Analysis of Experimental Asymmetries using Uncertainty Quantification: Inertial Confinement Fusion (ICF) & its Applications

Transcription

1 Analysis of Experimental Asymmetries using Uncertainty Quantification: Inertial Confinement Fusion (ICF) & its Applications Joshua Levin January 9, 2009 (Edited: June 15, 2009) 1

2 Contents 1. Uncertainty Quantification (UQ) Abstract 2. Inertial Confinement Fusion (ICF) a. Overview of ICF b. Lawson Criterion and Relevance to ICF c. ICF Application to Star-Center Energies 3. Uncertainty in ICF 4. Future Work 5. References 2

3 Uncertainty Quantification (UQ) Abstract Uncertainty Quantification (UQ) is, in essence, a statistical method of measuring uncertainties in a numerical model of a complex system. Specifically, Uncertainty Quantification using Ensembles of Models (UQ-EM) explores a variety of contributions to the uncertainty in extrapolating from complex models that involve simulation. UQ-EM contains multiple components, all of which must be analyzed in order to produce reliable results: calibration data, a simulator, an interpolator, a sampler, and a tester. Each experimental system has a given set of inputs, from which if we knew internal variables that characterize the system we could compute various outputs, within some stochastic error and some approximation error. In extremely complex systems, calculating system response can be expensive, so approximations are used. In particular, a simple interpolation representation of the simulator is more economical. However, one must be aware of the danger of error layering, in which layers of approximations upon approximations become increasingly delicate, due particularly to the multiple dependencies of the parametric uncertainties. In a general system, common sources of uncertainty include errors in the calibration data, theoretical approximations to the physical model, numerical approximations of the theoretical in the simulator, approximation error in interpolating the results from the simulator, and the choices of sampling-testing model. In what will be discussed, the theoretical approximations to the physics of the experiment are of prime importance, as well as the choice of parameters to be tested. These issues with uncertainty can be addressed theoretically from the choice of models, as well as the modulus of continuity of the physical predictions relative to the changes in said model. Inertial Confinement Fusion (ICF) Overview of ICF The physical model we will be examining is that of Inertial Confinement Fusion (ICF), an experimental method of recreating nuclear fusion through the heating and compression of a fuel target, often shaped as a spherical pellet, composed of a mixture of Deuterium and Tritium. This target is located within a hollow cylindrical chamber, called a hohlraum, which reflects light to the Deuterium/Tritium compound. The aim is to heat the mixture (denoted as a DT mixture ) to high enough temperatures to overcome the charge repulsion of the high-energy plasmas surrounding the hohlraum in the experimental chamber, thus initiating fusion. The chamber aforementioned is built in the shape of a large metallic sphere, about 10 meters in diameter, containing 192 heavy-ion lasers, all aimed towards the stable hohlraum (which is no larger than a common pencil eraser). Once the lasers are activated and shot towards the hohlraum (simultaneously, in an ideal situation), the target inside will compress to the necessary size and condition to emit shockwaves of radiation to the surrounding plasma, eventually returning to the ablated pellet, forcing a fusion reaction within the holhraum. This process is often referred to as the ignition, and results from the high densities of the DT mixture, as well as the increasing thermonuclear energy produced by the waves of radiation interacting with the highly ionized plasmas. At high densities, α-particles produced from these DT reactions move away from the center spark (of the ignition), causing a faster propagation of the energy out into the surrounding cold fuel. This entire process occurs within several billionths of a second. Lawson Criterion and Relevance to ICF Specific to this experiment is the importance of a power balance between the thermonuclear energy and heat energy output needed for ignition. This condition is called the Lawson Criterion, named after J.D. Lawson, a 1950s particle physicist. Key to understanding this concept is the measurement of confinement time,, which measures the rate at which the system loses energy 3

4 where W is the energy content of the ignition process, and is the power loss during said experiment. In a simple ICF experiment, the volume rate, f, measures the number of reactions per unit volume per time where is the product of the number of Deuterium and Tritium reactions, is the fusion cross section of the reaction, is the relative velocity, and is the average over the Maxwellian velocity distribution at a specified temperature T. Thus, we can see that the volume rate is a function of temperature, a key to understanding the ablation process. Using this relationship, one may compute the Lawson Minimal Constant a function of the temperature output of the experiment. This allows for the estimation of the lower limit of the Lawson Criterion. In a more complex experimental procedure, the triple product function of the reaction density, temperature, and confinement time, is much more applicable, but for our purposes, it is superfluous. In the ICF experiment, is an order of time for sound propagation across the confined plasma, thus addressing a key asymmetric issue in the experimental analysis of the modern ICF procedure, performed through the National Ignition Facility (NIF). ICF Application to Star-Center Energies On a larger scientific spectrum, the study of ICF is vital to the understanding of the production of core energy by stars, as well as the stability of the star structure. Stars are powered by the high-velocity fusions of elements lighter than iron, particularly hydrogen, deuterium and tritium. The fusion of elements heavier than iron would create reactions that consume energy rather than produce (these reactions are the bases of nuclear fission, the energy source of modern nuclear weaponry). Due to the fact that elements with higher rest energies per nucleon make ignition increasingly difficult, hydrogen is the most suitable for producing the greatest available energy. The key parallel in the powering of stars and the ICF experiments is that of repulsion energies. In the stellar situation, the thermonuclear energy produced by the fusion reactions within the core of the star is repelled by the gravitational energy of the highly-massive star structure. Once the fuel at the center of the star is exhausted (usually over the span of many hundreds of millions of years, depending upon the mass of the star, measured in solar units), the gravitational energy of the star collapses the structure, in time resulting in a mass implosion, producing what we know as a black hole (assuming the star is massive enough to meet the collapse criteria). In order to recreate this effect on earth, the inertia of the heated plasma must be strong enough to contain the thermonuclear energy produced by the hohlraum fusion reaction, but weak enough to allow for the reaction to occur. Uncertainty in ICF The uncertainty quantification parameters of the ICF experiment are of prime concern. During early experiments, the delivery time of the laser (which was not a heavy-ion beam at that time), determined by the synchronicity of the laser shots, produced an unavoidable asymmetry. However, with the invention of fiber optics, the asymmetry of the delivery time has been minimized to several nanoseconds, an excusable delay for this experiment. A key asymmetry, though, is that of the 4

5 propagation of radiation waves to the surrounding plasma following the ignition of the target within the hohlraum, which is vital to the completion of the reaction. Density asymmetry is also of key importance, wherein certain areas of the highly-ionized plasma are denser than others, resulting in premature ablation and faster propagation of radiation waves. An imbalance, known as a beam-beam imbalance, in which certain lasers impose higher photon energies upon the target (i.e. an energy differential between the heavy-ion lasers), causes concern as well. Finally, a distinct fear lies in a process known as beam anisotropy, where there exist hot spots of energy in the beam cross section, resulting in an uneven compression of the hohlraum surface, producing high amounts of propagation asymmetry (specifically referred to as Rayleigh-Taylor (RT) instabilities ). Therefore, we can see that mitigating these asymmetries in the experimental procedure is crucial to the success of ICF. Future Work At this point, in order to fully understand these uncertainties in the ICF experiment, a more reliable and economic computer program should be constructed in order to better run simulations and analyze the most damaging asymmetries in the experimental model. It is also important to further understand exactly how each asymmetry fluctuates according to the point in the simulation. Using this program, one may adjust the experiment according to the determined points of uncertainty, and better produce a successful test for inertial confinement fusion. A contour mapping program is in the process of production, but in order for it to be as accurate as possible, more data from previous experiments and simulations must be collected. 5

6 References Duderstadt, James J. Inertial Confinement Fusion. New York: Wiley, Glenzer, S. H., D. H Froula, L. Divol, M. Dorr, R. L. Berger, S. Dixit, B. A. Hammel, C. Haynam, J. A. Hittinger, J. P. Holder, O. S. Jones, D. H Kalantar, O. L. Landen, A. B Langdon, S. Langer, B. J. MacGowan, A. J. Mackinnnon, N. Meezan, E. I. Moses, C. Niemann, C. H. Still, L. J. Suter, R. J. Wallace, E. A Williams, and B. Young. Experiments and multiscale simulations of laser propagation through ignition-scale plasmas. Publication. Vol. 3. Nature Group, Lawson, J. D. Some Criteria for a Useful Thermonuclear Reactor. Publication. Culham, United Kingdom: Atomic Energy Research Establishment, Lindl, John. Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition and gain. Publication. 11th ed. Vol. 2. Livermore, CA: Lawrence Livermore National Labs, Stark, Philip B. Uncertainty Quantification: LLNL Ensemble Models Approach. Tech. 4th ed. Berkeley, CA: Department of Statistics, University of California, Berkeley, "Understanding the Universe: NIF & Photon Science." National Ignition Facility & Photon Science - The Power of Light National Research Council. 09 Jan < 6