Scattering matrix symmetries

We first point out a few scattering matrix symmetries for single particles based on results in previous chapters. 

We have about exhausted possible scattering matrix symmetries obtained by appealing to approximate or exact solutions to specific scattering problems. But more can be said about particles, regardless of their shape, size, and composition, without explicit solutions in hand. The scattering matrix for a given particle implies those for particles obtained from this particle by the symmetry operations of rotation and reflection. We shall consider each of these symmetry operations in turn. 

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.,... 

To check (13.13) we can invoke the solution to the scattering problem for an optically active sphere. Such a sphere is symmetric under all rotations, but the off-diagonal elements of its scattering matrix (13.7) do not vanish identically. As required by symmetry and the matrices (13.12) and (13.13), (13.7) is invariant with respect to interchange and sign reversal of its off-diagonal elements. 

This is the form of the scattering matrix for any medium with rotational symmetry even if all the particles are not identical in shape and composition. A collection of optically active spheres is perhaps the simplest example of a particulate medium which is symmetric under all rotations but not under reflection. Mirror asymmetry in a collection of randomly oriented particles can arise either from the shape of the particles (corkscrews, for example) or from optical activity (circular birefringence and circular dichroism). 

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. 

For incident unpolarized light to be (partially) circularly polarized upon scattering by a collection of particles, the scattering matrix element S4l must not be zero. It was shown in Section 13.6 that the scattering matrix for a collection (with mirror symmetry) of randomly oriented particles has the form... 

Clary, D.C. (1994) Four-atom reaction dynamics, J. Phys. Chem. 98, 10678-10688. Pack, R.T. and Parker, G.A. (1987) Quantum reactive scattering in three dimensions using hyperspherical (APH) coordinates. Theory, J. Chem. Phys. 87, 3888-3921. Truhlar, D.G., Mead, C.A. and Brandt, 5I.A. (1975) Time-Reversal Invariance, Representations for Scattering Wavcfunctions, Symmetry of the Scattering Matrix, and Differential Cross-Sections, Adv. Chem. Phys. 33, 295-344. 

In particular, the T-matrix approach as formulated by WATERMAN [4.21] and STROM [4.22] explicitly incorporates the symmetry of the scattering matrix due to the time-reversal invariance, and is sufficiently general to allow variations which yield different approximation schemes. An important feature of these methods is their applicability to scattering by objects of arbitrary shape. For some alternative formulations and numerical results see BARBER and YEN [4.23] and WATERMAN [4.24]. 

Thus, once the open part of the scattering matrix is known, so are all the reaction probabilities at the same total energy. From the symmetry of S we have... 

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