Evaluation of misindexing of EBSD patterns in a ferritic steel

Authors


T. Karthikeyan. Physical Metallurgy Group, Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India. Tel: +91-44-27480306; fax: +91-44-27480202; e-mail: tkarthi@igcar.gov.in

Summary

The systematic misindexing caused by pseudo-symmetry Kikuchi diffraction patterns in automated Electron Backscatter Diffraction analysis has been studied in a 9Cr-1Mo ferritic steel. Grains with its [1 1 1] directed towards detector centre were found to be prone to misindexing, and the solutions exhibit a relative orientation of ±30° and 60° about the common [1 1 1] axis (as compared to the true orientation). Fictitious boundaries were detected within such grains, which satisfy the Σ3 or Σ13b type coincidence site lattice boundary criteria. Misindexing rate was reduced with more than six detected bands, but 30° rotated solution was comparatively more persistent, as the additional bands of (3 1 0)-type exhibited a nearly good pattern match. Increase in detector collection angle to 0.96 sr or number of detected bands to nine were found to be beneficial in preventing the misindexing problem.

Introduction

The Scanning Electron Microscope (SEM) based Electron Backscatter Diffraction (EBSD) technique has become an important tool for characterising the microstructure of materials (Schwarzer et al., 2009). It has been widely used in study of metallic materials to assess microtexture, orientation relationships and grain boundary character distribution (GBCD) (Randle, 2004; Watanabe, 2011). The procedure of matching the experimentally acquired pattern with the simulated patterns (of the suspected phase) to confirm the phase and its orientation is known as ‘indexing’, and this is automated by an inbuilt-algorithm available in EBSD software (Wright & Adams, 1992; Wilkinson & Hirsch, 1997; Dingley, 2004). Once the calibration settings, such as the location of Pattern Centre (PC) and detector-specimen distance d have been fine-tuned/ensured, the automated EBSD scan experiment over the region of interest can be readily performed. However, optimal choices of various experimental steps are important in obtaining reliable EBSD data (Randle, 2009).

When different crystal orientations exhibit similar Kikuchi diffraction patterns, there is a tendency to misindex (Lloyd, 2000), which is termed as ‘pseudo-symmetry patterns’ in EBSD literature. This needs to be distinguished from ‘pseudo symmetry operation’ used in general crystallography (IUCr Online dictionary, 2011). When a n-fold rotation axis of a crystal is detected as a 2n-fold axis, then there are two variants that are equally likely and this forms the origin of pseudo-symmetry patterns in EBSD and the related misindexing (Vaudin, 2005). Failure to detect line positions with sufficient accuracy to distinguish small differences in the ratio of unit cell is another cause of pseudo-symmetry artefacts in tetragonal materials with c/a∼1 (Vaudin, 2005; Wright, 2006). Geological samples due to their lower crystal symmetry and lower signal strength are more prone to misindexing compared to metals (Prior et al., 2009). The effect of algorithm parameters on the indexing ambiguities have been studied in face-centred cubic (fcc) nickel, trigonal alumina and tetragonal zirconium oxide systems and pseudo-symmetry problems did not occur in fcc crystals for calculations based on eight bands (Nowell & Wright, 2005). In case of body-centred cubic (bcc) crystal structures, nonunique solutions were observed for a small range of orientations, when insufficient number of detected bands or smaller solid angle was covered by the detector (Adams et al., 1993). A roughly etched sample has been reported to exhibit higher misindexing (Takebayashi et al., 2001). In bcc steels, the pseudo-symmetric relationship of 60° or 30° misorientation about the [1 1 1] rotation axis causes misindexing and a ‘checker board pattern’ within grains (Ryde, 2006). When <1 1 1> zone axis is in the middle of the pattern, wrongly indexed solutions with 30° <1 1 1> rotation about the right orientation were observed (Gourgues et al., 2000). Errors in EBSD grain boundary analysis arising out of pseudo-symmetry solutions have been observed/recognized in bcc ferrous alloys, and an unusually high amount of coincidence site lattice (csl) boundaries of Σ3 or Σ13b are observed in such cases (Gupta et al., 2004; Gazder et al., 2008; Karthikeyan et al., 2009; Gazder et al., 2011). Decrease of sample-detector distance, increase of minimum number of bands for indexing, a higher resolution in the Hough space, indexing procedure based on band edges instead of band centre (and decrease of accelerating voltage) have been suggested as possible methods for reducing the incidence of such pseudo-symmetry solutions (Ryde, 2006) and are recommended in EBSD user manual. Misindexing errors are reported to reduce by well-calibrated EBSD system (with accurate PC location and d) and with a higher number of detected bands (at least more than seven for hcp and cubic materials) (Wagner et al., 2002). The methods for overcoming misindexing are known in literature, and this study is aimed at a quantitative assessment of the different strategies for overcoming the misindexing problem in a bcc system of 9Cr-1Mo ferritic steel.

Experiments

A polygonal ferrite microstructure in 9Cr-1Mo (P9 grade) steel obtained by isothermal annealing treatment (1323 K/1 h/step cool to 1023 K/4 h/air cool to room temperature), was utilized in the study. Metallographic specimen with a final electropolishing step (20% perchloric acid in methanol solution, 248 K, 15 V, 20 s) was prepared and SEM-EBSD examination was conducted in a W-filament Philips XL 30 instrument equipped with Oxford Instruments EBSD system (NordlysS detector—40 mm × 30 mm screen size, and Channel 5 software). The specimen was loaded in a pretilt 70° holder, and probed using 30 kV electron beam of about 500 nm size with the working distance held at about 15 mm. Detector camera settings were kept at ‘2 × 2 binning’ (that gives 672 × 512 pixel resolution) and ‘high gain’ (for increased sensitivity). The Hough resolution parameter was set at 80. The EBSD image obtained after background subtraction was solved by comparing it with simulated Kikuchi patterns of bcc iron phase. The number of diffracting planes of bcc iron considered for simulation was set at a default value of 50. Calibration settings were fine adjusted based on 10 reliable solutions in the region of interest before setting up automated mapping scans. As pseudo-symmetry patterns and associated misindexing arise only for a specific narrow range of orientations, several trial scans were performed over different regions in the specimen to first identify the grain that would be susceptible to the problem. The region encompassing such a grain was then selected for further investigation. Several EBSD map scans with a step size of 0.6 μm were carried out in this region, by altering the algorithm settings used in pattern analysis software, and these are listed in Table 1. The scan A denotes the reference scan used for comparing the other scans; detector was inserted 158.5 mm (from its rest position) into the SEM specimen chamber, and the eight bands detected based on band centre lines was used for indexing. In scan B, solid angle covered by the detector was increased by positioning the detector closer to the specimen (by additionally advancing the detector by 2.5 mm). In scan C, the band edges were detected and used in the indexing procedure. In scans D, E and F, the number of bands used for detection and solving of pattern was set as 6, 7 and 9, respectively. The average angular deviation between the observed and the simulated diffracting planes denoted by mean angular MAD is used in qualifying the solution, and the tolerance MAD criteria for acceptance of solution was set as 3°. Channel 5 software was used to process the scan data to output various microtexture results.

Table 1.  Experimental settings used in the different EBSD scans, and the indexing rates.
Scan labelDetector position (mm)Band detectionNo. of bandsIndexing%Mean band contrast
A158.5Centre894102
B161Centre896161
C158.5Edge891 99
D158.5Centre698148
E158.5Centre794 92
F158.5Centre996129

Theory/Calculation

Simulation of pseudo-symmetry patterns

The geometry of a Kikuchi pattern can be interpreted as a gnomonic projection of the crystal lattice on the flat phosphor screen, with band centre lines being the trace of the diffracting plane and the band edges corresponding to the diffracting cones (Schwarzer et al., 2009). Figure 1 schematically illustrates the geometry of specimen and detector positions, and the simulation of diffraction band centres from the spherical Kikuchi map construction. The simulated gnomonic projection of the diffracting planes for bcc is shown in Figure 2, for the case of [1 1 1] direction pointing at the screen centre and the orthogonal direction of [1 –1 0] aligned with the positive horizontal axis. The pattern centre was considered to be at the screen centre, and the solid angle covered within the rectangular detector window is 1.04 sr. The area of spherical triangles associated with the pair of neighbouring corners of the window and the pattern centre were summed to get the solid angle. The reflector intensities based on kinematical theory was used to select the six prominent {h k l} diffracting planes in bcc iron (Table 2), and their positions alone have been considered in forming the simulated pattern.

Figure 1.

Illustration of specimen and detector geometry, spherical Kikuchi map construction of crystal plane trace and its gnomonic projection as straight lines collected over a small solid angle covered by the detector screen.

Figure 2.

Calculated gnomonic projection of the six prominent diffracting planes at the detector when the [1 1 1] zone axis points towards the screen centre for bcc crystal. The [1 –1 0] direction is considered to lie along the horizontal direction, and the lines represent the Kikuchi band centres. The pseudo-symmetry is caused by the symmetrically intersecting bands of (1 1 0) and (2 1 1) at centre.

Table 2.  List of prominent diffracting planes in bcc iron.
Plane familyRelative intensity (%)MultiplicityCumulative count
{1 1 0}100 6 6
{2 0 0} 51 3 9
{2 1 1} 321221
{2 2 0} 23 627
{3 1 0} 171239
{2 2 2} 13 443
{3 2 1} 102467

Cubic crystal possesses a 3-fold rotational symmetry about its [1 1 1] axis, but a higher symmetry is perceived from the intersecting strong lines at the [1 1 1] zone axis of the simulated patterns in Figure 2. In bcc, three lines corresponding to strongly diffracting planes of (1 –1 0), (0 1 –1) and (1 0 –1), termed as (1 1 0)-type lines for brevity hereafter, intersect symmetrically at the centre giving an apparent 6-fold symmetry. Similarly, the three (2 1 1)-type lines also intersect symmetrically at the centre and also bisect the (1 1 0)-type lines. Two cubic crystal orientations that share a common [1 1 1] axis but are misoriented by 60° have such identical distributions at the [1 1 1] zone axis and this forms the origin of pseudo-symmetry (Ryde, 2006). Additional lines nearby must be considered to find the correct orientation, and these are (2 2 2)-type, (3 1 0)-type and (2 1 1)-type that form at a distance of 19.5°, 21.4° and 28.1° respectively from [1 1 1]. Although (3 2 1)-type lines exhibit distribution not prone to pseudo-symmetry, they are less intense (Table 2) and are usually not considered in indexing.

Misindexing due to pseudo-symmetry patterns

In the simulated line pattern of Figure 2, the (1 1 0)-type and (2 1 1)-type lines intersecting at [1 1 1] zone axis look alike, as they are symmetrically distributed with a separation angle of 30°. The (1 1 0) plane has larger inter-planar spacing compared to (2 1 1) and thus would exhibit a smaller band width in EBSD patterns. But, band width differences are usually not considered in indexing procedures, in which case the (1 1 0) and (2 1 1) bands intersecting at [1 1 1] are not distinguished. This gives an apparent 12-fold rotational symmetry, compared to the actual 3-fold symmetry of the [1 1 1] axis. Thus, indexing based on the six (band centre) lines at the centre can give four solutions, that all have a common [1 1 1] axis and the three wrong solutions are rotated by +30°, –30° and 60° compared to the true orientation. Figure 3(a) shows a simulated line position again with [1 1 1] zone axis at screen centre but with [1 -1 0] direction inclined at +10° from the horizontal (this specific orientation was experimentally observed & described in next section). The pattern centre is assumed to be at the screen centre, and the region within the dotted line covers a smaller solid angle of 0.87 sr compared to the outer rectangular window of 1.04 sr. Pertinent lines and zone axis have been labelled in the figure, and the angle between the (3 –1 0) and (3 0 –1) lines is 27.8°. The angular distance of zone axis [1 3 2]/[1 2 3] and [1 3 3] from the [1 1 1] centre is nearly same at 22.2° and 22.0° respectively.

Figure 3.

(a) Calculated gnomonic projection of prominent Kikuchi band centres for bcc crystal with an orientation similar to the grain that showed pseudo-symmetry solutions in EBSD experiment; the inner dotted window region denotes the solid angle captured for the larger detector–sample distance. The pseudo-symmetry solutions are illustrated by superimposed Kikuchi patterns of thin lines, got by angular rotations of (b) 60° (c) +30° and (d) –30° about the central [1 1 1] zone axis. The six lines at the centre are identically positioned for the pseudo-symmetry solutions. The additional bands have a different configuration in case of 60° pseudo-symmetry solution, whereas three of the six (3 1 0)-type lines exhibit a near identical configuration for the ±30° rotations.

The superimposed line patterns obtained on rotation about [1 1 1] axis by +60°, +30° and –30° angles are depicted in Figures 3(b–d). In case of 60° rotated solution, the six central bands have identical distribution, but the positions of the outer bands centres are changed and uniquely different. Thus, indexing based on more than six bands can annul the wrong pseudo-symmetry solution associated with 60° rotation. In case of ±30° rotated solutions, the positions of (1 1 0)-type lines and (2 1 1)-type lines passing thorough [1 1 1] zone axis are exchanged. Although none of the outer bands exhibited identical positions, three of the six (3 1 0)-type lines are seen to assume a nearly identical position after the ±30° rotation, and these lines are marked in the figures. When any of these lines are additionally picked up (apart from central six lines) for indexing, it could still be difficult to find the correct solution. For example, if the additional line was that corresponding to (3 0 –1) in Figure 3(a), it could be falsely identified as (3 –1 0 as its 30° rotated position shows a near match in Fig. 3c), and the angular difference between these lines is 2.2° (30°– 27.8°) only. Similarly, if three additional (3 1 0)-type lines are those of the triplets marked in Figure 3(c) or (d), then the tendency for wrong solution can persist. However, if the outer line of (2 1 1)-type or (2 0 0)-type or the less intense line of (2 2 2) is detected, then the ambiguity in indexing could be resolved. Misindexing problem in bcc materials is thus closely linked with the inherent nature of Kikuchi band distributions—pseudo-symmetry of six intersecting bands/lines at the [1 1 1] zone axis, additional band formation at larger angular distances from [1 1 1] and near similarity of (3 1 0)-type band distributions for the pseudo-symmetry solutions. The effectiveness of different methods to circumvent misindexing problem are presented in the following sections.

Results

Pseudo-symmetry EBSD patterns in 9Cr-1Mo steel

An illustrative experimental EBSD pattern with pseudo-symmetry from the selected grain is shown in Figure 4(a). Although the pattern centre was not at the screen centre, a good match with the simulated output of Figure 3(a) could be observed. When the pattern was solved based on centre line positions of 8 bands, two competing solutions were observed, and these are shown by overlain simulated lines in Figures 4(b) and (c). The first solution appears genuine, as all the 20 lines match with the experimental pattern, whereas the second solution which is relatively rotated by +30° about the [1 1 1] axis shows good match for nine bands only, and these could be recognized using Figure 3(c). Figure 5 shows the observed EBSD patterns when the detector was brought closer by 2.5 mm distance to the specimen. Backscatter electron signals covering more solid angle could be collected and the bands of higher intensity at the periphery regions become accessible. Additional bands (apart from six central bands) corresponding to unambiguous planes of (2 1 1)-type and (2 0 0)-type are preferentially detected instead of (3 1 0)-type and a definitive orientation solution could be obtained. This shows the benefit of collecting the EBSD pattern over a larger solid angle. The solid angle within the window shown in Figures 4 and 5 was 0.83 and 0.96 sr, respectively. If the pattern centre coincided with centre of the screen, then the solid angle subtended by the specimen point at the detector screen would have been larger at 0.87 and 1.04 sr for the two detector positions considered in the study.

Figure 4.

(a) Experimental EBSD pattern (after background subtraction) showing the symmetric band pattern around the (1 1 1) zone axis formed near the centre of the detector screen. Software indexing procedure based on band centre lines of 8 bands giving (b) correct solution and (c) pseudo-symmetry solution.

Figure 5.

(a) Experimental EBSD pattern (after background subtraction) when the detector is placed closer to specimen (scan B settings) and detection of eight bands and (b) Unambiguous solution obtained by indexing procedure based on band centre lines.

Effect of indexing parameters on automated scans

Six areal scans covering the grain that displayed pseudo-symmetry patterns was carried out for evaluating the role of detector-specimen separation distance, band centre/edge detection methods and the number of bands in indexing. The scan settings are listed in Table 1, along with the indexing success rate and mean contrast value of the EBSD patterns. As the different scans were not obtained during a single SEM experiment, they exhibited different mean band contrast values. A better indexing rate as well as enhanced band contrast value was obtained (scan B) when the detector was positioned closer to the specimen. A general trend of higher indexing rates was observed with decrease in stipulated number of detected bands or with better band contrast.

Crystal orientation maps.  The crystal orientation map of the unprocessed EBSD data obtained from the six scans is depicted in Figure 6. The horizontal direction of the specimen was chosen as the reference direction for depicting the crystallographic orientations (using the standard inverse pole figure colouring). The un-indexed points denoted by black pixels were seen at grain boundaries, patchy blemish regions and sometimes as isolated dispersed points. The grain at the centre (marked by grey coloured boundary line in Fig. 6a) exhibited pseudo-symmetry patterns, and misindexing could be readily observed from the sporadic appearances of pseudo-symmetry solutions (blue or purple pixels) within the uniform green coloured grain. The incidence of blue shade pixels did not reduce by using band edge algorithm (Fig. 6c), and the numbers of un-indexed points was also high. On the other hand, uniform (green) grain orientations devoid of pseudo-symmetry solutions was obtained in the scan based on higher solid angle signal collection (Fig. 6b) and scan using nine detected bands (Fig. 6f). Also, the misindexing problem is seen to clearly worsen when the number of bands stipulated for indexing changed from eight bands (Fig. 6a) to seven bands (Fig. 6e) to six bands (Fig. 6d). Pseudo-symmetry orientations (purple) appeared in large numbers in the scan based on six-band algorithm but was absent in the reference scan.

Figure 6.

Crystal orientation map results of the different EBSD scans from the region containing a grain with pseudo-symmetry Kikuchi pattern—(a) scan A (b) scan B (c) scan C (d) scan D (e) scan E and (f) scan F. The colour code denotes the crystallographic direction which is aligned parallel with the X-axis of sample (also the horizontal direction of detector screen).

Pole distributions of (1 1 1).  The orientation data from the grain of interest (identified by grey outline in Fig. 6a) was partitioned, and the distribution of (1 1 1) pole points are depicted in Figure 7. As the misindexing is related to the pseudo-symmetry patterns perceived by the detector, the detector reference axis was used for plotting the pole figure instead of the specimen reference axis. In Figure 1, the line joining pattern centre to specimen point has an inclination of 30° from the specimen normal. Further, pattern centre was positioned approximately 10° vertically away from the screen centre in EBSD patterns (Figs 4 and 5). Thus, a net relative rotation by 40° about the horizontal axis was used to transform reference axis from specimen to detector. The centre of pole figures in Figure 7 thus corresponds to vector direction joining screen centre and specimen, whereas the peripheral point at horizontal horizon of pole figure circle is parallel to the horizontal direction of the detector. A single orientation with a (1 1 1) pole at the centre of the circle could be deduced from the pole figures of Figures 7(b) and (f), whereas the other pole figures exhibited additional points arising from pseudo-symmetry solutions. The pseudo-symmetry solutions share a common (1 1 1) pole at centre of the circle, with the other pole points formed systematically at rotated positions of +30° (blue), –30° (purple) and 60° (again green shade) positions compared to the genuine orientation (green). Only one additional pseudo-symmetry solution of +30° kind was observed for the reference scan (Fig. 7a) and the scan based on band widths (Fig. 7c). A high incidence of the three pseudo-symmetry solutions was observed for the scan based on relaxed algorithm of six bands (Fig. 7d), and the +30° pseudo-symmetry solution was found to persist with increase of number of bands to 7–8.

Figure 7.

Stereographic projection of (1 1 1) poles from the grain exhibiting pseudo symmetry orientation for the different scans—(a) scan A (b) scan B (c) scan C (d) scan D (e) scan E and (f) scan F. The colours correspond to the crystal orientation map of Figure 5.

Grain boundary maps.  The microtexture data from the individual scans was processed using the EBSD software to identify and classify the grain boundaries. Standard noise reduction steps were used to remove isolated misindexed points, and an average orientation was assigned to the un-indexed points (based on its surrounding points). The misorientation between neighbouring pixels was used to identify the presence of grain boundary (misorientation angle ω>15°) and the occurrence of specific coincidence site lattice boundaries of Σ3 and Σ13b based on Brandon's criteria (Brandon, 1966). The ideal misorientation axis-angle set for Σ3 and Σ13b are [1 1 1]-60° and [1 1 1]-27.8° respectively, with 8.7° and 4.2° as the associated tolerances from ideal misorientations. The grain boundary map results are shown in Figure 8, and fictitious boundaries with ‘checker board patterns’ were observed within the grain for the scans prone to misindexing (Ryde, 2006). These false boundaries were also identified to be a coincidence site lattice boundary of Σ13b type (Fig. 8a, Figs 8(c–e)) or as Σ3 (Figs 8d and e). The scanned region contained genuine Σ3 and Σ13b boundaries at other locations, which were correctly identified in all the scans. The interface between the points with a correct solution and a 60° pseudo-symmetry solution automatically satisfies to be a Σ3 boundary. Similarly, the interface between the genuine solution and the 30° pseudo-symmetry solutions satisfies the Σ13b criteria, as the deviation angle of 2.2° is within the Brandon limit. Thus, artificially large amounts of Σ3 and Σ13b boundaries are seen in the scans containing pseudo-symmetry solutions. Experiments with a shorter detector-specimen distance (scan B) or 9 band algorithm (scan F) are found to be useful in avoiding the pseudo-symmetry solution and obtaining the correct estimation of coincidence site lattice (Σ3 and Σ13b) boundaries.

Figure 8.

Grain boundary map results of the different EBSD scans from the region containing a grain with pseudo-symmetry Kikuchi pattern—(a) scan A (b) scan B (c) scan C (d) scan D (e) scan E and (f) scan F. (red-Σ13b, blue-Σ3 and black-other boundaries).

Discussion

Automated indexing procedure consists of several steps: detection of prominent bands using Hough transform, calculation of inter-band angles based on pattern centre and detector distance values, comparison with look-up table of inter-planar angles (of suspected crystal) to identify possible solution(s), simulation of band centres for the suspected solution and measurement of MAD from the experimental pattern to qualify the solution (Wright & Adams, 1992; Adams et al., 1993; Wilkinson & Hirsch, 1997). Although minimum number of detected bands required in automatic indexing is three, a larger number of bands are generally used in the procedure. More bands are particularly useful in finding the genuine solution among the apparent solutions, as in the case of pseudo-symmetry patterns.

In bcc materials, though correct indexing could be achieved based on six bands for most grain orientations, spurious solutions are obtained for orientations that show pseudo-symmetry pattern. Consider the case of crystal orientation with [1 1 1] zone axis close to screen centre and coverage angle of detector is such as to collect only the six central bands and any of the six nearest prominent bands of (3 1 0)-type. If indexing is based on 6 bands, then the intense bands (of (1 1 0)-type and (2 1 1)-type) at the centre would be preferentially selected, and four solutions could be obtained with equal likelihood (scan D). When seven bands are detected for indexing, the additional band of (3 1 0)-type could eliminate the 60° pseudo-symmetry solution, but ±30° pseudo-symmetry solution can persist. Considering Figure 3(a), the angular deviation between the actual (3 0 –1) plane normal and the misidentified (3 –1 0) plane normal (after the 30° rotation) can be calculated as 2.05°. The MAD of genuine solution would be 0°, whereas the MAD of the 30° pseudo-symmetry solution would be 0.29° (2.05° deviation for a (3 1 0)-type band divided by the seven bands considered), and the correct solution should prevail under ideal conditions. However, due to experimental errors (in evaluation of precise location of band centres, calibration setting values at the point of interest), the MAD would have a finite nonzero value even for the correct solution. Then, the wrong pseudo-symmetry solution could sometimes stochastically acquire a smaller MAD and get qualified as a solution of the pattern. When the detection band settings is increased to eight, then depending on the specific choice of two additional bands of (3 1 0)-type, the ±30° pseudo-symmetry solution could recur. As seen from Figures 3(c and d), two (3 1 0)-type bands of the same type (arrow marked) can be picked up with 50% probability. The associated MAD of the competing pseudo-symmetry solution would be 0.51° (2×2.05°/8), and would have a reduced chances of being selected as solution. If 9 bands are considered, the specific set of three bands among the six (3 10)-type bands could be picked up as the additional bands with 10% probability. The associated MAD of the 30° pseudo-symmetry solution is 0.68° (3×2.05/9), and the relative chances for picking up this solution is even less. This explains the reduction of pseudo-symmetry solutions with increasing number of bands. In the above analysis, the (2 2 2)-type bands have been ignored as their intensity is lesser and are thus less likely to be detected as compared to (3 1 0). However, if a (2 2 2)-type band is detected, misindexing would not occur. Figure 9(a) compares the average MAD values obtained within the grain prone to misindexing and the remaining region, for the six scans. Average MAD within the grain was only marginally higher. A typical MAD map (corresponding to scan E based on nine detected bands) is shown in Figure 9, and misindexed points did not show any distinctively higher MAD value. Thus, MAD could not be used to gauge the correctness of solution, as the MAD of the pseudo-symmetry solutions is not high.

Figure 9.

(a) Chart showing the average MAD value obtained within and outside the grain of interest for the six scans (b) MAD distribution map of scan E.

Although (2 2 2)-type band is closer to [1 1 1] zone axis than (3 1 0)-type band, intensity of (3 1 0) bands is 1.3 times higher and hence would be preferentially detected. On this premise, it is possible to assess the set of crystallographic orientations that would be prone to misindexing. Figure 10(a) depicts the simulated gnomonic projection of Kikuchi line pattern with [1 1 1] zone axis at pattern centre and the circular region considered for analysis. Usually, a line segment of certain minimum distance is required for its detection, and for simplification, this distance is assumed to be equal to radius of circle itself here. Then, the angular radius associated with the detection of neighbouring bands of (3 1 0)-type and (2 1 1)-type could be calculated as 24.4° and 31.7°, respectively. Solid angle Ω covered by the spherical cap of angular radius θ is,

image(1)
Figure 10.

(a) Simulated pseudo-symmetry line patterns with [1 1 1] zone axis coinciding with the pattern centre, and the circles of different sizes denoting the region covered around the pattern centre. Circles with angular radius of 24.4° and 31.7° covers the (3 1 0) and (2 1 1)-type lines respectively (when the detected line length is equal to radius) (b) Circle with intermediate angular radius of θ about the pattern centre, and relative position of a (2 1 –1) line and [1 1 1] zone axis. Orientations with [1 1 1] zone axis positioned within the inner circle would be prone to misindexing.

The solid angles covered within the two circular regions of Figure 10(a) are 0.56 and 0.94 sr. If the solid angle covered is more than 0.56 sr, then the 60° pseudo-symmetry solution could be avoided although ±30° pseudo-symmetry solutions may persist. However, if solid angle is greater than above 0.94 sr, then one of the (2 1 –1)-type bands can get detected to give unambiguous correct solution. If the solid angle of the spherical cap is between 0.56 and 0.94 sr, then detection of (2 1 –1)-type band depends on the location of [1 1 1] zone axis from the pattern centre, and the limiting condition for its detection is illustrated in Figure 10(b). The value of α for a specific θ can be solved from the below equations (got from the spherical triangle formulae for angle between its sides),

image(2)
image(3)

When the crystal is oriented such that the [1 1 1] zone axis lies within the inner circle of angular radius α, then the (2 1 –1) line segment would become shortened below the threshold limit for detection and indexing based on (3 1 0)-type lines would be susceptible to ±30° pseudo-symmetry solutions. The probability for any of the eight <1 1 1> directions to lie within this inner spherical cap then represents the fraction of random orientations exhibiting pseudo-symmetry patterns, and is given by,

image(4)

Using the Eqs (1)(4), the probability for occurrence of crystal orientations with pseudo-symmetry as a function of solid angle could be obtained, and Figure 11 depicts the result. It is about 2.5% at a solid angle of 0.56 sr, and reduces to 0% for the solid angle of 0.94 sr. Although the theoretical estimation is based on some assumptions and considers solid angle calculated from circular regions about pattern centre, a good correlation with the experiments is seen. Although the detector placed at farther distance (with a 0.83 sr solid angle within rectangular screen) exhibited misindexing, it was absent when the detector was brought closer to specimen (when solid angle was 0.96 sr).

Figure 11.

Estimated probability for a random orientation of bcc crystal to exhibit a pseudo-symmetry pattern and associated misindexing, as a function of solid angle covered by a circular window region around the pattern centre.

Only a small fraction of crystal orientations are prone to misindexing and is thus generally not observed in most scan experiments, but randomly seen in some scans. It particularly manifests in grain boundary character distribution analysis as a pronouncedly large amounts of Σ13b or sometimes Σ3 boundaries. Elimination of ±30° pseudo-symmetry solution in EBSD experiment is comparatively more difficult compared to 60° pseudo-symmetry solution. Detection of more than nine bands or signal collection from a larger solid angle (>0.94 sr) of Kikuchi sphere is necessary to overcome the problem. Detection of 10 or more bands is possible from EBSD patterns exhibiting good band contrast, but may not be feasible from diffused patterns. Thus, indexing based on such large number of bands would be prone to lower indexing rates. Decrease of specimen-detector distance also has drawbacks in terms of limitation on specimen size and decrease of angular resolution of orientation. The band detection based on band edges was not helpful in preferential selection of (2 1 1)-type bands over (3 1 0)-type bands, and did not reduce the misindexing rate in this study.

Summary

The bcc ferrite grains with its [1 1 1] direction oriented towards the detector centre are identified to be prone to misindexing problem, and the wrong solutions are misoriented by ±30°, 60° about the common [1 1 1] axis. In such patterns, as the outer bands of (3 1 0)-type exhibit a nearly good match with the ±30° pseudo-symmetry solutions, it becomes difficult to overcome misindexing based on 7–8 detected bands. Scan settings that use reduced specimen-detector distance (with associated solid angle of 0.96 sr) or increased number (9) of bands were found to be beneficial in overcoming misindexing.

Acknowledgements

Authors would like to thank Mr. S.C. Chetal, Director, Indira Gandhi Centre for Atomic Research and Dr. T. Jayakumar, Director-Metallurgy and Materials Group, IGCAR for their support and encouragement to pursue this study.

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