Stopping, blooming, and straggling of directed energetic electrons in hydrogenic and arbitrary-z plasmas

Transcription

1 Stopping, blooming, and straggling of directed energetic electrons in hydrogenic and arbitrary-z plasmas This model Monte Carlo 1 MeV e 1 MeV e C. K. Li and R. D. Petrasso MIT 47th Annual Meeting of the America Physics Society Division of Plasma Physics Denver, CO October 24-28, 25

2 Motivation Stopping in solids Bohr, Bethe, Molière, Seltzer Fast ignition Electron penetration and straggling Energy deposition profile Beam blooming Preheat to determine tolerable levels Astrophysics (e.g. relativistic astrophysical jets) Companion presentations: R. Petrasso et al., GO1.5.e preheat C. Chen et al., QP1.137 simulations

3 Summary Fundamental elements of this plasma stopping model For hydrogenic plasmas, binary e e and e i scattering are comparable, and must be treated on an equal footing Energy loss, penetration, and scattering are inextricably coupled together Blooming and straggling effects, a consequence of scattering, lead to a non-uniform, extended region of energy deposition Whenever the Debye length is smaller than the gyro radius, binary interactions will dominate penetration, blooming, and straggling effects

4 Summary Additional elements of this plasma stopping model The results: Are insensitive to plasma density and temperature gradients Depend largely on ρ<x> Have strong Z dependence Applies to degenerate plasmas

5 Outline Introduction Relevance to fast ignition Models Discussions

6 Electron energy loss along the path (continuous slowing down) does not include the effects of scattering Binary interaction + plasma oscillation de ds = de ds Binary + de ds Oscillation mathematical equivalence Dielectric response function de ds 3 d k ik v 4πZe = Ze 3 2 ( 2π ) v k ( k,k v) ε L

7 Electron scattering must be included in calculating the energy deposition µm 1 MeV e ΣB ~ 4% x 14.1 µm µm 18.8 MeV p 2.2 µm 2.8 µm ΣB x ~. 5% 14.1 µm

8 Scattering reduces the electron linear penetration, and it results in longitudinal straggling and beam blooming Σ R Σ R e de ds Σ B Σ B de dx plasma <x> Combine all these effects, the energy deposition profile is modified

9 Outline Introduction Relevance to fast ignition Models Discussions

10 For the interior of a FI capsule, scattering dominates other mechanisms in affecting energy deposition, beam blooming, and straggling Laser ρ n b /n e > 1-2 : self fields, instabilities,. n b /n e ~1-2 n b /n e < 1-2 : scattering

11 This talk focuses on dense, deeply collisional regimes for which self-field corrections are unimportant B(r) B field 1-MeV e t p =1ps r (µm) r (µm) r b =1 µm L >> D L >> r G Current (Amp) 1.E+1 1.E+9 1.E E+12 1.E+11 1.E+1 B field (Guass) Beam energy (kj)

12 When λ D < r G, blooming, straggling, and penetration are determined by (collisional) binary interactions T e =5 kev; ρ=3g/cm 3 DT plasma n e ~ 1 26 /cm 3 1.E-6.1 rg (cm) 1.E-7 1.E-8 r b = 4 µm 1 µm λ D 3 µm r G 2 µm λ D Ratio.1 n b /n e (r G = λ D ) Scattering dominant Self fields important 1.E Beam energy (kj) Beam energy (kj) Only for very small deposition regions (beam size) and very large beam energy does r G approach λ D

13 Outline Introduction Relevance to fast ignition Models Discussions

14 This plasma model includes necessary effects that were previously untreated: Couples directly scattering and energy loss Includes the effects of longitudinal momentum loss [ v=v(1-cosθ)] upon energy loss Treats the case of full energy loss Results in both blooming and straggling In addition: This model is insensitive to plasma screening, and applies to degenerate plasmas (e.g. for e preheat)

15 Electron angular and spatial distributions are calculated by solving a diffusion equation V X θ V f s + [ f ( x, v', s) f ( x, v, s) ] ( v v' ) dv' v f = N σ v x = -v(1-cosθ) Angular distribution mean deflection angle, <cosθ> Longitudinal distribution penetration and straggling Lateral distribution beam blooming

16 The effects of energy loss and scattering must be treated with a unified approach 1.E+1 r o -2 sin 4 (θ /2 )(d σ /d Ω ) 1.E+ 1.E-1 Moller Rutherford f ( θ, E) E 1 = ( 2l + 1) P l(cosθ )exp k 4π l= E l ( ) de 1 ' E de' ds 1.E Scattering E (%) Energy loss Where: dσ kl( E) = ni 1 l θ ) dω [ P (cos ] dω

17 For hydrogenic plasmas, e-e scattering is comparable to e-i scattering (for γ <1) and must be included ~ 1.5 R = d σ d Ω ee d σ d Ω ei R 1..5 Z=3 Z=2 Z= γ

18 Scattering are insensitive to plasma screening models κ 1 ( E ) κ 1 (E) de ds Normnized κ1(e) 1 κ1(e)(de/ds) Debye interpartice TF Debye interparticle TF κ scattering 1 ( E) de ds 1 lnλ ei Plasma screening: λ D λ d Energy loss ---- Debye length ( γ + 1) 4 + ( γ lnλ 2 )/ inter-particle distance 4 lnλ ee E/E λ TF ---- Thomas-Fermi ρ = 3 g/cm 3 ; T e = 5 kev

19 Multiple scattering enhances electron linear-energy deposition MeV g -1 cm de dx de ds de 1 =< cosθ > dx de ds Electron energy (MeV) ρ = 3 g/cm 3 ; T e = 5 kev where E < cos θ >= exp κ1 E ' ( E ) de ds ' 1 de '

20 For 1-MeV electrons, straggling and blooming are proportional to when the energy loss > 4% x STDV (µm) E ~ 4% <x> 1/2 Σ B Σ R <x>: ~ 14 µm Σ B : ~ 5 µm Σ R : ~ 3 µm Assumption of uniform energy deposition is reasonable when E < 4%, as little straggling and blooming has occurred

21 For electrons with low energies, blooming and straggling become important even with little energy loss ( E) 4 Σ B STDV (µm) 2 6 E ~6% Σ R MeV Σ B 4 E ~4% Σ R 1 MeV E ~25% Σ B.1 MeV Σ R <x> 1/2

22 An effective Bragg peak results from the effects of blooming and straggling MeV g -1 cm 2 de/d(ρr) Σ R Σ R ρr (g/cm 2 ) Conventional Bragg peak results from the velocity match dσ dω Effective Bragg peak Conventional Bragg peak ~ 1 v e v th 4

23 The qualitative features of this model --- penetration, blooming and straggling --- are replicated by Monte Carlo calculations for solid DT ~2.6 µm ~2.6 µm > 3 µm 3 µm 1 MeV e 5 µm 1 MeV e 4.3 µm 5 µm 4.3 µm ~14 µm 14.6 µm This model Monte Carlo Calculated by Cliff Chen

24 Combine all these effects, electron energy deposition profile is modified 1 MeV e r b = 1 µm ε b = 1 kj ρ This model 4 2 Density (g/cm 3 ) Distance (µm) Uniform model Simultaneous coupling of penetration, blooming and straggling requires a rigorous numerical calculation

25 Outline Introduction Relevance to fast ignition Models Discussions

26 Penetration, blooming and straggling are insensitive to the plasma density and temperature g/cm 2.3 ρ<x> Σ B /<x> Σ R /<x>.3 Ratio ρ<x> (g/cm 2 ).4.2 ρ<x> 5 1 ρ g/cm T e (kev) This insensitivity indicates that density and temperature gradients will not impact these results

27 Penetration, blooming and straggling have a strong dependence upon the plasma Z Distance (µm) 2 1 R Z=1 Z=4 Z=13 Z=29 Assuming all elements are fully ionized and have same n e equal R --- the total path length 1.2 Σ B x.8 Z=29 Z=13 Z=4.4 Z=1 5 1 E (%)

28 Penetration drops with decreasing electron energy, while blooming and straggling increase Σ B x Σ B/<x >.4.2 Σ B x <x> Penetration (µm) Distance (µ m) 4 2 Σ B Σ R Electron Energy (MeV) Electron Energy (MeV)

29 Scattering will be important for setting the requirements of Fast Ignition (E ig, W ig and I ig ) Designed by R. Betti ~ 16 µm 6 15 D en sity (g /cm 3) 4 2 ρ ρr ~2.5 g/cm Temperature (ev) 3 MeV e ~ 15µm 1 µm Distance (µm) For NIF fast ignition capsules, 2-3 MeV electrons are required <x>: ~ 6 µm Σ R : ~ 8 µm Σ B : ~ 15 µm ( > r b =1 µm)

30 Summary Fundamental elements of this plasma stopping model For hydrogenic plasmas, binary e e and e i scattering are comparable Energy loss, penetration and scattering are inextricably coupled together Blooming and straggling effects, a consequence of scattering, lead to a non-uniform, extended region of energy deposition Whenever the Debye length is smaller than the gyro radius, binary interactions will dominate penetration, straggling and blooming effects This model is insensitive to the plasma screening, and applies to degenerate plasmas (e.g. for e preheat )