Multi-layer model of the electron magnetic insulation in vacuum coaxial lines

In order to increase the ion diode voltage and to improve the focusing properties of the ion beam, KALIF will be extended with a high energy linear induction accelerator (HELIA) voltage adder. For a stationary case, the conservation of the Lagrange invariant L=B_φ/rn_e and the cathode side potential V, along the electron trajectories makes it possible to connect the parameters of the electron layer in the neighboring sections. This results in a possibility of the matching of the successive adder sections by making use of the minimum property of the internal energy for the magnetic insulation equilibrium. The theory developed can be extended to a large number of the electron layers N, thus giving the simplified resulting equations for the description of the electron multi-layer structure