computer programs
Crystallographic analysis of the lattice metric (CALM) from single electron backscatter diffraction or transmission Kikuchi diffraction patterns
^{a}Federal Institute for Materials Research and Testing (BAM), Unter den Eichen 87, 12205 Berlin, Germany, ^{b}TU Bergakademie Freiberg, Institut für Mineralogie, Brennhausgasse 16, 09599 Freiberg, Germany, ^{c}AGH University of Science and Technology, Academic Centre for Materials and Nanotechnology, Mickiewicza 30, 30059 Krakow, Poland, and ^{d}Department of Physics, SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom
^{*}Correspondence email: gert.nolze@bam.de, blochwave@gmail.com
A new software is presented for the determination of
parameters from the positions and widths of Kikuchi bands in a diffraction pattern. Starting with a single wideangle Kikuchi pattern of arbitrary resolution and unknown phase, the traces of all visibly diffracting lattice planes are manually derived from four initial Kikuchi band traces via an intuitive graphical user interface. A single Kikuchi bandwidth is then used as reference to scale all point distances. Kikuchi band detection, via a filtered Funk transformation, and simultaneous display of the band intensity profile helps users to select band positions and widths. Bandwidths are calculated using the first derivative of the band profiles as excessdeficiency effects have minimal influence. From the the metrics of possible types are derived for all crystal systems. The measured lattice parameters achieve a precision of <1%, even for good quality Kikuchi diffraction patterns of 400 × 300 pixels. This bandedge detection approach has been validated on several hundred experimental diffraction patterns from phases of different symmetries and random orientations. It produces a systematic lattice parameter offset of up to ±4%, which appears to scale with the mean or the backscatter coefficient.1. Introduction
Up to now, the determination of
or structure parameters from Kikuchi diffraction (KD) patterns has been a topic that has rarely moved into the focus of interest. The reason for this is the high expectation of users who compare results with the standard method for lattice parameter determination, Xray diffraction (XRD). Meanwhile the high automation of XRD techniques makes one forget that even this method has only been automated in the past 2–3 decades to such an extent that users should be able to obtain highly accurate and reliable data after some practice.Bragg angles extractable from electron backscatter diffraction (EBSD) or transmission Kikuchi diffraction (TKD) patterns are significantly smaller and less precise than those of the frequently used Cu Kα radiation in XRD owing to the considerably shorter wavelength of commonly used electron energies. However, the application of electron diffraction is not to be seen as in competition with XRD, but rather as an option when XRD as a method is no longer able to provide the desired results. This case occurs mainly when the fraction of the phase to be determined falls below a reliably measurable value. It is only when minimal precipitates are to be characterized that electron diffraction becomes interesting, and even general statements about the symmetry of the phase or the order of magnitude of the lattice parameters are often sufficient to draw conclusions about the nature of the precipitating phase.
This task has been covered for decades by ; Zou et al., 2004, 2011; Lábár, 2005; Lin, 2014; Yang et al., 2017; Li, 2019). Several of these point patterns as sections through the are necessary to estimate the type, the expected from it and, roughly, the lattice parameters. TEM makes it possible to obtain diffraction information from even the smallest crystals, which in XRD would produce broadened reflections (increased caused by the small grain size, not to mention the small phase amount) due to the longer wavelength, making XRD unsuitable as an investigating method.
(TEM), which with diffraction methods such as (SAED) produce a point pattern equivalent to the distribution of lattice points in (Dorset, 1995But TEM also has certain disadvantages, for example the need to take several diffraction patterns of the same crystallite from different angles, which quickly becomes extremely experimentally challenging. One solution is electron diffraction tomography (Kolb et al., 2008; Zhang et al., 2010).
Another requirement is that the samples have to be electron transparent, i.e. a preparation of very thin samples (about 100 nm) is necessary, which reduces the probability of finding the desired phase in sufficient quantity in the remaining specimen volume. This problem at least could be solved by target preparation techniques like the use of a focused ion beam.
Therefore, even during the early research on EBSD, there was hope that this technique could also be used for locally resolved phase analysis or identification. Over the years there have been repeated attempts to reevaluate the possibilities of lattice parameter or symmetry determination, often after introduction of improved detectors (BabaKishi & Dingley, 1989; Goehner & Michael, 1996; Michael & Eades, 2000; Dingley & Wright, 2009; Saowadee et al., 2017), but these were never followed by commercial success, so that a further dissemination of the developed approaches failed to materialize. In contrast to this, the belief has prevailed that EBSD is a technique which can be used very well for the fast analysis of orientation and phase distribution but which seems completely unsuitable for a crystallographic phase characterization. This is astonishing, because the diffraction patterns of different structures can look very diverse. One reason for the lack of faith may be that a physically based simulation of diffraction patterns was published very late (Winkelmann et al., 2007). It took another 12 years until the intensity information of such simulations was used for the first time to improve the indexing of KD patterns (Wright et al., 2019). So far, all EBSD systems have worked with purely geometrically derived band positions and have been based on the kinematic diffraction theory, which is actually unsuitable for electrons because individual reflectors are used for indexing instead of the entire band information.
The most promising approach for lattice parameter determination and EBSDL (Li et al., 2014) that with only information on the electron wavelength and a single KD pattern a derivation of the is possible (Li & Han, 2015; Han et al., 2018; Han & Zhao, 2018a,b). This includes not only the lattice parameters and types derived from the band positions and widths but also the simultaneous determination of the projection centre. If the projection centre position is correct, the fit between the band positions and widths given on the diffraction image and the crystal and measurement quantities derived from them are perfect.
type derivation so far was provided by a Chinese group around Ming Han. They showed with the softwareWe take up this idea and show that the derivation of the lattice metric from a single KD pattern can be greatly simplified by the specific application of some basic crystallographic principles. This reduces possible errors and is furthermore associated with significantly lower requirements on the resolution of the diffraction patterns as well as the monitor used.
In addition, new visualization tools make the systematic relationships between diffraction patterns and crystallographic laws and rules easier to understand than the approach used in EBSDL of minimizing deviations between individually and independently described band positions and widths and a possible crystal lattice.
We assume, in contrast to EBSDL, that the projection centre is sufficiently known to a first approximation, for example determined from a pattern corresponding to a known phase of the surrounding matrix.
2. General functionality
2.1. Diffraction pattern projection
CALM is a C++ program with its own graphical user interface. It displays the KD pattern as it was originally recorded by the detector and as backgroundcorrected by the used acquisition software. Although the image can be captured by either a round or a rectangular detector screen, the latter has the advantage of using the entire image area of a CCD or CMOS chip [see Fig. 1(a)]. Therefore, the captured sector is effectively larger.
Besides the (b)]. Among other advantages, this presentation makes the captured sector size visible.
as the central projection from a point source onto a flat plane, the KD pattern is also displayed in using the projection centre (PC) position as origin [see Fig. 1A fully automated detection of bands as equivalents for traces of diffracting lattice planes is only possible to a limited extent from KD patterns. The search for an acceptable tool ended with the introduction of the Hough transform, which modifies the linelike into pointlike intensity features whose maxima are comparatively easy to determine (Krieger Lassen, 1992). Unfortunately, typical crystallographic properties such as the affiliation of lattice planes to a zone are no longer so obvious. Therefore, we decided to use an alternative transformation, which to our knowledge was first applied to EBSD patterns by Day (2008). It is based on the Funk transform (Funk, 1915), often also called the spherical Radon transform, which integrates the diffracted intensity projected on a sphere along great circles. Displayed as a and modified by appropriate edge filtering and intensity adjustment, it makes the bands appear as rings (cf. Fig. 2).
In the case of an incomplete diffraction signal such as a KD pattern, the bands form degenerated rings, which are, however, still lined up along a great circle when they belong to a zone. The degeneration becomes stronger the smaller the solid angle detected by the camera. When the bands in the diffraction pattern become even shorter at corners or outer edges, the typical (half) rings look more like an eye that increasingly degenerates into two parallel lines (Fig. 1). But their distance is still equal to the diameter of the ideal circle which indicates the bandwidth.
A major advantage of the adapted Funk transform is that up to twice as many bands can be detected, especially wide bands that are difficult to identify in
but very beneficial for the determination of lattice parameters.2.2. Fixing the lattice plane position
It is generally accepted that the centres of Kikuchi bands mark the imaginary intersection of infinitely extended diffracting lattice planes (hkl) with the detector screen. This is not entirely true, because the gnomonic distortion leads to the fact that with increasing distance from the socalled pattern centre (orthogonal projection of PC onto the detector screen) the displacement of the intersection line from the local band centre increases. This means that the assumption that the Hough peak represents the normal direction of the diffracting lattice plane is suitable for a rough orientation determination but too vague for more advanced analytical techniques like the derivation of lattice parameters.
On the other hand, the point of intersection of lattice plane traces marks a lattice direction [uvw], which is the Lattice planes belong to the same zone when they have an identical i.e. their traces meet at a point. This concept describes the trace definition using the fourline approach formulated by Nolze & Winkelmann (2017) for EBSD. It states that by fixing only four traces, no three of which are tautozonal, the traces of all other visible bands can be inferred without doubt. Thus the error in the trace position is determined by only these four initial traces. Conversely, the setting of the four initial traces can be significantly enhanced by improved matching of all other traces in this way, and most importantly, gnomonic distortions are easy to recognize and to correct. This approach is further extended by another tool which represents the intensity summation and gives the intensity profile of the band as function of the diffraction angle θ (cf. the grey curve in Fig. 3).
The band edge positions (extrema of the red curve indicated by the green lines) are only symmetric to the trace if the trace is centred with respect to the diffraction angle θ. The shift out of the centre is given in Fig. 3 by the value Th.cent., which is for the displayed band +0.012°. Both conditions together, the qualitative match of all band positions and the quantitative deviation Th.cent., allow a sufficient description of the lattice plane traces by shifting and tilting the initial traces only, even for lowresolution patterns.
The match between experimentally derived and ideal lattice planes in 3D depends not only on the trace description but also on a correct position of the projection centre PC. Since PC is always subject to small errors, lattice planes can never be exactly described, even if the trace positions appear to be perfect.
The manual setting of the four initial lattice planes is possible in the
where four straight lines must be drawn as their traces, as well as in the filtered Funk transformation, where the centres of circles can be roughly described as the direction of the normal vector by a doubleclick on the mouse button.The derivation of other lattice planes from the four initial traces, however, is possible in all three projections: gnomonic and stereographic projections of the pattern as well as in the Funk transformation. This is done by combining declared lattice directions [uvw], whereby newly usable [uvw] are created by the intersections of already fixed lattice planes. In the in Fig. 4(b), main zones are characterized by lowindexed zone axes aligned along great circles. These zone axes describe again zones of several bands. From Fig. 4(b) it is also clear that a KD pattern contains practically all information about the entire since the majority of the used zone axes and lattice planes lie far outside the region represented by the KD pattern. Thus, possible symmetries can be analysed even if they are not visible directly in the KD pattern.
In the edgefiltered Funk transform in Fig. 4(c) planes and directions appear opposite. Lattice planes are represented by directions (blue dots), while zone axes become great circles (thin dotted lines) along which all bandforming lattice planes of this zone are aligned. Asymmetric positions of the extrema of the first derivative are displayed as black lines for each given lattice plane. The direction of the black line points to the projection centre, whereas the sign of the angular shift determines whether the line points to or away from the centre. Systematic deviations, as visible in the lower right sector where all lines are directed toward the centre, indicate a slightly imperfect positioning of at least one of the four initial traces. A slight readjustment can help to reduce such systematic shifts.
All three available possibilities of fixing the lattice planes have their advantages and disadvantages. The (a)] similarly aligned bands can easily be mistaken for each other.
allows a rather simple description of lattice plane traces but is unbeatable especially during visual alignment and readjustment of the four initial traces. The human brain is optimized to recognize deviations from parallel alignments so that the is perfect for finetuning. In the case of many overlapping bands, however, most errors occur there, because despite the adjustable magnification of the angular range [zoomout compared with Fig. 1The stereographic projections in Figs. 4(b) and 4(c), on the other hand, offer more systematic approaches, because there zones whose zone axes lie far outside the segment covered by the EBSD detector screen can still be reliably recognized and used. The KD pattern shown in Fig. 4(b), however, only helps to a limited extent owing to its small size, so that a combined use with the is recommended.
The Funk transformation in Fig. 4(c) is unusual because of its inverted character: traces become points and zone axes (points) become great circles. Apart from the lower suitability for the alignment of the four initial traces, it is, however, the most universal possibility for an unequivocal detection of lattice planes. In particular, it shows the bands of lattice planes that one would never find in a even if there was no strong overlapping of bands. In experimental KD patterns of complex but nearly perfect crystal structures (e.g. metal alloys) we discovered up to 160 bands. For the pattern in Fig. 4 more than 130 bands can be identified.
Misinterpretations can be almost completely eliminated by displaying the band profiles simultaneously (cf. Fig. 3). If Th.cent. is greater than for adjacent bands, the probability is very high that an incorrect lattice plane has been assigned to a zone. Similar situations can occur if a pattern is unusually noisy. Therefore, the derived band profile is also displayed so that the user can visually decide whether the deviation is of a systematic nature or results from a band profile shape that is influenced by other strong bands [cf. the profile drawn in black in Fig. 5(d)] which affect the band profile by superimposition.
2.3. Bandwidth selection
The relative metric of the ). This means that the ratios of the lattice parameters as well as the angles between the basis vectors are already now invariant. All directions of the and of the are fixed, although their specific indexing is not yet known since the basis vectors still have to be found. For fixing the absolute lengths of the basis vectors we only need a single reference bandwidth, which is equivalent to the distance of a point from the origin (Nolze & Winkelmann, 2017). The necessary conditions for all other point positions are given by the combination of lattice directions and the translation symmetry, which is of course also valid for the All previously fixed directions and planes must, without exception, be describable by newly discovered lattice and points. From the derived vectors the can be deduced, which enables the determination of the vectors of the e.g.
is in practice already fixed by the trace positions and the assumed projection centre PC (Nolze & Winkelmann, 2017From the basis vectors , , and the angles α_{p}, β_{p}, γ_{p} between them the most probable types and their lattice parameters a, b, c, α, β, γ are derivable [see e.g. Gruber (1973, 1989)].
In CALM the reference bandwidth can be set at any band and can be adjusted at any time, for example to a band that seems more suitable than the band originally selected.
The determination of the band edge by the first derivative was proposed by Saowadee et al. (2017) for an automated tracking of bandwidths. In Figs. 3 and 5(d) the first derivative is displayed as a red curve. In CALM it mainly serves as a decision criterion for the description of a point. If correct, the extrema of the first derivative show a perfect match with the expected positions θ_{B} derived from the width of the reference band and drawn as vertical dashed lines.
In Fig. 5 the crystallographic derivation of a bandwidth is demonstrated for a selected zone defined by intersecting lattice plane traces. The red point marks the projection of the [uvw]. To select the width of the band with the white trace, the bands with the yellow and blue coloured traces are used. The developed procedure searches for the combination of vectors which, on the one hand, satisfies (or adapts) the translation symmetry of the lattice already known in this zone and, on the other hand, correctly represents the interference order(s) visible through the band profile (Nolze & Winkelmann, 2017). There are only a very limited number of combinations that fulfil both conditions. Zone axes with only three intersecting lattice planes are very successful, since it is highly likely that in the case of one known bandwidth the other bandwidths can be immediately derived by vector addition or subtraction. For a better visualization the selected can be virtually rotated into the pattern centre as shown in Fig. 5(b). The corresponding plane is now located in the image plane [Fig. 5(c)]. It contains all points of which we only know the normal directions to (hkl). However, we do not know the distances of the points from the origin, but we know that they must all satisfy the conditions of a translation lattice. The regular line grid, just visible in the background of Fig. 5(c), is derived from the shortest known vectors that have been discovered up to this point in time in this specific plane. The intersection points of the grid indicate possible positions of points. They already describe the translation symmetry either in the plane or of a Newly derived vectors must therefore either match these lattice points or create new lattice point layers, which have to be compatible with the already existing lattice points. In Fig. 5(c) a yellow vector (given by the position of the left yellow sphere) and a blue direction have already been selected. The resulting intersection of the blue with the white line after a virtual shift of the blue line into the yellow sphere is marked as a pink sphere. The distance from the pink sphere to the origin represents the resulting θ_{B} and is given as the dashed lines in Fig. 5(d). In the case of a match with the extreme positions (green lines) of the first derivative the probability is very high that the bandwidth is correctly chosen. Moreover, higher interference order(s) are easy to recognize (cf. Fig. 3). Starting from a band that contains several zone axes and thus ensures access to all other bands, all bandwidths and thus the can be derived step by step.
2.4. type and lattice parameter determination
After transformation of the i.e. possibly matching types need to be tested. This procedure uses purely metric conditions, which are described in detail elsewhere (Le Page, 1982; Macíček & Yordanov, 1992; Grimmer, 2015). Because of small errors in fixing the band traces and PC position, often a whole series of basis vector descriptions results. The number of presented descriptions depends on the maximal deviations entered for α, β and γ as well as for the respective lattice parameter ratios. Although the highestsymmetry description is often the most probable (Müller, 2013), simultaneously also all lowersymmetry solutions are listed in real time. For the KD pattern of cassiterite (Fig. 5) all solutions are given in Table 1.
into a realspace translation lattice its classification is still to be determined,Owing to experimental errors (fixing the initial traces, PC) no angle between the base vectors results exactly in 90°, and the ratio between a and b is not 1 as expected but shows only a tiny deviation of −0.1%. However, even under ideal conditions, solutions can differ considerably from the common lattice descriptions. For orthorhombic crystals, for example, the basis vectors can be permuted without restriction, which naturally leads to different sets of lattice parameter ratios a/b:1:c/b. This seems surprising when used to identify minerals in databases.
Therefore, CALM contains a tool that is valuable for the identification of potentially erroneous solutions caused by wrong band descriptions and therefore incorrect points. It displays all derived points in 3D. Fig. 6 shows the of an allfacecentred F*, which is outlined by the yellow frame. F* always corresponds to a bodycentred point lattice in real space: F* ↔ I. If single bandwidths are incorrectly described, the derived points do not fit the translation symmetry of all other lattice points. Incorrectly assigned bandwidths that do not fit the translation symmetry of the inevitably produce a larger unit cell.
Fig. 6, however, only displays correct points. Missing points are irrelevant. They simply indicate nondetected bands. The omnipresent translation symmetry is assumed to fill all the gaps where points are missing.
2.5. Reducible and unavoidable errors
2.5.1. Experimental errors
The different indicate relationships between the lattices (Müller, 2013). The errors given in brackets are only estimates. They are based on a possible misalignment of the lattice plane traces and simply describe the variation of the lattice parameters when the bandwidth and band alignment are changed. Since the four initial traces are set manually, future improvements can only be expected by automatically adjusting the traces to minimize the errors.
solutions in Table 1A modification of the projection centre position of, for example, ΔPCx = 0.001 would cause a change in the lattice parameters of up to Δa = 0.004 Å, which results in a variation of the lattice parameter ratios of Δ(a/b) = 0.001. The derived lattice parameters or lattice parameter ratios change by a comparable amount in response to a shift or tilt of the lattice plane traces. Owing to the large number of bands taken into account, the precision of the lattice parameter ratios is virtually unchanged even with less well resolved Kikuchi patterns.
2.5.2. Accuracy and precision
To better estimate the accuracy and precision, 34 higherresolution (800 × 576 pixel) and 41 lowerresolution (400 × 288 pixel) KD patterns of corundum (Al_{2}O_{3}, rhombohedral) were examined, all showing arbitrary orientations. The results in terms of both the lattice parameter ratios (a) and the absolute lattice parameters (b) are shown in Fig. 7. With purely manual trace positioning, higherresolution KD patterns do not automatically lead to a significant improvement in the precision of the lattice parameters [Fig. 7(a)]. The reason is the already discussed bandwidth, which can be evaluated for corundum almost loss free even with lowerresolution patterns. Furthermore, it should not be forgotten that only a single bandwidth is necessary, while the others are only used for the discrimination of the type. This means that also for phases with narrower bands only a single wide band affects the absolute scaling of the lattice parameter. Since the widths of the reference bands in all patterns were comparable, the uncertainty in the derived lattice parameters is practically identical. The accuracy of the lattice parameter ratios is in all probability even highe, since position and slope are in principle better fixable for narrow bands than for wide bands.
While the precisions of the lattice parameter determinations on corundum performed on 75 samples are comparable to one another and better than expected for both pattern resolutions, cf. the deviation from the diagonal in Fig. 7(b) of approximately ±0.025 Å (∼ ±0.5%), the accuracy of approximately ±0.1 Å (∼ ±2%) is somewhat lower. But it is also much better than expected, especially if we consider the low pixel resolution of some of the patterns.
More worrying is the size of the absolute lattice parameter determined, which is consistently ∼3% short. This observation suggests that the bandwidth determination by the first derivative does not correlate exactly with the uvw] was applied (Nolze et al., 2020). This actually reduced the deviation from the ideal lattice metric but did not eliminate the observed shift.
Further reasons for the observed offset could be excessdeficiency effects, which might influence the accuracy of the projection centre determination systematically. Therefore, an alternative determination of the projection centre based on four indexed lattice directions [2.5.3. Lattice parameter offset
In order to prevent any uncertainty resulting from varying lattice parameter ratios, for the investigation of the nature of the observed offset only experimental KD patterns of cubic phases and random orientation were analysed. Table 2 lists the 23 cubic phases for which the lattice parameters were determined from KD patterns, some of them with considerably different image resolutions.

The phases are ordered according to the backscatter coefficient η, which is calculated by a polynomial using the Z (Reuter, 1972; Reimer, 1998):
For compounds, the mean value Z_{m} has been taken, which is corrected by the of the respective elements and the stoichiometry. As an alternative correlation, the mean Z_{at} derived from stoichiometry only is also listed in Table 2 but not shown as a separate graphic.
In Fig. 8, the relative deviations between Bragg angles determined experimentally from the first derivative and computed theoretically from databases are shown as a function of the backscatter coefficient η.
Fig. 8(a) indicates that the deviation between the extremum positions of the first derivative θ_{max} and the θ_{B} computed from lattice parameters in Table 1 is different from phase to phase but apparently scales with η. Only for phases with η ≃ 0.35 like GaAs (η = 0.324), BaSO_{4} (η = 0.356) or SnO_{2} (cassiterite, η = 0.372) does the first derivative approximately deliver the absolute lattice parameters correctly. For phases with lower backscatter coefficients the first derivative slightly overestimate θ_{B}, i.e. a < a_{o}, while for stronger scattering phases θ_{B} is slightly underestimated and the derived lattice parameters a, b and c thus appear proportionally larger. Fig. 8(b) proves that the averaged lattice parameter ratio is not affected, which is no surprise since the ratios do not depend on the bandwidths. Moreover, compared with Fig. 7(a) the deviations tend to be smaller because of the higher redundancy for phases of cubic symmetry. Slightly misaligned traces are typically easier to recognize.
Fig. 8 again shows that a high KD pattern resolution does not necessarily lead to an improved accuracy of the lattice parameter determination. It is definitely an advantage, if only because of the resulting lower noise in the band profiles, but smaller image formats obviously provide sufficiently accurate lattice parameters. However, if the image quality is improved by a suitable averaging of lowresolution KD patterns, the results are practically equivalent to those derived from highresolution KD patterns.
2.5.4. Pseudosymmetry
Table 1 suggests that the derived lattice point arrangement can be described by different types. The geometrically necessary conditions are discussed in detail by Grimmer (2015) and Flack (2015) and explain why in Table 1 not only tI but also the subgroups oF, oI, mS and aP are listed. It cannot be excluded that, for example, an apparently hexagonal pattern is not actually formed by an orthorhombic phase. In order to control the consideration of different lattice descriptions, the maximum accepted deviations for angles between basis vectors and for symmetryequivalent basis vector lengths can be varied in CALM.
Comparing the XRD reference data with the derived tetragonal cell dimensions in Table 1, CALM obtained lattice parameters which are about 1% larger than expected: Δa/a ≃ 1.1% and Δc/c ≃ 0.9%. Since for SnO_{2} η = 0.366, this deviation is also expected from the linear approach in Fig. 8(a).
In contrast to the absolute lattice parameters, their ratio shows a much better match:Δ(c/a)/(c/a) ≃ −0.16%. Therefore, for noncubic phases, lattice parameter ratios are often much better suited for phase identification with EBSD than the derived lattice parameters themselves (Dingley & Wright, 2009). It is therefore only important to know the projection centre position and to have very well described lattice plane traces. Bandwidths are of minor worth.
Although the lattice parameters for cassiterite are sufficiently extracted by CALM, a careful look at Table 1 shows that the type is not correctly identified. Instead of the primitive tP, a bodycentred tetragonal tI lattice has been identified (cf. Fig. 6). This misinterpretation has its origin in the of cassiterite (see Fig. 9).
Sn (Z = 50) is much heavier and occupies the same position as the lattice points in tI. Only the remaining O atoms are responsible for the clearly lower symmetry. O atoms (Z = 8) have, however, a significantly lower scattering power so that the visible diffraction signal used in CALM is actually dominated by Sn but considers of course the whole structure of cassiterite. The dynamical simulation of Kikuchi patterns reported by Winkelmann et al. (2007) allows one to separate the intensity contributions of each individual element to the overall pattern. The extracted intensity profiles of Sn and O in Fig. 10(b) confirm the suspicion of a weak signal of oxygen compared with Sn. As expected, the simulated pattern in Fig. 10(a) reveals the correct type tP.
The example of SnO_{2} shows that a diffraction signal can fake higher translation symmetry if lowintensity bands are difficult to identify and therefore points are systematically hidden. In this specific case the lattice parameters are only accidentally not influenced, because all undiscovered translation vectors describe without exception centred points and therefore the basis vectors for tI and tP are identical. However, the resulting primitive, reduced cells are different. In the cassiterite pattern for approximately 1/4 of bands the interference order was wrongly assigned. Their first order actually has only half the as actually fixed.
Similar effects are known for ordered structures where independent atomic positions are occupied by atoms with comparable Z. Typical examples are pseudocubic chalcopyrites and stannites or intermetallics like TiAl or Nibase superalloys. Often the ordering causes multiplied basis vectors, which results in larger unit cells. This results in bandwidths that are narrower than the base vector and longer by the same factor. However, the resulting new first order is often so weak in intensity that it is hardly detectable in the centre of the highintensity band of the now higher interference order.
2.5.5. Sample charging
Additionally to all uncertainties related to band positioning, CALM uses the acceleration voltage and the relativistic de Broglie relationship for the computation of the electron wavelength (Nolze & Winkelmann, 2017). Local charging, however, can easily reduce the energy so that the wavelength λ_{e} might be larger than assumed. If the bandwidth 2θ is taken as a measured constant, then according to Bragg's equation for EBSD,
approximation and projection centre position, one effect is not considered at all: the electron energy which delivers the wavelength.the derived d_{hkl} increases by the same factor as λ_{e} changes by the decrease of the electron energy and suggests larger lattice parameters. This means that the real lattice parameters could be proportionally shorter than those calculated by CALM.
3. Summary and conclusions
CALM is a Windows software that allows the recognition of the type as well as the determination of lattice parameters from a single wideangle Kikuchi diffraction pattern, such as those obtained during EBSD or TKD measurements. The requirements for such analyses are
(i) an EBSD pattern of highest possible quality, but not necessarily of high resolution,
(ii) knowledge about the projection centre (PC) position (alternatively, a reference pattern of a cubic phase collected under identical conditions is suitable to derive PC in CALM directly),
(iii) the applied acceleration voltage, which is also used as electron energy, and
(iv) a monitor resolution of at least 1920 × 1080 pixels.
CALM has a graphical user interface and mainly uses combined keyboard and mouse interactions to fix and derive lattice planes and directions. In this way, all further planes are derived from four selected nontautozonal planes by connecting two already existing zone axes. The derivation of the lattice planes and directions is possible in the original KD pattern, but also in the stereographic projections of the pattern or its Funk transformation. For the manual adjustment of lattice plane and direction alignment the original pattern is best, whereas for a fast and reliable fixing of trace positions the stereographic projections are recommended.
Regardless of the crystal symmetry, only one single bandwidth is required for the calculation of the lattice parameters. All other bandwidths must be describable from this bandwidth, which ultimately determines the
type. This and the lattice parameters are displayed during the selection of the bandwidths.The software has been tested on hundreds of experimental and some simulated patterns for different image resolutions and for phases of different symmetries. The experience gained leads to the following initial conclusions:
(1) The developed algorithms work reliably for any crystal symmetry. Although complicated crystal structures make it difficult to fix the band positions, the selection of the bands is still straightforward.
(2) CALM also enables less experienced users to develop a feeling for the relationship between crystal and Fundamental laws and rules of crystallography can be experienced visually through consistent use.
(3) The applicable image resolution starts at about 400 × 300 pixels, but also smaller images like 160 × 120 pixels enable the recognition of the
type with errors still <5%. Larger images are of course better, but not as much better as their size might suggest.(4) The projection centre position is the key information. The better it is, the higher the correctness of the derived crystal lattice.
(5) With the help of the Funk transformation, far more than 100 (especially wider) bands can be identified in a single highquality pattern.
(6) The band positions in combination with the position of the projection centre already fix the lattice parameter ratios as well as α, β and γ. However, the actual values still require knowledge of any single bandwidth in order to determine the lattice and thus the basis vectors from all subsequently selected bandwidths.
(7) Fortunately, the first derivative of a band profile allows the bandwidth to be determined automatically in a suitable way. Subjective errors are thus also reduced to a single value, similar to the drawing of the band positions using a fourline approach, which is valid for all bands.
(8) The double Bragg angles are not perfectly described by bandwidths derived from the first derivative (Δ2θ < 5%) but can be satisfactorily corrected by the backscatter coefficient η, which is roughly derivable from the background intensity of the original raw EBSD pattern.
(9) Owing to the small sector covered by the detector compared with the total signal, there is a risk that a number of band profiles will be more or less influenced by neighbouring bands or dominant zone axes as well as by excessdeficiency effects in an unfavourable way. Therefore, it must be assumed that the determined lattice parameters will always contain deviations from the ideal values.
(10) Patterns with exclusively narrow visible Kikuchi bands are a challenge because it is difficult to determine the
type and thus the lattice parameters from a single KD pattern. The band edge profile practically has the width of the which makes its correct derivation considerably more difficult.(11) As with all other diffraction techniques,
can become a serious problem. reflections of ordered phases are more likely to be detected in exceptional cases, so that in such cases SAED in a transmission electron microscope is recommended for clarification. In such cases a will be derived.(12) Also problematic are phases in which nonequivalent positions are occupied by different elements with almost the same scattering power. But also clearly different scattering powers can lead to wrong
types, if the significantly stronger scattering element occupies a special and thus suggests a higher symmetry of the lattice.(13) The detectability of minor lattice distortions is limited by the uncertainty in the position of the projection centre and very small errors in the trace positions.
(14) Local charging needs to be prevented. It not only results in an unpredictable overestimation of the lattice parameters but, together with local magnetic fields, can also lead to image distortions.
Interested persons can obtain the software from the authors on request.
Supporting information
Zipped program with manual. DOI: https://doi.org//10.1107/S1600576721004210/nb5295sup1.zip
Acknowledgements
The authors thank Romeo Saliwan Neumann and Michaela Buchheim (BAM, Germany) for their continuous support in compiling the extensive collection of Kikuchi diffraction patterns. Special thanks also go to Austin P. Day and Joseph R. Michael, who contributed to the success of the paper with their valuable comments and advice on the text. Open access funding enabled and organized by Projekt DEAL.
References
BabaKishi, K. Z. & Dingley, D. J. (1989). Scanning, 11, 305–312. CAS Google Scholar
Day, A. P. (2008). J. Microsc. 230, 472–486. Web of Science CrossRef PubMed CAS Google Scholar
Dingley, D. J. & Wright, S. I. (2009). J. Appl. Cryst. 42, 234–241. Web of Science CrossRef CAS IUCr Journals Google Scholar
Dorset, D. L. (1995). Structural Electron Crystallography, 1st ed. New York: Springer US. Google Scholar
Downs, R. T. & HallWallace, M. (2003). Am. Mineral. 88, 247–250. Web of Science CrossRef CAS Google Scholar
Flack, H. D. (2015). Acta Cryst. A71, 141–142. CrossRef IUCr Journals Google Scholar
Funk, P. (1915). Math. Ann. 77, 129–135. CrossRef Google Scholar
Goehner, R. P. & Michael, J. R. (1996). J. Res. Natl Inst. Stand. Technol. 101, 301–308. CrossRef CAS PubMed Web of Science Google Scholar
Grimmer, H. (2015). Acta Cryst. A71, 143–149. Web of Science CrossRef IUCr Journals Google Scholar
Gruber, B. (1973). Acta Cryst. A29, 433–440. CrossRef IUCr Journals Web of Science Google Scholar
Gruber, B. (1989). Acta Cryst. A45, 123–131. CrossRef Web of Science IUCr Journals Google Scholar
Han, M., Chen, C., Zhao, G., Li, L., Nolze, G., Yu, B., Huang, X. & Zhu, Y. (2018). Acta Cryst. A74, 630–639. CrossRef IUCr Journals Google Scholar
Han, M. & Zhao, G. (2018a). Microsc. Microanal. 24, 670–671. CrossRef Google Scholar
Han, M. & Zhao, G. (2018b). Microsc. Microanal. 24, 672–673. CrossRef Google Scholar
Kolb, U., Gorelik, T. & Otten, M. (2008). Ultramicroscopy, 108, 763–772. CrossRef PubMed CAS Google Scholar
Krieger Lassen, N. C. (1992). Micron Microsc. Acta, 23, 191–192. Google Scholar
Lábár, J. L. (2005). Ultramicroscopy, 103, 237–249. Web of Science PubMed Google Scholar
Le Page, Y. (1982). J. Appl. Cryst. 15, 255–259. CrossRef CAS Web of Science IUCr Journals Google Scholar
Li, L. & Han, M. (2015). J. Appl. Cryst. 48, 107–115. Web of Science CrossRef CAS IUCr Journals Google Scholar
Li, L., Ouyang, S., Yang, Y. & Han, M. (2014). J. Appl. Cryst. 47, 1466–1468. Web of Science CrossRef CAS IUCr Journals Google Scholar
Li, X.Z. (2019). Micron, 117, 1–7. CrossRef CAS PubMed Google Scholar
Lin, K.L. (2014). Microsc. Res. 02, 57–66. CrossRef Google Scholar
Macíček, J. & Yordanov, A. (1992). J. Appl. Cryst. 25, 73–80. CrossRef Web of Science IUCr Journals Google Scholar
Michael, J. R. & Eades, J. A. (2000). Ultramicroscopy, 81, 67–81. Web of Science CrossRef PubMed CAS Google Scholar
Müller, U. (2013). Symmetry Relationships Between Crystal Structures: Applications of Crystallographic Group Theory in Crystal Chemistry. Oxford University Press. Google Scholar
Nolze, G., Tokarski, T., Cios, G. & Winkelmann, A. (2020). J. Appl. Cryst. 53, 435–443. CrossRef CAS IUCr Journals Google Scholar
Nolze, G. & Winkelmann, A. (2017). J. Appl. Cryst. 50, 102–119. Web of Science CrossRef CAS IUCr Journals Google Scholar
Reimer, L. (1998). Scanning Electron Microscopy: Physics of Image Formation and Microanalysis, 2nd ed. Berlin, Heidelberg, New York: SpringerVerlag. Google Scholar
Reuter, W. (1972). 6th International Conference on Xray Optics and Microanalysis, p. 121. University of Tokyo Press. Google Scholar
Saowadee, N., Agersted, K. & Bowen, J. R. (2017). J. Microsc. 266, 200–210. CrossRef CAS PubMed Google Scholar
Winkelmann, A., TragerCowan, C., Sweeney, F., Day, A. P. & Parbrook, P. (2007). Ultramicroscopy, 107, 414–421. CrossRef PubMed CAS Google Scholar
Wright, S. I., Singh, S. & De Graef, M. (2019). Microsc. Microanal. 25, 675–681. Web of Science CrossRef CAS PubMed Google Scholar
Yang, Y., Cai, C., Lin, J., Gong, L. & Yang, Q. (2017). Micron, 96, 9–15. CrossRef CAS PubMed Google Scholar
Zhang, D., Grüner, D., Oleynikov, P., Wan, W., Hovmöller, S. & Zou, X. (2010). Ultramicroscopy, 111, 47–55. Web of Science CrossRef CAS PubMed Google Scholar
Zou, X., Hovmöller, A. & Hovmöller, S. (2004). Ultramicroscopy, 98, 187–193. Web of Science CrossRef PubMed CAS Google Scholar
Zou, X., Hovmöller, S. & Oleynikov, P. (2011). Electron Crystallography: Electron Microscopy and Electron Diffraction, IUCr Texts on Crystallography, Vol. 16. Oxford University Press. Google Scholar
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