CHAPTER 4

4.1 X-RAY DIFFRACTION METHOD

WHY XRD..??Most effective methods for determining the crystalline structure of materials.It can identify chemical compounds from their crystalline structure, not from compositions of chemical elements.

• It means that differentcompound (phase)that have same compositioncan be identified!

4.1.1 X-RAY RADIATION

4.1.1.1 GENERATION OF X-RAY

To generate X-ray, we need X-ray tube.

X-ray radiation from the anode is guided by the windows of the tube to producean X-ray beam for diffraction

• X-ray tube contained a source of electrons and two metal electrodes in a vacuum tube.

• The high voltage maintained across these electrodes rapidly draws the electrons to the anode (target).

• X-ray produced at the point of impact on the target surface and radiated at all directions.

• There are windows to guide X-rays out of tube.• Extensive cooling is necessary for the X-ray tube because

most of the kinetic energy of electrons are converted into heat.

• The X-ray tube generates X-ray radiation with range of wavelengths starting from minimum λ, called continuous X-ray or white x-ray.

• 5, 10, 15, 20 and 25 kV is acceleration voltage.

• The continuous X-rays have short wavelength limit (SWL)

• The sharp intensity maxima at certain wavelength is the characteristic x-rays

• XRD method always require a monochromatic source (single wavelength) that come from the characteristic X-rays by filtering out other radiation.

4.1.1.2 X-RAY ABSORPTION

To obtain monochromatic characteristic radiation (Kα), a filtersystem is needed to filter out continuous X-rays.X-ray filters are made from material that strongly absorbs X-raysother than Kα.It has a function of linear absorption coefficient (μ) and massDensity (ρ).So, the intensity (I) passing through an absorption layer withThickness (x) is: Ix = Ioe-(μ/ρ)ρx

4.1.2 THEORITICAL BACKGROUND OF DIFFRACTION

4.1.2.1 DIFFRACTION GEOMETRY

BRAGG’S LAW

X-ray diffraction is based on the wave interference phenomena.When they have phase difference of nλ (n is integer), is called‘in phase’ so constructive interference occur.If phase difference nλ/2, is called completely destructiveInterference.X-ray beams incident on a crystalline solid will be diffracted bythe crystallographic planes.

The deflected waves will not be in phase unless the relationship above is satisfied.Based on Braggs law, the information on the spacing between planes of crystal is obtained when constructive interferences detectedat a given incident angle and a wavelength of the incident beam.

4.1.2.2 RECIPROCAL LATTICE

A crystallographic plane (hkl) is represented as a light spot ofconstructive interference when Braggs equation is satisfied.The diffraction spot of crystallographic plane in a crystal formA 3D array which is the reciprocal lattice of crystal .

A direction in reciprocal lattice defined by a vector d*hkl with Reciprocal unit vectors a*,b* and c* in reciprocal space.

So, d*hkl = ha* + kb* + lc*

Actually, Braggs law describes the necessary condition to detectcrystal planes by diffraction. The condition can also be graphically expressed by Ewald sphere.

4.1.2.3 EWALD SPHERE

It is an imaginary sphere with radius of λ-1 in reciprocal space.Center of Ewald sphere is located at a crystal to be examined.

incident beam represented by a line that passes through a crystalat the sphere centre and targets the origin of the reciprocal lattice.

4.1.3 X-RAY DIFFRACTOMETRY (XRD)

widely used in X-ray diffraction technique.The basic function of a diffractometer is to detect X-ray diffractionfrom materials and record the diffraction intensity as a function ofthe diffraction angle (2θ)

4.1.3.1 INSTRUMENTATION OF XRD Three main parts – X-ray source, specimen and detectorThe soller slits – commonly used in the diffractometer & made froma set of closely spaced thin metal plates and act to prevent beamdivergence.

1. X-ray beam passing through slits strike the sample (form of flat plate)2. X-rays diffracted by the sample and form convergent beam at receiving slits before enter a detector.3. Diffracted x-ray beam must pass through monochromatic filter.

4.1.4 UNDERSTANDING THE LATTICE PLANE & MILLER INDICES General label is (h k l) which intersects at a/h, b/k, c/l

(hkl) is the MILLER INDEX of that plane (round brackets, no commas).

Plane perpendicular to y cuts at , 1, (0 1 0) plane

This diagonal cuts at 1, 1, (1 1 0) plane

an index 0 means that the plane is parallel to that axis

• Miller indices define the orientation of the plane within the unit cell

• The Miller Index defines a set of planes parallel to one another (remember the unit cell is a subset of the “infinite” crystal

• (002) planes are parallel to (001) planes, and so on

IN SUMMARY…

d-spacing formula

For orthogonal crystal systems (i.e. = = =90) :- 2

2

2

2

2

2

2 cl

bk

ah

d1

For cubic crystals (CC) (special case of orthogonal) a=b=c :- 2

222

2 alkh

d1

e.g. for (1 0 0) d = a(2 0 0) d = a/2(1 1 0) d = a/2 etc.

Diffraction from crystals

XY

Z

d

Incident radiation “Reflected” radiation

Transmitted radiation

1

2

X-ray Tube Detector

We normally set n=1 and adjust Miller indices, to give 2dhkl sin =

e.g. X-rays with wavelength 1.54Å are reflected from planes with d=1.2Å. Calculate the Bragg angle, , for constructive interference.

= 1.54 x 10-10 m, d = 1.2 x 10-10 m, =?

d2nsin

nsind2

1

n=1 : = 39.9°n=2 : X (n/2d)>1

Use Bragg’s law and the d-spacing equation to solve a wide variety of problems

2d sin = n

or

2dhkl sin =

X-rays with wavelength 1.54 Å are “reflected” from the (1 0 0) and (2 0 0) planes of a cubic crystal with unit cell a = 6 Å. Calculate the Bragg angle, , for all orders of reflection, n.

Combining Bragg and d-spacing equation

2

222

2 alkh

d1

d (1 0 0)= 6 Å

d (2 0 0)= 3 Å

Substitute the value in 2d sin = n

(1 0 0) reflection, d=6 Ån=1, =7.37o

n=2, =14.87o

n=3, =22.64o

n=4, =30.87o

n=5, =39.92o

n=6, =50.35o N=7, θ=63.94°no reflection for n>7

(2 0 0) reflection, d=3 Å

n=1, =14.87o

n=2, =30.89o

n=3, =50.35o

no reflection for n4

We can imagine planes within a crystal Each set of planes is uniquely identified by its

Miller index (h k l) We can calculate the separation, d, for each set

of planes (h k l) Crystals diffract radiation of a similar order of

wavelength to the interatomic spacings We model this diffraction by considering the

“reflection” of radiation from planes - Bragg’s Law

FOR CONCLUSION…

QUIZ 2

BAND TOTAL

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