Rotation spherical

The most convincing physical model [63] explains electron spin in terms of spherical rotation, another way of rotating a solid object, different from the well-known mode of rotation about an axis. It starts as a slight wobble, which, by continuous exaggeration of the motion, develops into a double 

Should a fermion represent some special distortion, or knot, in the aether, spherical rotation allows it to move freely through the space-time continuum without getting entangled with its environment, which consists of the same stuff as the fermion. While rotating in spherical mode adhesion to the environment is rythmically stretched and relaxed as the fermion moves through space. This half-frequency disturbance of the wave-field, that supports the fermion in space, constitutes the effect observed as spin. 

The peculiarity of spherical rotation is that rotation by 2ir fails to return the rotating object to its initial orientation. Evidently there is an additional aspect to the state of orientation that needs to be taken into account. Two versions are said to be associated with each orientation. The quaternion operator 

In order to take the version into account spherical rotation is represented symbolically by 

This operation represents a quarter turn on a great circle or rotation of ir radians about an axis. An intermediate position is given by the spinor  

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Although Ki and Ki are defined by physical quantities of different nature, their time evolution is universally determined by orientational relaxation. This discussion is restricted to linear molecules and vibrations of spherical rotators for which / is a symmetric tensor / = fiki- In this case the following relation holds... 

Coincidence with the exact expression is caused by the fact that terms of higher order in do not contribute to the second and fourth moments. Correspondingly, for spherical rotators (r = 3) we have... 

With formulae (3.58), (3.59) and (3.66) Q-branch contours are calculated for CARS spectra of spherical rotators at various pressures and for various magnitudes of parameter y (Fig. 3.14). For comparison with experimental data, obtained in [162], the characteristic parameters of the spectra were extracted from these contours half-widths and shifts of the maximum subject to the density. They are plotted in Fig. 3.15 and Fig. 3.16. The corresponding experimental dependences for methane were plotted by one-parameter fitting. As a result, the cross-section for rotational energy relaxation oe is found ... 

V = equivalent volume of a spherically rotating molecule, R = the gas constant, and T = temperature in Kelvins. 

The value of tan A depends upon the modulation frequency, the excited state lifetime, and the rate of rotation. The value decreases to zero when the rotation period is either longer or shorter than the excited state lifetime and is a maximum when the two times are comparable in magnitude. Tan A also increases as the modulation frequency increases. For spherical rotators, the measured value of tan A for a given modulation frequency and excited state lifetime allows the rotational rate to be calculated from... 

For spherical rotators, the measured value of tan Amax is independent of the rate of rotation. However, for non spherical rotators, the measured value of tan Amax depends on the molecular shape and is always smaller than the measured value of tan Amax for spherical molecules. Nonsphericity can be detected by calculating the tan... 

Polycrystal-type (rings) electron diffraction patterns (Fig.6) are especially valuable for precision studies - checking on the scattering law, identification of the nature of chemical bonding, and refinement of the chemical composition of the specimen - because these patterns allow the precision measurements of reflection intensities. The reciprocal lattice of a polycrystal is obtained by spherical rotation of the reciprocal lattice of a single crystal around a fixed 000 point it forms a system of spheres placed one inside the other and has the symmetry co oo.m. It is also important for structure... 

There are a number of other designs that have advantages in particular circumstances. Many are just variants on the theme of maximally spanning the factor space. The Box-Behnken design is a two-level, spherical, rotatable design (Box and Behnken 1960). For three factors it has experiments at the center and middle of each edge of a cube (figure 3.14). 

Clarkson et al. investigated molecular dynamics of vanadyl-EDTA and DTPA complexes in sucrose solution or attached to PAMAM dendrimers by EPR [74,75]. The motion-sensitive EPR data of the dendrimeric system have been fitted to an anisotropic model which is described by an overall spherical rotation combined with a rotation around the axis of the arm branching out of the central core. The motions around the axis of the branch connecting the chelate to the central core were found to be very rapid, whereas the overall tumbling was slow. 

To understand the appearance of spin it is necessary to consider a fermion as some inhomogeneity in the space-time continuum, or aether. In order to move through space the fermion must rotate in spherical mode, causing a measurable disturbance in its immediate vicinity, observable as an angular momentum of h/2, called spin. The inertial resistance experienced by a moving fermion relates to the angular velocity of the spherical rotation and is measurable as the mass of the fermion. 

The spinor that describes the spherical rotation satisfies Schrodinger s equation and specifies two orientations of the spin, colloquially known as up and down (j) and ( [), distinguished by the allowed values of the magnetic spin quantum number, ms = . The two-way splitting of a beam of silver ions in a Stern-Gerlach experiment is explained by the interaction of spin angular momentum with the magnetic field. 

The second term on the left in (20) is explained as a quantum torque not associated with the motion and although L2 = 0 for a stationary particle, the angular momentum may be non-zero, as in a pz-state [35]. Although the electron has angular momentum in three dimensions the projection thereof on an axis, is zero. The quantum torque can therefore not be due to rotation about an axis and arises from spherical rotation about a point, to be described in chapter 4.7. 

Rotation by 2tt rad, normally considered to be an identity operation, fails to return a spherically rotating object to its initial state of entanglement with the environment. Evidently something, a version of an orientation - is commonly overlooked. As the object has two possible versions of the spin matrix associated with a rotation, the quantity... 

Spin refers to rotation that is a linear function of time. A spherically rotating object the initial configuration of which is given by a spinor ... 

The nature of spin as revealed with a model of spherical rotation is fully consistent with all known attributes of electron spin [97]. It represents an intrinsic magnetic moment of one bohr magneton8... 

The conservation of angular momentum is a consequence of isotropy or spherical rotational symmetry of space (1.3.1). An alternative statement of a conservation law is in terms of a nonobservable, which in this case is an absolute direction in space. Whenever an absolute direction is observed, conservation no longer holds, and vice versa. The alignment of spin, that allows of no intermediate orientations, defines such a direction with respect to conservation of angular momentum. One infers that space is not rotationally symmetrical at the quantum level. 

Attempts to formulate a causal description of electron spin have not been completely successful. Two approaches were to model the motion on either a rigid sphere with the Pauli equation [102] as basis, or a point particle using Dirac s equation, which is pursued here no further. The methodology is nevertheless of interest and consistent with the spherical rotation model. The basic problem is to formulate a wave function in polar form E = RetS h as a spinor, by expressing each complex component in spinor form... 

Hence, it ould be stressed, a single expoirentially decaying emission anisotropy does not uniquely correspond to a spherical rotating body. On the other hand, the observation of non-exix)nential behaviour indicates deviations from spherical behaviour. 

The model that we are going to consider in this section is given by two spherical rotators, simply called body 1 and body 2. Body 1 is the solute molecule, whereas body 2 is the instantaneous structure of solvent molecules in the immediate surroundings of the solute. The rest of the solvent is described as a homogeneous, isotropic and continuous viscous fluid. In the overdamped regime, the system is described by a Smoluchowski equation in the phase space where ft, and ftj... 

We start with the two-body Smoluchowski model (2BSM) the details of the formulation (matrix and starting vector) are discussed in Section II.C. A stochastic system made of two spherical rotators in a diffusive (Smoluchowski) regime has been used recently to interpret typical bifurcation phenomena of supercooled organic liquids [40]. In that work it was shown that the presence of a slow body coupled to the solute causes unusual decay behavior that is strongly dependent on the rank of the interaction potential. 

The next model that we have treated in order of complexity is a three-body Smoluchowski model (3BSM). A field X has been included, coupled exclusively through first rank (dipole-field) interactions to the two spherical rotators. No direct coupling has been taken to exist... 

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