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.