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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.
A Brief Introduction of the Crystal Structure
Published in Prem Pal, Kazuo Sato, Silicon Wet Bulk Micromachining for MEMS, 2017
A symmetry operation is an operation that can be performed either physically or imaginatively that results in no change in the appearance of an object, that is, a crystal remains invariant under a symmetry operation [1,2,3,4–5]. The symmetry operations performed about a point (or a line) are called point group, for example, rotations, reflections, and inversions (no translations). If the symmetry operations performed by translation as well (i.e., point group operations + translations) are categorized as space group. Crystals exhibit both types of symmetries independently and in compatible combinations. These operations are briefly described below: Translation: A move of one cell in each of three axis directions restores the structure.Rotations: A lattice is said to possess the rotation symmetry if its rotation by θ about an axis (or a point in a two-dimensional lattice) transform the lattice into itself. If an object can be rotated about an axis and repeats itself every 90° of rotation then it is said to have an axis of 4-fold rotational symmetry. The axis along which the rotation is performed is an element of symmetry referred to as a rotation axis. The possible rotational symmetry which are compatible with the requirement of translation symmetry to build the long-range order of a crystalline solid are 1-fold, 2-fold, 3-fold, 4-fold, and 6-fold. Long-range order is synonymous with periodicity, requiring some unit structure which repeats itself by translation in all directions infinitely. Although objects themselves may appear to have 5-fold, 7-fold, 8-fold or higher-fold rotation axes, these are not possible in crystals as it is not possible to fill the area of a plane with a connected array such as pentagons, octagons, etc., as shown in Fig. 1.5. It means a single molecule can have any degree of rotational symmetry, but an infinite lattice cannot. Reflection: Reflection across a plane restores the structures.Inversion: A symmetry operation in which each point of an object is converted to an equivalent point by projecting through a common center (called center of inversion or center of symmetry) and extending an equal distance beyond this center. If the center of symmetry is at the origin of the coordinates, every point (x,y,z) becomes (−x, −y, −z). The number of point groups (i.e., combination of certain symmetry operations, such as rotation, reflection, and inversion) in two and three dimensions is 10 and 32, respectively.The number of distinct space groups (i.e., group of all symmetry operations) possible in two and three dimensions is 17 and 230, respectively.
A kinetic model and parameters estimate for the synthesis of 2-phenyloctane: a starting material of bio-degradable surfactant
Published in Indian Chemical Engineer, 2023
Sudip Banerjee, Md Aurangzeb, Amit Kumar
The symmetry element and operation define the symmetry of molecules. The symmetry element is a theoretical concept that includes the line, plane and point about which the symmetry operation can be performed. The symmetry operation includes rotation, reflection, inversion and rotation-reflection movement about the symmetry element such that the object is indistinguishable after the procedure. The present work is limited to the operation of rotation. It is designated as Cn, which indicates that the rotation by 2π/n brings the object into an equivalent position. It is commonly referred to as n-fold rotation and the corresponding symmetry element as an n-fold rotational axis. For instance, C4 indicates a 4-fold rotation, 4-fold rotational axis and rotation through 2π/4 (=90°) brings the object into an equivalent position. In general, for linear alkene and phenylalkane, there is no internal axes rotation except methyl bond. Also, during protonation and de-protonation of alkene, the shifting of the methyl group does not occur, so the internal symmetry of reactant and activated complex is identical.