Azimuthal setting

From a nanoscale point of view, /3-PP crystallizes in a hexagonal form with a = b = 11.01 A and c = 6.49 A (the space group is probably P3i) [86,96-98]. It belongs to a class of polymers which displays a frustrated structure, referring to the fact that the helices in the unit cells have different azimuthal settings [37,99-101]. The relative amounts of a- and /3-forms in a sample are controlled by intrinsic molecular parameters and extrinsic factors. They will be discussed in detail in Sect. 3.2. 

The unit cell of odPP phase is of particular interest. It is monoclinic, with one distinctive and essential feature its a and c parameters are nearly equal (=0.65 nm). The threefold helices are organized in iso-chiral layers in the ac plane, but the helical hand alternates in successive layers, as well as the azimuthal setting of the chains. As a consequence, different ac contact faces exist in the structure [28]. They differ by the density of methyl groups, as illustrated in Figure 8.4. 

It is convenient to employ two sets of coordinate systems. Spherical polar coordinates r, Q, A) are defined with the origin at the vertex of the cone the axis is 0=0, the surface of the conical portion of the cyclone is the cone 0 = 0% and the azimuthal coordinate is A. Using the same origin, cylindrical polar coordinates (R, A, Z) are defined, where R = r sin 0 and the Z-axis coincides with the axis 0=0. 

Here are the salient features from a typical Gaussian98 run on ethene at the HF/6-311G level of theory, using the Z-matrix option (Figure 14.3). I have forced D2h symmetry by setting all the C-H bonds equal, all HCC bond angles equal and all azimuthal angles equal. 

A problem with the early MWD mud pulse systems was the very slow rate of data transmission. Several minutes were needed to transmit one set of directional data. Anadrill working with a Mobil patent [100] developed in the early 1980s a continuous wave system with a much faster data rate. It became possible to transmit many more drilling data, and also to transmit logging data making LWD possible. Today, as many as 16 parameters can be transmitted in 16 s. The dream of the early pioneers has been more than fulfilled since azimuth, inclination, tool face, downhole weight-on-bit, downhole torque, shocks, caliper, resistivity, gamma ray, neutron, density, Pe, sonic and more can be transmitted in realtime to the rig floor and the main office. 

With the modern accelerometers and solid-state magnetometers, a complete set of data is available for inclination, tool face and azimuth calculation. Magnetic corrections can be done. Inclination can be calculated with Equations 4-174 and 4-175. The gravity tool face angle can be calculated with Equation 4-176. 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

An inference of fundamental importance follows from Eqs. (2.3.9) and (2.3.11) When long axes of nonpolar molecules deviate from the surface-normal direction slightly enough, their azimuthal orientational behavior is accounted for by much the same Hamiltonian as that for a two-dimensional dipole system. Indeed, at sin<9 1 the main nonlocal contribution to Eq. (2.3.9) is provided by a term quadratic in which contains the interaction tensor V 2 (r) of much the same structure as dipole-dipole interaction tensor 2B3 > 0, B4 < 0, only differing in values 2B3 and B4. For dipole-dipole interactions, 2B3 = D = flic (p is the dipole moment) and B4 = -3D, whereas, e.g., purely quadrupole-quadrupole interactions are characterized by 2B3 = 3U, B4 = - SU (see Table 2.2). Evidently, it is for this reason that the dipole model applied to the system CO/NaCl(100), with rather small values 0(6 25°), provided an adequate picture for the ground-state orientational structure.81 A contradiction arose only in the estimation of the temperature Tc of the observed orientational phase transition For the experimental value Tc = 25 K to be reproduced, the dipole moment should have been set n = 1.3D, which is ten times as large as the corresponding value n in a gas phase. Section 2.4 will be devoted to a detailed consideration of orientational states and excitation spectra of a model system on a square lattice described by relations (2.3.9)-(2.3.11). 

The initial conditions for the velocity components are set up so that there is a tubular shear layer aligned along the 2 -direction at time t = 0. The tv-velocity has a top-hat profile with a tan-hyperbolic shear layer. Stream wise and azimuthal perturbations are introduced to expedite roll-up and the development of the Widnall instability. The details can be found in [7]. The initial velocity field is made divergence-free using the Helmholtz decomposition. The size of the computational domain (one periodic cubical box) is 4do on each side. 

Figure 7 Incident electron energy dependence of the X v = 0, 1, 2, 3 vibrational and the a Ag (v = 0) electronic loss scattered intensities from a 10-layer film of O2 condensed on Pt(lll). was set at 10° with 6 at 45° and the azimuth at 10°. Also shown is the energy dependence of the inelastic background intensity located just before the v = 1 loss peak onset at Aif = 0.16 eV along with that contributing to each energy-loss profile (dashed lines). (From Ref. 118.)...

Similarly, within any one electronic shell, the set of orbitals with azimuthal quantum number I corresponds to a subspace V of dimension... 

Other than linear molecules. If molecules of symmetry other than axial are considered, it is not possible to describe their orientation by an azimuthal and polar angle, Euler angles, Q = a pi, y and Wigner rotation matrices are then needed as Eq. 4.8 suggests. In that case, besides the set of parameters X, X2, A, L that has been used for linear molecules, two new parameters, u, with i = 1,2, occur that enter through the rotation matrices. These must be chosen so that the dipole moment is invariant under any rotation belonging to the molecular symmetry group. The rotation matrix is expressed as a linear combination of such... 

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