Vibration ellipse

Equation (2.78) describes an ellipse, the vibration ellipse (Fig. 2.11). If A = 0 (or B = 0), the vibration ellipse is just a straight line, and the wave is said to be linearly polarized , the vector B then specifies the direction of vibration. (The term plane polarized is also used, but it has become less fashionable in recent years.)If A = B and A B = 0, the vibration ellipse is a circle, and the wave is said to be circularly polarized. In general, a monochromatic wave of the form (2.77) is elliptically polarized. 

In addition to its handedness, a vibration ellipse is characterized by its ellipticity, the ratio of the length of its semiminor axis to that of its semimajor axis, and its azimuth, the angle between the semimajor axis and an arbitrary reference direction (Fig. 2.13). Handedness, ellipticity, and azimuth, together with irradiance, are the ellipsometric parameters of a plane wave. 

Figure 2.13 Vibration ellipse with ellipticity b/a and azimuth y.

Although a strictly monochromatic wave, one for which the time dependence is exp( — itot), has a well-defined vibration ellipse, not all waves do. Let us consider a nearly monochromatic, or quasi-monochromatic beam ... 

If the incident light is obliquely polarized at an angle of 45° to the scattering plane, the scattered light will, in general, be elliptically polarized, although the azimuth of the vibration ellipse need not be 45°. The amount of rotation of the azimuth, as well as the ellipticity, depends not only on the particle characteristics but also on the direction in which the light is scattered. 

The azimuth y and ellipticity tan rj of the vibration ellipse for an arbitrary beam can be determined from the Stokes parameters by (2.82). 

P vanishes for unpolarized light and is equal to unity for fully polarized light. For a partially polarized beam (0 < P < 1) with V 0, the sign of V indicates the preferential handedness of the vibration ellipses described by the endpoint of the electric vector. Specifically, a positive V indicates left-handed polarization and a negative V indicates right-handed polarization. By analogy... 

We now proceed to relate the complex amplitudes Beo,/ and Eeo,a to the ellipsometric parameters Eq, ip and y. Representing the semi-axes of the vibration ellipse as... 

Kreutz T G and Flynn G W 1990 Analysis of translational, rotational, and vibrational energy transfer in collisions between COj and hot hydrogen atoms the three dimensional breathing ellipse model J. Chem. Phys. 93 452-65... 

Fig. 12. Tentative flow curves of low-density polyethylene with MFI = 2.0 g/10 min extruded at 170 °C through channels with a two-angle ellipse Wber cross section with a length of 50 (a), 75 (b), and 100 (c) mm with reciprocating-rotary vibration of the element in the zone upstream of the inlet to the channel (according to the diagram given in Fig. 9) ...

Figure 7.19 Pictorial descriptions of the phase difference between bound and continuum vibrational wavefunctions. The top part of the figure shows the crossing bound and repulsive potential curves and the two paths between which the phase shift is to be determined. The lower part of the figure represents the classical phase-space trajectories for motion on Vj (ellipse) and V2 (parabola). The shaded area is the phase difference between the two paths, (o) Outer wall crossing. Path I (single arrows) is the most direct dissociation path ai to Rc on Vj, Rc to oo on V2. Path II (double arrows) is the shortest indirect path 01 to i to Rc on Vi, Rc to a2 to oo on V2. (6) Inner wall crossing. The phase difference is between the shortest ( i to Rc on V, Rc to a2 to oo on V2) and next longer ( i to 01 to Rc on Vi,Rc to oo on V2) path.

Figure 7.19a is a pictorial description of Ev) for an outer wall curve crossing. In phase space, bound motion in the uth vibrational level appears as an ellipse in the harmonic approximation. Motion on a linear unbound potential is represented as a parabola. The shaded area is 2unbound motion parabola shifts to the left so that the minimum value of R at P = 0 occurs at V2 Rmin) = Ev consequently, d> increases with u for E > Ec Whenever the value of maximum value [except for the first maximum, v = 0, at which Eq. (7.6.11) is invalid] (Child, 1980b). 

Figure 14 Comparison of vibrational and electron transfer hopping frequencies. Closed ellipses based on kinetic data open rectangles based on Equation (75). See Table 6.

Fig. 7.7. A ball oscillating in a potential energy well (scheme), (a) and (b) show the normal vibrations (normal modes) about a point /fo = being a minimum of the potential energy function V(/ o + ) of two variables = (xj, X2). This function is first approximated by a quadratic function i.e., a paraboloid V X, X2)- Computing the normal modes is equivalent to such a rotation of the Cartesian coordinate system (a), that the new axes (b) xj and x become the principal axes of any section of V by a plane V = const (i.e., ellipses). Then, we have V(xi,X2) = V Rq = 0) + j/ti (xj) + k2 The problem then becomes equivalent to the two-dimensional harmonic oscillator (cf.,...

But we can do better. The thermal motions cause a problem because the amplitudes are so large, which is in part due in turn to the close spacing of the vibrational levels. The problem can be most easily understood with reference to the two ellipses shown at the top of Figure 2.4. These two ellipses represent the electron densities of two... 

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