transmission electron microscopy skills:diffraction lecture 9

8/8/2019 Transmission Electron Microscopy Skills:Diffraction Lecture 9

1/25

(an embarrassingly short discussion of)an embarrassingly short discussion of)Diffractioniffraction

Lecture 8ecture 8


2/25

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


3/25

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


4/25

Diffractioniffraction - one slitne slit

http://www.micro.magnet.fsu.edu/primer/java/diffraction/basicdiffraction/index.htm

d

I= Io

sin

2

=dsin


5/25

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


6/25





7/25




d

L

r

Diffraction and interferenceiffraction and interferencecombineombine


8/25





9/25

Diffraction fromiffraction frommultiple slitsultiple slits

I=

Io

N

sin

2

sin2 N

sin2


10/25

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


11/25

d k i

kd

0 G2

k i

g2GG 3GG

-G-2GG

4GG

kd - ki = KKkd - ki = ggK = g= g


12/25

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


13/25

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


14/25

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


15/25

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


16/25

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


17/25

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


18/25

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


19/25

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


20/25

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


21/25

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


22/25

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


23/25

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


24/25

Kikuchi diffractionikuchi diffraction


25/25

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.

transmission electron microscopy skills:diffraction lecture 9

Documents