xrd intensity factors

8/9/2019 XRD Intensity Factors

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XRD Intensity Factors


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What defines the intensity and

presence/absense of some beams?

•

To understand how intensities varies for the different peaksto look at the interaction of X-Rays with the crystal.

• This can be understood by considering the interaction of X-Ran electron, then an atom (group of electrons since X-rays dinteract with the nucleus), and finally a crystal (group of atoadded periodicity)


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Scattering by electron

•

X-Rays are electromagnetic radiation, this is, they have mutperpendicular electric and magnetic felds. Electrons are negcharged particles that are affected by these electric and magfelds. When X-Rays interact with electrons, they cause it toThe oscillating electron will in turn emit X-rays. This is calledscattering.


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Scattering by electron

•

There are 2 kinds of scattering:

1. Coherent scattering: emitted X-rays have the same wavele(same energy) as the incident X-rays. This is also called elascattering. The scattered radiation has a definite phase re

the incident X-rays.2. Incoherent scattering: the X-rays lose some energy during

scattering, so the final energy is lower (wavelength is highthe incident energy. There is no definite phase relation toincident radiation. This is also called inelastic scattering.


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Elastic Scattering

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For elastic scattering, the scattered radiation is phase-shiftethe incident beam by π/2 or has a path difference of λ/2.

• The scattering takes place in all directions, yet the intensityon the angle of scattering.


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Elastic Scattering

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The expression for this is given by J.J. Thompson.

• where, is the intensity at a distance r from the electron athe angle between the recoiling electron and the scattered xthe case of diffraction we want the intensity in terms of thebetween the incident and scattered radiation (2).


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Elastic Scattering

• This is the Thompson equation for the scattering of x-rays belectron. The intensity decreases as square of the distance f

electron. Scattering is high in the forward or backward direc(2 = 0° 180°). The term (1 + 2/2 ) is called thepolarization factor.


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Structure factor with a single atom basis

• Using equation 7 it is possible to calculate the structure fact(ℎ) plane of any unit cell, once the atom scattering factorpositions are known. The structure factor is related to the in

the x-ray diffracted from that plane. Thus if the structure facgiven plane is zero then there will be no diffracted intensity,though Bragg's law is satisfied. Thus, Bragg's law only decidevalue for a given plane while structure factor decides the int


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Simple Cubic SC)

•

In a simple cubic unit cell there is only one atom per unit ceat (000). So = 1 and () is (000). So using equationstructure factor for a (ℎ) plane simply becomes = for a simple cubic structure diffraction is possible for all posplanes since the structure factor is a constant, equal to the ascattering factor. Thus, all planes can be seen in the diffracti

pattern of a simple cubic structure.

• NOTE. The intensities are still not the same since there are ofactors involved.


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Body Centered Cubic BCC)

•

In a body centered cubic structure there are 2 atoms per un(000) and (½,½,½). These represent the fractional coordithe atoms in the unit cell. Substituting them in equation 7 wthe expression for the structure factor as:

• Thus, diffraction intensities are only found for specific ℎ pbcc which satisfy the condition in equation 10 (i.e. ℎ + + be even).


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Face centered cubic FCC)

•

In the face centered cubic structure there are 4 atoms per uone at the corner and 3 on the face centers. Once again usinequation 7, the expression for the structure factore simplifie

• Thus, for fcc structure, diffraction lines are seen for planes l(111), (200), (220), (311), (222) where the indices are allodd.


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Hexagonal close packed (HCP)

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In the case of the hcp structure, the fractional coordinates fare 000 and ( ,

,

). The equation for the structurethen becomes:


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X-Ray Intensity

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In the absence of all other factors the intensity of the diffracis equal to the square of the structure factor. But there are aof other factors that affect the intensity.


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Multiplicity Factor

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Relative proportion of the planes contributing to diffractionmore the number of planes, greater is the intensity.

• Compare diffraction from the (100) and (111) set of planes ocrystal. The (100) family has 3 planes (100), (010), and (001)same d-spacing (hence same 2Θ) but different orientations.family has 4 planes with the same d-spacing but different or

(1,1,1), (-1,1,1), (1,-1,1), (1,1,-1).


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Multiplicity Factor

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Thus, everything else being same, the ratio of the intensities(100) and (111) planes will be in the ratio of 3/4. Different phave different multiplicity values, which are usually tabulatemultiplicity factor depends on the crystal structure since themust have the same d-spacing.


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Lorentz-polarization Factor

• The Lorentz factor is a trigonometric factor that relates to thdistribution of planes as a function of 2.

• For a powder simple, with randomly oriented particles, theintensity of a reflection at a given Bragg angle depends on tnumber of particles with that orientation.


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• Even though the particles are oriented at random, this numa constant but depends on the value of the Bragg angle. Thidependence is called the Lorentz factor and is given by:


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• Usually, the Lorentz factor is combined with the polarizationform the Lorentz-polarization Factor:


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• The Lorentz-polarization factor as a function of a Bragg anglby:


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Absorption Factor

• The X-ray intensity also depends on the absorption of the spbeing investigated. This also depends on the geometry of thIn the case of the diffractometer, the absorption factor is indof the Bragg angle and does not affects the relative intensitidifferent lines.


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Temperature Factor

• Atoms in the lattic are constantly vibrating about their equilposotion. As the temperatura increases, this vibration amplso. One effect of this vibration is that lattice spacing constanchanges, thus the overall intensity of a line decreases by incthe temperature.

• For a given temperature, the effect is pronounced at higher

angles since the d-spacing is smaller. The temperature effecdetermined experimentally and written in the form exp −2


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X-ray Intensity

• Putting the various other factors together, the intensity of thdiffraction line is given by:

• refers to the relative integrated intensity, is the Structurefor the ℎ plane and is the Multiplicity Factor. The othe

are the Lorentz-polarization and temperature factors. The afactor is not included for the relative intensisties since it doedepends on the Bragg angle.

• NOTE. If there is any preferred orientation due to external clike growth or stress, then this equation is no longer valid.


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Reference:

• http://mme.iitm.ac.in/swamnthn/sites/default/files/Lec6.p

xrd intensity factors

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