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Orientational order Euler angles

Euler angles Q define the e orientation, and an orientational correlation function of 1th order is introduced in the usual way ... [Pg.242]

Fig. 7. The nature of information concerning the mean orientation and dynamics of an internuclear vector r, which can be obtained from RDC analysis. Upon diagonali-zation of the Cartesian dipolar interaction tensor R, described in the text, the mean vector orientation, r, will be described by the Euler angles a and /3. The eigenvalues will correspond to the axial and rhombic order parameters which describe the amplitude of motion. If the motion is asymmetric, as reflected in a nonzero rhombic order parameter, then the principal direction of asymmetry is described by the Euler angle y. Fig. 7. The nature of information concerning the mean orientation and dynamics of an internuclear vector r, which can be obtained from RDC analysis. Upon diagonali-zation of the Cartesian dipolar interaction tensor R, described in the text, the mean vector orientation, r, will be described by the Euler angles a and /3. The eigenvalues will correspond to the axial and rhombic order parameters which describe the amplitude of motion. If the motion is asymmetric, as reflected in a nonzero rhombic order parameter, then the principal direction of asymmetry is described by the Euler angle y.
N coordinates are needed in order to completely specify the position of the nuclei. Three coordinates are needed in order to specify the position of the center of mass.1 Thus, 3N — 3 coordinates account for the internal degrees of freedom, that is, overall orientation and internuclear coordinates. The overall orientation can be specified by two coordinates for a linear molecule, say by the two polar angles (6, ). For a nonlinear molecule three coordinates are needed in order to specify the orientation. These coordinates are often chosen as the so-called Euler angles. Thus, for a molecule with N atomic nuclei,... [Pg.36]

The residual anisotropy A describes the orientation assumed by heavy water molecules. It is expressed as a function of the Saupe order tensor and of the Euler angles describing the orientation of the D20 water molecule with respect to the local director (29) ... [Pg.401]

Clearly, situations intermediate between perfect order and random distributions occur in arrays of absorbing chromophores, and a treatment is required that allows expression of the orientational distribution of structural units such as crystallites or segments which may be fluorescent in a bulk sample having uniaxial or biaxial symmetry. A complete mathematical approach using a herical harmonic expansion technique has been developed which expresses the distribution as spherical harmonics of various orders in terms of the Euler angles which specify the orientation of the coordinate system in a fixed structural unit with respect to the coordinate system in the bulk sample This is of use in solid systems, where time dependence is not observed. [Pg.77]

For uniaxial molecules, the polar order depends only on the polar Euler angle 0. We consider only linear polarizations of the writing beams. Variation of the orientational distribution of molecules N(fllo) initially oriented in the solid angle Qo(0o>9o) after summing up all the contributions to the light-induced polar and axial excitations is given by ... [Pg.349]

The macroscopic property observed in sum-frequency experiments, Xs . is a sum of the molecular hyperpolarizabilities, over all vibrational modes and all of the molecules at the interface, which takes into account the orientation of each molecule. Orientational information is obtained from the experimental spectra through consideration of the relationship between the observed Cartesian components of the macroscopic second-order susceptibility Xuk, the corresponding spectroscopically active components of the molecular hyperpolarizability, This is accomplished through an Euler angle rotation of the molecular axis system into the laboratory axis system as defined through the use of the rotational matrix iiuK imn- The general expression for the transformation from a molecular-fixed axis system to a laboratory-fixed system is... [Pg.30]

A sufficient number of relative initial orientations must be considered in order to achieve a convenient grid in the space of Euler angles so that an accurate average can be taken to yield a cross section for a particular process /, e.g., direct scattering or electron charge exchange... [Pg.258]

Due to anisometric (particularly elongated) shape of molecules, these crystals possess both the translational and orientational order. The latter is determined by Euler angles 9,0 such molecules form with respect to selected coordinate frame as shown in the right part of Fig. 2.11. The third Euler angle describing rotation of a molecule about its longest axis is not shown for simplicity. The point group symmetry includes this orientational order. [Pg.16]

To determine the orientation of the fixed local reference to the body, Euler angles or Cardan angles can be used. Every rotation can be reconstructed by using three consecutive rotations about three specific axes. In particular, in order to define the transformation of the global reference to the local one, an initial rotation must be defined 0°, about the z-axis of the 0 global reference giving the 0 frame, a sec-... [Pg.334]

Steele[2] generalized this type of expansion to deal with nonlinear molecules. Here the orientation is described by Euler angles (a,3,Y), and a describing the direction of an arbitrary axis within the molecule (usually the highest-order symmetry axis) like 6 and respectively in Figure 2, and Y describing the angle of... [Pg.5]

Yet another solution to the problem, known as multiple-quantum magic-angle spinning (MQ-MAS) relies on the fact that B2 and B4 are functions of 7, p, Yj, a and 8, where p is the order of the multiquantum coherence and a and P are the Euler angles corresponding to the orientation of each crystallite in the powder with respect to the rotor axis. Under fast MAS, the chemical shift anisotropy, heter-onuclear dipolar interactions and the term proportional to P2 in Equation [20] are removed, so that... [Pg.381]


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See also in sourсe #XX -- [ Pg.246 , Pg.247 ]




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