Motion of Electrons in a Magnetic Bottle II

Abstract

A theoretical investigation of the motion of a single charged particle in an external magnetic field is given. The present description is slightly different in approach from the usual drift theory, though up to the first order the same results are obtained. Our method is based on the canonical formalism and easier to develop.to higher order approximation with some cost of intuitive aspects.
The static and axially symmetric nature of the field is assumed, and this leads to the magnetic lines of force and magnetic equi-potential lines, which, as well as rotating angle around the axis, make an orthogonal curvilinear coordinate system (ξ2). This system is suitable to describe the motion of the particle. The position and momentum 'it coordinates of the particle are transformed canonically into new variables (Q_i. P_i.) (i=1, 2, 3), whose components represent gyration, longitudinal drift motion, and azimuthal drift motion, respectively.. Equations of motion for these variable, are derived in ξ3. In solving the motion the Hamilton-Jacob, formalism is applied throughout. If we confine ourselves to the region near the magnetic axis, the expansion, of quantities around the axis is useful, and the curved nature of the lines of force and the equi-potential lines is considered as perturbation. In the case of the homogeneous field (ξ4), of course, the problem is soluble completely. As the lowest (zeroth) approximation of the perturbation expansion, we consider the case of parallel but non-uniform field along the axis, in which case a separable equation is obtained. The angle and action variables (W_i ⁽⁰⁾ , J_i ⁽⁰⁾ ) are introduced. Action variables J₁ ⁽⁰⁾ and J₂ ⁽⁰⁾ correspond to the magnetic moment associated with the gyration and the longitudinal invariant respectively, which are adiabatic invariants, while J₃ ⁽⁰⁾ the angular momentum around the axis, conserves absolutely. Following Born's prescription, we go ahead into the first order approximation (ξ5).The variations of J₁ ⁽⁰⁾ and J₂ ⁽⁰⁾ are calculated, where use of Born-Oppenheimer's approximation is made. As a secular motion for w₃, the azimuthal drift motion appears.