Alignment symmetry coordinates

If there are two orthogonal planes of material property symmetry for a material, symmetry will exist relative to a third mutually orthogonal plane. The stress-strain relations in coordinates aligned with principal material directions are... 

Figure 1 shows how we defined the geometry and reference frame for a conductor in space, where the inherent symmetry is assumed to be characterized by the cylindrical nature of the system. This also allowed us to simplify the problem to two coordinate directions. We assume the z-axis to be along the longitudinal direction of the system aligned with the magnet field (Bq), and to be the direction in which samples are inserted into the scanner. 

We assume that the stellar magnetic field is dipolar ( m d), and has axial symmetry everywhere. We use cylindrical coordinates (w,, z) centered on the neutron star and aligned with the stellar rotation axis. This configuration is sketched in Figure 1. We obtain the nondimensionalized equations which construct a complete set for the dynamics of reservoir ring, as following,... 

Consider a lamellar mesophase, being macroscopically aligned so that the symmetry axis, referred to as the director, has the same direction throughout the sample. If the transformation from the molecular coordinate system to the laboratory system is performed via the director coordinate system (D), Equation 2 reads... 

Vra / ft ) is the quadrupole coupling constant. The matrix of S values represents the order parameters, and they give the alignment of the compound with respect to the applied magnetic field. They can be, and usually are, defined in terms of a molecular-fixed coordinate system. S is a symmetrical 3x3 matrix, and the sum of the diagonal elements of S is zero, so that in a molecular-fixed coordinate system, the number of components of the S matrix varies from 5 for compounds with no elements of symmetry, such as chiral species, to 1 for entities with a C3 or higher axis of symmetry. 

Fig. 6.6. Schematic of realization of alignment-orientation Stark conversion, (a) Choice of coordinate systems, (b) Possible realization scheme for AB molecules seeded in a free jet of X atoms, (c) Symbolic polar plot of J distribution (dashed line refers to initial cylindrical symmetry over beam axis z ).

Such order can be described in terms of the preferential alignment of the director, a unit vector that describes the orientation of molecules in a nematic phase. Because the molecules are still subject to random fluctuations, only an average orientation can be described, usually by an ordering matrix S, which can be expressed in terms of any Cartesian coordinate system fixed in the molecule. S is symmetric and traceless and hence has five independent elements, but a suitable choice of the molecular axes may reduce the number. In principle, it is always possible to diagonalize S, and in such a principal axis coordinate system there are only two nonzero elements (as there would be, for example, in a quadrupole coupling tensor). In the absence of symmetry in the molecule, there is no way of specifying the orientation of the principal axes of S, but considerable simplification is obtained for symmetric molecules. If a molecule has a threefold or higher axis of symmetry, its selection as one of the axes of the Cartesian coordinate system leaves only one independent order parameter, with the now familiar form ... 

Therefore, zero, one, two or all three coordinates change their signs, but this only holds for symmetry elements of the first and second order when they are aligned with one of the three major crystallographic axes. Symmetry operations describing both diagonal symmetry elements and symmetry elements with higher order (i.e. three-, four- and six-fold rotations) may cause permutations and more complex relationships between the coordinates. For example ... 

Weighted Holistic Invariant Molecular (WHIM) indices [77] represent a different 3D approach to overcome the molecular 3D alignment problem, as they are invariant to molecular rotations and translations. These indices encapsulate information about the molecular 3D structure in terms of size, shape, symmetry and atom distribution, solely derived from Cartesian coordinates. 

Other whole molecule descriptors that do not require alignment include the Weighted Holistic Invariant Molecular (WHIM) indices developed by Todeschini et al. [70]. These indices are calculated from the 3D coordinates, which are weighted and centered to make them invariant to translation principal component analysis (PCA) is applied to obtain three principal components. These are used to produce new coordinates, which can be analyzed to obtain a series of 10 descriptors based on eigenvalues and the third-order and fourth-order moments of the three score column vectors. These descriptors are related to molecular size, shape, symmetry, and atom distribution and density. 

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