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Mathematics and Crystals

Mathematics and Crystals. By Carrie Barkhouse. What is Mathematical Crystallography?.

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Mathematics and Crystals

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  1. Mathematics and Crystals By Carrie Barkhouse

  2. What is Mathematical Crystallography? Mathematical crystallography deals with the fundamental properties of the symmetries of crystals, as well as the physical properties of crystals. It is used to classify crystals according to their symmetry, and geometric development. It is also used to help solve crystallographic problems, by using mathematical methods and/or concepts. In some cases, mathematical problems can be solved due to the contribution of mathematical crystallography by using crystallographic methods. Crystal: a solid body enclosed by symmetrically aligned plane surfaces, that intersect at angles.

  3. Systems of Crystals Crystal systems classify crystals based upon their observed or external symmetry, and according to the geometric form in which crystals grow. Christian Weiss (1780-1856) created crystal systems by grouping crystals into 7 different crystal systems. There are exactly 230 types of crystallographic groups, known as the 230 space groups. 32 types of subgroups, the 32 point groups, correspond to all crystallographic groups. Each crystal class, also known as point group, has different symmetry.

  4. The 7 Systems of Crystals Monoclinic Cubic Triclinic Tetragonal Hexagonal Orthorombic Rhombohedral

  5. The 7 Systems of Crystals

  6. The 32 Point Groups A point group is a set of symmetry operations where as each atom of a crystal is moved to the location of another atom, a point remains. Therefore, after any operations in its point group, a crystal would keep the same appearance. In crystallography, there are many 3D point groups. Though, there are restrictions, known as the crystallographic restriction theorem, which result in 32 crystallographic point groups. The 32 point groups can be placed in the 7 crystal systems.

  7. Schönflies’ notation Point groups are noted by their symmetries. Schönflies’ notation can be used as a notation of crystallography. Schönflies’ notation: point groups are noted by a letter and a subscript. O (octahedron): the group has the symmetry of an octahedron or cube. T (tetrahedron): the group has the symmetry of a tetrahedron. Td: contains improper operations. T: does not contain improper operations Th: T with an inversion. Cn(cyclic): the group has an n-fold rotation axis. Cnh: Cn with a mirror plane perpendicular to the axis of rotation. Cnv: Cn with a mirror plane parallel to the axis of rotation. Sn (Spiegel, German for mirror): the group only contains an n-fold improper rotation - angle of rotation is 360°/n. Dn (dihedral): the group contains an n-fold rotation axis and a two-fold axis perpendicular to that axis. Dnh: includes a mirror plane perpendicular to the n-fold axis. Dnv: includes mirror planes parallel to the n-fold axis.

  8. Crystallographic Restriction Theorem Crystallographic restriction theorem: n (fold) = 1, 2, 3, 4, or 6 in 2 or 3 dimension space. D4d and D6d: include improper rotations - combination of a rotation about an axis and a reflection in a plane perpendicular to the axis, therefore they are forbidden. The 27 point groups in the table, along with T, Td, Th, O and Oh are the 32 crystallographic point groups.

  9. Mathematical ProofCrystallographic Restriction Theorem It has been proved that the only possible values of α in the 0° to 180° range are 0°, 60°, 90°, 120°, and 180°. The only allowed rotations are given by 2π/n, where n = 1, 2, 3, 4, 6. This matches to 1-, 2-, 3-, 4-, and 6-fold symmetry. Therefore 5-fold/greater than 6-fold symmetry is not possible.

  10. The 230 Space Groups Space group: describes the symmetry of a crystal. Space groups define how symmetry elements are aligned in space. They help to determine a crystal’s structure atomically. There are 230 types of crystallographic space groups; a crystal can include one of the 230 types. Space groups represent all existing types of crystals and 3D structures.

  11. Works Cited • http://mathforum.org/alejandre/geometrylist.html • http://www.emis.de/journals/BJGA/10.1/bt-tam.pdf • http://www.staff.uni-marburg.de/~fischerw/mathcryst.htm • http://www.materials.ac.uk/elearning/matter/Crystallography/3dCrystallography/7crystalsystems.html • http://en.wikipedia.org/wiki/Crystal_system • http://en.wikipedia.org/wiki/Crystallographic_point_group • http://en.wikipedia.org/wiki/Space_group

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