Differential rotational motion

In the following section, we only consider the integration of the equation of linear motion Eq. (20) the procedure for the equation of rotational motion, Eq. (21), will be completely analogous. Mathematically, Eq. (20) represents an initial-value ordinary differential equation. The evolution of particle positions and velocities can be traced by using any kind of method for ordinary differential equations. The simplest method is the first-order integrating scheme, which calculates the values at a time t + 5t from the initial values at time t (which are indicated by the superscript 0 ) via ... 

Based on equation (8) one can obtain alternative methods for the calculation of the electronic contribution to the g tensor by differentiating expressions for the electronic energy in the presence of an ejrternal magnetic flux density B and the coupling with the rotational motion Using equation (6) the... 

Using the equation of motion for a system with differential rotation taking into account the inertial, frictional, torsional and electrostatic forces ... 

Within the rotational sudden approximation we assume that the interaction time is much smaller than the rotational period of the fragment molecule so that the diatom BC does not appreciably rotate from its original position while the two fragments separate. In terms of energies, this requires the rotational energy, Erot, to be much smaller than the total available energy. If that is true, the operator for the rotational motion of BC, hrot, can be neglected in (3.16). The partial differential equation thus becomes an ordinary differential equation,... 

The rotational motion of such a rod by diffusional reorientation without shear was solved in Section 11.4.4. With shear, we have the differential equation... 

Vertical convectively driven turbulence. While differential rotation may inhibit convective motions in the radial direction in a disk, motions parallel to the rotation axis are relatively unaffected by rotation. In a disk where heat is being generated near the midplane, and where dust grains are the dominant source of opacity, the disk is likely to be unstable to convective motions in the vertical direction, which carry the heat away from the disk s midplane and deposit it close to the disk s surface, where it can be radiated away. Convective instability was conjectured to lead to sufficiently robust turbulence for the resulting... 

The Langevin approach has been used by many authors in order to treat nonlinear systems. This is of importance to us since the equations of rotational motion are intrinsically nonlinear. The concept of a nonlinear Langevin equation is also subject to a number of criticisms. These have been discussed extensively by van Kampen [58] (Chapters 8 and 14). In our calculations, we shall encounter stochastic differential equations of the form... 

So, if the interfacial regions have to be considered, we must differentiate the amorphous/crystal polymer interface from the amorphous polymer/mineral interface (29). By DMA measurements, a decrease in Tg matrix from 7°C for the PP/talc composites and also for the neat PP processed under similar conditions while a decrease upto 13°C was found for PP/mica composites. A higher fraction of free amorphous phase on the PP/mica system than on the PP/talc composites was evidenced. This free amorphous phase appeared to participate in the cooperative segmental free-rotation motion, well accepted (30) to be responsible for glass transition for the polymer matrix as fully discussed in Reference 29. 

The scattering from a molecule will be more complicated than for a single atom because the other molecular motions of rotation and vibration come into play. If there are no inelastic features in the measured energy transfer range studied, the vibrational term will only affect the measured intensities in the QENS domain through a Debye-Waller factor. On the other hand, the influence of the rotation on the observed profiles has to be treated in more detail. Sears has derived analytical expressions for the total differential cross-section of a molecular system, where the rotational motion is isotropic [12]. From his work, a simplified expression (Eq. 22) for the double-differential cross-section can be obtained it is spht into three terms ... 

We use here the Ericksen-Leslie continuum theory to describe the effect. The rotational motion of the director (i.e., molecular reorientation) is driven by the pump laser pulse, but it is also coupled with the translation motion (flow) of the fluid through viscosity. Thus, with a finite pump beam, a rigorous theoretical calculation would require the solution of a set of coupled three-dimensional nonlinear partial differential equations for the angle of... 

If we consider the variables 0 and (/> in equation 11.46 independently, we see that only one term in the differential contains , the last term. If 0 were held constant, then the first two differential terms would be identically zero (derivatives are zero if the variable in question is held constant), and the Schrodinger equation would have the same form as that for 2-D rotational motion. Therefore, the first part of the solution contains only the variable (f> and is the same function derived for the 2-D rotating system ... 

Spectroscopic methods can work with the chiral selector associated with the ligand either in solid state or in solution. The chiroptical spectroscopies, circular dichroism, and optical rotatory dispersion, represent an important means for evaluating structural properties of selector-ligand adducts [14]. NMR can specifically investigate proton or carbon atom positions and differentiate one from the other. X-ray crystallography is a powerful technique to investigate the absolute configuration of diastereoisomeric complexes but in the solid state only. Fluorescence anisotropy is a polarization-based technique that is a measure, in solution, of the rotational motion of a fluorescent molecule or a molecule + selector complex [15]. 

The determination of the good actions describing vibration-rotation motion requires the solution of the molecular Hamilton-Jacobi equation, which is a nonlinear partial differential equation in 3Na"5 variables (including rotation), where is the number of atoms. Even for = 3 (a triatomic molecule) an exact solution to this equation is extremely complex computationally, and it is not practical for collisional applications. Several approximations can be used to simplify this treatment, however, including (i) the separation of vibration from rotation (valid in the limit of an adequate vibration-rotation time scale separation), and (ii) the use of classical perturbation theory (in 2nd and 3rd order) to solve the three-dimensional vibrational Hamilton-Jacobi equation which remains after the separation of rotation. Details of both the separation procedures and the perturbation-theory solution are discussed elsewhere. For the present application, the validity of the first... 

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