OIM

Do not try this at home: Microwave-Rubies

M. Sc. Julia Mausz, Application Scientist, Gatan/EDAX

Synthetic gemstone quality rubies are commonly manufactured with the Verneuil process, which got its name from its ”father ” Dr. A.V.L. Verneuil. This process was designed to produce single crystalline synthetic rubies and can now be used to melt a variety of high melting point oxides. The details of this flame fusion process were already published in 1902-1904 [1]. As I have neither a ruby mine nor a flame fusion device handy, I aimed to manufacture rubies using a different approach. However, I was unsure if it was possible to form single crystals or even large grains with this technique.

Like in the Verneuil process, the starting point of my synthetic rubies was Al2O3 and Cr2O3 powder. Those were homogeneously mixed, aiming at 1 – 2 at. % chromium content. Considering the melting point of Al2O3 (2,038 °C) [2] and Cr2O3(2,435 °C) [3], the maximum local temperature required to melt a powder mixture of both is 2,435 °C.

A microwave-induced plasma will supply the heat. With an operational frequency of about 2.450 GHz, kitchen microwaves can create high temperature plasmas, even at atmospheric pressure [4]. While bulk metals undergo little heating from microwaves due to the reflection of the waves, it is possible to heat fine-grained metal particles with dielectric heating. However, there is a more effective phenomenon to heat metal with microwaves. Electric discharge can occur due to changes in the distribution of charges when a conductive material with a sharp edge or tip is exposed to microwaves in that frequency regime. The heat resulting from the discharge dissipates very locally into the conductive material, resulting in temperature hot spots able to melt metals and metal oxides in direct contact with the metal, as shown later [5] [6] [7].

The main gases relevant for the plasma will be nitrogen (approx. 78%) and oxygen (approx. 21%) from the surrounding air. The electron source to ignite the plasma will be fine, sharp aluminum edges. Therefore, the powder mixture was placed in a glass crucible and covered with a network of fine aluminum stripes. The crucible was shallow and closed with a glass lid to prevent the hot gas from rising away from the powder. Then, the microwave was operated at 900 W and could sustain the plasma for 60 s. Then, the fused parts were collected from the powder, cleaned, and mounted onto an aluminum stub for observation in the SEM. The resulting fused particles were in the order of 0.5 – 2 mm and already showed the expected pink to purple color, which can be seen in Figures 1a and 1b. The fluorescence yield of rubies can be seen under black light. Without blacklights available, I needed to rely on the 8 kV argon ion beam from the Gatan PECS™ II, and the resulting fluorescence is shown in Figure 1c.

a) Various rubies mounted on a carbon tape. b) Detailed view of the rubies under an optical microscope. c) Fluorescing ruby in an argon ion beam in the PECS II using stationary single beam from one side.

Figure 1. a) Various rubies mounted on a carbon tape. b) Detailed view of the rubies under an optical microscope. c) Fluorescing ruby in an argon ion beam in the PECS II using stationary single beam from one side.

The Zeiss Sigma 500 VP SEM was set to 12 kV acceleration voltage, 120 μm aperture, and 3 Pa low vacuum to prevent charging. The microstructure was then analyzed on the unpolished surface using the EDAX Velocity Super EBSD detector. After fusion of the powder, the resulting ruby has a smooth surface with the crystal structure extending all the way to the surface. Therefore, the ruby could be indexed without any polishing step. It is fascinating with how much ease and speed an unpolished, charging material could be analyzed.

Hough indexing already achieved high indexing rates, considering the dirt and the shadowing on the sample. To bring back even more shadowed points and to refine the grain boundaries, I reprocessed the dataset using Neighbor Pattern Averaging & Reindexing (NPAR™) [8] and spherical indexing [9]. For spherical indexing, a dynamic simulation of trigonal Al2O3 was used. For each, the image quality (IQ) map [10] and confidence index (CI) map, an overlay of the orientation map is shown in Figure 2.

Figure 2. Ruby surface. a) IQ map, b) IQ map + IPF map with CI > 0.2 filter and CIS, c) CI map, and d) CI map + IPF map with CI >0.2 and CIS.

The dataset clearly shows a polycrystalline structure. Note that although the grains can be easily recognized, the shape and size of the grains are distorted due to the variation in surface topography.

In contrast to the grain shape, misorientation and texture analyses are unaffected. The detected bands in the EBSD patterns are direct projections of the lattice planes. As the active lattice planes are independent of the surface structure, the measured crystal orientation is not affected by the surface orientation.

The orientation map is displayed in Figure 3a after applying the confidence index standardization (CIS) procedure and a CI filter of 0.2. Figure 3b shows the overlay of this orientation map with its corresponding CI map and the grain boundaries with a minimum misorientation angle of 5° marked in black.

Figure 3. Ruby surface. a) IPF map with CI >0.2 and CIS and b) overlay of IPF map with CI >0.2 and CIS with grain boundary (>5°) in black and CI Map after CIS.

Interestingly, the as-fused state of the ruby showed a clear spike in the misorientation angle of 60°, as shown in Figure 4a. The twin boundaries of 60° with a tolerance angle of 2° are marked in black on top of the detail orientation map in Figure 4b. The crystal wire figure is schematically shown on both sides of the twin boundary, showing a 60° rotation along the c-axis.

Figure 4. Ruby surface. a) Misorientation chart with black highlighting and b) orientation map with black twin boundaries and crystal visualization of both sides.

In Figure 5, the (0001) texture pole figure reveals a weak texture. The orientation maximum is shifted somewhat towards the top-right, corresponding to the surface’s slanting in the same direction. This suggests that there is a weak preferred orientation of the (0001) planes parallel to the surface of the ruby aggregate particle.

Figure 5. Ruby surface. Texture Pole Figure.

It is possible to form synthetic rubies using microwave-induced plasma in a commercial microwave oven. However, the resulting rubies are small, of unpredictable shape, and due to their polycrystalline nature, not of high clarity. While ruby production in the microwave did not qualify to open a gemstone side business, it is a reliable source for making interesting EBSD samples, and we might see some more gemstone blogs in the future.

References

  1. NASSAU, K. Dr. AVL Verneuil: The man and the method. Journal of Crystal Growth, 1972, 13. Jg., S. 12-18.
  2. SCHNEIDER, Samuel J.; MCDANIEL, C. L. Effect of environment upon the melting point of Al2O3. Journal of Research of the National Bureau of Standards. Section A, Physics and Chemistry, 1967, 71. Jg., Nr. 4, S. 317.
  3. GIBOT, Pierre; VIDAL, Loïc. Original synthesis of chromium (III) oxide nanoparticles. Journal of the European Ceramic Society, 2010, 30. Jg., Nr. 4, S. 911-915.
  4. KOCH, Helmut; WINTER, Michael; BEYER, Julian. Optical Diagnostics on Equilibrium and Non-equilibrium Low Power Plasmas. In: 48th AIAA Plasmadynamics and Lasers Conference. 2017. S. 4158.
  5. SUN, Jing, et al. Review on microwave–metal discharges and their applications in energy and industrial processes. Applied Energy, 2016, 175. Jg., S. 141-157.
  6. LIU, Wensheng; MA, Yunzhu; ZHANG, Jiajia. Properties and microstructural evolution of W-Ni-Fe alloy via microwave sintering. International Journal of Refractory Metals and Hard Materials, 2012, 35. Jg., S. 138-142.
  7. ZHOU, Chengshang, et al. Effect of heating rate on the microwave sintered W–Ni–Fe heavy alloys. Journal of Alloys and Compounds, 2009, 482. Jg., Nr. 1-2, S. L6-L8.
  8. WRIGHT, Stuart I., et al. Improved EBSD Map Fidelity through Re-indexing of Neighbor Averaged Patterns. Microscopy and Microanalysis, 2015, 21. Jg., Nr. S3, S. 2373-2374.
  9. LENTHE, W. C., et al. Spherical indexing of overlap EBSD patterns for orientation-related phases–Application to titanium. Acta Materialia, 2020, 188. Jg., S. 579-590.
  10. WRIGHT, Stuart I.; NOWELL, Matthew M. EBSD image quality mapping. Microscopy and Microanalysis, 2006, 12. Jg., Nr. 1, S. 72-84.

EBSD in a vacuum

Dr. Stuart Wright, Senior Scientist, Gatan/EDAX

I recently co-authored a paper with my colleagues Will Lenthe and Matt Nowell that focused on our parent grain reconstruction tool in OIM Analysis™ [1]. As part of that paper, we show the results from a little round-robin we did. I also showed some results in my webinar on parent microstructure reconstruction in January 2021.

Participating in a round-robin is always a bit unnerving as you are never completely sure how your work will stand up relative to others – especially for those well-recognized experts. This was not an officially moderated round-robin, but rather, me asking other researchers in the area that I happen to have had the good fortune of interacting with in the past if they would be willing to contribute. For the round-robin, the same input EBSD dataset was used for each algorithm. The EBSD dataset was obtained from a low-carbon steel rolled-sheet sample with a fully transformed ferrite body-centered cubic (bcc) microstructure, as shown in Figure 1.

Figure 1. a) Crystal orientation (IPF) map for a ferrite microstructure in a low carbon steel, b) color scheme for the IPF map.

This dataset was used as the input to the parent reconstruction tool in OIM Analysis, as well as several other reconstruction tools. Figure 2 shows the reconstruction results.

Figure 2. IPF Maps of the parent austenite microstructure reconstructed using a) OIM Analysis [2], b) Merengue [3], c) Graph Cutting [4] and d) ROPA [5].

Generally, the results are in reasonable agreement, e.g., the grain sizes and orientations (colors) are in general agreement. The results suggest that if these algorithms were applied to the input dataset obtained from a larger area, then the textures and grain size statistics would all be expected to be quite similar. The differences tend to be in the details, particularly at the boundaries between grains. Our paper discusses some of the nuances of the different algorithms that lead to the differences in reconstruction results.

In the paper, we briefly acknowledge each of those who were kind enough to provide us with the reconstruction results using the different algorithms. However, I want to add a little more detail about the contributors.

The original dataset came to me via Stephen Cluff when he was a Ph.D. researcher in Professor David Fullwood’s group at Brigham Young University working on austenite reconstruction (https://scholarsarchive.byu.edu/etd/9051/). Stephen is now a Materials Engineer at the U.S. Army Research Lab.

The original dataset was collected and shared by Matt Merwin at U. S. Steel. Matt and I co-organized a symposium on EBSD analysis of steel for the 2009 TMS meeting.

The dataset was used in a paper by Chasen Ranger and co-workers on austenite reconstruction (Ranger, C., Tari, V., Farjami, S., Merwin, M.J., Germain, L. and Rollett, A., 2018. Austenite reconstruction elucidates prior grain size dependence of toughness in a low alloy steel. Metallurgical and Materials Transactions A, 49, pp.4521-4535.).

Anthony Rollett is the last author on this paper. I have known Tony for many years – he was my ‘boss’ in my first job out of school as a post-doc at Los Alamos National Lab and is now the U.S. Steel Professor of Metallurgical Engineering and Materials Science at Carnegie Mellon University. I reached out to Tony for data from the paper, and he kindly supplied reconstruction results on the austenite dataset obtained using Lionel Germain’s Merengue code. Lionel is at the University of Lorraine in France, which is where the next ICOTOM will be held (https://icotom20.sciencesconf.org/).

I saw a presentation on parent reconstruction using Graph Cutting by Stephen Niezgoda of Ohio State University (OSU), so I asked if he would apply his algorithm to this dataset. He kindly responded, and his student Charles Xu supplied me with the results from their algorithm. I have known Stephen for many years and have had the opportunity to visit his research group at OSU.

I was also aware of an algorithm from Goro Miyamoto called ROPA. I asked my Japanese colleagues Seichii Suzuki and Tatsuya Fukino of TSL Solutions KK, who are familiar with the ROPA software, if they would run the same dataset, and they kindly obliged. I’ve had the good fortune of enjoying many trips to Japan to visit with my colleagues at TSL Solutions and had the opportunity to host them in Utah.

Why the shameless “name-dropping”?

First, it is good to see the agreement between the different algorithmic approaches to the reconstruction problem. While there are certainly differences between the results, the overall reconstructed microstructures are quite similar.
Second, I have interacted with many of these researchers through a shared interest in EBSD and personal connections that started during my graduate school research under Professor Brent L. Adams. I did a Master of Science degree at BYU and a Ph.D. at Yale University under Brent’s guidance, both of which focused on EBSD. Many of the researchers listed here have worked and published with Brent Adams.

So, my second point is to emphasize that while EBSD is performed in a vacuum – science is much more fruitful and enjoyable when not performed in a vacuum. The connections we build through our interactions with others in the research community are essential to moving science forward – it is good to attend conferences again after the COVID-enforced hiatus.

References

  1. Wright, S.I., Lenthe, W.C., Nowell, M.M. Parent Grain Reconstruction in an Additive Manufactured Titanium Alloy, Metals, 2023, 14, 51. DOI: https://doi.org/10.3390/met14010051.
  2. Ranger, C.; Tari, V.; Farjami, S.; Merwin, M.J.; Germain, L.; Rollett, A. Austenite reconstruction elucidates prior grain size dependence of toughness in a low alloy steel. Metall Mater Trans A 2018, 49, 4521-4535. DOI: https://doi.org/10.1007/s11661-018-4825-7.
  3. Germain, L.; Gey, N.; Mercier, R.; Blaineau, P.; Humbert, M. An advanced approach to reconstructing parent orientation maps in the case of approximate orientation relations: Application to steels. Acta Mater 2012, 60, 4551-4562. DOI: https://doi.org/10.1016/j.actamat.2012.04.034.
  4. Brust, A.; Payton, E.; Hobbs, T., Sinha, V.; Yardley, V.; Niezgoda, S. Probabilistic reconstruction of austenite microstructure from electron backscatter diffraction observations of martensite. Microsc Microanal 2021, 27, 1035-1055. DOI: https://doi.org/10.1017/S1431927621012484.
  5. Miyamoto, G.; Iwata, N.; Takayama, N.; Furuhara, T. Mapping the parent austenite orientation reconstructed from the orientation of martensite by EBSD and its application to ausformed martensite. Acta Mater 2010, 58, 6393-6403. DOI: https://doi.org/10.1016/j.actamat.2010.08.001.

Being more precise, again

Dr. Stuart Wright, Senior Scientist, Gatan/EDAX

In my last blog posting, I was excited to show results from version 9 of EDAX OIM Analysis™ for refining EBSD orientation measurements. However, two questions have been gnawing at me since that post. (1) How much does the size of the patterns affect the results? and (2) How sensitive is the refinement to noise in the patterns? To explore these two questions, I will use data from the same silicon single crystal I used in my previous post – a 1 x 1 mm scan with a 30 µm step size. The patterns were 480 x 480 pixels and of excellent quality.

I added two levels of Poisson noise to the patterns, as shown in Figure 1, and will term these noise levels 1 and 2 for the subsequent analysis.

Figure 1. Si single crystal patterns processed with adaptive histogram equalization [1]. (a) initial pattern, (b) pattern after a moderate level of added noise, and (c) pattern after a significant level of added noise.

The next step was to bin the patterns, index them using spherical indexing, and then apply orientation refinement as implemented in version 9 of EDAX OIM Matrix™. To perform the experiments, I binned the patterns to 360 × 360, 240 × 240, 160 × 160, 120 × 120, 96 × 96, 80 × 80, 60 × 60, and 48 × 48. After binning, I re-indexed them using spherical indexing and then calculated kernel average misorientations (KAM). I used the average KAM value as a measure of precision and plotted that against the binned pattern size for all three noise levels (0, 1, and 2). Figure 2 shows the results of the experiments.

Figure 2. Plot of average KAM values vs. pattern width for all three noise levels.

I have a couple of observations from these results.

  • In general, the first level of noise has only a minimal impact on the precision, whereas the higher level of noise has a more significant impact.
  • For noise levels 0 and 1, the average KAM values remain relatively constant until the pattern size dips below 120 × 120 pixels. Surprisingly, good results can be obtained until the smallest size of 48 × 48 pixels is reached. For noise level 2, the precision drops off significantly at a pattern size of 96 × 96. Those using Velocity cameras have probably noticed that the default pattern size is 120 × 120 pixels. Similar results to those I’ve presented here lead us to choose 120 × 120 pixels as the default. These results confirm the soundness of that choice.

I hope these results can guide the expectations for what orientation refinement can achieve in your samples. We will announce the official release of EDAX OIM Analysis 9 in the next few weeks. We hope you are excited to apply it to your materials. The orientation refinement tools are part of EDAX OIM Matrix, which is an add-on module. While you wait for your copy of version 9, make sure you save the patterns you plan to apply orientation refinement measurements to. No, I’m not getting paid by the hard drive manufacturers 😉.

Figure 3. Screenshot of EDAX APEX showing where the check-box to save patterns is located within the software.

[1] Pizer, S.M., Amburn, E.P., Austin, J.D., Cromartie, R., Geselowitz, A., Greer, T., ter Haar Romeny, B., Zimmerman, J.B. and Zuiderveld, K., 1987. Adaptive histogram equalization and its variations. Computer vision, graphics, and image processing 39: 355-368.

It runs (or rolls) in the family

Matt Nowell, EBSD Product Manager, Gatan/EDAX

I have two sons graduating this year. My oldest son is graduating college with a Materials Science and Engineering degree and is interested in materials characterization. My middle son is graduating high school and has grown up refining ores in Minecraft, casting characters from Dungeons and Dragons, and 3D printing school projects. I’m glad they are both interested in materials and how they can affect daily living. I’ve also been a little sentimental and nostalgic thinking about how we have tried to learn more about materials in our household.

One activity they have always enjoyed is collecting pressed coins. These machines squeeze a coin between two rollers, one of which has an engraving on its surface that is then imprinted onto the stretched and flattened surface of the deformed coin. We have collected these coins from around the world. One example is shown in Figure 1, which is a pressed coin from Universal Studios. This was the most recent addition to the collection. I decided to press a second coin that we could prepare and characterize with electron backscatter diffraction (EBSD) to see the microstructural developments that occur during the pressing process.

Figure 1. A pressed coin from Universal Studios.

The pressed coin was mechanically polished down to 0.02 µm colloidal silica and then analyzed using the new EDAX Velocity Ultra EBSD detector. This new detector allowed for high-speed data collection at acquisition rates of 6,500 indexed patterns per second. Figure 2 shows the inverse pole figure (IPF) orientation map collected from a 134 µm x 104 µm area with a 100 nm step size, with the coloring relative to the orientations aligned with the sample’s surface normal direction. At these speeds, the acquisition time was less than five minutes. A copper blank was used instead of the traditional penny for this sample. This was noticeable when indexing the EBSD patterns. Since 1982, pennies have been made of zinc coated with copper. Zinc has a hexagonal crystal structure, while the EBSD patterns from this coin were face-centered cubic (FCC). EDS analysis confirmed that the material was copper.

Figure 2. An IPF orientation map collect from a 134 µm x 104 µm area of the pressed coin with a 100 nm step size. The coloring is relative to the orientations aligned with the sample’s surface normal direction.

The IPF map shows a significant amount of deformation. This can be seen in the IPF maps with the color variation within each grain. This is, of course, expected, as the elongation and thinning of the coin are easily observed while watching the machine. EBSD is an ideal tool for characterizing this deformation within the material. While there are several different map types to visualize local misorientations and deformation, Figure 3 shows one of my favorites, the grain reference orientation deviation (GROD) map. In this map, the grains are first calculated by grouping measurements of similar orientation using a 5° tolerance angle. Next, the average orientation of each grain is calculated. Finally, each pixel within a grain is colored according to its misorientation from the average orientation of its grain. The microstructure’s largest GROD angular value is 61.9°, indicating a large spread of orientations. This map also shows the grain boundaries as black lines to indicate the original grain boundary positions.

Figure 3. A GROD map of the pressed coin.

Figure 4 shows a fascinating view of how the material is deformed within a selected grain. This chart was created by drawing a line within a grain and plotting the point-to-point and point-to-origin misorientations along this line. The point-to-point distribution shows that each step is typically a small misorientation value below the grain tolerance angle. The point-to-origin distribution shows an accumulation of misorientations within this grain, with the overall misorientation changing more than 30° over the 25 µm distance within the grain. This type of result always gets me thinking about what a grain really is in a deformed material.

Figure 4. A view of how the material is deformed within a selected grain. This chart was created by drawing a line within a grain and plotting the point-to-point and point-to-origin misorientations along this line.

Figure 5. The (001), (111), and (110) pole figures calculated from the measured orientations.

Figure 5 shows the (001), (111), and (110) pole figures calculated from the measured orientations. These pole figures are incomplete and resemble what is expected for a rolled FCC material. This is due to the small number of grains sampled in this area. A second map was collected over a 1,148 µm x 895 µm area with a 2 µm step size in under a minute to get a better sampling of the entire microstructure. The pole figures for this data are shown in Figure 6. Comparing Figures 5 and 6 shows that the additional sampling within the second scan adds more symmetry to the pole figures.

Figure 6. The pole figures for the second map that was collected over a 1,148 µm x 895 µm area with a 2 µm step size.

This was a fun example to show the different data types that can be derived from EBSD measurements. In materials science, understanding the relationship between materials processing and the resulting microstructure is critical to understanding the material’s final properties. It’s clear that pressing a coin causes significant deformation within the material, which can then be measured and quantified with EBSD. Maybe the next time we go to the zoo, we will vary the speed at which we roll the coins and see what effect that has on the data.

Being more precise

Dr. Stuart Wright, Senior Scientist, Gatan/EDAX

The precision and accuracy of orientation measurements by electron backscatter diffraction (EBSD) have been of interest since the advent of EBSD [1, 2]. In contrast, reliability (in terms of correctly identifying the orientation at least within 5°) was of greater concern when indexing was first automated (there is a section of my thesis [3] devoted to precision, as well as Krieger Lassen’s thesis [4]). I’ve written a few papers on the subject [5 – 7], and there have been several more by other authors [8 – 11]. High-resolution EBSD (HREBSD) has shown success in markedly improving precision [12]. Now that dictionary indexing (DI) has become more common; there has been a resurgence in papers on the precision that can be achieved using DI [13 – 15]. I know that is a lot of references for a blog post, but I wanted to give you an idea of how many different research groups have studied angular precision in EBSD measurements – the references given are only a sampling; there are certainly more.

Will Lenthe and I have been working hard to improve the dictionary indexing capabilities in the EDAX OIM Matrix™ add-on module to EDAX OIM Analysis™. In addition, Will has added the ability to perform spherical indexing within OIM Matrix [16 – 17] (see Will’s “New Tools for EBSD Data Collection and Analysis” webinar for more information). These new capabilities will be available soon in OIM Analysis 9. I’m excited about the progress we’ve made. You will find OIM Matrix much easier to use and more robust. In addition, we’ve sped up many aspects of OIM Analysis, which will help with the big datasets routinely obtained with the EDAX Velocity™ cameras.

The precision of indexing via spherical indexing has recently been explored [18]. Using OIM Analysis 9, we’ve been exploring what we can achieve in terms of orientation precision with orientation refinement [19 – 21] applied to initial indexing results obtained by Hough transform-based indexing, dictionary indexing, and spherical indexing. We haven’t quantified our results yet. Still, the KAM maps (which indicate the orientation precision) we’ve obtained are so promising that I want to show our preliminary results. Our refinement method is essentially a hybrid of that proposed by Singh, Ram, and De Graef [19] and Pang, Larsen, and Schuh [21]. But for the spherical indexing, we also have implemented an additional refinement in the harmonic frequency space. Figure 1 shows some results I am excited to share.

Figure 1. KAM maps from nickel [22]. (Top row) As-indexed, (middle row) with NPAR for Hough-based indexing and refinement in the spherical harmonics for spherical indexing, and (bottom row) after real-space refinement. The first column is for Hough-based indexing, columns 2 – 4 are for dictionary indexing with different dictionary target disorientations, and columns 5 – 6 are for SI with different harmonic bandwidths.

It is pretty interesting that the KAM maps after refinement are all nearly the same, no matter which type of indexing was used to obtain the initial orientation measurements. We do not expect much plastic strain or permanent deformation in these samples, so the reduced KAM values are more of what we expect for the sample.

Here is another set of results for a silicon single crystal. The scan is approximately 1 x 1 mm with a 30 m step size. You can see the dramatic improvement in these results. Unfortunately, the two points with the largest KAM values are due to some dust particles on the sample’s surface.

Figure 2. KAM maps were constructed using Hough-based indexing, SI, and SI followed by refinement.

We are very excited to get these advancements into your hands and are putting in extra hours to get the software ready for release. We hope you are as precisely excited as we are to apply it to your samples!

[1] Harland CJ, Akhter P, Venables JA (1981) Accurate microcrystallography at high spatial resolution using electron backscattering patterns in a field emission gun scanning electron microscope. Journal of Physics E 14:175-182
[2] Dingley DJ (1981) A Comparison of Diffraction Techniques for the SEM. Scanning Electron Microscopy IV: 273-286
[3] Wright SI (1992) Individual Lattice Orientation Measurements Development and Applications of a Fully Automatic Technique. Ph.D. Thesis., Yale University.
[4] Krieger Lassen NC (1994) Automated Determination of Crystal Orientations from Electron Backscattering Patterns. Ph.D. Thesis, Danmarks Tekniske Universitet.
[5] Wright S, Nowell M (2008) High-Speed EBSD. Advanced Materials and Processes 66: 29-31
[6] Wright SI, Basinger JA, Nowell MM (2012) Angular precision of automated electron backscatter diffraction measurements. Materials Science Forum 702: 548-553
[7] Wright SI, Nowell MM, de Kloe R, Chan L (2014) Orientation Precision of Electron Backscatter Diffraction Measurements Near Grain Boundaries. Microscopy and Microanalysis 20:852-863
[8] Humphreys FJ, Huang Y, Brough I, Harris C (1999) Electron backscatter diffraction of grain and subgrain structures – resolution considerations. Journal of Microscopy – Oxford 195:212-216.
[9] Demirel MC, El-Dasher BS, Adams BL, Rollett AD (2000) Studies on the Accuracy of Electron Backscatter Diffraction Measurements. In: Schwartz AJ, Kumar M, Adams BL (eds) Electron Backscatter Diffraction in Materials Science. Kluwer Academic/Plenum Publishers, New York, pp 65-74.
[10] Godfrey A, Wu GL, Liu Q (2002) Characterisation of Orientation Noise during EBSP Investigation of Deformed Samples. In: Lee DN (ed) ICOTOM 13, Seoul, Korea, Textures of Materials. Trans Tech Publications Inc., pp 221-226.
[11] Ram F, Zaefferer S, Jäpel T, Raabe D (2015) Error analysis of the crystal orientations and disorientations obtained by the classical electron backscatter diffraction technique. Journal of Applied Crystallography 48: 797-813
[12] Wilkinson AJ, Britton TB (2012) Strains, planes, and EBSD in materials science. Materials Today 15: 366-376
[13] Ram F, Singh S, Wright SI, De Graef M (2017) Error Analysis of Crystal Orientations Obtained by the Dictionary Approach to EBSD Indexing. Ultramicroscopy 181:17-26.
[14] Nolze G, Jürgens M, Olbricht J, Winkelmann A (2018) Improving the precision of orientation measurements from technical materials via EBSD pattern matching. Acta Materialia 159:408-415
[15] Shi Q, Loisnard D, Dan C, Zhang F, Zhong H, Li H, Li Y, Chen Z, Wang H, Roux S (2021) Calibration of crystal orientation and pattern center of EBSD using integrated digital image correlation. Materials Characterization 178:111206
[16] Lenthe W, Singh S, De Graef M (2019) A spherical harmonic transform approach to the indexing of electron backscattered diffraction patterns. Ultramicroscopy 207:112841
[17] Hielscher R, Bartel F, Britton TB (2019) Gazing at crystal balls: Electron backscatter diffraction pattern analysis and cross-correlation on the sphere. Ultramicroscopy 207:112836
[18] Sparks G, Shade PA, Uchic MD, Niezgoda SR, Mills MJ, Obstalecki M (2021) High-precision orientation mapping from spherical harmonic transform indexing of electron backscatter diffraction patterns. Ultramicroscopy 222:113187
[19] Singh S, Ram F, De Graef M (2017) Application of forward models to crystal orientation refinement. Journal of Applied Crystallography 50:1664-1676.
[20] Winkelmann A, Jablon BM, Tong V, Trager‐Cowan C, Mingard K (2020) Improving EBSD precision by orientation refinement with full pattern matching. Journal of Microscopy 277:79-92
[21] Pang EL, Larsen PM, Schuh CA (2020) Global optimization for accurate determination of EBSD pattern centers. Ultramicroscopy 209:112876
[22] Wright SI, Nowell MM, Lindeman SP, Camus PP, De Graef M, Jackson MA (2015) Introduction and comparison of new EBSD post-processing methodologies. Ultramicroscopy 159:81-94

Grain Analysis in OIM Analysis

Dr. Sophie Yan, Applications Engineer, EDAX

Recently, we held a webinar on Grain Analysis in OIM Analysis™. After the webinar, many users mentioned that the basic operation overview was very helpful. Since there was a very enthusiastic response, I want to take this opportunity to share these fundamental tips and tricks with the greater electron backscatter diffraction (EBSD) community.

Perhaps the most popular EBSD application is grain analysis, as it’s fundamental to characterizing many materials. Because the results of grain analysis are sometimes consistent or inconsistent with other tests, it’s great to start with a basic understanding of a grain with respect to EBSD and how grain analysis works.

The definition of a grain in OIM Analysis differs from the strict academic definition, which refers to the collection of pixels within a certain orientation range. This orientation range, namely grain tolerance angle, can be changed in OIM Analysis, which is generally set to 5° by default. You can also vary the number of pixels in a grain (the default is 2). These parameters affect the result of grain size, so we should pay attention to them in the analysis. The prerequisite of grain analysis is that the data is statistically valuable. Sometimes this requires a lot of tests to achieve the goal, repetitive studies to diminish errors, or the data should be filtered or processed before the analysis (per relevant standards, accordingly).

Figure 1. A typical grain map.

A standard display for grain size analysis is the Grain Size (diameter) chart. First, the grain is fit to a circle, and then the software calculates the diameter. The data distribution range and average grain size are on the chart’s right side. The most frequent question users ask is, “What is the formula to calculate the average grain size?”. In fact, two results of the average grain size, which are calculated by two different methods, are shown. The ‘number’ method calculates the average area of each grain first (the sum area is divided by grain number values first) before it determines the diameter. In contrast, it considers different weights due to different areas for the ‘area’ method. Since large grains have larger weight percentages, it first calculates the average grain area using different weight percentages, then calculates the average grain size.

In addition to the average grain size, OIM Analysis offers a variety of charts and plots to characterize grain shape. The most popular one is the grain shape aspect ratio, an essential parameter to display the columnar grain property (grains are fit as an ellipse). In addition to the shape aspect ratio, the Grain Shape Orientation in OIM Analysis shows the angle between the long axis and the horizontal direction, which is suitable for grains with a specific growth direction.

OIM Analysis offers numerous functions. Concerning grain analysis, there are six different charts for grain size and eight for grain shapes. Some charts are not common, but they have corresponding application scenarios. If you do not know the meaning of those charts, you can query the OIM Analysis Help file to get relative information.

Grain analysis is a very common function of EBSD applications. As a webinar speaker, I enjoyed digging up some less familiar details so users could gain a deeper understanding of software operations. I look forward to continually introducing webinar topics to meet the EBSD community’s needs and make greater progress in the new year.

关于晶粒那些事儿

Dr. Sophie Yan, Applications Engineer, EDAX

最近我们办了一期OIM Analysis如何进行晶粒分析的直播，效果颇出我意料。大家对于基本操作的热情令我始料未及，在播出之后联系我的人中，也往往会提到这一场直播对他们有所帮助。我本以为，这一类关于基本操作的直播并不太吸引人，充其量也不过是成为大家在日常操作中备查的工具视频而已；但，果然是我灯下黑，事实并不是如此。我也借此机会，给大家分享一些基本的原则或窍门。

可能在EBSD的各种应用场景中，最常碰到的是就是粒度分析。绝大多数的人都会有这个需求。EBSD粒度分析的结果，有些会与其它测试的结果吻合，有些会有出入。这时，我们就必须了解EBSD的晶粒是怎么定义，粒度又是怎么测得的，才可能更好的分析EBSD测试的结果。

OIM中的晶粒不同于学术上严格的定义，是指在一定取向范围内的像素点的集合。这个取向范围，即容差角，是可以设置的，一般默认为5度。像素点的个数，也同样可以设置，默认个数为2; 这些参数的设置，其实都会影响我们统计晶粒粒径的结果，因此需要在粒径分析中予以注意。当然，进行粒径分析的前提是数据具有统计意义，有时需要进行大量的重复性的测试来减少误差，也需要在测试之前对数据进行筛选或处理（可参照相关标准进行操作）。

典型的晶粒图

最常见的粒度分析的结果是将晶粒拟合成圆，计算其平均直径，即常见的Grain Size(diameter)曲线。曲线（或柱状图）的右边是数据部分，有不同粒度分布的占比，也列出了平均粒径。这是我被客户问得最多的部分，即，我们的平均粒径的结果是怎么计算的：这个平均粒径其实列出了两种结果，由两种方法分别计算。“number”方法是指按数数目的方法，先计算每个颗粒平均的面积(总面积除以晶粒数)，然后再计算直径；”area”则要考虑因面积不同而带来的不同权重，大颗粒占权重较大，按权重计算出颗粒的平均面积，再计算平均粒径。

除平均粒径外，OIM Analysis还提供了多种表征颗粒形状的图表。最常见的是短长比（grain shape aspect ratio）,是描述非等轴晶粒径的重要参数(晶粒被拟合成椭圆)。当然，除了短边长边的比值，有些颗粒有形状还有明显的择优，按特定的方向生长，针对这一点，Grain Shape Orientation表征长边与水平方向的角度，可以描述这一特性。

OIM Analysis提供非常丰富的功能，晶粒粒径有6种不同图表，晶粒形状有8种。有些图表可能不太常用，但都有对应的应用场景。对于这些相对冷门的图表，一般用户，如不了解其功能，可以在OIM Analysis提供的帮助文件中查询，可以得到关于其功能或定义的相关信息。

粒度分析是EBSD应用非常常用的功能。这一次，作为主讲人，在准备过程中我也挖掘出一些平常不注意的细节，对软件操作也有了更深的了解。相信，随着我们随后不定期推出的一系列培训，我们能更加贴合用户的需求，协助大家把软件用得更好，在新的一年里，获得更大的进步。

Reaching Out

Dr. René de Kloe, Applications Scientist, EDAX

2022 was a year of changes. In the beginning, I set up a desk in the scanning electron microscope (SEM) lab where, without truly reaching out, I only needed to turn in my chair to switch from emails and virtual customers on my laptop to the live energy dispersive spectroscopy (EDS) and electron backscatter diffraction (EBSD) system and real data on the microscope. As travel restrictions gradually eased worldwide, we were all able to start meeting “real” people again. After almost two years of being grounded, I finally met people face to face again, discussing their analysis needs, and answering questions do not compare to online meetings. We restarted in-person training courses, and I participated in many external courses, exhibitions, and conferences, reaching out to microscopists all over Europe.

And as always, I try to correlate real life with some nice application examples. And what is similar to reaching out to people in the microanalysis world? Reaching out to things! So, what came to mind are remote thermal sensors, which most of us will have at home in the kitchen: a thermostat in an oven and a wired thermometer that you can use to measure food temperatures. And I just happened to have a broken one that was ready to be cut up and analyzed.

Figure 1. a) A food thermometer and b) an oven thermostat sensor.

On the outside, these two sensors looked very similar; both were thin metal tubes connected to a control unit. Because of this similarity, I was also expecting more or less the same measuring method, like using a thermocouple in both thermometers. But to my surprise, that was not quite the case.

The long tube of the food thermometer was mostly empty. Right at the tip, I found this little sensor about 1 mm across connected to copper wires that led to the control unit. After mounting and careful sectioning, I could collect EDS maps showing that the sensor consisted of a central block of Mn-Co-Fe-oxide material sandwiched between silver electrodes soldered to the copper-plated Ni wires.

Note that in the image, you only see one of the wires, the other is still below the surface, and I did not want to polish it any deeper.

Figure 2. The temperature sensor taken out of the tube of the food thermometer.

Figure 3. A forward scatter SEM image of the polished cross-section showing the central MnCoFe-oxide material and one of the connecting wires.

This was no thermocouple.

Figure 4. The element distribution in the sensor.

Figure 5. The EDS spectrum of the central CoMnFe-oxide area.

Instead, the principle of this sensor is based on measuring the changing resistivity with temperature. The EBSD map of the central Co-Mn-De oxide area shows a coarse-grained structure without any preferred orientation to make the resistivity uniform in all directions.

Figure 6. An EBSD IPF on Image Quality map of the sensor in the food thermometer.

Figure 7. (001) pole figure of the MnCoFe oxide phase, showing a random orientation distribution.

And where the tube of the food thermometer was mostly empty, the tube of the oven thermostat sensor was completely empty. There were not even electrical connections. The sensor was simply a thin hollow metal tube that contained a gas that expands when heated. This expansion would move a small disk with a measurement gauge that was then correlated with a temperature readout. Although this sounded very simple, some clever engineering was needed to prevent the tube from pinching shut when bending and moving it during installation.

I cut and polished the tube, and an EBSD map of the entire cross-section is shown below.

Figure 8. a) EBSD IQ and b) IPF maps of a cross-section through the entire tube of the oven thermostat sensor.

The tube is constructed out of three layers of a Fe-Cr-Ni alloy with fine-grained multiphase chromium phosphide layers in between. This microstructure is what provides corrosion protection, and it also adds flexibility to the tube. And this, in turn, is crucial to prevent cracks from forming that would cause the leaking of the contained gas, which is critical in getting a good temperature reading.

The detailed map below shows a section of the phosphide layer. There are two chromium phosphide phases, and in between, there are dendritic Ni grains that link everything together.

Figure 9. EDS maps showing the composition of one of the phosphide layers.

Figure 10. EBSD IPF maps of the different phases. a) All phases on a PRIAS center image, b) CrP, c) Fe matrix, and d) Ni dendrites, Cr3P.

When you look at the microstructure of both sensors in detail, it is possible to determine how they work, and you can appreciate why they have been designed as they are. The two devices are efficient and tailored to their intended use. The oven thermostat is designed to be mounted in a fixed position to be secure so that it can be used for a very long time. The food thermometer is very flexible and can easily be moved around.

In that respect, I feel there is another similarity between these sensors and the different kinds of meetings between people we have experienced over the past year. It does not matter how you do it; you can always reach out and feel some warmth.

I wish everybody a very happy and peaceful 2023.

Improved IPF Color Palettes

Will Lenthe, Principal Software Engineer, EDAX

Most IPF color schemes have several shortcomings:

  1. Although red, green, and blue are placed at a high symmetry axis, the remaining colors are not uniformly distributed
  2. Saturated rainbow palettes are not perceptually uniform, so the same orientation gradient will have different apparent intensities when centered around different orientations
  3. Groups with two or four high symmetry directions do not have a natural mapping to three principal colors
  4. Choosing red and green as principal colors result in poor contrast for individuals with red-green color vision deficiency (CVD)

OIM Analysis™ v9 implements four new Inverse Pole Figure (IPF) color palettes to address these issues, as shown in Figure 1. For fundamental sectors with three principal directions, CVD colors replace green with yellow for the second principal color. For fundamental sectors with four principal directions, red, yellow, green, and blue are used for traditional colors, and red, yellow, cyan, and blue are used for CVD colors. Notice that the new legends distribute colors smoothly while the old ones have large patches of red, green, and blue extending from the corners and sharp bands of yellow, cyan, and magenta.

Figure 1. The m3m (top) and m3 (bottom) IPF legend is shown from left to right for OIM Analysis v8 colors, new saturated colors, perceptually uniform colors, CVD saturated colors, and perceptually uniform CVD colors.

Figure 2. A nickel dataset is IPF colored with saturated (left) and perceptually uniform (right) color maps using traditional (top) and CVD (middle) colors. Notice that some significant orientation gradients in the KAM map (bottom left) are visible with perceptually uniform colors but may be invisible if the orientation falls in a low contrast region of the saturated color map. OIM Analysis v8 coloring is shown in the bottom right.

Figure 3. A partially recrystallized steel dataset is IPF colored with saturated (left) and perceptually uniform (right) color maps using traditional (top) and CVD (middle) colors. Notice that orientation gradients are over-emphasized in darker regions of the saturated color maps (blue and purple) and under-emphasized in brighter regions (green, yellow, and cyan).

Perceptual Uniformity

Perceptually uniform color maps are designed so that a constant size step in the data being colored results in an apparent color change of constant magnitude regardless of the starting value. The uniformity of a color map can be visualized by imposing a ripple onto a ramp, as shown in Figure 3 and described by Kovesi [1]. The ripple disappears in brighter regions of traditional saturated color maps but has a uniform relative intensity in perceptually uniform maps, as shown in Figure 4. The new perceptually uniform IPF colors in OIM Analysis v9 extend perceptually uniform cyclic color maps to a hemisphere by adding a white center point.

Figure 4. A perceptually uniform ramp is modified by a sine wave to create a test signal (green). The test signal is colored with a perceptually uniform black to white color map with maximum sine wave amplitude at the top of the image and minimum amplitude at the bottom. Note that the relative intensity of the ripple is the same at every gray level near the top edge and the ramp appears extremely smooth near the bottom edge. Figure adapted from Kovesi [1].


Figure 5. Traditional saturated color maps (top) are shown for heat (left) and rainbow (right) colors. Notice that the ripples are nearly invisible near red on both maps, yellow on the heat map, and green on the rainbow map. Perceptually uniform equivalents (bottom) sacrifice some color saturation/vividness to achieve a uniform sensitivity response across the entire map. Legends from Kovesi [1].


CVD Colors

Deuteranomaly (red-green CVD) is the most common form of CVD and is simulated in Figure 6 to illustrate how much ambiguity is introduced in traditional colors. CVD impacts roughly 1 in 12 men and 1 in 200 women, so CVD colors should be preferred for papers and presentations.

Figure 6. Deuteranomaly is simulated with increasing severity from left to right (normal, 30%, 70%, 100%/Deuteranopia) for the traditional (top) and CVD (bottom) saturated palettes. Notice that in the far-right column, the traditional map has different directions with the same color, while the CVD map is significantly less ambiguous.

Enhanced IPF saturated color palettes maintain a similar look and feel while more uniformly distributing the available gamut. Perceptually uniform IPF color palettes sacrifice the full use of the RGB gamut to render crystal directions with increased precision, and CVD colors avoid red-green ambiguity. Together these new palettes enable visualization and accurate interpretation of orientation data for the widest range of audiences.

References

  1. Kovesi, P. (2015). Good colour maps: How to design them. arXiv preprint arXiv:1509.03700.
  2. Nolze, G., & Hielscher, R. (2016). Orientations–perfectly colored. Journal of Applied Crystallography, 49(5), 1786-1802.

Spherical Indexing

Will Lenthe, Principal Software Engineer, EDAX

Dictionary indexing compares experimental electron backscatter diffraction (EBSD) patterns against a dictionary of simulated patterns for each orientation on a uniform grid in orientation space [1,2]. Synthetic patterns are generated by rotating the Kikuchi sphere by the crystal orientation and projecting onto a plane using the experimental geometry. Comparison against a physics-based forward model gives excellent precision and noise tolerance at the cost of significant computational overhead. Spherical harmonic-based indexing uses the same Kikuchi sphere or ‘master pattern,’ but back projects experimental patterns onto the sphere instead. The orientation is indexed using the maximum spherical cross-correlation between the back-projected pattern and the Kikuchi sphere [3,4]. Mathematically, dictionary and spherical indexing are extremely similar, but the spherical approach is more numerically efficient since it can leverage fast Fourier transforms for the computations. In practice, spherical indexing provides similar precision [5] and noise tolerance to dictionary indexing but at much faster speeds.

A GPU implementation of spherical harmonic-based EBSD indexing implemented in OIM Analysis™ as part of the OIM Matrix module provides excellent indexing quality at hundreds or thousands of patterns per second. Here, we applied it to a range of scans to demonstrate the indexing quality and user parameters.

Spherical harmonic indexing has two parameters: bandwidth and grid size. Bandwidth is how far in frequency space to compute harmonics (analogous to a low pass filter on the EBSD pattern). Grid size is the correlation resolution with an Euler angle cube of (grid size)3 used for correlation (i.e., 0 – 360 for phi1, Phi, and phi2). In general, computation time scales with the number of Euler angle grid points, and a reasonable bandwidth is one less than half the grid size. For example, the following are some reasonable pairs of values:

BandwidthGrid Size
63128
95192
127256

Once the best Euler grid point (maximum cross-correlation) is selected, subpixel resolution can be achieved through a refinement step.

Ni Sequence

This dataset is a scan of the same region at different camera gains to intentionally produce corresponding sets of low and high-quality patterns.

Figure 1. Shows a) the result of indexing high-quality patterns, b) spherical harmonic indexing at a bandwidth of 63 and Euler grid of 1283 without refinement, and c) at a bandwidth of 63 with refinement.

Figure 1 shows a) the result of indexing high-quality patterns, b) spherical harmonic indexing at a bandwidth of 63 and Euler grid of 1283 without refinement, and c) at a bandwidth of 63 with refinement. Note that since grid point spacing is ~2.8° (360° / 128), the unrefined result has a stepped appearance due to the discrete orientation possibilities. After refinement, any orientation is possible, providing smooth results.

Figure 2. KAM maps are shown for the same region at a) 0°, b) 1°, and c) 2°.

In Figure 2, KAM maps are shown for the same region at a) 0°, b) 1°, and c) 2°. Notice that without refinement, there is no misorientation within a patch and a sharp spike between them. Even though both the Hough and refined spherical appear smooth, the slight orientation noise in the Hough indexing is visible using KAM.

Figure 3. With low-quality patterns, Hough indexing a) starts to fail, but b) spherical indexing still provides robust solutions and c) accurately captures continuous orientation gradients after refinement.

With low-quality patterns, Hough indexing a) starts to fail, but b) spherical indexing still provides robust solutions and c) accurately captures continuous orientation gradients after refinement (Figure 3).

Figure 4. a) bandwidths of 63, b) 95, and c) 127 are compared before (a – c) and after (d – f) refinement.

For very low-quality patterns, higher bandwidths may be required for better indexing results. In Figure 4, a) bandwidths of 63, b) 95, and c) 127 are compared before (a – c) and after (d – f) refinement. Note that the discrete steps in orientations before refinement become smaller with increased Euler angle grid resolution, but they refine to similar orientations. For all three bandwidths, the grid size is 2 * (bandwidth + 1).

Figure 5. 4. a) Raw pattern and b) NPAR pattern using Hough indexing and c) raw pattern and d) NPAR pattern using spherical indexing with a bandwidth of 127.

With spherical indexing integrated into OIM Analysis, existing image processing algorithms can be used for especially difficult patterns. At extremely high noise levels, Hough indexing cannot index any points, and the spherical indexing begins to fail for some points. NPAR trades spatial resolution for pattern quality by averaging each pattern with its neighbors. The improved patterns can be indexed reliably by both methods but Hough indexing struggles with the resulting overlap patterns near grain boundaries (Figure 5).

Hot Rolled Mg

Figure 6. Hough indexing struggles to index when pattern quality is reduced by a) high deformation, but b) spherical indexing is robust against significantly degraded pattern quality. Note that the d) spherical indexing confidence index strongly correlates with c) image quality but is high even in some regions with extremely low IQ.

Hough indexing struggles to index when pattern quality is reduced by a) high deformation, but b) spherical indexing is robust against significantly degraded pattern quality. Note that the d) spherical indexing confidence index strongly correlates with c) image quality but is high even in some regions with extremely low IQ (Figure 6).

Rutile

Figure 7. Excellent results are possible even with a single pattern center used for the entire dataset. Vignetting is visible in a) an IPF+IQ map of Hough indexing with a fixed pattern center. The field is flat over the entire area for b) an IPF+CI map of spherical indexing with a fixed pattern center.

Spherical indexing can use a unique pattern center for each point at no extra cost for large fields of view. Excellent results are possible even with a single pattern center used for the entire dataset, as shown in Figure 7. 

Deformed Duplex Steel

Figure 8. Phase discrimination depends on the similarity of the phases with a two-phase steel. In addition to the quality in orientation results with d – f) spherical indexing vs. a – c) Hough indexing, b – c & e – f) phase discrimination is improved with spherical BCC and FCC iron well separable.

Spherical indexing can be applied to multiple phases in the same way as any other indexing technique. Phase discrimination depends on the similarity of the phases with a two-phase steel shown in Figure 8. In addition to the quality in orientation results with d – f) spherical indexing vs. a – c) Hough indexing, b – c & e – f) phase discrimination is improved with spherical BCC and FCC iron well separable. Real space refinement may be required for particularly difficult cases in addition to the spherical harmonic refinement shown.

Figure 9. a) spherical CI + IPF shows similar trends as b) Hough IQ + IPF.

Again, spherical indexing’s confidence index correlates well with pattern quality. In Figure 9, a) spherical CI + IPF shows similar trends as b) Hough IQ + IPF.

References

  1. Callahan, P. G., & De Graef, M. (2013). Dynamical electron backscatter diffraction patterns. Part I: Pattern simulations. Microscopy and Microanalysis, 19(5), 1255-1265.
  2. Callahan, P. G., & De Graef, M. (2013). Dynamical electron backscatter diffraction patterns. Part I: Pattern simulations. Microscopy and Microanalysis, 19(5), 1255-1265.
  3. Lenthe, W. C., Singh, S., & De Graef, M. (2019). A spherical harmonic transform approach to the indexing of electron backscattered diffraction patterns. Ultramicroscopy, 207, 112841.
  4. Hielscher, R., Bartel, F., & Britton, T. B. (2019). Gazing at crystal balls: Electron backscatter diffraction pattern analysis and cross-correlation on the sphere. Ultramicroscopy, 207, 112836.
  5. Sparks, G., Shade, P. A., Uchic, M. D., Niezgoda, S. R., Mills, M. J., & Obstalecki, M. (2021). High-precision orientation mapping from spherical harmonic transform indexing of electron backscatter diffraction patterns. Ultramicroscopy, 222, 113187.