Amplitude Scattering Matrix Elements

If the amplitude scattering matrix elements (5.4) for a homogeneous, isotropic sphere of radius a are divided by the volume v, the resulting quotients approach finite limits as the sphere radius tends to zero  

Because of condition (6.2) it is customary (but not necessary) to write the scattering matrix elements (6.1) as 

Consider an arbitrary particle illuminated by a plane wave propagating in the z direction (Fig. 6.1). The contribution of a volume element At located at a point O to the field scattered by the particle in a direction specified by the unit vector er is 

Imphcit in the derivation of (6.9) is the assumption that the particle is homogeneous. This is not necessary, however the particle may be composed of several distinct regions. The generalization of (6.9) to a heterogeneous particle is straightforward  

The optical theorem yields the absorption cross section 

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In Chapter 8 we shall derive the field scattered by an infinite cylinder of arbitrary radius and refractive index we shall also consider scattering by a finite cylinder in the diffraction theory approximation. Although the finite cylinder scattering problem is not exactly soluble, we can obtain analytical expressions for the amplitude scattering matrix elements in the Rayleigh-Gan s approximation. 

The amplitude scattering matrix elements correct to terms of order x2 are... 

As a check on the amplitude scattering matrix elements, we compute Qcxt in BHMIE from the optical theorem (4.76), whereas Qsca is computed from the series (4.61). POL, the degree of polarization, must vanish for scattering angles of 0 and 180°, as must 34- Also, the 4x4 scattering matrix elements must satisfy... 

SUBROUTINE BHCYL CALCULATES AMPLITUDE SCATTERING MATRIX ELEMENTS AND EFFICIENCIES FOR EXTINCTION AND SCATTERING FOR A GIVEN SIZE PARAMETER AND RELATIVE REFRACTIVE INDEX THE INCIDENT LIGHT IS NORMAL TO THE CYLINDER AXIS PAR .ELECTRIC FIELD PARALLEL TO CYLINDER AXIS PERIELECTRIC FIELD PERPENDICULAR TO CYLINDER AXIS sc sc sc scsc scsc sc sc sc sc ... 

To calculate the amplitude scattering matrix elements, the incident and the scattered fields are decomposed into the components parallel and perpendicular to the scattering plane as shown in Figure 4. 

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... 

In Section 4.4 we showed that the off-diagonal elements of the amplitude scattering matrix (3.12) are zero for a nonactive sphere. If the sphere is optically active, however, the matrix elements are... 

Thus, the difference between the diagonal elements of the forward amplitude scattering matrix in the circular polarization representation has a simple physical interpretation. Although we considered identical particles for conveni-... 

The off-diagonal elements of the amplitude scattering matrix for an optically active sphere (Section 8.3)... 

We close this section by noting that the destructive interference has its origin in the spatial wavefunction, which gives cancellations in the scattering matrix element if the amplitudes for particle a (or j3) has opposite signs when the neutron scatters from sites 1 and 2. Fig. 5 is an attempt to illustrate the scattering from a particle that is delocalized over two different sites. 

The scattering amplitude Is related to the body-fixed scattering matrix element according to... 

Here the distortion (diagonal) and back coupling matrix elements in the two-level equations (section B2.2.8.4) are ignored so that = exp(ik.-R) remains an imdistorted plane wave. The asymptotic solution for ij-when compared with the asymptotic boundary condition then provides the Bom elastic ( =f) or inelastic scattering amplitudes... 

This matrix element represents the amplitude for the scattering of two negatons from an initial state... 

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