China
Africa
Explore chapters and articles related to this topic
Geometry of Rotation
Published in Kenichi Kanatani, 3D Rotations, 2020
There exist linear transformations that preserve lengths and angles but not the sense. Such a transformation is a composition of a rotation and a reflection (Fig. 2.1(b)). A reflection is a mapping into a position symmetric with respect to some plane passing through the origin. If a rotation is composed with a reflection, a right-handed system {a, b, c} is mapped to a left-handed system {a′, b′, c′}. The volume of the parallelepipeds they define have reversed signs; |a′, b′, c′| = −|a, b, c|. Composition of a rotation and a reflection is sometimes called improper rotation, while the usual rotation is called proper rotation. In this book, rotations always mean proper rotations.
Crystalline Structure of Different Semiconductors
Published in Jyoti Prasad Banerjee, Suranjana Banerjee, Physics of Semiconductors and Nanostructures, 2019
Jyoti Prasad Banerjee, Suranjana Banerjee
A particular symmetry operation is rotation about an axis through a lattice point that leaves crystal lattices invariant. In this case, the character of the object remains unchanged by the operation, i.e., a right-handed object is repeated or self-coincident as a right-handed object after the rotational operation. The objects forming such a set are said to be congruent. The axis about which the rotational operation brings the object into self-coincidence is called a rotation axis, and such rotation is called proper rotation, and the rotation axis is called as a proper rotation axis; on the other hand, when a rotational operation relates enantimorphous objects such that right-handed object becomes a left-handed one, then the operation is called improper rotation and the corresponding symmetry element is called as improper rotation axis.
The Group Theory
Published in Mikhail G. Brik, Chong-Geng Ma, Theoretical Spectroscopy of Transition Metal and Rare Earth Ions, 2019
Mikhail G. Brik, Chong-Geng Ma
Let us consider a cube (Fig. 7.1) as an example of a highly symmetric object. The following symmetry elements can be distinguished: The mirror planes (or the reflection planes). They divide the cube into two equal halves, and each half is a mirror image of its counterpart. The mirror planes shown in Fig. 7.1 are in the xy plane (a horizontal mirror plane), in the zx plane (a vertical mirror plane) and a diagonal mirror plane, which goes through the z axis and diagonals of the cube bases. The standard notations for these mirror planes are σh, σν, and σd, where the horizontal, vertical, and diagonal planes are denoted by the h, v, and d subscripts, correspondingly. Figure 7.2 illustrates the difference between these types of the reflection planes.The axes of rotation. The axis of rotation is an axis, which transforms the cube into itself after rotation through a definite angle about this axis. The angle of rotation is 2π/n radians (n is an integer number). Such an axis is called an n-fold rotation axis and is denoted as Cn. The axes of the fourth, third, and second orders are shown in Fig. 7.1.The inversion center. In this case, it is the center of the cube. Each point of the cube becomes another point of the cube at the opposite side after inversion through this center. This symmetry element (denoted as I) is of particular importance, since the presence of the inversion center allows to classify the wave functions of the considered system as the odd and even functions.Improper rotation. This is a combination of the rotation through 2π/n radians about an axis with subsequent reflection in the plane perpendicular to this axis (Fig. 7.3). The improper axes are denoted as Sn (the capital letter S comes from the word “Spiegel,” which means a “mirror” in German). It is easy to see that the order, in which the rotation and reflection are performed, is not important for a final result of the improper rotation. The improper rotation concept allows for representing inversion I as an improper rotation of the second order (rotation through 180° and reflection in the plane perpendicular to the axis of rotation): I = C2σh = σhC2.
Understanding ground and excited-state molecular structure in strong magnetic fields using the maximum overlap method
Published in Molecular Physics, 2023
Meilani Wibowo, Bang C. Huynh, Chi Y. Cheng, Tom J. P. Irons, Andrew M. Teale
In the presence of a magnetic field, the unitary symmetry point group of the molecule may be restricted to one of its subgroups since only symmetry operations that leave the combined molecule and field unchanged remain. In fact, the unitary symmetry group of the system in a uniform magnetic field is given by the intersection where is the zero-field unitary symmetry point group of the molecule and is the well-known unitary symmetry group of the uniform magnetic field in the absence of any other external potentials [68,69]. In general, only proper or improper rotation axes parallel to the field, mirror planes perpendicular to the field, and the centre of inversion, if present, will remain. As such, the reduction depends on the orientation of the molecular frame relative to the direction of the applied magnetic field. Furthermore, it can be shown that all possible unitary symmetry groups in the presence of a magnetic field are Abelian [68,69].
Symmetry arguments and the totalitarian principle in the physics of liquid crystals and other condensed matter systems
Published in Liquid Crystals, 2023
Tianyi Guo, Xiaoyu Zheng, Peter Palffy-Muhoray
Symmetry is defined in terms of operations which leave something unchanged. In chemistry, the traditional perspective is that a symmetry operation is an action which leaves an object – typically a molecule – unchanged. The operation may be the identity , an fold rotation about an axis, reflection about a plane of symmetry , inversion through the centre, an improper rotation (a proper rotation followed by a reflection) and combinations of these. In physics, a useful approach is to define a symmetry operation as an action that leaves a representation unchanged. Representations are specifications in a coordinate system – such as the components of a vector, or elements of a tensor. In the simplest case, the operations would be translation, rotation and inversion about the origin. In this paper, we focus on rotation and inversion – with emphasis on the latter.
On the possibility of frozen nuclei
Published in Molecular Physics, 2021
The structure of the groups , and is more difficult. They belong to MS groups of type-II and cannot be written as a direct product of their permutation subgroup and the inversion group, like Equation (13a). Yet, these groups decompose according to [54, 74] In all cases, Equations (19a), (19b), and (19c), the permutation subgroups are, again, isomorphic to the groups or . MS groups that are isomorphic to the improper rotation groups with even are exceptional: They can be written neither as a direct nor a semi-direct product of their permutation subgroup and one other subgroup of order two. Since these groups are cyclic, however, their permutation subgroups are necessarily cyclic as well; they are isomorphic to . As we show in Section 3, this is the only argument we need to prove that the frozen-nuclei approximation is physically incorrect.