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Chemical Bond III: Complements
Published in Franco Battaglia, Thomas F. George, Understanding Molecules, 2018
Franco Battaglia, Thomas F. George
In the case of a crystal, it has both point and (discrete) translational symmetry properties, and these properties are used upon classifying crystals according to their structure. To explain how crystal lattices are classified, we need to introduce the concept of a Bravais lattice. To a given structure constituted by an elementary unit repeated in space, it is associated a lattice as follows. An arbitrary point in the structure is chosen and then it is considered the set of all points equivalent to the one chosen: the set of these points are, by definition, the Bravais lattice of the given structure. Therefore, peculiar to the Bravais lattice is that each point has a surrounding equal to the surrounding of any other lattice point. The real crystal structure is obtained from its Bravais lattice equipping each Bravais lattice point with a basis, i.e., with an atom or a set of atoms which, once located on each Bravais lattice point, generate that structure, as we shall readily clarify with examples.
Metal Crystals—I Periodicity
Published in Alan Cottrell, An Introduction to Metallurgy, 2019
Because of its periodicity, a crystal has translational symmetry. Let us take a space lattice and somewhere in this infinite framework mount the crystal which it represents. We now slide the whole crystal, without rotation, to another part of the lattice by moving it along lattice vectors, as defined in eqn. 17.1. Then the association of the crystal sites with lattice points is exactly the same as before. This translational symmetry appears here as a purely mathematical property but it also has great practical importance in connection with the plastic properties of crystals.
Symmetry of Crystals, Point Groups and Space Groups
Published in Dong ZhiLi, Fundamentals of Crystallography, Powder X-ray Diffraction, and Transmission Electron Microscopy for Materials Scientists, 2022
Translational symmetry describes the periodic repetition of a motif or basis across a length or through an area or volume. Crystallographic point symmetry, on the other hand, describes the periodic repetition of atoms (ions) around a point. As presented in many textbooks, the symmetry elements in point symmetry are discussed first. By definition, a symmetry element is a geometrical entity about which a symmetry operation is performed.
Unique solutions, stability and travelling waves for some generalized fractional differential problems
Published in Applied Mathematics in Science and Engineering, 2023
Mahdi Rakah, Yazid Gouari, Rabha W. Ibrahim, Zoubir Dahmani, Hasan Kahtan
Traveling waves can exhibit different types of symmetry, depending on the characteristics of the wave and the medium through which it travels. Here are some examples of symmetry in travelling waves: Reflection symmetry: A wave has reflection symmetry if it looks the same when reflected in a line or plane. For example, a sinusoidal wave that travels along a string has reflection symmetry because it looks the same when reflected in the string.Translational symmetry: A wave has translational symmetry if it looks the same after it has been shifted by a fixed distance. For example, a wave that travels along an infinitely long string has translational symmetry because it looks the same at any point along the string.Rotational symmetry: A wave has rotational symmetry if it looks the same after it has been rotated around a fixed point. For example, a circular wave that travels outward from a point source has rotational symmetry because it looks the same when viewed from any angle around the source. Time-reversal symmetry: A wave has time-reversal symmetry if it looks the same when time is reversed. For example, an electromagnetic wave that travels through free space has time-reversal symmetry because it looks the same whether it is moving forward or backward in time. In addition, the Duffing equation is symmetric under the transformation , which is known as the parity transformation. This means that if the displacement x of the oscillator is replaced with its negative, the equation remains unchanged.