Researching | High-order corrections to the radiation-free dynamics of an electron in the strongly radiation-dominated regime

Journals >Matter and Radiation at Extremes >Volume 8 >Issue 1 >Page 014402 > Article

Matter and Radiation at Extremes
Vol. 8, Issue 1, 014402 (2023)

High-order corrections to the radiation-free dynamics of an electron in the strongly radiation-dominated regime

A. S. Samsonov^a), E. N. Nerush, and I. Yu. Kostyukov

Author Affiliations

Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanov St., Nizhny Novgorod 603950, Russia

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DOI: 10.1063/5.0117504 Cite this Article

A. S. Samsonov, E. N. Nerush, I. Yu. Kostyukov. High-order corrections to the radiation-free dynamics of an electron in the strongly radiation-dominated regime[J]. Matter and Radiation at Extremes, 2023, 8(1): 014402 Copy Citation Text

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Abstract

A system of reduced equations is proposed for electron motion in the strongly radiation-dominated regime for an arbitrary electromagnetic field configuration. The approach developed here is used to analyze various scenarios of electron dynamics in this regime: motion in rotating electric and magnetic fields and longitudinal acceleration in a plane wave and in a plasma wakefield. The results obtained show that this approach is able to describe features of electron dynamics that are essential in certain scenarios, but cannot be captured in the framework of the original radiation-free approximation [Samsonov et al., Phys. Rev. A 98, 053858 (2018) and A. Gonoskov and M. Marklund, Phys. Plasmas 25, 093109 (2018)]. The results are verified by numerical integration of the nonreduced equations of motion with account taken of radiation reaction in both semiclassical and fully quantum cases.

a_{0} = \frac{e E_{0}}{m c ω},

(1)

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χ = \frac{γ}{E_{S}} \sqrt{{(E + v \times B)}^{2} - {(v E)}^{2}},

(2)

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W_{rad} \approx α \frac{m c^{2}}{γ ℏ} \times \{\begin{cases} 1.4 χ, & χ ≪ 1, \\ 0.7 χ^{2 / 3}, & χ ≫ 1, \end{cases}

(3)

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\frac{λ_{W}}{λ} \approx \frac{1}{α a_{0}} \times \{\begin{cases} 1, & χ ≪ 1, \\ χ^{1 / 3}, & χ ≫ 1 . \end{cases}

(4)

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λ_{f} ≪ λ_{W} ≪ λ,

(5)

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\frac{d p}{d t} = - E - v \times B - F_{rr} v,

(6)

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\frac{d γ}{d t} = - v E - F_{rr} v^{2},

(7)

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F_{rr} = \frac{α a_{S}}{3 \sqrt{3} π} \int_{0}^{\infty} \frac{4 u^{3} + 5 u^{2} + 4 u}{{(1 + u)}^{4}} K_{2 / 3} (\frac{2 u}{3 χ}) d u,

(8)

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F_{rr} \approx α a_{S} \times \{\begin{cases} 0.67 χ^{2}, & χ ≪ 1, \\ 0.37 χ^{2 / 3}, & χ ≫ 1 . \end{cases}

(9)

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\frac{d γ}{d t} = - v E - F_{rr} v^{2},

(10)

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\frac{d v}{d t} = - \frac{1}{γ} (E + v \times B - v (v E) + \frac{F_{rr} v}{γ^{2}}) .

(11)

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E + v_{0} \times B - v_{0} (v_{0} E) = 0 .

(12)

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\frac{d r}{d t} = v_{0} (E (r, t), B (r, t)) .

(13)

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v = (1 - \frac{δ^{2}}{2}) v_{0} + v_{1},

(14)

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δ^{2} \approx v_{1}^{2} + γ^{- 2} .

(15)

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\frac{d v_{1}}{d t} = \frac{F_{1}}{γ} - \frac{d v_{0}}{d t} - v_{0} (v_{1} \frac{d v_{0}}{d t}),

(16)

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\frac{d γ}{d t} = - v_{0} E (1 - \frac{v_{1}^{2}}{2} - \frac{1}{2 γ^{2}}) - v_{1} E - F_{rr} (χ),

(17)

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F_{1} = (v_{0} E) v_{1} + (v_{0} B) [v_{0} \times v_{1}] + O (δ^{2}),

(18)

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χ = \frac{γ | F_{1} |}{a_{S}} .

(19)

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\frac{d v_{0}}{d t} = \frac{\partial v_{0}}{\partial t} + (v \nabla) v_{0} .

(20)

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\frac{1}{2} \frac{d v_{1}^{2}}{d t} = - v_{1} \frac{d v_{0}}{d t} + \frac{v_{1}^{2} (v_{0} E)}{γ} .

(21)

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τ_{v} = \frac{γ}{| v_{0} E |} \sim \frac{γ}{a_{0}} .

(22)

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Ω \times v_{1} = - \frac{E}{γ} v_{1} + \frac{B}{γ} {e \times v}_{1} + Ω \times e + v_{1} e .

(23)

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v_{1} = v_{⊥} Ω \times e + v_{x} Ω .

(24)

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v_{x} = \frac{γ B}{E^{2} + B^{2}},

(25)

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v_{⊥} = \frac{γ E}{E^{2} + B^{2}} .

(26)

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E = F_{rr} (χ) = F_{rr} (\frac{γ^{2}}{a_{S}}) .

(27)

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v_{0} = \frac{E \times B}{E^{2}},

(28)

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\frac{d γ}{d φ} = - 2 \frac{p γ E}{1 + p^{2}} - 2 A_{rr} E^{2} (1 + p^{2}),

(29)

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\frac{d p}{d φ} = - E - 2 A_{rr} E^{2} \frac{p}{γ} (1 + p^{2}),

(30)

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\begin{align} p_{pw} & = - a_{0} \sin φ, \end{align}

(31)

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\begin{align} γ_{pw} & = γ_{0} (1 + {p_{pw}}^{2}), \end{align}

(32)

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γ \approx γ_{0} (1 + a_{0}^{2} \sin^{2} φ) (1 + \frac{A_{rr} a_{0}^{2}}{γ_{0}} φ),

(33)

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v \approx - \frac{a_{0} \sin φ}{γ} .

(34)

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⟨ χ ⟩, ⟨ v_{1} ⟩, ⟨ γ^{- 1} ⟩ \propto {(A_{rr} t)}^{- 1 / 3} .

(35)

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χ = \frac{E (φ)}{E_{S}} (γ - p_{x}) .

(36)

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\frac{d ⟨ χ^{2} ⟩}{d t} = 0,

(37)

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χ = \frac{γ E v_{1}}{a_{S}} .

(38)

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⟨ F_{rr} ⟩ ⟨ v_{1}^{2} ⟩ + γ ⟨v_{1} \frac{d v_{0}}{d t}⟩ = 0 .

(39)

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\frac{〈 F_{rr} 〉}{γ} (3 〈 F_{rr} 〉 - 2 E_{acc}) 〈 v_{1}^{2} 〉 + γ ⟨v_{1} \frac{d^{2} v_{0}}{d t^{2}} - {(\frac{d v_{0}}{d t})}^{2}⟩ = 0 .

(40)

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\frac{d r}{d t} = v_{0} + v_{1},

(41)

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v_{0} = - \frac{- z_{0} E_{acc} + y E_{foc}}{E} \approx z_{0} - y \frac{E_{foc}}{E_{acc}} \equiv z_{0} - κ y .

(42)

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y = y_{0} \cos ω t,

(43)

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v_{1, y} = y_{0} (κ \cos ω t - ω \sin ω t) .

(44)

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\begin{align} ⟨v_{1} \frac{d^{2} v_{0}}{d t^{2}} - {(\frac{d v_{0}}{d t})}^{2}⟩ \\ = y_{0}^{2} ω^{2} κ (κ ⟨ \cos^{2} ω t - \sin^{2} ω t ⟩ - ω ⟨ \sin ω t \cos ω t ⟩) = 0 . \end{align}

(45)

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⟨ F_{rr} ⟩ = \frac{2}{3} E_{acc} .

(46)

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⟨ F_{rr} ⟩ = \frac{12}{19} E_{acc},

(47)

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A. S. Samsonov, E. N. Nerush, I. Yu. Kostyukov. High-order corrections to the radiation-free dynamics of an electron in the strongly radiation-dominated regime[J]. Matter and Radiation at Extremes, 2023, 8(1): 014402

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