tem diffraction
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MATE580 Spring2010 CLJ Lecture 6
Phase Contrast and High-resolution TEM
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MATE580 Spring2010 CLJ Lecture 6
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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MATE580 Spring2010 CLJ Lecture 6
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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MATE580 Spring2010 CLJ Lecture 6
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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MATE580 Spring2010 CLJ Lecture 6
)'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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MATE580 Spring2010 CLJ Lecture 6
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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MATE580 Spring2010 CLJ Lecture 6
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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MATE580 Spring2010 CLJ Lecture 6
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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MATE580 Spring2010 CLJ Lecture 6
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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MATE580 Spring2010 CLJ Lecture 6
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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MATE580 Spring2010 CLJ Lecture 6
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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MATE580 Spring2010 CLJ Lecture 6
),(),(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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MATE580 Spring2010 CLJ Lecture 6
)()( 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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MATE580 Spring2010 CLJ Lecture 6
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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