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• W4229701164 abstract "<strong class=journal-contentHeaderColor>Abstract.</strong> This paper provides a complete generalization of the classic result that the radius of curvature (<i>&#x03C1;</i>) of a charged-particle trajectory confined to the equatorial plane of a magnetic dipole is directly proportional to the cube of the particle's equatorial distance (&#x03D6;) from the dipole (i.e. <i>&#x03C1;</i> &#x221D; &#x03D6;³). Comparable results are derived for the radii of curvature of all possible planar charged-particle trajectories in an individual static magnetic multipole of arbitrary order <i>m</i> and degree <i>n</i>. Such trajectories arise wherever there exists a plane (or planes) such that the multipole magnetic field is locally perpendicular to this plane (or planes), everywhere apart from possibly at a set of magnetic neutral lines. Therefore planar trajectories exist in the equatorial plane of an axisymmetric (<i>m</i> = 0), or zonal, magnetic multipole, provided <i>n</i> is odd: the radius of curvature varies directly as &#982;<i>ⁿ</i>⁺². This result reduces to the classic one in the case of a zonal magnetic dipole (<i>n </i>=1). Planar trajectories exist in 2<i>m</i> meridional planes in the case of the general tesseral (0 < <i>m</i> < <i>n</i>) magnetic multipole. These meridional planes are defined by the 2<i>m</i> roots of the equation cos[<i>m</i>(<i>&#934;</i> – <i>&#934;_n^m</i>)] = 0, where <i>&#934;_n^m</i> = (1/<i>m</i>) arctan (<i>h_n^m</i>/<i>g_n^m</i>); <i>g_n^m</i> and <i>h_n^m</i> denote the spherical harmonic coefficients. Equatorial planar trajectories also exist if (<i>n</i> – <i>m</i>) is odd. The polar axis (<i>&#952;</i> = 0,<i>&#960;</i>) of a tesseral magnetic multipole is a magnetic neutral line if <i>m </i> > 1. A further 2<i>m</i>(<i>n</i> – <i>m</i>) neutral lines exist at the intersections of the 2<i>m</i> meridional planes with the (<i>n</i> – <i>m</i>) cones defined by the (<i>n</i> – <i>m</i>) roots of the equation <i>P_n^m</i>(cos <i>&#952;</i>) = 0 in the range 0 < <i>&#952;</i> < <i>&#960;</i>, where <i>P_n^m</i>(cos <I>&#952;</I>) denotes the associated Legendre function. If (<i>n</i> – <i>m</i>) is odd, one of these cones coincides with the equator and the magnetic field is then perpendicular to the equator everywhere apart from the 2<i>m</i> equatorial neutral lines. The radius of curvature of an equatorial trajectory is directly proportional to &#982;<i>ⁿ</i>⁺² and inversely proportional to cos[<i>m</i>(<i>&#934;</i> – <i>&#934;_n^m</i>)]. Since this last expression vanishes at the 2<i>m</i> equatorial neutral lines, the radius of curvature becomes infinitely large as the particle approaches any one of these neutral lines. The radius of curvature of a meridional trajectory is directly proportional to <i>rⁿ</i>⁺², where <i>r</i> denotes radial distance from the multipole, and inversely proportional to <i>P_n^m</i>(cos <i>&#952;</i>)/sin <i>&#952</i>;. Hence the radius of curvature becomes infinitely large if the particle approaches the polar magnetic neutral line (<i>m</i> > 1) or any one of the 2<i>m</i>(<i>n</i> – <i>m</i>) neutral lines located at the intersections of the 2<i>m</i> meridional planes with the (<i>n</i> – <i>m</i>) cones. Illustrative particle trajectories, derived by stepwise numerical integration of the exact equations of particle motion, are presented for low-degree (<i>n</i> &#x2264; 3) magnetic multipoles. These computed particle trajectories clearly demonstrate the "non-adiabatic'' scattering of charged particles at magnetic neutral lines. Brief comments are made on the different regions of phase space defined by regular and irregular trajectories." @default.
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• W4229701164 date "1997-01-01" @default.
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• W4229701164 title "Planar charged-particle trajectories in multipole magnetic fields" @default.
• W4229701164 doi "https://doi.org/10.1007/s005850050433" @default.
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