The electromagnetic Darwin model for intense charged particle beams

Lee, W. Wei-li; Davidson, Ronald C.; Startsev, Edward A.; Qin, Hong

doi:10.1016/j.nima.2005.01.233

Cited by 6 publications

(4 citation statements)

References 17 publications

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“…The computation of canonical momentum P (x, v), which depends on the positions and velocities of other particles, may cause problems in this case because of the fact that the bounded quantities P , v, and A do not evolve consistently, so that the iteration does not converge for at least one of them. The same numerical difficulty was reported in [18].…”

Section: F Integration Schemesupporting

confidence: 80%

“…The derivation of expressions for electromagnetic fields is provided in the following subsections. This approach has a direct and intuitive character and has some advantages compared to the Hamiltonian formulation, but in contrast to the numerical solution in the fully electromagnetic case, some numerical difficulties reported by previous authors [5]- [7], [9], [10], [18] appear. They will be discussed briefly in Section II-F. We plan to incorporate this formulation to the magnetoinductive version of the PEPC code; however, in the following text, we concentrate on the Hamiltonian formulation only.…”

Section: B Lagrangian Formulationmentioning

confidence: 99%

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Mesh-Free Magnetoinductive Plasma Model

Mašek

Gibbon

2010

IEEE Trans. Plasma Sci.

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To study the transport phenomena of energetic particles in plasmas, a new type of mesh-free plasma simulation model is introduced. The model is implemented within the parallel tree code Pretty Efficient Parallel Coulomb-solver, which uses an O(N log N ) Barnes-Hut tree algorithm to speed up the force calculation. In this paper, we present an extension of this approach to include magnetic fields within the so-called Darwin or magnetoinductive approximation, which neglects the transverse part of the displacement current in Ampère's law. This effectively omits radiation, which means that the timestep is no longer limited by the Courant condition, as in the fully electromagnetic case. The ability of the model to give a valid description of an N-body system with self-consistently evolving E-and B-fields is demonstrated by preliminary tests.Index Terms-Darwin approximation, hierarchical tree code.

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Section: F Integration Schemesupporting

confidence: 80%

Section: B Lagrangian Formulationmentioning

confidence: 99%

Mesh-Free Magnetoinductive Plasma Model

Mašek

Gibbon

2010

IEEE Trans. Plasma Sci.

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“…The widely used particle-in-cell (PIC) simulation in the framework of this model for both 1D and 2D was developed in [1,9] and can be found in the review papers [3,5] or textbook [13]. New analytical investigations and applications of the Darwin model can be found in, e.g., [7,8]. Vlasov-Darwin codes have also been proposed more recently (see e.g., [10]).…”

Section: Introductionmentioning

confidence: 99%

Darwin model in plasma physics revisited

Xie¹,

Zhu²,

W.³

2014

Phys. Scr.

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Dispersion relations from the Darwin (a.k.a., magnetoinductive or magnetostatic) model are given and compared with those of the full electromagnetic model. Analytical and numerical solutions show that the errors from the Darwin approximation can be large even if phase velocity for a low-frequency wave is close to or larger than the speed of light. Besides missing two wave branches associated mainly with the electron dynamics, the coupling branch of the electrons and ions in the Darwin model is modified to become a new artificial branch that incorrectly represents the coupling dynamics of the electrons and ions.

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“…The magnetic interaction described by the Darwin Lagrangian is essential in relativistic many-electron calculations as noted by Breit and others [12][13][14][15]. It has found applications in nuclear physics [16,17], and especially in plasma physics, for numerical simulation [18][19][20][21][22], thermodynamics and kinetics [23][24][25][26][27], as well fundamental theory [28][29][30]. Barcons and Lapiedra [31] noted that the Darwin approach is not valid for a relativistic plasma and therefore used a different approach to its statistical mechanics.…”

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confidence: 99%

The exact Darwin Lagrangian

Essén

2007

Europhys. Lett.

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Darwin (1920) noted that when radiation can be neglected it should be possible to eliminate the radiation degrees-of-freedom from the action of classical electrodynamics and keep the discrete particle degrees-of-freedom only. Darwin derived his well known Lagrangian by series expansion in v/c keeping terms up to order (v/c) 2 . Since radiation is due to acceleration the assumption of low speed should not be necessary. A Lagrangian is suggested that neglects radiation without assuming low speed. It cures deficiencies of the Darwin Lagrangian in the ultra-relativistic regime.PACS numbers: 03.50. De, 11.10.Ef When radiation can be neglected the Lagrangian of classical electrodynamics, putting β = v/c, can be written,(1) In 1920 Darwin [1] expanded the Liénard-Wiechert potentials to second order in β = v/c and thus found that,and (hats are used for unit vectors),give the correct Lagrangian to this order. More recent derivations can be found in a few textbooks [2][3][4]. In particular Jackson [4] notes that using the Coulomb gauge (∇ · A = 0) makes the electrostatic Coulomb potential φ exact and moves all approximation to the vector potential A which obeys the inhomogeneous wave equation with the transverse (divergence free) current as source. [7]), and it is useful in various fundamental studies of electrodynamics [8][9][10][11]. The magnetic interaction described by the Darwin Lagrangian is essential in relativistic many-electron calculations as noted by Breit and others [12][13][14][15]. It has found applications in nuclear physics [16,17], and especially in plasma physics, for numerical simulation [18][19][20][21][22], thermodynamics and kinetics [23][24][25][26][27], as well fundamental theory [28][29][30]. Barcons and Lapiedra [31] noted that the Darwin approach is not valid for a relativistic plasma and therefore used a different approach to its statistical mechanics.Corrections to the Darwin Lagrangian have been discussed. Since a system of particles with identical charge to mass ratio does not dipole radiate a higher order expansion should be meaningful for such systems [32][33][34]. To that order, however, acceleration inevitably enters and must be handled in some way. Others have argued that since radiation is due to acceleration, v/c expansion is irrelevant, and further that radiation can be negligible even if the particle speeds are considerable (Trubnikov and Kosachev [35], Frejlak [36]). We will pursue that lead here.One frequently encounters the statement that the Darwin approach neglects retardation. This may be due to the fact that the, nowadays best known, elegant derivation by Jackson [4] hides the complications due to retardation. Nevertheless it is wrong. The derivations by Darwin [1] and by Landau and Lifshitz [2] show that the contribution of retardation to the Coulomb potential in the Lorenz gauge, is quite large. The main acceleration dependent part, however, vanishes either, as in Darwin's derivation, because it gives a total time derivative term in the Lagrangian, or, as in Landau ...

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The electromagnetic Darwin model for intense charged particle beams

Cited by 6 publications

References 17 publications

Mesh-Free Magnetoinductive Plasma Model

Mesh-Free Magnetoinductive Plasma Model

Darwin model in plasma physics revisited

The exact Darwin Lagrangian

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