For an EBSD pattern taken at 20 kV, the lattice parameters and supposed symmetry are to be determined. The projection center was derived from the pattern of a cubic phase near this position. However, it was not the identical position, so PC can only be considered an approximate description:

PC= [0.575, 0.314, 0.863]

For the manual definition of the initial traces, the gnomonic projection (left) was used. If possible, the bands should be chosen in such a way that the next bands derived from them are also clearly visible. Obviously, the crystal structure of the phase is quite complicated, because at first sight there are no obvious ways to fulfill this recommendation. Thus, with the upper, almost horizontal (green) trace, a band was chosen that is practically invisible. However, there must be a lattice plane there that connects the two very visible zone axes. Two [uvw] always define one (hkl).

The (filtered) Funk transformation (right) represents the same lattice planes by their normal directions [hkl]* (reciprocal lattice directions) in a stereographic projection. At the same time, the zone axes [uvw] from the gnomonic projection (left) turn into magenta-colored great circles in the Funk transformation, corresponding to perpendicular planes with the indices (uvw)* in reciprocal space.*Depending on the PC, the approximately centered empty area in the Funk transformation represents the directional space where a great circle on the projection sphere does not cross the area covered by the detector screen, so that any information is missing.*

Bars and profiles

The black bar that starts in some [hkl]* poles in the Funk transformation describes the asymmetry of the edge positions of the respective Kikuchi band. If the bands were perfect, no bars would appear. On the other hand, if the bars are long, a high asymmetry of the respective band edges is indicated. For the two [hkl]* at the bottom in the Funk transformation that have long bars, CALM displays the following band profiles:

The left [hkl]* with the very long bar represents a very intense Kikuchi band that actually looks quite inconspicuous in the gnomonic projection. However, the intensity profile (white curve) shows a large asymmetry, where the band edges as such cannot really be seen with the naked eye. The maximum, on the other hand, lies quite centrally, but is of no use for edge determination.

For the automatic determination of the band edges CALM uses the 1^{st} derivative of the profile curve and outputs it as a red curve. The red curve also displays the reason for the length of the bar. First and second order reflection are visible on both sides, but on the **left** the **global** **maximum** is the first order, while on the **right** the second order describes the **global minimum**. These extrema are **marked by the green** vertical **lines**.

The right profile is from the hypothetical band, where only the two zone axes were connected to define one of the initial lattice plane traces. This explains why the profile is barely visible. Thus, the proposed band edges (green lines) may not be real at all, or at least overlaid by other bands to such an extent that the effective displacement occurs. However, this is crystallographically unrealistic and only the result of the real measurement.

*The numbers shown in the profile diagrams are self-explaining. The first one (I_ampl) makes a statement about the intensity of the profile. So far, this information is not used further. Q_aver describes the half bandwidth used as experimental Bragg angle. From it the lattice parameter aCALM is derived. The third value q_asym describes the asymmetry. The last two values are given in degrees.*

Obviously, the remaining four bands are nearly perfectly symmetric since no bars are displayed.

Geometric derivation of bands using Goldschmidt’s complication rule

Now we can start connecting previously declared zone axes described by white dots or magenta-colored great circles to define the positions of further bands by their lattice planes. We can and must also declare further zone axes as intersections of already defined (hkl) in order to describe as far as possible all visible bands with symmetrical profiles and to be able to use them during lattice parameter determination. The more bands, the better and also more reliable the result.

*For the definition of [uvw] and (hkl) we initially also used the gnomonic and stereographic projection of the Kikuchi pattern, before we realized that the Funk transformation is much more suitable. There, many more bands become visible through the ring- or eye-shaped features.To connect two points in the Funk transformation – in this case planes in real space – and thus define a new zone axis [uvw], Ctrl+LMB (left mouse button) must be pressed at each position. A connecting great circle will then be drawn automatically, but it must still be confirmed by a double-click close to one of the points. *

*To define new (hkl), the intersection of two selected great circles is neccessary. The selection happens with Shift+LMB. The intersection of the red colored great circles must finally confirmed a double mouse click. Then a new pole is drawn.*

Some minutes later by the use of 69 [uvw] only, where one some are visible in the gnomonic projection but all in the Funk transform as magenta lines, 113 (hkl) have been defined. Some of them are not well or not at all presented by a visible band (see the long bars), but this is not important since these bands are automatically excluded during analysis by CALM.

*Please note: All 113 (hkl) – blue dots in the Funk transformation (right) – are also visible in the gnomonic projection (left) as blue straight lines. Only those four, which are in the empty region of the Funk transformation, do not touch the Kikuchi pattern and are only used for the definition of zone axes required for other (hkl).*

*Please note:* *Indicating the sign of the asymmetric shift, the comparatively long bars either show towards or away from the center. They are a typical sign of experimental patterns where excess-deficiency effects, non-predictable interactions between adjacent bands, but also pattern distortions exist, e.g. caused by local charging on the sample surface. The asymmetry is displayed up to θ_asym=(θ _{max}+ θ_{min})=±0.2°. For bigger deviations the line is red colored and displayed with a fixed (angular) length.*

Althouh there are apparently many bands which are not well described (what is not problematic since they are all ignored and do not affect the result), there is a sufficient number which well match to a perfect crystal lattice.

Bandwidths

After definition of lattice direction [uvw] and reciprocal lattice directions [hkl]* we only know that there are lattice points along them but we do not know where. However, this is exactly what we need to know in order to be able to derive the Bravais lattice type. After definition of the basis vectors we automatically know their lengths, i.e., the lattice parameters.

Therefore, it is now time to derive the positions of lattice and reciprocal lattice points. Presently, we only know the conditions: They both have to match to each other, and they are described by a perfect translation symmetry. To find these positions we select a single band (for [hkl]*: Ctrl+LMB or for (hkl): Shift+LMB) and press F5. *For high-symmetic phases now nearly all bands are described already. Here we have to add some bands manually since the automated procedure is not yet that successful. *

The following two images display the recognition of 91 bandwidths selected by purely crystallographic conditions. The vertical bars display the deviation for each considered band as difference between the geometrically proposed and experimentally derived bandwidth by the 1^{st} derivative of the band profile.

For some bands the profiles are noisy and practically they are all very different from the ideal profiles represented in presentations or books. The more surprising is the fact that the mean lattice parameter a_{CALM} is quite stable, despite the spread observed from individual bands.

*Lattice parameter a _{CALM} =5.78 ± 0.10 Å derived from 51 bands (red dots) which match the angular region 2 ≤ θ ≤ 4.5°. All other bands are excluded since with decreasing bandwidth the edge profiles interfere and deliver an apparent Bragg angle θ_{hkl} which systematically increases the lattice parameter*

The point clowd in the upper diagram might surprize but it simply describes the spread of bandwidths derived by the 1^{st} derivative. Fortunately and in contrast to the spread, the mean lattice parameter a_{CALM} is quite stable even if single bands are removed. The displayed standard deviation is, however, not a good estimation for the error since it only reflects the low precision of single band widths.

The analysis of two further Kikuchi patterns of totally different orientation deliveres 5.79(08) Å and 5.75 (05) Å.

it indicates that the precision of the mean lattice parameter is smaller but this requires also a very excellent knowledge about the projection center position which was for the upper patterns only approximately the same.

*Please note: The accuracy of the basis vector lengths, however, seems to depend on the chemical composition which requires an additional correction.*

Bravais lattice evaluation

After selecting the checkbox in “Lattice solution”, for the first analysed pattern the following candidates for a description of all lattice points are displayed, from higher (top) to lower symmetry (bottom).

Here we have cut out the three solutions and displayed them side by side:

Default is always a primitive triclinic cell as description. Additionally, CALM analysis whether there are higher-symmetric descriptions possible. Maximum deviations from symmetry-specific conditions are typically 1° for angles and 1% for lattice parameter ratios (i.e. a=b is still accepted if 0.99< a/b< 1.01). The maximum deviations should be (clearly) bigger, when PC might be not very good.

Even with the default deviations, CALM discovers for the derived lattice a monoclinic description which is in fact the same as the triclinic, although axes and angles are exchanged or supplementary angles are used. The orthorhombic description keeps one axis and since two basis vectors are defined differently, the angles between have to be different.

Please note: *Since the solutions are related to the same arrangement of reciprocal lattice points, there must be a relationship between all three solutions ! *

Since numbers are often not very helpful and images are more intuitive, respective lattice projections can be used. The best way is to select specific projectins. In crystallography this usually means directions of symmetry axis. Since there is an orthorhombic solution proposed and the lattice has always the highest symmetry, all basis vectors indicate a 2-fold rotation axis with mirror planes perpendicular to each other. The symmetry mmm in its long form expresses this better: 2/m 2/m 2/m.

If we select the orthorhombic solution by double click, it turns into red letters (double click at an empty place in the solution window colors it back to white). Switching off the check box “display” in the Funk transform menu, the possibly hidden stereographic projection becomes visible. There the main axis are indicated by extra circles: [RGB] refers to [a,b,c], i.e. red marks [100], green is [010] and blue [001.

If we select [001] by Ctrl+LMB (left mouse button) and press “c” (for “center”) at the keyboard, [001] will be rotated into the center of the stereographic projection. [100] can be then aligned horizontally or vertically by using the mouse wheel. The projection should then look like the images below, if all checkboxes for traces, poles and bands in the maun menu were unselected. The centered image displays the Kikuchi patterns as stereographic projection, accordingly aligned to the discovered symmetry, i.e. mirror planes are now assumed to be vertically and horizontally aligned. This means that we have practically no real chance to check whether the pattern fullfills the assumed symmetry since there are only two symmetric parts.

monoclinic (triclininc) solution mP mmm orthorhombic solution oA

Regarding the reciprocal lattices, it is clear that the monoclinic/triclinic arrangment (left) shows the same points as the orthorhombic (right). In case of non-primitive lattices (oA) CALM displays for a better clarity centered lattice points slightly smaller and darker, so that both images finally look different since we can recognize which points are assumed to be centered and which not.

The small frames in both images indicate size and alignment of the respective unit cell. Now the relationship between both lattice descriptions is more obvious: the orthorhombic basis vectors can be described by the monoclinic by

**a**_{o}=**c**_{m} , **b**_{o}=½(**a**_{m}+**b**_{m}) and **c**_{o}=½(**a**_{m}–**b**_{m}).

What is real?

The final question is: Is this pattern really orthorhombic or not?

In this case, the diffracting phase is arsenopyrite: FeAsS. The free available American Mineralogist Crystal Structure Database (**AMCSD**) lists five structure descriptions with different lattice parameters for arsenopyrite:

*Am. Mineral.*

**46**(1961) 1448-1469

[2] Bindi, L. et al.

*Can. Mineral.*

**50**(2012) 471-479

[3] Fuess, H. et al.

*Z. Krist.*

**179**(1987) 335-346

[4] Buerger, M.

*Z. Krist.*

**95**(1936) 83-113

A multiple descriptions of apparently the same phase with often deviating descriptions of lattice and structure is actually common. This Table shows, however, impressively what all can happen using a database. *C2 _{1}/d* as well as

*B2*are no space-group descriptions in International Tables for Crystallography, i.e., it is very unlikely that any EBSD software is able to derive the correct structure and thus the correct intensity necessary for the selection of the most dominant bands for indexing. At least the description of [3] has been transformed into the primitive standard setting

_{1}/d*P2*(No.14).

_{1}/cThe tables displays at least that a the result of a centered orthorhombic phase is reasonable, although this is actually a pseudosymmetry since the crystal structure is either monoclinic or even triclinic. Its also demonstrates, that even with XRD a discrimination is not simple, or even under discussion.

Often, multiple descriptions of the apparently same phase only indicates that this phese was either imperfect or, what happens very often in nature, minor contaminations with other elements are responsible for the differences observed.

However, finally it cannot be excluded, that experimental errors or input errors during database import are responsible for incorrect crystal structure descriptions.