Abstract

A solution to the phase problem for diffraction by two-dimensional crystalline slabs is described, based on the application of a compact support normal to the slab. Specifically we apply the iterative Gerchberg-Saxton-Fienup algorithm to simulated three-dimensional transmission electron diffraction data from monolayer organic crystals. We find that oversampling normal to the monolayer alone does not solve the phase problem in this geometry in general. However, based on simulations for a crystalline monolayer (tetracyanoethylene), we find that convergence is obtained if phases are supplied from a few high-resolution electron microscope images recorded at small tilts to the beam direction. Since current cryomicroscopy methods required a large number of images to phase tomographic diffraction data, this method should greatly reduce the labor involved in data acquisition and analysis in cryo-electron microscopy of organic thin crystals by avoiding the need to record images at high tilt angles. We discuss also the use of laser tweezers as a method of supporting nanoparticles in TEM for diffractive imaging

© 2003 Optical Society of America

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References

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Acta Crystallogr. (1)

D. Sayre, "Some implications of a theorem due to Shannon", Acta Crystallogr. 5, 843 (1952).
[CrossRef]

Acta Crystallogr. A (3)

P. A. Doyle and P. S. Turner, "Relativistic Hartree-Fock X-Ray and Electron Scattering Factors", Acta Crystallogr. A 24, 390 (1968).

R. J. Read, "Improved Fourier Coefficients for Maps Using Phases from Partial Structures with Errors," Acta Crystallogr. A 42, 140-149 (1986).
[CrossRef]

H. He, S. Marchesini, M. Howells, U. Weierstall, G. Hembree, and J. C. H. Spence, "Experimental lensless soft-X-ray imaging using iterative algorithms: phasing diffuse scattering," Acta Crystallogr. A 59, 143-152 (2003).
[CrossRef] [PubMed]

Acta Crystallogr. B (1)

D. Belemlilga, J. M. Gillet, and P. J. Becker, "Charge and momentum densities of cubic tetracyanoethylene and its insertion compounds," Acta Crystallogr. B 55, 192-202 (1999).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Ashkin and J. M. Dziedzic "Optical levitation in high vacuum", Appl. Phys. Lett. 28, 333-335 (1976).
[CrossRef]

J. Mol. Biol. (1)

N. Grigorieff, T. A. Ceska, K. H. Downing, J. M. Baldwin, and R. Henderson, "Electron-crystallographic refinement of the structure of bacteriorhodopsin," J. Mol. Biol. 259, 393-421 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (2)

J. Struct. Biol. (1)

J.C.H.Spence, U. Weierstall, T.T.Fricke, K.H.Downing and R.M.Glaeser, "Three-dimensional diffractive imaging for crystalline monolayers with one-dimensional compact support," J. Struct. Biol., submitted (2003).
[CrossRef] [PubMed]

Nature (3)

E. Nogales, S. G. Wolf, and K. H. Downing, "Structure of the alpha beta tubulin dimer by electron crystallography," Nature 393, 191-191 (1998).
[CrossRef]

J. W. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens," Nature 400, 342-344 (1999).
[CrossRef]

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond X-ray pulses," Nature 406, 752-757 (2000).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. B (2)

L.D. Marks, �??General solution for three-dimensional surface structures using direct methods,�?? Phys. Rev. B 60, 2771-2780 (1999).
[CrossRef]

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and Spence,, "Inversion of X-ray diffuse scattering to images using prepared objects," Phys. Rev. B 67, 174114 (2003).
[CrossRef]

Science (1)

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, "Atomic resolution imaging of a carbon nanotube from diffraction intensities," Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Ultramicroscopy (1)

U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, and R. R. Panepucci, "Image reconstruction from electron and X-ray diffraction patterns using iterative algorithms: experiment and simulation," Ultramicroscopy 90, 171-195 (2002).
[CrossRef] [PubMed]

Other (4)

S. Lindaas, B. Calef, K. Downing, M. Howells, C. Magowan, D. Pinkas, and C. Jacobsen, "X-ray Holography of Fast-Frozen Hydrated Biological Samples," in X-ray Microscopy and Spectromicroscopy, J. Thieme, ed. (Springer, Berlin, 1998), pp. II-75.

H. H. S. Marchesini , H. N. Chapman , S. P. Hau-Riege A. Noy, M. R. Howells , U. Weierstall , J.C.H. Spence, "Imaging without lenses," <a href=http://arxiv.org/abs/physics/0306174 (2003).>http://arxiv.org/abs/physics/0306174 (2003).</a>

H. Stark, Image recovery : theory and application (Academic Press, Orlando, 1987).

M. J. Buerger, Vector space, and its application in crystal-structure investigation (Wiley, New York, 1959).

Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1.

Geometry of “oversampling” for a two-dimensional crystalline slab. The compact support constraint is applied in real space normal to the molecular monolayer. In reciprocal space the intensities along the relrods are assumed known from a diffraction pattern tilt series. Known phases are supplied on a plane normal to the relrods and two planes tilted by 45 degrees. The HIO-algorithm then reconstructs the phases along the relrods.

Fig. 2.
Fig. 2.

Arrangement of molecules in the cubic unit cell of a tetracyanoethylene (TCNE) crystal. Additional molecules are generated by a center of inversion at (1/4, 1/4, 1/4). The center of inversion for the molecule lies at the midpoint of the cell axes

Fig. 3.
Fig. 3.

Correlation coefficient CC (solid line) between the estimated and true charge density and the HIO rms-error εk (dashed line) against the iteration number for TCNE with known phases supplied on one (green) and three planes (red). The algorithm didn’t converge for phases known only on one plane (green). For phases known on three planes, the algorithm converged to values of εκ of 0.047 after 70 iterations. The correlation coefficient CC was then CC=0.995, which means almost perfect agreement between estimated and true charge density. The algorithm consists of cycles of 25 HIO iterations followed by 10 error-reduction iterations, which explains the periodic structure in the rms-error. When the HIO-iterations start after 10 error reduction iterations (at iteration number 35 and 70) the rms-error gets worse but improves soon after. For details see [6].

Fig. 4.
Fig. 4.

(300kB) Movie of HIO iterations. Isopotential view of the electrostatic potential of TCNE as reconstructed by the HiO algorithm from simulated diffracted intensities (to 0.35 Å) combined with 3 images (to 0.35 Angstroms). The computational supercell is shown, the size in the z (horizontal) direction is three times the thickness of the monolayer, providing for the compact support. The size in the x and y direction is the size of one unit cell (0.97nm). 70 iterations were needed for the 30×30×90 voxel data set. The movie is assembled from the resulting images after every 5th iteration.

Equations (1)

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CC = V ρ t ( r ) ρ e ( r ) d r [ V ρ t ( r ) 2 d r V ρ e ( r ) 2 d r ] 1 2 = h F h t F h e cos ( ϕ h t ϕ h e ) [ h F h t 2 h F h e 2 ] 1 2 = h F h 2 cos ( ϕ h t ϕ h e ) h F h 2

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