Scattering matrix measurements

Note that the sums are restricted to the portion of the frill S matrix that describes reaction (or the specific reactive process that is of interest). It is clear from this definition that the CRP is a highly averaged property where there is no infomiation about individual quantum states, so it is of interest to develop methods that detemiine this probability directly from the Scln-ddinger equation rather than indirectly from the scattering matrix. In this section we first show how the CRP is related to the physically measurable rate constant, and then we discuss some rigorous and approximate methods for directly detennining the CRP. Much of this discussion is adapted from Miller and coworkers [44, 45]. 

Thus, if we have in hand the scattering coefficients an and bn, we can determine all the measurable quantities associated with scattering and absorption, such as cross sections and scattering matrix elements. 

The maximum amount of information about scattering by any particle or collection of particles is contained in all the elements of the 4x4 scattering matrix (3.16), which will be treated in more generality later in this chapter. Most measurements and calculations, however, are restricted to unpolarized or linearly polarized light incident on a collection of randomly oriented particles with an internal plane of symmetry (no optical activity, for example). In such instances, the relevant matrix elements are those in the upper left-hand 2x2 block of the scattering matrix, which has the symmetry shown below (see, e.g.,... 

Equality holds for a single sphere or a collection of identical spheres inequality holds if they are distributed in size or composition. This inequality was used by Hunt and Huffman (1973), for example, as an indicator of dispersion in suspensions of spherical particles. It was pointed out by Fry and Kattawar (1981) that the inequalities they derived are useful consistency checks on measurements of all 16 scattering matrix elements. 

The simplest, and probably most obvious, way to measure scattering matrix elements is with a conventional nephelometer (Fig. 13.5) and various optical elements fore and aft of the scattering medium. Recall that we introduced Stokes parameters in Section 2.11 by way of a series of conceptual measurements of differences between irradiances with different polarizers in the beam. Although we did not specify the origin of the beam, it could be light scattered in any direction. Combinations of scattering matrix elements can therefore be extracted from these kinds of measurements. There are now, however, two beams—incident and scattered—and many possible pairs of optical elements these are discussed below. 

Table 13.1 Combinations of Scattering Matrix Elements That Result from Measurements with a Polarizer Ps Forward of the Scattering Medium and an Analyzer A s aft"...

The possible outcomes of measurements—combinations of scattering matrix elements—listed in Table 13.1 follow from multiplication of three matrices those representing the polarizer, the scattering medium, and the analyzer. If U is an element in the optical train, then the measured irradiance depends on only two matrix elements. In general, however, there are four elements in a combination, so that four measurements are required to obtain one matrix element. 

Few measurements or calculations of all 16 scattering matrix elements have been reported. There are only four nonzero independent elements for spherical particles and six for a collection of randomly oriented particles with mirror symmetry (Section 13.6). It is sometimes worth the effort, however, to determine if the expected equalities and zeros really occur. If they do not, this may signal interesting properties such as deviations from sphericity, unexpected asymmetry, or partial alignment some examples are given in this section. But we begin with spherical particles. 

Sekera (1957) and Rozenberg (1960) emphasized the importance of measuring all matrix elements for atmospheric aerosols, and a few such measurements have been reported (Pritchard and Elliot, 1960 Beardsley, 1968 Golovanev et al., 1971). With sensitive modulation techniques it should indeed be possible to probe atmospheric particles remotely using the complete scattering matrix to infer not only size distributions but also refractive indices. Care must be exercised, however, because nonsphericity can lead to false inferences about absorption analysis based on Mie theory cannot disentangle the two effects. 

One of the few sets of measurements of all scattering matrix elements for nonspherical particles was made by Holland and Gagne (1970), who used various combinations of polarizers and retarders (see Section 13.7). They studied quartz (sand) particles with a fairly broad range of sizes. To investigate further the effects of nonsphericity on all matrix elements Perry et al. (1978),... 

Measurements by Thompson et al. (1978) on cultured populations of phytoplankton, such as might be expected to scatter light in ocean waters, revealed only a scattering matrix of the form (13.5), characteristic of particles for which either the Rayleigh (Chapter 5) or Rayleigh-Gans (Chapter 6) approximations are valid. 

Bottiger, J. R., E. S. Fry, and R. C. Thompson, 1980. Phase matrix measurements for electromagnetic scattering by sphere aggregates, in Light Scattering by Irregularly Shaped Particles, D. Schuerman (Ed.), Plenum, New York, pp. 283-290. 

It is rarely addressed in the literature that for molecular versions of circuit elements to be useful, there has to be the possibility to connect them together in a way where their electrical characteristics — measured individually between electrodes — would be preserved in the assembled circuit. However, it has been recently shown that such a downscaling of electrical circuits within classical network theory cannot be realized due to quantum effects, which introduce additional terms into Kirchhoff s laws and let the classical concept of circuit design collapse [16]. Circuit simulations on the basis of a topological scattering matrix approach have corroborated these results [34]. 

Let LkJk denote the diagonal matrix that contains the k eigenvalues I of the MCD scatter matrix, sorted from largest to smallest. Thus /, > /2 >. .. > Ik. The score distance of the ith sample measures the robust distance of its projection to the center of all the projected observations. Hence, it is measured within the PCA subspace, where due to the knowledge of the eigenvalues, we have information about the covariance structure of the scores. Consequently, the score distance is defined as in Equation 6.6 ... 

The measured current from the source to the drain electrode is proportional to the energy integral of squared modulus of the scattering matrix S(E) 2, whose integrand is called the quantum dot transparency T(E) = (E1) 2, overlapped with the difference of Fermi functions of the electrons in the right and the left leads... 

The Compton scatter matrix correction is based on the observation, referred to above, that the intensity of the Compton scatter peak is inversely proportional to the bulk matrix attenuation correction factor. Matrix corrections may then be applied by simply normalizing all fluorescence measurements from a sample to the intensity of the Compton scatter line derived from one of the characteristic fluorescence lines from the X-ray source. This procedure is, however, subject to an important restriction. Corrections are only valid providing no significant absorption edge intervenes between the energy of the Compton scatter peak and the fluorescence line... 

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