Towards sub-Å atomic resolution electron diffraction imaging of metallic nanoclusters: A simulation study of experimental parameters and reconstruction algorithms

Wave field reconstruction and phase imaging by electron diffractive imaging

In electron diffractive imaging, the phase image of a sample is reconstructed from its diffraction intensity through iterative calculations. The principle of this method is based on the Fourier transform relation between the real-space wave field transmitted by the sample and its Fraunhofer diffraction wave field. Since Gerchberg's experimental work in 1972, various advancements have been achieved, which have substantially improved the quality of the reconstructed phase images and extended the applicable range of the method. In this review article, the principle of diffractive imaging, various experimental processes using electron beams and application to specific samples are explained in detail.

When holography meets coherent diffraction imaging

The phase problem is inherent to crystallographic, astronomical and optical imaging where only the intensity of the scattered signal is detected and the phase information is lost and must somehow be recovered to reconstruct the object's structure. Modern imaging techniques at the molecular scale rely on utilizing novel coherent light sources like X-ray free electron lasers for the ultimate goal of visualizing such objects as individual biomolecules rather than crystals. Here, unlike in the case of crystals where structures can be solved by model building and phase refinement, the phase distribution of the wave scattered by an individual molecule must directly be recovered. There are two well-known solutions to the phase problem: holography and coherent diffraction imaging (CDI). Both techniques have their pros and cons. In holography, the reconstruction of the scattered complex-valued object wave is directly provided by a well-defined reference wave that must cover the entire detector area which often is an experimental challenge. CDI provides the highest possible, only wavelength limited, resolution, but the phase recovery is an iterative process which requires some pre-defined information about the object and whose outcome is not always uniquely-defined. Moreover, the diffraction patterns must be recorded under oversampling conditions, a pre-requisite to be able to solve the phase problem. Here, we report how holography and CDI can be merged into one superior technique: holographic coherent diffraction imaging (HCDI). An inline hologram can be recorded by employing a modified CDI experimental scheme. We demonstrate that the amplitude of the Fourier transform of an inline hologram is related to the complex-valued visibility, thus providing information on both, the amplitude and the phase of the scattered wave in the plane of the diffraction pattern. With the phase information available, the condition of oversampling the diffraction patterns can be relaxed, and the phase problem can be solved in a fast and unambiguous manner. We demonstrate the reconstruction of various diffraction patterns of objects recorded with visible light as well as with low-energy electrons. Although we have demonstrated our HCDI method using laser light and low-energy electrons, it can also be applied to any other coherent radiation such as X-rays or high-energy electrons.

Keyhole electron diffractive imaging (KEDI)

Electron diffractive imaging (EDI) relies on combining information from the high-resolution transmission electron microscopy image of an isolated kinematically diffracting nano-particle with the corresponding nano-electron diffraction pattern. Phase-retrieval algorithms allow one to derive the phase, lost in the acquisition of the diffraction pattern, to visualize the actual atomic projected potential within the specimen at sub-ångström resolution, overcoming limitations due to the electron lens aberrations. Here the approach is generalized to study extended crystalline specimens. The new technique has been called keyhole electron diffractive imaging (KEDI) because it aims to investigate nano-regions of extended specimens at sub-ångström resolution by properly confining the illuminated area. Some basic issues of retrieving phase information from the EDI/KEDI measured diffracted amplitudes are discussed. By using the generalized Shannon sampling theorem it is shown that whenever suitable oversampling conditions are satisfied, EDI/KEDI diffraction patterns can contain enough information to lead to reliable phase retrieval of the unknown specimen electrostatic potential. Hence, the KEDI method has been demonstrated by simulations and experiments performed on an Si crystal cross section in the [112] zone-axis orientation, achieving a resolution of 71 pm.

Electrons for single molecule diffraction and imaging

We demonstrate the potential of electrons for single molecule diffraction and imaging using C₆₀ molecules confined inside single-walled carbon nanotubes (C₆₀s@SWCNT or peapod) as a model system. A 25 nm diameter electron beam from a field emission gun source is used to record diffraction patterns from individual peapods using imaging plates. The electron beam illuminates about 25 C₆₀ molecules. Experimentally, we found that the molecules diffract inside ∼15% of the host nanotubes. With the help of simulations, we explore the limits of electron molecular diffraction and its sensitivity to the molecular configurations. We show that the combination of electron diffraction and electron direct imaging provides the best approach to single molecule imaging.

A method for correcting the effect of specimen drift on coherent diffractive imaging

Coherent diffractive imaging involves the inversion of a diffraction pattern to find the wave function at the exit-surface plane of the specimen. It is a promising technique for imaging, for example, nanoparticles with electrons and biological molecules with X-rays. If the illumination is not a plane wave of infinite extent, then a relative drift between the illumination and the object introduces errors into the diffraction pattern; an issue which is often overlooked. This may be of particular importance for applications with electron microscopes which use nanoscale probes. Here we show that beams which are uniform over a sufficiently large region can be used to pose a phase retrieval problem that is immune from specimen drift, provided suitable analysis of the diffraction data is undertaken. The method only applies to objects contained within a support that is smaller than a uniform region of the beam.

Low-dose performance of parallel-beam nanodiffraction

We evaluate the low-dose performance of parallel nano-beam diffraction (NBD) in the transmission electron microscope as a method for characterizing radiation sensitive materials at low electron irradiation dose. A criterion, analogous to Rose's, is established for detecting a diffraction spot with desired signal-to-noise ratio. Our experimental data show that a dose substantially lower than in high-resolution bright-field imaging is sufficient to determine structure and orientation of individual nanoscale objects embedded in amorphous matrix. In an instrument equipped with a cold field-emission gun it is possible to form a probe with sub-3 nm diameter and sub-0.3 mrad convergence angle with sufficient beam current to record a diffraction pattern with less than 0.2 s acquisition time. The interpretation of NBD patterns is identical to that of selected area diffraction patterns. We illustrate the physical principles underlying good low-dose performance of NBD by means of a phase grating. The electron irradiation dose needed to detect a diffraction peak in NBD is found proportional to 1/N2, where N is the number of lattice planes contributing to the peak.