Individual particle motion and the effect of scattering in an axially symmetric magnetic field

Garren, A.; Riddell, Robert J.; Smith, Lloyd H.; Bing, George F.; Roberts, John Η.; Northrop, T. G.; Henrich, Louis R.

doi:10.1016/0891-3919(58)90136-0

Cited by 19 publications

(13 citation statements)

References 5 publications

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“…In this paper we pursue the question of nonadiabatic motion and derive an analytical expression for A#, the change in magnetic moment # = vñ2/2B, of particles as they cross the model Jovian equator. Such changes have been observed previously in calculations which follow numerically the orbits of charged particles in magnetic fields modeled after those in laboratory devices [Garren et al, 1958;Grad and Van Norton, 1962;Leftel and Gray, 1969;Howard, 1971;Cohen et al, 1978], the earth's radiation belts [Taylor and Hastie, 1971], and the terrestrial magnetotail [Gray and Lee, 1982]. In each instance, particles move along field lines, and v•_2/2B is nearly constant except at the point of weak field intensity.…”

Section: Introductionsupporting

confidence: 60%

Pitch angle diffusion in the Jovian magnetodisc

Birmingham¹

1984

J. Geophys. Res.

120

131

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Currents in the Jovian middle magnetosphere draw out magnetic field lines radially into a configuration which has high equatorial curvature and weak field strength. Charged particles spiral along field lines into this equatorial region, where, depending on their energies, they may have gyroradii comparable to or larger than the scale length of the field. Magnetic moment violation resulting in a stochastic change in pitch angle may thus occur locally at the equator. Using a linear approximation to the field variation and a mathematical technique for distorting the orbit integration into the complex gyrophase plane, we derive a formula for the nonadiabatic change Δµ in magnetic moment. Δµ varies as cos Ψr where Ψr is particle gyrophase at the equator, so that if a particle traverses the equator several times with different values of Ψr, it suffers randomly correlated values of Δµ. The algebraic formula for Δµ is in excellent agreement with previous numerical computations. The phase space density, averaged over several bounce periods, satisfies a diffusion equation in pitch angle. The diffusion coefficient depends on particle energy, pitch angle, and the field line along which the particles are moving. Energetic sulfur ions (hundreds of keV and higher) are diffused in a few tens of bounce periods beyond 20 RJ, and the behavior of protons is not too dissimilar. Electrons, which at the same energy have much smaller gyroradii, become nonadiabatic and diffusive only near 28 RJ, where the magnetic model predicts a combination of especially weak field and large spatial variation. Pioneer and Voyager particle results are summarized in the light of this diffusion mechanism; while there is evidence for magnetic moment violation, an utter absence of pitch angle structure in the outer magnetodisc region (which should thus follow from the magnetic model) has not been confirmed (or looked for, to a large degree). Implications for other magnetospheric mechanisms are discussed. Finally, the possibility of a similar diffusion in the Saturnian ring current region is addressed, and it is found that only at much higher energies (>100‐MeV protons) is this type of nonadiabatic behavior to be expected.

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Section: Introductionsupporting

confidence: 60%

Pitch angle diffusion in the Jovian magnetodisc

Birmingham¹

1984

J. Geophys. Res.

120

131

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“…. The distinction behreen these tHo possibilities~ cumulative and oscillatory, may not ahrays be sharp, though in one geometry it seemed to be quite sharp for the magnetj_c moment [see Garren, et · al., 1958]. …”

Section: T\pplication Of Adll\bi\tic Theory To Plasmasmentioning

confidence: 99%

Adiabatic charged‐particle motion

Northrop¹

1963

Reviews of Geophysics

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603

185

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The adiabatic theory of charged‐particle motion is developed systematically in this review. We present the essentials of the theory without giving all the analysis in detail. The general expressions for guiding‐center motion and particle energy change are given, with application to the Van Allen radiation and to Fermi acceleration. It is shown that Fermi acceleration and betatron acceleration should not be regarded as distinct processes. Modifications of the nonrelativistic theory that are necessary when the particle is relativistic are discussed. Proofs are given of the invariance to lowest order of the first and second adiabatic invariants for the case of static fields. Finally, applications are made to the theory of plasmas.

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“…As the perturbation grows and ε 1, a dramatic change in the character of the motion, from bounded to unbounded, will occur, as the adiabaticity breaks down, the third constant of motion is lost, the motion becomes non-integrable, and the orbits become chaotic. This process was first investigated numerically [3] and later fully explained by the rigorous KAM (Kolmogorov, Arnold & Moser) theory [4]. Subsequently, Chirikov provided an approximate, but more practical, approach when he introduced his semi-empirical resonance-overlap criterion and the standard mapping [5].…”

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confidence: 99%

Breakdown of adiabatic invariance in spherical tokamaks

Carlsson

2001

Physics of Plasmas

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ABSTRACT. Thermal ions in spherical tokamaks have two adiabatic invariants: the magnetic moment and the longitudinal invariant. For hot ions, variations in magnetic-field strength over a gyro period can become sufficiently large to cause breakdown of the adiabatic invariance. The magnetic moment is more sensitive to perturbations than the longitudinal invariant and there exists an intermediate regime, super-adiabaticity, where the longitudinal invariant remains adiabatic, but the magnetic moment does not. The motion of super-adiabatic ions remains integrable and confinement is thus preserved. However, above a threshold energy, the longitudinal invariant becomes non-adiabatic too, and confinement is lost as the motion becomes chaotic. We predict beam ions in present-day spherical tokamaks to be super-adiabatic but fusion alphas in proposed burning-plasma spherical tokamaks to be non-adiabatic.Consider a charged particle, with mass m and charge q, gyrating with speed v in a magnetic field, with field strength B. Assume axi-symmetric toroidal geometry with coordinates (R, φ, Z). The motion preserves the energy E (because the Lorentz force performs no work) and canonical toroidal momentum P φ (because the toroidal coordinate is ignorable). There is no third exact constant of motion, but the normalized magnetic momentwhere v ⊥ is the velocity component perpendicular to the magnetic field (v 2 ⊥ = v 2 −v 2 , v = v·B/B) and B 0 is the on-axis magnetic field, is an adiabatic invariant, i.e. as long as the variation in the magnetic field experienced by the particle during one gyro period is small, Λ oscillates at the gyro frequency around a constant average. If we introduce the adiabaticity parameter, ε, the condition of slow (adiabatic) variation becomes ε = ̺ |∇B| B ≪ 1 , where the gyro radius ̺ = v ⊥ /Ω, with the gyro frequency Ω = |q|B/m. The approximate constancy of the magnetic moment was first pointed out by Alfvén [1]. Just as the magnetic moment is associated with the gyro motion, a second adiabatic invariant is associated with the slower drift motion:When ε ≪ 1, there are thus three constants of motion, one for each degree of freedom, i.e. the motion is integrable, the orbits are quasi-periodic and, in the absence of collisions, a particle will be eternally confined. As the perturbation grows and ε 1, a dramatic change in the character of the motion, from bounded to unbounded, will occur, as the adiabaticity breaks down, the third constant of motion is lost, the motion becomes non-integrable, and the orbits become chaotic. This process was first investigated numerically [3] and later fully explained by the rigorous KAM (Kolmogorov, Arnold & Moser) theory [4]. Subsequently, Chirikov provided an approximate, but more practical, approach when he introduced his semi-empirical resonance-overlap criterion and the standard mapping [5]. In the Chirikovian model, the breakdown of adiabaticity is caused by strong nonlinear resonance between the periodic motions occurring on different timescales.

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Individual particle motion and the effect of scattering in an axially symmetric magnetic field

Cited by 19 publications

References 5 publications

Pitch angle diffusion in the Jovian magnetodisc

Pitch angle diffusion in the Jovian magnetodisc

Adiabatic charged‐particle motion

Breakdown of adiabatic invariance in spherical tokamaks

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