ABSTRACT
Abstract ................................................................................................... 80 2.1 Introduction .................................................................................... 81 2.2 A Brief History of Crystallography: The Influence in
Art and Philosophy ......................................................................... 83 2.3 The Symmetry of the Isolated Bodies ............................................ 88 2.4 The Crystals’ Metrics ..................................................................... 99 2.4.1 Mathematical Point Networks: Ideal Crystals .................... 99 2.4.2 Crystallographic Systems ................................................. 105 2.4.3 The Reducing Cell Method ...............................................110 2.4.4 The Bravais Lattice ...........................................................115 2.5 Point Groups .................................................................................118 2.5.1 Symmetry Formulas: Symmetry Classes ..........................118 2.5.2 Schoenflies Notation ........................................................ 121 2.5.3 The International (Hermann-Mauguin) Notation ............. 126 2.5.4 The Crystals Morphology and Its Structural Laws .......... 129 2.5.4.1 The Crystalline Habitus .................................... 129 2.5.4.2 The general Laws of the Crystalline Shape ........133 2.5.5 The Crystallographic Indexing ......................................... 138 2.5.5.1 Miller Indices .................................................... 138 2.5.5.2 The Weiss Zone Law of Crystal Faces .............. 142 2.5.5.3 Crystallographic Formulae ............................... 147 2.5.5.4 The Miller-Bravais Indices ............................... 149
2.5.6 Stereographic Projection. The Wulff Map ....................... 152 2.5.7 The Crystallographic Groups ........................................... 158 2.5.7.1 Holohedric and Merihedric Classes .................. 158 2.5.7.2 The Crystallographic Forms ............................. 160 2.5.7.3 Correlated crystallographic systems and classes ... 166 2.5.8 Group Symmetry Perturbation ......................................... 176 2.5.8.1 The Crystallographic Anisotropy ...................... 176 2.5.8.2 The Curie Principle of Symmetry ..................... 186 2.5.8.3 Optical Activity by Crystals .............................. 187 2.6 Space Groups ............................................................................... 192 2.6.1 Introduction to the Space Groups ..................................... 192 2.6.1.1 Helicals and the Glide Planes ........................... 192 2.6.1.2 The Symbolistics for the Space Groups ............ 199 2.6.1.3 The Pearson Classification ................................ 213 2.6.2 The Crystallographic Description of the Spatial Groups ....... 215 2.6.2.1 The Symmetry Analysis of the Space Groups.......215 2.6.2.1 Determination of a Space Group Symmetries .......221 2.6.3 Extensions of the Space Groups ....................................... 242 2.6.3.1 The Structural Class .......................................... 242 2.6.3.2 The Magnetic/Colored Groups ......................... 246 2.7 Conclusion ................................................................................... 249 Keywords .............................................................................................. 251 References ............................................................................................. 252 Author’s Main Reference ............................................................. 252 Specific References ...................................................................... 252
ABSTRACT
The crystal (motif + lattice) system is a analyzed on its stable ground state by means of geometrical characterization of axes, inversion center and mirroring planes, as the main symmetrical elements driving the
allowed operation in such ordered giant molecular structure: they where so provided the crystallographic systems (7), then extended to Bravais lattice (14), then enlarged to the point groups (32) when all elements of symmetry that intersects on a point are comprehensibly combined, and finally to the spatial groups (230) when the translation comes into the play too; these classification schemes correspond in fact with the most resumed way of chemical classification of substances, by their crystalline groups (points or spatial) and corresponds with the maps obtained by X-ray diffraction or by other optical action (optical activity by laser action or by magnetization) especially through the involved reciprocal space (with the allied theorems of faces represented by normal axes and axes represented by nodes in reciprocal space and of its projective maps) – there where the quantum behavior is recovered by means of wave vector, reciprocally related with the acting yet in resonance with electronic wave functions, specific to living electrons in crystals and ordered solids.
