To view this presentation, you'll need to enable Flash.

Show me how

After you enable Flash, refresh this webpage and the presentation should play.

Loading...

PPT – X-ray Scattering PowerPoint presentation | free to download - id: 7f6e69-ZjIxM

The Adobe Flash plugin is needed to view this content

View by Category

Presentations

Products
Sold on our sister site CrystalGraphics.com

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

X-ray Scattering

- Peak intensities
- Peak widths

Second Annual SSRL Workshop on Synchrotron X-ray Scattering Techniques in Materials and Environmental Sciences Theory and Application

Tuesday, May 15 - Thursday, May 17, 2007

Peak Intensities and Widths

Silicon

Basic Principles of Interaction of Waves

- Periodic wave characteristics
- Frequency ? number of waves (cycles) per unit

time ?cycles/time. - ? 1/sec Hz.
- Period T time required for one complete cycle

T1/?time/cycle. T sec. - Amplitude A maximum value of the wave during

cycle. - Wavelength ? the length of one complete cycle.

? m, nm, Å.

A simple wave completes cycle in 360 degrees

Basic Principles of Interaction of Waves

- For t 0
- For x 0

Basic Principles of Interaction of Waves

- Consider two waves with the same wavelength and

amplitude but displaced a distance x0. - The phase shift

Waves 1 and 2 are 90o out of phase

Waves 1 and 2 are 180o out of phase

When similar waves combine, the outcome can be

constructive or destructive interference

Superposition of Waves

- Resulting wave is algebraic sum of the amplitudes

at each point

Small difference in phase

Large difference in phase

Superposition of Waves

Difference in frequency and phase

The Laue Equations

- If an X-ray beam impinges on a row of atoms, each

atom can serve as a source of scattered X-rays. - The scattered X-rays will reinforce in certain

directions to produce zero-, first-, and

higher-order diffracted beams.

The Laue Equations

- Consider 1D array of scatterers spaced a apart.
- Let x-ray be incident with wavelength ?.

Lattice Planes

Braggs Law

- If the path AB BC is a multiple of the x-ray

wavelength ?, then two waves will give a

constructive interference

Braggs Law

- The incident beam and diffracted beam are always

coplanar. - The angle between the diffracted beam and the

transmitted beam is always 2?. - Rewrite Braggs law

Since sin? ? 1

For n 1

UV radiation ? ? 500 Å Cu K?1 ? 1.5406 Å

For most crystals d 3 Å

? ? 6 Å

Braggs Law

Silicon lattice constant

aSi 5.43 Å

Scattering by an Electron

- Elementary scattering unit in an atom is electron
- Classical scattering by a single free electron

Thomson scattering equation

The polarization factor of an unpolarized primary

beam

If r few cm

Another way for electron to scatter is manifested

in Compton effect. Cullity p.127

1 mg of matter has 1020 electrons

Scattering by an Atom

Cu

Atomic Scattering Factor

Scattering by an Atom

- Scattering by a group of electrons at positions

rn

Scattering factor per electron

Assuming spherical symmetry for the charge

distribution ? ?(r ) and taking origin at the

center of the atom

For an atom containing several electrons

f atomic scattering factor

Calling Z the number of electrons per atom we

get

Scattering by an Atom

- The atomic scattering factor f Z for any atom

in the forward direction (2? 0) I(2?0) Z2 - As ? increases f decreases ? functional

dependence of the decrease depends on the details

of the distribution of electrons around an atom

(sometimes called the form factor) - f is calculated using quantum mechanics

Scattering by an Atom

Scattering by a Unit Cell

For atoms A C

For atoms A B

phase

If atom B position

For 3D

Scattering by a Unit Cell

We can write

Scattering by a Unit Cell

Useful expressions

Hint dont try to use the trigonometric form

using exponentials is much easier

Scattering by a Unit Cell

- Examples

Unit cell has one atom at the origin

In this case the structure factor is independent

of h, k and l it will decrease with f as sin?/?

increases (higher-order reflections)

Scattering by a Unit Cell

- Examples

Unit cell is base-centered

h and k unmixed

h and k mixed

(200), (400), (220) ? ? (100), (121), (300) ? ?

forbiddenreflections

Scattering by a Unit Cell

- Examples

For body-centered cell

when (h k l ) is even

when (h k l ) is odd

(200), (400), (220) ? ? (100), (111), (300) ? ?

forbiddenreflections

Scattering by a Unit Cell

- Examples

For body centered cell with different atoms

when (h k l) is even

Cs

Cl

when (h k l) is odd

(200), (400), (220) ? ? (100), (111), (300) ? ?

Scattering by a Unit Cell

- The fcc crystal structure has atoms at (0 0 0),

(½ ½ 0), (½ 0 ½) and (0 ½ ½)

- If h, k and l are all even or all odd numbers

(unmixed), then the exponential terms all equal

to 1 ? F 4 f - If h, k and l are mixed even and odd, then two

of the exponential terms will equal -1 while one

will equal 1 ? F 0

16f 2, h, k and l unmixed even and odd 0, h, k

and l mixed even and odd

The Structure Factor

- The structure factor contains the information

regarding the types ( f ) and locations (u, v, w)

of atoms within a unit cell. - A comparison of the observed and calculated

structure factors is a common goal of X-ray

structural analyses. - The observed intensities must be corrected for

experimental and geometric effects before these

analyses can be performed.

Diffracted Beam Intensity

- Structure factor
- Polarization factor
- Lorentz factor
- Multiplicity factor
- Temperature factor
- Absorption factor
- ..

The Polarization Factor

- The polarization factor p arises from the fact

that an electron does not scatter along its

direction of vibration - In other directions electrons radiate with an

intensity proportional to (sin a)2

The polarization factor (assuming that the

incident beam is unpolarized)

The Lorentz - Polarization Factor

- The Lorenz factor L depends on the measurement

technique used and, for the diffractometer data

obtained by the usual ?-2? or ?-2? scans, it can

be written as - The combination of geometric corrections are

lumped together into a single Lorentz-polarization

(Lp) factor

The effect of the Lp factor is to decrease the

intensity at intermediate angles and increase the

intensity in the forward and backwards directions

The Temperature Factor

- As atoms vibrate about their equilibrium

positions in a crystal, the electron density is

spread out over a larger volume. - This causes the atomic scattering factor to

decrease with sinq/l (or S 4psinq/l) more

rapidly than it would normally.

The temperature factor is given by

where the thermal factor B is related to the mean

square displacement of the atomic vibration

Scattering by C atom expressed in electrons

This is incorporated into the atomic scattering

factor

The Multiplicity Factor

- The multiplicity factor arises from the fact that

in general there will be several sets of hkl

-planes having different orientations in a

crystal but with the same d and F 2 values - Evaluated by finding the number of variations in

position and sign in ?h, ?k and ?l and have

planes with the same d and F 2 - The value depends on hkl and crystal symmetry
- For the highest cubic symmetry we have

p100 6

p110 12

p1118

The Absorption Factor

- Angle-dependent absorption within the sample

itself will modify the observed intensity

Absorption factor for infinite thickness specimen

is

Absorption factor for thin films is given by

where µ is the absorption coefficient, t is the

total thickness of the film

Diffracted Beam Intensity

where K is the scaling factor, Ib is the

background intensity, q 4sin?/? is the

scattering vector for x-rays of wavelength ?

Diffraction Real Samples

- Up to this point we have been considering

diffraction arising from infinitely large

crystals that are strain free and behave like

ideally imperfect materials ( x-rays only

scattered once within a crystal) - Crystal size and strain affect the diffraction

pattern - we can learn about them from the diffraction

pattern - High quality crystals such as those produced for

the semiconductor industry are not ideally

imperfect - need a different theory to understand how they

scatter x-rays - Not all materials are well ordered crystals

Crystallite Size

- As the crystallites in a powder get smaller the

diffraction peaks in a powder pattern get wider. - Consider diffraction from a crystal of thickness

t and how the diffracted intensity varies as we

move away from the exact Bragg angle - If thickness was infinite we would only see

diffraction at the Bragg angle

Crystallite Size

Suppose the crystal of thickness t has (m 1)

planes in the diffraction direction. Let say ?

is variable with value ?B that exactly satisfies

Braggs Law

- Rays A, D, , M makes angle ?B
- Rays B, , L makes angle ?1
- Rays C, , N makes angle ?2

Crystallite Size

- For angle ?B diffracted intensity is maximum
- For ?1 and ?2 intensity is 0.
- For angles ?1 gt ? gt ?2 intensity is nonzero.

The Scherrer Equation

Subtracting

?1 and ?2 are close to ?B, so

Thus

The Scherrer Equation

- More exact treatment (see Warren) gives

Scherrers formula

Suppose ? 1.54 Å, d 1.0 Å, and ? 49o for

crystal size of 1 mm, B 10-5 deg. for crystal

size of 500 Å, B 0.2 deg.

Interference Function

- We calculate the diffraction peak at the exact

Bragg angle ?B and at angles that have small

deviations from ?B. - If crystal is infinite then at ? ? ?B intensity

0. - If crystal is small then at ? ? ?B intensity ? 0.

It varies with angle as a function of the number

of unit cells along the diffraction vector (s

s0). - At deviations from ?B individual unit cells will

scatter slightly out of phase. - Vector (s s0)/? no longer extends to the

reciprocal lattice point (RLP).

Interference Function

- ?(1) gt ?B(1) for 001 and ?(2) gt ?B(2) for 002
- If Hhkl H is reciprocal lattice vector then (s

s0)/? ? H.

Real space

Reciprocal space

Interference Function

- We define
- as deviation parameter

Interference Function

- In order to calculate the intensity diffracted

from the crystal at ? ? ?B, the phase differences

from different unit cells must be included. - For three unit vectors a1, a2 and a3

ni particular unit cells Ni total number of

unit cells along ai

From the definition of the reciprocal lattice

vector

Interference Function

since

Converting from exp to sines

Interference Function

- Calculating intensity we lose phase information

therefore

interference function

Maximum intensity at Bragg peak is F2N2 Width of

the Bragg peak ? 1/N N is a number of unit cells

along (s - s0)

Interference Function

Interference Function

Small crystallites (nanocrystallites)

- Consider our sample as any form of matter in

which there is random orientation. - This includes gases, liquids, amorphous solids,

and crystalline powders. - The scattered intensity from such sample

where

takes all orientations

Small crystallites (nanocrystallites)

- Average intensity from an array of atoms which

takes all orientations in space

Debye scattering equation

where

It involves only the magnitudes of the distances

rmn of each atom from every other atom

Small crystallites (nanocrystallites)

- Consider material consisting of polyatomic

molecules. - It is not too dense there is complete

incoherency between the scattering by different

molecules. - Intensity per molecule
- Lets take a carbon tetrachloride as example.
- It is composed of tetrahedral molecules CCl4.
- Then

correction factor

C Cl4

Small crystallites (nanocrystallites)

- For tetrahedral CCl4 molecule
- The intensity depends on just one distance r(C

Cl). - Peaks and dips do not require the existence of a

crystalline structure. - Certain interatomic distances that are more

probable than others are enough to get peaks and

dips on the scattering curve.

Intensity (in e.u. per molecule) for a CCl4 gas

in which the C Cl distance is r 1.82 Å.

(Warren)

Small crystallites (nanocrystallites)

- Lets treat crystal as a molecule
- FCC has 14 atoms
- 8 at corners of a cube
- 6 at face center positions
- a is the edge of a cube

Reciprocal Lattice and Diffraction

It is equivalent to the Bragg law since

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Home About Us Terms and Conditions Privacy Policy Presentation Removal Request Contact Us Send Us Feedback

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "X-ray Scattering" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!