Choi, C K

Choi, M

Kothe, D B

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Particle Accelerators, 1992, Vols. 37-38, pp. 513-518Reprints available directly from the publisherPhotocopying permitted by license only© 1992 Gordon and Breach Science Publishers, S.A.Printed in the United KingdomNON-UNIFORM HEAVY-ION BEAMTARGET INTERACTIONSC. K. CHOI, D. B. KOTHEt and M. CHOIPurdue University, School of Nuclear Engineering, West Lafayette, IN 47907-1290,U.S.A.( Received 20 December 1990)The stability and symmetry of a direct-driven inertial confinement fusion (lCF) target irradiated by aheavy-ion beam is studied. in planar geometry. Non-uniformities consisting of target surface perturbations(stability) and beam intensity perturbations (symmetry), causing highly distorted flows in the targetabsorption layers, are considered. The two-dimensional, two-temperature numerical hydrodynamic modelFLIP (Fluid-Implicit-Particle), was developed to analyze the beam-driven target implosions. Perturbationamplitudes obtained with FLIP indicate that perturbation wavelengths close to the target shell thicknessresult in the largest implosion asymmetries.INTRODUCTIONThe main purpose of this study is to understand the hydrodynamic response of adirect driven planar heavy-ion ICF target which undergoes non-uniform heating byhigh-intensity ion beams. The physics included in the numerical model comprises 2-dcompressible hydrodynamics, 2-T energy transport for electrons and ions, averagecharge states (Saha relation), Sesame equation of state with equilibrium mix, and theion beam energy deposition with a unified slowing-down theory. Radiation transportis not included in the model. Neither are atomic-physics processes that are not inlocal thermodynamic equilibrium (LTE). Burn physics and convergence effects arealso neglected, as the hydrodynamic response to perturbations in target materialinterfaces and beam intensities is the primary emphasis of this study.A simplifying planar rather than spherical geometry is adopted which is in manycases justified for the early implosion stage. Convergent effects in spherical geometriesare likely to amplify any perturbations in planar geometry. The problem is attackedfrom ·both a stability and symmetry point of view. The stability point of view is tostudy the effects of a uniform deposition layer on an initially perturbed target (Figurela), while the symmetry point of view is to study the effects of a non-uniform beamdeposition layer on an initially smooth target (Figure 1b). The actual situation is, ofcourse, a combination of the two, but the effects are studied separately in hopes ofsingling out their contributions.t Present Address: Los Alamos National Laboratory, Theoretical Division, Los Alamos, NM 87545.513514Perturbed:brfac:eEnergyDepositionProfileE(x,z,t.)ION BEAM(A)C. K. CHOI, D. B. KOTHE AND M. CHOIDT(B)FIGURE 1 Planar model of an ion beam-driven ICF target: (a) uniform beam deposition on an initiallyperturbed target, and (b) non-uniform beam deposition on an initially smooth target. This planar targethas configurations relevant to the HIBALL target!.2 PHYSICS AND SIMULATIONS OF BEAM-DRIVEN TARGETIMPLOSIONThe hydrodynamics, energy transport, and equations of state for the target plasmasare described in Equations (1) through (7):dp-=-pV'V, (1)dtdVPat = -Vp, (2)Pe = Pe(P, Ge),Te = Te(p, Ge), ~ = ~(p, GJ,(5)(6)(7)where p and V are the density and velocity of the fluid, respectively. Since discontin-uous solutions are not allowed in the transport of momentum, an artificial viscosityis added to the pressure gradient term in Eq. (2) in regions of compression such asshocks in the flow. Eqs. (3) and (4), the specific internal energies for ions and electrons,include thermal diffusion (with diffusivities K i and Ke), collisions (with collisionNON-UNIFORM TARGET INTERACTIONS 515frequency Wei), pdV work (with ion and electron pressures Pi and Pe), viscousdissipation (with coefficients of a shear viscosity J.1 and bulk viscosity A, and adissipation function <1». There is also an arbitrary source (beam-plasma energyexchanges Si and Se),I(x) (dE(X))Si(e)(X) = E(x) a;;: i(ej'where I(x) is beam intensity, and E(x) and dE(x)/dx are ion beam energy and energyloss per unit path length, respectively. Separate energy transport for the ions andelectrons requires separate ion and electron equations of state (EOS) in order to closethe system of conservation equations. The SESAME EOS data library from the LosAlamos National Laboratory2 is used~ Energy loss, dE/dx, to free electrons andplasma ions is described by a unified theory that includes both friction and diffusionin velocity space3 . This unified formalism, which combines both the binary collisionand collective wave phenomena, is capable of handling an arbitrary stopping medium(electrons or ions) without introducing the Coulomb logarithm, and thus is valid forall interaction ranges.Conventional numerical approaches (Lagrangian or Eulerian) can not accuratelymodel the physics of distorted flows, and thus the Particle-in-Cell (PIC) model4 forfluid flow needs to be revisited. However, the original (or "classical") PIC has itsshortcomings; it is noisy, and it has higher numerical viscosity and heat conductionthan would be acceptable today 5. Many improvements of the "classical" PIC modelhave been made recently in an algorithm known as FLIp6, which eliminates thenumerical viscosity and thermal conduction by using a full Lagrangian representation'of the fluid (i.e., each particle is attributed all of the properties of the fluid includingmomentum and energy). The present fluid model for heavy-ion beam-plasma interac-tions, FLIP-PHD (PlasmaHydroDynamics)7,8 is based on the FLIP model for fluid flowwith the following characteristics: (1) Electric and magnetic fields are neglected; (2)plasmas are assumed to be collisional; (3) ions dominate momentum transport andelectrons dominate energy transport; (4) ion beam energy is partitioned differentlybetween plasma ions and electrons, forcing T e -# T i ; and (5) partial ionization andnon-ideal gas EOS are included.Both zero- and one-dimensional simulations have been performed by FLIP7with theproton beam on aluminum foil. The pressure profiles as function of time from thezero-dimensional model of Evans9 , and the simulations by FLIP were compared forboth 15 Ilm and 30 Ilm thick aluminum foils that were irradiated by a 16-TW/cm2 ,I-MeV proton beam with the average range of 2.8 mg/cm2 . 10,11 The model predictsreasonably well the initial rise in pressure as well as the approximate maximum,calculated by FLIP for 15-llm foil, but breaks down thereafter when a rarefactionpropagates back through the foil. When a 30 Ilm foil, twice as thick as the originalthickness, was simulated, a smaller pressure drop was found. This observationexplained the original discrepancy that was due to the uniform density assumptionmade by Evans, and suggested that the agreement between the models was good.The proton-foil interaction recently studied at the Naval Research Laboratory l0516 C. K. CHOI, D. B. KOTHE AND M. CHOIwas compared with the FLIP calculation for I-D planar implosion7. The FLIPcalculation used 50 zones with 16 particles per cell and incorporated physics discussedin this section. The grid boundary was allowed to move with a semi-Lagrangian inorder to follow the expansion of foil. Temperature, density, and energy depositionprofiles are compared and they represent the same functional behavior with somenumerical discrepancies, which were caused mainly by using different stopping powertheories.3 2-D STABILITY AND SYMMETRY ANALYSIS FOR NON-UNIFORMTARGETSThe "planar" HIBALL (Heavy Ion Beams And Lithium Lead)1 target configuration(i.e., single-shell, multilayered target with an infinite radius as in Figure 1) was chosenfor this assessment since it currently represents the prototype design of a target tobe imploded by direct driven heavy-ion beams in future ICF reactors. The planartarget consists of a tamper layer of lead (L\Pb = 140 ,urn), an absorber/pusher layer oflithium seeded with lead (L\LiPb = 500,um), and cryogenic liquid DT fuel layer(L\OT = 155 ,urn). Since convergence effects are not modeled here, the void on the innerside of the DT layer will be treated as an infinite vacuum in planar geometry. The10 GeV Bt ions depositing 4.37 MJ of energy in the target with 4 mg of cryogenicDT over a period of 30 ns are considered to simulate a HIBALL-like target. This designgrew from a design originally proposed by Bangerter and Meeker12, which to dateremains the generic single-shell design (tamper, absorber/pusher, and fuel layer) forion-driven ICF targets.Two cases of perturbed non-uniform beam-target interactions are considered; CaseI, the perturbation on target surface for stability studies, and Case II, the perturbedincoming ion beams for symmetry analyses. Case I corresponds to the implosionphase of a non-uniform planar target initially having a single-mode Pb/LiPb surfaceperturbation, with target thickness 795 ,urn, and dimensionless wave number kd = 20.The target is uniformly irradiated for 30 ns by a normally incident, 10 GeV Bt beamwith an intensity 10 equal to 1 TW/cm2. Case II is related to a uniform planar targetdriven for 30 ns by a normally incident, 10 GeV Bt perturbed beam having anintensity variation l(x, t) = 10(t) [1 + a cos(kx)], where x is defined in the lateraldirection of the target with 0 :::; x :::; 120 ,urn, 10 = 1 TW/cm2, a = 0.05, and kd= 20.Figure 2 shows the results of the time history of various perturbation amplitudesin Case I. The density perturbation amplitudes in Figure 2 deviate from a straightline, indicating a departure from linear growth (at an amplitude-to-wavelength ratioof about 0.2). This result is in qualitative agreement with the simulations oflaser-driven, ablatively accelerated thin shells by Verdon et al. 13. They found thatRayleigh-Taylor (R-T) bubble growth is the primary cause of shell rupture. A morerealistic spherical target having a Pb/LiPb surface perturbation similar to the oneimposed in this implosion, admittedly severe and larger than fabrication tolerances,would most likely fail to ignite.Computational results for Case II are also illustrated in Figure 3. It shows theNON-UNIFORM TARGET INTERACTIONS 51710=1012 W/cm2DODDE ODOoDODO 0 00:t lcf EBEBEBEBEBEBEBEBEBffiEBEBEBEBEB 0 0"--'"0 0(l)"d 0 0::s 0 0~.~0 0 6-6-r-l 6-6.6.6.6.6.6.6.6-6.p.. 6.6.S 06.6.6.6.0 6.< ld 6.6-~ OOOOO6.~~ 0 6. \6- Pb/LiPb interface 6.0 LiPb/DT interface 6- 6. phase0 Density change10° ffi De osition0 10 30 40 50 60time (ns)FIGURE 2 Computed average perturbation amplitudes (one-half of the peak-to-valley value) of variousquantities versus time for a uniform 1 TW/cm 2 irradiation on the planar target having an initially imposedPb/LiPb surface perturbation of amplitude 22.5 /lm and wavelength 240/lm.0 00 0 00 0 0 0 0 nit0 0 0 0 0 00 0 0 nlr0 nbd6. 6- 6- 6- 6-~ 6-6- 6-6- ~ 6. 6.~o 24 J.Lffio 240 p,rn~ 2400 p,m10 =1012 W/cm211=0.0510504530 35 40time (ns)2510° -+-__-~__,r--- - _ __.,..___r---__.__-...---_-_ __120FIGURE 3 Computed average perturbation amplitudes (one-half of the peak-to-valley value) of densityof a uniform planar target for three beam intensity perturbation wavelengths, each with a 5% amplitudeabout 1 TW/cm 2 .518 C. K. CHOI, D. B. KOTHE AND M. CHOIpattern of density amplitude growth with three distinct slopes visible through thepoints denoting growth rates nbd ("beam driven" for 22 to 30 ns), nt ("transitional"for 30-40 ns), and nif ("interfacial" for 40-50 ns). Density amplitudes for A = 2400,umare very small, almost at the level of computational noise. The short-wavelengthcalculation (A = 24,um) is terminated at 32 ns because of problems with the finitegrid instability, which rendered the results suspicious at later times. It is apparent inFigure 3 that the maximum density perturbation growth rate occurs at A = 240,urn.4 DISCUSSIONThe two-dimensional stability and symmetry analysis for non-uniform heavy-ionbeam-target interactions was investigated. At low intensities for Case I, surfaceperturbations at the Pb/LiPb interface appear to be more disastrous than similarperturbations at the LiPb/DT interface because the LiPb/DT interface is beyond thebeam ion range. Perturbations at the Pb/LiPb and LiPb/DT interfaces grow in atargets appear to be extremely sensitive to beam non-uniformities, with the targetsensitivity going up with the beam intensity. The beam intensity perturbations withwavelengths on the order of payload (i.e., DT plus fraction of LiPb absorber/pusher)shell thickness result in higher implosion asymmetries.REFERENCES1. B. Badger et al., HIBALL-A Conceptual Heavy Ion Beam Driven Fusion Reactor Study, University ofWisconsin report UWFDM-450 (1981); also, B. Badger et al. HIBALL-II-An Improved HeavyIon Beam Driven Fusion Reactor Study, UWFDM-625 (1984).2. K. S. Holian, ed., Los Alamos National Laboratory report LA-I0160-MS (1984).3. C. K. Choi 'and M. Hsiao, Nucl. Technol.! Fusion 3 273-279 (1983).4.F. H. Harlow and M. Evans, "A machine calculation method for hydrodynamics problems," LosAlamos National Laboratory report LAMS-1956 (1955); also, F. H. Harlow, J. Assoc. Comput. Mach.4 137-142 (1957).5. A. Nishiguchi and T. Yabe, J. Comput. Phys. 52 390 (1983).6. 1. U. Brackbill and H. M. Ruppel, J. Comput. Phys. 65 314-343 (1986).7. D. B. Kothe, Ph.D. thesis, Purdue University, 1987.8. D. B. Kothe, C. K. Choi, and 1. U. Brackbill, Laser Interaction (ed. by H. Hora and G. Miley) (Plenum,1988), Vol. 8, pp. 701-721; also, D. B. Kothe, J. U. Brackbill, and C. K. Choi, Phys. Fluids B 2 (1990)1898-1906. .9. R. G. Evans, Laser and Particle Beams 1231-239 (1983).10. J. E. Rogerson, R. W. Clark, and 1. Davis, Phys. Rev. A 31 3323-3331 (1985).11. B. P. Goel, G. A. Moses, and R. R. Peterson, Laser and Particle Beams 5 133-154 (1987).12. R. Bangerter and D. Meeker, "Ion beam inertial fusion target designs," Lawrence Livermore NationalLaboratory report, UCRL-78474 (1976).13. C. P. Verdon, R. L. McCrory, R. L. Morse, G. R. Baker, D. I. Meiron, and S. A. Orszag, Phys. Fluids25 1653-1674 (1982).

Non-uniform heavy-ions beam target interactions

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