Philip N. H. Nakashima

The bonding electron density in aluminum

A combination of microscopy and first-principle calculations is used to study the bonding charge density in aluminum. Aluminum is considered to approach an “ideal” metal or free electron gas. The valence electrons move freely, as if unaffected by the presence of the metal ions. Therefore, the electron redistribution due to chemical bonding is subtle and has proven extremely difficult to determine. Experimental measurements and ab initio calculations have yielded substantially different results. We applied quantitative convergent-beam electron diffraction to aluminum to provide an experimental determination of the bonding electron distribution. Calculation of the electron distribution based on density functional theory is shown to be in close agreement. Our results yield an accurate quantitative correlation between the anisotropic elastic properties of aluminum and the bonding electron and electrostatic potential distributions.

Measuring the PSF from aperture images of arbitrary shape-an algorithm

A new algorithm for determining the point spread function (PSF) of digital imaging systems is presented. The input is an image of an aperture whose shape need not be regular. The aperture shape is refined to an effective sub-pixel resolution and the PSF of the system is determined by de-convolution, assuming uniform illumination and a step function edge. The method has been tested on theoretical aperture images of varying shape and PSF, with and without noise. Depending on the degree of noise, a known PSF can be recovered to an accuracy of between 0.2 and 0.8%. Some typical results are given for a Gatan Image Filter with a 794 YAG multiscan camera on a Philips EM 430 transmission electron microscope at 200 and 300 kV. An example of a de-convoluted convergent beam electron diffraction pattern is included. The algorithm tolerates a small amount of de-focus.

Charge density analysis from complementary high energy synchrotron X-ray and electron diffraction data

Abstract To accurately measure the low order structure factors of α-Al2O3, two-dimensional convergent beam electron diffraction (CBED) data from parallel-sided platelets at various accelerating voltages, thickness and orientations have been matched using the Bloch-wave method. Middle and high angle extinction-free data has been obtained by the extrapolation of multi-wavelength synchrotron X-ray measurements to zero wavelength. The combination of high-energy synchrotron X-ray diffraction and CBED allows the extinction-free absolute-scale measurements and improves the reliability of the electron charge density maps in α-Al2O3.

Optimization of exit-plane waves restored from HRTEM through-focal series

Atomic-resolution transmission electron microscopy has largely benefited from the implementation of aberration correctors in the imaging part of the microscope. Though the dominant geometrical axial aberrations can in principle be corrected or suitably adjusted, the impact of higher-order aberrations, which are mainly due to the implementation of non-round electron optical elements, on the imaging process remains unclear. Based on a semi-empirical criterion, we analyze the impact of residual aperture aberrations on the quality of exit-plane waves that are retrieved from through-focal series recorded using an aberration-corrected and monochromated instrument which was operated at 300kV and enabled for an information transfer of approximately 0.05nm. We show that the impact of some of the higher-order aberrations in retrieved exit-plane waves can be balanced by a suitable adjustment of symmetry equivalent lower-order aberrations. We find that proper compensation and correction of 1st and 2nd order aberrations is critical, and that the required accuracy is difficult to achieve. This results in an apparent insensitivity towards residual higher-order aberrations. We also investigate the influence of the detector characteristics on the image contrast. We find that correction for the modulation transfer function results in a contrast gain of up to 40%.

Structural phase and amplitude measurement from distances in convergent-beam electron diffraction patterns.

The structure of a periodic object, such as a crystal, may be described by an infinite series of Fourier coefficients and phases. In associating this with scattering theory appropriate to any radiation, a classic problem arises, namely, the determination of phases from the resulting discrete diffraction pattern. The solution to this phase problem is presented in this paper in which the first direct measurement of structural phase by inspection of convergent-beam electron diffraction patterns is described.

Improved quantitative CBED structure-factor measurement by refinement of nonlinear geometric distortion corrections

A new method that accounts for small but significant geometric distortions in quantitative convergent beam electron diffraction (QCBED) is briefly introduced. A summary of preliminary results obtained with this method shows an average three- to fourfold improvement in structure-factor measurement precision by QCBED. In the present work this method is applied to α-\rm Al_{2}O_{3}, a benchmark compound for charge density studies. Experimental uncertainty is reduced to a level three times smaller than differences between density functional theory and periodic Hartree–Fock calculated structure factors.

Differential quantitative analysis of background structure in energy-filtered convergent-beam electron diffraction patterns

Measurements of electronic structure in solids by quantitative convergent-beamelectrondiffraction(QCBED)willnotreachtheirultimateaccuracyorprecisionuntil the contribution of the background to the reflections in energy-filteredCBED patterns is fully accounted for. Apart from the well known diffusebackground that arises from thermal diffuse scattering of electrons, there is acomponent that has a much higher angular frequency. The present work reportsexperimental evidence that this component mimics the angular distribution oftheelastically scatteredelectrons within each reflection. A differentialapproachto QCBED is suggested as a means of quantitatively accounting for thebackground in energy-filtered CBED data.1. IntroductionThe high accuracy and precision of electronic structuremeasurements in inorganic solids with a high degree of crystalperfection by quantitative convergent-beam electron diffrac-tion (QCBED) is now well established (Zuo et al.,1988;BirdSZuo,1993;Deiningeretal.,1994;Holmestadetal.,1995;PengZSaundersetal.,1995,1996,1999;Zuo et al.,1997,1999;TsudaTStreltsovet al.,2001, 2003; Tsuda et al.,2002;Jianget al.,2003;Ogataet al.,2004; Friis, Madsen et al.,2003;Friis,Jianget al.,2003;Jiangetal.,2004;Friiset al.,2004,2005;Nakashima,2005,2007).Suchmeasurements are precise enough to allow meaningfulcomparisons of experimentally measured electronic structurewith different ab initio theoretical models (Zuo et al.,1997,1999; Saunders et al.,1999;Friis,Madsenet al.,2003;Jiangetal.,2003,2004;Friiset al.,2004,2005;Nakashima,2005),including those derived from density functional theory andperiodic Hartree–Fock, Dirac–Fock and linear combination ofatomic orbitals calculations.QCBED originated in 1940 (MacGillavry, 1940) withsporadic application until the late 1980s (Goodman L Voss et al.,1980)whenthedevelopmentofenergy-filtering optics for transmission electron microscopesresulted in a strong revival of the technique (Zuo et al.,1988,1997, 1999; Bird & Saunders, 1992; Zuo, 1993; Holmestad etal.,1995;PengZSaunderset al.,1995,1996,1999;Tsuda & Tanaka, 1999; Streltsov et al.,2001,2003;Tsudaet al.,2002; Jiang et al.,2003;Ogataet al.,2004;Friis,Madsenet al.,2003; Friis, Jiang et al.,2003;Jianget al.,2004;Friiset al.,2004,2005; Nakashima, 2005). The ability to exclude almost all ofthe inelastic signal by energy filtering, coupled with highdynamic range digital signal detection (via CCDs) and thecontinuous expansion in computing power, allowed unprece-dented analysis of experimental CBED data via patternmatching based on elastic scattering theory. As a result, therefinement of Fourier coefficients of the crystal potential(structure amplitudes or structure factors) during the pattern-matching process became sufficiently precise and accurate tobecomparabletothemostpreciseandaccuratemeasurementsever made by X-ray diffraction (Dawson, 1967; Kato, 1969;Hart & Milne, 1970).Even so, QCBED has not reached its full potential becauseenergy filtering does not remove all components of a CBEDpattern that are unaccounted for by elastic scattering theory.Thermaldiffuse scattering (TDS) ofelectrons results in energylossesoflessthan0.1 eV,wellbelowthe resolution ofthemostmodern energy filters and also below the spread in energiesfrom the latest monochromated electron sources. Argumentsagainst applying calculations incorporating a full treatment ofTDS because of their nearly prohibitive cost in computingpower and time (compared with purely elastic scatteringcalculations) will eventually fade with the advent of newsupercomputing technologies such as graphics processingunits. A growing number of CBED calculations incorporateTDS (Rossouw et al.,1990;Loaneet al.,1991;Wang,1992;Muller et al.,2001;Omotoet al.,2002;Dwyer,2003,2005;Dwyer & Etheridge, 2003), with Omoto et al. (2002) illus-tratinghowTDScalculationscanbeincluded(non-iteratively)in QCBED. However, there is very little information in theliterature about the structure of the background associatedwith each disc in a CBED pattern. Most calculations deal onlywiththetotalsignalinthepattern,withoutseparatingtheTDSand elastic components. Omoto et al. (2002) examined thebackground due to TDS experimentally and theoretically inthe absence of the CBED pattern, but without comment as to

Quantum Crystallography: Current Developments and Future Perspectives

Crystallography and quantum mechanics have always been tightly connected because reliable quantum mechanical models are needed to determine crystal structures. Due to this natural synergy, nowadays accurate distributions of electrons in space can be obtained from diffraction and scattering experiments. In the original definition of quantum crystallography (QCr) given by Massa, Karle and Huang, direct extraction of wavefunctions or density matrices from measured intensities of reflections or, conversely, ad hoc quantum mechanical calculations to enhance the accuracy of the crystallographic refinement are implicated. Nevertheless, many other active and emerging research areas involving quantum mechanics and scattering experiments are not covered by the original definition although they enable to observe and explain quantum phenomena as accurately and successfully as the original strategies. Therefore, we give an overview over current research that is related to a broader notion of QCr, and discuss options how QCr can evolve to become a complete and independent domain of natural sciences. The goal of this paper is to initiate discussions around QCr, but not to find a final definition of the field.

Mechanisms of void shrinkage in aluminium

Voids can significantly affect the performance of materials and a key question is how voids form and evolve. Voids also provide a rare opportunity to study the fundamental interplay between surface crystallography and atomic diffusion at the nanoscale. In the present work, the shrinkage of voids in aluminium from 20 to 1 nm in diameter through in situ annealing is imaged in a transmission electron microscope. It is found that voids first shrink anisotropically from a non-equilibrium to an equilibrium shape and then shrink while maintaining their equilibrium shape until they collapse. It is revealed that this process maximizes the reduction in total surface energy per vacancy emitted. It is also observed that shrinkage is quantized, taking place one atomic layer and one void facet at a time. By taking the quantization and electron irradiation into account, the measured void shrinkage rates can be modelled satisfactorily for voids down to 5 nm using bulk diffusion kinetics. Continuous electron irradiation accelerates the shrinkage kinetics significantly; however, it does not affect the energetics, which control void shape.

Projected thickness reconstruction from a single defocused transmission electron microscope image of an amorphous object.

Single defocused transmission electron microscope phase contrast images are used to reconstruct the projected thickness map of a single-material object. The algorithm is non-iterative and stable, and we extend it to account for the presence of spherical aberration in the objective optics. The technique can reconstruct the projected thickness map of general single-material objects in the strong phase/weak amplitude regime. It is sensitive to any excursions in the projected thickness from the average, and ideal for examining voids and free volume accumulation in amorphous/glassy materials at the nanometer scale. The resolution of the technique depends on the choice of defocus and the thickness of the specimen. In a certain regime, we demonstrate that variations in the transverse projected thickness with a lateral diameter of ∼ 0.25 nm may be detected. We use our algorithm to quantitatively reconstruct the projected thickness of latex sphere test specimens from single defocused electron micrographs. We demonstrate that the reconstruction has a large tolerance for error in the input parameters. Simulations confirm that the technique is quantitative, and demonstrate that the origin of low-frequency artifacts is an instability due to noise. We show that the autocorrelation of the projected thickness map may be used to measure the size of open structures in the object using both simulation and latex sphere data.