On thermonuclear ignition criterion at the National Ignition Facility

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1 On thermonuclear ignition criterion at the National Ignition Facility Baolian Cheng, Thomas J. T. Kwan, Yi-Ming Wang, and Steven H. Batha Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Sustained thermonuclear fusion at the National Ignition Facility remains elusive. Although recent experiments approached or exceeded the anticipated ignition thresholds, the nuclear performance of the laser-driven capsules was well below predictions in terms of energy and neutron production. Such discrepancies between expectations and reality motivate a reassessment of the physics of ignition. We have developed a predictive analytical model from fundamental physics principles. Based on the model, we obtained a general thermonuclear ignition criterion in terms of the areal density and temperature of the hot fuel. This newly derived ignition threshold and its alternative forms explicitly show the minimum requirements of the hot fuel pressure, mass, areal density, and burn fraction for achieving ignition. Comparison of our criterion with existing theories, simulations, and the experimental data show that our ignition threshold is more stringent than those in existing literature and that our results are consistent with the experiments. PACS numbers: 5.57, -z, r, b I. INTRODUCTION Recent National Ignition Campaign (NIC) experiments [] at the National Ignition Facility (NIF) achieved nearly 95% of the required peak implosion velocity, and the total areal density of the fuel was greater than the ignition threshold of g/cm [, 3]. Yet, the nuclear performance of the capsule was well below that needed for ignition: neutron outputs was at least two orders of magnitude below expectation, the energy observed in the hot deuterium-tritium (DT) was only /7th the total energy in the capsule [4, 5]. Such discrepancies motivate us to reexamine the physics of ignition. Researchers [,,6-4] have for decades developed criteria for the onset of thermonuclear ignition in inertial confinement fusion (ICF) approaches. The Lawson criterion is the classic example, expressed through physically measurable quantities such as the hot spot ion temperature T, pressure P, confinement time τ c, and areal density ρr of the DT fuel. Here, we derive a general thermonuclear burn criterion for sustained burn in the hot spot of the NIC capsule design and, in turn, an ignition threshold in terms of measurable physical quantities. To facilitate comparison with experimental results, we also derive alternative expressions for the ignition threshold in terms of other quantities, for example, the minimum required hot fuel pressure, mass, and burn fraction. The fundamental differences between our model and other theories and simulations are the distinction between the areal density of the hot spot (ρr) hs and the total areal density of the capsule (ρr) tot, and our definition of confinement time for thermonuclear (TN) ignition. We take the confinement time to be the hydrodynamic disassembly time, i.e., the hot spot radius divided by the effective sound speed in the hot spot. Other models [, 4, ] use a deceleration time taken as the ratio of the radius of the cold DT fuel and the peak implosion velocity[4]. The use of the deceleration time as the confinement time overestimates the TN burn time in the capsule leading to an overly optimistic criterion because the peak implosion velocity is less than the sound speed and the radius of the cold DT is nearly twice the radius of the hot spot at maximum implosion. II. IGNITION CRITERION The ignition condition for ICF capsules can be determined by energy balance principles, specifically, by two time scales[5]: the nuclear fusion reproduction time (τ rep = E T /Ė) and the hydrodynamic disassembly time (τ H = R hs /C s ), where E T = (3/)(n D +n T )kt +E rad is the total energy density of the hot DT fuel in which k is the Boltzmann constant and E rad the radiation energy density. Ė = n T n D < σv > DT W α is the energy deposition rate by fusion reactions, where W α is the energy deposited into the hot DT per fusion, which normally equals to a fraction (f α ) of the α-particle kinetic energy of 3.5 MeV. In this work, we choose f α =. The neutron energies are assumed escaped because the mean free path of the 4 MeV neutrons is much larger than the hot spot radius R hs. For the disassembly time, C s is the adiabatic sound speed. To account for the tamping effect by the cold fuel, we replace the sound speed with an effective sound speed Cs C s /f T, where f T ρ p /ρ hs [0] is the tamping factor, and ρ p and ρ hs are the mass density of the pusher and the hot spot at the interface between the hot and cold fuel, respectively. Other discussions on tamping factor can be found in []. In practice, energy losses are inevitable, and the net energy deposition rate can be written as Ė = n T n D < σv > DT W α i dqi l /dt, where superscript i denotes various losses, such as the energy loss by electron bremsstrahlung, dq b l /dt = C ρ DT T / and the heat outflux by electron conduction, dq e l /dt = C T 3.35 /Rhs [7], where C 9. 0 erg-cm 3 /(kev / g s), C 6.9

2 0 Power index n <σv> ~ T n T (kev) FIG.. The power index fitting of < σv > DT T n vs. temperature. 0 9 erg/(kev 3.35 cm-s), and T is in kev. These energy loss expressions were obtained in the approximation of the total yield of the bremsstrahlung[7], which are slightly different from the general bremsstrahlung radiation loss, C = Z 3 /A DT ergcm 3 /(kev / g s)[0], and the general heat conduction loss rate Q e l C T 3.5 /Rhs, C 8.87 ln Λ 09 erg/(kev 3.5 cm-s)[0] in use. Here Z is the ionization state and ln Λ is the Coulomb logarithm depending on the cut-off parameters. These general expressions would lead to a higher ignition threshold. Ignition occurs when τ rep /τ H, the condition for sustained TN burn of the hot spot. This condition gives a threshold for the areal density of the hot DT (ρr) hs [( + d) /d][3kt + E rad /n DT ]C sa DT /N A < σv > DT W α ( Q b l + Q e l )( + d) /(dn DT ), () where n DT n D + n T = ρ DT N A /A DT, Q i l dqi l /dt (i = b, e), d = D/T is the D to T ratio, N A Avogadro s number, and A DT the atomic weight mass of the DT mixture. Nuclear reactivity < σv > DT [6] can be approximated by a power law of the temperature, T n, with the power index n dependent on the temperature as shown in Fig.. In the temperature range 0. 6 kev, n (0.3/T /3 )/3[7]. For temperature range kev, < σv > DT C DT T 4 is a good approximation as shown in Fig., where C DT cm 3 /s/kev 4. It is evident from Eq. () that any energy loss increases the areal density threshold for ignition. Furthermore, with the denominator being positive and definite, it leads to two important observations: () for d =, the energy loss by bremsstrahlung emission prevents ignition at temperatures below T min = {[4C /(C DT W α )](A DT /N A ) } /7 3 kev at any (ρr) hs, and () the energy loss through electron heat conduction makes ignition impossible at any temperature when (ρr) hs < {[4C /(C DT W α )](A DT /N A ) T 0.65 } / 0.3 g/cm. These constraints may only be improved if the burn is in equilibrium when the bremsstrahlung and inverse-bremsstrahlung emission balance such that the associated energy loss is small relative to the heat <σv> /(at b ) a=.37e-0, b=4 a=.5e-0, b=3.9 a=.3e-0, b=4 a=.3e-0, b= T (kev) FIG.. The power law at b fit of < σv > DT vs. temperature. The black solid line represents a = and b = 4, red dot-dashed line for a = and b = 3.98, black dashed line for a = and b = 3.9, and blue dotted line for a = and b = 4, where T is in kev. capacity of the blackbody radiation loss. For simplicity and comparison purpose, hereafter, our theoretical model assumes spherical symmetry with no mix, no energy loss from bremsstrahlung or electron heat conduction, and a blackbody radiation energy loss (E rad at 4, where a is the radiation energy constant) from the hot DT into a compressing shell. It is obvious that any asymmetry will make ignition more difficult. Thus, Eq. () becomes (ρr) hs ( + d) d [3kT + at 4 /n DT ]C s A DT < σv > DT W α N A, () which gives ignition curves on the ρr-t plane under various conditions, as shown in Fig. 3. The general threshold depends on the ratio of D to T and is minimum at d =. Equation () suggests two ways to achieve ignition: () low ρr at high temperature, or () high ρr at low temperature. The NIC ignition design point is at (ρr) hs 0.3 g/cm and T 4 kev []. For d =, substituting the approximation < σv > DT T 4 into Eq. () and ignoring the radiation energy, Eq. () becomes (ρr) hs 4 κc f T ( T kev ).5 g/cm (3) for 3 kev < T < 5.5 kev, where κ c /C DT 5.54 is a constant depending on the power law fitting approximation for < σv > DT. Any departure from the ideal condition, such as asymmetry, a decrease in α-particle energy deposition, or radiation and other energy losses would lead to higher TN threshold and therefore greater difficulty in achieving ignition. Figure 3 shows various ignition curves at f T = on the ρr-t plane, showing that inclusion of the radiation heat capacity significantly raises the ignition threshold at temperatures above 3 kev. Below 3 kev, achieving ignition in an equilibrium mode is only feasible at high areal density. For reference, we have also plotted the ignition curve with the contributions of electron bremsstrahlung

3 3 III.a Lawson criterion and ignition parameter Substituting ρ hs = A DT P/RT (R is the gas constant) and R hs C s τ H into Eq. () gives the Lawson criterion P (Gbar)τ H(µs) > κ c ( + d) A DT d γg / T (kev), (4) FIG. 3. The ignition curve on the ρr-t plane for f T =. The ignition curve on the ρr-t plane above separates out the ignition region. The black curve represents the ignition criterion in Eq. () when the radiation term is neglected. Both the red and green curves show the impacts of the radiation term with ρ hs = g/cm 3 and ρ hs = 3 g/cm 3, respectively. The maroon dot-dashed line includes the contributions from both electron bremsstrahlung and heat conductions. Clearly, inclusion of radiation will make ignition more difficult. The dashed blue curve is the analytic solution in Eq. (3) for the NIC design. The red dots denote the D numerical calculations performed by Betti et al. [3]. and heat conductions in Fig. 3, which shows a more stringent ignition space in temperature (i.e., T min > 3. kev). We emphasize that, for a given temperature, the most critical metric characterizing the ICF ignition threshold is the areal density of the hot fuel, not the total areal density of the fuel. The total areal density is the sum of the areal densities of the hot spot and the surrounding cold DT fuel. It is obvious that a sustained TN burn of the hot spot must be maintained in order to light the cold fuel. The formation of hot spot in a capsule is complicated and depends on experimental conditions. Any relation between the hot spot mass and the total fuel mass is at best approximate. Therefore, it is misleading to use the total areal density of the fuel to characterize the ignition threshold or as an optimal design parameter for ignition. III. APPLICATIONS The physical applications of our ignition criterion are straightforward. From Eq. (), we can derive various alternative requirements, such as the minimum hot spot fuel pressure, mass, and burn fraction required for ignition. which is independent of f T. Here we have used C s (γ g RT/A DT ) / γ g T (kev )cm/s. For d = and γ g = 5/3, substituting κ c = 5.54 into the above equation gives P (Gbar)τ H (µs) 0.475/T (kev). At T = 4 kev and R hs 30µm, for example, the required minimum hot spot pressure would be 70 Gbar, which is more stringent than (nearly three times) the ignition conditions given by Betti et al. [4]. The essential difference between our model and that developed by Betti et al. is the physical definition of the confinement time. Using Eq. (3), one can define the ignition parameter for the NIC design χ d κ c( + d) f T (ρr) hs T (kev).5. (5) Ignition requires χ. For d =, the expression becomes 0.045f T (ρr) hs T (kev).5. At T = 4 kev, the ignition condition becomes (ρr) hs 0.69f T g/cm. To compare with NIC experiments, we apply the scaling relationship between (ρr) hs and the total areal density of the fuel, (ρr) tot, derived in [5], (ρr) hs = (ρr) tot ψ/(+ψ). Letting the adiabatic index of the cold fuel (γ p ) and hot fuel (γ g ) be 5/3, Eq. (5) becomes χ = d κ c ( + d) f ψ T + ψ (ρr) tott (kev).5, (6) where ψ ηvimp /ϵ hs is the ratio of the specific implosion energy (ηvimp /) to the specific internal energy (ϵ hs = 3RT/A DT ) of the hot fuel. Here, V imp is the peak implosion velocity and η the implosion energy efficiency[5]. For V imp 370 km/s and T=4keV, ψ 0.48, (ρr) tot 5.35 g/cm for ignition if f T = d = η =. This is in sharp contrast to the NIC ignition design requiring (ρr) tot > g/cm [, ] and the Betti et al. D criterion from D simulations χ (ρr) 0.8 tot[t (kev)/4.4].8 [4], which gives (ρr) tot.4 g/cm at T = 4 kev. Again, our model shows a significant increase in the ρr requirement for ignition, which is consistent with the NIC data where < (ρr) tot < 5.35g/cm were achieved. In the NIC experiments, the total areal density of the fuel is a measurable quantity through the measured down scattering neutron ratio (DSR), (ρr) tot DSR [4, 8]. For example, at DSR and T 3. kev, as in shot N03, (ρr) tot.3g/cm, the inferred (ρr) hs 0.47g/cm and P τ 4.4 atm-s according to the formula in [4]. For shot N39, DSR 0.04, T 5 kev[9], and χ.. These results exceed the ignition thresholds and would predict ignition

4 4 of the capsule. However the neutron yield of the capsule (Y n and ) was far below ignition. In contrast, according to our model the areal density of the hot spot for N03 was only 0. g/cm, and the ignition parameter χ was at f T =. Similarly, for N39, (ρr) hs 0.077g/cm and χ These values are far below our ignition criterion, (ρr) hs 0.69 g/cm for f T = and (ρr) hs 0.35 g/cm for f T =, which is consistent with the low output from the capsule. Function ψ is sensitive to the adiabat and equation state of the pusher (mainly cold fuel). For the same required hot spot areal density for ignition, if the pusher were cold and hard, for example, γ p =.0 and γ g = 5/3, ψ 0.944, then the required total areal density for achieving ignition at the same conditions (V imp = 370 km/s and T = 4 kev) can be reduced from 5.35 g/cm to 4.4 g/cm. One of the most important difference between the criterion from our model and the one used in NIC is the use of the fuel areal density: (ρr) hs vs. (ρr) tot. Our model focuses on the hot fuel and uses the areal density of the hot spot as the fuel areal density in the derivation because the necessary and sufficient condition for achieving ignition is to have a sustainable TN burn in the hot spot. The sustained burn can then light the cold fuel under the right conditions resulting in TN ignition. III.b Minimum hot spot mass and burn fraction Sustaining TN burn in the hot spot and achieving ignition for a given design (e.g., laser energy and convergence ratio) require a minimum hot fuel mass. This minimum mass can be obtained from the ignition condition (), and for T = kev, can be expressed M min hs 4π 3 κ c ( + d) R0F f T T (kev).5 d Cf, (7) where C f R 0F /R hs is the geometric convergence ratio, and R 0F and R hs are, respectively, the initial inner radius of the fuel and the final radius of the hot spot. The ignition parameter given by Eq. (5) provides a measure of how far a given design is from ignition. At d =, T = 4 kev, R 0F = 000 µm, and C f 35, as in the NIC design, the minimum hot spot mass from Eq. (7) is about 4 µg if f T =. For C f 7, as in the recent high-foot shots at NIF, the minimum hot spot mass is 40. µg if f T =. The minimum hot spot mass for ignition given here is nearly four times the hot spot mass observed in NIF experiments. For example, M hs was only 3.9 µg in the low-foot shot N03 [4] and 7. µg in the highfoot shot N3097 [0]. Again, our model is consistent with observation of non-ignition. We point out that although the required minimum hot fuel mass is decreasing dramatically when the convergence ratio is increasing, the instabilities at the interface Burn parameter ζ T (g/cm ) T (kev) τ c ~τ H τ c ~τ H /3 τ c ~τ H /4 FIG. 4. The burn parameter vs. temperature.. ϕ ϕ between the cold and hot fuel grow nonlinearly with C f [3]. Thus, for high-convergence capsules, the greatest challenge is to achieve a hot, clean spot in the fuel. The burn fraction that characterizes the TN burn is defined as the ratio ϕ N fus /NDT 0, with N fus being the total number of fusion reactions and NDT 0 the number of DT pairs initially present in the fuel, and can be expressed as = N 0 DT < σv > τ c, where the nuclear confiement time τ c is defined as the time during which the fuel is maintained at a temperature above the critical ignition temperature to sustain the fusion reaction. If we assume τ c to be at least the hydrodynamic disassembly time τ H, the required burn fraction of the hot fuel for sustained TN burn is derived from ignition condition () and given by ϕ min DT (ρr) hs ζ T (T ) + (ρr) hs, (8) where ζ T (T ) ( +d d )( A DT C s N A <σv> ) is a burn parameter that varies with the temperature of the hot fuel. The burn parameter, ζ T, has values of 99 at T = 4 kev and between 4 and 8 g/cm at 0 kev < T <5 kev for f T = as shown in Fig. 4. The minimum burn fraction can also be approximately expressed in terms of the ion temperature of the hot spot, ϕ min DT ( + d)(3kt + at 4 /n)/w α. Table I lists the required minimum burn fractions at different temperatures for d =. At 4 kev, the hot spot burn frac- T (kev) ζ T (g/cm ) (ρr) hs (g/cm ) ϕ min DT (%) TABLE I. Burn parameter, (ρr) hs d = f T = with no radiation. and ϕ min DT vs. T at tion needed for ignition is ϕ > 0.677%. NIF experiments reached only a small fraction of this value. For example, the low-foot shot N5, which had the highest yield of the NIC shots ( neutrons) only achieved % of the required burn fraction, while the burn frac-

5 where M p is the mass of the pusher at peak implosion velocity[5] and τ B is the burn width. We have applied this formula to a series of NIC and NIF experiments. Table II lists the shots that we considered in this paper and the physical parameters associated with each shot. Table III summarizes the results, showing Shot # M p V imp T hs τ B ν (µg) (cm/s) (kev) (ns) N N N N FIG. 5. The minimum required burn fraction and areal density of the hot spot for ignition vs. the ion temperature. tion of the high-foot shot N3097 ( neutrons) was only 4% of that required for ignition [9]. Fig. 5 shows the minimum burn fraction and the corresponding minimum areal density of the hot spot required for ignition as a function of the ion temperature. Clearly, both the burn fraction and burn parameter are very sensitive to the fuel temperature and areal density. The lower the hot spot areal density, the larger the burn fraction required. Again, we emphasize that the areal density ρr in the burn fraction equation, Eq. (8), is the areal density of the hot fuel (ρr) hs, not the total areal density (ρr) tot. The latter overestimates the burn fractions in NIC capsules. The two areal densities are equivalent only when the burn front has propagated through and heated all the fuel. III.c Neutron yield The neutron yield of the capsule can be estimated by the formula Y n = τ B V 0 0 n Dn T < σv > dv dt, where V is the volume of the hot fuel. For NIF experiments, V corresponds to the volume of the hot spot. Substituting the implosion scaling laws derived in [5] into the above equation yield Y n d ( + d) ( N A (γ p )(γ g ) R ) (3γ p ) ηc DT M p V impp τ B T kev, in the c.g.s system units except for T is in kev. For convenience, letting d = η = Eq. (9) can be further reduced to Y n (γ p )(γ g ) ( M p (3γ p ) µg )( V imp P τ B km/s ) ( Gbar.ns )T kev, (0) (9) TABLE II. Physical parameters in the NIF shots. Where ν A DT Vimp/(RT ) 0.96Vimp/T is a model parameter, in which V imp is in 00km/s and T is in kev. good agreements between Eq. (0) and the experimental data, where γ g = 5/3, η = and Table II were used. Shot # γ p DSR (ρr) hs P τ B Y n(0 4 ) Y n(0 4 ) (%) (g/cm ) (Gb.ns) (theo.) (obs.) N005 5/ N03 4/ N3050 5/ N3097 5/ TABLE III. Yield comparison between theory and experimental data at NIF. Here we have used the upper bound of DSR for shot N3097 to match the inferred hot spot areal density (ρr) hs 0. g/cm [0]. CONCLUSION In summary, we developed a general analytical ignition criterion in Eq. () for ICF capsules under the assumptions of perfect symmetry and no mix. Our thermonuclear burn criterion was derived by balancing the DT fusion heating time with the hydrodynamic disassembly time. By ignoring energy loss processes and radiation, we arrive at a simplified expression of the ignition criterion for NIF design in Eq. (3). Our recent D LASNEX calculations starting with an assembly of hot fuel with predetermined areal densities and temperatures but including losses (electron heat conduction and bremsstrahlung) during its TN reaction show that our ignition threshold under ideal conditions is optimistic as we anticipated. However we found that the ignition criterion in Eq. (3) is still more stringent than that used in the NIF design. The threshold values given by Eqs. (3-8) are consistently more than three times higher than those in existing literature. The fundamental differences between these criteria are reflected in the relevant use of (ρr) hs vs. (ρr) tot

6 6 for ignition and the definition of confinement time. Note that the general expression of the ignition criterion given in Eq. () includes energy losses. A numerical solution of Eq. () taking into account of the energy losses will further enhance the predictive capability of our analysis. ACKNOWLEDGEMENTS The authors wish to thank the referee for many insightful and constructive comments that led to notable improvement of our manuscript. The authors are grateful to P. Amendt, C. Cerjan, D. Clark, S. Haan, P. Patel, H. Robey, B. Tipton for sharing the NIC data, analysis, and to B. Albright, N. Hoffman, J. Mercer-Smith, K. Molvig, J. Pedicini, A. Simakov, C. Snell and D. Wilson for helpful discussions and to C.S. Carmer for editing this article. This work was performed under the auspices of the U.S. Department of Energy by the Los Alamos National Laboratory under Contract No. W-7405-ENG-36. [] J. Lindl, O. Landen, J. Edwards, E. Moses and NIC Team (with erratum pending), Phys. 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8 T (kev) 9 8 <σv> ~ T n Power index n

9 T (kev) a=.37e-0, b=4 a=.5e-0, b=3.9 a=.3e-0, b=4 a=.3e-0, b=3.98 <σv> /(at b )

10 (ρr) DT (g/cm ) Matter dominated (T i =T e ) + radiation (T i =T e =T r, ρ hs =) + radiation (T i =T e =T r, ρ hs =3) Analytic solution (T i =T e ) D simulations (Zhou & Betti) + bremsstrahlung + e-conduction T peak (kev)

11 T (kev) Burn parameter ζ T (g/cm ) 00 0 τ c ~τ H τ c ~τ H /3 τ c ~τ H /4

12 Areal density (g/cm ) or Burn fraction Minimum ρr (g/cm ) Burn fraction φ=ρr/(ζ T +ρr) Peak no-burn temperature (kev)