Measuring Density Functional Parameters from Electron Diffraction Patterns.

We integrate density functional theory (DFT) into quantitative convergent-beam electron diffraction (QCBED) to create a synergy between experiment and theory called QCBED-DFT. This synergy resides entirely in the electron density which, in real materials, gives rise to the experimental CBED patterns used by QCBED-DFT to refine DFT model parameters. We use it to measure the Hubbard energy U for two strongly correlated electron systems, NiO and CeB_{6} (U=7.4±0.6 eV for d orbitals in NiO and U=3.0±0.6 eV for f orbitals in CeB_{6}), and the boron position parameter x for CeB_{6} (x=0.1992±0.0003). In verifying our measurements, we demonstrate an accuracy test for any modeled electron density.

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.

Direct atomic structure determination by the inspection of structural phase.

A century has passed since Bragg solved the first atomic structure using diffraction. As with this first structure, all atomic structures to date have been deduced from the measurement of many diffracted intensities using iterative and statistical methods. We show that centrosymmetric atomic structures can be determined without the need to measure or even record a diffracted intensity. Instead, atomic structures can be determined directly and quickly from the observation of crystallographic phases in electron diffraction patterns. Furthermore, only a few phases are required to achieve high resolution. This represents a paradigm shift in structure determination methods, which we demonstrate with the moderately complex α-Al2O3. We show that the observation of just nine phases enables the location of all atoms with a resolution of better than 0.1 Å. This level of certainty previously required the measurement of thousands of diffracted intensities.

Observations of specimen morphology effects on near-zone-axis convergent-beam electron diffraction patterns.

This work presents observations of symmetry breakages in the intensity distributions of near-zone-axis convergent-beam electron diffraction (CBED) patterns that can only be explained by the symmetry of the specimen and not the symmetry of the unit cell describing the atomic structure of the material. The specimen is an aluminium-copper-tin alloy containing voids many tens of nanometres in size within continuous single crystals of the aluminium host matrix. Several CBED patterns where the incident beam enters and exits parallel void facets without the incident beam being perpendicular to these facets are examined. The symmetries in their intensity distributions are explained by the specimen morphology alone using a geometric argument based on the multislice theory. This work shows that it is possible to deduce nanoscale morphological information about the specimen in the direction of the electron beam - the elusive third dimension in transmission electron microscopy - from the inspection of CBED patterns.

In situ quantification of noise as a function of signal in digital images.

An efficient and accurate algorithm for determining the magnitude of noise as a function of signal in an arbitrary digital image is presented and demonstrated. The algorithm is robust and largely independent of the form of the image, returning the noise function with subcount error across the full dynamic range of a synthetic test image where noise of a known form has been added. The noise performance of a CCD under different image recording and processing conditions is examined using the algorithm. The effect of different noise functions on pattern-matching measurements of electronic structure by quantitative convergent beam electron diffraction is investigated.

Identification of the impurity phase in high-purity CeB6 by convergent-beam electron diffraction.

The rare earth hexaborides are known for their tendency towards very high crystal perfection. They can be grown into large single crystals of very high purity by inert gas arc floating zone refinement. The authors have found that single-crystal cerium hexaboride grown in this manner contains a significant number of inclusions of an impurity phase that interrupts the otherwise single crystallinity of this prominent cathode material. An iterative approach is used to unequivocally determine the space group and the lattice parameters of the impurity phase based on geometries of convergent-beam electron diffraction (CBED) patterns and the symmetry elements that they possess in their intensity distributions. It is found that the impurity phase has a tetragonal unit cell with space group P4/mbm and lattice parameters a = b = 7.23 ± 0.03 and c = 4.09 ± 0.02 Å. These agree very well with those of a known material, CeB4. Confirmation that this is indeed the identity of the impurity phase is provided by quantitative CBED (QCBED) where the very close match between experimental and calculated CBED patterns has confirmed the atomic structure. Further confirmation is provided by a density functional theory calculation and also by high-angle annular dark-field scanning transmission electron microscopy.

A practical guide to the measurement of structure phases and magnitudes by three-beam convergent beam electron diffraction.

The practical details for the direct measurement of structural phases and magnitudes from centrosymmetric crystals using three-beam convergent beam electron diffraction patterns are given. These details are presented as a step-by-step guide for easy implementation. Each step is illustrated with worked examples from alpha-Al(2)O(3). Twenty-nine measurements of 6 independent three-phase invariants and 87 measurements of 10 independent structure magnitudes have been made.

Three-beam convergent-beam electron diffraction for measuring crystallographic phases.

Under almost all circumstances, electron diffraction patterns contain information about the phases of structure factors, a consequence of the short wavelength of an electron and its strong Coulombic interaction with matter. However, extracting this information remains a challenge and no generic method exists. In this work, a set of simple analytical expressions is derived for the intensity distribution in convergent-beam electron diffraction (CBED) patterns recorded under three-beam conditions. It is shown that these expressions can be used to identify features in three-beam CBED patterns from which three-phase invariants can be extracted directly, without any iterative refinement processes. The octant, in which the three-phase invariant lies, can be determined simply by inspection of the indexed CBED patterns (i.e. the uncertainty of the phase measurement is ±22.5°). This approach is demonstrated with the experimental measurement of three-phase invariants in two simple test cases: centrosymmetric Si and non-centrosymmetric GaAs. This method may complement existing structure determination methods by providing direct measurements of three-phase invariants to replace 'guessed' invariants in ab initio phasing methods and hence provide more stringent constraints to the structure solution.

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.

Topologically Enclosed Aluminum Voids as Plasmonic Nanostructures.

Recent advances in the ability to synthesize metallic nanoparticles with tailored geometries have led to a revolution in the field of plasmonics. However, studies of the important complementary system, an inverted nanostructure, have so far been limited to two-dimensional sphere-segment voids or holes. Here we reveal the localized surface plasmon resonances (LSPRs) of nanovoids that are topologically enclosed in three-dimensions: an "anti-nanoparticle". We combine this topology with the favorable plasmonic properties of aluminum to observe strongly localized field enhancements with LSPR energies in the extreme UV range, well beyond those accessible with noble metals or yet achieved with aluminum. We demonstrate the resonance tunability by tailoring the shape and size of the nanovoids, which are truncated octahedra in the 10-20 nm range. This system is pristine: the nanovoid cavity is free from any oxide or supporting substrate that would affect the LSPRs. We exploit this to infer LSPRs of pure, sub-20-nm Al nanoparticles, which have yet to be synthesized. Access to this extreme UV range will allow applications in LSPR-enhanced UV photoemission spectroscopy and photoionization.

A new approach to structure amplitude determination from 3-beam convergent beam electron diffraction patterns.

The intensity distribution in three-beam CBED patterns from centrosymmetric crystals can be inverted analytically to enable the direct measurement of crystal structure amplitudes and three-phase invariants. The accuracy of the measurements depends upon the accuracy and precision with which specific loci within the discs can be identified. The present work exploits the equivalence in form of the intensity distribution along these loci to provide an algorithm for their automated location, enabling the rapid and unequivocal identification of their position. Moreover, it demonstrates how the loci can be used to determine directly the relative magnitudes of structure amplitudes with superior accuracy and without recourse to complex pattern-matching calculations.

The bonding electron 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.

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%.

Thickness difference: a new filtering tool for quantitative electron diffraction.

A new way of filtering electron diffraction patterns has been discovered. Patterns from slightly different specimen thicknesses beyond the mean free path for inelastic scattering are subtracted. Only thickness sensitive information (dominantly elastic) remains. Thermal diffuse scattering and Borrmann effects are removed in addition to the inelastic signal eliminated by conventional energy filtering. One application is quantitative convergent beam electron diffraction without an energy filter. Structure factors for alpha - Al(2)O(3) have been measured with an average uncertainty of 0.25%.

Crystal structure study of a beta'-copper vanadium bronze, Cu(x)V2O5 (x = 0.63), by X-ray and convergent beam electron diffraction.

The single-crystal structure of a beta'-copper vanadium bronze, Cu0.63V2O5, has been studied at room temperature and 9.6 K, and compared with that of the beta-sodium vanadium bronze, Na0.33V2O5, structure. No convincing evidence to oppose an assignment of centrosymmetric C2/m symmetry to the structure was identified using the X-ray data. A subsequent convergent beam electron diffraction (CBED) experiment was performed and confirmed the C2/m space group. The oxygen-vanadium atom framework of Cu0.63V2O5 is close to that of Na0.33V2O5. However, in the copper compound the Cu atoms are located in two positions: Cu1 in the 4(i) position with x=0.541, y=0 and z=0.345, and Cu2 in the 8(j) position with x=0.529, y=0.038 and z=0.357. The crystal structure changes little with temperature. Disorder of the Cu ion over two sites is seen at 9.6 K. This suggests that distribution of the Cu atoms over two sites is of a more static than dynamic nature.

A combination method of charge density measurement in hard materials using accurate, quantitative electron and X-ray diffraction: the alpha-Al2O3 case.

Current X-ray diffraction techniques intended for "ideally imperfect" specimens provide structure factors only on a relative scale and ever-present multiple scattering in strong low-angle Bragg reflections is difficult to correct. Multiple scattering is implicit in the quantitative convergent beam electron diffraction (QCBED) method, which provides absolutely scaled structure factors. Conventional single crystal X-ray diffraction has proved adequate in softer materials where crystal perfection is limited. In hard materials, the highly perfect nature of the crystals is often a difficulty, due to the inadequacy of the conventional corrections for multiple scattering (extinction corrections). The present study on alpha-Al2O3 exploits the complementarity of synchrotron X-ray measurements for weak and medium intensities and QCBED measurement of the strong low-angle reflections. Two-dimensional near zone axis QCBED data from different crystals at various accelerating voltages, thicknesses, and orientations have been matched using Bloch-wave and multislice methods. The reproducibility of QCBED data is better than 0.5%. The low-angle strong QCBED structure factors were combined with middle and high-angle extinction-free data from synchrotron X-ray diffraction measurements. Static deformation charge density maps for alpha-Al2O3 have been calculated from a multipole expansion model refined using the combined QCBED and X-ray data.