The discrete dipole approximation (DDA) was used to compute the scattering cross-section of a variety of hexagonal prisms from long columns through to thin plates. Here we focus on the Rayleigh regime which is appropriate for particles which are small relative to the wavelength. In this case the scattered electric field can be expressed in terms of an induced dipole moment which scales in proportion to the volume of the crystal, and this means that the problem is simplified to deriving the elements of a "polarisability tensor" which relates the applied electric field from the radar to the induced dipole moment. For particles with hexagonal symmetry this tensor is, like the case of a spheroid, diagonal and turns out to contain only 2 independent elements.
These polarisabilities are compared to Gans's theory. We find that the spheroid approximation in fact significantly overestimates the polarimetric signal from oriented plate crystals: for an aspect ratio in the range 2--10 the differential reflectivity is 1--1.5dB too high using the spheroid model.
To improve on this situation, a simple empirical modification of Gans's theory is proposed. The same functional dependence on volume and permittivity is assumed, but a new pair of geometrical functions L as a function of aspect ratio are derived from the numerical results. These new empirical formula give excellent agreement with the DDA data (RMS difference <1%) and are actually simpler to apply than Gans's original equations.
Having analytical formulae for the polarisabilities is highly advantageous, since the scattered field for particles in any desired orientation (or distribution of orientations) and transmitted polarisation can be computed immediately, without the need to rerun the computationally expensive DDA simulations. Forward scattering (ie. differential phase shift) is captured as well as the backscatter quantities.
DDA calculations of the scattering from more complex branched and dendritic plate crystals have also been performed. These indicate that such crystals may be accurately modelled as an enclosing hexagonal prism, but but with a reduced permittivity dependent on the volume of ice within that enclosing boundary, prescribed by the Maxwell-Garnett dielectric mixture theory.
Supplementary URL: http://www.met.reading.ac.uk/~sws04cdw/polarisability_new.pdf