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- Internal Structure of Crystals III

indicate mirror lines Small red solid ellipses indicate 2 fold rotation axes perpendicular to the plane of the drawing So we now have the first Plane Group on which the Class 2mm can be based The primitive Rectangular Net can also accommodate for motifs having a lesser degree of point symmetry provided that their symmetry elements are aligned with the corresponding symmetry elements of the net Such a motif can either have a point symmetry of 2 i e having a 2 fold rotation axis as its only symmetry element or have a point symmetry of m i e having a mirror line as its only symmetry element We ll start with the latter type of motif Two such motifs are placed in a mesh of a primitive rectangular net as follows Figure 4 Two motifs each having point symmetry m are placed in a mesh of a primitive rectangular lattice as indicated The next Figure depicts a regular periodic pattern of motifs that have a symmetry m resulting in the Plane Group P2mg Figure 4a If we place two motifs having a symmertry of m in each mesh of the primitive net as indicated in Figure 4 then we will obtain a periodic pattern of motifs representing the Plane Group P2mg Figure 4b For the pattern of Figure 4a a unit mesh is chosen yellow This unit mesh has point symmetry 2 and is primitive P Point n is not equivalent to point a because the orientations of the motifs bottom left and top right to them are different So indeed the unit mesh is primitive The next Figure gives the motif s str and the motif s l of the pattern of Figure 4a The motif s l is repeated indefinitely across the two dimensional plane Figure 4c The motif s str black and the motif s l black blue of the pattern representing Plane Group P2mg It is indefinitely repeated along the directions of the 2 D lattice To determine the translation free residue i e to determine the point group symmetry of the pattern representing the Plane Group P2mg we contract the whole structure together till all translations are zero The figure shape pattern we end up with is depicted in the next Figure Figure 4d Eliminating all translations yields a figure that has a point symmetry 2mm which represents the translation free residue of the Plane Group P2mg In the pattern representing the Plane Group P2mg we can detect 2 fold rotation axes mirror lines and glide lines The next Figure indicates some of the 2 fold rotation axes Figure 4e Some of the 2 fold rotation axes belonging to the symmetry content of the Plane Group P2mg are indicated The colored lines serve to show that there indeed are 2 fold rotation axes perpendicular to the plane of the drawing Figure 4f The pattern representing Plane Group P2mg has mirror lines parallel to the y direction One of them is depicted Figure 4g The

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Open archived version from archive - Internal Structure of Crystals IV

Pm See Figure 2 Figure 2 When we place motifs possessing a point symmetry m at the nodes of a primitive rectangular net we get a pattern that represents Plane Group Pm Figure 2a For the repetitive pattern of Figure 2 a unit mesh is chosen yellow This unit mesh has a point symmetry m and is primitive When all translations are eliminated all the motifs coincide and the resulting figure has a point symmetry m so the Point Group Crystal Class associated with this Plane Group is m The only symmetry elements besides simple translation of this Plane Group are mirror lines parallel to the y direction The total symmetry content of the Plane Group Pm is depicted in the next Figure Figure 2b The total symmetry content of the Plane Group Pm The only symmetry elements this Group has are mirror lines parallel to the y direction They are indicated by red lines For clarity the nodes of the lattice are indicated black dots So we have found the first Plane Group Pm belonging to the planar Point Group m If we place asymmetric motifs in fact motif units that are symmetric with respect to each other in a primitive rectangular 2 D lattice net such that we let them alternate along the y edges of the unit mesh and all repetitions of it then we obtain a pattern that represents the Plane Group Pg See Figure 3 Figure 3 Asymmetric motifs in fact motif units placed in a primitive rectangular net such that they alternate along the y edges of the meshes produce a pattern that represents the Plane Group Pg Figure 3a A unit mesh is chosen yellow This unit mesh is primitive and has a point symmetry 1 i e it is asymmetric When we remove all translations the individuals of the same type of motif unit will be superimposed upon each other resulting in only one individual per type In the present case this means that only two individuals remain one of each type the two types of comma in our Figures and these two individuals become aligned to each other because of the disappearance of translational distances and in this way together form a figure that has point symmetry m Thus the Point Group to which our Plane Group belongs is m See Figure 3b Figure 3b The translation free residue of the pattern representing Plane Group Pg has m symmetry i e the only symmetry element it has is one mirror line and represents the Point Group to which the Plane Group Pg belongs The next Figure shows what the very motif s str actually is and that it is constituted out of two motif units This motif together with its surroundings such that together they make up the motif s l tiles the 2 D plane completely shown in Figure 3d Figure 3c The real motif constituting the pattern of Figure 3 Figure 3d The real motif s l tiles the

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Open archived version from archive - Internal Structure of Crystals V

indicated by small blue solid triangles 2 fold rotation axes are indicated by small blue solid ellipses Glide lines are indicated by red dashed lines Mirror lines are indicated by solid lines red and black One should not worry about the small discrepancies at some places in the drawing So we have found the only Plane Group P6mm belonging to Class Point Group 6mm of the 2 D Hexagonal Crystal System Next we will consider the Class 6 Placing motifs with point symmetry 6 i e motifs having only a 6 fold rotation axis into a hexagonal net such that the motifs are associated with the lattice nodes yields a repeating pattern representing Plane Group P6 See Figure 2 Figure 2 Inserting motifs with point symmetry 6 into a 2 D hexagonal lattice yields a periodic pattern representing Plane Group P6 Figure 2a A unit mesh is chosen yellow This unit mesh has a point symmetry 2 and is primitive If the pattern representing Plane Group P6 is contracted such that all translations are eliminated then we end up with a figure that has 6 symmetry It is the translation free residue of that Plane Group and as such represents the Point Group to which that Plane Group belongs See Figure 2b Figure 2b The translation free residue of the Plane Group P6 The symmetry of this residue is 6 i e it has as its only symmetry element a 6 fold rotation axis going through its center and perpendicular to the plane of the drawing The translation free residue is in the present case also the motif s l which means that as such it tiles the 2 D plane completely just like the translation free residue of the pattern of Figure 1 A pattern representing Plane Group P6 has no mirror lines and also no glide lines The distribution of rotation axes is however exactly the same as in Plane Group P6mm See Figure 2c for the total symmetry content of Plane Group P6 Figure 2c Total symmetry content of Plane Group P6 So we have found the only Plane Group P6 belonging to Class Point Group 6 of the 2 D Hexagonal Crystal System Two distinct Plane Groups P3m1 and P31m belong to Point Group 3m They have the same total symmetry content and shape but the motifs differ in orientation with respect to the edges of the unit mesh Let s start with Plane Group P3m1 Placing motifs into a hexagonal net but now motifs having a point symmetry 3m meaning that each motif has a 3 fold rotation axis going through its center and perpendicular to the plane of the drawing and three equivalent mirror lines such that their point of intersection coincides with the center of such a motif and oriented in the net such that their mirror lines do not coincide with the connecting lines of the net yields a pattern of repeated motifs representing Plane Group P3m1 See Figure 3 Figure 3

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Open archived version from archive - Internal Structure of Crystals VI

a mirror line If the pattern representing Plane Group P31m is contracted such that all translations are eliminated then we end up with a figure that has 3m symmetry It is the translation free residue of that Plane Group and as such represents the Point Group to which that Plane Group belongs See Figure 1c Figure 1c The translation free residue of the pattern representing Plane Group P31m of Figure 1a The point symmetry of this residue is 3m and as such it represents the Point Group 3m to which the present Plane Group belongs The translation free residue is at the same time the motif s l and tiles the 2 D plane completely The symmetry elements involved in a pattern representing Plane Group P31m are 3 fold rotation axes mirror lines and glide lines See Figures Figure 1d A pattern representing Plane Group P31m has mirror lines One of them is depicted here Figure 1e A pattern representing Plane Group P31m has glide lines One of them is depicted here Figure 1f A pattern representing Plane Group P31m has 3 fold rotation axes One of them is depicted here The total symmetry content of the Plane Group P31m is given in the next Figure Figure 1g Total symmetry content of Plane Group P31m Mirror lines are indicated by solid lines red and black So we now have discussed the second and last Plane Group P31m belonging to the Class Point Group 3m Finally we have arrived at the last 2 D Crystal Class namely 3 The only Plane Group that belongs to it is P3 If we place motifs having a point symmetry 3 i e the only symmetry element each one of them has is a 3 fold rotation axis in a hexagonal net then we obtain a periodic pattern of motifs representing the Plane Group P3 See Figure 2 Figure 2 Arranging motifs with point symmetry 3 in a hexagonal 2 D lattice yields a pattern that represents Plane Group P3 The symmetry of the motifs is indicated by their shape and by their coloration Figure 2a A unit mesh is chosen yellow it is primitive and has point symmetry 1 i e it has no symmetry whatsoever If we contract the pattern of Figure 2 representing Plane Group P3 such that all translations are eliminated then we end up with a figure having a point symmetry 3 This figure then represents the Point Group Crystal Class 3 to which that Plane Group belongs The figure is at the same time the motif s l of the pattern It tiles the 2 D plane completely See Figure 2b Figure 2b Translation free residue of the pattern of Figure 2 It represents the Point Group of the Plane Group P3 The symmetry elements involved in a pattern representing Plane Group P3 are 3 fold rotation axes only Through every node of the net there is such an axis and a pair of them is situated in each

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Open archived version from archive - Internal Structure of Crystals VII

Summary of the 2 D lattices planar Point Groups and Plane Groups In what follows are the 17 2 D patterns representing the 17 Plane Groups as discussed in the previous Parts They are given here in the order in which they were discussed This concludes our summary of the 17 Plane Groups and implicitly of the 10 planar Point Groups referring to two dimensional periodic patterns In the next

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Open archived version from archive - Internal Structure of Crystals VIII

be smooth macroscopically The next Figure shows the same 2 D crystal without indication of crystallographic axes and the 2 fold rotation axis The crystal shape emerges from a combination of two Forms one Form consisting of the two faces d the other Form consisting of the faces e Figure 11 Two dimensional crystal Heavy solid blue lines indicate faces Figure 12 shows the same crystal without the auxiliary lines indicating faces The nodes of the lattice are at the intersections of the lines outlining the building blocks These nodes indicate the periodic repetition of motifs and are equivalent Figure 12 The two dimensional crystal drawn without any auxiliary features The faces d are associated with a higher density of nodes than the faces e are When everything else would be the same the faces d would grow slower than the faces e because more material forming the motifs must be imported from the growing environment Figures 11 and 12 show that if the faces d were chemically and physically like the faces e they would then grow slower than the faces e because more nutrient material must be imported from the growing environment to complete a layer of motifs on the growing crystal The next two Figures give two more possible Forms Figure 13 A possible Form of the 2 D Crystal Class 2 of the 2 D Oblique Crystal System The axial system and the 2 fold rotation axis representing the point symmetry of the Form are indicated Figure 14 A possible Form of the 2 D Crystal Class 2 of the 2 D Oblique Crystal System The axial system and the 2 fold rotation axis representing the point symmetry of the Form are indicated Figure 15 Construction of a 2 D crystal from a combination of four Forms First Form The two faces d i e the Form of Figure 7 Second Form The two faces e i e the Form of Figure 9 Third Form The two faces f i e the Form of Figure 13 Fourth Form The two faces g i e the Form of Figure 14 The axial system and the 2 fold rotation axis representing the point symmetry of the crystal are indicated The final result of the construction is given in the next Figure Figure 16 A two dimensional crystal constructed from the four Forms mentioned above The final habit of the crystal i e its intrinsic shape is determined by several factors The nature of the crystal lattice which determines the shape of the empty building block The chemical composition i e the nature of the motifs placed in the lattice These two factors determine the relative intrinsic growth rates of the faces we can also say the relative intrinsic growth rates of the Forms And these intrinsic growth rates directly determine the intrinsic shape of the crystal Accidental irregularities in the growing environment concerning local concentrations or other local factors differing from place to place in that environment can

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Open archived version from archive - Internal Structure of Crystals IX

block considered as empty determines the 2 D Rectangular Crystal System It has two Classes m and 2mm In contrast to the previously considered Classes each Rectangular Class can refer to several different Plane Groups which means that when two 2 D crystals belonging to the 2 D Rectangular Crystal System have the same point symmetry they can nonetheless differ with respect to their total symmetry content i e they can differ when we also consider their translational symmetries The reason for this possible difference is a difference in the chemical contents in our drawings representing by motifs and motif units of the respective lattices A rectangular net 2 D rectangular lattice can be either primitive which means that there are lattice nodes nodes are equivalent points at the corners of the meshes only or centered which means that nodes are not only present at the corners of the meshes but in the center of each mesh as well A primitive lattice is denoted by the letter P a centered lattice by the letter C In Figure 9 a possible rectangular primitive net is drawn and the axial directions are indicated Figure 9 A possible rectangular net 2 D lattice This net can accommodate several types of motifs such that they are periodically repeated across the 2 D plane according to the geometry of such a net The axial directions i e crystallographic axes are indicated The two axes are perpendicular to each other but are not equivalent The nodes of the net are at the intersections of the lines These nodes are equivalent points In constructing 2 D crystal faces we know that only those faces are possible that can be constructed by the periodic stacking of the building blocks the meshes of the net We re now going to consider possible crystal faces Forms and combinations of Forms from the same Crystal Class in order to determine the intrinsic shapes and symmetries of 2 D crystals belonging to the 2 D Rectangular Crystal System Let us start with the Point Group Crystal Class m Crystals belonging to this Class exhibit a point symmetry m which means that macroscopically the only symmetry element they possess is a mirror line When we introduce a face parallel to the longer crystallographic axis that face will be duplicated in virtue of the action of the mirror line resulting in a Form consisting of two faces that are symmetrical to each other and in the present case moreover parallel to each other See Figure 10 Figure 10 An introduced face parallel to the longer crystallographic axis implies a second face in virtue of a mirror line m In order to make descriptions simple the mirror line as the only point symmetry element is chosen such that it is aligned with one of the crystallographic axes This can also be restated as follows There is only one point symmetry element namely a mirror line and we choose the axial system such that one axis coincides with that mirror line The resulting face pair is a possible Form of the present Class It is an open Form Another possible Form can be derived from a possible face having a different orientation with respect to the crystallographic axes Figure 11 An introduced face not parallel to either crystallographic axis implies a second face in virtue of the action of the mirror line m The resulting face pair is a possible Form of the present Class It is an open Form It is imaginable that the location of the system of crystallographic axes was chosen such that the horizontal axis cuts through the horizontal middle lines of the relevant building blocks instead of coinciding with their horizontal edges The horizontal axis should then like before in order to let the description of the crystals be simple coincide with the horizontal mirror line See Figure 11a Figure 11a The same initial and implied faces as in Figure 11 Because of the chosen location of the axial system the corner formed by the two faces consists of one building block only instead of two in the case of Figure 11 Although the corners become sharper in this way and as such more neatly expressed in drawings we prefer the crystallographic axes to coincide with the edges of the building blocks unit meshes instead of somehow going through their interior Another possible face parallel to the shorter crystallographic axis does not imply a second face because it is perpendicular to the mirror line which now maps the face onto itself See Figure 12 Figure 12 An introduced face perpendicular to the mirror line is mapped onto itself resulting in a Form consisting of a single face Also this is an open Form Introducing a possible face having yet another orientation with respect to the axial system leads to its reflection by the mirror line resulting in yet another Form consisting of two faces that are symmetrical to each other See Figure 13 Figure 13 An introduced face that is inclined to the mirror line implies a second face symmetrically related to the initial face The resulting face pair is a possible open Form of the present Class Recall that the faces as drawn must be imagined to be extended indefinitely because only their orientations are being considered In order to constitute a crystal the open forms must combine in such a way that a closed face configuration is the result like the one depicted in the next two Figures Figure 14 A 2 D crystal is formed from a combination of four individual Forms a a b b c d The final result is shown in the next Figure Figure 14a The 2 D crystal emerging from the construction in Figure 14 It belongs to the Class m of the 2 D Rectangular Crystal System The next two Figures give yet another possible combination of Forms resulting in a crystal Figure 15 construction of a 2 D crystal by combining

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Open archived version from archive - Internal Structure of Crystals X

of the undoing of the horizontal shift indicated in Figure 4 is given in the lower half of the present Figure This then is directly compared with the pattern of Figure 2 which is again depicted in the upper half of the present Figure We can clearly see that the two halves top and bottom halves are fully symmetric meaning that the faces a and b are equivalent Another orientation of a possible face effects yet another atomic aspect to the environment See Figure 6 Figure 6 Atomic aspect of the initial face of Figure 13 in Part Nine in the present Figure indicated by c Figure 7 Atomic aspect of the implied face of Figure 13 in Part Nine in the present Figure indicated by d The atomic aspect which this face presents to the environment looks somewhat different from that of the face c of Figure 6 But the difference only consists of a horizontal shift of a microscopical magnitude of half the length of the unit mesh and is macroscopically not visible When we shift the pattern of motifs of Figure 7 to the left by half a horizontal unit mesh dimension i e when we eliminate the glide translational difference of this pattern with that of Figure 6 then we like in the previous case obtain two symmetrical halves indicating that the faces c and d are equivalent See Figure 8 Figure 8 After removing the glide translational difference between the pattern of Figure 6 and that of Figure 7 we can see that they are symmetrically related to each other with respect to a horizontal mirror line expressing the point symmetry of the present Class m Now we will consider the atomic aspects in cases where we have to do with the third and last Plane Group belonging to the Point Group m It is the Plane Group Cm The patterns representing this Plane Group are based on a rectangular lattice that has the same point symmetry as the rectangular lattices discussed so far but is nevertheless fundamentally different It is the Centered Rectangular 2 D Lattice Net Unlike the previous discussed rectangular lattice which was primitive the centered rectangular lattice has not only nodes at the corners of the unit meshes but also one at their center This means that the point in the center of whatever unit mesh is equivalent to the points representing the corner nodes and thus the center point is a real node of the lattice We will now consider the atomic aspects that are presented to the environment by the possible faces that we have derived earlier in Part Nine i e faces that can be constructed by the regular stacking of rectangular building blocks To begin with let us again give a pattern of motifs representing the Plane Group Cm Figure 9 A pattern of periodically repeated motifs representing Plane Group Cm The periodicity of the pattern is based on i e described by a centered rectangular net

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