Kinetic simulations of fusion ignition with hot-spot ablator mix
James D. Sadler, Yingchao Lu, Benjamin Spiers, Marko W. Mayr, Alex Savin, Robin H. W. Wang, Ramy Aboushelbaya, Kevin Glize, Robert Bingham, Hui Li, Kirk A. Flippo, Peter A. Norreys
aa r X i v : . [ phy s i c s . p l a s m - ph ] A ug Kinetic simulations of fusion ignition with hot-spot ablator mix
James D. Sadler,
1, 2
Yingchao Lu, Benjamin Spiers, Marko W. Mayr, Alex Savin, Robin H. W. Wang, Ramy Aboushelbaya, Kevin Glize, Robert Bingham,
3, 4
Hui Li, Kirk A. Flippo, and Peter A. Norreys
2, 3 Los Alamos National Laboratory, P.O. Box 1663, Los Alamos, New Mexico 87545, USA Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK Central Laser Facility, STFC Rutherford Appleton Laboratory, Didcot, OX11 0QX, UK Department of Physics, University of Strathclyde, 107 Rottenrow East, Glasgow, G4 0NG, UK
Inertial confinement fusion fuel suffers increased X-ray radiation losses when carbon from thecapsule ablator mixes into the hot-spot. Here we present one and two-dimensional ion Vlasov-Fokker-Planck simulations that resolve hot-spot self heating in the presence of a localized spike ofcarbon mix, totalling 1 . INTRODUCTION
Fusion reactions between light elements may yield aclean source of nuclear energy, with few long-lived wasteproducts. To achieve sufficient energy gain with a con-tainable yield, the fuel must be compressed to muchgreater than solid density. This has been achieved usinglaser ablative compression, for example at the national ig-nition facility [1–3]. Achievement of ignition would havefar reaching applications in nuclear physics and experi-mental stockpile stewardship.In the typical scheme, ignition occurs in a hot cen-tral region of the compressed fuel, known as the hot-spot. The burn wave then propagates into the colderand denser surrounding fuel. Sufficient fusion reactiv-ity requires hydrogen temperature
T > . ρr > . − , fast alpha parti-cles are mostly confined to the hot-spot and conductionlosses are tolerable [4].Previous kinetic numerical studies [5, 6] found thatparticle collisions keep the fuel ion distributions closeto Maxwellian, despite depletion of the faster ions dueto fusion. The Maxwellian averaged fusion reactivity isaccurate to better than 1%. However, a more seriousreactivity reduction occurs due to dense fuel boundaries[7–11]. This is because the fast ions have a long meanfree path and may be absorbed by the boundary layer, re-ducing yield [8]. Kinetic effects during the implosion canalso lead to deuterium tritium species separation [12, 13].Another significant consideration is the incursion ofthe surrounding carbon ablator material into the hot fu-sion core [14, 15], introducing further boundaries withinthe hot-spot. Bremsstrahlung radiation increases rapidlywith ion atomic number, lowering the temperature andfusion yield in carbon mix regions.In this work, we adapt the Fokker-Planck method ofreference [16] to simulate the evolution of the hot-spotwith localized regions of ablator mix. Increased electron bremsstrahlung radiation causes rapid cooling and leadsto steep temperature gradients and non-local effects. Themix region Knudsen layer reduces the fusion reactivity.We find that radiative contraction of the mix region in-creases its density and therefore its radiative losses. Act-ing as a barrier to alpha particle transport, it reducesthe effective hot-spot areal density. These effects makea localized mix spike more detrimental than uniformlydistributed mix. The contraction also contributes to thebroadening of the fusion neutron spectrum, leading tooverestimates in the inferred ion temperature.Experimental presence of ablator mix was confirmedwith a corresponding increase of X-ray yield on lower fu-sion yield shots [17–19]. Ablator mix may occur due todiffusion near the time of stagnation [20, 21] or target en-gineering defects exacerbated by the non-linear Rayleigh-Taylor instability [22, 23]. These asymmetries can leadto jets of carbon or tungsten dopant through the hot-spot. Localized spikes of mix have been confirmed withX-ray self emission images [24], showing increased emis-sion from the tent and fill tube positions.Three-dimensional radiation-hydrodynamic simula-tions also produced jets of ablator mix [23, 25]. Radiativecooling produces steep temperature gradients around themix region. The estimated hydrogen ion mean free pathwas λ mfp = 50 nm and the temperature gradient scale-length was 5 µ m, giving a Knudsen number λ mfp |∇ T | /T of around 0 .
01, requiring kinetic corrections to heat flowand fusion reactivity [8, 26, 27]. The development ofthe fast alpha particle spectrum may also be influencedby the mix region and its rapid evolution. As a result,although a fluid simulation can predict the existence ofthese mix regions, they incur additional hot-spot physicsthat can only be treated with a kinetic model.At the ignition threshold with T = 5 keV and den-sity ρ = 100 gcm − , the Coulomb logarithm ln(Λ) =0 . λ /b ) ≃
3, where b min is the minimum im-pact parameter and λ D is the Debye length [28]. Parti-cle collisions will affect the plasma evolution and can bemodelled using the Fokker-Planck operator [29]. Typicalcollision timescales are sub-picosecond, much less thanthe fuel stagnation timescale of 100 ps, so the electronand hydrogen ion distributions are expected to be nearMaxwellian. This permits a simplified kinetic model al-lowing small deviations from Maxwellian distributions,allowing assessment of the kinetic corrections to fusionreactivity.In the first section of this paper, we describe the trun-cated ion kinetic model and implementation. The secondsection analyses a one-dimensional simulation and the ad-ditional insights from the kinetic model. The third sec-tion discusses the kinetic effect that reduces yield aroundthe mix region. The fourth section introduces a two-dimensional simulation and the fifth section analyses itto show how the mix region broadens the neutron spec-trum. The final section summarises the results. TRUNCATED KINETIC MODEL
The hot-spot is modelled in one or two Cartesian spa-tial dimensions, with periodic boundaries. The physicalmodel follows that of Keskinen [30], in which the dis-tribution functions f a (where a labels the species) aredecomposed in spherical coordinates in velocity spacewith a Cartesian tensor expansion. Since the collisionswill cause f a to tend towards isotropy, the higher orderterms will quickly decay and the expansion can be ac-curately truncated. The truncation is valid so long asthe mean fluid velocity is less than the thermal velocity,which should be true after efficient conversion of the im-plosion energy to hot-spot internal energy at stagnation.This diffusive approximation yields f a ( t, x , v ) = f a ( t, x , v ) + ˆv . f a ( t, x , v ) , (1)where v = | v | , ˆv = v /v and v is the particle veloc-ity. This decomposition decreases the dimensionality ofthe f a while retaining arbitrary anisotropic deviation inany Cartesian direction. The quantity f a is the isotropicpart of the distribution function and f a is the anisotropicpart. Although the angular resolution of the anisotropyis reduced by this approximation, it still allows emergentkinetic phenomena such as Landau damping and flux lim-ited heat flow.The moments of the distribution functions give thenumber density n a , fluid velocity u a , energy density U a and temperature T a , for ion species a of mass m a andcharge q a , via the integrals n a = 4 π Z ∞ f a v dv, (2) u a = 4 π n a Z ∞ f a v dv, (3) U a = 2 πm a Z ∞ f a v dv, (4) U a = 12 n a m a | u a | + 32 n a k B T a . (5) The diffusive approximation is valid so long as thecontribution of the first term in equation (5) is smallcompared to the second term. This rules out mod-elling of the implosion (as previously achieved with one-dimensional ion-kinetic codes [31]) but does allow effi-cient two-dimensional modelling of the stagnated hot-spot. As such, simulations will be initialized in the timeof peak fuel compression. This approximation also breaksdown during the later stages of ignition and burn wavepropagation, so the simulations will be curtailed once theapproximation (1) is no longer valid.Inserting equation (1) into the non-relativistic Vlasov-Fokker-Planck equation and averaging over angles leadsto the kinetic equations for each ion species [32, 33], ∂f a ∂t + v ∇ . f a + q a m a v ∂∂v ( v E . f a ) = C a + F a , (6) ∂ f a ∂t + v ∇ f a + q a m a E ∂f a ∂v = C a . (7)In addition to these ion equations, the electrons aremodelled as a fluid, with density and velocity found byassuming quasi-neutrality and electrostatic approxima-tions, neglecting magnetic effects. This fluid approachis valid since the electron collision time is approximately1 fs, much less than self-heating timescales. However,the ion collision timescales are not negligible, requir-ing the kinetic treatment. The quasi-neutral assumptiongives the electric field from the electron pressure gradient E = −∇ p e / ( n e e ), where p e = n e k B T e and is found fromequation (5). The electron energy density is evolved us-ing a diffusion equation with Spitzer conductivity κ S anda flux limiter at 5% of the free streaming value, ∂U e ∂t + ∇ . [( p e + U e ) u e − κ S ∇ T e ] = 0 . (8)The collision term C a was given by the Rosenbluth-Fokker-Planck form [29, 33], where the collision operatorfor each ion species a is summed over collisions with allspecies b , C a ( x , v ) = X b Γ ab v ∂∂v (cid:18) m a m b I b + ( I b + J b − ) v ∂∂v (cid:19) f a , (9) I bp ( x , v ) = 4 πv p Z v f b ( x , w ) w p dw, (10) J bp ( x , v ) = 4 πv p Z ∞ v f b ( x , w ) w p dw, (11)Γ ab = q a q b ln(Λ)4 πǫ m a . (12)The collision operator was implemented with the im-plicit, mass conserving method of Chang and Cooper [34],allowing a time-step similar to the ion collision time. The I and J integrals for electrons were performed analyti-cally by assuming a Maxwellian distribution, whereas theintegrals were performed numerically for each ion species.Since the energy gained by ions must be lost by electrons,energy was conserved by updating U e accordingly.The collision operator C a is responsible for spreadingof the particle direction. When the diffusive approxima-tion (1) is substituted in to the Fokker-Planck form givenby equation (2) in ref. [8], it yields the velocity dependentKrook form [30, 33], C a = − X b Γ ab n b v ( f a − f M a ) , (13) f M a = − u DT ∂f a ∂v . (14)Although this is an approximation to the full Fokker-Planck form, it is most accurate for the supra-thermalions responsible for the fusion reactivity reduction [8],a key kinetic effect. There is also a correction to thecollision operator so that it more closely conserves mo-mentum, tending towards f M a , the anisotropic part of aMaxwellian with fluid velocity equal to the deuterium-tritium fluid velocity.The fusion term F a in equation (6) used theMaxwellian fusion reactivity formula [35], giving the vol-umetric reaction rate r as a function of temperature.There may be kinetic corrections to this formula, so themore accurate reactivity will be post-processed by inte-grating the full hydrogen distribution function from thesimulation. The hydrogen ion density was reduced andthe alpha particle distribution was increased using thelocal r at each time-step. Alpha particles were initializedwith energy of 3 . Z a . This means radiative losses will begreater within regions of carbon mix. The electron volu-metric emission rate was calculated in reference [4], as afunction of electron temperature in eV and the ion num-ber densities in cm − , W ( T e ) = 1 . × − n e p T e X a n a Z a Wcm − . (15)The sum is taken over all ion species, which are assumedfully ionized. Since equation (15) is only valid for weaklycoupled plasma and not the dense fuel shell, the modelimposes a smooth exponential cut-off of W for regions ofplasma with temperature below 1 . ONE-DIMENSIONAL SIMULATION
Conditions were chosen close to the ignition threshold,with uniform initial pressure and a hot-spot with peaktemperature 4 . − and arealdensity ρr = 0 .
35 gcm − . It was surrounded by a densefuel shell with density five times that of the hot-spot. Theone-dimensional Cartesian simulation domain had width120 µ m, 288 cells and periodic boundaries. The velocitygrid consisted of 3000 points extending to 1 . × ms − .The deuterium and tritium ions were modelled as a singlespecies with mass equal to 2 . t = 0 ,
20 and 40 ps, for two separatesimulations. The first simulation initialized the carbonions in a localized Gaussian region of waist 5 µ m and peaknumber density 8 × m − , constituting 1 . n .Once it cools and contracts, the mix region acts asan effective heat sink and barrier for the alpha particles,partially separating the two sides of the hot-spot and re-ducing the effective hot-spot areal density to below thethreshold Lawson value. Although radiative losses areincreased in the uniform case, the mix does not act asa barrier and there is no reduction of the hot-spot arealdensity. The uniform carbon mix acts purely to reducethe self heating rate across the hot-spot, whereas the lo-calized mix region acts as a partial barrier and heat sinkfor alpha particles.Fig. 2 shows the f distribution function of the alphaparticles across the hot-spot at two separate times, for n ( m − ) ×10 (ai) localizeduniform −50 0 50x (µm)0246 T ( k e V ) (aii) (bi)−50 0 50x (µm) (bii) (ci)−50 0 50x (µm) (cii) FIG. 1. Results of two one-dimensional kinetic simulations, the first with carbon ablator mix in a Gaussian profile of waist5 µ m in the centre of the simulation domain and the second with the same carbon mass spread uniformly across the hot-spot.The (i) panels in the top row show the number density for the combined deuterium-tritium ion species at three separate times(a) 0 ps, (b) 20 ps and (c) 40 ps. The bottom row (ii) panels show the deuterium-tritium temperature. Data for other speciesare not shown. The carbon ions, assumed fully ionized, constituted 1 . × m − in the localized case and 10 m − in the uniform case. the localized case. The distributions have been multi-plied by the spherical velocity coordinates Jacobian v .The fusion alpha particles are created at a fixed speed,then slow down via collisions. The fuel temperature, andtherefore the fusion reactivity, have decreased at the latertime-step, so the supra-thermal phase space density islower in Fig. 2b than Fig. 2a. The high phase spacedensity for v < ms − is a result of the alpha parti-cles that have slowed to form a thermal distribution with T ≃ T e . The alpha particle stopping time is several pi-coseconds, so the thermal particles form the majority ofthe alpha particle population at the time-steps shown.It is clear from Fig. 2 that the central mix region ismore effective at slowing the alpha particles, as thereare more alpha particles at lower velocity within the mixregion than outside of it. Since the radiative power scalesas n , the alpha energy is more effectively radiated oncethe mix region becomes denser. This means the effectsof the mix region become worse over time, an effect thatis not observed for uniformly distributed mix.Fig. 2c shows the alpha particle number density for thetwo separate time-steps. There is a surplus of alpha par-ticles within the dense mix region, even though the fusionreaction rate is lower here. This shows that a large frac-tion of alpha particles have travelled from other parts ofthe hot-spot into the mix region. They then slowed downand deposited their energy here, where it is radiated awayand wasted. By efficiently slowing the alpha particles, themix region acts in a similar way to the dense plasma shell.This effect becomes more pronounced as the mix regionbecomes denser at later times. These results show thatlocalized carbon mix is more damaging and requires agreater ignition temperature than uniformly distributedcarbon. KINETIC REDUCTION OF FUSIONREACTIVITY
Temperature gradients at the edge of the hot-spot areknown to cause reductions to fusion reactivity through apurely kinetic effect [8]. At the temperatures simulated,supra-thermal fuel ions are more likely to fuse, but alsohave a longer collisional mean free path, increasing thelikelihood of their absorption by a nearby dense boundarylayer. This can deplete the tail of the distribution andreduce fusion reactivity, even several thermal mean freepaths from the boundary.The simulations in the present work show that thissame mechanism also affects the edge of the mix region.Fig. 3 shows the ratio of the simulated fusion rate withthat of an equivalent Maxwellian distribution with equalfluid moments, calculated at the 40 ps time-step shownin Fig. 1. The volumetric fusion reaction rate was calcu-lated by numerically integrating the fusion cross sectionacross the distribution function of the fuel ions in threevelocity space dimensions, r ( x ) = Z d v ′ d v f D ( x , v ′ ) f T ( x , v ) σ ( v r ) v r , (16)where v r = | v − v ′ | is the relative speed of the two inter-acting particles and σ is the fusion cross section [35].The fusion rate is reduced in Knudsen layers at theedge of the hot-spot and near the mix region. The de-pletion of the tail of the distribution is strongest at theposition of steep temperature gradients. The reductionis more severe at the edge of the hot-spot than aroundthe mix region.The region of fusion reactivity reduction is wider thanthe carbon mix profile, shown by the dashed line, show- v ( m s − ) ×10 (a)0.00.51.01.5 v ( m s − ) ×10 (b) 036 v f ( m − s ) ×10 −50 0 50x (µm)0123 n ( m − ) ×10 (c) FIG. 2. Distribution function v f ( x, v ) of the supra-thermalalpha particles in the localized mix case, shown at times (a)6 ps and (b) 20 ps. Fusion alpha particles are created withspeed 1 . × ms − . The thermalized alpha particles in thered stripe (exceeding the color scale) at the bottom of eachplot have a thermal velocity v th ≃ p k B T e /m α = 3 × ms − and constitute the bulk of the alpha particle number density.The central mix region is more effective at slowing the alphaparticles. Equation (2) shows that the integral of these dis-tribution functions along the v axis gives the alpha particlenumber density, shown in panel (c) for both time-steps. −60 −40 −20 0 20 40 60 x (µm) r / r M FIG. 3. Fusion reactivity of the deuterium-tritium plasmafrom the one-dimensional simulation at 40 ps, calculated froma numerical integral of equation (16) using the simulated dis-tribution function with localized carbon mix. The reactivityis normalized to the reactivity of a Maxwellian distributionwith identical fluid moments. The normalized profile of thecarbon mix is also overlaid with the dashed line. ing that the carbon has an extended effect beyond itsboundaries. This is due to the long mean free path of thefaster fuel ions. Within the mix region itself, the plasmais denser and more collisional and so the distribution iscloser to Maxwellian.
TWO-DIMENSIONAL SIMULATION
Localized mix regions may result from the Rayleigh-Taylor instability, resulting in an extended jet shape intwo dimensions. The simulation was repeated in twoCartesian dimensions using equivalent parameters and acircular shaped hot-spot with radius 35 µ m. The mix re-gion was initialized using a Gaussian jet with waist 5 µ min the azimuthal direction and waist 20 µ m in the ra-dial direction. The centre of the Gaussian was offset inthe y direction by 25 µ m from the centre of the simula-tion domain. The peak carbon ion number density was8 × m − .Fig. 4a shows the deuterium-tritium temperature pro-file after 40 ps. The temperature is reduced in the mixregion by radiative losses. This alters the shape of thehot-spot to become more elongated.Fig. 4b shows the ratio of the deuterium-tritiumkinetic fluid energy to the total energy, given by mnu / (2 U ). This is a measure of the fluid flows in thehot-spot, and also indicates the applicability of the ap-proximation given by equation (1). Due to the radiativecooling around the edge of the hot-spot, there is a pres-sure gradient which causes fluid flow. There is also a flowof similar magnitude towards the mix region, which hassteep pressure gradients due to the radiative cooling fromthe increased carbon concentration.Due to its lower temperature, the fusion reaction rate r is reduced in the mix region. Furthermore, inflow of thefuel to the mix region reduces the fuel density in regionsaround the mix spike. This means the reduction of fusionrate extends beyond the boundaries of the mix region.Fig. 4c shows the fusion reaction rate r , as calculatedusing equation (16). Due to the decreased density andthe kinetic effect, the shape of the hot-spot, as measuredby the fusion burn profile, is more flattened than theshape measured by the temperature profile.There may be experimental signatures from the local-ized carbon region, since the fusion neutron hot-spot im-age [similar to Fig. 4c] is a considerably different shapeto the hot-spot X-ray emission and temperature profile[Fig. 4a]. This may allow increased diagnostic confidencewhen differentiating between uniform or localized mix.Fig. 4d shows the ratio of the fusion reactivity to thatof an equivalent Maxwellian fuel distribution with equalfluid moments. The Knudsen layers reduce the reactionrate around the regions of high temperature gradients.The reduction is maximal around the edge of the hot-spot, reaching up to 20 %. A similar reduction is alsopresent at the edge of the mix region. The reduction isalso more severe at the transverse edges of the jet, ratherthan at its tip.The reactivity reduction of approximately 10 % willalso be expected in regions between the multiple mixjets predicted by hydrodynamic simulations [23], a vol-ume comprising a large fraction of the hot-spot. The10 % value obtained in these simulations is in approxi-mate agreement with theoretical estimates [8]. −20 0 20x (µm)−20020 y ( µ m ) (a)0.0 2.5 5.0T DT (keV) −20 0 20x (µm) (b)0.00 0.05 0.10E K /U −20 0 20x (µm) (c)0.0 0.5 1.0r (10 m −3 s −1 ) −20 0 20x (µm) (d)0.8 0.9 1.0r/r M FIG. 4. Results of the two-dimensional kinetic simulation after 40 ps, with carbon ablator mix localized in a Gaussian spike ofwaist 5 µ m in the x direction, waist 20 µ m in the y direction and centred at y = − µ m. The peak carbon number density was8 × m − and all other parameters are equivalent to Fig. 1. (a) The temperature of the deuterium tritium species, showingthe radiative cooling in the mix region. (b) The ratio of the deuterium-tritium fluid kinetic energy to the total energy, given by mn | u | / (2 U ). (c) The volumetric fusion reaction rate. (d) The ratio of the fusion reaction rate to that of a Maxwellian withequal density, temperature and fluid velocity. NEUTRON SPECTRUM
The contraction of the mix region may help to explainwhy the measured experimental ion temperature (fromthe width of the neutron spectrum) is anomalously highcompared to that expected from fluid simulations and themeasured fusion yield [3, 36]. This could be due to a sig-nificant fraction of the hot-spot energy residing in fluidkinetic energy rather than internal energy [37–39]. Thereis an intrinsic neutron spectrum width due to ion ther-mal motion, but fluid flows will also cause broadening.There is residual fluid motion due to incomplete shockstagnation. On top of this, the simulations here showthat localized regions of ablator material in the hot-spotmay radiatively cool, causing rapid contraction and ad-ditional fluid flows. Furthermore, the mix spike is asym-metric, leading to line of sight variations in the measuredneutron spectrum.To investigate this effect, we calculated the simulatedDT fusion neutron spectra from the simulation shown inFig. 4, at the t = 40 ps time-step. There is a large inflowtowards the cooler mix region. The neutron spectrumhas a thermal broadening around the central energy E ≃
14 MeV, given by [40] f n ∝ Z d x r exp (cid:18) − ( E − E ) k B T E m n + m α m n (cid:19) , (17) E ( x , v ) = 12 m n ( v − u DT . ˆr ) , (18)where ˆr is a unit vector along the line of sight, r is thereaction rate defined in equation (16), u DT is the fuelfluid velocity and m n is the neutron mass. The neutronspectrum around v ≃ . × ms − is broadened by theion thermal motion and shifted by the local ion fluid ve- E (MeV) f n ( E ) ( a r b . ) Controlxy
FIG. 5. Synthetic normalized neutron spectra generated fromthe 40 ps time-step of the two-dimensional simulation shownin Fig. 4. The solid line shows the case where the broadeningfrom fluid motion is neglected, the dashed line shows the cal-culation with the broadening due to fluid motion for a line ofsight in the x direction, and the dotted line shows the same forthe y direction. The fitted temperatures are 4026 eV, 4464 eVand 4526 eV respectively. Neutron scattering was neglected. locity, which has a peak value | u DT | ≃ × ms − . Thespatially integrated neutron spectrum from the wholefuel must be weighted by the fusion reaction rate r ateach point. Since u DT . ˆr is both positive and negative,the spatially integrated spectral shift equates to an addi-tional spectral broadening. Equation (17) assumes thatthe neutrons are not scattered by collisions as they es-cape.The calculation was performed to find the broadeningfor lines of sight in the x and y directions [Fig. 5]. Thenumerical integration of equation (17) for the simulationdata at t = 40 ps was compared to the same integrationwith u DT artificially set to zero. This negates the spectralshift from the fluid motion and leaves only the thermalbroadening.The spectra have an increased width as a result of theintegrated fluid motion. This is a result of the motion atthe edge of the hot-spot and is also partially due to thecontraction of the mix region. There is a slight differencein broadening in the x and y directions, as well as a smallshift of the spectrum central energy for the y directioncase. This is due to the contraction of the mix region inthe y direction. Since the effects from the hot-spot edgeare isotropic, these differences between the x and y linesof sight are purely due to the mix region.If the width of the spectra were to be used as an ex-perimental ion temperature diagnostic, a fit of equation(17) to the spectra yields T = 4026 eV in the control case, T = 4464 eV along the x direction and T = 4526 eV inthe y direction. Therefore the inflow of hot fuel into themix region and dense fuel shell is enough to significantlybroaden the fusion neutron distribution. The magnitudeof this effect is similar to the known discrepancy at theNational Ignition Facility [3]. The variance in tempera-tures from different lines of sight are also consistent withexperiments. Although the fluid flows towards the mixregion have a greater magnitude in the x direction thanthe y direction, the y neutron spectrum is broader be-cause the fusion reaction rate and neutron emission ismuch greater at the tip of the mix spike than at its sides. SUMMARY
We have used an ion kinetic model to compare thecases of uniformly distributed and localized carbon mix.The bremsstrahlung effects of localized mix were foundto be more severe than the same carbon mass spread uniformly across the hot-spot. The localized mix re-gion radiatively cools and contracts, increasing the ra-diative losses which scale as n . This contraction inducesstrong flows which broaden the fusion neutron spectrum,possibly affecting experimental ion temperature measure-ments. The cool, low density mix region acts as a bar-rier to alpha particles, separating different parts of thehot-spot and decreasing the effective hot-spot areal den-sity. Kinetic depletion of the distribution tail around themix region reduces the fusion reactivity below that of anequivalent Maxwellian distribution. Although all carbonmix will raise the ignition threshold, the present work in-dicates that localized jets of mix are especially damaging. ACKNOWLEDGMENTS [1] S. Le Pape, L. F. Berzak Hopkins, L. Divol, A. Pak, E. L.Dewald, S. Bhandarkar, L. R. Bennedetti, T. Bunn, J.Biener et al ., Phys. Rev. Lett. , 245003 (2018).[2] R. Betti, A. R. Christopherson, B. K. Spears, R. Nora,A. Bose, J. Howard, K. M. Woo, M. J. Edwards andJ. Sanz, Phys. Rev. Lett. , 255003 (2015).[3] O. A. Hurricane, D. A. Callahan, D. T. Casey, P. M. Cel-liers, C. Cerjan, E. L. Dewald, T. R Dittrich, T. D¨oppner,D. E. Hinkel et al. , Nature , 343 (2014).[4] S. Atzeni and J. Meyer-ter Vehn,
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