Symmetries of the Transition Matrix

Symmetry properties of the transition matrix can be derived for specific particle shapes. These symmetry relations can be used to test numerical codes as well as to simplify many equations of the T-matrix method and develop efEcient numerical procedures. In fact, the computer time for the numerical evaluation of the surface integrals (which is the most time consuming part of the T-matrix calculation) can be substantially reduced. The surface integrals are usually computed in spherical coordinates and for a surface defined by 

For an axisynunetric particle, the equation of the surface does not depend on ip, and therefore dr jdip = 0. The vector r has only r- and e -components and is independent of ip. The integral over the azimuthal angle can be performed analytically and we see that the result is zero unless m = m.  

Therefore, the T matrix is diagonal with respect to the azimuthal indices m and mi, i.e.. 

We consider a particle with a principal Wfold axis of rotation symmetry and assume that the axis of symmetry coincides with the 2 -axis of the particle coordinate system. In this case, r, drjdO and drjdif are periodic in with period 2-k/N. Taking into account the definition of the matrices, we see that the surface integrals are of the form 

As a consequence of the rotation symmetry around the 2-axis, the T matrix is invariant with respect to discrete rotations of angles Ofc = 2irk/N, k = 1,2. N - 1. A necessary and sufHcient condition for the equation 

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