Axis system, definition

Figure Al.4.4. The definition of the Euler angles (0, ( ), x) that relate the orientation of the molecule fixed (x, y, z) axes to the (X, Y, Z) axes. The origin of both axis systems is at the nuclear centre of mass O, and the node line ON is directed so that a right handed screw is driven along ON in its positive direction by twisting it from Z to z through 9 where 0 < 9 < n. ( ) and x have the ranges 0 to In. x is measured from the node line.

Figure 5 Definition of the polar angles in the principal axis system of the diffusion-rotation tensor.

In order to extract some more information from the csa contribution to relaxation times, the next step is to switch to a molecular frame (x,y,z) where the shielding tensor is diagonal (x, y, z is called the Principal Axis System i.e., PAS). Owing to the properties reported in (44), the relevant calculations include the transformation of gzz into g x, yy, and g z involving, for the calculation of spectral densities, the correlation function of squares of trigonometric functions such as cos20(t)cos20(O) (see the previous section and more importantly Eq. (29) for the definition of the normalized spectral density J((d)). They yield for an isotropic reorientation (the molecule is supposed to behave as a sphere)... 

This definition occurs in several textbooks, perhaps because it is easier to appreciate physically. However, it is not so convenient mathematically because it refers to rotations in different axis systems. 

Nuclear quadrupolar interaction arises from the coupling between the nuclear quadrupole moment Q and the EFG at the nuclear position. The EFG varies in space and is described by a traceless second-rank tensor. The EFG tensor is diagonal and its three principal components are VXXr Vyy and Rzz with the definition of VZZ > Vyy > VX < Such a principal-axis system for the EFG tensor is defined with the direction of the external magnetic field, as illustrated in Figure 3(A). It is convenient to express such quadrupolar interactions by using the following two parameters ... 

Figure 2 Examples of global and local axis systems, (a) Molecular axis system for a homonuclear diatomic. Wth this system, all central multipoles with k 0 otl odd will be zero, and no S functions with k 0 oxl odd can appear in a molecule-molecule expansion of U(R, Q). The atomic multipoles Q q (all / 0 allowed) on the two atoms will be related by Qio = (-l) Qio- ( ) Local atomic axis system for a homonuclear diatomic molecule. With this definition QJq = Qio- (c) Molecular axis system for water. The nonzero atomic multipole moments for the O atom would be QoO> QlO) QzOj Qz2c = (Q22 + Qz-2)1 > QsOJ Q32c = (Qs2 + on the hydrogen atoms Qjo = Qoo. Qio = Qio> Qiic = (-Q11 + Qi-i) = -Qiic...

Figure 1. Definition of dipole operator directions and axis system (a) long-axis chelate transitions (b) short-axis chelate transitions.

Fig. 1. Definition of the set of Euler angles fl/j = ajj, Pu, 7/j relating axis system I to axis system J. Boldface arrows denote the respective axis of rotation and x,y,z are defined in the text.

Fig. 7.9 The definition of the orientation of the applied magnetic field Bq relative to the principal-axis system x,y,z of the magnetic dipole-dipole interaction.

Fig. 8.9 Definition of experimental (XYZ) and molecular (xyz) systems for pyridine adsorbed on a metal surface. The two axis systems can be connected with three Euler-ian angles 6, x stid . Symmetry species fli, fa, and <>2 have dipole derivatives along z, x and y directions, respectively.

It is straightforward to establish the relationship between the rotational constants A, B, C in the principal axis system and the constants, 4ram. 5ram. Cram, and Dab in the rho-axis system using the definition of p or by diagonalizing the 3x3 matrix of the RAM rotational constants. In the particular case of an a b) symmetry plane, it gives... 

The system of Cartesian axes chosen is dierefore die principal axis system (PAS) of the tensor V. Using the definitions given in die text, we have dierefore ... 

Figure 1.4. Reference for the position ofpoint M using its coordinates (r,0, z) in a cylindrical coordinate system of axis and definition of vectors 6r, ee and...

Now Qq/3 is traceless by definition. In the principal axis system of the susceptibility tensor, Qa/3 can be written in terms of two order parameters Q and P,... 

Figure 5.15 Definition of the Euler angles Q = a, P, y) that relate the molecular axis system xyz to the laboratory axis system XYZ and thus specify the orientation of a non-cylindrically symmetric molecule...

Fig. 7. Definition of the DNA backbone torsion angles. The torsion angles are defined as the clockwise rotation about B—C that aligns A—B with C—D in the sequence of atoms A—B—C—D w is for 0-3 —P—0-5 —C-5 (/> is for P—0-5 —C-5 —C-4 is for 0-5 —C-5 —C-4 —C-3 y/ is for C-5 —C-4 —C-3 —0-3 0 is for C-4 —C-3 — 0-3 —P (o is for C-3 —0-3 —P—0-5. The principal axis system is illustrated for rthe P chemical-shift tensor. From Keepers and James (1982). Copyright 1982 American Chemical Society.

Figure 11, (a) Definition of interlayer cation sets I and II relative to a fixed axis system. Double circles represent lower hydroxyl groups of interlayer sheet (anion plane no. 5 of Figure 10) (b) superposition of interlayer sheet on 2 1 layer in position a or b. 

The same methods apply also to ternary systems, the only difference being the use of a third axis in the definition of the composition, and the consequent substitution of surfaces for curves. 

Let us mentally represent this body as an infinite system of elementary masses dm, located at different distances r from the axis and, first, we obtain an equation for the rotation of some mass dm. By definition, the linear velocity, v, of each mass is related to the angular velocity by... 

First, we introduce a Cartesian system of coordinates with an origin 0 at the center of mass, Fig. 3.11b. The x and y axes are directed to north and east, respectively, and the z-axis is along the vertical. By definition, the moment of the force F(x, y, z) with components F, Fy, and F with respect to point 0 is... 

We also see another common definition—bounded input bounded output (BIBO) stability A system is BIBO stable if the output response is bounded for any bounded input. One illustration of this definition is to consider a hypothetical situation with a closed-loop pole at the origin. In such a case, we know that if we apply an impulse input or a rectangular pulse input, the response remains bounded. However, if we apply a step input, which is bounded, the response is a ramp, which has no upper bound. For this reason, we cannot accept any control system that has closed-loop poles lying on the imaginary axis. They must be in the LHP. 1... 

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