Post on 09-Apr-2018
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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(an embarrassingly short discussion of)an embarrassingly short discussion of)Diffractioniffraction
Lecture 8ecture 8
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Fresnel diffraction at an edgeresnel diffraction at an edgeFresnel diffraction isresnel diffraction is near fieldear fielddiffractioniffractionSee the effects in imagesee the effects in imagesIn LaBn LaB6, number of visiblenumber of visibleimages diminished due to:mages diminished due to:
Lack of coherence / finiteack of coherence / finitesource sizeource size Incident convergence anglencident convergence angle
In FEG, multiple Fresnel fringesn FEG, multiple Fresnel fringesobservedbserved
Greater source coherencereater source coherence
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Fraunhoferraunhofer DiffractioniffractionFar fieldar field diffractioniffraction
Looking at the same event from an infinite distanceLooking at the same event from an infinite distance
i.e. the resultant pattern is independent of distance from screei.e. the resultant pattern is independent of distance from screenn
This is also the pattern observed in the back focal planehis is also the pattern observed in the back focal planeof the lensf the lens The back focal plane is effectively at infinite distanceThe back focal plane is effectively at infinite distance
Wavelength is 10Wavelength is 10s ofs ofpicometerspicometers, image planes are on order of, image planes are on order ofmillimetersmillimeters
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Diffractioniffraction - one slitne slit
http://www.micro.magnet.fsu.edu/primer/java/diffraction/basicdiffraction/index.htm
d
I= Io
sin
2
=dsin
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Diffraction from two slitsiffraction from two slits
Very narrow slits select just two of theery narrow slits select just two of theHuygenuygens waveletsavelets These must have the same phase at the slitshese must have the same phase at the slits
Path difference L = d sinath difference L = d sinFor an arbitrary direction r, phase differenceor an arbitrary direction r, phase difference = 22L//Constructive interference when d sinonstructive interference when d sin() =) = n
d
L
r
Diffraction and interferenceiffraction and interferencecombineombine
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Diffraction from two slitsiffraction from two slits
Very narrow slits select just two of theery narrow slits select just two of theHuygenuygens waveletsavelets These must have the same phase at the slitshese must have the same phase at the slits
Path difference L = d sinath difference L = d sinFor an arbitrary direction r, phase differenceor an arbitrary direction r, phase difference = 22L//Constructive interference when d sinonstructive interference when d sin() =) = n
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Diffraction from two slitsiffraction from two slits
Very narrow slits select just two of theery narrow slits select just two of theHuygenuygens waveletsavelets These must have the same phase at the slitshese must have the same phase at the slits
Path difference L = d sinath difference L = d sinFor an arbitrary direction r, phase differenceor an arbitrary direction r, phase difference = 22L//Constructive interference when d sinonstructive interference when d sin() =) = n
d
L
r
Diffraction and interferenceiffraction and interferencecombineombine
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Diffraction from two slitsiffraction from two slits
Very narrow slits select just two of theery narrow slits select just two of theHuygenuygens waveletsavelets These must have the same phase at the slitshese must have the same phase at the slits
Path difference L = d sinath difference L = d sinFor an arbitrary direction r, phase differenceor an arbitrary direction r, phase difference = 22L//Constructive interference when d sinonstructive interference when d sin() =) = n
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Diffraction fromiffraction frommultiple slitsultiple slits
I=
Io
N
sin
2
sin2 N
sin2
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Braggraggs LawLaw
Here path difference is AB + BC = 2ere path difference is AB + BC = 2dsininConstructive interference when:onstructive interference when:
n = 2dsindsin
dA CB
Scattered plane wavecattered plane waveIncident plane wavencident plane wave
Parallelarallelreflectingeflectingplaneslanes
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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d k i
kd
0 G2
k i
g2GG 3GG
-G-2GG
4GG
kd - ki = KKkd - ki = ggK = g= g
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Geometry of diffraction patternseometry of diffraction patternsFor small angles (such asor small angles (such aswe have in the TEM), Bragge have in the TEM), BraggLaw approximates as:aw approximates as:
Geometry of diffractioneometry of diffractionshown to right indicateshown to right indicatesthat:hat:
Manipulate to get:anipulate to get:
= 2d
R / L = 2
Rd = L Key equation
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Braggraggs LawLaw
Here path difference is AB + BC = 2ere path difference is AB + BC = 2dsininConstructive interference when:onstructive interference when:
n = 2dsindsin
dA CB
Scattered plane wavecattered plane waveIncident plane wavencident plane wave
Parallelarallelreflectingeflectingplaneslanes
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Geometry of diffraction patternseometry of diffraction patternsFor small angles (such asor small angles (such aswe have in the TEM), Bragge have in the TEM), BraggLaw approximates as:aw approximates as:
Geometry of diffractioneometry of diffractionshown to right indicateshown to right indicatesthat:hat:
Manipulate to get:anipulate to get:
= 2d
R / L = 2
Rd = L Key equation
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Diffraction pattern types: Spot (Selected area diffraction) Ring (multiple crystal - polycrystals) Amorphous (Diffuse rings) CBED (Convergent beam) Kikuchi (Inelastic scattering) HOLZ (High order Laue Zones)From diffraction patterns we can: measure the average spacing between layers or rows of atoms determine the orientation of a single crystal or grain find the crystal structure of an unknown material measure the size, shape and internal stress of small crystallineregions
Diffraction pattern types & usesiffraction pattern types & uses
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SAD patternAD patternformationormationDiffraction pattern formediffraction pattern formedin back focal plane ofn back focal plane ofobjective lensbjective lens
Location of back focal planeocation of back focal planedetermined by strength ofetermined by strength ofobjective lens (i.e. lensbjective lens (i.e. lenscurrent)urrent)Intermediate lens mustntermediate lens mustfocus at this pointocus at this point
Need to adjust exacteed to adjust exactintermediate lens focus /ntermediate lens focus /currenturrent DP magnification thusP magnification thus
depends on objective lensepends on objective lenscurrenturrent
SampleSample
Objective lensObjective lens
Back focal planeBack focal plane
ImageImage
ProjectorProjector
Back focal planeBack focal plane
Intermediate
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Single crystal materialsingle crystal materialsIf we know the index for
closest set of diffraction
spots it is possible to indexthe rest of the spots by
using vector addition
Every diffraction spot can
be reached by a
combination of these two
vectors
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Indexing single crystal patternsndexing single crystal patternsTilt sample until in zone axisilt sample until in zone axis i.e. until crystal is tilted so that.e. until crystal is tilted so thatthe beam is parallel to a zone ofhe beam is parallel to a zone ofplaneslanes
Zone axis:one axis: Weiss Zone Laweiss Zone Law
hUU + kV +kV + lWW = 00 Thus (hus (hklkl) // to // to
Use Rd =se Rd = L = constant= constant R1d1 = RR2d2 = RR3d3 = Ratio of datio of d2 / dd1, dd3 / dd1, etc.etc.characteristic of the zoneharacteristic of the zone Compare withompare with knownsnowns (e.g. Figse.g. Figs18.178.17 - 18.19, or computed8.19, or computedpatterns)atterns)
Example: bcc
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Single crystal materialsingle crystal materialsIf we know the index for
closest set of diffraction
spots it is possible to indexthe rest of the spots by
using vector addition
Every diffraction spot can
be reached by a
combination of these two
vectors
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Indexing single crystal patternsndexing single crystal patternsTilt sample until in zone axisilt sample until in zone axis i.e. until crystal is tilted so that.e. until crystal is tilted so thatthe beam is parallel to a zone ofhe beam is parallel to a zone ofplaneslanes
Zone axis:one axis: Weiss Zone Laweiss Zone Law
hUU + kV +kV + lWW = 00 Thus (hus (hklkl) // to // to
Use Rd =se Rd = L = constant= constant R1d1 = RR2d2 = RR3d3 = Ratio of datio of d2 / dd1, dd3 / dd1, etc.etc.characteristic of the zoneharacteristic of the zone Compare withompare with knownsnowns (e.g. Figse.g. Figs18.178.17 - 18.19, or computed8.19, or computedpatterns)atterns)
Example: bcc
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Polycrystalline materialsolycrystalline materialsIn small grained samples,n small grained samples,random distribution resultsandom distribution resultsin rings in DPn rings in DPIndexing is simple:ndexing is simple:
Rd =d = LMeasure R on patterneasure R on patternCamera contrastamera contrast L isiscalibratedalibrated Strongly dependent ontrongly dependent onobjective lens currentbjective lens current Must be certain you use theust be certain you use the
same current whename current whencalibrating with respect to aalibrating with respect to aknown standardnown standard
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Texture in ring patternsexture in ring patternsTexture: Distribution ofexture: Distribution oforientations is not random,rientations is not random,but preferredut preferredReadily visible in ringeadily visible in ringpatterns as arcs ofatterns as arcs ofintensityntensity
8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9
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Amorphous diffraction patternsmorphous diffraction patternsAmorphous materials do notmorphous materials do nothaveave randomandom placement oflacement ofatomstomsInstead distance betweennstead distance betweenneighbors follows a probabilityeighbors follows a probabilityfunctionunction Radial distribution functionadial distribution function Can be recorded and measuredan be recorded and measured
New techniqueew technique - FluctuationluctuationMicroscopyicroscopy Second order correlationsecond order correlations Medium range orderedium range order
DP from amorphous TiO2 andmeasured RDF
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Kikuchi diffractionikuchi diffraction
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Kikuchi diffractionikuchi diffraction
The Kikuchi lines
pass straight through
the transmitted and
diffracted spots. The
diffracting planes are
therefore tilted at
exactly the Bragg
angle to the optic axis
The crystal has now
been titled slightly
away from the Bragg
angle, so that the
Kikuchi lines no
longer pass through
the transmitted and
diffracted spots.
Here the crystal is
tilted so that more
that one set of
planes are
diffracting. Each set
of diffracting planes
has its own pair of
Kikuchi lines.