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  • Internal Structure of Crystals XI
    that vector the pattern will map onto it self The same applies with respect to the other vector The next Figure analyses these vectors further Figure 8 1 The glide translation vectors from Figure 7 2 Addition of the two vectors yields one vector equivalent to those two vectors together 3 Sign red equivalent to and indicating the set of two vectors of 1 As such this set will be indicated in the next Figure The existing difference between the asymmetric situation of Figure 6 and a possible symmetric situation can be imagined to be caused by a diagonal shift of half the diagonal of the unit mesh according to the set of two vectors described above of one half of the pattern of Figure 6 say the bottom half with respect to its other half So if we make this shift undone by means of the indicated vectors we expect to obtain two symmetrical halves Let me explain this If we execute this shift but now followed by reflections as described above then we in fact subject the structure to two glide line operations and the pattern will then map onto itself So if we omit the reflections then the result will be a mirror reflection of the the pattern Consequently if we do that to say the lower half only it will become a mirror reflection of the other half Figure 9 Indication of the diagonal shift of the bottom half of the pattern of Figure 6 in order to make it fully symmetrical with respect to its top half Because we shift only the bottom half its uppermost motifs only partially visible will not themselves be the result of any shift because otherwise some motifs of the upper half of the pattern would have been shifted while ex hypothesi they shouldn t So we have three types of motifs 1 Motifs that have not been shifted i e not been copied onto other motifs and replacing the latter and are themselves also not the result of any such shift 2 Motifs that have been shifted i e they have been copied onto other motifs while replacing the latter and are themselves also the result of such a shift 3 Motifs that are copied onto other motifs imposing their shape upon them while they themselves are not the result of any such copying from other motifs The motifs of 1 are from the upper half of the pattern and the motifs of 2 are from the lower half of the pattern The motifs of 3 do not in this respect have a definite association with one or the other half of the pattern These motifs will therefore not comply with the symmetry of the two halves with respect to each other But one line of deviating motifs will not be distinguished macroscopically Also the mentioned shift is not visible macroscopically that s why such structures look perfectly symmetrical Macroscopically we see the glide lines as mirror lines The

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  • Internal Structure of Crystals XII
    now be chosen naturally namely along the edges of the unit mesh while the horizontal axis coincides with a mirror line Figure 5 The axial system indicated in blue chosen along the edges of the unit mesh The horizontal axis coincides with a mirror line m of the pattern This mirror line also expresses part of the point symmetry of the Class 2mm Now we can again concentrate on atomic aspects while using the new position of the net with respect to the motif pattern Figure 6 Atomic aspects presented to the environment by the faces derived in the Figures 13 and 14 of Part Ten of a two dimensional crystal using the new position of the net To make this crystal fully comparable with the one depicted in Figure 2 one horizontal layer of building blocks should be added to the latter Figure 7 Atomic aspects presented to the environment by two of the faces of Figures 16 16a and 16b in Part Ten here indicated as a and b The horizontal mirror line m expressing one of the point symmetry elements of the Point Group 2mm goes right between the two middle rows of relevant building blocks and coincides with horizontal edges of them The horizontal crystallographic axis should be chosen such as to coincide with this mirror line Compare with Figure 3 From a microscopic view this position of the mirror line with respect to the edges of the building blocks unit meshes looks natural The atomic aspects of the faces a and b are symmetric with respect to the mentioned mirror line Figure 8 Atomic aspect of one of the faces of the Figures 18 and 18a in Part Ten The next Figure is like Figure 6 but now a somwhat smaller mesh is used According to the Class to which this pattern belongs there is a vertical mirror line such that the left and right faces are symmetrical with respect to that line But as can be seen from the Figure the internal structure does not support this The cause of this asymmetry is however related to just a microscopical shift translation of part of the motif pattern and is consequently not visible at the macroscopical level See The next Figures Figure 8a A pattern representing Plane Group P2mg depicted as a 2 D crystal The vertical mirror line as one of the symmetry elements of the present Class is not evident at the microscopic level Figure 8b When we shift the right hand half of the pattern of Figure 8a according to the vector indicated these halves will become mirror symmetric with respect to a vertical line separating those halves also symmetric according to the internal structure proving that the asymmetry is caused just by a translational shift of microscopic dimensions This asymmetry is consequently not visible at the macroscopic level Figure 8c Indication of the shift of motifs as specified above Figure 8d Result of the above specified shift of motifs The pattern

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  • Internal Structure of Crystals XIII
    the construction Figure 6b Removal of indented angles indented angles will not occur in single crystals The result is two squares chopping off each other s corners Figure 6c Final result of the construction representing a 2 D crystal of the Class 4 consisting of a combination of two Forms A combination as the one above can be such that the two Forms balance each other or that one or the other dominates The crystals so far considered are considered to be built up by empty building blocks To conceptually generate genuine 2 D crystals those building blocks must be furnished with content i e with motifs representing chemical units And indeed if a certain lattice is given and we insert motifs into it motifs compatible with that lattice this lattice then describing the specific periodic repeat of those motifs then the total symmetry of the resulting motif pattern is representing a certain Plane Group Only one Plane Group belongs to the Class 4 of the 2 D Tetragonal Crystal System It is the Plane Group P4 Let us display a pattern given earlier representing this Plane Group Figure 7 A pattern representing Plane Group P4 We can now consider the different atomic aspects presented to the environment by the possible faces and Forms of crystals of the Class 4 and representing Plane Group P4 we will use a square net with smaller meshes than we saw in Figure 7 Figure 8 Atomic aspect of the faces as derived in the Figures 2 3 and 3a of a 2 D crystal of the Class 4 of the 2 D Tetragonal Crystal System The incompleteness of motifs at the faces symbolizes unsaturated or distorted chemical valences In Figure 8 one can clearly see that the opposite faces of this crystal are not symmetric with respect to one or another mirror line The true symmetry of the crystal namely 4 and not say 4mm is revealed by its internal composition and this composition will reflect itself in certain physical or chemical differences by means of which one can actually determine that true symmetry imagining 2 D crystals to exist in reality Figure 9 Atomic aspect of one of the faces in the Figures 4 4a and 4b here indicated by a Figure 10 Atomic aspect of one of the faces in the Figures 5 5a and 5b here indicated by c This concludes our discussion of the Class 4 and its only Plane Group P4 The second and last Crystal Class of the 2 D Tetragonal System is the Class 4mm It admits of two types of motifs resulting in two different types of periodic pattern according to two different Plane Groups P4mm and P4gm We will derive possible faces Forms and combinations of Forms within this Class 4mm Figure 11 Introducing a face parallel to one of the crystallographic axes and consequently perpendicular to the other axis implies three more faces resulting in a closed Form having the shape of a

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  • Internal Structure of Crystals XIV
    mapped onto itself The translations involved in such operations are as such macroscopically not visible i e when we observe macroscopically we as it were ignore translations when i e because their lengths are very small indeed only of the order of the unit mesh dimensions thereby we cannot and will not ignore the cumulative effect of the translational repetition of the motifs because they lead to a macroscopically visible effect namely the macroscopical dimensions of the crystal itself In the next Figures we show that those mirror symmetries as demanded by the point symmetry of the present Crystal Class 4mm that are not expressed by the internal organization can be restored such that they now will be expressed by the internal organization We accomplish this restoration by means of translations of the above mentioned sort A part of the pattern is shifted according to some of those translations and then as a consequence the internal structure will be such as to be symmetrical with respect to the mentioned mirror lines of its point symmetry This proves that the mentioned asymmetries are indeed caused just by some translational differences And because the translations involved in these differences have microscopical dimensions the asymmetry will not be visible macroscopically The fact that the asymmetries are just caused by some very small translations is shown in the next Figures to begin with for the case of the horizontal mirror line as one of the point symmetry elements of the present Class 4mm by means of restoring the symmetry by appropriate microscopical translations Figure 8 A shift of the lower half of the pattern of Figure 4 in virtue of a diagonal vector indicated by its two components a horizontal an a vertical component effects the whole pattern i e lower plus upper half to be mirror symmetrical with respect to the horizontal mirror line m as one of the point symmetry elements See the next Figure Figure 9 According to Figure 8 we shift the motifs as indicated in that figure The shifted motifs are shown in red But the motifs lying on the horizontal mirror line m and indicated in blue are it is true also shifted but a copy of them remains at the original locations So they are not themselves results of any shift The faces a and b are symmetrical and now also according to the internal structure except for one row of motifs indicated in blue lying on the mirror line itself Such a discrepancy of just one row is however macroscopically invisible The next Figure depicts the 2 D crystal of Figure 2 and shows that a mirror symmetry with respect to a vertical mirror line and as such macroscopically expressed by the symmetry of the faces d and e supposing that the depicted crystal is extended to macroscopical dimensions is not supported by the internal organization Figure 10 A two dimensional crystal of the Class 4mm and belonging to the Plane Group P4gm A vertical mirror

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  • Internal Structure of Crystals XV
    an equilateral triangle The outline of a crystal representing this Form is given Figure 10 A two dimensional crystal of the Class 3 of the 2 D Hexagonal Crystal System representing the above derived Form An introduced face with yet another general orientation with respect to the crystallographic axes is multiplied by the 3 fold rotation axis resulting again in a closed Form having the shape of an equilateral triangle See the next Figure Figure 11 Yet another initial face implies two more faces yielding a Form having a point symmetry consistent with that of the present Class 3 The Form again has the shape of an equilateral triangle and can represent a crystal of this Class See next Figure Figure 12 A two dimensional Crystal of the Class 3 of the 2 D Hexagonal Crystal System as a result of the above construction These Forms not only can each for themselves represent crystals of the Class 3 but also combinations of them can represent crystals of this same Class See Figures 13 14 and 15 for an example Figure 13 Combining the Forms of the Figures 3 blue and 6 green results in a new crystal indicated in the next Figure Figure 14 The outline of the crystal consisting of the two mentioned Forms is indicated The next Figure depicts the whole crystal Figure 15 A two dimensional crystal of the Class 3 of the 2 D Hexagonal Crystal System It is a combination of two Forms as specified above The present Class 3 admits of one Plane Group only namely P3 Considering the Plane Group means placing motifs in the empty building blocks When this is done we can investigate the possible atomic aspects presented to the environment by the possible crystal faces of this Class Let us first give a pattern of motifs representing Plane Group P3 Figure 16 A pattern of repeated motifs as given earlier representing Plane Group P3 The point symmetry of each motif is 3 indicated by their shape and coloration which means that the only symmetry element that such a motif possesses is a 3 fold rotation axis The next Figure is a pattern equivalent to the one in the previous Figure but based on a net with smaller meshes We will use this for the exposition of the possible atomic aspects Figure 17 A pattern of repeated motifs the same motifs as in Figure 16 representing Plane Group P3 now based on a net with smaller meshes than the one used in Figure 16 coloration of motifs slightly changed for convenience The next Figure shows the atomic aspects presented to the environment by the faces of Figures 3 and 4 Figure 18 Atomic aspects presented to the environment by some possible faces together making up a possible Form of the Class 3 of the 2 D Hexagonal Crystal System In the next Figure the net which is only a device to describe the ordering of the motifs and as such just

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  • Internal Structure of Crystals XVI
    be used to construct possible crystal faces and Forms a larger meshed version of it will be used to show the possible atomic aspects presented to the environment by those faces An introduced face parallel to one of the equivalent crystallographic axes is multiplied by 3 in virtue of the presence of the 3 fold rotation axis as the only symmetry element of our Class i e the Crystal Class 3 The resulting Form is an equilateral triangle regular trigon and its point symmetry complies with the symmetry of the Class See Figure 7 Figure 7 An initial face parallel to one of the crystallographic axes implies two more equivalent faces together making up a closed Form that has the shape of an equilateral triangle and can as such represent a crystal of the present Class 3 The 3 fold rotation axis as the only symmetry element of our Class is situated in the center of the system of crystallographic axes Figure 8 A crystal based on the Form derived above is indicated The triangular pattern of hexagons complies with a central 3 fold rotation axis and so is consistent with the symmetry of our Class But also mirror lines are clearly present in that pattern because at this stage we have not yet furnished the net with compatible motifs only then should the absence of mirror lines be appearant The yellow meshes serve to emphasize the 3 fold rotational symmetry of the triangular pattern See the next Figure for the final result of the construction of the crystal Figure 9 A two dimensional crystal of the Class 3 of the 2 D Hexagonal Crystal system the crystal as specified above Because the building blocks are still empty the crystal as depicted has the point symmetry of the most symmetric trigonal crystals The latter have a point symmetry 3m and indeed we see this point symmetry to be present The crystal is mirror symmetric with respect to three mirror lines perpendicular to the crystallographic axes But as soon as we furnish the building blocks with motifs compatible with the Class 3 and consequently obtain a pattern representing its only Plane Group the absence of the mirror lines will become appearant Introduction of a face that is perpendicular to one of the crystallographic axes will lead to a multiplication of that face by 3 in virtue of the action of the 3 fold rotation axis resulting in a closed Form having the shape of an equilateral triangle having a different orientation than the one derived above This triangle can represent a 2 D crystal of the present Class See the next Figures Figure 10 An initial face perpendicular to one of the equivalent crystallographic axes implies two more faces resulting in a closed Form having the shape of an equilateral triangle The system of crystallographic axes red is indicated This triangle can represent a 2 D crystal of the present Class See next Figure Figure 11 A two dimensional crystal of

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  • Internal Structure of Crystals XVII
    types of angles which means that two angular magnitudes with respect to angles between the sides are involved that alternate regularly around the figure As can be seen in our figure there are no receding angles involved So this is a genuine Form of the Class It moreover can represent all by itself a crystal of that Class See next Figure Figure 5f A two dimensional crystal of the Class 3m of the 2 D Hexagonal Crystal System Its ditrigonal shape is emphasized in the next Figure Figure 5g The same as Figure 5f but now with emphasis on the trigonal shape of the crystal The derived Forms can enter in combinations with each other See for an example the next Figure Figure 6 Indication of a possible crystal of the Class 3m as a combination of two Forms the one depicted in Figure 3 and the one depicted in Figure 5 The latter cuts three corners from the former as indicated The result is presented in the next Figure Figure 7 A two dimensional crystal consisting of two Forms as specified above The present Class 3m admits of two Plane Groups P3m1 and P31m which means that two types of motifs are possible motifs that can furnish the empty building blocks of the hexagonal net and be with respect to symmetry consistent with the point symmetry of the Class In fact it turns out that the difference between the two types of motifs mentioned is only their orientation with respect to the crystallographic axes Let us start with the Plane Group P3m1 the last symbol is a one A pattern representing this Plane Group as depicted earlier is the following Figure 8 A periodic pattern of repeated motifs representing Plane Group P3m1 The periodic repetition is indicated by a hexagonal net drawn as a stacking of unit meshes having the shape of a rhombus Figure 9 Again a periodic pattern of motifs representing Plane Group P3m1 Now a net with smaller meshes is chosen and lines going NW SE added in order to outline h e x a g o n a l building blocks For the latter see next Figure Figure 9a As Figure 9 but now with hexagonal building blocks indicated in several ways These hexagonal building blocks tile the plane completely The next Figure indicates the atomic aspect presented to the growing environment by the face on the right side of the Form derived in Figures 4 and 5 Figure 10 The atomic aspect of the face specified above is indicated but not yet fully constructed See Figure 11 The pattern represents the Plane Group P3m1 Figure 11 Atomic aspect presented to the environment white of the face on the right side of the Form crystal derived in Figures 4 and 5 The next Figure depicts the same situation as in the previous Figure but now with some coloration omitted in order to highlight the motifs Figure 12 The atomic aspect as given in Figure 11

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  • Internal Structure of Crystals XVIII
    a closed Form having the shape of an equilateral triangle This triangle can represent a crystal of the present Class See Figure 7 Figure 7 A two dimensional crystal of the Class 3m of the 2 D Hexagonal Crystal System It has the same shape as the one depicted in Figure 5 in Part Seventeen but its orientation with respect to the system of crystallographic axes is different Introducing yet another possible face namely one that is neither perpendicular nor parallel to one of the crystallographic axes leads to five more faces in virtue of the symmetry elements of our Class See the next Figures Figure 7a An initial face that is neither perpendicular nor parallel to one of the crystallographic axes We will concentrate on the blue part of it Figure 7b When an initial face as specified above is subjected to the symmetry elements of the present Class 3m five more faces will be generated The resulting closed Form however turns out to have receding angles The figure as can be seen is indented It cannot therefore represent a crystal because of energetic reasons So we must look for a face that is as general with respect to its orientation as this one but not causing receding angles Such a face is depicted in the next Figure Figure 7c An initial face that is neither parallel nor perpendicular to any of the crystallographic axes but with a different orientation than the one in Figure 7a From the face given in Figure 7c we now will derive the Form by subjecting it to the symmetry elements of our Class 3m See the next Figures Figure 7d When the initial face as specified above Figure 7c is subjected to the symmetry elements of the present Class a ditrigonal Figure will be the result a figure without receding angles Here only the first stage of this generation is shown The initial face is reflected in a mirror line as one of the symmetry elements of our Class resulting in two faces that are symmetrically situated with respect to the mirror line Figure 7e When the initial face as specified above is introduced the actions of the mirror lines will generate five more faces The 3 fold rotation axis is then implied resulting in a closed Form having the shape of a ditrigon i e a six sided figure with equal sides but with alternating angles As can be seen there are no receding angles present This closed Form can represent a crystal of our Class See next Figure Figure 7f A two dimensional crystal of the Class 3m of the 2 D Hexagonal Crystal System Its ditrigonal shape is emphasized in the next Figure Figure 7g The same as Figure 7f but now with emphasis on the ditrigonal shape of the crystal We can now study the atomic aspects presented to the growing environment by the faces and Forms of our Class 3m In our case the orientation and symmetry of

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