TY - JOUR
AU - Tsuda, Kenji
AB - Abstract The accuracy of electron density distribution analysis using large-angle convergent-beam electron diffraction (LACBED) patterns is evaluated for different convergence angles. An orbital ordered state of FeCr2O4 is used as an example of the analysis. Ideal orbital-ordered and non-ordered states are simulated by using orbital scattering factors. LACBED patterns calculated for the orbital-ordered state were used as hypothetical experimental data sets. Electron density distribution of the Fe 3d orbitals has been successfully reconstructed with a higher accuracy from LACBED patterns with convergence angles larger than 15.2 mrad, which is 4 times as large as that for conventional convergent-beam electron diffraction patterns. Excitation of particular Bloch waves with the aid of LACBED patterns has a key role in the accurate analysis of electron density distributions. convergent-beam electron diffraction, electrostatic potential analysis, large-angle rocking, crystal structure factor, orbital-ordered state, orbital scattering factor Introduction Convergent-beam electron diffraction (CBED) is a powerful tool for analyzing electrostatic potential and electron density distributions within a unit cell of the crystalline specimen [1–19]. Low-order structure factors are determined as structural parameters by fitting intensity distributions of experimental CBED patterns and theoretical ones. The low-order structure factors are sensitive to charge transfer, bonding and deviation of the electron density associated with each atom from a spherical distribution [7]. This is because electrons are scattered by the electrostatic potential which is formed by the balance of positively charged nuclei and negatively charged electrons. So far, we applied the quantitative analysis method of CBED patterns to an orbital-ordered state of FeCr2O4 [20] where low-order structure factors were refined within the range of g < 0.6 Å−1. A clear anisotropy of the electrostatic potential was observed at Fe sites, which was assigned to a ferrotype-ordered state of Fe |$3{d}_{3{\mathrm{z}}^2-{\mathrm{r}}^2}$| orbitals. A larger number of low-order structure factors than are available from conventional CBED patterns are required to visualize the electron density distribution of the Fe 3d orbital-ordered state, and this requires the measurement of structure factors with larger reciprocal lattice vectors. However, such reflections, which appear in the relatively outer areas of conventional zone-axis CBED patterns, do not have enough intensity because those reflections are far from their Bragg conditions over the greater parts of their reflection disks. One promising way to overcome this problem is large-angle convergent-beam electron diffraction (LACBED) using an incident beam with a much larger convergence angle, which enables the observation of intensities of outer reflections at their Bragg conditions simultaneously [21]. LACBED patterns have rich information for a wide range of reciprocal spaces. Recently, large-angle rocking beam electron diffraction (LARBED) techniques were reported which make it possible to take LACBED patterns from a nanoscale area without disk overlap [22–25]. In this paper, we demonstrate the effectiveness of LACBED patterns for the accurate determination of electrostatic potential and electron density distribution analysis using hypothetical experimental data sets of LACBED patterns for the orbital-ordered state of FeCr2O4 as examples. An examination of the orbital ordered state is a suitable example for the evaluation of the accuracy of analysis because ideal spherical and orbital ordered states can be simulated using orbital scattering factors [26]. Fig. 1 Open in new tabDownload slide (a) A schematic picture of the crystal structure of spinel oxide FeCr2O4. (b) The electronic state of the 3d orbital at Cr and Fe sites. (c) Electrostatic potential distribution of the Fe site in an orbital-ordered phase determined by CBED method [20] and (d) simulated by using orbital scattering factors. Fig. 1 Open in new tabDownload slide (a) A schematic picture of the crystal structure of spinel oxide FeCr2O4. (b) The electronic state of the 3d orbital at Cr and Fe sites. (c) Electrostatic potential distribution of the Fe site in an orbital-ordered phase determined by CBED method [20] and (d) simulated by using orbital scattering factors. Fig. 2 Open in new tabDownload slide LACBED pattern of 020 reflection with [100] incidence. Each pattern is simulated by using orbital-ordered and spherical states, respectively. Colored solid circles indicate convergence angles of 3.8, 15.2 and 30.4 mrad. Fig. 2 Open in new tabDownload slide LACBED pattern of 020 reflection with [100] incidence. Each pattern is simulated by using orbital-ordered and spherical states, respectively. Colored solid circles indicate convergence angles of 3.8, 15.2 and 30.4 mrad. Methods We chose a spinel oxide FeCr2O4 as an example for an analysis of an orbital-ordered state. FeCr2O4 undergoes a structural phase transformation at 135 K from the cubic phase of a spinel structure to the tetragonal phase with a ferrotype orbital ordering of 3d electrons of Fe atoms [27–29]. Figure 1a shows a crystal structure of the cubic phase of FeCr2O4. Fe is located at the center of oxygen tetrahedra, and Cr is located at the center of oxygen octahedra. Accompanying the structural transition, 3d electrons of Fe sites show an orbital-ordered state. Figure 1b shows the electronic state of 3d orbitals at Cr and Fe sites. A trivalent Fe has six 3d electrons. In the tetrahedral configuration, t2 and e orbitals are energetically separated due to a crystal field splitting. One electron in the e orbital has an orbital degree of freedom. This orbital-ordered state in spinel systems is explained in some other papers [30–38]. Table 1 Summary of reflections with g-vector smaller than 0.83 Å−1 Observable reflections at each incidence are indicated. Crystal structure factors [Å] calculated for spherical (non-orbital ordered) and orbital-ordered phases are shown. Open in new tab Table 1 Summary of reflections with g-vector smaller than 0.83 Å−1 Observable reflections at each incidence are indicated. Crystal structure factors [Å] calculated for spherical (non-orbital ordered) and orbital-ordered phases are shown. Open in new tab Fig. 3 Open in new tabDownload slide (a) The difference LACBED pattern of the 020 reflection between the orbital-ordered and spherical state. (b) Dispersion surface of electron with [100] incidence. A horizontal axis corresponds to a dashed line in (a). The eight Bloch state shows higher excitation amplitude at the area marked by dotted circles. (c) A schematic picture of a crystal structure with [100] incidence. (d) A projected electron density of the eight Bloch state. Fig. 3 Open in new tabDownload slide (a) The difference LACBED pattern of the 020 reflection between the orbital-ordered and spherical state. (b) Dispersion surface of electron with [100] incidence. A horizontal axis corresponds to a dashed line in (a). The eight Bloch state shows higher excitation amplitude at the area marked by dotted circles. (c) A schematic picture of a crystal structure with [100] incidence. (d) A projected electron density of the eight Bloch state. Fig. 4 Open in new tabDownload slide Deviation of goodness of fit for changes in crystal structure factors. Fig. 4 Open in new tabDownload slide Deviation of goodness of fit for changes in crystal structure factors. Fig. 5 Open in new tabDownload slide Residual value by subtraction from ideal orbital-ordered state from the fitting results. Fig. 5 Open in new tabDownload slide Residual value by subtraction from ideal orbital-ordered state from the fitting results. Fig. 6 Open in new tabDownload slide Subtracted electron density distribution in the (110) slice. The subtraction is obtained from fitting results to a spherical state. (a) A subtraction between the ideal orbital-ordered state and the spherical state as a reference. (b–d) Subtracted electron densities between fitting results and the spherical state for convergence angles of 30.4, 15.2 and 3.8 mrad, respectively. Fig. 6 Open in new tabDownload slide Subtracted electron density distribution in the (110) slice. The subtraction is obtained from fitting results to a spherical state. (a) A subtraction between the ideal orbital-ordered state and the spherical state as a reference. (b–d) Subtracted electron densities between fitting results and the spherical state for convergence angles of 30.4, 15.2 and 3.8 mrad, respectively. Fig. 7 Open in new tabDownload slide Changing of the position of intensity inside the disks for a larger g-vector. Fig. 7 Open in new tabDownload slide Changing of the position of intensity inside the disks for a larger g-vector. Figure 1c shows the electrostatic potential distribution of the orbital-ordered state obtained from a quantitative analysis of conventional zone-axis CBED patterns [20]. In the analysis, 11 low-order structure factors with reciprocal vectors of g < 0.6 Å−1 were refined using three CBED patterns taken at three zone-axis incidences of [100], [110] and [210]. The figure shows an electrostatic potential distribution with color at an isosurface of electron density of 4 e Å−3. Red regions indicate relatively high electrostatic potential compared with other parts and correspond to slight elongation of electron density in the c-direction. This deviation corresponds to the orbital-ordered state. The electron density distribution of the ideal orbital-ordered state can be simulated by using orbital scattering factors calculated by Zheng et al. [26]. Crystal structure factors for electrons can be calculated from the orbital scattering factors, and CBED patterns for the ideal orbital-ordered state and non-orbital-ordered state, which correspond to the spherical distribution of electron density, can be simulated from the calculated crystal structure factors. Figure 1d shows the electrostatic potential simulated for the ideal orbital-ordered state, where crystal structure factors with g-vectors less than 0.68 Å−1 were obtained using the orbital scattering factors and those with g-vectors larger than this were set to be those of the independent atom model. The deviation of electrostatic potential along the c-axis shown in Fig. 1c and d indicating a similar tendency but clear differences in detail. This probably comes from a limitation of the number of refined crystal structure factors. The software, MBFIT [4], developed by Tsuda and co-workers was used for dynamical calculations based on the Bloch wave theory. MBFIT can simulate dispersion surfaces and Bloch states inside crystals and refine structural parameters such as crystal structure factors and atom positions using a nonlinear least squares method. The effect of anomalous absorption effect due to thermal diffuse scattering is taken into account by the use of an imaginary (absorption) potential [39]. It is noted that in the present case, values of the structure factors of the real (crystal) potential are pure real numbers and those of the absorption potential are pure imaginary numbers because the crystal structure is centrosymmetric. Only the real parts of the structure factors are refined. Refinements were conducted on a Linux work station with AMD Opteron multi CPUs with 96 cores. The reconstructed electron density and electrostatic potential distributions were visualized by software VESTA [40]. Results and discussion Table 1 summarizes the sets of reflections observable for each incident direction, which are indicated by shaded areas. The calculated crystal structure factors using orbital scattering factors [26] for spherical (non-orbital ordered) and orbital-ordered states are displayed. We focused on the [100] and [111] incidences and structure factors of 16 reflections with g < 0.68 Å−1, which are surrounded by a red rectangle in Table 1. First, the intensity differences of LACBED patterns simulated for the spherical (non-orbital ordered) and the orbital ordered states were examined. Figure 2 shows the intensity difference map of the 020 reflections as an example. The specimen thickness was set to 100 nm. The colored solid circles indicate convergence angles of the incident beam of 3.8, 15.2 and 30.4 mrad, respectively. The convergence angle of 3.8 mrad corresponds to the case of a conventional CBED pattern without overlap between neighboring disks. It should be noted that the large intensity differences are found mainly in the angles between 3.8 and 15.2 mrad. From this point, it is expected that the CBED patterns with a convergence angle of 15.2 mrad have higher sensitivity for orbital-ordered state than those of 3.8 mrad. To examine the origin of high sensitivity of the LACBED patterns to the orbital-ordered state of FeCr2O4, Bloch waves in the specimen were simulated. Figure 3b shows squared excitation amplitudes |${\big|{\varepsilon}^{(j)}\big|}^2$| of the j-th Bloch wave, where the excitation amplitudes were calculated for the incidence condition along the dashed line in Fig. 3a. It is seen that at the positions of the maxima of the intensity differences in Fig. 3a, the excitation amplitudes of the eight Bloch wave in Fig. 3b have peaks, as indicated by red dotted ovals. Figure 3d shows the projected electron density distribution of the eight Bloch wave. A schematic picture of crystal structure with this incidence condition is shown in Fig. 3c. It is seen that the Bloch wave is concentrated at Fe and O sites. The |$3{d}_{3{\mathrm{z}}^2-{\mathrm{r}}^2}$| orbital of Fe atoms is expected to expand to the O sites [6]. In order to examine sensitivities of the LACBED patterns to the low-order structure factors, hypothetical experimental patterns were prepared by adding artificial random noise according to the Poisson statistics. The LACBED patterns were calculated using crystal scattering factors for the ideal orbital-ordered state where 3z2−r2 orbitals are occupied at all Fe sites. In Fig. 4, ΔFg means the deviation from an ideal orbital-ordered condition. Goodness of fit (GOF) is an indicator of the accuracy of fitting, and a lower value means a better fitting result. The GOF is calculated as |${\big[\sum_i{\big({I}_i^{\mathrm{exp}}-{I}_i^{\mathrm{cal}}\big)}^2/\big(N-M\big)\big]}^{1/2}$|⁠, where N is the number of data points and M is the number of parameters to be refined. Using the hypothetical experimental patterns, GOF was evaluated with changes in the structure factors of two reflections, 020 and 004, from −5 to +5%. It is obvious that a GOF change becomes rapid for the data set of 15.2 mrad. This is mainly because observable intensities are stronger for a larger convergence angle because of the satisfaction of the Bragg condition as shown in Fig. 2. As a result, we can conclude that LACBED patterns with a convergence angle of 15.2 mrad have a higher sensitivity to changing Fg than the patterns with a conventional convergence angle of 3.8 mrad. We performed refinements of low-order structure factors using the hypothetical experimental LACBED patterns. Three data sets with different convergence angles of 3.8, 15.2 and 30.4 mrad were used. Fg’s of the 16 low-order reflections shown in Table 1 were refined using the hypothetical data of the [110] and [111] incidences. Initial values of the Fg’s for the refinement were set to those of the ideal spherical (IAM) case shown in Table 1. ΔFg in Fig. 5 means a residual difference from those of the ideal orbital-ordered state. The horizontal axis is the length of the g-vector for each reflection. It should be noted that the deviations from the value of the ideal orbital-ordered state were significantly reduced for the data with larger convergence angles of 15.2 and 30.4 mrad. Figure 6 shows the obtained electron density distributions of the (110) slice across the Fe atoms, where the difference electron densities are shown between the fitting results and that of the spherical (non-orbital ordered) state. The result for the difference between the ideal orbital-ordered state and the spherical state is shown in Fig. 6a as a reference. The color scale is set from −0.03 to 0.03 e Å−3, and the contours are drawn from −0.10 to 0.10 at 0.01 e Å−3 interval. The spatial distribution of 3z2−r2 orbital of Fe atoms is visualized, and the distribution is well overlapped with the projected electron density of the eight Bloch state as shown in Fig. 3d. Figure 6b–d show the difference electron densities between the fitting results and the spherical state for convergence angles of 30.4, 15.2 and 3.8 mrad, respectively. The elongated electron density distribution due to the orbital-ordered state at the Fe sites shows the same tendency for all results. As expected, Fig. 6b and c are seen as being close to the ideal case of Fig. 6a. In contrast, Fig. 6d is seen to have some ghost densities (indicated by red arrowheads) as artifacts. It was confirmed that the accuracy of a small convergence angle is lower than the other results. From the fitting results, we may conclude that the refinement result with a convergence angle of 15.2 mrad is sufficiently accurate to obtain an electron density map of this orbital-ordered state. An accurate determination of electrostatic potential and electron density distribution is governed by two factors. One is the accuracy of the determined crystal structure factors. The other is the number of low-order structure factors to be refined. The structure factors with higher g-vectors are more important for obtaining the detailed distributions. Figure 7 explains the effect of using a large convergence angle for intensity recording. The 004, 006 and 008 reflections are shown as examples. Each reflection corresponds to a g-vector of 0.484, 0.726 and 0.969 Å−1, respectively. In the case of fitting reflections with g-vectors lower than 0.68 Å−1, as in the present work, the 004 reflection is a typical example, and with a convergence angle of 15.2 mrad, it includes enough intensity inside the disk. However, the 006 reflection does not show strong intensity in the range of a convergence angle of 15.2 mrad. This means that the 006 reflection with a convergence angle of 15.2 mrad cannot be used for a determination of F006 because the sensitivity of the reflection intensities to changes in F006 is not enough for the refinement of this structure factor. In the case where F008 is to be determined, a convergence angle of 30.4 mrad should be used. Larger convergence angle requires longer calculation time for obtaining a final fitting result. Thus, before performing an experimental analysis, the required precision of the final electron density distribution needs to be well examined. To achieve the required precision, the number of Fg can be estimated. From this estimation, a required convergence angle can be determined. Concluding remarks We have demonstrated the effectiveness of LACBED patterns for accurate determination of electrostatic potential and electron density distributions. The low-order structure factors of the orbital-ordered phase of FeCr2O4 up to g < 0.68 Å−1 were refined using hypothetical LACBED data sets obtained from simulation with orbital scattering factors. The electron density distribution showing the ordering of Fe |$3{d}_{3{\mathrm{z}}^2-{\mathrm{r}}^2}$| orbital was successfully reconstructed from the LACBED data with convergence angles of 15.2 and 30.4 mrad, which are 4 and 8 times larger than that of conventional CBED data, respectively. The LACBED pattern with convergence angle larger than 15.2 mrad can excite a particular Bloch wave which is localized around the Fe and O atoms. The Bloch wave plays a key role in the accurate determination of electron density distribution of the orbital-ordered state. 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TI - Evaluation of accuracy in the determination of crystal structure factors using large-angle convergent-beam electron diffraction patterns
JF - Microscopy
DO - 10.1093/jmicro/dfaa041
DA - 2020-07-21
UR - https://www.deepdyve.com/lp/oxford-university-press/evaluation-of-accuracy-in-the-determination-of-crystal-structure-LbtmRxuMXR
SP - 1
EP - 1
VL - Advance Article
IS - 
DP - DeepDyve
ER -