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The seven lattice systems and their Bravais lattices in three dimensions. In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (), [1] is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
The oblique lattice is one of the five two-dimensional Bravais lattice types. [1] The symmetry category of the lattice is wallpaper group p2. The primitive translation vectors of the oblique lattice form an angle other than 90° and are of unequal lengths.
The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 2 1 is a 180° (twofold) rotation followed by a translation of 1 / 2 of the lattice vector. 3 1 is a 120° (threefold) rotation followed by a translation of 1 / 3 of ...
These lattices are classified by the space group of the lattice itself, viewed as a collection of points; there are 14 Bravais lattices in three dimensions; each belongs to one lattice system only. They [ clarification needed ] represent the maximum symmetry a structure with the given translational symmetry can have.
The 17 wallpaper groups, with finite fundamental domains, are given by International notation, orbifold notation, and Coxeter notation, classified by the 5 Bravais lattices in the plane: square, oblique (parallelogrammatic), hexagonal (equilateral triangular), rectangular (centered rhombic), and rhombic (centered rectangular).
However, in real materials there are deviations from this in some metals where the unit cell is distorted in one direction but the structure still retains the hcp space group—remarkable all the elements have a ratio of lattice parameters c/a < 1.633 (best are Mg and Co and worst Be with c/a ~ 1.568).
The rectangular lattice and rhombic lattice (or centered rectangular lattice) constitute two of the five two-dimensional Bravais lattice types. [1] The symmetry categories of these lattices are wallpaper groups pmm and cmm respectively. The conventional translation vectors of the rectangular lattices form an angle of 90° and are of unequal ...
Leave out the Bravais lattice type. Convert all symmetry elements with translational components into their respective symmetry elements without translation symmetry. (Glide planes are converted into simple mirror planes; screw axes are converted into simple axes of rotation.) Axes of rotation, rotoinversion axes, and mirror planes remain unchanged.