Right-handed coordinate

It should be noted that the positive sign of this result depends on the choice of a right-handed coordinate system in which the angle is acute. The relation developed here for the volume of a parallelepiped is often employed in crystallography to calculate the volume of a unit cell, as shown in the following section. 

When treating CF parameters in any of the two formalisms, non-specialists often overlook that the coefficients of the expansion of the CF potential (i.e. the values of CF parameters) depend on the choice of the coordinate system, so that conventions for assigning the correct reference framework are required. The conventional choice in which parameters are expressed requires the z-direction to be the principal symmetry axis, while the y-axis is chosen to coincide with a twofold symmetry axis (if present). Finally, the x-axis is perpendicular to both y- and z-axes, in such a way that the three axes form a right-handed coordinate system [31]. For symmetry in which no binary axis perpendicular to principal symmetry axis exists (e.g. C3h, Ctt), y is usually chosen so as to set one of the B kq (in Wybourne s approach) or Aq with q < 0 (in Stevens approach) to zero, thereby reducing the number of terms providing a non-zero imaginary contribution to the matrix elements of the ligand field Hamiltonian. Finally, for even lower symmetry (orthorhombic or monoclinic), the correct choice is such that the ratio of the Stevens parameter is restrained to X = /A (0, 1) and equivalently k =... 

Fig. 2.6 Translation and deformation of a fluid element in the r-z plane. For the right-hand coordinate system (z, r, 9), note that the positive 6 direction is into the page. The displacements in the figure are grossly exaggerated to facilitate annotation. In the limit dr - 0, on which the analysis is based, the two elements approach being colocated.

The shear-stress convention is a bit more complicated to explain. In a differential control volume, the shear stresses act as a couple that produces a torque on the volume. The sign of the torques defines the positive directions of the shear stresses. Assume a right-handed coordinate system, here defined by (z, r, 9). The shear-stress sign convention is related to ordering of the coordinate indexes as follows a positive shear xzr produces a torque in the direction, a positive xrg produces a torque in the z direction, and a positive x z produces a torque in the r direction. Note also, for example, that a positive xrz produces a torque in the negative 6 direction. 

Figure 14.11 Schematic design of a quadrupole magnet. The arrangement of the poles will provide x focusing in the usual right-handed coordinate system with the z axis emerging from the page. [From O. Chumberlin, Ann. Rev. Nucl. Sci. 10, 161 (I960).]...

A helix is the figure generated by the motion of a point around and along a line — the helix axis. Archetypical y, this axis is a straight line and the two kinds of motion are, respectively, circular and linear, at a constant distance from the axis. The helix is dextroverse (right-handed) if the overall motion is clockwise and away from the observer (or counterclockwise and toward) and sinistroverse if counterclockwise and away. In a right-handed coordinate system such a helix, if dextroverse, obeys the equations ... 

Overlap of d orbitals leading to the formation of a quadruple bond between two metal atoms. Note that the z axis of each metal atom is taken to point toward the other, such that if a right-handed coordinate system is used for the atom on the left, a left-handed coordinate system must be used for the atom on the right. 

A cover is a horizontal string of at least p such objects, where the distance between two neighboring objects must be less than q points (p and q are parameters typical of a metaphase cell). The starting coordinates of a cover (xs, ys) is the left-hand coordinates of its left most object, and the ending coordinate of this cover (Xe, ye) is the right-hand coordinates of its right most object, and... 

Its primary use in our applications is for expressing various commutation relations in compact form and performing calculations involving them. The Levi-Civita symbol also arises naturally in vector analysis. Thus, if et, i = 1, 2, 3, are three mutually perpendicular unit vectors defining a right-handed coordinate system, then... 

Lin (145) has carried out an extensive theoretical investigation of the radiative and nonradiative mechanisms involving vibronic, spin-orbit, and vibronic-spin-orbit coupling in formaldehyde. Earlier, Yeung (254) calculated the SVL values of Tg, and Yeung and Moore (255) calculated the SVL values of x g. Lin used the left-hand Cartesian coordinate system in which planar formaldehyde lies in the x-z plane rather than in the y-z planes for the right-hand coordinate, which is accepted as the standard spectroscopic convention. Here, we adhere to the latter... 

A standard set of reference axes and equations to describe spontaneous strains is now well established (Schlenker et al. 1978, Redfern and Salje 1987, Carpenter et al. 1998a). The orthogonal reference axes, X, Y and Z, are selected so that Y is parallel to the crystallographic y-axis, Z is parallel to the normal to the (001) plane (i.e. parallel to c ) and X is perpendicular to both. The +X direction is chosen to conform to a right-handed coordinate system. Strain is a second rank tensor three linear components, cn, 622 and 33 are tensile strain parallel to X, Y and Z respectively and co, 623, eu are shear strains in the XZ, YZ and XY planes, respectively. The general equations of Schlenker et al. (1978) define the strains in terms of the lattice parameters of a crystal (a, b, c, a, P, y, where P is the reciprocal lattice angle) with respect to the reference state for the crystal ( , bo, Co, cto, Po,Yo) ... 

For the diffractometer with Bragg-Brentano geometry the following right-hand coordinate systems related to the source, sample and receiving slit are used (Figure 6.5). 

It is noted that among the Rg bases used in Eq. (65) the function and the set ( nc. S ) transform in the standard way according to the Ai and Ei = representations of Doo (Table 11), whereas the set is not standard under C. The function is symmetrical under Cl and the function antisymmetrical. However, in order to transform by the standard matrices, i. e. by those of [ c. ]. one of the p-functions will have to change its sign. We take the set [— as the standard choice so that the set ( , — sc) = (z. —y, x) forms a right-handed coordinate system. By this choice, the two last columns of the Racah lemma matrix are completely specified. One has, returning to the middle element,... 

The structural differences of the catalytic domains are also reflected in differences in subunit interactions. For LDH, s-MDH, and GAPDH these have been expressed in terms of an orthogonal, right-handed coordinate system (P,Q,R) deflned by Rossmann et al. (57). In LDH they form the three mutually perpendicular molecular 2-fold axes. These axes can be defined with respect to the direction of the strands in the 3-pleated sheet. 

CONVENTION. A right-handed coordinate system is chosen Le. a, b and c are taken in the order of the thumb, index and middle finger of the right hand a is the angle between b and c, p is the angle between a and c, y is the angle between... 

Figures 1.8(b), 1.8(c) and 1.8(d) all represent the same structure and the same translational lattice as Fig. 1.8(a). In Fig. 1.8(c) we have chosen another origin for the coordinate systto. In Fig. 1.8(b) we have chosen another base a = 2a + b, b = a + b. The area of the cell in Fig. 1.8(b) is the same as in Fig. 1.8(a) a X b = a X b. If the determinant of the transformation matrix between the systems (a, b ) and (a,b) is equal to 1, the area of the cell remains unchanged. Analogously, if the determinant of the transformation matrix between the systems (a, b, c ) and (a,b,c) in a three-dimensional structure is equal to 1, the volume of the cell remains unchanged. If the determinant is negative, we pass from a right-handed coordinate system to a left-handed one or vice versa. This determinant is equal to 2 in Fig. 1.8(d) where a" = a + b and b" = — a + b. The corresponding cell has double the area. The coordinates of the lattice points with respect to (a",b") are (u v) and (m + l/2 i + 1/2) where u and v are integers, i.e. (a" +b")/2 is a translation.

In Fig. 15.5, the rotation of a right-handed coordinate system around the z-axis is shown. In this particular rotation, the coordinates will be transformed as follows ... 

Fig. 15.5 Rotation of a right-handed coordinate system around the z-axis...

This equation was derived previously for the case of 1,2 dichloroethylene (equation 2.17). The possible permutations for neighboring bonds with a threefold potential barrier can be represented by a statistical weight matrix composed of elements based on individual statistical weights. The matrix methods involved are described in Appendix 2E. The reference bond angle Q, is placed in a right-handed coordinate system. Transformation matrices A[ (see for example equation 2C.25) are then used to transform the coordinates of the Ql bond into that of the (2, one. The process is repeated until the coordinates of all the bonds in a given conformation are generated. 

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