Geometrical symmetry

For plane stress in the 1-2 plane of a unidirectional lamina with fibers in the 1-direction, < = T. 3 = r23 = 0- However, from the cross section of such a lamina in Figure 2-39, Y = Z from the obvious geometrical symmetry of the material construction. Thus, Equation (2.126) leads to... 

Lu Q, Hu J, Tang K, Qian Y, Zhou G, Liu X (2000) Synthesis of nanocrystaUine CuMS2 (M = In or Ga) through a solvothermal process. Inorg Chem 39 1606-1607 Wu C, Yu S-H, Antoniette M (2006) Complex concaved cuboctahedrons of copper sulfide crystals with highly geometrical symmetry created by a solution process. Chem Mater 18 3599-3601... 

The permeability coefficients, PD and PR, are influenced by hydrodynamics. Depending upon the geometric symmetry or asymmetry of stirring in the donor and receiver chambers, their values may be equal or unequal. To analyze these situations, let us define PAm, as the effective permeability coefficient of the ABLs therefore, the geometric average of the mass transfer resistance of the ABLs is... 

In those early days, when computer power was limited, often use was made of a symmetry assumption each quarter of the vessel containing one of the four baffles at the vessel wall was supposed to behave identically hence, a steady flow in the RANS approach was simulated in just a quarter vessel. Such strong simplifications are no longer in use. Precessing vortices moving around the vessel centerline contribute to flow unsteadiness and, therefore, exclude models that just assume flow steadiness or allow for domain reductions through geometrical symmetries. The most correct response to this flow unsteadiness is the concept of LES. 

In polyatomic molecules, the geometric symmetry of the molecule also plays a very important role. For example, the benzene molecule, which is the example we discuss in this book (Figure 6.1) has the point group symmetry D6h. 

The double degeneracy of NBMOs in m-[8] has nothing to do with the geometrical symmetry of the molecule, but, rather, with the connectivity of the two radical centres, or the phase relationship of the atomic orbitals in the conjugated system. Therefore, the term topological symmetry has been proposed to describe the connectivity of the carbon atoms carrying the n-electrons and the periodicity of the Jt-orbitals in this class of non-Kekule hydrocarbons. 

The author believes it is not wise to take advantage of only geometrical symmetry synthetic efforts should be directed towards triplet and quartet D or A molecules due to topological symmetry, e.g. [47] and [48] (Azuma et ai, 1974 Kuhn et al., 1966). Of special interest is a stable perchloro-derivative of the Schlenck hydrocarbon [11]. Due to restricted rotation of the aryl groups, the biradical exists in two stereoisomeric forms meso and dl. They... 

While crystal structures can be characterised by the geometric symmetries repeated throughout the lattice, the overall physical properties... 

It can be shown that for the cross-terms 221 = 2i2, 2si = 2b. and so on, so that of the initial 36 values, there are only 21 independent elastic constants necessary to completely define an anisotropic volume without any geometrical symmetry (not to be confused with matrix symmetry). The number of independent elastic constants decreases with increasing geometrical symmetry. For example, orthorhombic symmetry has 9 elastic constants, tetragonal 6, hexagonal 5, and cubic only 3. If the body is isotropic, the number of independent moduli can decrease even fmther, to a limiting... 

Some examples of "geometrical symmetries" encountered in classical physics might include ... 

If the interaction between two objects does not depend explicitly on the lime coordinate, then the actions lhal take place do nol depend on when one starts to measure lime i.e.. the properties of the system are invariant with respect to a translation of the origin of coordinates along the lime axis. This symmetry is associated with conservation of energy. Use of a 4-dimen.sional coordinate system allows one to associate conservation of momentum and energy in a unified manner with the geometrical symmetry of space-time. 

The application of quantum-mechanical methods to the prediction of electronic structure has yielded much detailed information about atomic and molecular properties.13 Particularly in the past few years, the availability of high-speed computers with large storage capacities has made it possible to examine both atomic and molecular systems using an ab initio variational approach wherein no empirical parameters are employed.14 Variational calculations for molecules employ a Hamiltonian based on the nonrelativistic electrostatic nuclei-electron interaction and a wave function formed by antisymmetrizing a suitable many-electron function of spatial and spin coordinates. For most applications it is also necessary that the wave function represent a particular spin eigenstate and that it have appropriate geometric symmetry. 

A common idea underlying particular forms of symmetry is the invariance of a system under a certain set (group) of transformations. The normally considered forms of symmetry are rotational symmetry, which is based on the equivalence of all directions in space, and permutation symmetry, which is caused by identical particles. The operations of the geometrical symmetry group are responsible for appropriate conservation laws. So, the rotational symmetry of a closed system gives rise to the law of conservation of angular momentum. 

Synthesis of the closely related acyclic (19) and macrocyclic (20) polyradicals has recently been reported (Figure 5.1).1231 The -conjugated carbanions (e.g., the calix[4]-arene-based tetraanion and the related calix[3]arene-based trianion) were synthesized and studied.1241 Oxidation of these tetra- and tri-anions gave the corresponding tetra- and tri-radicals, respectively. It has been shown in closely related systems that it is not the shape or overall geometric symmetry of the molecules, but rather it is the juxtaposition of the carbenic centers within the jt-cross-conjugated structure, that is most important in determining the spin multiplicity of the alternant hydrocarbon molecule.1251... 

Involvement of d orbitals, which offer more bonding modes and geometric symmetries than simple organic molecules... 

The geometric symmetry of a product can influence process selection. Both shape and design details are heavily process related. The ability to mold ribs, for example, may depend on material flow during a process or on the flowability of a plastic reinforced with glass. The ability to produce hollow shapes depends on the ability to use removable cores, including air, fusible or soluble solids, and even sand. Hollow shapes can also be produced using cores that remain in the product, such as foam inserts in RTM or metal inserts in IM. 

The spin selection rule, AS = 0, might be expected to be of universal applicability, since it does not require the molecule under consideration to have any geometrical symmetry. However, spin-forbidden transitions are also frequently observed. The spin rule is based again on the idea of separability of wavefunctions, this time of the spin and spatial components of the electronic wavefunction. However, the electron experiences a magnetic field as a result of the relative motion of the positive nucleus with respect to it, and this field causes some mixing of spatial and spin components, giving rise to spin-orbit... 

The partitioned grand resistance matrix in Eq. (7.13) is a function only of the instantaneous geometrical configuration of the particulate phase. This consists of the fixed particle shapes together with the variable relative particle positions and orientations. As such, geometrical symmetry arguments (where such symmetry exists) may be used to reduce the number of independent, nonzero scalar components of the coefficient tensors in Eq. (7.13) for particular choices of coordinate axes (e.g., principal axis systems). 

Calculations have thus far been performed for the three standard cubic arrays, namely simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fee). As a result of this geometric symmetry, the couple N and particle stress dyadic A are given by the configuration-specific relations... 

The application of a symmetry element is a symmetry operation and the symmetry elements are the symmetry operators. The consequence of a symmetry operation is a symmetry transformation. Strict definitions refer to geometrical symmetry, and will serve us as guidelines only. They will be followed qualitatively in our discussion of primarily non-geometric symmetries, according to the ideas of the Introduction. 

The geometrical symmetry of the plate configuration requires the derivative of the potential to vanish at the midpoint x = x,° of the region R i.e.,... 

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