Complex E-plane

Figure 12.9d shows the dielectric function of several metals that either have been discussed in Chapter 9 or will be discussed in connection with small particle extinction in Section 12.4. The energy dependence of the dielectric function is given in the form of trajectories in the complex e plane, similar to ihe Cole-Cole plots (1941) that are commonly used for polar dielectrics the numbers indicated on the trajectories are photon energies in electron volts. 

Therefore, a fixed order of the factors in PN is implied as indicated by Eq. (59). The symmetry of SSim(E) of Eq. (58) should be and is assumed to be enforced by an appropriate choice of vectors gv. In practice, this is difficult to express explicitly. However, determination of g by parameter fitting is impractical, anyway. The S matrix (58) is often very useful when gv need not be specified explicitly, as will be seen in the following, since it is quite a general representation that is unitary at real E and has poles at the right positions in the complex E plane. 

In the oo limit, the integration over E can be performed analytically by contour integration (see Fig. 2.4). To see this we note that in that limit the integrand on a large semicircle in the lower part of the complex E plane is zero, since, for... 

The form of Eq. (2.58) appears more complicated than that of Eq. (2.57) because the (E, m 0 vP (f) amplitude is composed of contributions from all degenerate n 0) states. Why use the outgoing states at all then The reason is that in " ordinary scattering events (e.g., the collision of two particles) we use states whose s S > paktis well known to us. These are the outgoing states because when t —> — oo it is ". " fhe contour on the semicircle in the upper half of the complex E plane that vanishes.. .. Thus, supplementing the real E integration by such a contour keeps the E — E — ie... 

A similar analysis may be carried out for the electron detachment poles as well and the Auger poles (Eq — E 1( )) will obviously be found in the first quadrant of the complex E-plane. With this brief discussion of the pole structure of the complex scaled electron propagator as a common background, its utility in direct and simultaneous treatment of resonances of N+l electron and N-l electron systems becomes manifest e.g., for Be both Be+ (Is-1) 2S Auger and 2P Be- shape resonances have been calculated simultaneously from a single calculation on Be/25,26/. 

The phase factor in Eq. 22 leads to significant complications for the two-photon calculation, as that factor leads to a cut in the complex E plane. This factor is in general complex, and while we are interested in the real... 

It may be used in the integral Eq. (3.14) instead of the form Eq. (3.17), provided the integration of the energy variable is performed as a contour integral in the complex E-plane. An appropriate contour is chosen such that it bypasses the singularities at E = e on the real axis. Figure 3.1 displays an acceptable contour for the case that fk > fi when ck < cj. 

Other empirical distributed elements have been described, which can be expressed as a combination of a CPE and one or more ideal circuit elements. Cole and Cole found that frequency dispersion in dielectrics results in an arc in the complex e plane (an alternative form of presentation) with its center below the real axis (Fig. 10a) [16]. They suggested the equivalent circuit shown in Fig. 10(b), which includes a CPE and two capacitors. For ft) —> 0, the model yields capacitance Co and for ft) —> oo the model yields capacitance Coo- The model can be expressed with the following empirical formula for the complex dielectric constant... 

Like the Cole-Davidson function, the Williams-Watts approach gives asymmetric plots in the complex e plane. A detailed comparison of these two forms has been made by Lindsey and Patterson [1980]. 

Coherent states and diverse semiclassical approximations to molecular wavepackets are essentially dependent on the relative phases between the wave components. Due to the need to keep this chapter to a reasonable size, we can mention here only a sample of original works (e.g., [202-205]) and some summaries [206-208]. In these, the reader will come across the Maslov index [209], which we pause to mention here, since it links up in a natural way to the modulus-phase relations described in Section III and with the phase-fiacing method in Section IV. The Maslov index relates to the phase acquired when the semiclassical wave function haverses a zero (or a singularity, if there be one) and it (and, particularly, its sign) is the consequence of the analytic behavior of the wave function in the complex time plane. 

The important point is that v E,k) and A0( ,fe) are multivalued, with a logarithmic branch point at fe = s = 0, while the residual functions m(e, k, zo) and w(e, k) are single valued. The result, as discussed in more detail later, is that the value of the quantum number v depends on the chosen location of the branch cut and on which Riemann sheet is taken, bearing in mind that the branch of arctan( /fe) must be taken according to the appropriate quadrant of the complex k, e) plane. Thus cj) = arctan( /fe) + ti/2 increases smoothly from zero to In around a counterclockwise circle in the k, e) plane, starting at = 0 and < 0. 

More insight into shape effects in absorption spectra of small particles can be acquired from contour plots in the complex c plane lines of constant dimensionless cross section 3(Cabs)/kt> are shown in Fig. 12.9a, b, c. Note that the curves are symmetric about the lines c = — 2, c = - 1, and e = 0 for the sphere, needle, and disk, respectively. Three points representing certain solids... 

The fields E, and % are orthogonal components in the complex phase plane for the oscillations due to the small displacement of the scalar field, which is thereby characterized completely. The scalar field Lagrangian becomes... 

Rct is better visualized in a complex admittance plane diagram (i.e. Yal plotted against V j) or a complex impedance plane diagram (i.e. Zal = Z" plotted against Z ei + Rn = Z )- The latter is known as the Sluyters plot and is widely used to determine the respective parameters. Some typical examples are represented in Fig. 18. 

It is shown in Section 7.10.1 that a continuous system is unstable if any root of the associated characteristic equation (i.e. any pole of the system transfer function) lies in the right half of the complex s-plane (Fig. 7.93a). If this root is s, then i, can be expressed in terms of its real and imaginary parts, i.e. ... 

In addition to looking at the position of the eigenvalues in the k-plane, we can also analyze their appearance on the complex energy plane due to the direct connection between the energy of the particle and its momentum at the asymptotes E = y. Figure 1.9 shows the distribution of the Siegert solutions on both the Energy and the wave vector (k) planes. 

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