The Classical Motion Of The Neutral Particles In The Ioffe Trap

The study of the magnetostatic traps for neutral particles is one of the most interesting studies at present . In this paper we study some basic and simple magnetostatic traps. In cylindrical coordinates frame, by using Biot-Savart law and other techniques, the general formulas of these fields are gi\en. From the exact expression of the field, we obtain a multipole polynomial expansion, and under the paraxial condition we furthermore obtain the approximate expression.The loffe trap, consisting of two coils with parallel currents and four straight conductors with currents in alternating directions, is one of the most important traps.We specially study the field structure of it by using both the exact expression and a multipole polynomial expansion that facilitates studies of classical or quantum orbits.If the region near the origin is of interest,we may obtain a simple expression of the field and this configuration may be called idealized loffe trap. We consider a neutral particle with magnetic moment antiparallel to the field.With the interaction potential energy between the magnetic moment of the particle and the magnetic field ,we obtain the classical motion equation of the neutral particles in the loffe trap.In some limit conditions,by using the perturbative method, the equations may take on concise forms.of which the two equations about x and y are Mathieu equations.If we properly set the parameters and have the condition A >> q > 0,we can solve the Mathieu equation with the traditional WKBJ method.As a new attemptation,with Fourier series expansion we solve the Mathieu equation and obtain the classical motion law of the neutral particles.