SAED3: simulation and analysis of electron diffraction patterns

Xing-Zhong Li, Nebraska Center for Materials and Nanoscience, University of Nebraska, Lincoln, NE 68588, USA.
X.-Z. Li works at the University of Nebraska. He manages an electron microscopy facility and joins with a research group focusing on the development of novel magnetic materials. He obtained his PhD degree in materials physics in 1993 and has had an interest in software development since 1999, especially in electron diffraction analysis and crystallography. 
The SAED3 software has been developed for interactive simulation and analysis of selected-area (and nano-beam) electron diffraction patterns. It can be used as a teaching aid on electron microscopy courses and a practical tool for materials science research. Both kinematical and dynamical theories were implemented in the SAED3 for calculation of diffraction intensities. Besides diffraction patterns of single crystalline phases, composite patterns of multiple phases can also be simulated. Other functions include basic image processing and zone-axis determination of experimental diffraction patterns. Application examples on various research topics are presented to show the features of the SAED3.
Application examples were selected from author’s previous research works. The author would like to thanks his colleagues on the collaborations. 
Corresponding author
Xing-Zhong Li, PhD
Nebraska Center for Materials and Nanoscience, 
University of Nebraska, 
Lincoln, NE 68588, USA. 
Tel: 402 472 8762
The selected electron diffraction (SAED) technique has been extensively used in materials science research, e.g., for phase identification, interpretation of twins and coexisted multiple phases and so forth.
The nano-beam electron nanodiffraction technique, which can be applied to even smaller sample area, provides diffraction spot patterns similar to SAED patterns. Electron nanodiffraction can be carried out in both a transmission electron microscope and a scanning transmission electron microscope. Simulation of SAED† patterns plays an important role to interpret experimental results. 
SAED patterns of a single crystal grain and of a polycrystalline sample are common in essence but different in many aspects, so we treated the two cases separately for advanced simulation and analysis of SAED patterns. Software PCED3, which was upgraded from PCED2.01, is designed for simulation of SAED patterns of polycrystalline samples while software SAED3 focuses on simulation of SAED patterns of single crystal phase, including composite patterns of multiple phases.
For advanced simulation and analysis of SAED patterns, software should be designed for interactive simulation applying not only on single crystalline phase but also on multiple phases, e.g., twining, coexisted phases with a fixed orientation, or convenient comparison of SAED patterns. In addition, a practical task in SAED analysis is to determine zone axes and indices of experimental SAED patterns, such analysis is necessary for phase identification and orientation determination. With the motivation to fulfill these needs, we have developed the SAED3, as the successor of the previous JECP/ED2 and SAED2.03
In this paper, first the theoretical background of the SAED3 was briefly covered and then the design of the SAED3 was described in detail. Next, application examples on various research topics were selected to illustrate the main features of the SAED3. Finally, a summary of the SAED3 is given.
The step-by-step usage of the SAED3 was given in a user manual and also in form of online video instruction, which will be not duplicated here. An executable file, example data files and other related documents are available on the author’s website4.  
Electron diffraction geometry and intensity
Kinematical and dynamical theories of high-energy electron diffraction have been well documented in the textbooks on electron microscopy, e.g. by Reimer, 19895, by De Graef, 20036 and by Peng et al., 20047 and in the review papers on electron diffraction theories, e.g., by Humphreys, 19708, by Metherell, 19759 and by Self, 198310.
Readers, who are interested in the formulae, are suggested to refer to the original books and papers for detailed derivations. For simplicity, inelastic scattering effects were not included in the SAED3. The electron atomic scattering factors can be derived from x-ray atomic scattering factors from Mott-Bethe relationship11 or directly obtained from parameterized table of electron atomic scatter factors12. The second method was used in the SAED3. 
As well known, the geometric construction of diffraction on crystal structures is clearly explained via Ewald sphere. For high-energy electron diffraction, the radius of the Ewald sphere is fairly large, thus electron diffraction pattern of the thin sample reveals the two dimensional distribution of reciprocal lattice points. Weiss zone law gives the relationship of a zone axis [uvw] and a reflection (hkl), an equation in a general form - hu+kv+lw=N(1) - where N is always an integer, 0, ±1, ±2, ···
The kinematical theory leads to the Bragg condition and to a description of the influence of the structure of a unit cell and of the external size of a crystal on the diffracted amplitude in terms of structure and lattice amplitudes, respectively. The dynamical theory considers the interaction between the primary and reflected waves. On taking into account the boundary condition at the surface and the crystal periodicity of the wave-field inside the crystal, the solution of the Schrödinger equation becomes a Bloch-wave field. 
Twins and coexisted multiple phases 
Twinned crystals may be grouped into three general categories13. Non-merohedral twins have two or more crystalline domains with reciprocal lattices that either do not overlap or only partially overlapped. In contrast, merohedral twins have domains with diffraction patterns that are completely overlapped. The symmetry operations relating the twinned domains are a part of the Laue group of the sample for true merohedral twins and it is otherwise for pseudo-merohedral twins. 
Multiple crystalline phases may coexist with a fixed orientation relationship in a complex system, e.g. spinodal decomposition, precipitates in matrix, and eutectic reaction mixture. Structure characterization includes the identification of the crystalline phases, their orientation relationship, and the structures of the grain boundaries. The composite SAED patterns of twins or coexisted phases can be simulated in the SAED3 by overlaying calculated patterns of each component according to their crystallographic relationship. 
Determination of zone axis and indices 
A practical task in SAED analysis is to determinate zone axes and indices of experimental SAED patterns, such analysis is necessary e.g., for identification of crystalline phases and orientation determination of crystal grains. The processes include: to load a candidate of crystalline structures; to select two basic reciprocal vectors on an experimental pattern; to match the calculated vectors to the experimental ones within a given tolerance; to sort a list of possible zone axes according the result; to compare the simulated patterns to the experimental SAED pattern. 
The SAED3 provides a tool for zone-axis determination of experimental SAED patterns using the above method. Instead of the angle between two basic reciprocal vectors, the reciprocal length of the third edge in the triangle formed by two vectors is used. The calibration of reciprocal lengths is prerequisite in the search process using the scale bar on the experimental SAED pattern or using a known crystal structure and the same experimental conditions.
As object-orientated software design, functional blocks and their interaction relations in the SAED3 are shown in Figure 1. Process core is the central block of the software. Calculated SAED patterns are generated using a list of crystal structures and input parameters. 

Figure 1
. A flowchart for software design of the SAED3
Diffraction intensities are calculated in the kinematical theory or the dynamical theory. The orientation of the calculated pattern can be changed by both rotation and mirror operations. The simulation is either calculated patterns of single crystal phases or the composite patterns of multiple crystal phases. Experimental SAED patterns can be loaded for display and go through basic image processing if necessary. Zone axes are first searched according to geometric fitting using a candidate of crystal structures and then simulated patterns are generated to check the distribution of reflection intensities with the experimental SAED pattern.
A flexible index scheme is provided to show either the indices of two basic reflections or the indices of reflections whose intensities higher than a user-defined threshold. The output image within a region of interested (ROI) can be saved as an image file in JEPG or TIFF format. 
Figure 2 shows two main graphical user interfaces in the SAED3. Figure 2(a) is a frame with a menu system and a panel for display of simulated and experimental SAED patterns. A calculated SAED pattern of a default crystal phase, f.c.c. aluminum, is shown in the display panel.

Figure 2.
 Main interface of the SAED3, (a) a frame with a menu and a display panel, (b) a panel for calculation parameters and display options. 
Figure 2(b) is a panel for calculation parameters and display options. The choice of kinematical or dynamical theory and the high voltage of the incident beam are listed on top of the panel. The next item is the file name of a structure currently used and followed by a group of basic parameters for calculation, including, e.g., small tilting angle of incident beam. The rest of the panel is options for display, including, e.g., diffraction spots of first order Laue zone (FOLZ). Other panels can be activated from the drop-down menu or the graphic menu bar for further input parameters and operations. As an example, the panel for preparation of new structure data is shown in Figure 3. This is a template for speeding up the preparation process of crystal structure data with computer assistance.
Figure 3. A template for preparation of a data file in the SAED3.
In default setting simulation is carried out in the kinematical theory, the other choice is the Bloch-wave dynamical theory. Since the calculation on a large unit cell in the dynamical theory could become extremely intensive, it is only suitable for crystals with relative small lattice parameters. The overall intensities of each calculated pattern can be adjusted according to their mass quantity in the case of multiple phases. This operation can be used to roughly estimate the mass ratio of each crystal phase using an experimental diffraction pattern of multiple phases. Adjust the mass proportion mp(i) of each crystalline phase in the calculated pattern to match the relative intensities of the SAED pattern to the experimental pattern, then the final mass ratio of phase i is,
In analysis of experimental results, two methods are available for loading experimental SAED patterns in the SAED3, i.e., via file system or drag-and-drop. Basic image processing operations in the SAED3 include contrast inverted, image resize and rotation, alignment of pattern for centering incident beam. The prerequisite for zone axis determination is to calibrate the matching-factor for experimental SAED patterns. An application example on the zone axis determination is presented in next section. Snap-shots of selected panels for analysis of experimental SAED patterns are shown in Figure 4.
Figure 4. Selected panels for analysis of experimental SAED patterns in the SAED3.
The SAED3 was implemented in Java SE development kit (JDK 8), and was used in author’s home institute on multiple PCs with Microsoft Windows 7 and up operating systems and the Java runtime environment (JRE 8). The SAED3, also in the previous versions, i.e., SAED2.0 and JECP/ED, have been distributed to more than 30 universities/research institutes worldwide so far. A launcher was designed as an organizer to run the SAED3 and other related software4, e.g., SVAT, PCED3 and SPICA314, conveniently. SVAT is for display and analysis of crystal structure in a unit-cell. PCED3 is for simulation and analysis of SAED patterns on polycrystalline samples. SPICA is for the application of stereographic projection. The same format of crystal structure files is used.
The purpose of the SAED3 is twofold, as a teaching aid and a research tool. Application examples were selected from author’s previous research works to illustrate the features of the SAED3. In each example below, we make a short introduction of research field first and then present the usage of the SAED3. The application examples cover the following aspects, a) on simulation in kinematical theory; b) on simulation in dynamical theory; c) as a tool for phase identification; d) on composite patterns for twinned crystals; e) on composite patterns for coexisted crystals; f) on determination of grain orientations. 
Simulation in kinematical theory: crystalline approximant of Mn-Al decagonal quasicrystal, MnAl3
A quasiperiodic crystal, or quasicrystal, is an ordered structure but without periodic translation order. A decagonal quasicrystal has a 10 or 105 fold axis of rotational symmetry. An approximant is a crystalline phase, which is structurally related to the quasicrystal, similar structural blocks packed in periodically in the former and aperiodically in the latter. MnAl3 is a typical crystalline approximant of the Mn-Al decagonal quasicrystal15.
Figure 5 shows two calculated SAED patterns of the MnAl3 phase along the zone axes of (a) [001] and (b) [100] using the SAED3. Indices are labelled for reflections whose intensities higher than a threshold value.
Figure 5. Simulated SAED patterns of the MnAl3 phase along the zone axes of (a) [001] and (b) [100] using the SAED3
Although the diffraction intensities were calculated in the kinematical theory, the resemblance of the SAED patterns to those of a decagonal phase is revealed, e.g., pseudo-tenfold symmetry can be clearly seen in Figure 5 (a). Structural model of the Mn-Al decagonal quasicrystal was deduced from the structure of the MnAl3 phase and high-resolution electron microscopy images of the Mn-Al decagonal quasicrystal16. The model is in good agreement with those obtained by the single-crystal X-ray structure analysis of the Mn-Al decagonal quasicrystal. Readers, who are interested on this topics, may follow the related publication and reference therein.
Simulation in dynamical theory: magnetic materials for recording media, L10 FePt
L10 FePt alloy with a face-centered tetragonal layer structure possesses high magnetic anisotropy energy, which is about an order of magnitude larger than that of the currently used CoCr-based alloys.
This strong magnetic anisotropy can suppress the superparamagnetic fluctuation of magnetization at room temperature down to a particle size of about 3nm. Thus, FePt nanoparticles are expected as one of the most promising candidates for the future recording media with ultra-high densities17
Figure 6 shows simulated SAED patterns of the L10 FePt phase using the SAED3, in which diffraction intensities were calculated in (a) the kinematical theory and (b-d) the Bloch-wave dynamical theory. For very thin crystal sample, the SAED pattern calculated in the dynamical theory, as shown in Figure 6(b), is similar to the one calculated in the kinematical theory, as shown in Figure 6(a). In the simulation in the dynamical theory, the patterns show intensity distribution varying with the change of the crystal thickness, as shown in Figure 6(b-d). The SAED technique is useful to characterize alloy samples of the L10 FePt phase, e.g., the chemical ordering and the texture of nanoparticles or thin films. 
Figure 6. SAED patterns of the L10 FePt phase calculated in (a) the kinematical theory and (b-d) the dynamical theory using the SAED3. The crystal thickness in the calculation are posted in (b-d)
Phase identification: FULL-Heulser intermetallic phase, Ni-Mn-In
Heusler compounds are intermetallics with intriguing magnetic properties18. A typical Heusler compound is Cu2MnAl in the L21 structure. Surprisingly, the compound is ferromagnetic, even though all its elemental shows zero net magnetic moment by themselves. The Heusler compound commonly has a stoichiometric composition of X2YZ, where X and Y are transition metal elements, and Z is a group III, IV or V element. If one X site is vacant, a half-Heusler compound in the C1b structure is obtained with a composition of XYZ.
In investigation of magnetic skyrmions in Ni-Mn-In alloy, structural characterization was carried out thoroughly in samples of rapidly quenched ribbons19. Figure 7 shows (a) a TEM image and (b) a corresponding SAED pattern of a compound with a composition of Ni2-xMn1+yIn1-y where x = 0.483; y = 0.206, simulated SAED patterns were calculated on the basis of (c) the full-Heusler ideal structure and (d) the full-Heusler structure with defects. The defects are partial vacancy in Ni atom sites and a mixture of Mn and In in In atom sites. The experimental SAED pattern matches better to the calculated pattern in Figure 7(d). Thus the grain under examination can be viewed as a full-Heusler compound with defects along [001] zone axis. 
Figure 7. (a) A TEM image and (b) an experimental SAED pattern of the full-Heusler compound with defects in composition of Ni2-xMn1+yIn1-y where x = 0.483; y = 0.206.  Simulated SAED patterns were on the basis of (c) full-Heusler ideal structure and (d) full-Heusler structure with defects.
Twinned and Coexisted crystals: highly textured Pt-Bi thin films, Bi2Pt and BiPt
There has been considerable interest in understanding various properties of Pt-Bi based compounds because of their high activity as fuel-cell anode catalysts for formic acid (HCOOH) or methanol (CH3OH) oxidation20. Three common intermetallic compounds in the Pt-Bi alloy system are PtBi, Pt2Bi3 and PtBi2. PtBi and Pt2Bi3 adopt the hexagonal NiAs structure, PtBi: a = 0.4324 and c = 0.5501 nm and Pt2Bi3: a = 0.413 and c = 0.558 nm. There are three polymorphs of PtBi2, which crystallize in the AuSn2 type orthorhombic, α-PtBi2: a = 0.6732, b = 0.6794 and c = 1.3346 nm; in the FeS2 cubic pyrite type β-PtBi2: a = 0.6701 nm; and in a trigonal γ-PtBi2: a = 0.657, and c = 0.616 nm. 
Pt-Bi thin films were synthesized on glass and thermally oxidized silicon substrates by e-beam evaporation and annealing21. The films prepared by post-deposition annealing at 300 °C and 400 °C are mostly the polymorphic PtBi2 with a small trace of the PtBi. TEM analysis shows that the γ-PtBi2 is the dominant phase with a small amount of the β-PtBi2 in these samples.
The films have highly texture in which the c-axis of the γ-PtBi2 is along the normal of the film plane. Figure 8 shows (a) an experimental SAED pattern of the Pt-Bi thin film, which consists of a merohedral twin of the γ-PtBi2 together with the PtBi phase, (b) a simulated composite SAED pattern for interpretation of the experimental result and (c) a simulated SAED pattern using a single phase of the γ-PtBi2 for comparison. The orientation relationship is [001] of the γ-PtBi2 // [001] of the PtBi and (100) of the γ-PtBi2 // (110) of the PtBi. This example also shows that the mass ratio can be adjusted for a composite diffraction pattern of multiple phases in the SAED3. 
Figure 8. SAED analysis of the Pt-Bi thin film, (a) an experimental SAED pattern, (b) a composite SAED pattern simulated using a twinned γ-PtBi2 and a coexisted PtBi, (c) a simulated SAED pattern using a single grain of the γ-PtBi2
Orientation determination: hierarchical structures of nanowires on microfiber, Cu2S
Hierarchical micro- and nano-structures are increasingly attractive for application in optics, electronics, sensing, and so forth. Specifically, core-branch heterostructures, having core and branches composed of different materials, allow targeted properties of the wires and the base, offer high surface-to-volume ratio and wire-to-base ratio, and bring promise of novel functional membranes. 
A novel hierarchical architecture, inorganic Cu2S wires standing on organic polyacrylonitrile (PAN) fibers, has been produced by combining the electrospinning technique and room-temperature gas-solid diffusion-assisted chemical growth22
The produced nanowires are 1.5~1.8 µm long with uniform diameter of about 80 nm, as shown in Figure 9 (a). The wires consist of the monoclinic Cu2S phase. In order to determine the growth direction of the Cu2S crystals, the two basic reciprocal vectors on an experimental SAED pattern were measured, as shown in Figure 10.
The most possible zone axis was found to be [2 1 1]. The simulated pattern along [2 1 1] zone axis is shown in Figure 9 (c), which is well in good agreement with the experimental SAED pattern. The nanowires growth direction is determined to be perpendicular to the (2, 0, -4) crystal plane, i.e. parallel to the c axis of the monoclinic crystal, as shown in Figures 9 (b) and 9 (c). 
Figure 9. (a) A TEM image of Cu2S nanowires on a polyacrylonitrile fiber, (b) an experimental SAED pattern of a Cu2S nanowire with an arrow in growth direction, and (c) a simulated SAED pattern of the Cu2S along the zone axis of [2 1 1].
Figure 10. (a) Measurement of two basic reciprocal vectors in the experimental SAED pattern of a Cu2S nanowire and (b) a panel of the lengths and angles of the basic reciprocal vectors. The zone axis of the experimental SAED pattern is determined as [2 1 1].
Software SAED3 has been developed for simulation and analysis of electron diffraction patterns. The SAED3 can be used as a teaching aid on electron microscopy courses and a practical tool for materials science research. 
Theoretical background of the SAED3 is briefly covered and the design of the software was described in detail. The main features of the SAED3 include, i) interactive simulation of SAED patterns; ii) intensity calculation in kinematical theory and Bloch-wave theory; iii) SAED pattern simulation and analysis of single phase and multiple phases; iv) image processing of experimental SAED patterns; v) determination of zone axes for experimental SAED patterns.
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