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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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The Ewald Construction
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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The Ewald Construction
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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The Ewald Construction
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 Ewald Construction
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
A second approach is to grindthe crystal into a finepowder and use amonochromatic probe.
Each crystal grain willcorrespond to a recipricallattice in a differentorientation.
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Powder Diffraction
A second approach is to grindthe crystal into a finepowder and use amonochromatic probe.
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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Electron Diffraction
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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Electron Diffraction
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%3DN7/30/2019 Quantum Probes of Matter
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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.
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