Trojan Wave Packets in the Circularly Polarized and the Magnetic Fields on the multi-layer Langmuir type $(1)$ Helium trajectories extended for odd number of electron atoms

We extend the concept of the Langmuir type $(1)$ ``Hoop Earrings" rotating Helium-like model 

trajectories [1] used in the early attempts to impose the Hydrogen Bohr atom 

quantization  from the even $2N$ electron atoms to the $2N+1$

odd number or electrons. While the $2N$ electron 

Helium trajectories consist of two layers of electrons moving in phase on the two 

parallel circles with electron configurations placed at 

the vertexes of angles of regular polygons parallel in space those for $2N+1$ electrons must consist 

of k layers where k is the smallest divisor of the $2N+1$ with the one layer 

embedded in the plane containing the atom nucleus. When the $2N+1$ 

is a prime there is only one layer forming the regular polygon. Additionally the

polygons on one side of this plane can be of the different size so the 

whole configuration forms a ``Wigner Diamond".

Similarly to the $2N$ case the addition of the Circularly 

Polarized electromagnetic field with the electric field rotating in planes of 

the field free electrons is causing the shape polarization distortion from 

discrete rotational symmetry of the resulting polyhedron. The classical stabilization of the 

trajectories by the combination of fields further leads to the existence of 

non-dispersing localized wave packets moving around the trajectories. 

The time dependent Hartree simulations of existence of such Wave Packets as well as those with the Time 

Dependent Quantum Diffusion Monte Carlo Method are conducted.

[1] M. Kalinski, et al., Phys. Rev. Lett. {bf 95}, 103001, 

(2005).