Low-order Structure Factors Research Articles

The Application of 3D-ED to Distinguish the Superstructure of Sr1.2Ca0.8Nb2O7 Ignored in SC-XRD

Compared to X-rays, electrons have stronger interactions with matter. In electron diffraction, the low-order structure factors are sensitive to subtle changes in the arrangement of valence electrons around atoms when the scattering vector is smaller than the critical scattering vector. Therefore, electron diffraction is more advantageous for studying the distribution of atoms in the structure with atomic numbers smaller than that of sulfur. In this work, the crystal structure of Sr1.2Ca0.8Nb2O7 (SCNO-0.8) was analyzed using single-crystal X-ray diffraction (SC-XRD) and three-dimensional electron diffraction (3D-ED) techniques, respectively. Interestingly, the superstructure could only be identified by the 3D-ED technique, while no signal corresponding to the superstructure was detected from the SC-XRD data. The superstructure in SCNO-0.8 was disclosed to be caused by different tilting of NbO6 octahedra and the displacements of Sr/Ca atoms in the different atomic layers perpendicular to the a-axis. Therefore, the application of 3D-ED provides an effective method for studying superstructures caused by ordered arrangements of light atoms.

Improvement of precision in refinements of structure factors using convergent-beam electron diffraction patterns taken at Bragg-excited conditions

A local structure analysis method based on convergent-beam electron diffraction (CBED) has been used for refining isotropic atomic displacement parameters and five low-order structure factors with sin θ/λ ≤ 0.28 Å-1 of potassium tantalate (KTaO3). Comparison between structure factors determined from CBED patterns taken at the zone-axis (ZA) and Bragg-excited conditions is made in order to discuss their precision and sensitivities. Bragg-excited CBED patterns showed higher precision in the refinement of structure factors than ZA patterns. Consistency between higher precision and sensitivity of the Bragg-excited CBED patterns has been found only for structure factors of the outer zeroth-order Laue-zone reflections with larger reciprocal-lattice vectors. Correlation coefficients among the refined structure factors in the refinement of Bragg-excited patterns are smaller than those of the ZA ones. Such smaller correlation coefficients lead to higher precision in the refinement of structure factors.

Reassessment of the electron density in Cu2O using γ-ray diffraction.

The electron-density distribution in Cu2O has been critically reexamined to test controversial conclusions from earlier experimental and theoretical studies. The electron density is derived via multipole refinement of high-quality single-crystal diffraction data, collected at room temperature with 316.5 keV gamma radiation. Four γ-lines in the energy range 200-600 keV have been used to extrapolate extinction-free low-order structure factors. The remaining extinction corrections refine to a crystal mosaicity identical to the observed one. There is no support for anharmonic contributions to the thermal parameters. Important features of the derived electron density are (i) a partially filled d_{z^2} orbital, (ii) an incomplete ionization of Cu and O, and (iii) no interstitial Cu-Cu charge pileup, thereby refuting the covalent bonding hypothesis.

Validation of density functionals for transition metals and intermetallics using data from quantitative electron diffraction

Accurate low-order structure factors (Fg) measured by quantitative convergent beam electron diffraction (QCBED) were used for validation of different density functional theory (DFT) approximations. Twenty-three low-order Fg were measured for the transition metals Cr, Fe, Co, Ni, and Cu, and the transition metal based intermetallic phases γ-TiAl, β-NiAl, and γ1-FePd using a multi-beam off-zone axis QCBED method and then compared with Fg calculated by ab initio DFT using the local density approximation (LDA) and LDA + U, and different generalized gradient approximations (GGA) functionals. Different functionals perform very differently for different materials and crystal structures regarding prediction of low-order Fg. All the GGA functionals tested in the paper except for EV93 achieve good overall agreement with the experimentally determined low-order Fg for BCC Cr and Fe, while EV93 performs the best for FCC Ni and Cu. The LDA and GGA functional fail to predict accurately the low-order Fg for β-NiAl and γ1-FePd. The LDA + U approach, through tuning of U, can achieve excellent matches with the experimentally measured Fg for all the metallic systems investigated in this paper. The use of experimentally accessible low order Fg as an additional set of metrics in approaches of validation of DFT calculations is discussed and has potential to assist in and to stimulate development of improved functionals.

Comparison of convergent beam electron diffraction methods for simultaneous structure and Debye Waller factor determination

The measurement of accurate and precise structure factors and Debye Waller (DW) factors by quantitative convergent beam electron diffraction (QCBED) permits experimental determination of the electron density distribution and probing of interatomic bonding in crystals. The three QCBED methods used successfully for high precision measurements of low order structure factors to date, namely the zone axis pattern (ZAP) method, the excited row ER method and the multi-beam off-zone axis (MBOZA) technique, differ from each other regarding the crystal orientation relative to the incident electron beam. Consequently, the details of their respective dispersion surface representations differ regarding the number, relative amplitudes and phases of excited Bloch wave branches. Under the same experimental setup conditions, the factors most important to the degree of accuracy and precision achievable in electron density determination for crystals with QCBED methods ultimately depend on the sensitivity of the excited Bloch wave branches and the resultant contrast in the respective CBED patterns to changes in both structure and DW factors. In general, a QCBED pattern will be more sensitive to changes in both structure and DW factor, if it contains more and stronger excited Bloch wave branches, as dynamic interactions of the Bloch waves increase the sensitivity of the pattern. In this work we analyzed Bloch wave excitation and dispersion surfaces for the three most popular QCBED methods. The analysis indicates, that the QCBED patterns obtained using the MBOZA orientation generally contain more and stronger excited Bloch wave branches. Hence, MBOZA diffraction patterns are more sensitive than the ZAP and the ER patterns to changes in both DW and structure factors and therefore allow in differences to the other two methods simultaneous refinements effectively and robustly.

Simultaneous determination of highly precise Debye–Waller factors and multiple structure factors for chemically ordered tetragonal FePd

Accurate Debye-Waller (DW) factors and low-index structure factors up to 222 of chemically ordered FePd have been measured at 120 K. Ordered FePd has a simple tetragonal unit cell (tP2, P4/mmm) with Fe and Pd atoms at 0, 0, 0 and at ½, ½, ½, respectively, requiring the measurement of four different DW factors. It was possible to simultaneously determine all four DW factors and several low-order structure factors using different, special off-zone-axis multi-beam convergent-beam electron diffraction patterns with high precision and accuracy. The different diffraction conditions exhibit different levels of sensitivity to changes in DW and structure factors. Here the sensitivity of different off-zone-axis convergent-beam electron diffraction patterns with respect to changes in DW factors and structure factors is discussed.

Electron density distribution in vanadium and niobium fromγ-ray diffraction

The electron density distribution in paramagnetic vanadium and niobium is derived via multipole refinement of high-quality single-crystal diffraction data, complete up to sin\ensuremath{\theta}/\ensuremath{\lambda} $=$ 1.9 \AA{}${}^{\ensuremath{-}1}$, collected at room temperature with 316.5 keV $\ensuremath{\gamma}$ radiation. Four $\ensuremath{\gamma}$ lines in the energy range 200--600 keV have been used to extrapolate extinction-free low-order structure factors. Both metals display charge asphericity due to preferential occupancy of the ${t}_{2g}$ orbital. The asphericity in V is in quantitative agreement with ab initio calculations, but not with earlier experimental results, laying at rest a long-standing debate. In both metals, the $d$-electron density is strongly contracted relative to the free atoms. The high-precision thermal parameters are in good agreement with some of the older results. No support is found in the data for anharmonic contributions to the thermal parameters, contrary to claims in previous reports on V. Attention is paid to the 3$d$-4$s$ occupation problem in V, indicating no change in electronic configuration between atom and metal. Relativistic form factor issues are also discussed. It is pointed out that for the analysis of crystallographic data nonrelativistic form factors are to be preferred over their relativistic counterparts.

Experimental Studies of Bonding Related Properties in Binary Intermetallics by Convergent Beam Electron Diffraction

ABSTRACTAccurate Debye-Waller (DW) factors of chemically ordered β-NiAl (B2, cP2, ${\rm{Pm}}\bar 3 {\rm{m}}$) have been measured at different temperatures using an off-zone axis multi-beam convergent beam electron diffraction (CBED) method. We determined a cross over temperature below which the DW factor of Ni becomes smaller than that of Al of ~90K. Additionally, we measured for the first time DW factors and structure factors of chemically ordered γ1-FePd (L10, tP2, P4/mmm) at 120K. We were able to simultaneously determine all four anisotropic DW factors and several low order structure factors using different special off-zone axis multi-beam convergent beam electron diffraction patterns with high precision and accuracy. An electron charge density deformation map was constructed from measured X-ray diffraction structure factors for γ1-FePd.

Experimental and theoretical study of electron density and structure factors in CoSb3

We refine two low-order structure factors of the skutterudite CoSb₃ using convergent beam electron diffraction. The relatively large unit cell of this material causes the disks to overlap and introduces a series of challenges in the refinement procedure. These challenges and future work-arounds are discussed. The refined structure factors F₂₀₀ and F₆₀₀ are compared to X-ray diffraction and density functional calculated values, the latter calculated using two different functionals. Both relaxed and experimental lattice parameters are tested to explicitly highlight the impact of the lattice geometry and atomic position on the structure factors.

Charge density measurement and bonding character in LiNiO 2

Accurate low-order structure factors of LiNiO 2 were measured by quantitative convergent-beam electron diffraction (QCBED), and then transformed into X-ray structure factors with Mott formula. Combining the structure factors measured by electron diffraction with the structure factors from X-ray diffraction measurements, accurate charge density maps based on a multipole model were obtained. The parameters of the bond critical points (BCP) were calculated for topological analyses. It shows that closed-shell interactions exist between Ni and O atoms, and that the Ni–O and Ni–Ni bonds exhibit some covalent character. The calculated d-orbital occupancies show the charge deficiency at e g( e g) orbital and charge surplus at e g( t 2g) orbital. The remaining 29.12% population of e g( e g) is also an indication of covalent component in the Ni–O bond. The unusual small κ defv value of the O atom is also discussed.

Determination of the electrostatic potential and electron density of silicon using convergent-beam electron diffraction

A structure-analysis method using convergent-beam electron diffraction (CBED) developed by Tsuda et al. [Tsuda & Tanaka (1999), Acta Cryst. A55, 939-954; Tsuda, Ogata, Takagi, Hashimoto & Tanaka (2002), Acta Cryst. A58, 514-525] has been applied to the determination of the electrostatic potential and electron density of crystalline silicon. CBED patterns recorded at nine different incidences are simultaneously used to improve the accuracy of the refinement. The Debye-Waller factor and low-order structure factors of silicon have been successfully refined only using CBED data. The electrostatic potential and electron-density distribution have been reconstructed from the refined parameters. The latter clearly shows the bonding electrons between the nearest neighbor atoms. The obtained results are compared with the results of other CBED and recent X-ray diffraction experiments. The influence of the number of refined low-order structure factors on the electron density is discussed. The effect of the reduction of experimental data points on the accuracy of the refined parameters is also examined.

On the sensitivity of electron and X-ray scattering factors to valence charge distributions

The sensitivity of atomic scattering factors to valence charge distributions has been compared quantitatively for X-ray and electron diffraction. It is found that below a critical scattering vector,s(|s| = sinθ/λ), ranging typically from 0.2 to 0.6 Å−1depending on the atomic number, electron diffraction is more sensitive to valence charge densities than X-ray diffraction. Thus, electron diffraction provides crucial electronic structure informationviathe low-order structure factors, which are relatively insensitive to thermal vibrations, but sensitive to the charge distribution that characterizes the chemical bonding properties of the materials. On the other hand, the high-order structure factors, which are mainly influenced by atomic position and core charge, in many cases can be replaced by structure factors of a procrystal (superposition of neutral atoms), or by calculated structure factors from modern density functional theory (DFT), without losing significant accuracy. This is demonstrated by detailed analyses of an MgB2superconductor. The work reveals the importance of accurate determination of a very few low-order structure factors in valence electron density studies, and suggests the merit of the combined use of electron diffraction and DFT calculations for solids, especially those with large unit cells and nanocrystalline grains, unsuitable for X-ray studies.

Electron-density distribution in cubic SrTiO3: a comparative γ-ray diffraction study

The electron density and atomic displacements in the perovskite SrTiO(3) have been studied using extensive and accurate gamma-ray diffraction data (lambda = 0.0392 A) at room temperature. The six strongest low-order structure factors have been determined under extinction-free conditions. Gram-Charlier series expansion of the thermal parameters have revealed no evidence for anharmonicity. The population of the 3d subshell on Ti is found to be close to zero, in agreement with the observed magnetic behaviour. The electronic properties at the bond critical points indicate ionic Ti-O and Sr-O interactions of different strengths, which is corroborated by the net charges of the atomic basins [q(Sr) = 1.18 |e|, q(Ti) = 3.10 |e|, q(O) = -1.42 |e|]. A critical comparison is made with earlier experimental results from laboratory X-ray, synchrotron X-ray, electron and neutron diffraction studies. Agreement and discrepancies are identified and resolved.

A study of charge density in copper

Quantitative convergent-beam electron diffraction (QCBED) experiments allow absolute scale measurements of low-order structure factors with very high accuracy. In this paper, eight low-order structure factors for copper measured by QCBED have been combined with the higher-order gamma-ray structure factors in order to obtain a larger highly accurate experimental data set. The gamma-ray values were relativistically corrected and rescaled. The new data set was then used for studying the charge distribution in copper. Charge deformation maps have been produced and both a maximum-entropy and a multipole analysis have been applied to the data. The result is compared to density functional theory calculations. An almost spherical charge depletion is found around the atomic sites showing typical metal bonding in copper.

Electron densities in relation to magnetism: gamma-ray diffraction from transition-metal monoxides and difluorides.

In this article, results are described from an area of solid-state physics that heretofore has remained unexplored by Bragg charge-scattering experiments. It is shown that gamma-ray electron densities can provide detailed information on magnetic ground-state properties at a subatomic resolution. The use of 316 keV gamma radiation has allowed low-order structure factors at better than 0.1% accuracy to be determined and, at the same time, extended data sets to be collected up to high momentum transfer. Prototypic antiferromagnets such as the late 3d transition-metal monoxides and difluorides have been systematically investigated, in most cases both in the paramagnetic and in the fully ordered phase. At transitions with a lowering of symmetry, multidomain formation was suppressed by application of external mechanical stress or magnetic fields. The results are considered in the context of information provided by recent magnetic neutron and X-ray studies.

Charge density determination in icosahedral AlPdMn quasicrystal using quantitative convergent beam electron diffraction

Charge density distribution in icosahedral AlPdMn quasicrystal has been studied on a single-crystal specimen by using quantitative convergent beam electron diffraction (QCBED) technique. The QCBED systematic row method was used in the refinement of structure factors. To refine the low-order structure factors, the wave-mechanical formulation of electron diffraction dynamical theory was used in the calculation of electron diffraction intensities for the quasicrystal in fitting the experimental intensity line scan profiles. The shapes of atomic surfaces (occupation domains) were described with symmetry-adapted series of surface harmonics. An iterative procedure was used in determination of structure factors of the quasicrystal. The structure factors of nine strongest symmetry inequivalent reflections according to X-ray diffraction experiment were refined with QCBED technique. The average of refinement results for a given reflection performed on several CBED patterns, which were slightly different in orientation and sample thickness, and on different line scans, was taken as the value of structure factor for the reflection. The obtained structure factors for electrons were transformed into X-ray structure factors with Mott formula. The bonding charge density map for the quasicrystal was constructed with the obtained nine structure factors. Assuming that the atoms are spheres, the gain or loss of electrons for different atoms were calculated. It shows that identical atoms can have different valences at different kinds of positions. The bonding charge is localized along certain directions.

Magnesium: Comparison of density functional theory calculations with electron and x-ray diffraction experiments

Accurate experimental structure factors for Mg have been measured and compared with density functional theory (DFT) to test some commonly used functionals and self-interaction correction (SIC) schemes. Low order structure factors, free of extinction and on absolute scale, were measured accurately by quantitative convergent beam electron diffraction. In addition, a complete set of structure factors up to sin θ/λ=1.6 Å−1 was measured by x-ray diffraction at 10 K. The DFT calculations were performed using the full potential linearized augmented plane wave method. It was found that the agreement with experiment increases when going from the local density approximation (LDA) to the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhofer and further to the GGA of Engel and Vosko. Applying the SIC of Perdew and Zunger to the core states for LDA does not improve the agreement with theory, while applying the SIC of Lundin and Eriksson results in a significantly improved agreement. This implies that the main source of error in the LDA functional comes from the description of the core densities. Using the functional which agrees best with experiment, a non-nuclear maximum is established in the calculated electron density of beryllium but not of magnesium.

Quantitative Convergent Beam Electron Diffraction Measurements of Low-Order Structure Factors in Copper

Accurate low-order structure factors for copper metal have been measured by quantitative convergent beam electron diffraction (QCBED). The standard deviation of the measured structure factors is equal to or smaller than the most accurate measurement by any other method, including X-ray single crystal Pendellösung, Bragg gamma-ray diffraction, and high-energy electron diffraction. The electron structure factor for the (440) reflection was used to determine the Debye-Waller (DW) factor. The local heating of the specimen by the electron beam is determined to be 5 K under the current illumination conditions. The low-order structure factors for copper measured by different methods are compared and discussed. The new data set is used to test band theory and to obtain a charge density map. The charge deformation map shows a charge surplus between the atoms and agrees fairly well with the simple model of copper 2+ ions at the atomic sites in a sea of free uniformly distributed electrons.

Charge density and chemical bonding in rutile, TiO2.

The low-order structure factors of rutile (TiO(2)) have been measured with an accuracy of up to 0.09% by quantitative convergent-beam electron diffraction (QCBED). This error is an order of magnitude smaller than that in conventional Bragg X-ray diffraction and equivalent to the accuracy of the X-ray Pendellösung method. It is sufficient to distinguish atomic, covalent and ionic bonding. By refinement of the combined data of low-order reflections measured by electron diffraction with high-order reflections from X-ray diffraction, accurate charge-density maps are obtained and used to understand the role of the 3d electrons in Ti-O bonding. The results are combined with electron energy-loss spectra (EELS) in a study of the electronic structure.

Refinement of crystal structural parameters and charge density using convergent-beam electron diffraction--the rhombohedral phase of LaCrO3.

Atom positions and anisotropic Debye-Waller factors of the rhombohedral phase of LaCrO(3) have been refined simultaneously with the low-order structure factors, using a structure-analysis method of convergent-beam electron diffraction (CBED) proposed by Tsuda & Tanaka [Acta Cryst. (1999), A55, 939-954]. The method is based on the least-squares fitting between full dynamical calculations and energy-filtered intensities of two-dimensional higher-order Laue-zone (HOLZ) and zeroth-order Laue-zone (ZOLZ) CBED patterns. The positions of the oxygen atoms have been determined with a high precision. Clear anisotropy of the thermal vibrations of the oxygen atoms has been successfully determined by the CBED method for the first time. The charge transfer from the La and Cr atoms to the O atoms has been found from the deformation charge-density map.