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 processing39: 355-368.
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.
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 E14:175-182
[2] Dingley DJ (1981) A Comparison of Diffraction Techniques for the SEM. Scanning Electron MicroscopyIV: 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 Processes66: 29-31
[6] Wright SI, Basinger JA, Nowell MM (2012) Angular precision of automated electron backscatter diffraction measurements. Materials Science Forum702: 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 Microanalysis20:852-863
[8] Humphreys FJ, Huang Y, Brough I, Harris C (1999) Electron backscatter diffraction of grain and subgrain structures – resolution considerations. Journal of Microscopy – Oxford195:212-216.
[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 Crystallography48: 797-813
[12] Wilkinson AJ, Britton TB (2012) Strains, planes, and EBSD in materials science. Materials Today15: 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. Ultramicroscopy181:17-26.
[14] Nolze G, Jürgens M, Olbricht J, Winkelmann A (2018) Improving the precision of orientation measurements from technical materials via EBSD pattern matching. Acta Materialia159:408-415
[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 Characterization178:111206
[16] Lenthe W, Singh S, De Graef M (2019) A spherical harmonic transform approach to the indexing of electron backscattered diffraction patterns. Ultramicroscopy207:112841
[17] Hielscher R, Bartel F, Britton TB (2019) Gazing at crystal balls: Electron backscatter diffraction pattern analysis and cross-correlation on the sphere. Ultramicroscopy207:112836
[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. Ultramicroscopy222:113187
[19] Singh S, Ram F, De Graef M (2017) Application of forward models to crystal orientation refinement. Journal of Applied Crystallography50:1664-1676.
[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 Microscopy277:79-92
[21] Pang EL, Larsen PM, Schuh CA (2020) Global optimization for accurate determination of EBSD pattern centers. Ultramicroscopy209:112876
[22] Wright SI, Nowell MM, Lindeman SP, Camus PP, De Graef M, Jackson MA (2015) Introduction and comparison of new EBSD post-processing methodologies. Ultramicroscopy159:81-94
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.
Although red, green, and blue are placed at a high symmetry axis, the remaining colors are not uniformly distributed
Saturated rainbow palettes are not perceptually uniform, so the same orientation gradient will have different apparent intensities when centered around different orientations
Groups with two or four high symmetry directions do not have a natural mapping to three principal colors
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
Kovesi, P. (2015). Good colour maps: How to design them. arXiv preprint arXiv:1509.03700.
Nolze, G., & Hielscher, R. (2016). Orientations–perfectly colored. Journal of Applied Crystallography, 49(5), 1786-1802.
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:
Bandwidth
Grid Size
63
128
95
192
127
256
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
Callahan, P. G., & De Graef, M. (2013). Dynamical electron backscatter diffraction patterns. Part I: Pattern simulations. Microscopy and Microanalysis, 19(5), 1255-1265.
Callahan, P. G., & De Graef, M. (2013). Dynamical electron backscatter diffraction patterns. Part I: Pattern simulations. Microscopy and Microanalysis, 19(5), 1255-1265.
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.
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.
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.
Dr. Chang Lu, Application Specialist, Gatan & EDAX
In early 2022, Gatan and EDAX completed the integration, and our new division was named Electron Microscope Technology (EMT). As an EMT application scientist on the China applications team, I am responsible for almost all the Gatan and EDAX products for Northern China, on both Scanning Electron Microscopy (SEM) and Transmission Electron Microscopy (TEM) platforms. Therefore, I work with diversified products and diversified user groups that focus on different subject matters. In the first half of this year, I found that the data analysis software from EMT Gatan’s DigitalMicrograph® (DM) and EDAX’s OIM Analysis™ are not completely isolated, but in many cases, they can cooperate with each other to help our customers.
For instance, DM can do a series of electron microscopy-related data processing. For some energy dispersive spectroscopy (EDS) mapping data from the minor content, there are various methods to achieve smoothing and enhance the contrast. While in the MSA panel, the principal component analysis (PCA) function can be helpful in terms of high-resolution EDS mapping. However, in today’s EDAX blog, I will talk a little bit more about one feature in OIM Analysis that could potentially benefit a lot of Gatan camera users.
In northern China, there are a group of Gatan users who are focused on nanoscale phases and grains in the TEM. In most scenarios, they heavily employ electron diffraction or bright field imaging to make judgments. However, it is really difficult to determine the unknown (unidentified but has a known x-ray diffraction (XRD) pattern and chemical composition, so there is a potential for it) phase by simply relying on the minor changes of grayscale bright field images. You may say diffraction could help. Yes, a clean, beautiful diffractogram of a particular crystal direction can be helpful. But, no, you need to find the zone axis carefully. If this unknown phase has a crystal structure of low symmetry, most of the time, the effort will be in vain. Generally speaking, the Difpack tool in the DM software could help in determining d-spacing and angles, however, it is not intuitive enough to know the sample at first sight.
The solution is pattern simulation with OIM Matrix™. At first, I noticed this feature because it helped an EDAX user who was studying strains. It can easily export a theoretical Kikuchi pattern for a specific sample orientation with zero stress. Then one day, I had a sudden thought during my morning shower. Maybe I can change the acceleration voltage to 200 kV (typical for TEM), and the sample tilt angle to 0° (make it flat). After entering a specific orientation, we can get a Kikuchi pattern under TEM conditions! For example, take the simulated pattern from NdCeB. With Kinematic Color Overlay, we can also find out what crystal plane corresponds to a specific Kikuchi line. Now, when we start changing the zone axis in an unidentified sample, we can first simulate several orientations and compare them with what we see under TEM. In this way, the process of finding the Kikuchi pole turns out to be very convenient.
Figure 1. A simulated pattern from NdCeB using OIM Matrix.
Now, when some Gatan users bring in some “weird” unidentified samples and say they want to find various zone axis for doing diffractions. I don’t worry about it. I think from a problem-solving point of view, the powerful software from both Gatan and EDAX, like the integration of two companies, can also be combined to solve complex and difficult problems for our customers in the future.
It has been an interesting experience to build our OIM Matrix™ software package. As you may know, OIM Matrix is partially a front-end user interface to the EMsoft package developed by Professor Marc De Graef’s group at Carnegie Mellon University to make it convenient to use within the framework of OIM Analysis™. I have learned a lot in the process and am grateful for Marc’s patience with my many questions. Will Lenthe recently joined the EBSD group at EDAX. Will worked as a Post-Doc in Marc’s group, and his additional insights have been invaluable as we are striving to build the second generation of OIM Matrix. It will be easier to use, more robust, and provide some significant speed gains.
While our initial focus for OIM Matrix was on helping users improve the indexing of EBSD patterns from difficult-to-index materials, I’ve been surprised by how useful it has been for testing our software. It has also helped us in developing some of our new features. Having well-simulated patterns for known orientations and EBSD/SEM geometries is very helpful.
I used OIM Matrix for a study on feldspars. According to Wikipedia:
“Feldspars are a group of rock-forming aluminum tectosilicate minerals containing sodium, calcium, potassium, or barium. The most common members of the feldspar group are the plagioclase (sodium-calcium) feldspars and the alkali (potassium-sodium) feldspars. Feldspars make up about 60% of the Earth’s crust and 41% of the Earth’s continental crust by weight.”
Given that feldspars are relatively common, we are frequently asked to help index them. They are difficult, as a poster at the 2019 Quantitative Microanalysis (QMA) conference detailed [1]. I thought it might be interesting to see what we could learn about the limits of EBSD in characterizing these materials. I won’t give you all that we learned in that little study, but what I thought was an interesting snapshot. Figure 1 shows a phase diagram for the feldspar group of minerals.
Figure 1. Phase diagram for the feldspar group.
To start, I looked in the American Mineralogist Crystal Structure Database (AMCSD) for all the relevant entries I could find. There are a lot of variants. Here is a table:
Table 1. Number of entries in AMCSD for each feldspar.
I enjoy seeing pattern simulation results, but producing 149 master patterns [2] would take more patience than I have (each master pattern calculation can take several hours for these low-symmetry materials). So, I selected one entry for each mineral type. I tried to find one that seemed most representative of all the other entries in the set. After calculating the eight master patterns, I simulated one individual pattern at the same orientation for each mineral, as shown in Figure 2. Note that they are all similar, with the most deviation coming from the anorthite and sanidine end members of the series.
Figure 2. Patterns were simulated at Euler angles of (30°, 30°, 30°) for each feldspar.
To quantify the differences, I calculated the normalized dot-products [3] for all pattern pairs to get the following table. A value of “1” indicates the patterns are identical. As expected by the initial observation, the biggest difference is the sanidine to albite pair of patterns.
Table 2. Normalized dot products.
Of course, the next step would be to see how this holds up to experimental patterns and dictionary indexing [4]. I hope to eventually do this with samples Professor Rudy Wenk of Stanford University kindly gave me. Rudy has been one of the major contributors to the entries in the AMCSD for feldspars.
There was one more virtual experiment I thought would be interesting. I wanted to ascertain how much the chemical species in the feldspar series influenced the patterns. To do this, I created an average structure instead of using the lattice parameters for each feldspar. I then populated these structures with atoms to maintain the chemical composition ratios specified for each series. A master pattern for each ideal structure was calculated. Three hundred forty patterns were simulated uniformly, covering orientation space with a spacing of approximately 30° between orientations. The average normalized dot products were calculated for each pattern against the albite pattern at the same orientation. Figure 3 shows the results.
Figure 3. The normalized dot product of simulated patterns for idealized structures against the albite simulated patterns.
Clearly, the dot products are all very near 1, indicating that the differences in the simulated patterns due to chemical composition are small for these chemical species. This suggests that coupling EBSD with EDS is critical when trying to differentiate the different feldspar minerals. While this small study has not changed the world of feldspar indexing, it has, at least, been a stimulating study of simulating for me.
[1] B Schneider, and J Fournelle (2019) “Using Quantitative and Qualitative Analysis to Confirm Phase Identities for Large Area EBSD Mapping of Geological Thin Sections” Poster at Microanalysis Society Topical Conference: Quantitative Microanalysis, University of Minnesota, Minneapolis MN, June 2019.
[2] PG Callahan, and M De Graef (2013) “Dynamical electron backscatter diffraction patterns. Part I: Pattern simulations” Microscopy and Microanalysis, 19, 1255-1265.
[3] S Singh, and M De Graef (2016) “Orientation sampling for dictionary-based diffraction pattern indexing methods” Modelling and Simulation in Materials Science and Engineering, 24, 085013.
[4] K Marquardt, M De Graef, S Singh, H Marquardt, A Rosenthal, and S Koizuimi (2019) “Quantitative electron backscatter diffraction (EBSD) data analyses using the dictionary indexing (DI) approach: Overcoming indexing difficulties on geological materials” American Mineralogist: Journal of Earth and Planetary Materials, 102, 1843-1855.
![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.](https://i0.wp.com/edaxblog.com/wp-content/uploads/2023/09/Figure1.jpg?ssl=1)
