tem diffraction

7/30/2019 TEM Diffraction

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MATE580 Spring2010 CLJ Lecture 6

Phase Contrast and High-resolution TEM


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Student presentations

WEEK 1 MAR 29 NoneWEEK 2 APR 5 Stephanie and JJ HAADFWEEK 3 APR 12 Steven and Amalie Lorentz EM

Chris W and Asher CBED

WEEK 4 APR 19 Chris Dennison, Ertan Agar, Viral Chhasatiaand Eric Wargo Electron Tomography

WEEK 5 APR 26 Matt & Andrew Cs Correction & NCSIWEEK 6 MAY 3 Babak & Michael Strain Mapping

Kavan & Hasti TEM-CathodoluminescenceWEEK 7 MAY 10 Chris Barr & Michael Coster - ???

WEEK 8 MAY 17 Ioannis & Greg in situ nanoindentationWEEK 9 MAY 24 Ed & Tim - ???WEEK 10 MAY 31 Memorial Day (No Class)WEEK 11 JUN 7 ???WEEK 12 JUN 14 Finals Week (No Class)


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Phase contrast:

Multi-beamimaging

When electron waves pass through the sample their phase is shifted(diffraction) w.r.t. the incident wave in different ways. When severalwaves are allowed to interact the phase differences manifestthemselves in the 2-D interference pattern in the image plane: thephase-contrast image.

Phase contrast images can be difficult to interpret (even though theysometimes look very straight forward) because many factorscontribute to the phase shifts: thickness, orientation, scattering factor,focus, and astigmatism can all change the appearance of the image.


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Origin of lattice fringe

...32102

3

2

2

2

1

2 rgr

g

r

g

r

0GGG iiiiT eeee

rkg

rk

0

DI ii ezez 22 )()(

rewrite the wave equation for just two beams

'gksgkK g IID

Az )(0

iBez )(g

eff

eff

s

tst

B

sin

g

effts

2

somesubstitutions


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)'2(2

rgrk ii

BeAeI

)'2()'2(22 rgrg ii eeABBAI

)'2cos(222 rgABBAI take g to beparallel to x

)'2sin(222 stxgABBAI

Therefore, the intensity is a sinusoidal oscillation (this is the latticefringe!) normal to g, with a periodicity that depends on excitation error(s) and thickness (t)


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O

G

O

G

-G

s = 0 and g = g s 0

off-axis lattice

fringe imaging

on-axis 3-beam

imaging

O

G

-G

on-axis many-

beam imaging

s 0

Resist the temptation of interpreting the spots in the image as atoms!

All this is a some of the individual fringes. Proof on the next slide.


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d004 = 0.14 nm

0.13 nm fringe spacing in a0.25 nm resolution TEM?

generated by interference of

113 fringes

Not really atomic planes: 2nd order interference between 113 lattice fringes


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Gaussian image plane

Cs 0

Plane of least confusion

High-resolution TEM requiressampling high spatial frequenciesin the specimen

To sample high spatial

frequencies, we must includebeams scattered to (relatively)high angles

This means that off-axisaberrations (mainly Cs anddefocus) become very important

High frequencies = high aberrations


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High-resolution TEM

any point on the specimen function f(x,y) becomes an extended region (disk)g(x,y) in the image

yxf ,

rgyxg ,

optical system

Af Bf

Ag Bg

'

''

rrr

rrrrr

hf

dhfg

functionspreadpointrh

signals from fA and fBoverlap informationfrom fB in the sample ismixed with the signalfrom fA


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Specimen Transmission Function f(r)

),(exp),(),( yxiyxAyxf I

A(x,y) (=1) is the amplitude and I(x,y) is the phase (function of thickness)

A(x,y) = 1 for the incident wave amplitude and the phase changedepends on the specimen potential V(x,y,z)

t

t dzzyxVyxV0

),,(),( Projected potential

meEh

2

),,((2'

zyxVEmeh

wavelength of electron wavelength of electron in the sample


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Phase shift from the specimen

dzdzd 2

'2

dzzyxVE

d ),,(

meE

h

2

),,((2'

zyxVEme

h

wavelength of electron wavelength of electron in the sample

E

yxVdzzyxVd t where),(),,(

interaction constant


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),(),(exp),( yxyxViyxf t

Phase object approximation (POA) includes absorption term (x,y)

if the specimen is very thin then absorption can be ignored

for thinspecimens

),(1),( yxViyxf t Weak Phase ObjectApproximation (WPOA)

The WPOA states that (if the specimen is very thin)

the transmitted wave function is linearly related to theprojected potential of the specimen

Model for interpreting* what we see in HRTEM images*through image simulation and comparison


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Applying the WPOA

)(sin2)()()( uuuu EAT

new transfer function (objective lens transfer function in W&C) notequal to the contrast transfer function H(u)

Note: this is commonly also called the phase-contrast transferfunction almost everywhere (i.e., on the internet).

Whatever we call it, T(u) tells us what our HRTEM imagesshould look like

T(u) < 0 means positive phase contrast (dark atoms)

T(u) = 0 means no information in the image for this u

T(u) > 0 means negitive phase contrast (bright atoms)


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)()( uu AT first well ignore theenvelope function E(u)and the aberrationfunction sin X(u)

T(u)

0u1 u

Ideal form of T(u):

T(u) = 0 foru = 0

T(u) is negative and constant out to a frequency u1

T(u+u1) = 0

Aperture Function


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X(u) (phase distortion function) gives the phase shift caused byspherical aberration, defocus, and astigmatism

)(sin)( uu T

if X(u) = n /2 (where n is an odd integer), then CTF has maxima

if X(u) = n /2 (where n is an even integer), then CTF is zero

T(u)

0

u = /2

u

3/2

2 uA

aperture

5/2

3

Oscillating sign of T(u)not so good (someatoms dark and somebright)

Luckily sin X(u) is morecomplicated than sin x

simple sine wave

Aberration Function


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X(u) in terms of Cs and f (assuming corrected astigmatism)

fCs 3)(

24)()(

24

0

fCdD s integrate over arange of

gnd BB 2sin2

24

2)(

2)(

2244 uf

uCD s

u

432

2

)( uCuf s

u

point in the object becomesdisk with diameter() in theimage


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432

2)( uCuf s

u

T(u) = sin X(u)

0u

sin X(u) = 0 for u = 0

decreases with increasing u (for small u, f dominates)when Cs takes over the sin X(u) becomes zero thenpositivefor higher u, sin X(u) begins a sinusoidal oscillation thatcontinues to infinity

curve for negative fand positive Cs

interpretable contrast

balance between Csand f


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Scherzer Defocus: best CTF by balancing Cs and f

2/1)(2.1 sSch Cf

All diffracted beams have nearly constant phase out to the first zero

This is known as the instrumental resolution limit

Be careful when interpreting images if they include information atfrequencies higher than the frequency defined by the instrumentalresolution

Generally, its best not to include this information at all!


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3322 uCufdud

s derivative of X(u)

22

0 uCf s

432

23

2uCuf s

2/1

3

4

sSch Cf

Derivation of Scherzer Defocus

sin -/2 = -1 and-2/3 to - /3 is near -1

dX/du is zero when

sin X(u) is flat

Scherzer Defocus (4/3)1/2 = 1.2

43

41

66.0 sSch Cr Best resolution at fSch


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Envelope Damping Functions give the effects of finite spatial and temporalcoherence

)()()()( uuuu aceff EETT

spatial

temporal

Spatial coherence: limited by thedemagnified source size andconvergence

Temporal coherence: limited byenergy spread of the electrons

Envelope functions impose a virtualaperture in the BFP that limitsinformation transfer to the image

This is called the information limit


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Other defocus settings: Minimum contrast and Passbands

2/144.0 sMC Cf Minimum contrast focuscan be easily identified

2/1

2

38

sn

P Cn

f

sin X(u)

0u

Passband

Higher order passbands are used toget contrast at specific frequencies(specific Bragg reflections)


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Aberration-free focus condition

2

23.042 24

3

dd

Cmf snAFF

m = 0, 1, 2, 3, 4d = d-spacing

This gives maxima in the CTF for desired reflections andextends the resolution of the microscope beyond theinstrumental resolution

Caution: Using this or other higher order passband focussettings can give inaccurate images (must know the crystalwell and the crystal must be perfect)


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Simulations with CTF explorer: http://www.maxsidorov.com/ctfexplorer/


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Cautions for best HRTEM:

Most specimens are not weak phase objects (if you see a thicknessfringe then the WPOA does not apply)Fresnel effects can confuse imageInelastic scattering can confuse image

Considerations for best HRTEM:

Best instrument: low Cs and small

Well aligned and stable (electronics and moving parts)Repeatedly check beam tiltWork in thin, flat, clean areas of specimensAlign crystal to a zone axisRepeatedly check astigmatism correction

Find fMC and then record a focus series of imagesRecord the DP at the same condenser setting to calculate convergenceCompare images with simulations


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MATE580 Spring2010 CLJ Lecture 6thickness

Perfect alignmentScherzer defocus

Two-fold

astigmatism

Beam tilt

Beam tilt and 2-fold astigmatism

3-foldastigmatism

3-fold and 2-foldastigmatism

Crystal tilt


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Field Emission Sources and Delocalization

2max

2uCfuR s

1to0.75Mwhere2

max

2

uMCf sopt

3

max

3

min

4

1uCR s

As Cs decreases, delocalization decreasesAs decreases (voltage increases), delocalization decreases

Delocalization is always a problem in FEG unless you have Cs corrector


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Image simulation

image simulations dont agree with

experiment this time the theory wasright and the experiment wrong


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Model microscope:just like our drawing of the objective lens

I = incident wave

E = exit wave

D = diffractionpattern

Im = image

High voltage, e- gun,illumination system

all post specimenoptics reduced to an

objective lens


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Multislice Method

Incidentbeam

Projectionplane 1

Projectionplane 2

Projectionplane 3

Projectionplane 4

Projectionplane 5

propagate

propagate

propagate

propagate

calculatebeams

1. divide the sample up into thinslices

2. project the potential of a sliceonto a plane within that slice:this a phase grating

3. calculate the amplitudes andphases of all the beamsresulting from the incidentbeam interacting with theprojected potential

4. propagate these beamsthrough the microscope untilthey reach the next slice

5. Calculate a new set of beams

calculatebeams

calculatebeams

calculatebeams

calculatebeams


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Image Processing: Computer manipulation of HRTEM images

1. To improve the appearance of the image2. Quantify and/or extract data from the image

DigitalMicrograph (GATAN$) and ImageJ (free!) most common

image processing software for TEM

Always be careful to report any processing you have done on an image

and dont do it if you dont need to


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Fourier Filtering

Digital image FFT Apply mask FFT-1 Filtered image

Region of interest and masks act as virtual selected area andobjective apertures, respectively

Demo in DM


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Diffractogram (FFT) analysis: Objective lens astigmatism and drift

0 nm

14 nm

80 nm

0.3 nm

Astigmatism Drift

0.5 nm

0 nm

Whats wrong with this picture?


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432

2)( uCuf s u

sin X(u) = 1 when X(u) = n /2 where n is odd (bright ring)sin X(u) = 0 when X(u) = n /2 where n is even (dark ring)

Diffractogram (FFT) analysis: Spherical aberration and defocus

aberration function

fuCu

ns 2

23

2bmxy

1. collect image of amorphous material (with internal standard) at highmagnification

2. compute the digital diffractogram

3. assign n = 1 to first bright ring and n= 2 to first dark ring, and so on

4. calculate u from calibration standard

5. plot nu-2 against u2 and the slope = Cs

3 and intercept = 2f


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Diffractogram (FFT) analysis: Beam tilt

Making controlled (equal and opposite) beam tilts in the TEM whileobserving the live FFT of an amorphous material can give the beam tilt

Notice that beam tilt and astigmatism have similar effects on the

diffractogram

Typically correct beam tilt by modulating the objective lens current(current center) or voltage (voltage center)


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Through-focus Series Reconstruction: several images collected atdifferent defocus values and used to reconstruct exit surfacewave (computer aberration correction)

Crystallographic Image Processing: collect HRTEM and SAEDpatterns at different zone axes and combine to calculate the 3Dcrystal structure (analogous process to X-ray crystallography butincludes images and small areas)

Strain mapping: Measuring phase shifts (not of the beams) oflattice fringes in an image through Fourier processing (geometricphase analysis) or finding peaks in HRTEM images andcomparing shifts of those peaks around defects (peak-finding orpeak-pairs analysis) Babak and Michael will tell us more

Other image processing techniques for Quantitative HRTEM


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Moir fringe

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