DE102013102739B4 - Method of analyzing a multiphase, crystalline sample and a point-group-specific polyhedron - Google Patents

Abstract

A method for producing a polyhedron, comprising: - selecting a crystal class; - Creation of at least in a partial area asymmetrical patterns on the polyhedral surfaces of a regular cube (2) or a hexagonal bipyramid (3) or a corresponding pattern sheet, such that the arrangement of all symmetry elements of the crystal class is represented by corresponding symmetries of the patterns on the polyhedral surfaces, and that all crystallographically equivalent directions are described by the same, unique color according to a direction-dependent color coding (21).

Description

The invention is in the field of crystallography and materials science.

In crystallography one differentiates between seven crystal systems, which are mostly defined by the metric of a unit cell polyhedron and result from the combinatorics of three-dimensionally arranged symmetry elements. It is known that only 7 unit cell polyhedra or crystal systems (triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal and cubic) result from the 32 possibilities of the symmetry element arrangement (32 crystal classes). Each of these seven unit cell polyhedra therefore represents several symmetry element combinations. In mineralogy and crystallography, the resulting problem of ambiguous interpretability can be circumvented by using at least one so-called "general surface shape" for representation, which is then actually point-symmetry-specific. However, these surface models require bodies with up to 48 surfaces. However, use of such bodies is impractical for material analysis of multiphase, crystalline samples by means of EBSD measurements (electron diffraction of backscattered electrons) or X-ray structure analysis, since in addition to the large number of surfaces, the crystallographic indexing of the surfaces of the bodies can be very similar, which symmetrical, cubic phases can lead to misinterpretations. In addition, teaching aids with corresponding bodies (polyhedrons), as used for the training of crystallographers, mineralogists, physicists, chemists, materials scientists, etc., are very expensive.

Polyhedra with colored or patterned polyhedron faces are used in puzzles. This is how the document describes US Pat. No. 6,439,571 B1 a magnetic puzzle with cubes that have colored, truncated pyramidal protuberances on each of two diagonally arranged quadrants of their cube faces. In the publication US 3,788,645 is a mathematical cube puzzle with four separate cubes with colored cubes described. The publication US 4,600,199 describes a three-dimensional toy puzzle with colored polyhedron surfaces. In the publication US Pat. No. 6,422,560 B1 a puzzle cube is described with one image on each cube surface. A cube puzzle with z. B. 27 cubes that have two-color pattern on the cube surfaces, and can be assembled into a cube, is also from the document GB 2 461 888 A known. In addition, in the document GB 2 373 738 A a puzzle cube consisting of eight cuboids and whose cube sides may have images or patterns described. The publication WO 2010/005 635 A2 describes logical puzzles based on polyhedron elements whose polyhedral surfaces can have colored elements. The publication WO 2012/027 735 A1 describes a cube-puzzle game of interconnected tetrahedrons whose surfaces can be colored. However, the cited documents do not deal with crystallographic issues or the appropriate representation of the symmetry element arrangements.

Against this background, a method for producing a polyhedron according to claim 1, a punctiform specific polyhedron according to claim 3, a teaching aid set according to claim 4 and a use of point group specific polyhedra according to claim 6 are given.

A set of polyhedra can be used to train crystallographers, mineralogists, physicists, chemists, materials scientists, and so on. For each of the 32 crystal classes, a set of teaching aids comprises a polyhedron of polyhedron surfaces with patterns, the patterns being asymmetric at least in a partial area.

A set of teaching aids for colleges and colleges includes for each of the 32 crystal classes a polyhedron of polyhedral surfaces with patterns, the patterns being asymmetric in at least one subset. Typically, each of the polyhedra is characterized by a particular surface pattern on its polyhedron surfaces that describes all crystallographically equivalent orientations by the same unique color corresponding to directional color coding and alignment of all the symmetry elements of the particular crystal class by corresponding symmetries of the particular surface pattern. By means of such a set of polyhedra, the symmetry properties can be conveyed particularly well. In addition, the teaching set can be made much cheaper than the commonly used and preferably made of wood crystal models. Instead of the complex polyhedron body of the commonly used crystal models of the teaching set described herein comes with 25 cubes and seven achtflächigen bipyramids.

A method for generating a polyhedron comprises selecting a crystal class and generating at least in a subarea asymmetrical patterns for the polyhedral surfaces of a regular Cube or a hexagonal Bipyramide or a corresponding Schnittmusterbogens such that the arrangement of all the symmetry elements of the respective crystal class is represented by corresponding symmetries of the patterns on the polyhedral surfaces and that all crystallographically equivalent directions are described by the same, unique color according to a directional color coding. The generating may include applying the crystal class symmetry elements to the asymmetric patterns. The fact that only simple bodies such as the regular hexahedron (cube) or a hexagonal bipyramid with only twelve polyhedron faces - which can be traced back to the hexahedron in its shape as a sectional figure of two cubes twisted by the diagonal by 60 ° - can be used on one a simple and / or cost effective way of providing a real or virtual set of 32 point group specific or crystal class specific polyhedra.

The selection of the crystal class as well as the representation of the corresponding polyhedron can be made by a sample analysis system such as an EBSD system. For example, the polyhedron can be displayed on a connected screen of the sample analysis system.

According to yet another embodiment, the method comprises setting a direction-dependent color key for the selected crystal class, z. As a corresponding RGB color key, generating the asymmetric at least in a partial area pattern as patterns representing a crystal direction corresponding to the directional color key, and generating the asymmetric at least in a partial area pattern on the polyhedral surfaces of the regular hexahedron or the hexagonal bipyramid ,

In the production of real point group-specific polyhedra, the production of the asymmetrical patterns, at least in one subarea, on the polyhedral surfaces can take place, for example, by printing or laminating. As a result, the point-group-specific polyhedra can be produced very inexpensively. The generation of the asymmetric at least in a partial area pattern can also be on a the regular hexahedron or the hexagonal bipyramid corresponding cut arc, z. B. by imprinting done.

As a result, the point group specific polyhedra can be made extremely inexpensive and so, for example, students are provided as a set of 32 printed pattern sheets. However, it is also possible to generate the point group-specific polyhedra in a 3D printer.

A dot-group-specific polyhedron includes polyhedron surfaces with color-coded patterns, the patterns being asymmetric at least in a partial region. The patterns on the polyhedral surfaces are characteristic of the unique arrangement of the symmetry elements of a crystal class. Such a polyhedron can be displayed on a screen and assist in finding misallocations in the analysis of a multi-phase, crystalline sample. In addition, the point group specific polyhedron may be part of a teaching resource set. Depending on the crystal class (point symmetry group), the polyhedron is typically a cube or a hexagonal bipyramid. In the present case, the terms "crystallographic point group" "point symmetry group", "point group" and "crystal class" are used synonymously.

Further advantageous embodiments, details, aspects and features of the present invention will become apparent from the dependent claims, the description and the accompanying drawings. It shows:

1 in the left part of a schematic, perspective view of a cube 1 and the maximum possible number of symmetry elements, and in the right part of the figure an azimuthal projection of the symmetry elements according to an embodiment;

2A perspective views of dot-group specific polyhedra with color-coded patterns on the polyhedron faces as well as associated symmetry elements according to several embodiments;

2 B Pattern of polyhedron surfaces of dot-group specific cubes according to several embodiments;

3A perspective view of two point group specific polyhedron with color-coded patterns on the polyhedral surfaces according to two embodiments;

3B a perspective view of a point group specific polyhedron with color-coded patterns on the polyhedral surfaces according to another embodiment;

3C a perspective view of a point group specific polyhedron with color-coded patterns on the polyhedral surfaces and a coordinate system according to an embodiment;

4 perspective views of point group specific polyhedra with color-coded patterns on the polyhedral surfaces and associated symmetry elements according to several embodiments;

5 perspective views of point group specific polyhedra with color-coded patterns on the polyhedral surfaces and associated symmetry elements according to several embodiments;

6 perspective views of point group specific polyhedra with color-coded patterns on the polyhedral surfaces and associated symmetry elements according to several embodiments;

7A perspective views of point group specific polyhedra with color-coded patterns on the polyhedral surfaces and associated symmetry elements according to several embodiments;

7B perspective views of point group specific polyhedra with color-coded patterns on the polyhedral surfaces and associated symmetry elements according to several embodiments; and

7C perspective views of point group specific polyhedron with color-coded patterns on the polyhedral surfaces and associated symmetry elements according to several embodiments.

The symmetry of crystals is represented in different ways so far. The 32 point symmetry groups or crystal classes define the spatial orientation of symmetry elements, ie the axes of rotation, rotation inversion axes, mirror planes and the center of symmetry or inversion. Their representation is then usually in the form of an azimuthal, z. As the stereographic, projection of grid directions and planes that are parallel to the symmetry axes or mirror planes. The standard projections used are usually defined so that the reference directions of the projection coincide with as many base vectors of the grid as possible. However, since the determination of the basis vectors is determined by the position of the symmetry elements, the orientation of the symmetry elements in the standard projections is also fixed. For a cube 1 is the arrangement of possible mirror planes and axes of rotation (symmetry elements) in 1 shown. All symmetry elements run through the origin of coordinates (point symmetry group). In addition, in the left part of the figure, the rotation axes and planes of the cube 1 represented by assuming the maximum possible symmetry. However, the shown spatial representation of the symmetry alignment has disadvantages in terms of exact quantification and also has a whole series of occlusions. To limit these drawbacks, projections such as the conformal stereographic or equal area projection have been developed. The right part of the figure 1 shows the corresponding equal-area projection of spatial symmetry element alignment. In 1 the usual crystallographic symbols are used for the two-, three-, and fourfold axes, and in the left part of the figure they are also marked in color as they are for the point group m 3 m also be used for the color key introduced below. The mirror planes m are also according to their orientation - ie if m = 2 parallel to <110> (green) or to <100> (red) - colored differently.

The angles between the surfaces have proved to be much more characteristic than the area size, shape and formation of crystals (Steno's law of angular constancy). A distinction is made between special and general crystal surfaces. The surfaces whose normal with at least one symmetrical element coincide in such a way that the surface covers itself by the operation are considered to be special. In other words, the symmetry operation can not be explicitly recognized on this surface. Will each of the in 1 shown symmetry elements on the cube 1 applied, then the cube becomes 1 aligned with yourself. The example of a cube makes it clear that only 6 of the 48 symmetry elements are sufficient to produce the cube from a single cube surface. The other 42 symmetry elements are at least for the cube 1 not mandatory. On the other hand, with a general surface {hkl}, every existing symmetry operation leads to a new surface, ie in the case of 1 illustrated symmetry element arrangement would have the general surface shape 48 surfaces.

As can be seen from the respective symmetry element orientation, the cube exists as a special shape in all cubic point groups. Although the cube as a polyhedron gives itself, which symmetry could exist, it leaves more possibilities open. This will be explained using the example of pyrite which can also crystallize in cube form, but shows a characteristic of the pyrite stripe parallel to <100>, which does not match the 90 ° rotational symmetry of the cube face, which excludes all point groups with fourfold symmetry axis (m 3 m and 432). The respective twisted direction of the striation on the side surfaces, however, speaks at least for the existence of a threefold rotation axis parallel to the cube diagonal, which identifies the crystal as a cubic with high security. If that were not so, only an orthorhombic crystal class would remain as symmetry, or a very special tetragonal one. Thus, the point symmetry groups remain m 3 m, m 3 and 23 as candidates. Since further, external characteristics are missing, a distinction of these three crystal classes is not possible, so that following the rule the highest possible symmetry 4 3m which is wrong in this case, because pyrite crystallizes in symmetry m 3 , This example makes it clear that it is only by means of a highly symmetrical crystal metric that one can not unambiguously determine the point group symmetry and, indeed, the crystal system. For the representation of single-orientation measurements by means of EBSD, this means for lower-symmetric cubic phases that misinterpretations can not be distinguished on the basis of the previously used standard representation of cuboid edges or homogeneously colored cubic surfaces. The cube, as a visualization of the elementary cell orientation, as a special form always fakes a symmetry that applies only to the most symmetric cubic crystal class.

In addition, with the often extremely fast EBSD measurements, the distinctness of the five cubic point groups practically approaches zero because the band intensities are not used for the orientation determination. In most, albeit numerically few measurements, even between the centrosymmetric Laue groups m 3 m and m 3 practically indistinguishable because the visible differences can not be detected beyond doubt.

The use of elementary cell polyhedra leads in existing EBSD software packages to the two problems already mentioned, that visually not between the 11 Laue classes, but only between the 7 crystal systems can be distinguished and that misinterpretations can not be reliably detected in pseudo-symmetries.

As an alternative to the cubic or rather elementary cell representation for single orientation measurements, as already mentioned above, the use of a polyhedron corresponding to a general shape {hkl} would be conceivable. Although this would be a unique variant, as it would show completely different polyhedra for each point group, but their use would be very confusing because of the large number of sites, especially for the inexperienced. Especially with complex polyhedra, different orientations would be difficult to distinguish. In addition, the orientation of the crystallographic basis vectors will never be immediately apparent from the general form.

2A shows a schematic perspective view of cubes, each representing a cubic point symmetry group (Kristallkasse), and the spatial orientations of the associated symmetry elements, which are also referred to below as the symmetry coast. The number in brackets indicates the frequency of the area, the inverse value of which indicates the size of the fundamental zone (fundamental zone, asymmetric unit) in relation to the entire room.

dreizählig:

vierzählig: 4 ≡ ♦ und sechszählig: hier nicht vorhanden, und Drehinversionsachsen (dreizählig: 3 ≡ Dreieck mit Punkt, vierzählig: 4 ≡ Viereck mit Schrägstrich und sechszählig: hier nicht vorhanden) auf. Die Spiegelebenen ( 2 ≡ m) sind durch transparente Ebenen dargestellt.The symmetry towers are made up of rotation axes (twofold:

trifoliate:

fourfold: 4 ≡ ♦ and sixfold: not present here, and three-turn axes (threefold: 3 ≡ triangle with point, fourfold: 4 ≡ square with slash and sixteen: not here) on. The mirror planes ( 2 ≡ m) are represented by transparent planes.

The spatial orientation of the associated symmetry elements is reflected by the definition of the color gradients on the cube surfaces (polyhedron surfaces). The colors used correspond to the color coding of the direction of the respective symmetry element or the direction of the normal of the mirror plane. When the symmetry elements are described by one of the primary colors, they define the fundamental zone. Different colors mean that the respective symmetry element lies between these corner points on the edge of the fundamental zone. Therefore, in the example, they can only be either yellow, turquoise, or purple. If color jumps occur at the corners or edges, then one of the colors is assigned to the symmetry element, such that the color of the symmetry element does justice to the point symmetry.

However, the complementary representation of the symmetry elements to the cubes with polyhedral surfaces having color-coded patterns (color distributions) is only optional. The color distribution on cube, polyhedron or elementary cell surfaces is sufficient, different symmetries despite the same metric delimit each other, since the totality of all network levels z. B. by an RGB color code (RGB color key) or CMYK color code (cyan magenta yellow black) can be described. The respective symmetry then produces a distribution of the colors, which - like the general surface form - is unique for each symmetry. In other words, the alignment of all symmetry elements of a crystal class is represented by corresponding symmetries of the patterns of the polyhedral surfaces.

In the representations of the 2A , and 3A to 7C In most cases, the RGB color key described below was used, where R, G, and B represent the colors red, green, and blue, respectively. The in the 2A , and 3A to 7C framed point group are centrosymmetric groups (Laue groups), so have an inversion center.

wobei A die Kristallmatrix ist, die für die Definition, dass die Kristallachse a →₃ = c → ∥ e →₃ ist, die folgende Form annimmt:

In the exemplary embodiment, for the mathematical representation of the color of a crystal direction R → or of a reciprocal vector H → in a first step, a transformation of the vectors into a Cartesian reference system e → _i performed:

where A is the crystal matrix used for defining that the crystal axis a → ₃ = c → ∥ e → ₃ is, taking the following form:

With a, b and c, the lengths of the crystal axes (basis vectors) a → _i and with α, β, γ the angles between the crystal axes a → _i denoted (lattice constants).

Then the vertices of the color key, z. B. (h, k, l) ₁ ∥ G, (h, k, l) ₂ ∥ B, (h, k, l) ₃ ∥ R analogous to equation (1) as Cartesian coordinates converted and normalized, ie (h , k, l) ₁ → [x, y, z] ₁ .

die direkt für die Berechnung des RGB-Farbwertes eingesetzt werden kann:

wobei X ' / max den Maximalwert der drei Komponenten X ' / i des Vektors X →' beschreibt.From this a transformation matrix T can be created:

which can be used directly to calculate the RGB color value:

in which X '/ max the maximum value of the three components X '/ i of the vector X → ' describes.

Alternatively, other color keys can be used, which are usually more complicated, but can have a somewhat more homogeneous color sensitivity, which can further improve visual recognition.

In further embodiments, to set a directional color key to produce color-coded asymmetric patterns instead of the colors red, green and blue (additive), the colors yellow, cyan and magenta are used (subtractive).

In yet further embodiments, a two- or four-color color key (CMYK) using the respective crystal matrix A is used to produce color-coded asymmetrical patterns.

Due to the use of the respective crystal matrix A, it follows that the color value for a vector X → derived from a grating vector (hkl) or a reciprocal grating vector [uvw] depends not only on (hkl), respectively [uvw] and the point group symmetry, but is also a function of the lattice constants. Only for the often studied cubic case, or the {0 0 1} zone of tetragonal, hexagonal, and trigonal crystals, and the {1 1 1} zone of rhombohedral crystals are the lattice constants irrelevant. For the identical vector X → and a defined point group, however, the color value is always constant. The mentioned influence of {hkl} is compensated by the crystal matrix and is only mentioned here to illustrate that, unlike X →, the identically indexed lattice vectors of different phases are represented by different colors.

How out 2A As can be seen, three of the five point groups have the same surface frequency of 24. For the fundamental zones of the three point groups, this means that they have the same size. In general, this means that groups with the same surface frequency have the same size as the fundamental zones, but they can differ in shape and spatial distribution.

The size (area frequency), shape and orientation of the space or area segment (fundamental zone) is usually clearly defined by the Symmetrieelementausrichtung. It defines a space segment, which in many cases is bounded by symmetry elements at corners and / or edges and can not contain any further symmetry element in the interior. The color gradient character can be additionally set by assigning which corner to which color. Often, the higher symmetric point groups (ie, except triclinic and monoclinic) define red for the [001]. Depending on the vector to be described, each mixed color is exactly defined. Color keys that do not consist of three corner vectors - eg. B. two or four - are to be distorted accordingly.

It is also possible to use black and white patterns on the polyhedral surfaces instead of the color distributions. As a result, symmetry considerations can also be made or simplified for color deficient ones. This is exemplary in 2 B for each one cube area of three selected cubic point groups. In comparison to 2A It becomes clear that the fundamental zone used there may differ from that of the corresponding black and white pattern assignment, ie that it can not be imaged on them. Thus, for the point groups 432 and 23 in 2 B used surfaces deviate from the rectangular shape, since it depends only on the surface symmetry.

With uniform, symmetrical color patterns on the polyhedron surfaces or gray scale patterns, the maximum symmetry of the point group can be clarified. By contrast, the use of color-coded, asymmetric patterns for the fundamental zone also allows the general description of the direction of a vector and the anisotropy.

The comparison of 2A and 2 B also shows that the fundamental zones do not have to be one-to-one. Thus, for the groups of points 23 and 432, there are infinitely many fundamental zones which may even be limited by curved curves.

For the group of points 432, the colors selected in the 3A and 3B three cubes 2 . 2 ' . 2 '' with color-coded patterns on the polyhedral surfaces (selected space segments), since they can be generated by simple operations, ie applying the symmetry elements to subpatterns.

The difference in the color gradients of the two cubes 2 . 2 '' is that the asymmetric volume in the projection of the in 3A shown cube 2 is a square (that is, has four vertices, (dashed lines) while it is in 3B consists of an isosceles, right-angled triangle (dashed line). That both fundamental zones are equivalent from the point of view of symmetry can be seen from the fact that z. B. when using a 90 ° rotation to the surface normal in both cases, the color gradients come without overlap with itself to cover. Even a 120 ° rotation around the respective cube diagonal results in the same color distribution before and after the rotation. The same applies to the application of the twofold rotation axes ∥ {110}. Due to the lack of mirror planes ∥ {110} occur in the cubes 2 . 2 '' Color jumps on. However, the color jumps are in this case practically not relevant, especially not for EBSD representations, since the point group there automatically by the Laue group m 3 m is replaced. Within the currently achievable accuracy for fast measurements, this is generally accepted. In the case of the point group m 3 but that would not be the case. By suitable defined color keys, however, can at least largely avoid color jumps, as in 3A shown cubes 2 ' shows that like the cube 2 of the 3A has a square fundamental zone.

The hexagonal crystal system presents the only groups of points that can not be represented by a (complete) cube because there are no six-numbered rotation or rotation in the cube. This also results in additional directions for twofold axes and mirror planes that do not exist in the cubic system. Also, the mirror plane arranged perpendicular to the cube diagonal is characteristic of the hexagonal symmetry and does not occur in the cube itself. However, the description of the hexagonal point symmetry groups can be made by hexagonal bipyramids 3 , which use, for example, half of all polyhedral surfaces of a circumscribing cube, which was shown for didactic reasons. The number behind the respective point group symbol again indicates the area frequency. The framed representations mark the centrosymmetric point groups (Laue groups).

It is also in 4 for ease of understanding in addition to the point group specific bipyramids 3 again their respective Symmetrikelüst shown. As the color distributions for the individual symmetry groups show, five out of seven more than half of the point groups are characterized by discontinuities in the color gradient. This also applies to the Laue group 6 / m, so that for phases of this symmetry in the EBSD orientation maps locally apparent orientation jumps are displayed, which are none at all. By a parallel representation of the point group specific polyhedron and the orientation map, which are based on the same directional color key, apparent orientation jumps can be better recognized.

EBSD orientation maps can, for example, be used to visualize EBSD data. Since not the crystal with respect to the sample coordinate system, but - inverse (inverse) - a sample axis is described with respect to the crystal coordinate system, the orientation maps are also referred to as "inverse pole figure" map (IPF map). For each orientation measurement, only one reference direction of the sample is converted into crystal coordinates in order to express its crystallographic (anisotropic) character.

In contrast to the hexagonal system, the trigonal point groups can be easily retrieved from the cubic point group m 3 m derived by Symmetriereduktion and thus without contradiction by means of cubes 2 pose as in 5 will be shown. It is irrelevant that the elementary cell polyhedron is replaced by a rhombohedral element cell instead of the typically hexagonal one, since the cubic symmetry provides all the necessary symmetry elements in the desired orientations and the symmetry is thus complete.

In 5 are the five dice 2 with color-coded patterns on the polyhedron surfaces, each representing one of the five trigonal point groups. It was for the dice group representing the point group "3" 2 an alternative color key is selected at the bottom right, because its fundamental zone has no three defined corners. The front side and the right side surface describe the size of the fundamental zone for the point group "3". The corners of the sides of the cube are covered by the colors black-red-yellow-white-magenta-blue, but also from this, colors and crystal directions can be clearly assigned again.

In 6 are the cubes 2 with color-coded patterns on the polyhedral surfaces, each representing one of the seven tetragonal point groups. As 6 shows, occur in five of the seven point groups discontinuities in the color gradient (color jumps). Among them is the centrosymmetric Laue group 4 / m. A representative of this Laue group is the Schreibersit - a phosphide (Fe, Ni) ₃ P occurring in iron meteorites, which is in the space group I 4 crystallized. For this phase, a distinction can be made very well between the zone axes [uv 0] and [vu 0] even with fast measurements by means of EBSD. For example, in the in 9 Oriented map 32 of a sample with Schreibersit the [100] direction blue, and the actually symmetry equivalent [010] direction shown in green. For a grain for which a direction is practically parallel to [100], the experimental particular direction can be determined locally between [∞, 1, 0] and [∞, -1, 0] according to the measurement accuracy of z. B. 0.5 ° back and forth. This means that despite almost identical direction in the crystal lattice, the corresponding pixel in the same grain is then colored either green or blue. This applies to all grain orientations for which the reference vector to be colored is very close to the {100} plane. It follows that not the orientation determination is faulty, but only the presentation fails. The color distribution on the cube surfaces proves that these color jumps really must occur systematically.

For reasons of symmetry, an orthogonal crystal system such as the tetragonal or the orthorhombic is immediately obvious that the corresponding point groups can be represented very well by cubes with correspondingly patterned polyhedral surfaces.

In 7A are the three dice 2 with color-coded patterns on the polyhedron faces, each representing one of the three orthorhombic point groups.

The monoclinic crystal system is the first considered here, which completely frees at least one angle between the base vectors, ie it does not have to be 90 ° - as in all orthogonal crystal systems - nor 120 ° - as in the trigonal or hexagonal crystal system - amount. As 7B The three monoclinic point groups can also be shown with suitable point-group-specific cubes 2 represent. This is because the free angle can occur only perpendicular to the twofold rotation axis or to the mirror plane, ie perpendicular to one of the edges of the cube. The 180 ° rotational symmetry or mirror symmetry is not restricted by the cube. At most, a monoclinic angle between the monoclinic basis vectors causes the in 7B shown color gradients on the respective projection surface to rotate about the axis of symmetry.

In the triclinic crystal system, the symmetry reduction is compared to the cubic point group m 3 m particularly high, because there is at most one symmetry center for the triclinic point group 1 , In the point group 1, there is no symmetry equivalent to the actual object itself.

In 7C are two cubes 2 with color-coded patterns on the polyhedron faces, each representing one of the two triclinic point groups. The low symmetry in the triclinic point group 1 leads to the fact that the entire space must be color-coded. In the exemplary embodiment of the in 7C left dice 2 the bottom right corner is white and the left corner is black.

Related to the 2A , and 3A to 7C described polyhedra (cube 2 and bipyramids 3 ) have in common that their polyhedron surfaces color-coded patterns, typically color gradients, which are asymmetric and characteristic for the one-order arrangement of the symmetry elements of a group of points at least in a partial area.

The use of the color patterns or color gradients on the polyhedron surfaces not only has the advantage that it can be clarified with the help of the simple hexaders and the hexagonal bipyramid all symmetries, which offers extreme advantages in the crystallographic training. Another advantage derives from the way in which orientations are interpreted or presented using EBSD. In addition, the importance of the color key used as standard in EBSD can be better understood throughout the room. Still relatively simple to interpret for the highly symmetric point groups, the notion of aligning a vector in space by describing a color becomes increasingly difficult as the symmetry becomes smaller, or even discontinuities in the color transition that are currently unaware to most users and Therefore often lead to misinterpretation of their measurements or discarded as a measurement error.

This is because, for one and the same color of a measurement point, there are several directions in space that are completely symmetry-equivalent from the point of view of a property. Because z. For example, if each edge is equivalent to the other in one cube, each of these edges can also be used as the basis vector (as long as the result in terms of the symmetry of the color gradients leads to the same result). Thus, while maintaining a right-handed coordinate system for the highest symmetry m 3 m 24 ways to set the cube without changing its orientation by different basis vectors. This means that in each of these cases the physical or crystallographic description is correct, but the purely mathematical one is different in the form of a 3 × 3 matrix. If one uses color coding, which describes crystallographically a (physical or chemical) property change, each mathematically different description flows into the identical color description. In other words, where previously constantly changing vector definitions irritate the user, the consistent color definition clearly indicates an equivalent result. If the symmetry is reduced, either only half or 1/4 of all possible basis vector combinations are equivalent. Errors in the assignment are then immediately visible through a change in the color distribution on the surfaces, at least if no color jumps occur.

The color distribution in the orientation maps contains important information for the interpretation of EBSD measurements. Thus, the identical color at different points at the same time indicates all places which are characterized by identical characteristics for the direction shown (for which the color is displayed). Now, one direction does not characterize the orientation of a crystal. If one wants to see these, one simply generates orientation maps for several reference directions, usually perpendicular to one another, and can display their crystallographic directions in color. The point-group-specific polyhedra can help here, too, because they can be used to virtually define a direction and orient it in such a way that the colors resemble those in the orientation map. The orientation of the respective polyhedron produced in this way corresponds to the orientation of the crystal for most crystal descriptions. This will be in 3C clarifies that the in 2A shown below, the point group m 3 representing cubes 2 with an additional coordinate system with the reference directions x ₁ , x ₂ , x ₃ , the z. B.

Sample axes can be shows. At the point group m 3 m representing cubes (see 2A top left) it can be shown that if the color for the normal direction is coded red, the two perpendicular directions only have color components of red and green, but can never contain blue components. In contrast, almost identical directions of the point group m 3 representing cube 2 Switch between red and green, because with the color key used there, both colors adjoin one another in the center of the cube faces.

Particular applications of the described point group-specific polyhedra are in the field of materials science and materials science, in addition to EBSD measurements, for example in X-ray structure analysis and texture research, generally in the fields of quality control and basic research.

Due to the improved representation of the relationship between the crystal classes and Symmetriereduktion the point-group specific polyhedra are suitable 2 . 3 also very good for the training.

According to one embodiment, a teaching set consists of 32 point-group specific polyhedra, i. H. one for each of the 32 crystal classes, wherein the surfaces of the polyhedra have patterns, typically color patterns or color gradients, which are asymmetric at least in a partial area (fundamental zone). Typically, the polyhedron surfaces of the polyhedra have a special surface pattern that describes all crystallographically equivalent orientations by the same unique color according to directional color coding, and illustrates the alignment of all the symmetry elements of the particular crystal class by corresponding symmetries of the particular surface pattern. Such a teaching set comes with cubes and bipyramids, which can be produced inexpensively, as explained below.

According to one embodiment, a method for generating a polyhedron comprises selecting a crystal class and generating asymmetric patterns, at least in part, for the polyhedron faces of a regular hexahedron or a hexagonal bipyramid, such that the alignment of all the symmetry elements of the respective crystal class by corresponding symmetries of the patterns is represented. The generating may include applying the crystal class symmetry elements to the asymmetric patterns (fundamental zone). By using only simple bodies such as the regular hexahedron (cube) or a hexagonal bipyramid with only twelve polyhedron faces, a real or virtual set of 32 point group specific polyhedra can be provided in a simple and / or cost effective manner.

The production of the patterns on the polyhedral surfaces can be achieved by printing or by laminating on the polyhedron surfaces cubic or bipyramidal body or hollow body z. B. made of wood, glass, metal or plastic. As a result, the point-group-specific polyhedra can already be produced very inexpensively.

The patterning on the polyhedral surfaces can also be done by printing on cut sheets for cubes and bipyramids, which are folded into polyhedra and z. B. glued to edges. As a result, the point group specific polyhedra can be made extremely inexpensive and so, for example, students are provided as a set of 32 printed pattern sheets.

In another embodiment, the point group specific polyhedra are generated in a 3D printer.

Claims (6)

A method of producing a polyhedron comprising: - selection of a crystal class; Generating at least in a subarea asymmetrical patterns on the polyhedral surfaces of a regular cube 2 ) or a hexagonal bipyramid ( 3 ) or a corresponding cutting pattern sheet, such that the arrangement of all the crystal-class symmetry elements is represented by corresponding symmetries of the patterns on the polyhedron faces, and that all crystallographically equivalent directions are characterized by the same unique color corresponding to a direction-dependent color coding ( 21 ) to be discribed. The method of claim 1, comprising: - defining a directional color coding ( 21 ) according to the crystal class for producing color-coded asymmetric patterns; - generating the asymmetrical at least in a partial area pattern by applying the symmetry elements of the crystal class to the color-coded asymmetric pattern of a polyhedral surface; Generating the asymmetrical at least in a partial area pattern on polyhedral surfaces of a regular cube ( 2 ) or a hexagonal bipyramid ( 3 ); and / or - producing the asymmetrical pattern, at least in a subarea, on a regular hexahedron ( 2 ) or the hexagonal bipyramid ( 3 ) corresponding cut sheet. Point group specific polyhedra ( 2 . 3 comprising polyhedron surfaces with color-coded patterns, wherein the patterns are asymmetrical in at least a partial region, all crystallographically equivalent directions being characterized by the same, unambiguous color corresponding to a direction-dependent color coding ( 21 ), wherein the polyhedron ( 2 . 3 ) a cube ( 2 ) or a hexagonal bipyramid ( 3 ), and wherein the patterns on the polyhedral surfaces are characteristic of the unique arrangement of the symmetry elements of a crystal class. Teaching kit comprising a polyhedron for each of the 32 crystal classes ( 2 . 3 ) according to claim 3. A teaching aid set according to claim 4, comprising 25 cubes ( 2 ) and seven bipyramides ( 3 ). Use of polyhedra ( 2 . 3 ) according to claim 3 for the representation of EBSD measurements.

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