Azimuthal dependence of the Garton-Tomkins orbit in crossed magnetic and electric fields

Work on classical closed orbits in the diamagnetic Kepler problem is predominately focused on the chaos observed in the polar launch angle as opposed to the azimuthal launch angle. This is due to atomic systems, along with widely studied external-field geometries (parallel magnetic and electric fields or pure magnetic field), being uniform in azimuthal angle, rendering the azimuthal angle unimportant. In the case of crossed magnetic and electric fields, this is no longer the case, and closed orbits do present an azimuthal launch angle dependence. In atomic systems, due to their spherical symmetry, the electric-field orientation in the plane perpendicular to the magnetic field does not affect the spectrum of orbits. However, in shallow n-type donors in anisotropic semiconductors such as silicon, the orientation of the external fields with respect to conduction-band valleys will be important. In this work we examine the Garton-Tomkins orbit in crossed magnetic and electric fields, and analyze how it and its harmonics' azimuthal dependencies behave through variation of the scaled field or scaled energy. At low scaled fields, harmonics have either twofold or fourfold azimuthal dependencies determined by the rotational symmetry of the individual harmonics. As the scaled field or scaled energy is increased, several harmonics undergo significant bifurcations, resulting in large azimuthal angular regions of essentially closed orbits, which will lead to strong resonances in experimental work.