China
Africa
Explore chapters and articles related to this topic
Introductory Concepts
Published in Dragica Vasileska, Stephen M. Goodnick, Gerhard Klimeck, Computational Electronics, 2017
Dragica Vasileska, Stephen M. Goodnick, Gerhard Klimeck
The crystallographic point group or crystal class is the mathematical group comprising the symmetry operations that leave at least one point unmoved and that leave the appearance of the crystal structure unchanged. These symmetry operations can include reflection, which reflects the structure across a reflection plane; rotation, which rotates the structure a specified portion of a circle about a rotation axis; inversion, which changes the sign of the coordinate of each point with respect to a center of symmetry or inversion point; and improper rotation, which consists of a rotation about an axis followed by an inversion. Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively called symmetry elements. There are 32 possible crystal classes. Each one can be classified into one of the seven crystal systems.
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
The permitted combinations of point symmetry elements, or external symmetry elements, form crystallographic point groups. There are 32 crystallographic point groups, which can be classified into 7 crystal systems based on the presence of the characteristic symmetry elements.
Excited states of molecular and crystalline acetylene: application of TDHF and BSE via density fitting methods
Published in Molecular Physics, 2021
The T crystallographic point group of the space group to which the cubic phase of CH belongs, has A, E, T, A, E and T representations. The 14 lowest excitation energies for the cubic phase of CH are shown in Table 6, together with their degeneracy (as T-triply, E-doubly or A-singly degenerate) and their optical activity. The excited states are grouped as 12 low energy states (including degeneracy factors) around 7.3 and 7.8 eV and they split into 4 and 8 states at these energies. Only the triply degenerate state at 7.77 is (weakly) optically active in this energy range. This appears as a small feature in the optical absorption spectrum in Figure 1(a). The next optically active excitation is the strongest optical absorption predicted in the VUV energy range at 10.15 eV and there is a weaker absorption at 10.40 eV.
Effect of pressure on the mechanical and dynamical stability of NaAlSi
Published in Philosophical Magazine Letters, 2019
H. Y. Wu, Y. H. Chen, P. F. Yin, Z. R. Zhang, X. Y. Han, X. Y. Li
It is not sufficient to deduce structural stability from mechanical stability alone. A crystalline structure is stable if all its phonon modes have positive frequencies for all wave vectors [30]. To examine the dynamical stability of NaAlSi, the calculated phonon-dispersion curves along several high-symmetry directions in the Brillouin zone at ambient pressure for the relaxed crystal structure are shown in Figure 4a. The crystallographic point group of NaAlSi is D4h, and the vibrational modes at Γ are presented as: Γphonon = 3Eg + 3Eu + 3A2u + 2A1g + B1g, where g and u are Raman- and infrared-active modes respectively. Because there are six atoms in the primitive unit cell, there exist 18 normal modes for each q point in the Brillouin zone, where 3 are acoustic modes and 15 are optical modes. Our calculated frequencies of zone-center optical phonon modes, together with previous theoretical results, are presented in Table 3. It is clearly seen that our calculated results are consistent with other previous theoretical results [14,15]. Obviously, the full dispersion relations have positive frequencies at any vectors, suggesting that the structure of NaAlSi is dynamically stable at zero pressure.