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John Miao
Stanford Synchrotron Radiation LaboratoryStanford Linear Accelerator Center
Crystallography without Crystals and the Potential of Imaging Single Molecules
Detector
Coherent Beam Atoms
00 )(,)(),( * ii ee 00 rrrrr
The trivial phases:
The Phase Problem
Regular Sampling: Sampling at the Bragg-peak Frequency
)1()()(1
0
/2
N
NieFr
rkrk
1,2,1,0 Nk
complex:)(r
# of unknown variables
# of independent equations
1D N N/2
2D N2 N2/2
3D N3 N3/2
# of unknown variables
# of independent equations
1D 2N N
2D 2N2 N2
3D 2N3 N3
)(*)(real:)( kkr FF
Oversampling: Sampling at Twice of the Bragg-peak Frequency
)2()()(1
0
)2/(2
N
NieFr
rkrk
12,2,1,0 Nk
)(*)(real:)( kkr FF complex:)(r
# of unknown variables
# of independent equations
1D N N
2D N2 2N2
3D N3 4N3
# of unknown variables
# of independent equations
1D 2N 2N
2D 2N2 4N2
3D 2N3 8N3
Miao, Sayre & Chapman, J. Opt. Soc. Am. A 15, 1662 (1998).Miao & Sayre, Acta Cryst. A 56, 596 (2000).
)3(120
10)()(
NN
Ng
r
rrr
Eq. (2) )4()()(
12
0
)2/(2
N
NiegFr
rkrk
12,2,1,0 Nk
)5(regiondensityelectron
regiondensitynoregiondensityelectron
> 2: the phase information exists inside the diffraction intensity!
The Oversampling Method
NjkiN
jj
jejkF /21
0
)()(
Nki
jjezja /2),(
)()(1
0
1
0j
N
j
jN
jj zzzazF
1D Case: ( 2N multiple solutions)
2D & 3D Case: (No multiple solutions)
Mathematically, 2D and 3D polynomials usually can not be factorized.
Bruck & Sodin, Opt. Commun. 30, 304 (1979).
Multiple Solutions
)/1()/1()(*1
0j
N
jzzzFzF
.))()(( equivalentarezzFCandzF N
)/1()()(*)()(1
0jj
N
jzzzzzFzFzI
Coherence Requirements of the Oversampling Method
aO2
d
aO
Oversampling vs. temporal coherence:
sampleDafor
sampleDaforO
3
23
Oversampling vs. spatial coherence:
Miao et al., Phys. Rev. Lett. 89, 088303 (2002).
:a sample size
:d desired resolution
An Iterative Algorithm
(I)
(II) )(exp
' 1)()( kkk jeFFj
)0,0,0('j = 0(III)
)(' rj )(' kjF= FFT-1( )(IV)
(V)
0)()()(
0)(&)()(
''1
''
rrrr
rrrr
jjj
jj
jorSif
Sif
)(kjF(VI) )(rj= FFT( )
(VII) )(kjAdopt from )(kjF
Fienup, Appl. Opt. 21, 2758 (1982).
Miao et al., Phys. Rev. B 67, 174104 (2003).
)(exp kF with > 5
The First Experiment of Crystallography without Crystals
(a) A SEM image (b) An oversampled diffraction pattern (in a logarithmic scale) from (a).
(c) An image reconstructed from (b).
Miao, Charalambous, Kirz, Sayre, Nature 400, 342 (1999). (d) The convergence of the reconstruction.
Szyxj
Szyxj
,,
'
,,
'
)()1(
)(
γr
r
Phase Retrieval as a Function of the oversampling ratio ()
= 5 (180 x 180 pixels) = 4 (160 x 160 pixels)
= 2.6 (130 x 130 pixels) = 1.9 (110 x 110 pixels)
Imaging Buried Nanostructures
(a) A SEM image of a double-layered sample made of Ni (~2.7 x 2.5 x 1 m3)
(c) An image reconstructed from (b)
Miao et al., Phys. Rev. Lett. 89, 088303 (2002).
(b) A coherent diffraction pattern from (a) (the resolution at the edge is 8 nm)
3D Imaging of Nanostructures
The reconstructed top pattern The reconstructed bottom pattern
An iso-surface rendering of the reconstructed 3D structure
Determining the Absolute Electron Density of Disordered Materials at Sub-10 nm Resolution
(a) A coherent diffraction pattern from a porous silica particle
(b) The reconstructed absolute electron density
Miao et al., Phys. Rev. B, in press.
(c) The absolute electron density distribution
within a 100 x 100 nm2 area
Imaging E. Coli Bacteria
(b) A Coherent X-ray diffraction pattern
from E. Coli
(c) An image reconstructed from (b).
(a) Light and fluorescence microscopy images
of E. Coli labeled with manganese oxide
Miao et al., Proc. Natl. Acad. Sci. USA 100, 110 (2003).
3D Electron Diffraction Microscopy
Electron gun
Aperture
Lens
Sample
Detector
Computer Simulation of Imaging a 3D Nanocrystal ([Al12Si12O48]8) with coherent electron diffraction
(a) A section (0.5 Å thick) viewed along [100] at z = 0
(c) The reconstructed section (0.5 Å thick) of the nanocrystal viewed along [100] at z = 0
(b) One of the 29 diffraction patterns (the 0 projection), SNR = 3
Miao et al., Phys. Rev. Lett. 89, 155502 (2002).
Si Al O
The Linac Coherent Light Source (LCLS)
Peak and Time Averaged Brightness of the LCLS and Other Facilities Operating or Under Construction
TESLATESLA
A Potential Set-up for Imaging Single Biomolecules Using X-FELs
X-FEL Pulses
X-ray LensMolecular Spraying Gun
CCD
Two Major Challenges
Solemn & Baldwin, Science 218, 229 (1982).
Neutze et al., Nature 400, 752 (2000).
When an X-ray pulse is short enough ( < 50 fs), a 2D diffraction pattern could be recorded from a molecule before it is destroyed.
Use the methods developed in cryo-EM to determine the molecular orientation based on many 2D diffraction patterns.
Crowther, Phil. Trans. Roy. Soc. Lond. B. 261, 221 (1971).
Use laser fields to physically align each molecule.
Larsen et al., Phys. Rev. Lett. 85, 2470 (2000).
Orientation Determination
Radiation Damage
An Oversampled 3D Diffraction Pattern Calculated from 3 x 105 Rubisco Protein Molecules
(a) One section of the oversampled 3D diffraction pattern with Poisson noise
(b) Top view of (a)
The Reconstructed Electron Density from the Oversampled Diffraction Pattern
The reconstructed 3D electron density map The reconstructed active site
The 3D electron density map of a rubisco molecule The active site of the molecule
Miao, Hodgson & Sayre, Proc. Natl. Acad. Sci. USA 98, 6641 (2001).
Summary
• Proposed a theoretical explanation to the oversampling method.
• Carried out the first experiment of crystallography without crystals.
• Opened a door to atomic resolution 3D X-ray diffraction microscopy.
• Proposed 3D electron diffraction microscopy for achieving sub-atomic resolution.
• Future application with the LCLS – imaging non-crystalline materials nanocrystals, and large biomoleucles.
Acknowledgements
B. Johnson, D. Durkin, K. O. Hodgson, SSRL
J. Kirz, D. Sayre, SUNY at Stony Brook
R. Blankenbecler, SLAC
D. Donoho, Stanford University
C. Larabell, UC San Francisco & LBL
M. LeGros, E. Anderson, M. A. O’Keefe, LBL
B. Lai, APS
T. Ishikawa, Y. Nishino, Y. Kohmura, RIKEN/SPring-8
J. Amonette, PNL
O. Terasaki, Tohoku University
Schematic Layout of the Experimental Instrument
X-rays
Pinhole
Corner
Sample
BeamstopPhotodiodeCCD
743 mm12.7 mm 25.4 mm
Experimental Demonstration of Electron Diffraction Microscopy
Zuo et al., Science 300, 1419 (2003).
The recorded diffraction pattern from a DWNT. Left: The reconstructed DWNT image; Right: A structure model of the DWNT.