For the interpretation of large-angle Kikuchi pattern the periodicity or translation symmetry in crystals is of particular importance. Very advantageously, the **laws **developed in projective geometry for lattices **are** generally valid and **independent of** the particular **crystal symmetry**. All procedures implemented in CALM are thus universally applicable and differ only by the following crystal system-typical constraints.

*Constraints for lattice parameter ratios and the angles between the basis vectors for all seven translation lattices.**A blank space means that there is no condition, i.e., for example, a/b can take any value, but this also includes 1. From this followes: a/b=1 does not exclude orthorhombic, monoclinic and triclinic as lattice solution.* *It only means that if a phase is hexagonal, a/b must be 1.*

Compared to other crystal symmetries such as rotations and roto-inversions, **the translation symmetry** is so trivial that its importance is almost **totally** **underestimated**, although its effect is clear to everyone. Thereby, it often results in surprising consequences, as for example indexing and distribution of the zone axes in following picture shows.

*Regular distribution of projected zone axes [uvw] if the projection plane is parallel to any (hkl). Here a projection plane || (010) is used.**The pattern is simulat*ed.

If the projection plane is parallel to a lattice plane (hkl) the **translation symmetry** existing in the total lattice is responsible that **also in each** single **lattice plane** (hkl). This means, any lattice plane consists of a regular 2D arrangement of lattice points.

*A lattice plane (hkl) can be defined by infinitely many lattice directions [uvw] _{i} which are perpendicular to the normal direction [hkl]*=[uvw]_{p} x [uvw]_{q}. Along the directions [uvw]_{i}, lattice points occur at regular but [uvw]_{i}-specific distances. The regular network of lattice points in the plane must fit to the point distribution along all {uvw]_{i} but also to the whole crystal lattice. The lattice point density of a plane depends on the indexing of (hkl) and is proportional to the interplanar distance d_{(hkl)}.*

*Since the lattice point density is constant, lattice planes with a relatively big distance, i.e., low-indexed (hkl), also have a proportionally high lattice point density. The distance of zone axes in the projection is shorter.*

The same is **valid** for the **reciprocal lattice**, e.g. when in TEM spot patterns are used to illustrate symmetries and orientations, or determine the lattice type or parameters. Reciprocal lattice planes are then (uvw)*, and reciprocal lattice direction are [hkl]*. Both are perpendicular to the direction [uvw] or the plane (hkl) in the crystal lattice.

**Crystal lattice** T **and reciprocal lattice** T* **are inverse** to each other: T = (T*)^{-1}. This reciprocity is continuously used in CALM to optimize T, T* and ‒ if the lattice parameter ratios and angles are known ‒ also PC in order to find the best matching lattice type and parameters.

In Kikuchi patterns the crystal lattice **T** is **indicated by** all **band positions** and their intersections.

In combination **with** the projection center **PC** they define the **3D alignment of** lattice planes **(hkl)** **and** directions **[uvw]**, cf. the above shown pattern. *We know, both planes and directions are defined by lattice points but we have no idea where exactly they are or which distance they have.*

**First**, planes and directions of the reciprocal lattice **T*** can be derived **from (hkl) and [uvw]**:

[uvw]_i \times [uvw]_k = [hkl]^\ast \perp (hkl)\qquad\textrm{and}\qquad [hkl]_m^\ast\times [hkl]^\ast_n = [uvw]

i.e. we know again, where are directions and planes in the reciprocal lattice but also here we still have no idea where the reciprocal lattice points are. They describe the basis vectors and in combination the unit cell.

**Second**, distances in **T*** are **encoded in** the **bandwidths**, cf. Bragg’s law. *The bandwidths define the distances between reciprocal lattice points which described the lengths of the respective reciprocal lattice vectors. This means, the direction is given by the normal direction [hkl]* of the diffracting lattice plane (hkl), and the distance between two reciprocal lattice points is: *

|[hkl]^*|=1/d_{hkl}=\frac{2\theta_{hkl}}{\lambda_e}

Please note: **Only when** the projection center **PC is correct**, **T = (T*) ^{-1}** is valid, i.e., only then the derived [hkl]* describes a translation symmetric lattice in reciprocal space.

The derived distance should preferable by the distance between adjacent points along [hkl]*, but this is not mandatory. It is alsready helpful to know that this is the distance between two lattice points in T*. From the consideration of as many as possible bands the same amount of reciprocal lattice point positions is derived.

For the automated definition of bandedge positions the 1^{st} derivative of each profile is used. This will not deliver a perfect regular arangement of points since even in an perfect description of traces only the directions [hkl]* are exact and the pont positions vary slightly.

CALM now tries to minimize the deviation between a perfect translation lattice and the experimentally derived point positions. This is shown in the figure below.

*The left image displays the arrangement of reciprocal lattice points derived*

*from the Kikuchi pattern of arsenopyrite (*

*projected along [001])*. The centered image reflects the optimized point positions distinguishing already centered points (darker) and points at the unit cell corners (bright). The right displays all lattice points up to a certain d_{hkl}or Bragg angle.From the primitive unit cell of the perfect reciprocal lattice the primitive unit cell of the real lattice will be derived.

Finally, the translation lattice will be analysed, whether it can be described by one of the centered cell.