Planar Resonance Nanotrap of Electrons

The paper presents research about a planar penning trap with axial symmetry in the case of resonance. A uniform magnetic field slightly deviates from the axis of symmetry of the trap. The geometry of flat electrodes and the voltage applied to them, and the deflection angle of the magnetic field from the axis of the restrictions provide double resonance: in the main part of the Hamiltonian of the system and in the main part of perturbations at small deflection angle of the magnetic field. In this paper a double quantum averaging is done. The averaged Hamiltonian is explicitly expressed through the joint forming the algebra of symmetries of the two generators of averaging. Depending on the resonance ratio of the secondary resonance, the original Hamiltonian using coherent conversion, either written as an ordinary differential equation of the second order, or obtained the asymptotic of the spectrum in the first two orders of the magnitude of the magnetic field.