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7/30/2019 Lecture+8+Mak Xrd

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PHY 3201 FIZIK KEADAAN PEPEJAL

CHAPTER 2 : Wave Diffraction

and the Reciprocal Lattice

Diffraction of wave by crystals

Scattering wave amplitude

Brillouin Zones

Fourier analysis of the basisBertha Rntgens

Hand 8 Nov, 1895


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Learning outcomes :

By the end of this topic, student can able to : Explain Braggs Law.

Explain the diffraction of wave by crystal.

Explain and construct the reciprocal lattice.

Explain the use of x-ray diffraction method

for material characterization.


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X-Rays

X-rays were discovered in

1895 by the German

physicist Wilhelm Conrad

Rntgen and were so namedbecause their nature was

unknown at the time.

He was awarded the Nobelprize for physics in 1901.

Wilhelm Conrad Rntgen

(1845-1923)


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Diffraction of waves by crystal

The wavelength of X-rays is in the range of

few angstroms, which is comparable to theinteratomic spacing in crystals. Thus X-raysare diffracted by the periodically arrangedatoms in a crystal i.e. crystals act as a three-dimensional diffraction grating for X-rays.

Therefore crystal structures can bedetermined by the study of diffraction of X-rays

In 1913, W. L. Bragg presented a simple

explanation of the diffracted beams from acrystal.

Sir Will iam Henry Bragg (1862-1942),

Will iam Lawrenc e Bragg (1890-1971)1915, the father and son were awarded the

Nobel prize for physics "for their services in

the analysis of crystal structure by meansof Xrays".


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Experimental setup

(How do we produce x-rays?)

X-ray production is via Bremstrahlung radiation. The

electrons are produced typically from a tungsten source,

are accelerated towards a metal such as copper, and when

they hit the surface, they slow down. This .brakingradiation. is a broad band of light which is emitted as the

electron slows down (charged particles under acceleration

emit radiation)

(Eg:

Molybdenum)

35 KV


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X-Ray Spectrum of Molybdenum

X-Ray spectrum ofMolybdenum is obtained whenMolybdenum is used as targetmetal.

K and K are characteristic

of an element. For Molybdenum K occurs at

wave length of about 0.07nm.

Electrons of n=1 shell of targetmetal are knocked out bybombarding electrons.

Electrons of higher level dropdown by releasing energytoreplace lost electrons


7/18PHY 3201 FIZIK KEADAAN PEPEJAL

Consider a monochromatic, parallel beam of X-raysincident on a set of parallel crystals plane of Miller

indices



Now consider interference between reflections

from successive planes:

Constructive interference

occurs only when

n = AB + BC

AB = BCn = 2AB

Sin = AB/d

AB = d Sin

n = 2 d Sin

= 2 dhkl Sin hkl

Braggs law



What happens during a diffraction experiment?X-rays of a single wavelength (and therefore, energy) are

incident upon a crystal. The incoming rays are of the

proper wavelength for diffraction (on the order of the

interatomic spacing), and thus we see diffraction peaks

at certain values of , the scattered beam angle. Eachone of these peaks is from a plane of atoms within the

crystal. This is an elastic process.


10/18PHY 3201 FIZIK KEADAAN PEPEJALDiffraction peaks are observed as a function ofscattered angle



There are many types of X-ray camera

to sort out reflections from different crystal

planes. We will study only three types of X-

ray photograph that are widely used for the

simple structures.1.Laue photograph

2.Rotating crystal method

3.Powder photograph


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X-Ray Diffraction Method

Laue Rotating Crystal Powder

Orientation

Single Crystal

Polychromatic Beam

Fixed Angle

Lattice constant

Single Crystal

Monochromatic Beam

Variable Angle

Lattice Parameters

Polycrystal (powdered)

Monochromatic Beam

Variable Angle

X-RAY DIFFRACTIONMETHODS


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THE POWDERMETHOD

In powder or polycrystalline diffraction, if possible, we

normally grind the sample down to particles of about

0.002 mm to 0.005 mm cross section. The ideal sample is

homogeneous and the crystallites are randomly

distributed. The sample is pressed into a sample holder.

Ideally we now have a random distribution of all possible

h, k, l planes. If we have a truly random sample, each

possible reflection from a given set of h, k, l planes willhave an equal number of crystallites contributing to it.


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THE POWDERMETHOD


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THE POWDERMETHOD

An x-ray beam diffracted from a lattice plane can be

detected when the x-ray source, the sample and the

detector are correctly oriented to give Bragg

diffraction.A powder or polycrystalline sample contains an

enormous number of small crystallites, which will adopt

all possible orientations randomly

Thus for each possible diffraction angle there arecrystals oriented correctly for Bragg diffraction


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Debye ScherrerCamera

Each set of planes in a

crystal will give rise to a cone

of diffraction. These cones

intersect a strip of

photographic film located inthe cylindrical camera to

produce a characteristic set

of arcs on the film.


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Powder diffractionfilm

When the film is removed from the camera,

flattened and processed, it shows the diffraction

lines and the holes for the incident andtransmitted beams.


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There are a number of various setups for studying crystalstructure using x-ray diffraction.

In most cases, the wavelength of the radiation is fixedand the angle is varied to observed diffraction peakscorresponding to reflections from differentcrystallographic planes.

Using the Bragg law, one can then determine thedistance between the planes.

However, Bragg law is oversimplified, where it

Says nothing about intensity and width of x-raydiffraction peaks

Neglects differences in scattering from different atoms Neglects distribution of charge around atoms.

Hence, to fully understand, we need a deeper analysis todetermine the scattering intensity from the basis of atom.

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