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  • Orthorhombic Crystal System III
    the faces of the most general Form There are no mirror planes The three 2 fold rotation axes are indicated by small black solid ellipses These rotation axes are perpendicular to each other and coincident with the three crystallographic axes The face a b mc generates a rhombic sphenoid when subjected to the symmetry elements of the present Class First the face is duplicated by the vertical 2 fold rotation axis resulting in two upper faces Then one of the two horizontal 2 fold rotation axes generates for each face a corresponding lower face The result is a Form consisting of four faces two upper and two lower ones a sphenoid also called bisphenoid The other horizontal 2 fold rotation axis is now implied as can be seen in the next Figure Figure 9 1 Position of the face a b mc in the stereogram of the symmetry elements of the Rhombic bisphenoidic Crystal Class 2 The face is rotated about the vertical 2 fold rotation axis by 180 0 resulting in a second face this is equivalent to demanding that the resulting face configuration here the two faces has a 2 fold rotational symmetry Both faces are upper faces 3 One of the two horizontal 2 fold rotation axes generates for each face an additional corresponding lower one indicated by a small open red circle The final result then also already complies with the other horizontal 2 fold rotation axis so the resulting face configuration four faces has 2 2 2 symmetry i e the symmetry of our Class It is a Rhombic Sphenoid The face na b mc generates a rhombic sphenoid when subjected to the symmetry elements of the present Class The face is duplicated by the vertical 2 fold rotation axis giving two upper faces These faces get their respective lower faces by the action of a horizontal 2 fold rotation axis resulting in four faces making up a Rhombic Sphenoid The other 2 fold rotation axis is then implied See Figure 10 Figure 10 1 Position of the face na b mc in the stereogram of the symmetry elements of the Rhombic bisphenoidic Crystal Class 2 The face is rotated about the vertical 2 fold rotation axis by 180 0 resulting in a second face this is equivalent to demanding that the resulting face configuration here the two faces has a 2 fold rotational symmetry Both faces are upper faces 3 One of the two horizontal 2 fold rotation axes generates for each face an additional corresponding lower one indicated by a small open red circle The final result then also already complies with the other horizontal 2 fold rotation axis so the resulting face configuration four faces has 2 2 2 symmetry i e the symmetry of our Class It is a Rhombic Sphenoid The face a nb mc behaves in the same way as the ones above It generates a rhombic sphenoid when subjected to the symmetry elements of the present Class See Figure 11 Figure 11 1 Position of the face a nb mc in the stereogram of the symmetry elements of the Rhombic bisphenoidic Crystal Class 2 The face is rotated about the vertical 2 fold rotation axis by 180 0 resulting in a second face Both faces are upper faces 3 One of the two horizontal 2 fold rotation axes generates for each face an additional corresponding lower one The final result then also already complies with the other horizontal 2 fold rotation axis so the resulting face configuration four faces has 2 2 2 symmetry i e the symmetry of our Class It is a Rhombic Sphenoid The face a b c is vertical It generates a hemihedric rhombic protoprism when subjected to the symmetry elements of the present Class The face is duplicated by the action of the vertical 2 fold rotation axis giving two vertical faces Each face is then again duplicated by a horizontal 2 fold rotation axis resulting in four vertical faces making up a prism The other 2 fold rotation axis is then implied See Figure 12 Figure 12 1 Position of the face a b c in the stereogram of the symmetry elements of the Rhombic bisphenoidic Crystal Class 2 The face is rotated about the vertical 2 fold rotation axis by 180 0 resulting in a second face Both faces are vertical 3 One of the two horizontal 2 fold rotation axes generates for each face an additional one The final result then also already complies with the other horizontal 2 fold rotation axis so the resulting face configuration four vertical faces has 2 2 2 symmetry i e the symmetry of our Class It is a hemihedric Rhombic Protoprism The face na b c is also vertical It generates a hemihedric brachyprism when subjected to the symmetry elements of the present Class The face is duplicated by the vertical 2 fold rotation axis giving two vertical faces Each of these faces will be duplicated by a horizontal 2 fold rotation axis resulting in four vertical faces making up a prism The other horizontal 2 fold rotation axis is then implied See Figure 13 Figure 13 1 Position of the face na b c in the stereogram of the symmetry elements of the Rhombic bisphenoidic Crystal Class 2 The face is rotated about the vertical 2 fold rotation axis by 180 0 resulting in a second face Both faces are vertical 3 One of the two horizontal 2 fold rotation axes generates for each face an additional one The final result then also already complies with the other horizontal 2 fold rotation axis so the resulting face configuration four vertical faces has 2 2 2 symmetry i e the symmetry of our Class It is a hemihedric Rhombic Brachyprism The face a nb c is also vertical It generates a hemihedric macroprism when subjected to the symmetry elements of the present Class The face is duplicated by the vertical 2 fold rotation

    Original URL path: http://www.metafysica.nl/rhombic_3.html (2016-02-01)
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  • Monoclinic Crystal System I
    Of course also the negative derived orthohemipyramid a nb mc can represent the Orthodiagonal Series Like the monoclinic hemipyramids also the monoclinic vertical prisms come in three Series Vertical Prism of the Vertical Series This Series consists of only one member the monoclinic protoprism It can be derived from any member of the Vertical Series of hemipyramids Its Naumann symbol is Vertical Prisms of the Clinodiagonal Series the monoclinic clinoprisms They can be derived from the hemipyramids of the Clinodiagonal Series Vertical Prisms of the Orthodiagonal Series the monoclinic orthoprisms They can be derived from the hemipyramids of the Orthodiagonal Series From the monoclinic protohemipyramid as such belonging to the Vertical Series of hemipyramids can be derived the monoclinic protoprism by letting the derivation coefficient m become infinite The faces then become vertical resulting in a prism with its four faces parallel to the vertical axis c axis Every protohemipyramid positive or negative primary or non primary leads to the same monoclinic protoprism so there is only one such prism Its Weissian symbol is a b c where the sign stands for infinity its Naumann symbol is P and the Miller symbol is 110 With it we have found yet another basic face namely a b c for our list of all the basic faces compatible with the Monoclinic Crystal System The monoclinic protoprism is like the monoclinic hemipyramids an open Form consisting of four faces The straight lines indicating the upper and lower borders of the prism should not suggest upper and lower faces closing the prism If such faces are present then they are a separate Form the basic pinacoid consisting of two faces and combine with the prism resulting in a closed Form With the monoclinic protoprism we have yet another Form in our listing of monoclinic holohedric Forms i e Forms of the present Crystal Class In the next Figure we construct the monoclinic protoprism from the derived positive protohemipyramid Figure 11 1 The positive derived i e m is not equal to 1 Monoclinic Protohemipyramid 2 Construction of the Monoclinic Protoprism from the Monoclinic Protohemipyramid 3 Result of the construction The prism is drawn shortened along the vertical axis by reason of convenience which does not make any difference crystallographically 4 The Monoclinic Protoprism Axial system removed from the drawing From a clinohemipyramid we can derive a monoclinic clinoprism by letting the derivation coefficient m become infinite The resulting prism has faces each of which is parallel to the vertical axis cuts off a unit piece positive or negative from the ortho axis and cuts off a piece from the clino axis that is longer than the unit piece associated with that axis The Weissian symbol for these Forms is na b c the Naumann symbol is indicated in 2 of the above list of vertical prisms and the Miller symbol is kh0 With all this we have found yet another basic face for our listing of such faces namely na b c With the monoclinic clinoprism we have yet another Form for our listing of monoclinic holohedric Forms In the next Figure we will construct a monoclinic clinoprism from the derived positive clinohemipyramid of Figure 9 Figure 12 1 The positive derived Clinohemipyramid 2 Construction of the Monoclinic Clinoprism from the positive derived Clinohemipyramid The corresponding negative hemipyramid will give the same result 3 The resulting Monoclinic Clinoprism The prism is drawn shortened along the vertical axis by reason of convenience which does not make any difference crystallographically From an orthohemipyramid we can derive a monoclinic orthoprism by letting the derivation coefficient m become infinite The resulting prism has faces each of which is parallel to the vertical axis cuts off a unit piece positive or negative from the clino axis and cuts off a piece from the ortho axis that is longer than the unit piece associated with that axis The Weissian symbol for these Forms is a nb c the Naumann symbol is indicated in 3 of the above list of vertical prisms and the Miller symbol is hk0 With all this we have found yet another basic face namely a nb c With the monoclinic orthoprism we have found yet another Form for our listing of holohedric monoclinic Forms In the next Figure we will construct a monoclinic orthoprism from the derived positive orthohemipyramid of Figure 10 Figure 13 1 The derived positive Orthohemipyramid 2 Construction of a Monoclinic Orthoprism from the derived positive Orthohemipyramid The corresponding negative hemipyramid will give the same result 3 Result of the construction From the protohemipyramid we can also derive a prism that is not vertical but inclined according to the angle between the vertical and clino axes It can be obtained by letting the derivation coefficient n when it refers to the clino axis become infinite The result is a clinodome i e a Form consisting of four faces parallel to the clino axis Its Weissian symbol is a b mc its Naumann symbol is and the Miller symbol is 0hl With it we have found yet another basic face namely a b mc for our list of basic faces compatible with the Monoclinic Crystal System With the monoclinic clinodome we have found yet another Form for our list of holohedric monoclinic Forms See Figure 14 Figure 14 Construction derivation of the primary Clinodome from the primary positive Protohemipyramid 1 The primary positive Protohemipyramid 2 Construction of the primary Clinodome 3 Result of the construction 4 Crystallographic axes removed From any Protohemipyramid i e from any member of the Vertical Series of Hemipyramids and indeed from any Hemipyramid whatsoever Vertical Series Clino or Orthodiagonal Series a corresponding Clinodome can be derived From the protohemipyramid can also be derived the orthohemidome by letting the derivation coefficient n when it refers to the ortho axis and which is equal to 1 in all the protohemipyramids become infinite Then the two upper faces of the hemipyramid become one face parallel to the ortho axis and intersecting the vertical axis while the two lower faces of that same hemipyramid also become one face parallel to and opposite of the first face So the result is a face pair consisting of an upper and lower face parallel to the ortho axis If we had for the derivation taken a positive hemipyramid mP then the resulting new Form consists of two parallel faces one lower front face and one upper back face And if we had also taken the corresponding negative hemipyramid mP for such a derivation then we would get a face pair also consisting of two parallel faces but of which one is a lower back face and the other an upper front face If we combine these two face pairs then we get a horizontal prism parallel to the ortho axis an orthodome So this orthodome is not a simple Form it is a combination of two orthohemidomes a positive one and a negative one The Weissian symbol for the orthohemidome is a b mc more specifically it is the symbol for the negative hemidome for the positive hemidome it is a b mc its Naumann symbol is and the Miller symbol for the positive hemidome is h0l for the negative h0l The sign in the Naumann symbol refers to that orthohemidome of which the faces lie in the acute angle beta i e the upper back and lower front faces With this we ve found yet another basic face compatible with the Monoclinic Crystal system namely a b mc With the monoclinic orthohemidome we have found yet another Form for our list of monoclinic holohedric Forms In the next figure we will derive the primary m 1 positive orthohemidome from the primary positive protohemipyramid P See Figure 15 16 and 17 Figure 15 1 The primary positive Monoclinic Protohemipyramid 2 Construction derivation of the primary positive Monoclinic Orthohemidome from the primary positive Monoclinic Protohemipyramid The green lines are just visual aids 3 The resulting primary positive Monoclinic Orthohemidome with crystallographic axes 4 Crystallographic axes and visual aids removed From any positive Monoclinic Protohemipyramid can be constructed a corresponding positive Monoclinic Orthohemidome Indeed from any Hemipyramid can be derived a corresponding Orthohemidome The negative orthohemidome can be derived from the negative protohemipyramid Figure 16 1 The primary negative Monoclinic Protohemipyramid 2 Construction derivation of the primary negative Monoclinic Orthohemidome from the primary negative Monoclinic Protohemipyramid 3 Final result of the construction From any negative Monoclinic Protohemipyramid can be constructed a corresponding negative Monoclinic Orthohemidome And as has been said from any Hemipyramid can be derived a corresponding Orthohemidome When we combine the two constructed primary orthohemidomes we will get an orthodome See Figure 17 Figure 17 The primary Orthodome as a combination of a positive and negative primary Orthohemidome Any two corresponding positive and negative Orthohemidomes can combine to form a corresponding Orthodome From the monoclinic protohemipyramid we can also derive the clinopinacoid when we let the derivation coefficient referring to the clino axis as well as the one referring to the vertical axis become infinite We then obtain a pair of faces parallel to the clino and vertical axes It can close the orthodome at its left and right sides The Weissian symbol of this Form is a b c the Naumann symbol is and the Miller symbol is 010 With all this we ve found yet another basic face compatible with the Monoclinic Crystal System namely a b c With the monoclinic clinopinacoid we have found yet another Form for our list of monoclinic holohedric Forms See figure 18 Figure 18 1 The primary positive Monoclinic Protohemipyramid 2 Construction derivation of the Clinopinacoid from the primary positive Monoclinic Protohemipyramid 3 The result of the construction The crystallographic axes are shown by red lines This Clinopinacoid can be derived from any Protohemipyramid indeed from any Hemipyramid Also from any protohemipyramid can be derived the orthopinacoid when we let the derivation coefficient referring to the ortho axis as well as the one referring to the vertical axis become infinite It consists of a face pair parallel to the ortho and vertical axes and could close the near and far ends of the clinodome The Weissian symbol is a b c its Naumann symbol is and its Miller symbol is 100 With all this we ve found yet another basic face compatible with the Monoclinic Crystal System namely a b c With the monoclinic orthopinacoid we have found yet another Form for our listing of the monoclinic holohedric Forms See Figure 19 Figure 19 1 The primary positive Monoclinic Protohemipyramid 2 Construction derivation of the Orthopinacoid from the primary positive Monoclinic Protohemipyramid 3 The result of the construction The crystallographic axes are shown by red lines This Orthopinacoid can be derived from any Protohemipyramid indeed from any Hemipyramid Finally we can derive the monoclinic basic pinacoid from any protohemipyramid by letting the derivation coefficient referring to the clino axis as well as the one referring to the ortho axis become infinite It consists of two faces parallel to the clino and ortho axes and it can close any monoclinic vertical prism at its bottom and top The Weissian symbol of the monoclinic basic pinacoid is a b c its Naumann symbol is 0P and its Miller symbol is 001 With all this we ve found yet another basic face compatible with the Monoclinic Crystal System namely a b c With the monoclinic basic pinacoid we have found the last Form for our list of monoclinic holohedric Forms See Figure 20 Remark Instead of basic pinacoid we can also write basal pinacoid Figure 20 1 The primary positive Monoclinic Protohemipyramid 2 Construction derivation of the Basic Pinacoid from the primary positive Monoclinic Protohemipyramid 3 The result of the construction The crystallographic axes are shown by red lines in the center of the image They are for reasons of clarity mirrored above and below This Basic Pinacoid can be derived from any Protohemipyramid and indeed from any Hemipyramid This concludes our derivation of all the Forms of the Monoclinic prismatic Crystal Class Summarizing we can now list all these Forms as follows The Forms of the Monoclinic prismatic Crystal Class Holohedric division of the Monoclinic Crystal System Generalized Monoclinic Protohemipyramid Generalized Monoclinic Clinohemipyramid Generalized Monoclinic Orthohemipyramid Monoclinic Protoprism Generalized Monoclinic Clinoprism Generalized Monoclinic Orthoprism Generalized Monoclinic Clinodome Generalized Monoclinic Orthohemidome Monoclinic Clinopinacoid Monoclinic Orthopinacoid Monoclinic Basic Pinacoid All these Forms can and in this case must enter in combinations with each other in real crystals FACIAL APPROACH While deriving the above Forms from the Basic Form or equivalently from its parts we also found the corresponding basic faces compatible with the Monoclinic Crystal System We will shortly derive those same Forms by subjecting these basic faces one by one to the symmetry elements of the present Class the Monoclinic prismatic Class holohedric division of the Monoclinic System The derivations will be done with the aid of the stereographic projections of faces and symmetry elements Recall that these symmetry elements were the following One mirror plane One 2 fold rotation axis perpendicular to the mirror plane and coincident with the crystallographic ortho axis Center of symmetry From the above results we will now compose a list of all the basic faces compatible with the Monoclinic System a b mc na b mc a nb mc a b c na b c a nb c a b mc a b mc a b c a b c a b c The stereographic projection of the symmetry elements of the present Class and of all the faces of the most general Form a monoclinic orthohemipyramid is given in the next Figure Also the projections of the piercing points of the clino axis i e the projections onto the projection plane of the points of intersection of the clino axis with the projection sphere are given Recall that the clino axis is not horizontal but tilted by the angle beta This angle varies with the substance that is crystallized and so does the location of the piercing points See further explanation below Figure 21 Stereogram of the symmetry elements of the Monoclinic prismatic Crystal Class and of all the faces of the most general Form There is one mirror plane solid line and one 2 fold rotation axis indicated by a pair of small solid ellipses perpendicular to it The projection of the piercing points of the clino axis a axis is given in blue the lower piercing point is represented by an open square the upper piercing point by a solid square Stereographic projection of monoclinic crystals Figures 21a 21b and 21c With the Monoclinic System we for the first time have to do with an oblique system which here means that when we put one axis in a vertical direction then either the other two axes are perpendicular to it but their directions are non orthogonal not 90 0 with respect to each other or one of the axes is not orthogonal with that vertical axis these two possibilities depending on which axis is chosen to be the vertical one Here we choose the orientation of a crystal belonging to the present Class 2 m such that the ortho axis the axis that is perpendicular to both the other two axes coincides with the 2 fold rotation axis implying that the mirror plane is oriented vertically A second axis is now oriented vertically such that the front end positive end of the third inclined axis is pointing obliquely downward and the negative end obliquely upward That second axis is now called the vertical axis or c axis while the third axis is called the clino axis In the next Figure a vertical cross section through a monoclinic crystal is depicted and also the clino and vertical crystallographic axes The face bp is parallel to the clino and ortho axes Together with its opposite counter part it forms the monoclinic basic pinacoid The face op is parallel to the ortho and vertical axes and together with its opposite counterpart it forms the monoclinic orthopinacoid Figure 21a Vertical cross section through a monoclinic crystal The directions of the clino and vertical axes are indicated as well as the angle beta between them The direction of the ortho axis is perpendicular to the plane of the drawing Two faces are indicated bp and op The elongated direction of the crystal is chosen as the direction of the c axis vertical axis We will explain some features of the stereographic projection of monoclinic crystals by considering the stereographic projection of some elements of the above Figure namely the clino axis a axis the vertical axis c axis the face bp and the face op The next Figure is a vertical section through the projection sphere along the a c plane which means that this section contains the clino axis and the vertical axis The construction of 1 the stereographic projection of the piercing points of the clino axis i e of the points of intersection of the clino axis with the projection sphere and 2 of the stereographic projection of the face poles of the bp and op faces is shown Figure 21b Vertical section through the projection sphere showing the constuction of the stereographic projection of some elements in Figure 21a Indicated are the north pole N and the south pole S of the projection sphere and the trace of the projection plane Further we see the vertical axis c axis and the clino axis a axis The piercing points of the vertical axis through the projection sphere coincide with the north and south pole The piercing points of the clino axis are indicated as a and a The stereographic projections of the latter are indicated as a and a The projection of the face bp yields the corresponding face pole on the projection sphere this face pole is in fact the piercing point of the line starting from the center of the projection sphere

    Original URL path: http://www.metafysica.nl/monoclinic_1.html (2016-02-01)
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  • Monoclinic Crystal System II
    hemimorphous Monoclinic Orthohemidome from a negative holohedric Monoclinic Orthohemidome See Figure 9 Figure 9 1 The holohedric primary negative Monoclinic Orthohemidome 2 The hemimorphous primary negative Monoclinic Orthohemidome Also this Form consists of two faces parallel to the ortho axis with a lowered symmetry with respect to the corresponding holohedric Form From the holohedric monoclinic clinopinacoid can be derived a monoclinic pedion when subjected to hemimorphy The holohedric Form consisting of a vertical face pair parallel to the clino axis decays into two independent single vertical faces parallel to the clino axis i e into two Forms each consisting of just one face See Figure 10 Figure 10 1 The holohedric Monoclinic Clinopinacoid 2 The Monoclinic Pedion This Form consists of the left half of the holohedric Clinopinacoid A second possible Form consists of the right half of the holohedric Clinopinacoid When both occur on a real crystal then they still are two independent Forms From the holohedric monoclinic orthopinacoid can be derived the hemimorphous monoclinic orthopinacoid when hemimorphy is applied The Form does not change its external shape because the mirror plane which is to be suppressed is perpendicular to its faces See Figure 11 Figure 11 1 The holohedric Monoclinic Orthopinacoid 2 The hemimorphous Monoclinic Orthopinacoid This Form consists of two faces parallel to the vertical and ortho axes Its lowered symmetry is indicated by the coloring From the holohedric monoclinic basic pinacoid finally we can derive the hemimorphous monoclinic basic pinacoid when it is subjected to hemimorphy Because the mirror plane to be suppressed is perpendicular to its faces the Form does not change its external shape but the symmetry is lowered accordingly So the hemimorphous Form still consists of two faces parallel to the clino and ortho axes See Figure 12 Figure 12 1 The holohedric Monoclinic Basic Pinacoid 2 The hemimorphous Monoclinic Basic Pinacoid This Form consists of two faces parallel to the clino and ortho axes Its lowered symmetry is indicated by the coloring The red lines indicate crystallographic directions This concludes our derivation of all the Forms of the Monoclinic sphenoidic Crystal Class by means of the merohedric approach All these Forms can and in this case must enter in combinations with each other in real crystals FACIAL APPROACH We will now derive those same Forms from the basic faces compatible with the Monoclinic Crystal System by subjecting these faces one by one to the symmetry elements of the present Class The only symmetry element of this Class is a 2 fold rotation axis The derivations will be shown by means of stereographic projections Recall from Part One that the basic faces were the following a b mc na b mc a nb mc a b c na b c a nb c a b mc a b mc a b c a b c a b c The stereographic projection of the symmetry elements of the present Class and of all the faces of the most general Form a monoclinic sphenoid is

    Original URL path: http://www.metafysica.nl/monoclinic_2.html (2016-02-01)
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  • Monoclinic Crystal System III
    derived from the back faces of the orthoprism From the holohedric monoclinic clinodome can be derived the monoclinic clinohemidome when subjected to hemihedric The Holohedric Clinodome then decays into two independent halves clinohemidomes an upper one and a lower one Remark It does not decay into a left half and a right half because then not only the center of symmetry disappears but also the mirror plane containing the clino and vertical axes which would make the symmetry of the new Form inconsistent with that of our Class See Figure 7 Figure 7 1 The holohedric primary i e with m equal to 1 Clinodome 2 A hemihedric primary Clinohemidome derived from the upper faces of the clinodome A second Form can be derived from the lower faces of the clinodome Of course the non primary i e with m unequal to 1 Clinodomes give rise to the corresponding non primary Clinohemidomes From the holohedric monoclinic orthohemidome can be derived the monoclinic orthotetartodome when it is subjected to hemihedric The holohedric orthohemidome which consists of two faces parallel to the ortho axis then decays into two independent halves each consisting of just a single face See Figure 8 Figure 8 1 The holohedric positive primary Monoclinic Orthohemidome 2 The hemihedric positive primary Monoclinic Orthotetartodome derived from the lower face of the holohedric Orthohemidome A second such tetartodome can be derived from the upper face of the Orthohemidome In the same way such tetartodomes can be derived from the negative Orthohemidome See Figure 8a Of course the non primary i e with m unequal to 1 Orthohemidomes give rise to the corresponding non primary Tetartodomes Figure 8a 1 The holohedric Monoclinic negative primary Orthohemidome It consists of two faces parallel to the ortho axis and cutting the vertical axis at unit distances 2 The hemihedric Monoclinic negative primary Orthotetartodome derived from the upper face which intersects the vertical axis in its positive i e upper half of the negative Orthohemidome A second Form can be derived from the lower face of such a hemidome i e from the face that intersects the vertical axis in its negative i e lower half Of course the non primary Orthohemidomes yield the corresponding non primary Orthotetartodomes From the holohedric monoclinic clinopinacoid can be derived the hemihedric monoclinic clinopinacoid when subjected to hemihedric The external shape does not change but the symmetry is lowered by the loss of the center of symmetry So our new Form consists of two faces parallel to the clino and vertical axes See Figure 9 Figure 9 1 The holohedric Monoclinic Clinopinacoid 2 The hemihedric Monoclinic Clinopinacoid The coloring indicates that the center of symmetry is suppressed evident for example by the fact that x is not equivalent to y but the mirror plane is still present So indeed the resulting Form is compatible with the symmetry of the present Crystal Class From the holohedric monoclinic orthopinacoid can be derived the monoclinic orthohemipinacoid when hemihedric is applied to it The holohedric orthopinacoid then decays into two independent halves each one of which consists of a single face parallel to the ortho and vertical axes See Figure 10 Figure 10 1 The holohedric Monoclinic Orthopinacoid 2 The hemihedric Monoclinic Orthohemipinacoid derived from the front face of the holohedric Orthopinacoid The center of symmetry is suppressed but the mirror plane is still present A second Form can be derived from the back face of the holohedric Orthopinacoid From the holohedric monoclinic basic pinacoid can be derived the hemihedric monoclinic basic hemipinacoid hemihedric Monoclinic Pedion when it is subjected to hemihedric The holohedric Form then decays into two independent halves Pedions an upper one and a lower one See Figure 11 Figure 11 1 The holohedric Monoclinic Basic Pinacoid consisting of two faces parallel to the clino and ortho axes 2 The hemihedric Monoclinic Basic Hemipinacoid hemihedric Monoclinic Pedion derived from the upper face of the holohedric Basic Pinacoid The center of symmetry is suppressed but the mirror plane is still present A second Form can be derived from the lower face of the holohedric Basic Pinacoid This concludes our derivation of all the Forms of the Monoclinic domatic Crystal Class by means of the merohedric approach All these Forms can and in this case must enter in combinations with each other in real crystals FACIAL APPROACH We will now derive those same Forms from the basic faces compatible with the Monoclinic Crystal System by subjecting these faces one by one to the symmetry elements of the present Class The only symmetry element of this Class is a vertical mirror plane parallel to the clino axis The derivations will be shown by means of stereographic projections Recall from Part One that the basic faces were the following a b mc na b mc a nb mc a b c na b c a nb c a b mc a b mc a b c a b c a b c The stereographic projection of the symmetry elements of the present Class and of all the faces of the most general Form a monoclinic dome is given in the next Figure Also the projections of the piercing points of the clino axis i e the projections onto the projection plane of the points of intersection of the clino axis with the projection sphere are given Recall that the clino axis is not horizontal but tilted by the angle beta This angle varies with the substance that is crystallized and so does the location of the piercing points See for further explanation of the stereographic projection of monoclinic crystals Part One Figure 12 Stereogram of the symmetry elements of the Monoclinic domatic Crystal Class and of all the faces of the most general Form There is one mirror plane indicated by a solid line as being the only symmetry element of the Class The projection of the piercing points of the clino axis a axis is given in blue the lower piercing point is represented

    Original URL path: http://www.metafysica.nl/monoclinic_3.html (2016-02-01)
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  • Triclinic Crystal System I
    nb mc In due course we ll find more of such basic faces In the next Figures we have depicted respectively the derivation of a non primary prototetartopyramid Figure 5 a non primary brachytetartopyramid Figure 6 and a non primary macrotetartopyramid Figure 7 and 7a Figure 5 Derivation of a non primary Prototetartopyramid 1 The primary Triclinic Bipyramid a combination of four simple Forms 2 Emphasis of the Form a b c by extending its two faces Extension of its faces does not change the Form crystallographically 3 Removal of the other Forms The Form emphasized and isolated is the Triclinic primary Prototetartopyramid Crystallographic axes depicted by red lines 4 Construction derivation of a Triclinic non primary Prototetartopyramid by making m 1 5 Result of the construction of a Triclinic non primary Prototetartopyramid namely a b mc The Form consists of two parallel faces Remark In reporting on the crystallography of a new triclinic mineral or whatever new crystalline substance or one that has not been recorded in the literature the convention should be followed that c a b in which c is the length of the vertical axis a the length of the brachy axis and b the length of the macro axis In that case the Form depicted in 5 of Figure 5 should be rotated such that the shortest axis becomes vertical In the above case that would be the axis there signified as the brachy axis which now becomes the vertical axis The longest axis then should become the macro axis After having oriented the Form in that way it then is a macrotetartopyramid instead of a prototetartopyramid The next Figure shows the derivation of a triclinic non primary brachytetartopyramid Figure 6 Derivation of a Triclinic non primary Brachy tetartopyramid from a Triclinic non primary Prototetartopyramid 1 A Triclinic non primary Prototetartopyramid 2 Increasing the distance measured from the origin of the axial system of intersection of the brachy axis The intersection distances with respect to the vertical and macro axes are held constant 3 Construction of the faces of the new Form 4 Result of the construction derivation of a Triclinic non primary Brachytetartopyramid 5 The same as 4 Coloration applied The three crystallographic axes are given as red lines The next two Figures show the derivation of a triclinic non primary macrotetartopyramid Figure 7 Derivation of a Triclinic non primary Macrotetartopyramid from a Triclinic non primary Prototetartopyramid 1 A Triclinic non primary Prototetartopyramid 2 Construction of the Triclinic non primary Macrotetartopyramid by increasing the intersection distance with respect to the macro axis The intersection distances with respect to the brachy and vertical axes are held constant See further next Figure Figure 7a Result of the construction in Figure 7 a Triclinic non primary Macrotetartopyramid 2 Prisms and Domes These Forms can be derived from the tetartopyramids by letting one derivation coefficient become infinite This means that their faces intersect two crystallographic axes and are parallel to a third The prisms will be derived from the tetartopyramids by letting the derivation coefficient m become infinite resulting in their faces to be parallel to the vertical axis Because they only consist of two parallel faces they are hemiprisms The domes will be derived from the tetartopyramids by letting the derivation coefficient n referring either to the brachy or to the macro axis become infinite resulting in their faces to be parallel either to the brachy axis or to the macro axis Because they consist of only two faces they are hemidomes 2a Prisms From the triclinic tetarto pyramids we can derive three types of hemi prism The Protohemiprism with its Weissian symbol a b c and its Miller symbol 110 The Naumann symbols are given in the Figures The Brachyhemiprisms na b c The Macrohemiprisms a nb c From the triclinic pyramids of the Vertical Series we can derive the triclinic protohemiprism by letting the derivation coefficient m become infinite Of course it does not matter whether we derive such a hemiprism from a primary m 1 or from a non primary m unequal to 1 prototetartopyramid See Figure 8 Figure 8 Derivation of the Triclinic Protohemiprism 1 The derived Triclinic Prototetartopyramid 2 Construction derivation of the Triclinic Protohemiprism from a Triclinic Prototetartopyramid 3 Result of the construction The Form consists of two faces parallel to the vertical axis and cutting off unit distances from the other two axes From the pyramids of the Brachydiagonal Series can in the same way be derived triclinic brachyhemiprisms by letting the derivation coefficient m become infinite See Figure 9 Figure 9 Derivation of a Triclinic Brachyhemiprism from a Triclinic Brachytetartopyramid 1 A Triclinic non primary Brachytetartopyramid 2 Construction derivation of a Triclinic Brachyhemiprism from a Triclinic Brachytetartopyramid 3 Result of the construction Again the new Form is a verical face pair From the pyramids of the Macrodiagonal Series can in the same way be derived triclinic macrohemiprisms by letting the derivation coefficient m become infinite See Figure 10 Figure 10 The Triclinic non primary Macrotetartopyramid from which the Triclinic Macrohemiprism will be derived Figure 10a and 10b Figure 10a Construction derivation of a Triclinic Macrohemiprism from a Triclinic Macrotetartopyramid Figure 10b Result of the construction of a Triclinic Macrohemiprism Above we have derived the triclinic hemiprisms from non primary triclinic upper right tetartopyramids i e from the Forms a b mc na b mc a nb mc Of course those same hemiprisms can be derived from the respective primary versions of those tetartopyramids But still other triclinic hemiprisms can be derived from a b mc na b mc a nb mc I e from the upper left tetartopyramids The other tetartopyramids the lower left and the lower right do not yield new hemiprisms as is clear from the next Figure in this case concerning a primary triclinic bipyramid but that is immaterial for the derivation of hemiprisms Figure 11 Figure 4 The primary Triclinic Bipyramid It is a complex Form that is a combination of four simple Forms each consisting of two parallel faces The simple Forms tetartopyramids are denoted by their Naumann symbols The position of the little black squares indicates the position of the face representing the Form on the Triclinic Bipyramid The symbol for the combination of the four Forms making up the bipyramid is given below the drawing of that pyramid When we let the face representing the Form and its parallel counter face together making up that Form become vertical we get the right in contradistinction to left protohemiprism Exactly this same Form is obtained however when we let the face representing the Form and its parallel counter face become vertical When we let the face representing the Form and its parallel counter face become vertical we get the left protohemiprism Exactly the same Form is obtained however when we let the face representing the Form and its parallel counter face become vertical So only two protohemiprisms are possible a right one and a left one denoted by the Naumann symbols and The same applies with regard to the brachy and macrohemiprisms except that in their case more than one right and more than one left hemiprisms are possible because of the varying of the derivation coefficient n referring either to the brachy axis or to the macro axis With the derivation of the vertical prisms we have found three more basic faces a b c na b c a nb c 2b Domes Domes are in fact tilted prisms Also here such Forms consist of only two parallel faces which makes them hemidomes There are two types of triclinic hemi domes namely brachyhemidomes a b mc i e Forms consisting of two faces parallel to the brachy axis but cutting the vertical axis either at unit distance primary hemidomes or at a rational multiple of it non primary i e derived hemidomes The same goes for the macro axis Here we can set the face at the macro axis unit distance and then the value for m referring to the vertical axis is determined as some rational multiple of its corresponding unit distance i e the unit distance associated with the vertical axis The second type of triclinic hemi dome is represented by the macrohemidomes a b mc i e Forms also consisting of two faces but which are parallel to the macro axis Like the brachyhemidomes they are not parallel to the vertical axis When they intersect the brachy axis at unit distance then the value for m referring to the vertical axis is some rational multiple of its corresponding unit distance i e the unit distance associated with the vertical axis Recall that if we have a unit face chosen for the Crystal System concerned that cuts off distances measured from the origin of the axial system a b c from the respective crystallographic axes and if we also have another face that cuts off the distances p q r then the derivation coefficients of that second face are p a q b and r c where a b and c are set as unit distances So the derivation coefficient relating to say the vertical axis namely r c is equal to the number of times that the unit distance c associated with the vertical axis fits in the vertical axis cutt off distance r of the second face The Miller indices are the reciprocals of the derivation coefficients reduced to whole numbers The minus sign where it occurs is then placed above the relevant index On this website we then set an asterix after and above that index The Miller indices together constituting the Miller symbol characterizing a face are then placed between ordinary brackets When they characterize a Form they are placed between 2b1 Brachydomes From a triclinic non primary brachytetartopyramid can be derived a triclinic non primary brachyhemidome by letting the derivation coefficient n referring to the brachy axis become infinite See Figure 12 Figure 12 Derivation of a Triclinic non primary Brachyhemidome from a Triclinic non primary Brachytetartopyramid 1 Two parallel faces defg and hijk making up a Triclinic non primary Brachytetartopyramid Red lines including blue extensions indicate the three crystallographic axes 2 and 3 Construction derivation of a Triclinic Brachyhemidome from a Triclinic Brachytetartopyramid The two faces become parallel to the brachy axis 2 Both faces of the tetartopyramid are rotated such that they become parallel to the brachy axis Here in 2 the rotation of the face defg is emphasized It is rotated red arrow about the hinge ef till it is parallel to the brachy axis becoming the face d efg It is now the front face of the Brachyhemidome In 3 the rotation of the other face of the tetartopyramid is emphasized 3 Emphasizing the rotation black arrow of the face hijk about the hinge hk till it is parallel to the brachy axis It has then become the back face hi j k of the Brachyhemidome 4 The result of the construction of a Triclinic Brachyhemidome It is a Form consisting of two faces parallel to the brachy axis but intersecting the vertical axis and the macro axis Above we have constructed the brachyhemidome from a n upper right brachy tetartopyramid but of course the same brachyhemidome can be derived from any upper right tetartopyramid Besides this right brachyhemidome yet another brachyhemidome can be derived namely from the upper left tetartopyramid It will be the left brachyhemidome Two such hemidomes together make up a brachydome i e an inclined prism parallel to the brachy axis So such a prism is a combination of two simple Forms The lower right tetartopyramid does not yield a new Form because the rotation of its front face till it is parallel to the brachy axis results in a face that is identical to the parallel counter face of the front face of the left brachyhemidome The lower left tetartopyramid also does not yield any new Form because the rotation of its front face till it is parallel to the brachy axis results in a face that is identical to the parallel counter face of the front face of the right brachyhemidome 2b2 Macrodomes From a triclinic non primary macrotetartopyramid can be derived a triclinic non primary macrohemidome by letting the derivation coefficient n now referring to the macro axis become infinite See Figure 13 Figure 13 1 A Triclinic non primary Macrotetartopyramid 2 Construction derivation of a Triclinic non primary Macrohemidome from the tetartopyramid of 1 by rotating the faces of the tetartopyramid about the respective hinges one of them formed by the line connecting the intersection point on the positive vertical axis with the intersection point on the positive brachy axis while the other hinge is formed by the line connecting the intersection point on the negative vertical axis with the intersection point on the negative brachy axis The result of this construction is shown in Figure 13a Figure 13a A Triclinic non primary Macrohemidome a Form consisting of two faces parallel to the macro axis but intersecting the brachy as well as the vertical axis Above we constructed a macrohemidome by rotating the faces of the tetartopyramid till they were parallel to the macro axis Exactly the same Form will be obtained by totating the faces of the tetartopyramid The lower tetartopyramids and give identical macrohemidomes but different from those derived from the upper pyramids So we have in fact just upper macrohemidomes which can vary with respect to the derivation coefficient m referring to the vertical axis and lower macrohemidomes Going together a lower and an upper macrohemidome will constitute a macrodome i e a combination of two simple Forms Such a macrodome is an inclined prism parallel to the macro axis With the domes we have found yet two more basic faces a b mc a b mc 3 Pinacoids A triclinic pinacoid is a face pair parallel to two of the three crystallographic axes while from the other axis the faces cut off a unit distance in the positive or negative direction of that axis associated with that axis There are three types of pinacoids s str depending on which axis is intersected Brachypinacoid a b c resp 010 Macropinacoid a b c resp 100 Basic Pinacoid a b c resp 001 They can be derived from tetartopyramids 3a Brachypinacoid The triclinic brachypinacoid can be derived from a triclinic tetartopyramid by letting its faces become parallel to the brachy and vertical axes This is most simply done by taking a brachyhemidome which is itself derived from a tetartopyramid by letting the derivation coefficient m referring to the vertical axis become infinite as is done in the next Figure Figure 14 Derivation of the Triclinic Brachypinacoid from a Triclinic Brachyhemidome 1 A Triclinic non primary Brachyhemidome 2 Construction derivation of the Triclinic Brachypinacoid from a Triclinic Brachyhemidome by letting its faces become vertical 3 Result of the construction of the Triclinic Brachypinacoid 3b Macropinacoid The triclinic macropinacoid can be derived from a triclinic tetartopyramid by letting its faces become parallel to the macro and vertical axes This is most simply done by taking a macrohemidome which is itself derived from a tetartopyramid by letting the derivation coefficient m referring to the vertical axis become infinite as is done in the next Figure Figure 15 Derivation of the Triclinic Macropinacoid from a Triclinic Macrohemidome 1 A Triclinic non primary Macrohemidome 2 Construction derivation of the Triclinic Macropinacoid from a Triclinic Macrohemidome by letting its faces become vertical 3 Result of the construction of the Triclinic Macropinacoid 3c Basic Pinacoid From any triclinic tetartopyramid a b c a b mc na b mc a nb mc can be derived the basic pinacoid by letting the derivation coefficient for a and for b become infinite which implies that the two faces now are parallel to the plane in which the brachy and macro axes lie in other words the faces are parallel to the brachy as well as to the macro axis Consequently the Weissian symbol for the basic pinacoid is a b c its Naumann symbol is 0P and its Miller symbol is 001 In the next Figure we will derive the basic pinacoid from the primary prototetartopyramid a b c Figure 16 Derivation of the Triclinic Basic Pinacoid from a Triclinic Prototetartopyramid 1 The Triclinic primary Prototetartopyramid 2 Construction derivation of the Triclinic Basic Pinacoid from a Triclinic Prototetartopyramid by letting its faces become parallel to the brachy and macro axes 3 Result of the construction of the Triclinic Basic Pinacoid With the pinacoids we have found the last basic faces a b c a b c a b c This concludes our derivation of all the Forms of the Triclinic pinacoidal Crystal Class Holohedric division of the Triclinic System All these Forms can and in this case must enter into combinations with each other in real crystals such that closed complex Forms appear So a triclinic crystal belonging to the present Class the System s most symmetric Class consists of a set of parallel face pairs together making up a complex closed Form See the next Figure Figure 17 A sketch of a holohedric triclinic crystal of the mineral Axinite Its faces are marked with Miller symbols In the above Figure we see a combination of five Forms Each face visible in the Figure has a parallel counter face and as such i e as parallel face pair makes up a Form 110 is the front face of the right protohemiprism a b c 11 0 is the front face of the left protohemiprism which is independent of the right hemiprism a b c 111 is the front face of the primary upper right prototetartopyramid a b c 11 1 is the front face of the primary upper left prototetartopyramid a b c 201 is the front face of a non primary upper macrohemidome a b 2c With all these Forms we now also have a complete list of basic faces that

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  • Triclinic Crystal System II
    is being applied From the holohedric triclinic protohemiprism we can derive the triclinic prototetartoprism when it is subjected to hemihedric This is accomplished by dropping one of the faces See Figure 4 Figure 4 Derivation of the Triclinic Prototetartoprism from the Triclinic Protohemiprism 1 The Triclinic Protohemiprism 2 Suppression of a face here the lower one resulting in the Triclinic Prototetartoprism From the holohedric triclinic brachyhemiprism we can derive a triclinic brachytetartoprism when it is subjected to hemihedric This is accomplished by dropping one of its faces See Figure 5 Figure 5 Derivation of the Triclinic Brachytetartoprism from the Triclinic Brachyhemiprism 1 The Triclinic Brachyhemiprism 2 Suppression of a face here the lower one resulting in the Triclinic Brachytetartoprism From the holohedric triclinic macrohemiprism we can derive a triclinic macrotetartoprism when it is subjected to hemihedric This is accomplished by dropping one of its faces See Figure 6 Figure 6 Derivation of a Triclinic Macrotetartoprism from a Triclinic Macrohemiprism 1 The Triclinic Macrohemiprism 2 Suppression of a face here the lower one resulting in a Triclinic Macrotetartoprism From the holohedric triclinic brachyhemidome we can derive a triclinic brachytetartodome when it is subjected to hemihedric This is accomplished by suppressing one of its faces See Figure 7 Figure 7 Derivation of the Triclinic Brachytetartodome from the Triclinic Brachyhemidome 1 The Triclinic Brachyhemidome 2 Suppression of a face here the lower one resulting in the Triclinic Brachytetartodome From the holohedric triclinic macrohemidome we can derive the triclinic macrotetartodome when it is subjected to hemihedric This is accomplished by suppressing one of its faces See Figure 8 Figure 8 Derivation of the Triclinic Macrotetartodome from the Triclinic Macrohemidome 1 The Triclinic Macrohemidome 2 Suppression of a face here the lower one resulting in the Triclinic Macrotetartodome From the holohedric triclinic brachypinacoid we can derive the triclinic brachyhemipinacoid when it is subjected to hemihedric This is accomplished by suppressing one of its faces See Figure 9 Figure 9 Derivation of the Triclinic Brachyhemipinacoid from the Triclinic Brachypinacoid 1 The Triclinic Brachypinacoid 2 Suppression of a face here the left one resulting in the Triclinic Brachyhemipinacoid From the holohedric triclinic macropinacoid we can derive the triclinic macrohemipinacoid when it is subjected to hemihedric This is accomplished by suppressing one of its faces See Figure 10 Figure 10 Derivation of the Triclinic Macrohemipinacoid from the Triclinic Macropinacoid 1 The Triclinic Macropinacoid 2 Suppression of a face here the back face resulting in the Triclinic Macrohemipinacoid From the holohedric triclinic basic pinacoid finally we can derive the triclinic pedion when it is subjected to hemihedric This is accomplished by suppressing one of its faces See Figure 11 Figure 11 Derivation of the Triclinic Pedion from the Triclinic Basic Pinacoid 1 The Triclinic Basic Pinacoid 2 Suppression of a face here the lower one resulting in the Triclinic Pedion This concludes our derivation of all the Forms of the Triclinic asymmetric Crystal Class Every Form of this Class consists of a single face so they all are open

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  • Internal Structure of Crystals I
    Iron atoms by regular translations i e shifts along straight lines in three dimensions The 3 dimensional pattern is said to be homogeneous if the angles and distances from one motif to surrounding motifs in one location of the pattern are the same in all parts of the pattern This repetitive order can be expressed by certain symmetry operations Such a symmetry operation is any motion that brings the original motif into coincidence with the same motif elsewhere in the pattern When describing this repetitive i e periodic order such symmetry operations always include translations linear shifts which of course means that there is not any point in the structure that remains in place in contradistinction to point symmetry operations in which at least one point remains in place A one dimensional ordered pattern can be obtained by regularly repeating a motif in one dimension i e along a straight line by a translation vector t 1 over and over again When this repetition is imagined to have occurred for an infinite number of times the resulting pattern will map onto itself when translated along the vector t 1 i e the pattern has translational symmetry specified by the translation vector t 1 See Figure 1 top When we in turn translate such an infinite row of motifs over and over again in another direction specified by a translation vector t 2 making an angle gamma with t 1 we obtain an infinitely extended two dimensional periodic pattern of motifs This pattern will map onto itself if translated either by the vector t 1 or by the vector t 2 or by both See Figure 1 lower image When we in turn translate this 2 dimensional array of motifs over and over again in yet another direction t 3 that does not lie in the plane defined by t 1 and t 2 we obtain an infinite three dimensional periodic array of motifs This array will map onto itself when translated by one two or all of these vectors t 1 t 2 t 3 See Figure 2 In crystals such a pattern is not exactly infinite although it is generally considered so The magnitude of the translations in inorganic crystals is on the order of 1 to 10 angstroms where one angstrom is equal to 10 8 cm It can also be expressed in nanometers A nanometer is equal to 10 7 cm because that is the scale of ionic radii in crystals This means that a dimension of 1 cm in a crystal would contain approximately 100 million translations Indeed this can be considered infinite Of course such translations cannot be seen by the naked eye It is very important therefore to realize that the external form of a crystal although an expression of its internal structure is translation free The symmetry elements observable in the external form development of crystals are therefore also translation free Figure 1 Top image Translation by a vector t 1 of a motif represented by a comma over and over again results in an infinite one dimensional periodic pattern Bottom image Translation of this one dimensional periodic pattern over and over again by the vector t 2 results in an infinite two dimensional periodic pattern Figure 2 When a two dimensional periodic pattern is translated by a vector t 3 not lying in its plane over and over again in the Figure only one such translation is shown an infinite three dimensional periodic array results For reasons of clarity the added motifs are colored red It is often convenient to ignore the actual shape of the motifs in a pattern and to concentrate only on the geometry of the repetitions in space If the motifs are replaced by points we have a regularly pattern of points that is referred to as a lattice A lattice is therefore an imaginary pattern of points or nodes in which every point node has an environment that is identical to that of any other point node in the pattern A lattice has no specific origin a beginning as it can be shifted parallel to itself In order to fully understand the three dimensional lattices on which the structure of crystals is based and to understand the assignment of motifs to these 3 D lattices i e the replacement of their points nodes by motifs having a certain shape let us first develop two dimensional lattices and see how they can be provided with two dimensional motifs generating in this way two dimensional imaginary crystals When all this is done we ll turn our attention again to real i e three dimensional crystals The reason for this approach is the following Because we are not going to tackle the problem of understanding the structure of three dimensional crystals by means of a rigorous mathematical approach in terms of symmetry theory in Mathematics this will involve the Theory of Groups but by means of a more or less morphological approach this in order to be able to make a smooth transition to the understanding of the structure of Organisms it is absolutely necessary first to consider the structure of two dimensional crystals They will provide us with most of the basics for an understanding of the structure of three dimensional crystals In fact what we re going to undertake is a complete two dimensional crystallography along morphological lines and this will take twenty of the Parts on the internal structure of crystals The rest of the present Part and Part II XX Only after these Parts we will direct our attention to three dimensional crystals i e to real crystals Remark We ve just said that we will not rigorously derive the internal structure of three dimensional crystals In fact however we will let the reader choose among both possibilities the mathematical i e group theoretic approach and the morphological approach The latter approach is in a way less appropriate for dealing with the understanding of the internal structure of three dimensional crystals because drawings intended to illustrate this structure will become messy and unclear This in contradistinction to drawings depicting the internal structure of two dimensional imaginary crystals So with respect to three dimensional crystals we are in fact forced to follow the other approach the group theoretic approach We have done so in the sequel to this website which technically is a second website accessible by the following link Continuation of this Series This link can also be found at the bottom of the left frame This mathematical approach starts with a series of documents that introduces Group Theory with extra emphasis on the symmetry groups of two and three dimensional geometric Figures and also of two dimensional periodic patterns which can represent the internal structure of two dimensional imaginary crystals After that we use group theory to create a systematic exposition of all the symmetries of three dimensional crystals including their point symmetries point groups as well as their total symmetry i e including their translational symmetries In this whole approach although being mathematical we will not aspire to present all this in full mathematical rigor but nevertheless with enough rigor to serve our purposes But because we can understand that not every reader is willing to study the fairly long series of documents on group theory and its application to Crystallography and to the study of the basic stereometric forms of organisms Promorphology also laid down on the second website we will do our best to make it nevertheless possible for those readers to gain an understanding of the internal structure of three dimensional crystals without studying group theory For such an understanding the present and following documents on the internal structure of two dimensional crystals is of paramount importance If these are studied carefully then one certainly obtains some insight into the three dimensional case without studying group theory The present and following documents are about two dimensional crystals Such crystals involve precisely 10 point groups and 17 plane groups So in a way groups are involved anyway but not so much in a group theoretic context We could say that the two dimensional patterns involved in two dimensional crystals are representations of the corresponding groups which are the main mathematical entities in group theory and these representations are examined analytically which means that the mentioned patterns are as such given and what we do is to examine the symmetries inherent in those patterns In the mentioned group theoretic approach those patterns will be moreover studied from a synthetic point of view which means that such a pattern and with it the corresponding group will be generated by one or more basic motifs representing symmetry transformations and therefore representing group elements from some one initial motif representing the identity element of the relevant group This synthetic group theoretic approach will yield much insight in those patterns and paves the way for an understanding of three dimensional patterns as we see them in real crystals Two dimensional lattices Plane Lattices Intuitive derivation of the five Plane Lattices nets Crystals are of a translative periodic nature So their internal structure must have this translative character i e this structure must be based upon a lattice Likewise their 2 D analogues Thus a 2 D lattice not only should tile the 2 D space exhaustively i e without left over spaces it must also be translative The tiles must be repeated periodically in two directions implying that the sides of the tiles must be two by two parallel There are five and only five possible unique types of plane lattices or nets See Figures below These types are distinguished from each other by the equality or non equality of the sides of the individual repeated unit unit mesh which could serve to be a building block of a particular lattice and by the angle gamma obtaining between the two non parallel translation directions a and b of such a net This angle gamma can either be special namely 90 0 120 0 or 60 0 or such that cos gamma a 2b The latter angle is such that a rectangular net is produced that is centered i e having also a lattice point in the center of each rectangle or be general i e the angle gamma is neither 90 0 nor 60 0 nor 120 0 nor such that cos gamma a 2b An angle of 45 0 implies angles of 90 0 so it does not need to be considered Equal sides imply rhomb shaped tiles with angles of 120 0 or 60 0 or square tiles with angles of 90 0 Unequal sides imply rectangular tiles or tiles having the shape of a parallelogram This parallelogram is either such that cos gamma a 2b or a parallelogram with general angles not 90 0 not 60 0 not 120 0 Where the tile is a rhombus we have a hexagonal net There the smallest repeated unit is a rhombus A larger repeated unit is a centered hexagon We will see that concerning the hexagonal net this hexagon can be chosen as the preferred building block These five nets include four fundamentally different shapes general parallelogram rectangle rhombus or hexagon square These are the four fundamentally different 2 D building blocks and they represent the four 2 D Crystal Systems The point symmtries of the five nets without motifs are 2 oblique net 2mm rectangular net 2mm centered rectangular net 6mm hexagonal net 4mm square net The next Figures depict the five planar lattices nets In these Figures a is the translation vector in the x direction b is the translation vector making an angle gamma with a A unit mesh choice is indicated red The black solid dots are not motifs but nodes of the net i e just points The five nets represent the only possible ways to arrange points periodically in two dimensions Figure 3 Oblique Net Clinonet a is not equal to b gamma is not 90 0 Row 1 the upper row of nodes is in its entirety indefinitely repeated by translations along direction x with translation distance a Figure 4 Rectangular Net Orthonet a is not equal to b gamma is 90 0 Row 1 the upper row of nodes is in its entirety indefinitely repeated by translations along direction x with translation distance a Figure 5 Centered Rectangular Net centered Orthonet Diamond Net a is not equal to b b is measured from the origin of the net along the y direction until the first node gamma i e the angle between the x direction and the y direction is such that cos gamma a 2b Row 1 the nodes along the red line is in its entirety indefinitely repeated by translations along direction x with translation distance a A centered unit mesh choice is given in red Another choice of unit mesh is given in blue This is a primitive mesh with a b gamma i e the obtuse angle between a and b is not equal to 90 0 60 0 or 120 0 Because of the possibility of this diamond shaped unit mesh choice the net is also called Diamond Net Figure 6 Hexagonal Net Hexanet a is equal to b The angle gamma between the x direction and the y direction is 60 0 Row 1 is indefinitely repeated by translations along direction x with translation distance a The smallest unit mesh choice is indicated in red it is a rhombus Another larger and centered unit mesh choice is indicated in yellow Figure 7 Square Net Tetranet a b gamma 90 0 Row 1 is indefinitely repeated by translations along direction x with translation distance a The unit mesh choice is indicated in red If we place motifs in such a net such that each motif is associated with a lattice node note that the environment of a motif s str also belongs to the motif s l then we get a Plane Group i e we get a two dimensional periodic array of motifs which can be described as a Plane Group according to the mathematical theory of Groups These motifs however must be such that their symmetry is compatible with the point symmetry of the net lattice in which they are placed Derivation of the 10 two dimensional Crystal Classes 2 D Point Groups In Part One of the series of documents on The Derivation of the 32 Crystal Classes Section Derivation of the possible rotational symmetries in Crystals we proved that in any periodic array the only rotational symmetries that can occur are 1 2 3 4 and 6 In two dimensions horizontal rotation axes as well as inversion are not possible Mirrors mirror lines are of course possible From these data we can now derive all the possible 2 D Crystal Classes Planar Point Groups This means that we are going to derive all possible combinations of the symmetry elements 1 2 3 4 6 and m The symmetry elements may each be taken individually or may be combined to produce all the two dimensional Point Groups So from the outset we have already the following 2 D Point Groups 2 D Crystal Classes 1 2 3 4 and 6 Addition of a mirror line to the symmetry content of Class 1 yields Class m Which is equivalent to 1m See Figure 8 Figure 8 Derivation of the Class m from Class 1 by adding a mirror line Addition of a mirror line to the symmetry content of Class 2 yields Class 2mm See Figure 9 Figure 9 Derivation of Class 2mm from Class 2 1 Motif pattern consistent with Class 2 The 2 fold rotation axis perpendicular to the plane of the drawing in fact it is a 2 fold rotation point because we have to do with a 2 D pattern is indicated by a small solid red ellipse 2 Addition of a mirror line to the symmetry content of Class 2 This mirror line is indicated by a red solid line 3 The added mirror line generates two more motifs as indicated The resulting motif pattern implies a second mirror plane indicated by a blue solid line The symmetry configuration is now that of Class 2mm and is equivalent with respect to symmetry to the pattern of 4 4 2mm symmetry pattern equivalent to that depicted in 3 Addition of a mirror line to the symmetry content of Class 3 yields Class 3m See Figure 10 Figure 10 A mirror line added to the symmetry content of Class 3 duplicates the motifs as indicated The resulting motif pattern implies two more mirror lines and as such is consistent with the symmetry of Class 3m The 3 fold rotation axis perpendicular to the plane of the drawing is indicated by a small red solid triangle Addition of a mirror line to the symmetry content of Class 4 yields Class 4mm See Figure 11 Figure 11 A mirror line added to the symmetry content of Class 4 will be duplicated by the 4 fold rotation axis resulting in a second mirror line perpendicular to the added mirror line These two mirror lines will duplicate the motifs as indicated The resulting motif pattern now implies two more mirror lines precisely between the added and the implied mirror lines The symmetry configuration is now consistent with that of Class 4mm The 4 fold rotation axis perpendicular to the plane of the drawing is indicated by a small red solid square Addition of a mirror line to the symmetry content of Class 6 yields Class 6mm See Figure 12 Figure 12 A mirror line added to the symmetry content of Class 6 will be multiplied by the 6 fold rotation axis resulting in three mirror lines making angles of 60 0 with each other These three mirror lines will duplicate the motifs as indicated The emerging motif pattern implies yet three more

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  • Internal Structure of Crystals II
    Class supports a Plane Group P2 and P1 respectively Square Net The Square Net 2 D Square Lattice has point symmetry 4mm so it does support 4 fold rotation and two types of mirror line Consequently it surely can accommodate the insertion of a motif that has 4mm symmetry This results in a periodic pattern that represents the Plane Group P4mm See Figure 3 Figure 3 Placing 2 D motifs with point symmetry 4mm into a square 2 D lattice yields a pattern that represents the Plane Group P4mm Figure 3a A unit mesh choice is given in yellow This unit mesh contains four 1 4 motifs which is equivalent to it containing precisely one motif This unit mesh not containing a motif in its center is primitive and is denoted by the symbol P Its point symmetry is 4mm The motif s str and motif s l are depicted in the next Figure Figure 3b Repeated motif with point symmetry 4mm in the pattern of Figure 3 This is in the present case also the figure that emerges when all translations are eliminated Its point symmetry is 4mm and as such gives the point symmetry of the pattern of Figure 3 Besides simple translation the periodic pattern representing the Plane Group P4mm contains glide lines The next Figure depicts one of these glide lines Figure 3c One of the possible glide lines in the pattern of Figure 3 The total symmetry content of the Plane Group P4mm consists of simple translations glide lines 4 fold rotations and two types of mirror line See Figure 3d Figure 3d Total symmetry content of the Plane Group P4mm All continuous lines including those that represent the lines connecting the nodes of the net are mirror lines There are two types of them diagonal and non diagonal Dashed lines signify glide lines small solid ellipses signify 2 fold rotation axes perpendicular to the plane of the drawing and small solid squares signify 4 fold rotation axes perpendicular to the plane of the drawing There are still several other motifs having different symmetries that can be accommodated in a Square Net These motifs have lower symmetry than the one discussed above The resulting periodic pattern expresses the motif s lower symmetry One of these periodic patterns is especially instructive for a general understanding of two dimensional periodic arrays i e for an understanding of two dimensional crystals and with it an understanding of three dimensional crystals It is the Plane group P4gm So we will dwell a little longer upon one of the patterns representing this Plane Group Figure 4 A periodic pattern of motifs based on a 2 D square lattice net This particular pattern represents the Plane Group P4gm It must be imagined to extend indefinitely over the 2 D plane Figure 4a A unit mesh choice for the above pattern is given in yellow The pattern consists of some element i e a proper motif already present in but yet to be found and indicated which is repeated by translation vector a in the direction x and by translation vector b in direction y In the present case the length of a is equal to that of b When we eliminate all translations from the pattern of Figure 4 i e when we telescope the structure inward till all translations are zero the following figure shape emerges Figure 4b The symmetry of this figure expresses the point symmetry associated with the Plane Group P4gm It is the translation free residue of this Plane Group The point symmetry of this Figure is 4mm so the translation free symmetry of the pattern of Figure 4 i e the point symmetry associated with it is 4mm This point symmetry is the external symmetry of the structure It is the macroscopically visible symmetry because the translations are too small to be detected by the naked eye We here imagine 2 D crystals to exist in reality and consisting of atoms ions or complexes thereof like 3 D crystals do The pattern is supposed to be a regular array of motifs i e a repetition of a motif along two directions x and y with translation vectors a and b respectively As can be seen in Figure 4 the motif is not repeated because its orientation differs from place to place So we must still find out what in fact is the motif that has been exactly repeated throughout the structure In the next Figure the structure is again depicted but now some areas are highlighted in order to assist our search for the repeated motif i e the genuine motif Figure 4c Some areas in the pattern representing Plane Group P4gm are highligted in order to determine the actual motif that is being repeated throughout the structure Each highlighted area can be thought to be associated with a node of the net So there should be much more such areas each one of them associated with a node In this way the nodes remain equivalent to each other also when as in the present case motifs are inserted in the net The next Figure shows the four areas indicated above and their association with the nodes of the net Figure 4d The association of the four highlighted areas with the nodes red s t u v of the square net Each node must have such an associated area in order to remain equivalent Figure 4e determines the actual motifs that are being repeated These genuine motifs each consist of the motif s str itself consisting of several motif units and the motif s l the motif s str its proper surroundings One such motif is depicted Figure 4e A motif when repeated along the vectors a and b See Figure 4a generates the periodic pattern of Figure 4 The next Figure illustrates four such motifs Figure 4f Four genuine equivalent motifs highlighted with coloration each associated with a lattice node Not only these four lattice

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