Extinction matrix

It is useful to express the dichroic extinction in terms of elements of the extinction matrix [15,16,17], with the Stokes parameters I, Q, U V of the radiation being given for axisymmetric aligned particles by... 

As mentioned before, a scattering particle can change the state of polarization of the incident beam after it passes the particle. This phenomenon is called dichroism and is a consequence of the different values of attenuation rates for different polarization components of the incident light. A complete description of the extinction process requires the introduction of the so-called extinction matrix. In order to derive the expression of the extinction matrix we consider the case of the forward-scattering direction, = 6fc, and define the coherency vector of the total field E = Eg + E. by... 

The elements of the extinction matrix have the dimension of area and only seven components are independent. Equation (1-78) is an interpretation of the so-called optical theorem which will be discussed in the next section. This relation shows that the particle changes not only the total electromagnetic power received by a detector in the forward scattering direction, but also its state of polarization. 

The above relation is a representation of the optical theorem, and since the extinction cross-section is in terms of the scattering amplitude in the forward direction, the optical theorem is also known as the extinction theorem or the forward scattering theorem. This fundamental relation can be used to compute the extinction cross-section when the imaginary part of the scattering amplitude in the forward direction is known accurately. In view of (1.88) and (1.74), and taking into account the explicit expressions of the elements of the extinction matrix we see that... 

To compute the orientation-averaged extinction matrix it is necessary to evaluate the orientation-averaged quantities (S pq(e, e )). Taking into account the expressions of the elements of the amplitude matrix (cf. (1.97)), the equation of the orientation-averaged transition matrix (cf. (1.118) and (1.119)) and the expressions of the vector spherical harmonics in the forward direction (cf. (1.121)), we obtain... 

Inserting these expansions into the equations specifying the elements of the extinction matrix (cf. (1.79)), we see that the nonzero matrix elements are... 

In this specific case, the orientation-averaged extinction matrix becomes diagonal with diagonal elements being equal to the orientation-averaged extinction cross-section per particle, (K) = (Cext)J-... 

Ozin et al. 107,108) performed matrix, optical experiments that resulted in the identification of the dimers of these first-row, transition metals. For Sc and Ti (4s 3d and 4s 3d, respectively), a facile dimerization process was observed in argon. It was found that, for Sc, the atomic absorptions were blue-shifted 500-1000 cm with respect to gas-phase data, whereas the extinction coefficients for both Sc and Scj were of the same order of magnitude, a feature also deduced for Ti and Ti2. The optical transitions and tentative assignments (based on EHMO calculations) are summarized in Table I. 

The relative extinction-coefficients for Agi,2,s determined by pho-toaggregation procedures were found not to be strongly matrix-dependent (see Table VIII). Moreover, the results for Agj were in good agreement with those obtained by quantitative, metal-atom deposition-techniques. 

An isotropic extinction parameter, of type I and Lorentzian distribution (in the formalism of Becker and Coppens [16]), was also refined. The motions of the non-H atoms were described by anisotropic parameters, while those of the H atoms by isotropic B s. All these displacement parameters were included among the refinable quantities of the model, for a total of 1161 variables in a single least-squares matrix. 

The UV spectra for the azide in a diethylene glycol dimethyl ether solution and for the styrene resin film with 1.0 micron thickness are shown in Figure 5. The azide has an intense absorption at around 248 nm (molar extinction coefficient at 248 nm = 3.0xl04 1/M cm). The syrene resin used as matrix polymer exhibits a significant transparency at 248 nm (70%). 

It is difficult to measure the oscillator strengths of molecules embedded in a matrix. Despite this, good values of can be determined as a function of the temperature. A procedure we have used to extract the information from excitation spectra was to set the maximum of the excitation spectrum measured at room temperature equal to the extinction coefficient at the absorption maximum in solution. The integrals of the excitation spectra were then normalized to the integral of the corresponding spectmm at room temperature, which is reasonable because the oscillator strength / of a transition n <— m does not depend on the temperature. 

The second major contrast mechanism is extinction contrast. Here the distortion of the lattice arotmd a defect gives rise to a different scattering power from that of the surrotmding matrix. In all cases, it arises from a breakdown or change of the dynamical diffraction in the perfect ciystal. In classical structure analysis, the name extinction was used to describe the observation that the integrated intensity was less than that predicted by the kinematical theoiy. 

Several predicted features of infrared surface mode absorption by small spheres are verified by the experimental results shown in Fig. 12.13. The frequency of peak absorption by spheres is shifted an appreciable amount from what it is in the bulk solid the e" curve peaks at 1070 cm", whereas the peak of the small-sphere absorption is at 1111 cm-1, very close to the frequency where e is — 2em (— 4.6 for a KBr matrix). The absorption maximum (absorption is nearly equal to extinction for these small particles) is very strong Qabs for a 0.1-jum particle is about 7 at the Frohlich frequency. 

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