Tensor scattering amplitude

To introduce the concepts of tensor scattering amplitude and amplitude matrix it is necessary to choose an orthonormal unit system for polarization description. In Sect. 1.2 we chose a global coordinate system and used the vertical and horizontal polarization unit vectors Ba and e,g, to describe the polarization state of the incident wave (Fig. 1.9a). For the scattered wave we can proceed analogously by considering the vertical and horizontal polarization unit vectors and eg. Essentially, (e/t, 6/3,60.) are the spherical unit vectors of fee, while er,eg,e ) are the spherical unit vectors of fcs in... 

The tensor scattering amplitude or the scattering dyad is given by [169]... 

The tensor scattering amplitude satisfies a useful symmetry property which is referred to as reciprocity. As a consequence, reciprocity relations for the amplitude, phase and extinction matrices can be derived. Reciprocity is a manifestation of the symmetry of the scattering process with respect to an inversion of time and holds for particles in arbitrary orientations [169]. In order to derive this property we use the following result if H and E2 H2 are the total fields generated by the incident fields E i, ffei and E 2, He2, respectively, we have... 

To derive the expressions of the tensor scattering amplitude and amplitude matrix, we consider the scattering and incident directions and eu, and express the vector spherical harmonics as... 

To obtain the symmetry relation we proceed as in the derivation of the reciprocity relation for the tensor scattering amplitude, i.e., we consider the electromagnetic fields E, Hu generated by the incident fields E u, H u, with u= 1,2. The starting point is the integral (cf. (1.90))... 

B(t) is the scattering amplitude of a particle, which depends on the particle polarizability at given orientation. B(t) changes with time due to reorientation of the particle. If the scatterers are spherical, B(t) is constant and Cg(x) = 1. Note that Cg(x) does not depend on the scattering angle and can be calculated if the polarizability tensor and the rotational diffusion tensor of the particles are known. The calculation of CJi q,x) requires averaging of the translational diffusion tensor of the particle over all possible orientations to obtain the averaged translational diffusion coefficient. 

More recent interest has focused upon the interpretation of the relative intensities of the electronic Raman transitions. The theory of electronic Raman spectroscopy has been well-summarized elsewhere [63, 202], and the electronic Raman scattering amplitude from an initial i/rf) to a final jfj) vibronic state (where the phonon states are the same, and usually zero-phonon (i.e. electronic) states) is given by (i/ r czpCF In this expression, the cartesian polarizations of the incident photon (hcv) and the scattered photon (hcvs) are a and p, respectively. The Cartesian electronic Raman scattering tensor is written as... 

Thus a scattering center is characterized by the space and time dependent quantity (C/t) which is a second rank tensor. The amplitude of the scattered field from a given center observed under a given polarization geometry xj. /J3 is determined as far as the sample is concerned by the expreJsiSn ... 

Raman spectroscopy is an inelastic light scattering experiment for which the intensity depends on the amplitude of the polarizability variation associated with the molecular vibration under consideration. The polarizability variation is represented by a second-rank tensor, oiXyZ, the Raman tensor. Information about orientation arises because the intensity of the scattered light depends on the orientation of the Raman tensor with respect to the polarization directions of the electric fields of the incident and scattered light. Like IR spectroscopy, Raman... 

Accordingly, for a given Raman-active resonance r, the amplitudes of XijM can be expressed in terms of the isotropy and symmetric anisotropy invariants of the corresponding spontaneous Raman scattering tensor, a2 and 7S2, respectively. In the case of frequency-degenerate CRS considered here, the two relevant independent tensor components assume the following form [33, 34] ... 

Two points, however, should be taken into account. First, natural crystals can show significant variability that depends upon the growth conditions and locality (e.g., solid solutions and incorporation of impurities). It is necessary to measure the bulk crystal structure of such samples before it is possible to determine the surface structure using the CTR approach for such samples. Second, the CTR intensities can depend on the type of form factors (e.g., neutral or ionic form factors) used in the bulk structure analysis. At minimum, the calculated bulk Bragg reflectivities must reproduce the observed values precisely internal consistency requires that we use the same atomic form factors that were used in the determination of the bulk crystal structure. Similarly, the bulk vibrational amplitudes derived from the original bulk crystal structure analysis must be used. In many cases, vibrational amplitudes are anisotropic and are therefore described by a tensor. The appropriate projection of the vibrations for each scattering condition, Q, needs to be included in the expression for Fuc-... 

The theory of the method is rather complicated, because the amplitude of Raman scattering is described by a second-rank tensor, so it will not be discussed here. Just like for fluorescence, P2 cosff)) and (P4(cosd)), or cos 6) and (cos" 6), can, at least in principle, be obtained for the simplest type of uniaxial orientation distribution and these values now refer directly to the molecules of the polymer itself. In practice it is often necessary to make various simplifying assumptions. For biaxially oriented samples several other averages can be obtained. 

To evaluate the OBE amplitudes we use in this work the Bonn potential as it is defined by the meson parameters in Table A.2 of Ref. [7]. For further discussion of this topic, see e.g. Machleidt s contribution in these proceedings. There are three sets of meson parameters which then define three potentials, referred to as the Bonn A, B and C potentials. These potentials differ in the strength of the tensor force, which is reflected in the probability of the D-state of the deuteron. The significance of the tensor force for both nuclear matter and finite nuclei will be discussed in Section 2.3. The coupling constants, cutoffs and masses of the various mesons of Table A.2 of [7] are redisplayed in Table 1. These meson parameters are obtained through a solution of the scattering equation for... 

The Stokes wave is amplified if the gain g exceeds the losses /. The amplification factor g depends on the square of the laser amplitude Ei and on the term (da/dq). Stimulated Raman scattering is therefore observed only if the incident laser intensity exceeds a threshold value that is determined by the nonlinear term (daij/dq)o in the polarization tensor of the Raman-active normal vibration and by the loss factor / =... 

From Eq. (8.33) for the mean square amplitude of the director fluctuations, we can derive the amplitude of the fluctuations of the dielectric tensor and then find the cross-section for the light scattering, see Section 11.1.3. The de Gennes descriptirai of the director fluctuations in the continuous medium [12] was a strong argument against the so-called swarm models of liquid crystals. That model was based on the... 

In this expression, a is a factor proportional to the scattered light intensity, Sij q) is the amplitude of the Fourier component with wavevector q of the dielectric tensor fluctuation, while vectors f and i are polarizations of, respectively, reflected and incident light waves. In the context of Bragg reflection from the cholesteric helix, we know already from the expression (2.25) that there is just one Fourier component with wavevector 2q. Its amplitude is complex because the second term in the expression (2.25) can be written as... 

Here 5e r,t) is the Fourier transform of the fluctuating part of the dielectric tensor, /and I are unit vectors describing the polarization directions of the scattered and incident beams, respectively, c is the speed of light in the medium, and O) are the amplitude and the frequency of the incident light, respectively, and r is the distance between the sample and detector. The wave-vectors of the incident (, ) and scattered light kf) determine the wave-vector of the... 

Due to the fact that the order parameter Q is related to macroscopic observable quantities such as the susceptibility tensor % (Eq. [32]) and the dielectric tensor f, fluctuations in the components of Q are directly manifested as fluctuations in c and in % and are therefore experimentally measurable. In Section 5 we will show how the fluctuation amplitude, < Qiy( ) >, can be used to calculate cross section of light scattering by fluctuations in the isotropic phase of liquid crystals. 

The amplitude of the scattered field Eg varies with time due to the fluctuations in the director orientation and hence the dielectric tensor. Therefore, a very convenient way to observe the director dynamics is to study the dynamics of the scattered light, i.e. to use dynamic light scattering for measuring the autocorrelation function of the scattered light. The normalised intensity autocorrelation function (q, tf) is defined as [11]... 

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