Exit wave reconstructions using through focus series of HREM images

The through focus exit wave reconstruction technique uses a series of high resolution electron microscopy images to reconstruct the complex electron wavefunction at the exit plane of the specimen. The feasibility of the through focus exit wave reconstruction method and its most important limitations are discussed. It is shown that-provided all aberrations of the microscope are well corrected for-a large improvement in the interpretability of the information can be obtained.     

A method to determine the local surface profile from reconstructed exit waves

Reconstructed exit waves are useful to quantify unknown structure parameters such as the position and composition of the atom columns at atomic scale. Existing techniques provide a complex wave in a flat plane which is close to the plane where the electrons leave the atom columns. However, due to local deviation in the flatness of the exit surface, there will be an offset between the plane of reconstruction and the actual exit of a specific atom column. Using the channelling theory, it has been shown that this defocus offset can in principle be determined atom column-by-atom column. As such, the surface roughness could be quantified at atomic scale. However, the outcome strongly depends on the initial plane of reconstruction especially in a crystalline structure. If this plane is further away from the true exit, the waves of the atom columns become delocalized and interfere mutually which strongly complicates the interpretation of the exit wave in terms of the local structure. In this paper, we will study the delocalization with defocus using the channelling theory in a more systematic way.     

Determination of the orientation in small areas of the exit wave

Local differences in thickness and misorientation in a through-focus reconstructed exit wave hamper direct interpretation. These local thicknesses and misorientations can be refined on an atomic scale using the intensities of the Fourier components with the refinement procedure in which a least-squares refinement is combined with a multi-slice calculation. The method was applied to the superconductor La3Ni2B2N3. The crystal tilt and specimen thickness can be determined with an R-value of 8-25%, with a better R-value for thinner areas. Significant differences in the refined tilts, thicknesses and R-values are observed when reconstructed exit waves that are corrected for mechanical vibration are compared with reconstructed exit waves, which are uncorrected.     

Reconstruction of the projected electrostatic potential in high-resolution transmission electron microscopy including phenomenological absorption

The projected electrostatic potential is reconstructed from a high-resolution exit wave function through a maximum-likelihood refinement algorithm. The theory of an already existing algorithm [1] is extended to include the effects of phenomenological absorption. Various tests with a simulated exit wave function of YBa₂Cu₃O₇ in [1 0 0] orientation used as a source show that the reconstruction is successful, regardless of the strongly differing scattering power of atomic columns, even for the case of strong dynamical diffraction. Object thickness, the amount of absorption, and a residual defocus aberration of the wave function—parameters often unknown or difficult to measure in experiments—can be determined accurately with the aid of the refinement algorithm in a self-consistent way. For the next generation of instruments, with information limits of 0.05 nm and better, reconstruction accuracies of better than 2% can be expected, which is sufficient to measure and display the structural and chemical information with the aid of an accurate projected potential map.     

Resolution extension and exit wave reconstruction in complex HREM

Direct methods in real and reciprocal space are developed for structural reversion. The direct method in real space involves the use of a novel method to retrieve the phase in the image plane using transport of intensity equation/maximum entropy method (TIE/MEM) and exit wave reconstruction by self-consistent propagation. Since the exit wave is restored from the complex signal in the image planes, no image model between the exit wave and image is assumed. The structural information in the reconstructed exit wave is then further extended by a "complex" maximum entropy method as a direct method in reciprocal space to extrapolate the phase to higher frequencies.     

Object wave reconstruction by phase-plate transmission electron microscopy

A method is described for the reconstruction of the amplitude and phase of the object exit wave function by phase-plate transmission electron microscopy. The proposed method can be considered as in-line holography and requires three images, taken with different phase shifts between undiffracted and diffracted electrons induced by a suitable phase-shifting device. The proposed method is applicable for arbitrary object exit wave functions and non-linear image formation. Verification of the method is performed for examples of a simulated crystalline object wave function and a wave function acquired with off-axis holography. The impact of noise on the reconstruction of the wave function is investigated.     

The effect of mechanical vibration and drift on the reconstruction of exit waves from a through focus series of HREM images

Experimental HREM images can show a limited resolution as a result of mechanical vibration and drift. In this paper the effect of such mechanical vibrations on the accuracy of the through focus exit wave reconstruction method is investigated for different thicknesses of a test structure of La₃Ni₂B₂N₃. A through-focus series of HREM images for this structure is simulated for different kinds of mechanical vibration corresponding to an information limit g of about 7 nm⁻¹: (1) no mechanical vibration, (2) isotropic mechanical vibration, and (3) several anisotropic mechanical vibrations. From these through-focus series the reconstructed exit wave is calculated (Ultramicroscopy 64 (1996) 109). The above isotropic and anisotropic mechanical vibrations have a large effect on the reconstructed exit waves when compared with the reconstructed exit wave without mechanical vibration, i.e. the range of amplitudes and phases in a reconstructed exit wave decreases and the background intensity increases. The initial thickness and orientation can be obtained using a least-squares refinement procedure (Acta Crystallogr. A 54 (1998) 91) when there is no mechanical vibration present. In the case of isotropic or anisotropic vibration, the refined thickness and orientation are likely to give wrong results depending on the size of the vibrations and on the number of significant reflections (which is related to the size of the unit cell, the thickness and the misorientation).     

Exit wave reconstruction using a conventional transmission electron microscope

Exit wave reconstruction of a focus series of Ge in [110] using the PAMMAL algorithm was performed on a conventional electron microscope. The simulated images using the reconstructed object wave match very well with those obtained experimentally. Amplitudes from the complex wave function were extracted by means of local Fourier transformation. Crystal thickness and tilt were determined locally by quantitative comparison of the reconstructed amplitudes with amplitudes from multislice calculations. Detailed analysis yields the quasicoherent imaging approach used in the PAMMAL algorithm to produce the largest error in the analysis. For the Ge crystal specimen parameters were quantified to spatial frequencies of 5 nm¹. In the case of an object producing strong diffracted beams, the reconstruction may fail because the quasicoherent approximation will not describe correctly the nonlinear image formation.     

The effect of exit beam phase aberrations on parallel beam coherent x-ray reconstructions

Diffraction artifacts from imperfect x-ray windows near the sample are an important consideration in the design of coherent x-ray diffraction measurements. In this study, we used simulated and experimental diffraction patterns in two and three dimensions to explore the effect of phase imperfections in a beryllium window (such as a void or inclusion) on the convergence behavior of phasing algorithms and on the ultimate reconstruction. A predictive relationship between beam wavelength, sample size, and window position was derived to explain the dependence of reconstruction quality on beryllium defect size. Defects corresponding to this prediction cause the most damage to the sample exit wave and induce signature error oscillations during phasing that can be used as a fingerprint of experimental x-ray window artifacts. The relationship between x-ray window imperfection size and coherent x-ray diffractive imaging reconstruction quality explored in this work can play an important role in designing high-resolution in situ coherent imaging instrumentation and will help interpret the phasing behavior of coherent diffraction measured in these in situ environments.     

A study of deformation during cold rolling using visioplasticity

A study of the deformation occurring during cold rolling using the experimentally-based visioplasticity technique is described. Aluminium specimens of different work-hardening characteristics and of various thicknesses were rolled with different reductions and various back and/or front tensions. The geometry of the work-piece was such that plane strain was approximated. A grid on the side of the specimen was photographed and its deformation was analysed using the visioplasticity technique. Slip line fields constructed from this information accorded generally with the predictions of Firbank and Lancaster[1] and of Collins[3], showing similar profiles of roll pressure; similar variations in strain rate were observed.Plastic deformation producing an increase in the thickness of the specimen was found to occur on the exit side of the deformation zone.     

From thickness dependent exit waves to projected potential: Thickness derivative approach

In HREM, due to multiple scattering, the exit wave of the object is nonlinear thickness dependent so that there is no one-to-one relation between object structure and the exit wave. This feature hampers the direct retrieval of structural information from exit waves. In this paper we discuss the possibility to restore the object structure in a direct way using exit waves of different thicknesses. It is theoretically shown that the amplitude of the thickness derivative exit wave |∂ψ/∂z| may directly reflect the project potential in a simple way. Image simulations show that it can be applied to restore the projected potential.     

Error analysis on a generalized Feldkamp's cone-beam computed tomography algorithm

To overcome the limitations of Feldkamp's cone-beam reconstruction algorithm, we generalized it in the case of noncircular and nonplanar scanning. In this paper, an error analysis is performed on this generalized approximate algorithm. The study is based on a reconstruction error formula. Roughly speaking, the error is proportional to the distance from a voxel to the mid-plane in the planar scanning mode, to the pitch of a scanning locus in the helix-like scanning mode, and inversely proportional to the horizontal size of the scanning locus. The error also depends on longitudinal variation of a specimen function. Longitudinal partial derivative distributions of specimen transverse sections are modeled as stochastic fields. Practical factors are simulated as noise terms in the reconstruction error formula. The simulation results show that the generalized Feldkamp's algorithm is not sensitive to noise and allows more accurate reconstruction in the helix-like scanning mode.     

Shape reconstruction of three-dimensional flaw from backscattering data

Three dimensional Born and Kirchhoff inverse scattering methods are modified to convenient forms for a cylindrical specimen that includes three dimensional defect. One aluminum cylinder with flaw model is prepared and ultrasonic measurements are carried out. The measurement area in the modified methods is restricted in the plane perpendicular to the axis of cylindrical specimen. That is to say that the methods are modified to convenient form to use measured waveforms in the x₁–x₂ plane. The measured wave data are fed into the inversion method and cross-sectional images are obtained. Then, three dimensional shape reconstruction of flaw model in aluminum specimen is performed by piling up the cross-sectional images.     

Projected potential profiles across interfaces obtained by reconstructing the exit face wave function from through focal series

An iterative method for reconstructing the exit face wave function from a through focal series of transmission electron microscopy image line profiles across an interface is presented. Apart from high-resolution images recorded with small changes in defocus, this method works also well for a large defocus range as used for Fresnel imaging. Using the phase-object approximation the projected electrostatic as well as the absorptive potential profiles across an interface are determined from this exit face wave function. A new experimental image alignment procedure was developed in order to align images with large relative defocus shift. The performance of this procedure is shown to be superior to other image alignment procedures existing in the literature. The reconstruction method is applied to both simulated and experimental images.     

A simple channelling model for HREM contrast transfer under dynamical conditions

The application of electron channelling theory to dynamical exit wave calculations is briefly reviewed, and a comparison of channelling results with full dynamical calculations is presented. The channelling expression to the exit wave is combined with conventional imaging theory, and it is shown that a simple expression can be obtained for a dynamical contrast transfer function (D-CTF), which incorporates imaging aberrations and thickness-dependent dynamical scattering effects. The D-CTF can provide detailed insight into HREM images of a mixed cation oxide at thicknesses up to 200 Å, whereby an approximate correction for non-linear effects is utilized in the larger thickness regime.     

The interpolation of fourier transform data in tilted-view three-dimensional reconstruction

A method is described for interpolating the unevenly sampled Fourier transform data on lattice lines, collected during a tilted-view three-dimensional reconstruction of a plane lattice specimen, onto equally spaced sample points. The method performs local averaging of noisy data and inserts values into gaps in the data which are consistent with the spatial boundedness of the specimen. The method can be implemented computationally using a simple algorithm.     

Structure determination at the atomic level from dynamical electron diffraction data under systematic row conditions.

We discuss a method to obtain structural information on crystals at the atomic level in high-resolution transmission electron microscopy from dynamical diffraction data under systematic row conditions. Working at a fixed incident energy and within an N-beam approximation, data is required at a well defined set of N incident beam orientations to determine the scattering matrix, one orientation for each column in the matrix. At each orientation the corresponding column of the scattering-matrix is obtained by Fourier transformation of the exit surface wave function. Thus, in addition to each exit surface image, we must recover the phase of the wave function for that orientation in the image plane. We show that retrieval of the phase using algorithms based on conservation of flux, which assume continuity of the phase, can yield incorrect solutions for the phase. This is because singularities can occur in the phase of the wave field at points where the intensity is zero, which can lead to edge dislocations in the phase. We demonstrate, using a model example, how these edge dislocations arise. We will show that phase retrieval from a through focal series of measurements or using the Gerchberg-Saxton algorithm (starting from measurements of an image and the corresponding diffraction pattern), correctly retrieves the phase and hence the exit surface wave function for all the orientations required to obtain the scattering-matrix. The dynamical (multiple) scattering can then be inverted to uniquely obtain the projected potential.     

Three-dimensional reconstruction from a single-exposure,random conical tilt series applied to the 50S ribosomal subunit of Escherichia coli

We present a new reconstruction method that takes advantage of the fact that many biological macromolecular assemblies show a preferred orientation with respect to the plane of the specimen grid in the electron microscopic preparation. From one micrograph taken of such a specimen tilted by a large angle, a conical tilt series with random azimuthal angles can be extracted and used for a three-dimensional reconstruction. Our technique allows the determination of the molecular structure under low-dose conditions, which are not achievable with reconstruction methods that use conventional tilt series. The reconstruction method combines a number of existing image processing techniques with a newly developed weighted back-projection algorithm designed for three-dimensional reconstruction from projections taken with arbitrary projecting directions. The method is described as it was applied to the three-dimensional reconstruction of the structure of the 50S ribosomal subunit of Escherichia coli (E. coli).     

Optimal strategies for imaging thick biological specimens: exit wavefront reconstruction and energy-filtered imaging

In transmission electron microscopy (TEM) of thick biological specimens, the relationship between the recorded image intensities and the projected specimen mass density is distorted by incoherent electron–specimen interactions and aberrations of the objective lens. It is highly desirable to develop a strategy for maximizing and extracting the coherent image component, thereby allowing the projected specimen mass density to be directly related to image intensities. For this purpose, we previously used exit wavefront reconstruction to understand the nature of image formation for thick biological specimens in conventional TEM. Because electron energy-loss filtered imaging allows the contributions of inelastically scattered electrons to be removed, it is potentially advantageous for imaging thick, biological samples. In this paper, exit wavefront reconstruction is used to quantitatively analyse the imaging properties of an energy-filtered microscope and to assess its utility for thick-section microscopy. We found that for imaging thick biological specimens (> 0.5 μm) at 200 keV, only elastically scattered electrons contribute to the coherent image component. Surprisingly little coherent transfer was seen when using energy-filtering at the most probable energy loss (in this case at the first plasmon energy-loss peak). Furthermore, the use of zero-loss filtering in combination with exit wavefront reconstruction is considerably more effective at removing the effects of multiple elastic and inelastic scattering and microscope objective lens aberrations than either technique by itself. Optimization of the zero-loss signal requires operation at intermediate to high primary voltages (> 200 keV). These results have important implications for the accurate recording of images of thick biological specimens as, for instance, in electron microscope tomography.     

Numerical analysis of impulse waves discharged at the exit of a model tunnel

Impulse waves are micro-pressure waves, which always occur at the tunnel exit when a high-speed train is moving inside a train tunnel. The air around the train nose is compressed and compression waves are induced. The impulse wave is discharged at the exit of a train tunnel when a compression wave propagates outside of the tunnel exit. Impulse waves are weak-strength pressure waves, which lead to noise and other environmental problems. In order to efficiently control the impulse wave at the exit of a train tunnel, numerical studies on investigating the generation and propagation of the impulse wave were carried out. A 2-D axisymmetric model tunnel was simulated at different operating conditions. Different Mach numbers of compression waves were varied to induce different magnitudes of impulse waves at the tunnel exit. In addition, compression waves with different pressure gradients were assumed at the tunnel entry to check their effects on the generation of impulse waves. In order to observe impulse waves at far field, five monitor points were installed behind the tunnel exit to record pressure histories as impulse waves moved through these locations. The detailed magnitudes and characteristics of impulse waves were obtained in the present studies.