PSFC/JA-8-3 ffects of energy oss on interaction dynamics of energetic ctrons with pasmas C. K. Li and R. D. Petrasso November 8 Pasma Science and Fusion Center Massachusetts Institute of Technoogy Cambridge, MA 39 USA The work described here was performed in part at the LL Nationa Laser User s Faciity (NLUF), and was supported in part by US DO (Grant No. D-FG3-3SF69), LLNL (subcontract Grant No. B597), and LL (subcontract Grant No. 6-G). Submitted to Physics Review
ffects of energy oss on interaction dynamics of energetic ctrons with pasmas C. K. Li and R. D. Petrasso Pasma Science and Fusion Center, Massachusetts Institute of Technoogy, Cambridge, Massachusetts 39, USA An anaytic mode is deveoped for energetic ctrons interacting with pasmas. This mode rigorousy treats the effects of energy oss upon the Couomb interactions and reveas severa new and important features never before unreaized, incuding the inextricabe couping of scattering and energy oss which previous cacuations erroneousy treated as independent. The unique transparency and generaity of these cacuations aows for straightforward appications in the cases of partia and even tota energy oss of energetic ctrons: for exampe, the quantitative evauation of energy deposition of the energetic ctrons in various pasmas, incuding inertia confinement fusion pasmas. PACS numbers: No. 5..Mj, 5.5.Gj, 5.5.Tx The interaction of energetic ctrons with pasmas is a fundamenta probem with important impications for both basic physics and practica appications [-6]. Such an interaction invoves ctron energy oss and scattering, eading to ctron energy deposition and trajectory bending in the pasmas. In the context of a singe ctron interacting with pasmas, for exampe, such scatterings stochasticay cause ctron spatia distributions, consequenty resuting in modifications of the detaied energy deposition structure [7-9]. In addressing ctron scattering in pasmas, the conventiona assumption has bn that energetic ctrons scatter off pasma ions whie osing their kinetic energy to the pasma ctrons. Because of the significant mass difference betwn ctrons and background ions, the energy oss to the ions has bn negected. In addition, the two physics processes (i.e. energy oss and scattering) have bn treated independenty and subsequenty combined in a simpe way. For exampe, the mean-square of the defection ange has bn cacuated simpy by averaging over the soid ange θ = N c dσ θ dω, () dσ dω where N c = is the number of the coisions (which is a function of the ctron energy oss and can be independenty evauated) []. The treatment of the scattering is excusivey manifested by the integra θ (dσ/dω)dω. It has bn demonstrated that this approach is justified and is accurate for energetic ctrons interacting with thin soid foi [] since an ctron suffers ony a reativey sma number of coisions (~ - 3 ), and the energy oss of each individua coision is very sma compared to its tota kinetic energy due to the nature of sma-ange dominant Couomb interactions. Because of this, the energy-dependence in the scattering cross sections can be essentiay overooked. The same is true for high Z pasmas because e-ion scattering so dominates ( Z, and Z > for any meta fois) over e-e scattering [7-9]. However, such a thin approximation is unjustified and inaccurate when it is appied in the case where (for exampe, during pasma heating) an ctron oses a significant amount or a of its energy and suffers a very arge number (over ~ 6 coisions), or when an ctron interacts with hydrogenic sittings (Z=, for which the e-e scattering coud be comparabe with the e-ion scattering). An exampe of this is eucidated by Fig. where e-ion (Rutherford) and e-e (Møer) scattering cross sections are potted as a function of the energy oss [Δ=( -) / ] for -MeV ctrons in hydrogenic sittings. When Δ changes from beginning to end ( % of the energy oss), these cross sections increase over order of magnitudes, indicating that the effects of energy oss on scattering can not be ignored, and that a rigorous approach to the inextricabe couping of the energy oss to scattering is necessary. r o - sin (θ/)(dσ/dω).+.+.- Møer Rutherford.- 6 8 Δ (%) FIG.. The normaized Rutherford cross section (e-ion scattering) and Møer cross section (e-e scattering) are potted as a function of the fraction of the energy oss for -MeV ctrons. Both cross sections show the significant increase in scattering as an ctron oses energy. - -
In this paper, we expain the importance of the effects of energy oss upon scattering in the interaction regime described above based on a unified approach derived from fundamenta principes [7-9]. This mode naturay inks the inextricabe couping of scattering and energy oss, and wi revea severa of its new and important effects. In accordance with our approach [7-9], an integrodifferentia diffusion equation is soved to rigorousy determine the anguar and spatia distributions of the scattered ctrons: where f(x, v, s) is the ctron distribution function; n i the number density of fuy ionized, uniform time invariant background pasma ions of charge Z, x the position where scattering occurs; σ = σ ei +Zσ the tota scattering cross section with σ ei the Rutherford e-ion cross section [], and σ the Møer e-e cross section [3]. The equation is soved in cyindrica coordinates with the assumption that the scattering is azimuthay symmetric. Specificay, the anguar distribution is [7-9] ' ' d f ( θ, ) = ( + ) P (cosθ )exp κ d', π = (3) where P (cosθ ) is the Legendre poynomia. In this soution, the energy oss is manifested by the pasma stopping power [,5] d - π mc niz λd γ γ = nr + + ds β λc 8 γ γ.3 β n + n γ kte /mc, () where β = v/c and γ = (-β ) -/, r = e /m c is the cassica ctron radius, λ C =h /m c is ctron Compton wavngth, and λ D = (kt/πn e e) / is Debye ength. Note that q. () is vaid when β >>α (=/37), however, its cassica counterpart woud be accurate enough when β < α, such as in the case of ow-energy ctron preheating inertia confinement fusion (ICF) targets. Whie the effects of scattering are characterized by the macro transport cross sections κ dσ ( ) = ni [ P (cos θ )] d Ω. (5) The dominant terms are = r ( + ) ei γ κ = ( ) πn i Z n Λ + Z n Λ, (6) γβ ( γ + ) / - - which is reated to the sowing-down cross section and characterizes the oss of directed veocity (momentum) in the scattering []; and = r + ei γ, κ( ) = πn i Z nλ + Z nλ γβ ( γ + ) / (7) which is reated to the defection cross section and represents the mean-square increment in the transverse ctron veocity during the scattering process []. It f s + v f = n shoud be noted that such simpe anaytic versions of i f ( x, v,s) f ( x,v,s) σ( v v )d v, () () transport coefficients [qs. (6) and (7)] are ony vaid for γ [7-9], because in order to have a sma angedominant, Rutherford-ike Møer cross section, severa approximations have bn made. quation (8) gives the ratio of such a simpified Møer cross section [7-9] to Rutherford cross section ei ( γ + ) dσ dσ R( γ ) = Z d d. (8) Ω Ω ( γ + / ( ) ) Z This ratio is potted in Fig. for hydrogenic pasmas (Z=), (dσ/dω) is sighty arger (~%) (dσ/dω) ei for γ < 7 (consistent with Fig. ), whie significanty smaer for γ >. Figure aso shows that for a non-reativistic case (γ = ), one has (dσ/dω) (dσ/dω) ei []. This ceary indicates that directy appying a non-reativistic resut to the cases of reativistic ctron-pasmas interactions, such as fast-ignition ICF [7], resuts in significant inaccuracy. The inextricabe mutua coupings betwn energy oss and scatterings are expicity refected by the integrands [q. (3)] ' d' κ ( ) d'. (9) The integration is a function of ctron residua energy (). Because there is no restriction on ctron energy oss, q. (8) is vaid in the case of an arbitrary amount of even tota energy oss. As shown in Fig. 3, the anguar distribution converges rapidy to arge anges, and its R( γ ).5..5. Z= 5 5 γ FIG.. The ratio of e-e scattering cross section to e-ion scattering cross section is potted as a function of the γ.
Normaized probabiity.3. Δ 5%. Δ 9% 5 5 integration, the first-order approximation in terms of S κ ( s') ds' for an exponentia function resuts in d' ' κ ( s ') d ( + ) θ Av, () The θ is now ready to evauate based on the dominant contributions from = and = Ange (Degr) FIG. 3. The normaized anguar distribution is potted for two cases: energy oss Δ ~5% and Δ ~9%. shape is strongy dependent on the energy oss. Specificay, when energy oss is ~ 9%, for exampe, the resuting anguar distribution is characterized by a distribution with sma-ange mutipe scatterings pus arge-ange singe scatterings on the tai. In contrast, however, for energy oss 5%, the distribution is dominated by the sma-ange scatterings. We wi s how the thin approximation has the resut of decouping the effects of energy oss and scattering, as discussed in q. (9) ' d ' d ' κ ( ) d ' κ ( ) d ' = κ S κ t. () Where t is the thickness of the pasma and when it is thin, we find that S ' ' '. t S( ) = ds = ( d ds) d (The inkage of energy oss to scattering is impied by the reationship betwn the distance an ctron transverses and energy oss, since the father an ctron transverses, the more energy it oses and the more scatterings it suffers.) The approximation in q. () makes sense when Δ is very sma such that dσ/dω in q. (5) can be treated as independent of the energy, which in turn resuts in an energy-independent scattering parameter κ which factors out the integration in q. (9), indicating that scattering and energy oss have bn treated separatey. This approximation, as discussed above and shown by Fig., is of course unjustified in the case of tota or even significant energy oss of energetic ctrons in the pasmas which this paper is focused on. To further iustrate the effects of energy oss on scattering, we cacuate the mean-square defection ange <θ > from q. (9). For the sake of simpicity, a smaange scattering Fokker-Panck approximation is used by expanding the Legendre poynomia to the power of θ and kping ony the first two terms [6], i.e. P cos θ. 5( + ) θ. Using q. (3) and conducting the where and θ θ + θ = = θ. () ' d ' = = κ ( ) d '. (3) ' d ' θ = = κ ( ) 3 d ' () Figure compares the θ cacuated from q. () and q. (). As shown, a significant difference occurs when the ctrons have ost more energy. Another important resut from this unified mode is that the phenomenoogica ad hoc cutoffs (required to prevent mathematica divergence due to two-body Couomb interactions) has bn effectivey removed because of the incusion of energy oss in the ctron scatterings. In practica appications the choosing of a suitabe mode for pasma scrning and performing this phenomenoogica cutoff is a non-trivia undertaking. The ad hoc cutoffs directy refect the approximations made in the theoretica formuation. Depending on the different pasma densities and temperatures, for exampe, b max is usuay determined by either Debye ength, or Thomas- Fermi scrning ength (λ TF =.885a / Z /3 ) or mean interpartice distance (λ = λ Int = n -/3 ). The Debye ength from an exponentia scrned Couomb potentia [], <θ > thin approx. Unified approach 6 8 Δ (%) FIG.. Mean-square defection ange θ cacuated from the unified approach which has taken into account the effect of energy oss on ctron scattering (soid ine), and compared with the conventiona thin approximation (dashed ine). - 3 -
φ( r) r / λ D = φ e, (5) describes the shieding distance at which the potentia fas to its e-foding from its maximum. The Thomas- Fermi scrning ength, (a resuted derived originay from nucear scrning, with corrections for the effects of pasma temperature and density) is a reasonabe approximation for idea gas. Its accuracy requires that each Debye sphere has one singe ion (for reference, in reation to the typica pasma discussed here, ρ =3g/cm 3 and T e = 5 kev, with one Debye sphere having about 7 ions). Aso, the mean inter-partice distance is an approximation for dense pasmas when the Debye ength is even smaer than the mean inter-partice distance. However, such a mode constraint is argey reaxed due to the effective canceation embedded in q. (8). For exampe, the ctron defection is a function of product of energy oss (d/ds) with scatterings (κ ), ( γ + ) ei n Λ + n Λ ( γ + ) / d ( ) κ. (6) n Λ The effective canceation of the Couomb ogarithms in the numerator and denominator of q. (5) significanty reduces the sensitivity of the sction of pasma scrning modes. The physics behind such a canceation can be understood as the defection occurring simutaneousy with the sowing down and scattering-off in the encountering pasma mediums. The resut is iustrated in Fig. 5, where the normaized transport cross sections [κ ()(πn i ) - (r /γβ ) - ] are potted as a function of the energy oss [Fig. 5(a)], and differences exist for different modes. As shown in Fig. 5(b) where κ ()(d/ds) - is potted as a function of energy oss, negigibe differences wi sti render the effects of different modes on the choice of b max insignificant. In summary, we have used an anaytica mode to deineate the effects of energy oss on the interactions of energetic ctrons with pasmas. Our mode rigorousy examines the effects of energy oss upon the Couomb interactions and reveas severa new and important aspects never before reaized, incuding the inextricabe couping of scattering and energy oss which previous cacuations erroneousy treated as independent of each other. The unique transparency and generaity of these cacuations aows for straightforward appications in the cases of partia to even tota energy oss of energetic ctrons: for exampe, the quantitative evauation of the energy deposition of energetic ctrons in various pasmas, incuding inertia confinement fusion pasmas. This work was supported in part by U.S. Department of nergy Contract #D-FG3-99SF78, LL subcontract #PO5G, LLNL subcontract #B33975, and the Fusion Science Center at the University of Rochester. κ()(πn i ) - (r /γβ ) κ()(d/ds) - 8 - - -3 Debye interpartice TF Debye interpartice TF (a) (b) 6 8 Δ (%) FIG. 5. Using different scrning modes (Debye, Thomas-Fermi, and inter-partice distance), the normaized κ are potted as a function of the ctron energy in DT pasma (ρ = 3g/cm 3 and T e = 5 kev) (a). As is shown, the difference indicates that the importance of propery choosing the scrning parameters if the eastic scatterings are treated independenty. However, as is sn in (b), these differences are dramaticay reduced when we take the approach that energy oss and scattering couped. [] G. Moiere, Z. Naturforsch. 3a, 78 (98). [] H. A. Bethe, Phys. Rev. 89, 56 (953). [3] L. Spitzer, Physics of Fuy Ionized Gases (Interscience, New York, 96). [] B. Trubnikov, Review of Pasma Physics (consutants Bureau, New York, 965); [5] C. K. Li and R. D. Petrasso, Phys. Rev. Lett. 7, 359 (993). [6] C. K. Li and R. D. Petrasso, Phys. Rev. Lett. 7, 363 (993). [7] C. K. Li and R. D. Petrasso, Phys. Rev.. 7, 67 (). [8] C. K. Li and R. D. Petrasso, Phys. Rev.. 73, 6 (6). [9] C. K. Li and R. D. Petrasso, Phys. Pasmas 3, 563 (6). [] J. D. Jackson, Cassica ectrodynamics (Wiey, New York, 975). [] P. C. Hemmer et a., Phys. Rev. 68, 9 (968). [] R. D. vans, The Atomic Nuceus (McGraw-Hi, New York, 955). [3] C. Møer, Ann. Physik (Leipz), 53 (93). [] A. A. Soodov and R. Betti Phys. Pasmas 5 77 (8). [5] For the energy oss obtained in Ref. [7-9] we have used the cassica version of the maximum impact parameter for ad-hoc cutoff, which may have resuted in sighty overestimated inear energy oss. In the present paper we have corrected this inaccuracy (s q. ) by incuding quantum effects (simpy repacing the Debye ength λ D by the debroige wavngth λ deb within the ogarithmic term). [6] H. W. Lewis, Phys. Rev. 78, 56 (95). [7] M. Tabak et a., Phys. Pasmas, 66 (99). - -