The **difficulty** **of solving** **a** Kikuchi **pattern** **depends** primarily not **on the crystal** symmetry but on the **structure**.

**Because of** the always occurring **superposition of** one band by all other** bands**, whole zones of **lattice planes can hardly be separated** even in the Funk transformation. Thus, all bands are excluded from the analysis if their band edges are detected too asymmetrically or even show only one unique band edge.

**SiC:** A typical example are the patterns shown below collected in SiC. Many **bands** **in** the **dominant zones** in the Funk transformation are practically **unusable**. Fortunately, they are also **not necessary to derive** the **Bravais lattice type** and lattice parameters correctly so that it turned out: The phase is **SiC 6H**.

For the similarly looking pattern below the analysis in CALM shows that this pattern describes another phase: **SiC 15R**.

pattern of SiC 15R (800×600 pixels) Funk transformation and solved bandwidths

For such polymorphic phases many major bands do not change their angular relationships at all. Only minor bands require a definition of additional reciprocal lattice points between already existing ones which cause multiplies the translation periods in the crystal lattice. Practically, this often results in an apparent splitting of zone axes (e.g. the left of the two dominant in the 6H pattern). However, the term “splitting” is crystallographically misleading. The new visible bands are simply from physically different lattice planes, wheras the non-changing bands are from the same lattice planes which are, however, differently indexed.

Please note: The rhombohedral cell (**hR**) is** described** in CALM **by rhombohedral axes** **only**, i.e., the three-fold axis is parallel to [111]*. The reciprocal basis vectors ±**a*** || [100]*, ±**b*** || [010]*, ±**c*** || [001]*, are indicated by the red, green and blue circle only shown in the Funk transformation.

Difficult patterns

pyroxene enstatite Nd _{2}Fe_{14}B

Difficult patterns are such with a **minimum of shared zone axes** so that a simple combination of them is not so straightforward, even in a wide angle patterns (pyroxene, enstatite).

Another challenge is displayed by the right pattern where many interesting features are visible but **no**t so many **clearly defined bands**, cf. the pattern of Nd_{2}Fe_{14}B.

The power of point-group symmetry

**Quartz:** As visible from the left image below, even with patterns of 320 x 230 pixel resolution the determination of the Bravais lattice type is possible. The deviation between a and b is only 0.3%, i.e. a/b=1.003. The ratio c/a=1.107 is also only slightly different (1.101) similar to the angles which result to 90.5°, 89.9° and 120.1°.

There is only one small flaw: The symmetry of quartz is trigonal but the Bravais lattice reflects a hexagonal symmetry (hP) and suggests this higher symmetry for the studied phase. However, inspecting the Funk transformation, the trigonal symmetry is visible, despite the limited size captured in the Kikuchi pattern (only 10%…15% of the whole signal).

320×230 pixel pattern resolution “32”-aligned “32” aligned Funk transformation

Unfortunately, the symmetry is not available for all bands, even not for simulated patterns. The incomplete part affects some bands in a way that their profiles become imperfect and too asymmetric so that these bands are automatically ignored in CALM. Nevertheless, the remaining parts are often sufficient to make further conclusions regarding the lower-symmetric hemihedry compared to the **holohedry**** of the lattice**.

*Please note: You can see the three-fold rotation axis in the Funk transformation, although this is impossible in the identically rotated stereographic projection of the pattern*.

Larger lattice parameters?

Due to the smaller **Bragg angles** the bandwidths are **less precise** if the dimension of the unit cell turns out to be comparatively large. *However, this is not quite as tragic, because in principle only one band is needed for scaling. The only question is which one is the most trustworthy. For this purpose CALM uses as many bands as possible, analyzes the distribution and finally uses the mean value. *

As often in life, there is light and shadow. The **advantage of large unit cells** is that the bands hardly influence each other in the Funk transformation, which makes the **determination of** the **lattice parameter ratios** **and** the **angles** between the basis vectors much **easier**.

On the other hand, the **disadvantage** is that the uncertainty in the bandwidth determination increases, i.e. the relative **error in the absolute size of the cell** also **increases**.

*Please note: But it can also happen that misinterpretations of single bandwidths can suggest a wrong translation lattice. Therefore, caution is advised when analyzing narrow bands. All too narrow bands should therefore be distrusted. They are excluded anyway when averaging the scaling factor, but not when defining the Bravais lattice type.*

**Al _{3}Ni:** In the pattern of this orthorhombic phase (below)

**162 bands**

**with**sufficiently symmetric profile edge positions and

**matching**

**widths**have been

**discovered**, cf. Funk transformation.

After solving the Bravias lattice type all projections have been centered for [001]* (pole with a light blue circle in the center of the Fuk transformaton). Left is [100]* (red circled pole) and at the top [010]* (green circled pole).

Al _{3}Ni (orthorhombic), 800×576 pixelsFunk transformation, 169 traces, 162 bandwidths stereogr. projection, “mmm”-aligned 184 reciprocal lattice points, projection alon [001]*

From the stereographic projection of the pattern (bottom left) we recognize that we only see the mirror symmetry parallel (100): the vertical band. We neither see two fold rotation axes nor the two other mirror planes parallel to (010) and (001). Nevertheless, the so presented pattern looks indeed left-right mirror-symmetric. The derived part of the reciprocal lattice (bottom right) indicates the reciprocal unit cell in the center.