also contain crystals having the The sample will also contain crystals having the same family of planes oriented at same family of planes oriented at θθ to the beam to the beam but not at right angles to the plane of thebut not at right angles to the plane of the paper paper

• This will give rise to reflections coming out of This will give rise to reflections coming out of and going into the plane of the paper.and going into the plane of the paper.

• Thus if we have all possible orientations of this Thus if we have all possible orientations of this family of planes w.r.t to the plane of the paper family of planes w.r.t to the plane of the paper then a cone of reflected rays is produced of then a cone of reflected rays is produced of semi vertical angle 2semi vertical angle 2θθ..

• The sample will contain crystals having several The sample will contain crystals having several families of planes satisfying the Bragg eqn. families of planes satisfying the Bragg eqn.

• Since different families have different ‘d’ values Since different families have different ‘d’ values the diffraction cones will have different values the diffraction cones will have different values of the semi vertical angleof the semi vertical angle 2 2θθ..

• Thus the net result is the formation of a Thus the net result is the formation of a series of concentric cones.series of concentric cones.

POWDER DIFFRACTOMETER

• Here a flat specimen is mounted on a turntable around which Here a flat specimen is mounted on a turntable around which moves a detector.moves a detector.

• As the sample rotates , so the angle As the sample rotates , so the angle θθ between the incident beam between the incident beam and the sample changes.and the sample changes.

• Whenever the Bragg condition is fulfilled X-Rays are reflected to Whenever the Bragg condition is fulfilled X-Rays are reflected to the detector.the detector.

• The detector is connected to the specimen table and geared in The detector is connected to the specimen table and geared in such a way that when the table rotates through such a way that when the table rotates through θθ, the detector , the detector rotatesrotates through 2 through 2θθ degrees. degrees.

• This results in the detector always being in the correct This results in the detector always being in the correct position to receive X-Rays from the sample.position to receive X-Rays from the sample.

Scintag PAD V diffractometer (θ-2θ)

Application of XRD

1. Differentiation between crystalline and amorphous materials;

2. Determination of the structure of crystalline materials3. Determination of electron distribution within the

atoms, and throughout the unit cell;4. Determination of the orientation of single crystals;5. Determination of the texture of polygrained materials;6. Measurement of strain and small grain size

XRD is a nondestructive technique. Some of the uses of x-ray diffraction are;

Advantages and disadvantages of X-rays

Advantages;• X-ray is the cheapest, the most

convenient and widely used method.• X-rays are not absorbed very much by air,

so the specimen need not be in an evacuated chamber.

Disadvantage;• They do not interact very strongly with

lighter elements.

REFERENCES

• Instrumental Methods of Chemical Analysis by Gurdeep R. Chatwal and Sham K. Anand.

• X-Ray METHODS BY Clive Whiston.

• Internet source.

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