aberration correction and possible structural tomography in complex tem
DESCRIPTION
Aberration Correction and Possible Structural Tomography in Complex TEM Fu-Rong Chen(1) and Ji-Jung Kai(1) Dept. of Engineering and System Science, National Tsing-Hua University, Hsin Chu, Taiwan. -8.23.2003 Beijing TEM conference. Better electron microscopes. - PowerPoint PPT PresentationTRANSCRIPT
Aberration Correction and Possible Structural Tomography in Complex TEM
Fu-Rong Chen(1) and Ji-Jung Kai(1)
Dept. of Engineering and System Science, National Tsing-Hua University, Hsin Chu, Taiwan
-8.23.2003 Beijing TEM conference
It is very easy to answer many of these fundamental biological questions; you just look at the thing! Make the microscope one hundred times more powerful, and many problems of biology would be made very much easier.
It would be very easy to make an analysis of any complicated chemical substance; all one would have to do would be to look at it and see where the atoms are. The only trouble is that the electron microscope is one hundred times too poor.
Better electron microscopes
-Richard F. Feynman-12.29.1959 American Physics Society, CIT-(There’s plenty of room at the bottom)
Develop new tool to discover new science !
=/2(Cs34+2f2)
ElectronsSpecimen
g 2go-g-2g
Diffraction Plane
Image Plane
o
Iii *
Microscope transfer function (Cs ,Cc,Δf…)
The total phase shift due to spherical aberration and defocus is
)](exp[ iAe
)()exp(1 HPiei
Exit wave is blurred by the function exp(-iχ)
f
gg*=Ig
(phase lost)
(phase lost)
Phase
Reciprocal Space Real Space
=C∮V(r,z)dz-e/h∫∫B⊥(r).dA
V1(r,z) V2(r,z)
V2(r,z)
V1(r,z)
(h,k,l): atomic positions
Phase Retrieval
a) Transport Intensity Equation(TIE)
Kinematic Diffraction
Electron Dynamic Diffraction ?(oversampling, phase extension)
Resolution extension and exit wave reconstruction in complex TEMF.-R. Chen et. al., Ultramicroscopy (2003), PRL(submitted) JEM(1999), JEM(2001)
Non-interferometric Phase Retrieval MethodNon-interferometric Phase Retrieval Method
b) Electro-Static Phase Plate(Zernike was awarded Nobel prize at 1953)
I1 I2 I3Initialwave
Propagating wave
f
(I(r,0) )=-kzI(r,z)
Amplitude
phase
I=1+[AF-1(cosP(H))+ F-1(sinP(H))]
Reciprocal Space Real Space
Process Flow (Images 3D Structure)
Exit Wave Reconstruction(aberration correction)
Exit Wave->Structure(quantification of EW)
Structural Tomography ?
Process Flow (Images 3D Structure)
Exit Wave Reconstruction(aberration correction)
Exit Wave->Structure(quantification of EW)
Structural Tomography ?
CTF
PhaseModulus
PhaseModulus
Initial Phase of Imageusing TIE
under-focusunder-focus over-focusover-focusf = f = ±±60 nm60 nm
zI(r,z) Reconstructed Phase
Phase objectPhase object (I(r,0) )=-kzI(r,z)
Process Flow (Images 3D Structure)
Exit Wave Reconstruction(aberration correction)
Exit Wave->Structure(quantification of EW)
Structural Tomography ?
Refine the phase bySelf-consistent propagation
(Gerchberg-Saxton algorithm)
CTF
PhaseModulus
PhaseModulus
1i
2i
.
.
ni
e= (ni)N1
1i
2i
.
.
ni
√ Ini
Initial Phase of Imageusing TIE
)()exp(1 HPiei
Process Flow (Images 3D Structure)
Exit Wave Reconstruction(aberration correction)
Exit Wave->Structure(quantification of EW)
Structural Tomography ?
Refine the phase bySelf-consistent propagation
(Gerchberg-Saxton algorithm)
CTF
PhaseModulus
PhaseModulus
Initial Phase of Imageusing TIE
Aberration correction(linear imaging)e=Aoexp(io)
1i
2i
.
.
ni
W
exp((f1))P1(H)
exp((f2))P2(H)
.
. exp((f2))P2(H)
e
e=F-1( )W*
W* W
1i
Process Flow (Images 3D Structure)
Exit Wave Reconstruction(aberration correction)
Exit Wave->Structure(quantification of EW)
Structural Tomography ?
Refine the phase bySelf-consistent propagation
(Gerchberg-Saxton algorithm)
CTF
PhaseModulus
PhaseModulus
Initial Phase of Imageusing TIE
Aberration correction(linear imaging)e=Aoexp(io)
Refinement byNon-Linear imaging
A=Ao+dA=o+d
I(g)=∫e(g+g’) e*(g’)T(g+g’,g’)dg’
I(r)=ii*+()2n/n!(2ni2ni*)
2=fi2=(Ii(r)-I(r)i
exp)2
dA (FA)2 FAF FAFo
d FAF F)2 FFo
FA=fi
F=
fi
GGG
ee
ee
ee
dGGGGTGGG
GTG
GTG
GI
',0'
*
*
**
')','()'()'(
),0()()0(
),0()()0(
)(
Linear image
Non-linearimage
G2
G1
0
Part 1:Part 1:Consider Non-linear Contributions to Consider Non-linear Contributions to
the Imagethe Image
)','()','()'()'()','( * GGGEGGGEGpGGpGGGT s
GG
ee dGGGGTGGGGI'
* ')','()'()'()(
Temporal coherence spatial coherence
))(exp()( GiGp
})]'()'([)/(exp{)','( 22 GGGGGGEs
Pure phase transfer function
22 ))'(()'()'(2))'(( GGGGGG
For FEG TEM α<<1, this term can be ignored
]')'()exp[()',0()0,'()','( 222 GGGGEGGEGGGE
i *
i
Represent the diffraction wave in the regular Fourier optics approach
'')')('()'()'()'()!2
(
'')')('()'()'()'()(
')'()'()'()'()(
44**4
22**2
**
dGGGGGtGGtGGG
dGGGGGtGGtGGG
dGGtGGtGGGGI
ee
ee
ee
'
** )'()'()'()'()(G
ee GtGGGtGGGI
For FEG TEM, ~1
]')'()exp[()]'()'()(
2exp[ 2222
GGGGGG
Taylor expansion
)]()([)!2
(
)]()([)()()()(
*444
*222*
GGGG
GGGGGGGI
dd
dddd
)]()([)!
()()()( *222
1
* GGGGn
GGGI ddnn
ndd
)]()([)!
()()()( *222
1
* rrn
rrrI in
inn
nii
sei EEir )~exp()(
Fourier Transform
Linear image Non-linear image
where
Higher resolution information
)exp( iAe
These images were recorded using a JEOL 3000F FEGTEM, the lens aberration parameters are Cs=0.6mm at 300kV, focal spread f=4 nm, divergent angle =0.15 mrad and three-fold astigmatism a3=855.8 nm and a3=117.11 mrad.
f ( 30 im
ages)
Complex Oxide
(NbW)O3
Reconstructed exit wave
Amplitude
Phase
ie Ae
Cation: (Nb, W)
Anion: (O)
The Nb16W18O94 has a structure of M30-3xM4xO90+x with lattice constants a= 1.2251 nm, b= 3.6621 nm, and c=0.394 nm.
Amplitudea
b
Phase
Super High Resolution
• How to achieve Super How to achieve Super High ResolutionHigh Resolution phase extensionphase extension Exit wave → structureExit wave → structure
Information limit
Reconstructed Exit wave
The phase had been corrected inside The phase had been corrected inside the information limitthe information limit
Fourier Transform
1) The retrieved Exit Wave contains structural information up to
“information limit”
2) To extend the structural information beyond “information limit”,
we need diffraction intensities for “resolution extension”.
Phase extensionPhase extensionExit wave reconstructionExit wave reconstruction
For 200 kV FEG TEM phase transfer function (in Scherzer defocus)
PhasesPhases
AmplitudesAmplitudes
(1(1Å)Å)-1-1
(2(2Å)Å)-1-1
From images → From images → phases & amplitudesphases & amplitudes inside the small cycle inside the small cycleFrom diffraction intensities→ From diffraction intensities→ amplitudesamplitudes inside the big cycle inside the big cycle
For HRTEMFor HRTEMIt is possible to enhance the resolution by It is possible to enhance the resolution by phase extensionphase extension
Phase Phase ExtensionExtension
real spaceComplex Maximum Entropy(Complex Exit wave)
reciprocal spacereciprocal spaceGerchberg-Saxton algorithm(Electron diffraction)
Corrected Data
Resolution enhancement by phase extension
Maximum entropy image processing can extrapolatemissing information to yield a most-probable answer
I(blurred)=I(good image) O(blurring function)’
L=-I(good)log( )+(I(good) O-I(blurring))I(good)I(initial)
PhasePhaseextensionextension
““Correct” phase information was extrapolated to next Correct” phase information was extrapolated to next higher frequency after MEM deconvolutionhigher frequency after MEM deconvolution
Phase information before MEM deconvolutionPhase information before MEM deconvolution
The thickness of this particle is 6 nm. The lens aberration parameters used were Cs=0.6mmfocal spread, f =3 nm and divergence angle, =0.2 mrad.
-260 nm -268 nm -276 nm
-284 nm -294 nm -298 nm
CdSe nanoparticle viewed along the [112] directionCdSe nanoparticle viewed along the [112] direction
Information limit
Extrapolated information from MEM {444}
Phase of exit wavePhase of exit wave
After phase extensionAfter phase extension
{444} →0.87Å
Process Flow (Images Structure (positions and atomic type)
Exit Wave Reconstruction(aberration correction)
Structural reversionExit Wave->Structure(quantification of EW)
Structural Tomography ?
S-State model
n(R,z)≈{1+P3exp(-|P1|(R-Ri)2)[exp{-kP1P2/2Eo}-1]
P1=Eoo (energy of s-state)P2=tP3=Cj (excitation parameter)Eo=voltage of TEMk=1/
Analytic Multi-Slice
=qo=exp(iV) z
t=n*z
1=(qoPz)qo
n(R,z)≈qon-(iz/4)[qo
i(2qon-i)]
n(R,z)=Aexp[i(nV+)]
=tan-1[C1(2V)+iC2((V)2]A= |1+iC1(2V)+iC2((V)2|
C1=t(n-1/8C2=t(n-1)(2n-1)/24V=projected potential of a unit cell
Exit wave can be related tophase grating of one unit cell
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
100
200
300
400
500
600
AuPt
AgSr
Ni
Ca
Si
Al
foca
l len
gth
(A)
1/Z
Optical lens Atomic lens
Al Au
Classical Approach: Classical Approach: Multislice MMultislice Methodethod
q1(x,y)
q2(x,y)q3(x,y)
qn(x,y)
qn(x,y)=exp{iσVn(x,y)}
),(])],()],(),([),([[),(),( 2112 yxpyxpyxpyxqyxqyxqyx nne
specimen can be subdivided into thin slice The potential of each slice projected into a plane
From Exit wave to StructureFrom Exit wave to Structure
),(10 yxViq 01 q
][ 002 zpqq )exp( 2gcipz c
02
00 4qq
icq
)(4
)(4
02
02
0
2
2200
2233 000
qqqic
qqqqic
q
20
1
4
42
c
c
......)](
)()([4
)(4
220
20
0222
002
022
2
320
2220
2344
0
00
00000
qqq
qqqqqqic
qqqqqqic
q
1 unit cell
n
i
ininn qq
icq
1
2
000 4
Analytic Solution
Exit wave from analytical solution and Multislice (10nm)
MultiSlice
Modulus
Modulus
Modulus
Modulus
Phase Phase
PhasePhase
Au Si
Quantification of Exit Wave- (S-state Wave) Structure
Modulus
Phase
Cu segregates to Al grain boundary
Modulus Phase
Exit Wave of CdSe nano-particle
(reconstructed by Complex TEM)
Structural Tomography (Discrete)?
(112)
(111)
(110)
(112)(111)
(c)
f
1. Projection-Slice theoremBracewell, Aust. J. 198 (1956)2. Convolution-Backprojection AlgorithmProc. Nat. Acad. Sci. 68,2236-2240 (1971)
人生七十才開始
祝 郭先生有八十歲的智慧 和青春的健康
Conclusions
1.一流的設備≠一流的研究2.不要忽略人類的智慧才能讓發
揮功能