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A Measurable Lawson Criterion and Hydro-Equivalent Curves for Inertial Confinement Fusion C

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A Measurable Lawson Criterion and Hydro-Equivalent Curves for Inertial Confinement Fusion C A measurable Lawson criterion and hydro-equivalent curves for inertial confinement fusion C. D. Zhou and R. Betti Citation: Phys. Plasmas 15, 102707 (2008); doi: 10.1063/1.2998604 View online: http://dx.doi.org/10.1063/1.2998604 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v15/i10 Published by the American Institute of Physics. Related Articles Recover soft x-ray spectrum using virtual flat response channels with filtered x-ray diode array Rev. Sci. Instrum. 83, 113102 (2012) Average atom transport properties for pure and mixed species in the hot and warm dense matter regimes Phys. Plasmas 19, 102709 (2012) A novel technique for single-shot energy-resolved 2D x-ray imaging of plasmas relevant for the inertial confinement fusion Rev. Sci. Instrum. 83, 103504 (2012) Deuterium–tritium neutron yield measurements with the 4.5 m neutron-time-of-flight detectors at NIF Rev. Sci. Instrum. 83, 10D312 (2012) Characterizing time decay of bibenzyl scintillator using time correlated single photon counting Rev. Sci. Instrum. 83, 10D911 (2012) Additional information on Phys. Plasmas Journal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors Downloaded 29 Nov 2012 to 128.151.164.152. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions PHYSICS OF PLASMAS 15, 102707 ͑2008͒ A measurable Lawson criterion and hydro-equivalent curves for inertial confinement fusion C. D. Zhou1 and R. Betti1,2 1Fusion Science Center and Laboratory for Laser Energetics, University of Rochester, Rochester, New York 14623, USA 2Departments of Mechanical Engineering & Physics and Astronomy, University of Rochester, Rochester, New York 14623, USA ͑Received 16 May 2008; accepted 12 September 2008; published online 30 October 2008; publisher error corrected 4 November 2008͒ It is shown that the ignition condition ͑Lawson criterion͒ for inertial confinement fusion ͑ICF͒ can be cast in a form dependent on the only two parameters of the compressed fuel assembly that can ͑ h͒ be measured with existing techniques: the hot spot ion temperature Ti and the total areal density ͑␳ ͒ ␳ Rtot , which includes the cold shell contribution. A marginal ignition curve is derived in the Rtot, h Ti plane and current implosion experiments are compared with the ignition curve. On this plane, hydrodynamic equivalent curves show how a given implosion would perform with respect to the Ͻ͗ h͘ Ͻ ignition condition when scaled up in the laser-driver energy. For 3 Ti n 6 keV, an approximate ͑ ͒ ͗ h͘2.6 ͗␳ ͘ Ͼ 2.6 / 2 form of the ignition condition typical of laser-driven ICF is Ti n · Rtot n 50 keV ·gcm , ͗␳ ͘ ͗ h͘ where Rtot n and Ti n are the burn-averaged total areal density and hot spot ion temperature, respectively. Both quantities are calculated without accounting for the alpha-particle energy deposition. Such a criterion can be used to determine how surrogate D2 and subignited DT target implosions perform with respect to the one-dimensional ignition threshold. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2998604͔ I. INTRODUCTION ment of the surrounding cold shell, and ͑b͒ the hot spot areal density cannot be experimentally measured. 1,2 In inertial confinement fusion ͑ICF͒, a shell of cryo- A more accurate form of the hot spot ignition condition genic deuterium and tritium ice is imploded at high velocities is given in Refs. 1 and 4–6 with the alpha heating balancing ͓͑ϳ2–4͔ϫ107 cm/s͒ and low entropy to achieve high cen- all the hot spot power losses ͑thermal conduction and radia- tral temperatures and high areal densities. The final fuel as- tion losses͒. Our approach to ignition is somewhat different sembly consists of a relatively low density ͑ϳ30–80 g/cc͒ from that in Ref. 4. First, our ignition model is dynamic as it high temperature ͑ϳ4–8 keV͒ core ͑the hot spot͒ sur- includes both the compression and expansion phases of the rounded by a dense ͑ϳ300–1000 g/cc͒ cold ͑ϳ100 eV͒ fuel shell motion. Second, our ignition condition is given in terms 3 layer ͑the compressed shell͒. The Lawson criterion deter- of the total areal density rather than the hot spot areal den- mining the onset of thermonuclear ignition is usually ex- sity. Third, the ignition condition is viewed as an instability pressed through the product p␶Ͼ10 atm·s, where p is the of both the pressure and the temperature rather than only the plasma pressure in atm and ␶ is the energy confinement time temperature. This causes the heat conduction losses to enter in seconds. In magnetic fusion devices, both the pressure and the ignition condition in a fundamentally different way. A confinement time are routinely measured and the perfor- more detailed discussion of this point is provided in Sec. II. mance of each discharge can be assessed by comparing the It is important to emphasize that the presence of a cold dense value of p␶ with respect to the ignition value ͑10 atm·s͒.In shell surrounding the hot spot significantly alters the onset of inertial confinement fusion, both p and ␶ cannot be directly the thermonuclear instability ͑a similar point is made in Refs. measured, and the performance of subignited ICF implosions 4, 12, and 18͒. Since the heat conductivity is negligible in the cannot be assessed with respect to the ignition condition. cold shell, most of the heat leaving the hot spot is recycled Often, the Lawson criterion is extended to ICF by simply back into the hot spot in the form of internal energy and pdV restricting its application to the hot spot and by replacing p work of the plasma ablated off the inner shell surface. Much ␳ / ͑␳ with the ideal gas equation of state p=2 hTh mi h is the hot of the radiation losses are also recycled back through abla- spot mass density, Th is the hot spot temperature, and mi is tion since the cold shell is opaque to the low-energy portion the DT average ion mass͒, and ␶ with the sound wave trav- of the x-ray bremsstrahlung spectrum ͑only the high-energy ␶ϳ / ͑ eling time through the hot spot, Rh Cs here Rh is the hot x rays can penetrate the dense shell͒. As is argued in Ref. 12, ϳͱ ͒ spot radius and Cs is the hot spot sound speed, Cs Th . the heat conduction and, to some extent, the radiation losses ͑␳ ͒ͱ ͑ This leads to the hot spot ignition condition hRh Th do not appreciably change the hot spot pressure i.e., en- Ͼ ␳ ͒ const, where hRh is the hot spot areal density. There are ergy . Instead, those losses raise the density and lower the two problems with such a simple derivation: ͑a͒ The confine- temperature while keeping pϳ␳T approximately constant. ment time is incorrect since it neglects the inertial confine- Since the fusion rate scales as n2͗␴v͘, with ͗␴v͘ϳT3–4 for 1070-664X/2008/15͑10͒/102707/12/$23.0015, 102707-1 © 2008 American Institute of Physics Downloaded 29 Nov 2012 to 128.151.164.152. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions 102707-2 C. D. Zhou and R. Betti Phys. Plasmas 15, 102707 ͑2008͒ TϽ6–8 keV and ͗␴v͘ϳT2 for 6–8ϽTϽ25 keV, it follows can conclude that such an OMEGA implosion scales to one- that the alpha self-heating is degraded by heat conduction dimensional ignition on the National Ignition Facility11 and radiation losses only at low temperatures less than ͑NIF͒. 6–8 keV but unchanged at high temperatures TϾ6–8 keV. The paper is organized as follows. The analytic ignition This occurs because at high temperatures, the fusion rates model is described in Sec. II while its initial conditions are depend only on the hot spot pressure ͑n2͗␴v͘ϳp2͒ that is derived in Sec. III. The ignition condition from the analytic independent of the heat losses. While these recycling effects model is derived in Sec. IV, and compared with the results of ͑described in detail in Ref. 12͒ improve the ignition thresh- one-dimensional hydrodynamic simulations in Sec. V. The old, the expansion losses, that are often not included in the assumptions concerning the alpha-particle confinement are ignition condition, cause a transfer of internal energy to ki- discussed in Sec. VI, and comparison with previous forms of netic energy and degrade the ignition conditions. Since the the ignition condition are presented in Sec. VII. The hydro- hot spot expansion occurs against the dense shell, the igni- equivalent curves are derived in Sec. VIII and discussed in tion conditions depend on the inertia of the dense shell. Fur- the Conclusions ͑Sec. IX͒. thermore, the hot spot internal energy comes from the shell kinetic energy, which is also used to assemble the shell areal II. DYNAMIC MODEL OF THERMONUCLEAR IGNITION density. As shown in Ref. 13, there is a direct correlation between the hot spot areal density and shell areal density. The dynamic model described in this section includes Thus, one can expect that the ICF Lawson criterion depends the standard energy losses and sources ͑heat conduction, ra- on the shell areal density. diation losses, alpha heating͒ as well as the compression and In this paper, we derive a form of the Lawson criterion expansion dynamics of a hot spot surrounded by a dense that can be directly measured in ICF implosions and includes shell. The model describes the assembly phase of the hot the confinement of the surrounding cold shell.
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