Imaginary incident

That imaginary incident is the subject of a case study presented several times each year to groups of Dow supervisors and managers. The students represent a cross section of both experience and line and staff responsibilities. During a 2 -day Safety, Loss Prevention and Security Workshop, they will investigate the Dowville incident. 

To explain the law an imaginary incident at work is used which exemplifies aspects of the operation of our legal system. These issues will be identified and explained with differences of Scottish and Irish law being indicated... 

The sodium chloride (NaCl) crystal shows very high absorption and reflectivity in the infrared region, known as the Restrahlen region. The real and imaginary relative dielectric constants at 6000 nm are, respectively, ei = 16.8 and S2 = 91.4. At this wavelength, estimate (a) the refractive index and the extinction coefficient, and (b) the optical density and the reflectivity at normal incidence for a 1 mm thick NaCl sample, (c) If the previous sample is illuminated (at normal incidence) by a beam of intensity Iq at 6000 nm, estimate the intensity of this... 

For convenience we take the incident electric field E, = Eex to be x-polarized. Because the medium is nonabsorbing, Wa is independent of the radius r of the imaginary sphere. Therefore, we may choose r sufficiently large such that we are in the far-field region where... 

We showed in Section 2.3 that the real and imaginary parts of the electric susceptibility are connected by the dispersion relations (2.36) and (2.37). This followed as a consequence of the linear causal relation between the electric field and polarization together with the vanishing of x(<°) in the limit of infinite frequency to. We also stated that, in general, similar relations are expected to hold for any frequency-dependent function that connects an output with an input in a linear causal way. An example is the amplitude scattering matrix (4.75) the scattered field is linearly related to the incident field. Moreover, this relation must be causal the scattered field cannot precede in time the incident field that excited it. Therefore, the matrix elements should satisfy dispersion relations. In particular, this is true for the forward direction 6 = 0°. But 5(0°, to) does not have the required asymptotic behavior it is clear from the diffraction theory approximation (4.73) that for sufficiently large frequencies, 5(0°, to) is proportional to to2. Nevertheless, only minor fiddling with S makes it behave properly the function... 

The frequency dependence of the real and imaginary parts of the dielectric function (9.7) are shown schematically in Fig. 9.2b following this are the corresponding real and imaginary parts of the refractive index (Fig. 9.2 c). The reflectance at normal incidence (2.58) is also of interest and is shown in Fig. 9.2cl. Many features of the optical properties of materials are illustrated by... 

When kx < k the last term in the numerator and the denominator are real, and the reflectance function is real, so that there is no variation of phase with incident angle. When kx = k all terms except the first term in the numerator and the denominator vanish, and R(kx) = 1. When k square brackets depends only on kx and the properties of the solid. Provided that there is no dissipation in the solid, it vanishes when kx is equal to the Rayleigh wavenumber kR = co/vR, as can be seen by substituting X = (ks/kR)2, Y = (k /ks)2, and comparing with (6.55). 

The scattering function of eqn (12.13) can be extended to the more general case of different materials on either side of the boundary indeed it was originally derived in that form (Somekh et al. 1985). The two sides are denoted by subscripts 1 and 2, having Rayleigh wavenumbers kpi and kp2 with imaginary components ot and a2. Transmission and reflection coefficients Tri, Tr2, and Rri, Rr2 are defined for waves incident from sides 1 and 2, respectively. Then... 

Fig. IS. Angle-resolved photo emission from the edge surface of MoS2 single crystal (72), and imaginary sketch of reconstructed (T0I0) and (I0T0) surfaces. Incident angle of He—I was fixed at 30"C.

In case of vacuum (e = e0) the impedance is real and its absolute value is 377 fl (wave impedance of vacuum). In case of a metal an incident wave decays rapidly from the surface, thus the impedance of a metal is called surface impedance Zs = Rs + iXs. The real part of Zs is called surface resistance Rs, the imaginary part surface reactance Xs. 

The real and imaginary parts of the refractive index n quantify the scattering and absorption (or amplification) properties of a material- The refractive index is besfl derived from the susceptibility tensor y of the material, defined below, whi j describes the response of a macroscopic "system to incident radiation [212], Spe fically, an incident electric field E(r, t), where r denotes the location in the medium, tends to displace charges, thereby polarizing the medium. The change in dmd(r, the induced dipole moment, from point r to point r + dr is given in terms of th polarization vector P(r, t), defined as... 

Figure 7.S Real and imaginary parts of incident wave function leading to maximum a ( minimum outgoing flux for Ar + H2 (u = 0,7 = 2, ny = 0). (Taken from Fig. 15, Ref.

By far the greater proportion of incident X-radiation is transmitted by a crystalline sample. However, a small fraction is scattered (effectively reflected) in all directions by every motif in the material, without change in wavelength. The motif is the repeating unit of pattern in a crystal it is the TAG molecule in the case of a fat crystal. Motifs can be considered to be located at or near the intersections of an imaginary 3-dimensional grid called the crystal lattice and the intersections are called lattice points (Hammond, 1997). 

If n denotes a resonant (metastable state) of the anion, E +1(t ) = Ej,+1 — ir/2 and the corresponding pole (Ejf+1( )—Eq ) = (Ej,+1— Eq )—iT/2 and from the real part one may directly determine the kinetic energy of the incident electron for which resonance occurs and the imaginary part provides the half width. These poles are persistent once uncovered. 

The reflectivity is measured from a flat surface for normal incidence of the incoming light. Electromagnetic theory shows that the reflectivity R = IKa/ hn is given in terms of the real and imaginary parts n and k, respectively, of the complex index of refraction (cf. Bohren Huffman 1983),... 

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