A Conformally Invariant Derivation of Average Electromagnetic Helicity

The average helicity of a given electromagnetic field measures the difference between the number of left- and right-handed photons contained in the field. Here, the average helicity is derived using the conformally invariant inner product for Maxwell fields. Several equivalent integral expressions in momentum space, in ( r , t ) space, and in the time-harmonic ( r , ω ) space are obtained, featuring Riemann–Silberstein-like fields and potentials. The time-harmonic expressions can be directly evaluated using the outputs of common numerical solvers of Maxwell equations. The results are shown to be equivalent to the well-known volume integral for the average helicity, featuring the electric and magnetic fields and potentials.

Group-Theoretical Approach to Reduced Orientation Spaces for Crystallographic Textures

A unified group-theoretical approach to the reduction problem for the orientation space of a crystallographic texture is developed. After preliminary considerations of the three-dimensional rotation group SO(3) the concept of the invariant inner distance function in the group space has been introduced. Left and right group translations, inner auto-morphisms, motions of general form, and inversion transforms in the space SO(3) are analysed. The concept of Dirichlet-Voronoi partition dual to an arbitrary finite set of rotations has been considered. It is shown that the Dirichlet-Voronoi partition, dual to the proper point group for the grain lattice of original orientation, is regular with respect to the group of motions generated by elements of proper point group.It is demonstrated that the true orientation space of a texture (at the absence of specimen symmetry) may be obtained by passing to the topological closure of any Dirichlet-Voronoi domain with the next, identifying crystallographically equivalent rotations belonging to its topological boundary. Thus an invariant derivation for the reduced (true) orientation space is given that does not require using any particular parametrization for the group space SO(3).Symmetry properties of Dirichlet-Voronoi domains are studied in conclusion. It is shown that any domain of such kind admits a finite group of symmetries generated by elements of a proper point group and by an appropriate inversion of the group space SO(3). It is proved that only part of them may be extended onto the true orientation space.