Extinction matrix orientation-averaged

The interaction matrix in Eq. (3.2) does not depend on the incident wave orientation, so it is convenient to perform orientation averaging in the cluster coordinate frame. Using the inverse matrix B = A to solve Eq. (3.2), we obtain the following general eqnation for the extinction cross section... 

The above relation shows that the orientation-averaged extinction cross-section for macroscopically isotropic and mirror-symmetric media is determined by the diagonal elements of the transition matrix in the particle coordinate system. The same result can be established if we consider an ensemble of randomly oriented particles (with 0 and 0) illuminated by a linearly polarized plane wave (with real polarization vector Cpoi). 

Despite the derivation of simple analytical formulas, the above analysis shows that the orientation-averaged extinction and scattering cross-sections for macroscopically isotropic and mirror-symmetric media do not depend on the polarization state of the incident wave. The orientation-averaged extinction and scattering cross-sections are invariant with respect to rotations and translations of the coordinate system and using these properties, Mishchenko et al. [169] have derived several invariants of the transition matrix. 

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... 

In terms of the elements of the extinction matrix, the orientation-averaged extinction cross-section is (cf. (1.89))... 

The orientation-averaged extinction matrix K) is computed by using (1.125) and (1.126), and note that for macroscopically isotropic and mirror-symmetric media the off-diagonal elements are zero and the diagonal elements are equal to the orientation-averaged extinction cross-section per particle. 

Figure 14. Extinction and scattering spectra for ballistic RF aggregates with different conjugate numbers N = 1-100. All data are averaged over random orientations (T-matrix method) without statistical averaging. Parameters of conjugates are the shell thickness and refractive index s = 2.5 nm, =1.40, respectively, the gold core diameter <7 =15 nm (a,b) and 60 nm (c,d).

A thin specimen of an oriented polymer containing a deformation band observed between crossed polarisers is impossible to orient so that the material in the band and in the undeformed material outside the band (the matrix ) are at extinction simultaneously. The parameter of immediate interest is the angle a needed to rotate from extinction in the matrix to extinction in the band. (Note that the direction of maximum refractive index in many polymers coincides with the average direction of chain orientation.)... 

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