Lecture

Crystal Symmetry and Structures

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In course

CH-450: Solid state chemistry and energy applications

You will learn about the bonding and structure of several important families of solid state materials. You will gain insight into common synthetic and characterization methods and learn about the applicability of several classes of materials in energy rele ...

Description

This lecture covers point symmetry elements, crystal systems, Bravais lattices, and space groups. It discusses the 7 crystal systems, 14 Bravais lattices, and 32 point groups, including rotational axes, mirror planes, and inversion centers. The lecture also explores translational symmetry, glide reflections, screw axes, and symmetry identification in crystals. It delves into the 230 possible space groups, emphasizing the importance of finding the highest symmetry in crystal structures. The presentation concludes with a detailed analysis of solid state structures, focusing on close-packed arrangements and common crystal structures like rock salt.

Instructor

Wendy Lee Queen

Teaching & PhD PhD Students Pratap Narayan Soni, Pattaraphon Rodlamul, Matthieu Meoli, Olivier Thomas Smith, Sanjay Venkatachalam, Anne Faustine Sido Belin, Christopher William Gardner, Lorenzo Meneghin Past EPFL PhD Students Mehrdad Asgari, Daniel Teav Sun, Ilia Kochetygov, Vikram Vinayak Karve, Anita Justin, Till Marian Schertenleib, Jocelyn Richard Roth, Timo Markus Oskar Felder Courses Chemical engineering practice ChE-309 This course provides the students with practical experience wit

Official source

Ontological neighbourhood

Mathematics

Geometry: Symmetry

Related concepts (32)

Rotational symmetry

Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation. Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other spheroids.

Symmetry group

In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X). For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space.

Symmetry

Symmetry () in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations, such as translation, reflection, rotation, or scaling. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article.

Symmetry (geometry)

In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). Thus, a symmetry can be thought of as an immunity to change. For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable.

One-dimensional symmetry group

A one-dimensional symmetry group is a mathematical group that describes symmetries in one dimension (1D). A pattern in 1D can be represented as a function f(x) for, say, the color at position x. The only nontrivial point group in 1D is a simple reflection. It can be represented by the simplest Coxeter group, A1, [ ], or Coxeter-Dynkin diagram . Affine symmetry groups represent translation. Isometries which leave the function unchanged are translations x + a with a such that f(x + a) = f(x) and reflections a − x with a such that f(a − x) = f(x).

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