quantum probes of matter

7/30/2019 Quantum Probes of Matter

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Quantum Probes of Matter

Part 1

DiffractionThe Ewald Construction


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The Ewald Construction

It was the thesis of Paul Peter

Ewald that lead Max von Laue

to investigate the possibility of

X-rays diffracting off of crystals.

Later Ewald produced a simple

construct that predicted the

allowed diffraction spots


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Draw the reciprical latticepoints in k space. (Inthis case only the xz

plane is shown.)

The origin should alwayscoincide with one of

the lattice points.


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Find the center of the Ewaldsphere. This must be adistance kprobe=k= 2/away from the origin and in

the direction the probe iscoming from.

i.e. The vector AO describesthe momentum of theincoming probe. (dividedby)


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Positions such as B, (wherethe Ewald sphere intersectsa point on the recipricallattice) correspond to a

scattering event.Note: the vector G is the

spatial frequencyresponsible for thescattering while

The vector k* is the scatteredprobe.


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The role of the diagram is toensure that:

1) The magnitude of theprobes momentum does

not change. (energy isconserved)

2) The change in themomentum corresponds

to a spatial frequency thecrystal can provide.(conservation of pseudo-momentum)


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K space is BIG

Or another way to think aboutit is that the recipricallattice points are small andtypically sparse.


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K space is BIG

Or another way to think aboutit is that the recipricallattice points are small andtypically sparse.

Both the size (determined bythe probes momentum)and the direction

(determined by thecrystals orientation andprobes angle of incidence)have to be correct.


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Laue Diffraction

One way to overcome this isto use a broad spectrum orwhite probe beam. If thereis a range of energies for

the probe beam thenthere will also be a range ofmomentum.


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Laue Diffraction

One way to overcome this isto use a broad spectrum orwhite probe beam. If thereis a range of energies for

the probe beam thenthere will also be a range ofmomentum.

Now, the entire volumebetween the smallest and

largest Ewald spheresleads to scattering.


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Laue Diffraction

This image was recorded onfilm and corresponds to X-rays that were scattered indifferent directions.

source

appeturesample

film


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Laue Diffraction

A major advantage of theLaue approach is that it canimmediately provideinformation on the

symmetry of the crystal.

The image shown is for a BCCcrystal,(iron) with theincoming probe orientedalong the (1,0,0) direction.

Can you see the expected 4fold symmetry in thepattern?


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1,1,1

1,1,0

Laue Pattern of Silicon

Which corresponds to:

1,0,0


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1,1,1

1,1,0

Laue Pattern of Silicon

Which corresponds to:

1,0,0 4 fold symmetry

2 fold symmetry

3 fold symmetry


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Powder Diffraction

A second approach is to grindthe crystal into a finepowder and use amonochromatic probe.


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Powder Diffraction


Each crystal grain willcorrespond to a recipricallattice in a differentorientation.


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Powder Diffraction


Now the probe beam willsee all orientations of thereciprical lattice.


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Powder Diffraction

A second approach is to grindthe crystal into a finepowder and use amonochromatic probe.Now the probe beam will

see all orientations of thereciprical lattice.

This converts each of the

points to a circle, ensuringthat each reciprical latticekprobe or less away from theorigin provides a diffractionspot.


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Powder Diffraction

A weakness of powder diffraction is that all

of the symmetry information is lost,

however it is an extremely effective method

to make rapid comparisions to identify the

chemical phase(s) of a sample.


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Theta-2theta geometryEwald Construction

2 sin

2where

hkln d

k

d

k

ik

i

k

fk

k

fk

Braggs Law


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theta-2theta measurements

Th t 2Th t


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Theta-2Theta

Vary MAGNITUDE ofk while maintaining its orientation relative to

sample normal.

HOW? Usually rotate sample and detector with respect to x-raybeam.

Resulting data ofIntensity vs. 2 shows peaks at the detector (kf) for

kvalues satisfying the diffraction condition.

Detects periodicity of planes parallel to surface.

ikfk

ksmaller k

Th t 2Th t


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Polycrystalline sample has a number of peaks due to mixture of crystalorientations.

Allows rapid fingerprinting of samples

10 20 30 40 50 60 70 80 90 1000

2000

4000

6000Polycrystalline Silicon Powder

Intensity(counts/se

c)

Q

Theta-2Theta

Theta 2Theta


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10 20 30 40 50 60 70 80 90 1000

2000

4000

6000Polycrystalline Silicon Powder

Q

Theta-2Theta

2 sin

2where

hkln d

k

n = 2 d sin(2/2)

= .154 nm

d = .3134 nm.1919 nm

.1637 nm

.1357 nm

.1245 nm


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Rocking Curve Scan for epitaxial films

(a) (b)

Hexagonal CdTe nanowires produced on sapphire substrates

Question: How perfect is the alignment?


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Rocking Curve Scan for epitaxial films

Vary ORIENTATION ofk relative to sample normal while maintaining

its magnitude.How? Rock sample over a very small angular range.

Resulting data ofIntensity vs. Omega (w, sample angle) shows

detailed structure of diffraction peak being investigated.

ik fk

k

Rock Sample

k Sample normal


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Rocking Curve Example

Rocking curve of single crystal GaN around (002) diffraction

peak showing its detailed structure.

16.995 17.195 17.395 17.595 17.795

0

8000

16000

GaN Thin Film

(002) Reflection

Intensit

y(Counts/s)

Omega (deg)

Variation in d spacing 1%


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CdTe nanowire rocking curve

MA149,FML201,CdTd/Al2O3,ROCKING CURVE,Oct23,2006

Operations: Smooth 0.266 | Import

MA149,FML201,CdTd/Al2O3,ROCKING CURVE,Oct23,2006 - File: ma149.RAW - Type: Rocking curve - Start: 2.000 - End: 22.010 - Step: 0.030 - Step time: 4. s - Temp.: 25 C (Room) - Time St

Operations: Import

MA149,FML201,CdTd/Al2O3,ROCKING CURVE,Oct23,2006 - File: ma149.RAW - Type: Rocking curve - Start: 2.000 - End: 22.010 - Step: 0.030 - Step time: 4. s - Temp.: 25 C (Room) - Time St

Lin(Counts)

0

1000

2000

3000

4000

5000

6000

7000

Theta - Scale

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 2

Note the clever presentation


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X-Ray spectra (Copper)

Low accelerating voltage

High accelerating voltage

Can use filters and

monochromators

Specific to material


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Thermal Neutron Spectrum

Distribution of Neutrons versus Energy

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2

Energy (eV)

Fraction

Distribution of Neutrons versusWavelength

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Wavelength (nm)

Fraction

Compared to X-rays, neutrons used for diffraction havemany orders of magnitude less energy.

Surprisingly irradiation by a neutron beam can cause less

damage than via a beam of light!


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Neutrons vs. X-rays!

Neutrons allow easy access to atoms that are usually unseen in X-ray Scattering

Chatterji, Neutron Scattering from Magnetic Materials(2006)

H d t t ?


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How do we get neutrons?

Research Reactor Sources

Uses nuclear fission to

create neutrons

Continuous neutron flux Flux is dependent on

fission rate

Limited by heat flow in

from the reaction Creates radioactive

nuclear waste

Pynn, Neutron Scattering: A Primer (1989)


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Elastic Neutron Scattering

Determine length scales and

differentiate between nano-, micro-, and macro-

systems.

Utilizes position and

momentum correlation.

Mitchell et. al, Vibrational Spectroscopy with Neutrons(2005)

Pynn, Neutron Scattering: A Primer (1989)


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How are neutrons useful?

Mitchell et. al, Vibrational Spectroscopy with Neutrons(2005)


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Electron Diffraction

In principle electron diffraction

is similar to X-ray and neutron

scattering.

However the size and charge onthe electron conspire to make

electron diffraction very

different in practice. (although

the same conditions apply forthe Ewald sphere


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Electrons are charged so their detection can bevery straightforward. In many cases the charge iscollected on a piece of metal and converted to acurrent. (films and CCD arrays can also be used)

In part because electrons are light and repel eachother strongly, very high energy electron beamsare typical (150 KeV for example)

This brings their deBroglie wavelength down to avery small value. (


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The strong interactions of electrons with otherelectrons can lead to novel effects in thediffraction patterns of materials.

In particular, there are cases in which electronswhich have been scattered elastically andinelastically essentially become new probes i.e.generate new diffraction patterns that overlap the

original single-scattered patterns. It turns out that this can transform the pattern of

spots into lines called Kikuchi lines.

Traditionally this has rendered the use of electrondiffraction to a qualitative exercise.

Recently, motivated by nanoscience and enabled

by software and instruments improvments,quantitiative information has been extracted.
http://images.google.ca/imgres?imgurl=http://pwatlas.mt.umist.ac.uk/internetmicroscope/micrographs/microscopy/decr5b-a-big.jpg&imgrefurl=http://pwatlas.mt.umist.ac.uk/internetmicroscope/micrographs/microscopy/kikuchi.html&h=887&w=895&sz=81&hl=en&start=1&um=1&tbnid=mp2bPr0mEY6uCM:&tbnh=145&tbnw=146&prev=/images%3Fq%3Delectron%2Bmicroscopy%2Bkikuchi%26svnum%3D10%26um%3D1%26hl%3Den%26rls%3DGGLD,GGLD:2003-47,GGLD:en%26sa%3DN


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Channelling

While the electron clouds are diffuse, thenuclei in a material are very small positiveobjects.

In the lastest version of electron microscopes,the probe beams are much smaller than thesepaparation between atomic cores and can

be effectively guided along a column. Known as channelling, this can actually

reduce the effective size of the beam but italso limits where it can go.

McMaster will be using its advancedcomputer cluster to simulate this effect and

understand the detailed results on electronimaging of materials.

quantum probes of matter

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